Patrick Reany
Abstract: This page presents mathematics diversions, which are basically just
math problems of varying difficulty, which one can try to solve for fun or for
testing one's math skills, or for learning new methods and concepts.
Disclaimer: Many of the following problems were taken from SATs or old
Olympiad tests. My alternative solutions may not follow Olympiad rules, however.
The techniques I use may not be fit for use in Olympiads or entrance exams or the like.
I believe that it's good to see alternative solutions to better fill out one's
mathematical abililties. Four decades ago, I quipped that one cannot truly
understand a theorem until one is able to prove it in essentially two different ways.
It's meant to be a thought-provoking claim, perhaps containing a morsel of truth. But
when you take standard tests, use standard problem-solving practice.
Note 1: The comments 'Unipodal' or 'Not Unipodal' placed besides the problems
is there to indicate whether or not the problem solution has incorporated the
Unipodal algebra or not. If it has, the proof will be 'nonstandard', though
(hopefully) still correct.
Note 2: I have been endeavouring to update the problem solutions: a) correcting for spelling,
punctuation, and math errors, and b) improving the solutions. I have marked these notably
updated articles with an 'Update'
icon, downloaded from:
Update icons created by Freepik - Flaticon
[The icon won't be applied to articles containing only minor improvements.]
Note 3: Here are some short monographs on standard math that I use in my papers:
Link to my write-up on Basic Complex Numbers.
Link to my write-up on Pascal's Triangle.
Link to my write-up on Geometric Series.
Link to my write-up on logarithms over the real numbers.
Link to my write-up on Trigonometric functions.
Link to my write-up on Hyperbolic trig functions over the real numbers.
Link to my write-up on the Lambert W function.
Link to my write-up on Word Problem solving.
To go to Diversions2 page Link to Diversions2
To go to Diversions3 page Link to Diversions3
To go to Diversions4 page Link to Diversions4
To go to Diversions5 page Link to Diversions5
To go to Diversions6 page Link to Diversions6
Problem 1 [Not Unipodal]:
This problem is by
Source: https://www.youtube.com/watch?v=6auh7Sh1AL0
Title: Can You Solve This Challenging Olympiad Question?
Presenter: PreMath
Given the relation \begin{equation} x + \frac{1}{x} = 5\,, \end{equation} find the values of \begin{equation} \phi = x^5 + \frac{1}{x^5} \,. \end{equation}
Problem 2 [Not Unipodal]:
This problem is by
Source: PreMath -- December 23, 2021.
Title: ?
Presenter: PreMath
Given the relations \begin{align} \frac{x^3 + y^3}{x+y} &= 7\,,\\ \frac{x^3 - y^3}{x-y} &= 7 \,. \end{align} find the values of $x,y$.
Problem 3 [Not Unipodal]:
This problem is by
Source: https://www.youtube.com/watch?v=FPusvO63lI0
Title: Math Olympiad Question | Equation Solving
Presenter: Math Window
Given the relations \begin{align} a+b &= abc\,,\\ b+c &= abc \,,\\ c+a &= abc \,. \end{align} find the values of $a,b,c$.
Problem 4 [Not Unipodal]:
This problem is by
Source: https://www.youtube.com/watch?v=kN3AOMrnEUsStatement of the problem:
Title: Why did everyone miss this SAT Math question?
Presenter: MindYourDecisions
My solution is found here Solution to this problem.
Problem 5 [Not Unipodal]:
Statement of the problem: Find the perimeter of the following figure.
My solution is found here Solution to this problem.
Problem 6 [Not Unipodal]:
This problem is by
Source: https://www.youtube.com/watch?v=Y-dAhOmR_68
Title: can you solve this problem?
Presenter: Ankit Math Magics
Given the relation \begin{equation} 3^x\cdot 7^{x^2} = 21\,, \end{equation} find the values of $x$.
Problem 7 [Not Unipodal]:
This problem is by
Source: https://www.youtube.com/watch?v=ArVrjYmBhVwStatement of the problem: Solve for the measure of angle $x$.
Title: People are arguing about a simple geometry problem.
Presenter: MindYourDecisions
My solution to the problem Solution to this problem.
Problem 8 [Not Unipodal]:
This problem is by
Source: https://www.youtube.com/watch?v=82b0G38J35kStatement of the problem: Three men will start at point S and transit to point F, using a combination
Title: Google Interview Riddle - 3 Friends Bike and Walk
Presenter: Logically Yours
of walking and riding a scooter built for two. Find the minimum time that this can be done in. (Warning:
The figure below gives away a lot.) This problem demonstrates in problem solving the need for clear
organization and the creation of helpful visual aids.
My solution to the problem Solution to this problem.
Problem 9 [Not Unipodal]:
This problem is by
Source: https://www.youtube.com/watch?v=SUMLKweFAYk
Title: ``Steven Strogatz: In and out of love with math | 3b1b podcast #3"
Presenter: Grant Sanderson
Statement of the problem:
If the angle bisectors of two vertices of a triangle are congruent, show that the triangle is isosceles.
This is a well-known problem in the literature, and not without some considerable controversy,
that I
will not take up. It is known as the Steiner-Lehmus Theorem.
My solution to the problem is pending. I tried it, but haven't succeeded yet.
Problem 10 [Not Unipodal]: 
This problem is by
Source: https://www.youtube.com/watch?v=yyBOvZTRb7I
Title:An Interesting Exponential Expression
Presenter: SyberMath
Given the relation
\begin{equation}
\phi = e^{2^{\textstyle\frac{\ln(\ln 2)}{\ln 2}}} \,,
\end{equation}
simplify $\phi$ over the real numbers.
My solution to the problem Solution to this problem.
Link to my write-up on logarithms over the real numbers.
Problem 11 [Not Unipodal]:
This problem is by
Source: https://www.youtube.com/watch?v=tlUlPMaCY4k
Title: Solving A Differential Equation $|$ Two Methods
Presenter: SyberMath
Statement of the problem:
Solve for $y$ as a function of $x$ given the relation
\begin{equation}
\left(\frac{dy}{dx}\right)^2 - 1 = x^2.
\end{equation}
I offer an alternative solution that employs the unipodal algebra and
hyperbolic trig functions.
My solution to the problem Solution to this problem.
Link to my write-up on Hyperbolic trig functions over the real numbers.
Problem 12 [Not Unipodal]:
This problem is by
Source: https://www.youtube.com/watch?v=zLU5Cagv-Tc
Title: Solving a Quick and Easy Functional Equation
Presenter: SyberMath
Statement of the problem:
Solve for $f(x)$ as a function of $x$ given the relation \begin{equation} f(x + \sqrt{x^2 + 1}) = \frac{x}{x + 1}\,. \end{equation} I offer an alternative solution that employs hyperbolic trig functions.
My solution to the problem Solution to this problem.
Link to my write-up on Hyperbolic trig functions over the real numbers.
Problem 13 [Not Unipodal]:
This problem is by
Source: https://www.youtube.com/watch?v=3jnbBVpOf40
Title: A tricky problem with a 'divine' answer!
Presenter: MindYourDecisions
Statement of the problem:
Solve for real values of $x$ in the following equation: \begin{equation} \left(x - \frac{1}{x}\right)^{1/2} + \left(1 - \frac{1}{x}\right)^{1/2} = x\,. \end{equation}
My solution is found here Solution to this problem.
Problem 14 [Not Unipodal]:
This problem is by
Source: https://www.youtube.com/watch?v=k2QONlmEab0
Title: A very nice olympiad question | How to solve ....
Presenter: Spencer's Academy
Statement of the problem:
Solve for real values of $x$ in the following equation: \begin{equation} (4 + \sqrt{15})^x + (4 - \sqrt{15})^x = 62\,. \end{equation}
My solution is found here Solution to this problem.
Problem 15 [Unipodal]:
This problem is by
Source: ?
Title: ?
Presenter: ?
Statement of the problem:
Given the relation \begin{equation} a + \frac{1}{a} = 7\,,\label{eq:TheGiven} \end{equation} where $a$ is a positive real number, find \begin{equation} \sqrt{a} + \frac{1}{\sqrt{a}}\,, \end{equation} and \begin{equation} \sqrt{a} - \frac{1}{\sqrt{a}}\,. \end{equation}
My solution is found here Solution to this problem.
Problem 16 [Not Unipodal]:
This problem is by
Source: https://www.youtube.com/watch?v=A-SFQ0s4qQw
Title: This Oxford Integral Question STUMPED Students
Presenter: dy d Oscar
Statement of the problem:
Given the relation \begin{equation} 6 + f(x) = 2f(-x) +3x^2 \int_{-1}^1\! f(t) dt\,, \end{equation} we are asked to find the value of the integral
My solution is found here Solution to this problem.
Problem 17 [Unipodal]:
This problem is by
Source: https://www.youtube.com/watch?v=-70c48Fl-x4
Title: Evaluating x sqrt(x)+y sqrt(y)_
Presenter: Math is math
Given the relations \begin{align} x + y &= 19 \,,\\ x y &= 9 \,, \end{align} find \begin{equation} x \sqrt{x}+y \sqrt{y}\,, \end{equation} where $x,y$ are positive real numbers.
Problem 18: [Unipodal]
The YouTube video is found at :
https://www.youtube.com/watch?v=JTt0wjqG7vM
Titled: Harvard University Admission Question ||
Algebra Exam || 99\% Failed Entrance Test
Presenter: Super Academy
Given the relations \begin{align} x^5 + y^5 &= 152 \,,\\ x + y &= 2 \,, \end{align} find all solutions for $x,y$ over the complex numbers.
Problem 19: [Unipodal]
The YouTube video is found at:
https://www.youtube.com/watch?v=vhbLBruwDj0
Titled: A Nice math Olympiad Problem --
You should know this trick
Presenter: Learncommunolizer
Given the relation \begin{equation} \sqrt[3]{x+49} + \sqrt[3]{x-49} = 2 \,,\label{eq:TheGiven1}\\ \end{equation} solve for $x$ over the complex numbers.
Problem 20: [Unipodal]
The YouTube video is found at:
https://www.youtube.com/watch?v=C-0anvb3D4k
Titled: Japan | A nice Math Olympiad Algebra Problem
| Find x=? & y=?
Presenter: Super Academy
Given the relation \begin{align} x^2 - y^2 &= \sqrt{10} \,,\\ xy &= \sqrt{10} \,, \end{align} solve for $x+y$ over the complex numbers.
Problem 21: [Unipodal]
The YouTube video is found at :
https://www.youtube.com/watch?v=PaMbwZ1VCTY
Titled: A Mind-blowing Math Olympiad Equation | How to solve!!
Presenter: Master T Maths Class
Given the relations \begin{align} p + q &= 8 \,,\\ pq &= 20 \,, \end{align} solve for $p$ and $q$ over the complex numbers.
Problem 22: [Unipodal]
The YouTube video is found at:
https://www.youtube.com/watch?v=Iq-ToVDngHA
Titled: A nice Math Olympiad Problem || Find x=? & y=?
Presenter: Super Academy
Given the relations \begin{align} \sqrt{x} + \sqrt{y} &= 5 \,,\\ \sqrt{x+16} - \sqrt{y+5} &= 2 \,, \end{align} solve for $x$ and $y$ over the real numbers.
Problem 23: [Unipodal]
The YouTube video is found at:
https://www.youtube.com/watch?v=ubvYMrln5WA
Titled: Algebra | A Nice Radical Problem | Math Olympiad
Problem | How to Solve this | Find X and Y
Presenter: ilm PEDIA
Given the relations \begin{align} \sqrt{x} + \sqrt{y} &= 7 \,,\\ x - y &= 7 \,, \end{align} solve for $x>0$ and $y>0$ over the real numbers.
Problem 24: [Unipodal]:
The YouTube video is found at [Unipodal]:
https://www.youtube.com/watch?v=M5jU3a5-brA
Titled: Simplification | Can you Solve This | A Nice Math
Olympiad Algebra Problem | JEE | Find value of X+Y
Presenter: ilm PEDIA
Given the relations \begin{align} x^2+ y^2 &= 7 \,,\\ x^3+ y^3 &= 10\,, \end{align} solve for $x+y$ over the real numbers.
Problem 25: [Unipodal]:
The YouTube video is found at :
https://www.youtube.com/watch?v=WeOdqkh8tSs
Titled: Harvard University Simplification Tricks | Radical
Algebra Aptitude Test | Ivy League Entrance Exam
Presenter: Super Academy
Simplify the expression \begin{equation} (208 + 120\sqrt{3})^{1/6} \end{equation} over the real numbers.
Problem 26: [Unipodal]:
The YouTube video is found at:
https://www.youtube.com/watch?v=remIGvjSkV4
Titled:Harvard University Aptitude Test Tricks ||
Algebra Ratio Problem || 1 percent Passed Entrance
|| a+b/a-b=?
Presenter: Super Academy
Given the relation \begin{equation} a^4 + b^4 = 10a^2b^2\,, \end{equation} find the value of \begin{equation} \frac{a+b}{a-b} \,, \end{equation} over the real numbers.
Problem 27: [Unipodal]:
The source of this \problem\ is from G.\ Sobczyk's book (\cite{Sobczyk}, p.\ 36),
though I've already encountered many `Olympiad' problems to have a similar form.
\bibitem{Sobczyk}G.\ Sobczyk, {\it New Foundations in Mathematics, The Geometric
Concept of Number},
Birkhauser/Springer, New York, 2013.
Given the relation
\begin{equation}
z^n + \frac{1}{z^n}= \beta\,,
\end{equation}
find the value of $z$. (This special equation was called the
`dihedral equation' by Felix Klein.)
Problem 28: [Unipodal]:
The YouTube video is found at:
https://www.youtube.com/watch?v=mTwpkMCm--w
Titled: Olympiad Mathematics - Algebra Problem |
Vietnamese Junior Math Olympiad
Presenter: Math Booster
Given the relation \begin{equation} x + \frac{1}{x}= -1\,, \end{equation} find the value of \begin{equation} x^{25} + \frac{1}{x^{25}}=\ ? \end{equation}
Problem 29: [Unipodal]:
The YouTube video is found at:
https://www.youtube.com/watch?v=tpZRaMFagUg
Titled: Hardest Exam Question | Only 8 percent of
students got this math question correct
Presenter: Higher Mathematics
Simplify the expression \begin{equation} \left( \frac{1+\sqrt{5}}{2} \right)^{12}\,. \end{equation}
Problem 30: [Unipodal]:
The source of this problem is inspired from a test problem from
an `Olympiad'
problem. I changed it subtly.
Given the relations \begin{align} a + b &= 2\,,\\ a^3 + b^3 &= 6\,, \end{align} find the value of \begin{equation} a^3 - b^3 \,. \end{equation}
Problem 31: [Unipodal]:
This problem was inspired by the YouTube video is found at:
https://www.youtube.com/shorts/tna_mHHtChw
Titled: Typical SAT Question
Presenter: MrHTutoring
Given the relations \begin{align} x - y &= 10\,,\\ xy &= -20\,, \end{align} find the value of \begin{equation} \frac{1}{x} + \frac{1}{y} \,. \end{equation}
Problem 32: [Not Unipodal]:
This problem is from the YouTube video:
https://www.youtube.com/watch?v=LG0AgRNN1Po
Titled: Cambridge University Exponential Problem ||
Admission Interview tricks
Presenter: Super Academy
Given the relation \begin{equation} x^x = 2^{8+2x} \,, \end{equation} find the value of $x$.
Problem 33: [Not Unipodal]:
This problem was inspired by recent `olympiad' problems.
Perform the indefinite integral.
\begin{equation}
I = \int \ln\, (x+\sqrt{1+x^2}) dx \,.
\end{equation}
Solution to this problem.
Link to my write-up on Hyperbolic trig functions over the real numbers.
Problem 34: [Not Unipodal]:
https://www.youtube.com/watch?v=a4VP42kz7d0
Titled: Working with Logarithmic Expressions
Presenter: SyberMath
Given the relations \begin{equation} 6\log a = 4\log b = 3\log c \,. \end{equation} find the numeric value of \begin{equation} \log_{ab} c\,. \end{equation}
Link to my write-up on logarithms over the real numbers.
Problem 35: [Not Unipodal]:
https://www.youtube.com/watch?v=C5HTplGs1AQ
Titled: Evaluating A Nice Polynomial | Math Olympiads
Presenter: SyberMath
Given the relation \begin{equation} b^3 - b = 1\,. \end{equation} find \begin{equation} b^5 - b^4 \,. \end{equation}
Problem 36: [Not Unipodal]:
https://www.youtube.com/watch?v=LtuCtaW3CYw
Titled:A very tricky math question with factorial
Presenter: Higher Mathematics
Given the relation \begin{equation} 6!\times 7! = x!\,, \end{equation} find $x$.
Problem 37: [Not Unipodal]:
https://www.youtube.com/watch?v=Srn-PJwFZgg
Titled: A fun proof for an integer
Presenter: Prime Newtons
If $n$ is a positive integer, show that (I changed the problem slightly.) \begin{equation} \frac{n}{6} + \frac{n^2}{2}+ \frac{n^3}{3}\label{eq:TheExpression} \end{equation} is also a positive integer.
Problem 38: [Not Unipodal]:
https://www.youtube.com/watch?v=_jaPL00JUhU
Titled: Can You Simplify A Radical?
Presenter: SyberMath
Simplify the following radical. \begin{equation} R = \sqrt{a + \sqrt{a^2-1}} \,. \end{equation}
Solution to this problem.
Link to my write-up on Hyperbolic trig functions over the real numbers.
Problem 39: [Not Unipodal]:
https://www.youtube.com/watch?v=GX7MzC0_2oM
Titled: Harvard University Aptitude Test Tricks
|| Algebra Problem
Presenter: Super Academy
Given the relations \begin{align} 5^x + 5^y &= 15750\,,\\ x + y &= 9\,, \end{align} find the value of $x$ and $y$.
Problem 40: [Not Unipodal]:
https://www.youtube.com/watch?v=aYeuix9nGG0
Titled: Can you pass College Entrance Aptitude Test ?
|| Find x=?
Presenter: Super Academy
Given the relation \begin{equation} x^{\log_3 x} = 81\,, \end{equation} find the value of $x$.
Link to my write-up on logarithms over the real numbers.
Problem 41: [Unipodal]:
The YouTube video is found at:
https://www.youtube.com/watch?v=zCah09n9Zns
Titled: A very tricky Harvard University Admission
Algebra Exam | Entrance Aptitude Test | Find x=?
Presenter: Super Academy
Given the relation \begin{equation} (x+9)^4 + (x+11)^4 = 706\,, \end{equation} find the value of $x$.
Problem 42: [Unipodal]:
This problem is one I adapted from one I saw recently on YouTube
Presenter: Patrick
Given the relations \begin{align} \sqrt{a} + \sqrt{b} &= \sqrt{31}\,,\\ \sqrt{ab} &= 2\,, \end{align} find the value of $ \sqrt{a} - \sqrt{b} $, where $a,b>0$.
Problem 43: [Not Unipodal]:
https://www.youtube.com/watch?v=9v41Nxu9UMU
Titled: Harvard University Exponential Problem
Presenter: Super Academy
Given the relation \begin{equation} 2^x\cdot 3^{x^2} = 6\,, \end{equation} find the values of $x$.
Problem 44:[Not Unipodal]:
https://www.youtube.com/watch?v=FgUtVjfD4Vw
Titled: A nice Math Olympiad Problem | Algebra Equation
Presenter: Super Academy
Given the relation \begin{equation} \frac{(x+7)!}{(x+3)!} = 7920\,, \end{equation} find the value of $x$.
Problem 45:[Unipodal]:
This problem on YouTube at
https://www.youtube.com/watch?v=4SwFsYsTrms
Titled: Math Olympiad | A Nice Algebra Problem
| Find the values of X
Presenter: Learncommunolizer
Given the relation \begin{equation} (x+2)^4 + (x+1)^4 = 17\,, \end{equation} find the real values of $x$.
Problem 46: [Unipodal]: [This is an elegant solution that I stumbled upon. Enjoy.]
This problem on YouTube at
https://www.youtube.com/watch?v=JTt0wjqG7vM
Titled:Harvard University Admission Question
|| Algebra Exam
Presenter: Super Academy
Given the relations \begin{align} x^5 + y^5 &= 152\,,\\ x+y &= 2\,, \end{align} find the values of $x,y$
Problem 47: [Unipodal]:
This problem on YouTube at
https://www.youtube.com/watch?v=RcoaWR0esE0
Titled: Harvard University Aptitude Test Tricks
|| Algebra Problem
Presenter: Super Academy
Given the relations \begin{align} 4^x - 4^y &= 24\,,\\ 2^{x+y} &= 35\,, \end{align} find the values of $x-y$.
Problem 48:[Unipodal]:
This problem on YouTube at
Source: https://www.youtube.com/watch?v=bIHujxQSmmU
Title: China | Can you solve this ? | Math Olympiad
Presenter: Learncommunolizer
Given the relation \begin{equation} (x-6)^4 + (x-8)^4=16 \,, \end{equation} find the values of $x$.
Problem 49: [Unipodal]:
This problem on YouTube at
Source: https://www.youtube.com/watch?v=XxbsHOUQjfw
Title: A nice Algebra Problem || Harvard University
Aptitude Test Tricks
Presenter: Super Academy
Given the relations \begin{align} a^2 - b^2 -(a-b) &= 6\,,\\ 2ab- (a+b) &= 17\,, \end{align} find the values of $a,b$ in the real numbers.
Problem 50: [Unipodal]:
This problem on YouTube at
Source: https://www.youtube.com/watch?v=yEW3O_ToYDE
Title: France | Can you solve this ? | Solution to this problem
Presenter: Learncommunolizer
Given the relation \begin{equation} \sqrt[4]{x} + \sqrt[4]{97 - x} = 5 \,, \end{equation} find the values of $x$.
Problem 51: [Unipodal]:
This problem on YouTube at
Source:https://www.youtube.com/watch?v=YEwdgK8COlI
Title:Harvard University Exam Question ||
Algebra Exam
Presenter: Super Academy
Given the relation \begin{equation} (5x-6)^2 + (10-5x)^3 = 16 \,, \end{equation} find the values of $x$.
Problem 52: [Not Unipodal]:
This problem on YouTube at
Source: https://www.youtube.com/watch?v=8OxLe4BDJbU
Title: Maths Olympiad | How To Solve
Olympiad Maths faster
Presenter: Maths Atoka
Given the relation \begin{equation} (x+\sqrt{1+x^2}) (y+\sqrt{1+y^2}) = 1 \,, \end{equation} find the values of $ (x+y)^2$.
Solution to this problem.
Link to my write-up on Hyperbolic trig functions over the real numbers.
Problem 53: [Unipodal]:
This problem on YouTube at
Source: https://www.youtube.com/watch?v=LzBdHFx0A8g
Title: Solving Another Quartic Equation | Problem 357
Presenter: aplusbi
Given the relation \begin{equation} (z + 1)^5 = (z - 1)^5 \,, \end{equation} find the values of $z$.
Problem 54: [Not Unipodal]:
This problem on YouTube at
Source: ???
Title: ????
Presenter: ????
Given the relations \begin{align} \log y &= \log_x (2x-5)\,,\\\ \log x &= \log_y (x+15)\,, \end{align} find the values of $x,y$.
Link to my write-up on logarithms over the real numbers.
Problem 55: [Unipodal]:
This problem on YouTube at
Source: https://www.youtube.com/watch?v=hBx8RBkRVGg
Title: A Radical Equation With Radicals
Presenter: SyberMath
Given the relation \begin{equation} \sqrt{x+\sqrt{x}} + \sqrt{x-\sqrt{x}}= \sqrt{6} + \sqrt{2} \,, \end{equation} find the values of $x$.
Problem 56: [Unipodal]:
This problem on YouTube at
Source: https://www.youtube.com/watch?v=4FPdSXaydHA
Title: A very tricky Oxford University Exponential Question
Presenter: Super Academy
Given the relation \begin{equation} (\sqrt{2}+ 1)^x + (\sqrt{2} - 1)^x = 34 \,, \end{equation} find the values of $x$.
Problem 57: [Unipodal]:
This problem on YouTube at
Source: https://www.youtube.com/watch?v=fkmU5DGp5gM
Title: Stanford University Entrance Interview Tricks
| Find x=?
Presenter: Super Academy
Given the relation \begin{equation} \big(7 + \frac{1}{x}\big)^4 - \big(5 + \frac{1}{x}\big)^4 = 240 \,, \end{equation} find the real values of $x$.
Problem 58: [Unipodal]: [alternative solution]
This problem on YouTube at
Source: https://www.youtube.com/watch?v=fkmU5DGp5gM
Title: Stanford University Entrance Interview Tricks
| Find x=?
Presenter: Super Academy
Given the relation \begin{equation} \big(7 + \frac{1}{x}\big)^4 - \big(5 + \frac{1}{x}\big)^4 = 240 \,, \end{equation} find the real values of $x$.
Problem 59: [Not Unipodal]:
This problem on YouTube at
Source: https://www.youtube.com/watch?v=Q6vFA8FqKzA
Title: Indian l Olympiad Math Algebric Exponential
| Find x=?
Presenter: Math Master TV
Given the relation
\begin{equation}
6^x + 9^x = 2^{2x+1} \,,
\end{equation}
find the integer values of $x$. [As I saw no requirement to treat $x$
as real, I had no incentive to do so. I would hope that on an actual test,
one would be told what the domain is.]
Problem 60: [Unipodal]: [introduces using imaginary numbers from the start]
This problem on YouTube at
Source: https://www.youtube.com/watch?v=_Rk0zyqGhO0
Title: Math Olympiad | A Nice Algebra Problem |
A Nice Radical Equation
Presenter: SALogic
Given the relation \begin{equation} x^2 - 2 = \sqrt{x+2} \,, \end{equation} find the values of $x$.
Problem 61: [Not Unipodal]:
This problem on YouTube at
https://www.youtube.com/watch?v=XxSzU_YH0gI
Titled: A Nice Algebra maths olympiad problem |
math olympiad questions
Presenter: SouL Institution
Let $a$ be a positive real number. Then, if \begin{equation} a^2 - 17a = 16\sqrt{a} \,, \end{equation} find the values of \begin{equation} \sqrt{a-\sqrt{a}} \,. \end{equation}
Problem 62: [Unipodal]:
This problem on YouTube at
Source: https://www.youtube.com/watch?v=jnupmpMDBUI
Title: Stanford University Entrance Aptitude Test
Advanced tricks
Presenter: Super Academy
Given the relation \begin{equation} (x - x^3)^{1/2} + (x^2 - x^3)^{1/2} = 1 \,, \end{equation} find the values of $x$.
Problem 63: [Unipodal]:
This problem on YouTube at
Source: https://www.youtube.com/watch?v=VVM6SxLnRf0
Title: Harvard University Aptitude Test
Presenter: Super Academy
Given the relations \begin{align} x^4 -y^4 &= 16\,,\\ xy &= 2\,, \end{align} find the values of $x,y \in Reals$.
Problem 64: [Not Unipodal]:
This problem on YouTube at
Source: https://www.youtube.com/watch?v=_-CY3tbAnjA
Titled: Harvard University Entrance Exam tricks
Presenter: Super Academy
Given the relation \begin{equation} \sqrt{x+4} + \sqrt{-x-4} = 4 \,, \end{equation} find the values of $x$ over the complex numbers.
Problem 65: [Unipodal]:
This problem on YouTube at
Source: https://www.youtube.com/watch?v=wGyFflQj2X00
Title: Advanced Math from Cambridge University
Entrance Exam
Presenter: Super Academy
Given the relations \begin{align} 3^x -3^y &= 16\,,\\ 3^{x+y} &= 4\,, \end{align} find the values of $x,y \in$ Reals.
Problem 66: [Unipodal]:
This problem on YouTube at
Source: https://www.youtube.com/watch?v=W4YsEql1QYA
Title: A tricky Algebra Problem from Stanford
University Admission Interview
Presenter: Super Academy
Given the relation \begin{equation} (\sqrt{10}+3)^{x} + (\sqrt{10}-3)^{x} = 38 \,, \end{equation} find the values of $x$.
Problem 67: [Unipodal]:
This problem on YouTube at
Source: https://www.youtube.com/shorts/I0hQSOopzfs
Title:Challenging Math Olympiad Question
Presenter: Soul Institution
This has been one of the most revealing problems on the nature
of my using the unipodal algebra to solve these 'olympiad'
problems. I proved a simple but important lemma. Additionally,
I added some unipodal heuristics for consideration.
Given the relation \begin{equation} \sqrt{x}+\sqrt{x+1}+\sqrt{x+2} = \sqrt{x+7}\,, \end{equation} find the values of $x$.
Problem 68: [Not Unipodal]:
This problem on YouTube at
Source: https://www.youtube.com/watch?v=W0dJwFFGXRw
Title: maths olympiad question || An algebraic
exponential problem
Presenter: Maths Curiosity
Given the relation \begin{equation} 2^a+2^b+2^c = 148 \,, \end{equation} find the integer values of $a,b,c$.
Problem 69: [Unipodal]:
This problem on YouTube at
Source: https://www.youtube.com/watch?v=ep6a75cg6F8
Title: Germany | Can you solve this ? | Amazing
Olympiad Math Problem
Presenter: Learn Communolizer
Given the relation \begin{equation} \sqrt{2}+\sqrt{x} = 2\,, \end{equation} find the values of $x$.
Problem 70: [Unipodal]:
This problem on YouTube at
Source: https://www.youtube.com/watch?v=rbypW8CQaGQ
Title: A very tricky Question from Oxford
University Entrance Exam
Presenter: Super Academy
Given the relation \begin{equation} \left( \frac{x+1}{x}\right)^2 - \left( \frac{x+1}{x-1}\right)^2 = 1\,, \end{equation} find the values of $x$.
Problem 71: [Not Unipodal: Complex Numbers]:
This problem on YouTube at
Source: https://www.youtube.com/watch?v=lfxCDP7y_D8
Title: A tricky Solution from Stanford University
Admission Interview
Presenter: Super Academy
Given the relation \begin{equation} K^2 = 8 i\,, \end{equation} find the values of $K$.
Solution to this problem.
Link to my write-up on Basic Complex Numbers.
Problem 72: [Not Unipodal]:
This problem on YouTube at
Source: https://www.youtube.com/watch?v=QnBd2aIVEcQ
Title: A very tricky Question from Stanford
University Entrance Exam
Presenter: Super Academy
Given the relation \begin{equation} a^2 + 2ab + b = 44\,, \end{equation} find the positive integer values of $a,b$. I will add to this the constraint that \begin{equation} a + b \le 10\,. \end{equation}
Problem 73: [Unipodal]:
This problem on YouTube at
Source: https://www.youtube.com/watch?v=xSITkGFst_A
Title: South Africa Math Olympiad Question
Presenter: LKLogic
Given the relations \begin{equation} x^2 - y^2 = 24\,,\ \end{equation} and \begin{equation} xy = 35\,, \end{equation} find the solutions to $x,y$.
Problem 74: [Unipodal]:
This problem on YouTube at
Source: https://www.youtube.com/watch?v=YoocXFxYMlg
Title: Advanced Math from Harvard University Entrance Exam
Presenter: Asad International Academy
Given the relation \begin{equation} x^2 + \left( \frac{3x}{x-3}\right)^2 = 16\,, \end{equation} find the values of $x$.
The following is a conventional solution: \begin{equation} -\left(\frac{3x}{x-3}\right)^2 = x^2-16 \,. \end{equation} On multiplying this out and placing the result into standard form, we get \begin{equation} x^4 - 6x^3 + 2x^2 + 96x - 144 = 0 \,. \end{equation} And WolframAlpha.com claims the solutions are \begin{align} x &= 4\pm2\sqrt{2}\, i\,, \\ x &= -1\pm\sqrt{7}\,. \end{align}
Problem 75: [Not Unipodal]:
This problem on YouTube at
Source: https://www.youtube.com/watch?v=QnBd2aIVEcQ
Title: A very tricky Question from Stanford
University Entrance Exam
Presenter: Super Academy
Given the relation \begin{equation} x-3\sqrt{x} = 1\,, \end{equation} find the value of the objective function \begin{equation} x^2 + \frac{1}{x^2}\,. \end{equation}
Problem 76: [Unipodal]:
This problem on YouTube at
Source: https://www.youtube.com/watch?v=J87-vA7DTbk
Title: Everything is possible | A tricky math question
Presenter: Higher Mathematics
Given the relations \begin{align} x + y &= 1\,,\\ xy &= 1\,, \end{align} find the values of $x,y$.
Problem 77: [Unipodal]:
This problem is by
Presenter: Patrick
This is a problem I made up, and is similar
to other problems already solved earlier on this page.
Given the relation
\begin{equation}
\left(x - \frac{1}{x}\right)^2 + \left(x +\frac{1}{x}\right)^2 = 2\,,
\end{equation}
find the values of $x$.
Problem 78: [Not Unipodal]:
This problem is by
Source: https://www.youtube.com/watch?v=NfgEPsm9Jzw
Title: The Hardest Exam Question | Only 6% of
students solved it correctly
Presenter: Higher Mathematics
Find the value of $(\sqrt{2}-1)^{10}$.
Note: I think it helps to know a few lines of Pascal's Triangle.
Solution to this problem.
Link to my write-up on Pascal's Triangle.
Problem 79: [Not Unipodal: Complex]:
This problem is by
Source: https://www.youtube.com/watch?v=Aao4AhD9q3A
Title: Mastering The Oxford University Entrance Exam
With These Easy Tricks
Presenter: Super Academy
Given the relation \begin{equation} \frac{5}{x} \frac{5}{x} = \frac{x}{5}\,, \end{equation} find the values of $x$.
Problem 80: [Not Unipodal]:
This problem is by
Source: https://www.youtube.com/watch?v=0IA6Pqs5Rwk
Title: Can You Pass Harvard's Entrance Exam Question ?
Presenter: Asad Internmational Academy
Given the relation \begin{equation} 2^x + 4^x + 8^x = 155\,, \end{equation} find the values of $x$.
Problem 81: [Not Unipodal]:
This problem is by
Source: https://www.youtube.com/watch?v=KbELy5Crhf8
Title: Harvard University Admission Exam
|| Logarithms Problem Tricks
Presenter: Super Academy
Given the relation \begin{equation} (x-5)^{\log (5x-25)} = 2\,, \end{equation} find the values of $x$.
Link to my write-up on logarithms over the real numbers.
(More recently I solved this problem quite differently. Find it at: Solution to the problem.)
Problem 82: [Not Unipodal]:
Link to my write-up on Geometric Series.
This problem is by
Source: https://www.youtube.com/watch?v=xGvJkmmQ9XM
Title: Harvard Entrance Exams || No
Calculator Allowed
Presenter: Maths Explorer
Find the value of \begin{equation} 8^5 + 8^4 + 8^3 + 8^2 + 8^1 + 8^0\,.\ \end{equation}
Problem 83: [Not Unipodal: Lambert]:
This problem is by
Source: https://www.youtube.com/watch?v=lI8DAgFfh9c
Title: Math question for a "true" geniuses
Presenter: Higher Mathematics
Given the relation \begin{equation} x^{\sqrt{x}} = 3\,, \end{equation} find the values of $x$.
Solution to this problem.
Link to my write-up on the Lambert W function.
Problem 84: [Not Unipodal]:
This problem is by
Source: https://www.youtube.com/watch?v=cdRpsmKhqx8
Title: Germany - Math Olympiad Problem | Be Careful!
Presenter: Higher Mathematics
Given the relation \begin{equation} a^4 = (a-1)^4\,, \end{equation} find the values of $a$.
Problem 85: [Not Unipodal]:
This problem is by
Source: ?
Title: ?
Presenter: ?
Given the relations \begin{align} \sqrt{x} + y &= 7\,,\\ x + \sqrt{y} &= 11\,, \end{align} find the values of $x,y \in$ Reals.
Problem 86: [Unipodal]:
This problem is by
Source: https://www.youtube.com/watch?v=QZ0LaMSN1EM
Title: STANFORD UNIVERSITY Admission Interview
Secrets Revealed!
Presenter: Super Academy
Given the relations \begin{align} \log x + \log y &= 5\,,\\ \log x \cdot \log y &= 5\,, \end{align} find the values of $x,y$.
Link to my write-up on logarithms over the real numbers.
Problem 87: [Not Unipodal]:
This problem is by
Source: https://www.youtube.com/watch?v=nuy6sQckL24
Title: A beautiful Question from Harvard University Entrance Exam
Presenter: Super Academy
Given the relation \begin{equation} \sqrt{2}^{\sqrt{m}} - \sqrt{2}^{\sqrt{n}} = 32,768\,, \end{equation} find the values of $m,n$ as integers.
Link to my write-up on logarithms over the real numbers.
Problem 88: [Not Unipodal]:
This problem is by
Source: https://www.youtube.com/watch?v=Ok_rPUyimZc
Title: Math Olympiad - Exponential Trigonometric
Problem - find x!
Presenter: Math Master TV
Given the relation \begin{equation} 81^{\sin^2 x} + 81^{\cos^2 x} = 30\,, \end{equation} find the values of $x$ in (radians) between 0 and $2\pi$.
Solution to this problem.
Link to my write-up on Basic Complex Numbers.
Problem 89: [Not Unipodal]:
This problem is by
Source: https://www.youtube.com/watch?v=AJrZvwWpZZU
Title: Spain | A Nice Algebra Problem | Math Olympiad
Presenter: SALogic
Given the relation \begin{equation} \sqrt{1+ \sqrt{\strut1+x}} = \sqrt[3]{x}\,, \end{equation} find the values of $x$.
Problem 90: [Unipodal]:
This problem is by
Source: https://www.youtube.com/watch?v=3FRNl9Ry8cs
Title: France | A Nice Algebra Problem | Math Olympiad
Presenter: SALogic
Given the relation \begin{equation} \sqrt{\strut x-1} + \sqrt{\strut x+2} = 3\,, \end{equation} solve for $x$ values.
Problem 91: [Not Unipodal]:
This problem is by
Source: https://www.youtube.com/watch?v=mh5UPPaJLXY
Title: Can You Solve This 12-Year-Old’s Math
Problem from China?
Presenter: Global Maths
Given the relation \begin{equation} (x + 100)^2 = 2x + 199\,, \end{equation} find the values of $x$.
One possible substitution is to set $y=x+100$ (which exposes
a double root). I chose a different substitution to experiment
with another technique.
Problem 92: [Unipodal]:
This problem is by
Source: https://www.youtube.com/watch?v=9zkyOc3a2uo
Title: Olympiad Math Question || Nice Algebra Equation
Presenter: Pages by Aapi
Given the relations \begin{align} x + y &= 1\,,\\ x^4 + y^4 &= 881\,, \end{align} find the value of $xy$.
Problem 93: [Not Unipodal]:
This problem is by
Source: https://www.youtube.com/watch?v=X5RhYu88EYI
Title: Mexico | A Nice Algebra Problem
Presenter: SALogic
\begin{equation} \frac{(1+i)^{2024}}{(1-i)^{2023}}\,.\label{eq:Given.Expression} \end{equation}
Solution to this problem.
Link to my write-up on the Complex Numbers.
Problem 94: [Unipodal]:
This problem is by
Source: https://www.youtube.com/watch?v=NWxNdeiQ1Vw
Title: A nice mathematics algebra exponential problem
Presenter: Mathematics and Statistics Guru
Given the relation \begin{equation} x^6 + \frac{1}{x^6} = 488\,, \end{equation} solve for the values of \begin{equation} k = x + \frac{1}{x} \,, \end{equation} for $x$ over the positive reals.
Problem 95: [Not Unipodal]:
This problem is by
Source: https://www.youtube.com/watch?v=Xw8VBHXK81M
Title: Nice Exponential Math Problem |
Harvard Entrance Exam Question
Presenter: SchoolClass Math
Given the relation \begin{equation} 20^k\cdot 50^k = 8\,, \end{equation} find the values of $k$.
Problem 96: [Unipodal]:
This problem is by
Source: https://www.youtube.com/watch?v=Zly87p7fsqY
Title: Harvard University | Can you solve this ?
Presenter: Basic concept of Math
Given the relations \begin{align} 6^x +6^y &= 42\,,\\ x + y &= 3\,, \end{align} find the values of $x,y$.
Problem 97: [Unipodal]:
This problem is by
Source: https://www.youtube.com/watch?v=5pa1AryylpM
Title: believe in the math, not wolframalpha
Presenter: Blackpenredpen
Given the relation \begin{equation} x = \cuberoot{7+\sqrt{50}} + \cuberoot{7-\sqrt{50}} \,, \end{equation} solve for the (real) values of $x$ more simply.
Problem 98: [Unipodal]:
This problem is by
Source: ?
Title: ?
Presenter: ?
Given the relation \begin{equation} 9^{x+1} - 9^{x-1} = 20\,, \end{equation} solve for the (real) values of $x$.
Problem 99: [Not Unipodal]:
This problem is by
Source: https://www.youtube.com/watch?v=t6Vzq4DnQH8
Title: A tricky Stanford University Admission Algebra Interview
Presenter: Super Academy
Given the relation \begin{equation} \left(\frac{2}{3}\right)^x + \left(\frac{3}{2}\right)^x = 4\,, \end{equation} find the values of $x$.
Problem 100: [Unipodal]:
This problem is by
Source: https://www.youtube.com/watch?v=2DdEQWgBlIc
Title: A Nice System Of Logs
Presenter: SyberMath
Given the relations \begin{align} \log_yx + \log_xy &= \frac{26}{5}\,,\\ x y &= 64\,, \end{align} find the values of $x,y$ in the positive reals.
Link to my write-up on logarithms over the real numbers.
Problem 101: [Not Unipodal]:
This problem is by
Source: https://www.youtube.com/watch?v=S64w7YxWoTs
Title: France | Olympic math question - past exam
SAT MATH QUESTION
Presenter: Kmath addict
Given the relation \begin{equation} 5^x +35^{x/2} = 7^x\,, \end{equation} find the values of $x$.
Problem 102: [Not Unipodal]:
This problem is by
Source: https://www.youtube.com/watch?v=nL8SN1xaSFc
Title: Cambridge University Admission Exam Tricks
Presenter: Super Academy
Given the relations \begin{align} x^2 + xy + y^2 &= 96\,,\\ x + \sqrt{xy} + y &= 16\,, \end{align} find the values of $x,y \in$ Reals.
Problem 103: [Unipodal]:
This problem is by
Source: https://www.youtube.com/watch?v=5y7JXP7Ow2o
Title: ab= ? | Harvard MIT Math Tournament 2016 | HMMT
Presenter: Math Training
Given the relations \begin{align} a + b^{-1} &= 4\,,\\ a^{-1} + b&= \frac{16}{15}\,, \end{align} find the values of $ab$.
Problem 104: [Not Unipodal]:
This problem is by
Source: https://www.youtube.com/watch?v=ZlDFHJqUziE
Title: Solving a nice integral - Solution to past exam
Presenter: Kmath addict
Find the value of the following definite integral: \begin{equation} \int_0^1 \ln\, (1 + x^2)\, dx\,. \end{equation}
Problem 105: [Unipodal]:
This problem is by
Source: https://www.youtube.com/watch?v=nbWqIwTZrSg
Title: Harvad University Admission test
Presenter: Super Academy
Given the relation \begin{equation} \sqrt{x+7} - \sqrt{x-9} = 2\,, \end{equation} find the values of $x$.
Problem 106: [Unipodal]:
This problem is by
Source: https://www.youtube.com/watch?v=yl_0f9wJQeQ
Title: How to solve this nice math rational
Exponential algebra problem
Presenter: Mathematics & Statistics Guru
Given the relation \begin{equation} \frac{x}{y} + \frac{y}{x} = 1\,, \end{equation} find the value of \begin{equation} \left(\frac{x}{y}\right)^{153} + \left(\frac{y}{x}\right)^{153} = z\,, \end{equation} where $z$ is introduced now for convenience.
Problem 107: [Not Unipodal]:
This problem is by
Source: https://www.youtube.com/watch?v=bcG73tJvGz4
Title: Advance Algebra | Olympiad Mathematics
Presenter: Master T Maths Classes
Given the relation \begin{equation} 4^{x+1} - 4^{x-1} = 25\,, \end{equation} find the values of $x$.
Problem 108: [Not Unipodal]:
This problem is by
Source: https://www.youtube.com/watch?v=IGUh3Mv7DAI
Title: One Of The Most Difficult Harvard's
University Entrance Exam
Presenter: Maths with Chinwendu
Given the relation \begin{equation} x^{x} y^{y} = 8 x^{y} y^{x}\,, \end{equation} find the smallest positive integer values of $x,y$ with $x>y$.
Problem 109: [Not Unipodal: Lambert]: Updated
This problem is by
Source: https://www.youtube.com/watch?v=EO1d9GxEcaI
Title: Harvard MIT Math Tournament | HMMT 2018 |
Presenter: Math Gold Metalist
Given the relation \begin{equation} x^{2x^2} = 3\,, \end{equation} find the real values of $x$.
Solution to this problem.
Link to my write-up on the Lambert W function.
Problem 110: [Unipodal]:
The following problem is an adaptation of a YouTube problem
whose reference I lost.
Given the relations \begin{align} x^2 + y^2 &= 6\,,\\ x + y &= 3\,, \end{align} find the value of $x^{-1} - y^{-1}$.
Problem 111: [Not Unipodal]:
This problem is by
Source: https://www.youtube.com/watch?v=uofFyT0d1Wc
Title: A Nice Algebra Equations
Presenter: MathMinds
Given the relations \begin{align} x^2 + xy + x &= 10\,,\\ y^2+ xy + y &= 20\,, \end{align} find the values of $x,y$.
Problem 112: [Unipodal]:
This problem is by
Source: https://www.youtube.com/watch?v=MKsA2YkmVc4
Title: A Very Nice Math Olympiad Problem
Presenter: Spencer's Academy
Given the relations \begin{align} x + y &= 4\,,\\ x^5 + y^5 &= 464\,, \end{align} find the value of $x,y$.
Problem 113: [Not Unipodal]:
This problem is by
Source: https://www.youtube.com/watch?v=ZQZMu0SraSQ
Title: Can you Pass Harvard University Admission Interview
Presenter: Enjoy Math
Given the relation \begin{equation} \sqrt{\strut\frac{\strut 4^{20} - 2^{21} + 1}{\strut 2^{20} + 2^{11} + 1}} = 2^{x} - 1\,, \end{equation} find the values of $x$.
Problem 114: [Not Unipodal]:
This problem is by
Source: https://www.youtube.com/watch?v=VgIlmBmYKCE
Title: Harvard University Entrance Exam
Presenter: SchoolClass Math
Given the relation \begin{equation} m^2 - m^3 = 36\,, \end{equation} find the values of $m$
Problem 115: [Unipodal]:
This is one of my favorite problems so far. I solved it
in essentially three different ways.
This problem is by
Source: https://www.youtube.com/watch?v=Ib_sSaAEaUE
Title: Oxford University Entrance Exam Tricks
Presenter: Super Academy
Given the relations \begin{align} 4 x^2 - 4y^2 &= 1\,,\\ 4xy &= 1\,, \end{align} find the real values of $x+y$.
Problem 116: [Unipodal]:
This is one of my favorite problems so far. I solved it
in essentially three different ways.
This problem is by
Source: ?
Title: ?
Presenter: ?
Given the relation \begin{equation} x^2 = (5 - \sqrt{24})^x\,, \end{equation} find the values of $x \in$ Reals.
(More recently I solved this problem quite differently. Find it at: Solution to the problem.)
Problem 117 [Unipodal]:
This is one of my favorite problems so far. I solved it
in essentially three different ways.
This problem is by
Source: https://www.youtube.com/watch?v=9MogStH2q3Q
Title: A tricky Algebra from Stanford University
Advanced Aptitude test
Presenter: Super Academy
Given the relation \begin{equation} \sqrt{3+\sqrt{x+9}} = \cuberoot{x}\,,\ \end{equation} find the values of $x \in$ the positive reals.
Problem 118 [Not Unipodal]:
This problem is by
Source: https://www.youtube.com/watch?v=wuMld8B9d5k
Title: A Nice Math Olympiad Problem
Presenter: Maths Black Board
Given the relations \begin{align} \sqrt{a} + \sqrt{2-b} &= \sqrt{2}\,,\\ \sqrt{b} + \sqrt{2-a} &= \sqrt{2}\,, \end{align} find the values of $a,b$.
Problem 119 [Unipodal]:
This problem is by
Source: https://www.youtube.com/watch?v=yGmVy0FZdXY
Title: How to Solve Algebraic Equations
Presenter: Maths Black Board
Given the relations \begin{align} a^2 - ab &= 14\,,\\ b^2 + ab &= 60 \,,\ \end{align} find the values of $a,b$.
Problem 120 [Unipodal]:
This problem is by
Source: https://www.youtube.com/watch?v=nv-c8W6keJ0
Title: Portugal | A Nice Algebra Problem | Math Olympiad
Presenter: SALogic
Given the relation \begin{equation} (x+11)^4 - (x+9)^4 = 80\,, \end{equation} find the values of $x \in$ Reals.
Problem 121 [Unipodal]:
This problem is by
Source: https://www.youtube.com/watch?v=G3KLDzcvKYo
Title: Japanese | Can you solve this ?
Presenter: Learncommunolizer
Given the relations \begin{align} x + y &= 1\,,\\ xy &= 1\,, \end{align} find the real values of $x,y$.
Problem 122 [Not Unipodal]:
This problem is by
Source: https://www.youtube.com/watch?v=Ycyag8jY1TQ
Title: So, you want a HARD math question with exponents?
Presenter: Higher Mathematics
Given the relation \begin{equation} x^2 = 4^x\,, \end{equation} find the real values of $x$.
Problem 123 [Not Unipodal]:
This problem is by
Source: https://www.youtube.com/watch?v=ggDoEEcdBp0
Title: Crack This Olympiad Exponential Equation
Presenter: GT Academix
Given the relation \begin{equation} x^{x^{x^3}} = 3\,, \end{equation} find the real values of $x$.
Problem 124[Not Unipodal]:
This problem is by
Source: https://www.youtube.com/watch?v=nJwQGTPIyUM
Title: A Nice Math Olympiad Exponential Equation x^x^2 = 16
Presenter: MrMath
Given the relation \begin{equation} x^{x^{2}} = 16\,, \end{equation} find the real values of $x$.
Problem 125[Not Unipodal]:
This problem is by
Source: ?
Title: ?
Presenter: ?
Given the relation \begin{equation} x = \frac{15}{ 2+ \frac{15}{2+ \frac{15}{ 2+\frac{15}{\cdots}}}}\,, \end{equation} find the real values of $x$.
Problem 126 [Unipodal]:
This problem is by
Source: https://www.youtube.com/watch?v=D-TDK7pQafI
Title: Can you solve this? | Oxford entrance exam question
Presenter: Enjoy Math
Given the relations \begin{align} x &= \frac{\sqrt{7} + \sqrt{3}}{\sqrt{7} - \sqrt{3}}\,,\\ y &= \frac{\sqrt{7} - \sqrt{3}}{\sqrt{7} + \sqrt{3}}\,, \end{align} find the values of \begin{equation} x^4 + y^4 \,, \end{equation} over the reals.
Problem 127 [Not Unipodal]:
This problem is by
Source: https://www.youtube.com/watch?v=Uqk2MS5Lcsk
Title: Math Olympiad | Algebra Problem
Presenter: MathsFocus
Given the relation \begin{equation} \frac{2^{33}+2^{22}+2^{11}}{2^{33}-1}\,, \end{equation} find its value.
Problem 128 [Not Unipodal]:
This problem is by
Source: https://www.youtube.com/watch?v=VJmVcpfcy3Q
Title: A Tricky Math Olympiad challenge
Presenter: MathsFocus
Given the relation \begin{equation} x^{x^6} = \sqrt{2}^{\sqrt{2}}\,, \end{equation} find the values of $x$.
Problem 129 [Unipodal]:
This problem is by
Source: https://www.youtube.com/shorts/qAxXkDFRvHA
Title: Can you solve this Harvard
Presenter: EngineeringMathShorts
Given the relations \begin{align} x - y &= 4\,,\\ x^3 - y^3 &= 28\,, \end{align} find the values of $x,y$.
Problem 130 [Unipodal]:
This problem is by
Source: https://www.youtube.com/watch?v=NwbqN7myYhY
Title: Harvard University interview exponential math question
Presenter: JJ Online Maths Class
Given the relation \begin{equation} x^2 - x +1 = 0\,, \end{equation} find the values of \begin{equation} x^{2020} + x^{1010} - 1\,. \end{equation}
Problem 131 [Unipodal]:
This problem is by
Source: https://www.youtube.com/watch?v=nXK5scl4DJI
Title: A tricky Entrance Interview Question
from Harvard University
Presenter: Super Academy
Given the relations \begin{align} 3^{x-4}+ 3^{y-4} &= 244\,,\\ x+y &= 13\,, \end{align} find the values of $x,y$.
Problem 132 [Not Unipodal]:
This problem is by
Source: https://www.youtube.com/watch?v=oMhDv9tydW0
Title: Can you Solve a tricky Entrance Exam from
Cambridge University ?
Presenter: Super Academy
Given the relation \begin{equation} \frac{16}{x} - \frac{8}{x^2} + \frac{4}{x^3} - \frac{2}{x^4} + \frac{4}{x^5} = 32\,, \end{equation} find the values of $x$.
Problem 133 [Not Unipodal]:
This problem is by
Source: https://www.youtube.com/watch?v=XXFNxw-2bo8&list
=PLQG1LDIiB85gGzcpH00P_riNifgU_ZOQW&index=22
Title: A Nice Algebra Equation
Cambridge University ?
Presenter: MathMinds
Given the relation \begin{equation} x + \sqrt{x} = x\sqrt{x}\,, \end{equation} find the values of $x$ over the positive reals.
Problem 134 [Unipodal]:
This problem is by
Source: https://www.youtube.com/watch?v=vkdbgkAeVNk&list
=PLQG1LDIiB85gGzcpH00P_riNifgU_ZOQW&index=47
Title: A Nice Math Olympiad Algebra Problem
Presenter: MathMinds
Given the relation \begin{equation} x + \frac{1}{x} = \sqrt{3}\,, \end{equation} solve for the values of \begin{equation} x^{50} + \frac{1}{x^{50}} \,. \end{equation}
Problem 135 [Not Unipodal]:
This problem is by
Source: https://www.youtube.com/watch?v=67234_o6z-4
Title: Germany | A nice Logarithmic Math Olympiad Problem
Presenter: Super Academy
Given the relation \begin{equation} x^{\log 27} + 9^{\log x} = 36\,, \end{equation} find the values of $x$ over the real numbers. By the way, I interpret the logarithm as being in base 10.
Link to my write-up on logarithms over the real numbers.
Problem 136 [Not Unipodal: Lambert]:
This problem is by
Source: https://www.youtube.com/watch?v=MRTDBSwlH2I
Title: Harvard University Admission Exam Tricks
Presenter: Super Academy
Given the relation \begin{equation} 64^{x} = x^{192} \,, \end{equation} find the values of $x$ over the real numbers.
Solution to this problem.
Link to my write-up on the Lambert W function.
Problem 137 [Not Unipodal]:
This problem is by
Source: https://www.youtube.com/watch?v=nZmxQYLaH2g
Title: Germany | Can you solve this?
Presenter: Master T Maths Class
Given the relation \begin{equation} 25^{2x} = 50 \,, \end{equation} find the values of $x$ over the real numbers.
Problem 138 [Not Unipodal]:
This problem is by
Source: https://www.youtube.com/watch?v=7zwAqqH6C58&list
=PLMvuVeOn1Hd_KIT-dsvIVluQQN3pJrlmX&index=691
Title: Nice Algebra Math Simplification
Presenter: Master T Maths Class
Given the relation \begin{equation} \left(\frac{x}{5}\right)^x = 5^{5^2} \,, \end{equation} find the values of $x$ over the real numbers.
Problem 139 [Unipodal]:
This problem is by
Source: https://www.youtube.com/shorts/5Rfa6lDU3XY
Title: A Nice Olympiad Algebra Problem
Presenter: Numbers.Numbers
Given the relation \begin{equation} a + \frac{1}{a} =6 \,, \end{equation} find the values of \begin{equation} a - \frac{1}{a}\,. \end{equation}
Problem 140 [Not Unipodal]:
This problem is by
Source: https://www.youtube.com/watch?v=Klef16FHNK4
Title: Japanese | Can you solve this?
Presenter: Master T Maths Class
Given the relation \begin{equation} 4^x + 25^x = 10^{x+1} \,, \end{equation} find the values of $x$ over the real numbers.
Problem 141 [Not Unipodal]:
This problem is by
Source: https://www.youtube.com/watch?v=Gz6VLgvFLHs
Title: A nice Math Olympiad Simplification Problem
Presenter: Super Academy
Given the relation \begin{equation} 3^{3x-3x^2} = x^2 - x \,, \end{equation} find the values of $x$ over the real numbers.
Problem 142 [Unipodal]:
This problem is by
Source: https://www.youtube.com/shorts/5Rfa6lDU3XY
Title: A Nice Olympiad Algebra Problem
Presenter: Numbers.Numbers
Given the relation \begin{equation} a = \sqrt{5} + \sqrt{6} \,, \end{equation} find the values of the following expression over the real numbers. \begin{equation} a^2 + \frac{1}{a^2}\,. \end{equation}
Problem 143 [Not Unipodal]:
This problem is by
Source: https://www.youtube.com/watch?v=PvE7HiRakQk
Title: A Nice Algebra Problem | Math Olympiad
Presenter: SALogic
Given the relation \begin{equation} x^{x} = 4^{4+ x} \,, \end{equation} find the values of $x$ over the real numbers.
Problem 144 [Unipodal]: (One of my favorites!)
This problem is by
Source:https://www.youtube.com/watch?v=7KsIray2pXI&list
=PLMvuVeOn1Hd_KIT-dsvIVluQQN3pJrlmX&index=321
Title: China | Can you solve this ?
Presenter: Masters T Maths Class
Given the relations \begin{align} a^2 + b^2 &= 7\,,\\ a^3 + b^3 &= 10\,, \end{align} find the values of $a + b$.
Problem 145 [Not Unipodal]:
This problem is by
Source: https://www.youtube.com/watch?v=7zwAqqH6C58&list
=PLMvuVeOn1Hd_KIT-dsvIVluQQN3pJrlmX&index=691
Title: Nice Algebra Math Simplification
Presenter: Master T Maths Class
\begin{equation} \left(\frac{x}{5}\right)^{x} = 5^{5^2} \,, \end{equation} find the values of $x$ over the real numbers.
Problem 146 [Not Unipodal]:
This problem is by
Source: https://www.youtube.com/watch?v=CeLKQg9z7LM
Title: Nice Algebra Math Problem
Presenter: Master T Maths Class
Given the relation \begin{equation} x^{\sqrt{x}} = 4^{2} \,, \end{equation} find the values of $x$ over the real numbers.
Problem 147 [Unipodal]:
This problem is by
Source: https://www.youtube.com/watch?v=lSRh6UJdrlo
Title: Germany | Can you solve?
Presenter: Masters T Maths Class
Given the relations \begin{align} x^2 + y^2 &= 61\,,\\ x-y &= 11\,, \end{align} find the values of $x,y$.
Problem 148 [Not Unipodal: $\alpha$ substitution]:
This problem is by
Source: https://www.youtube.com/watch?v=X5iYlxCwGRo&list
=PLMvuVeOn1Hd_KIT-dsvIVluQQN3pJrlmX&index=46
Title: A Nice Exponential Equation
Presenter: Master T Maths Class
Given the relation \begin{equation} x^{x^{1+x}} = 256 \,, \end{equation} find the values of $x$ over the real numbers.
Problem 149 [Not Unipodal]:
This problem is by
Source: https://www.youtube.com/watch?v=9ljcSSCP8oE
Title: A Nice Algebra Problem | Math Olympiad
Presenter: SALogic
Given the relation \begin{equation} 2^x - 9^x = \sqrt{18^x-81^x} \,, \end{equation} find the values of $x$ over the real numbers.
Problem 150 [Not Unipodal]:
This problem is by
Source: https://www.youtube.com/watch?v=VayJZrGnbLI
Title: Russisa | A Nice Algebra Problem
Presenter: SALogic
Given the relation \begin{equation} \cuberoot{x} + \sqrt{x} = \frac{4}{27} \,, \end{equation} find the values of $x$ over the real numbers.
Problem 151 [Unipodal]:
This problem is by
Source: https://www.youtube.com/watch?v=dSwi5VSxMWI
Title: A Nice Olympiad Algebra Problem
Presenter: Master T Maths Class
Given the relations \begin{align} x^2 - y^2 &= 27\,,\\ xy &= 18\,, \end{align} find the values of $x+y$.
Problem 152 [Unipodal]:
This problem is by
Source: https://www.youtube.com/watch?v=06tHCsynl8w
Title: Nice Square Root Simplification
Presenter: Master T Maths Class
Given the relation \begin{equation} \sqrt{x} - \sqrt{x-2} = 1 \,, \end{equation} find the values of $x$.
Problem 153 [Unipodal]:
This problem is by
Source: https://www.youtube.com/watch?v=7P4eF0fV3gI
Title: Can You Crack This Radical Equation?
Presenter: InfyGyan
Given the relation \begin{equation} \sqrt{\cuberoot{33+x}} + \sqrt{\cuberoot{32-x}} = 3 \,, \end{equation} find the real values of $x$
Problem 154 [Unipodal]:
This problem is by
Source: https://www.youtube.com/watch?v=cCPdn0gaIVA
Title: China | Can you solve this?
Presenter: Master T Maths Class
Given the relations \begin{align} x^5 + y^5 &= 152\,,\\ x + y &= 2\,, \end{align} find the values of $x,y$.
Problem 155 [Unipodal]:
This problem is by
Source: https://www.youtube.com/watch?v=UwHSgLhjiHE
Title: Germany | Can you solve this? ?
Presenter: Master T Maths Class
Given the relation \begin{equation} (\sqrt{10}+3)^x + (\sqrt{10}-3)^x = 38\,, \end{equation} find the values of $x$.
Problem 156 [Not Unipodal: Lambert]:
This problem is by
Source: https://www.youtube.com/watch?v=U5Svh3S7Ziw
Title: A Nice Math Olympiad Algebra Problem
Presenter: Master T Maths Class
Given the relation \begin{equation} x^{ \sqrt{x}} = 10 \,, \end{equation} find the values of $x$ over the real numbers.
Solution to this problem.
Link to my write-up on the Lambert W function.
Problem 157 [Not Unipodal, Complex Numbers]:
This problem is by
Source: https://www.youtube.com/watch?v=3a0EF9j1jIk
Title: A Homemade Equation | Problem 292
Presenter: aplusbi
Given the relation \begin{equation} \obz + 2|z| = 13 - 4i \,, \end{equation} find the values of $z$ over the complex numbers.
Solution to this problem.
Link to my write-up on Basic Complex Numbers.
Problem 158 [Not Unipodal, Complex Numbers]:
This problem is by
Source: https://www.youtube.com/watch?v=3a0EF9j1jIk
Title: I Couldn't Solve This Equation - WA did! | 428
Presenter: aplusbi
Given the relation \begin{equation} \obz + z|z| = 24- 12i \,, \end{equation} find the values of $z$ over the complex numbers.
Solution to this problem.
Link to my write-up on Basic Complex Numbers.
Problem 159 [Unipodal]:
This problem is by
Source: https://www.youtube.com/watch?v=nBu-7ughFE8&list
=PLMvuVeOn1Hd_KIT-dsvIVluQQN3pJrlmX&index=158
Title:Unlocking the Secrets of a Mind-Bending Math
Olympiad Trigonometric Challenge!
Presenter: Master T Maths Class
Given the relation \begin{equation} \sin x + \cos x = \frac{1}{5}\,, \end{equation} find the values of $\tan x$ where \((0 < x < \pi)\).
Solution to this problem.
Link to my write-up on Trigonometric functions.
Problem 160 [Unipodal]:
This problem is by
Source: https://www.youtube.com/watch?v=nBu-7ughFE8&list
=PLMvuVeOn1Hd_KIT-dsvIVluQQN3pJrlmX&index=158
Title:Unlocking the Secrets of a Mind-Bending Math
Olympiad Trigonometric Challenge!
Presenter: Master T Maths Class
Given the relations \begin{align} a^2 + b^2 &= 74\,,\\ ab &= 35\,, \end{align} find the values of $a,b$.
Problem 161 [Not Unipodal]:
This problem is by
Source: https://www.youtube.com/watch?v=V1ye2oXNp-Y
Title: Russian | Can you solve this ?
Presenter: Master T Maths Class
Given the relation \begin{equation} x^{-x^x} = 2^{\sqrt{2}} \,, \end{equation} find the values of $x$ over the real numbers.
Problem 162 [Not Unipodal, Complex Numbers]:
This problem is by
Source: https://www.youtube.com/watch?v=iEWZXZ8XLew
Title: A Sum of Powers | Problem 421
Presenter: aplusbi
Given the relation \begin{equation} \phi = \left(\frac{1-i}{1+i} \right)^2 + \left(\frac{1-i}{1+i} \right)^3 + \left(\frac{1-i}{1+i} \right)^4 \,, \end{equation} find the values of $\phi$ over the complex numbers.
Solution to this problem.
Link to my write-up on Basic Complex Numbers.
Problem 163 [Not Unipodal]:
This problem is by
Source: https://www.youtube.com/watch?v=9ljcSSCP8oE
Title: A Nice Algebra Problem | Math Olympiad
Presenter: SALogic
Given the relation \begin{equation} 2^{x} - 9^x = \sqrt{ 18^{x} - 81^x} \,, \end{equation} find the values of $x$ over the real numbers.
Problem 164 [Unipodal]:
This problem is by
Source: https://www.youtube.com/watch?v=nbWqIwTZrSg
Title: Harvad University Admission test
Presenter: Super Academy
Given the relation \begin{equation} \sqrt{x+7} - \sqrt{x-9} = 2\,, \end{equation} find the values of $x$
Solution to this problem.
Link to my write-up on logarithms over the real numbers.
Problem 165 [Not Unipodal]:
This problem is by
Source: https://www.youtube.com/watch?v=fSfl-Ch1ujI
Title: Japanese | Can you solve this?
Presenter: Master T Maths Class
Given the relation \begin{equation} \log_2 x= \log_x4 \,, \end{equation} find the values of $x$ over the real numbers.
Link to my write-up on logarithms over the real numbers.
Problem 166 [Unipodal]:
This problem is by
Source: https://www.youtube.com/watch?v=MjmCfjALFdw
Title: Germany | Can you solve this ?
Presenter: Master T Maths Class
Given the relations \begin{align} x^2 + y^2 &= 61\,,\\ x - y &= 11\,, \end{align} find the values of $x,y$.
Problem 167 [Not Unipodal, Complex Numbers]:
This problem is by
Source: https://www.youtube.com/watch?v=A1shWBxiVns
Title: A Tangential Equation | Problem 361
Presenter: aplusbi
Given the relation \begin{equation} \tan z = -\frac{3i}{5}\,, \end{equation} find the values of $z$.
Solution to this problem.
Link to my write-up on Basic Complex Numbers.
Problem 168 [Unipodal]:
This problem is by
Source: https://www.youtube.com/watch?v=x-IHueYbzuQ
Title: Believe in triangles, not squaring both sides!
Presenter: blackpenredpen
Given the relation \begin{equation} x = \sqrt{x -\frac{1}{x}} + \sqrt{1 -\frac{1}{x}}\,, \end{equation}
Problem 169 [Not Unipodal]:
This problem is by
Source: https://www.youtube.com/watch?v=MAvbLgfzcww
Title: Nice Exponent Math Simplification
Presenter: Master T Maths Class
Given the relation \begin{equation} x^{x} = 3^{2x+27} \,, \end{equation} find the values of $x$ over the real numbers
Problem 170 [Not Unipodal]:
This problem is by
Source: ?
Title: ?
Presenter: ?
Given the relation \begin{equation} x^{7} + \frac{1}{x^{7}} = 6 \,, \end{equation} what is \begin{equation} \phi = x^{21} + \frac{1}{x^{21}}\,? \end{equation}
Problem 171 [Not Unipodal, Complex Numbers]:
This problem is by
Source: https://www.youtube.com/watch?v=uguB0e_8ygs
Title: A Natural Log Problem With Complex Numbers
Presenter: aplusbi
Given the relation \begin{equation} \ln \big(iz + \sqrt{1 - z^2} \big) = \frac{i\pi}{3}\,, \end{equation} find the values of $z$ over the real numbers.
Solution to this problem.
Link to my write-up on Hyperbolic trig functions over the real numbers.
Link to my write-up on Basic Complex Numbers.
Problem 172 [Not Unipodal, Complex Numbers]:
This problem is by
Source: https://www.youtube.com/watch?v=07Elc3M7Tk4
Title: A Proportional Problem of Ratios | Problem 294
Presenter: aplusbi
Given the relation \begin{equation} \frac{b+ai}{a+bi} = \frac{4+i}{1+4i} \,, \end{equation} find the values of $a,b$ over the real numbers.
Solution to this problem.
Link to my write-up on Basic Complex Numbers.
Problem 173 [Not Unipodal, Complex Numbers]:
This problem is by
Source: https://www.youtube.com/watch?v=JxCDIBqtf18
Title: Solving A Viewer Suggested Problem | Problem 238
Presenter: aplusbi
Given the relations $z=a+bi$ and \begin{equation} (a+bi)^2 = b+ia\,, \end{equation} find the values of $z$.
Solution to this problem.
Link to my write-up on Basic Complex Numbers.
Problem 174 [Unipodal]:
This problem is by
Source: https://www.youtube.com/watch?v=bmsMH_N7l_M
Title: Japanese | Can you solve this?
Presenter: Master T Maths Class
Given the relation \begin{equation} \frac{3}{9^{-x}} + \frac{10}{6^{-x}}= 4^{x} \,, \end{equation} find the real values of $x$.
Problem 175 [Not Unipodal, Complex Numbers]:
This problem is by
Source: https://www.youtube.com/watch?v=GnE_6JtOpio
Title: A Quatrodecic Equation | Problem 290
Presenter: aplusbi
Given the relations $z=a+bi = re^{i\theta}$ and \begin{equation} z^{14} = \obz\,, \end{equation} find the values of $z$.
Solution to this problem.
Link to my write-up on Basic Complex Numbers.
Problem 176 [Not Unipodal, Complex Numbers]:
This problem is by
Source: https://www.youtube.com/watch?v=WhS4BuIqlSc
Title: A Locus Problem | Problem 191
Presenter: aplusbi
Given the relations \begin{equation} \left|\frac{z}{z + 2}\right| = 3\,, \end{equation} find the values of $x,y$ in $z = x + iy$.
Solution to this problem.
Link to my write-up on Basic Complex Numbers.
Problem 177 [Not Unipodal, Complex Numbers]:
This problem is by
Source: https://www.youtube.com/watch?v=mPauY95OQLI
Title: Let's Solve A Problem With Complex Exponentiation
| Problem 164
Presenter: aplusbi
Given the relations \begin{equation} z^z= e^{\textstyle-\frac{\pi}{2}}\,, \end{equation} find the real values of $z$
Solution to this problem.
Link to my write-up on Basic Complex Numbers.
Problem 178 [Not Unipodal, Complex Numbers]:
This problem is by
Source: https://www.youtube.com/watch?v=ca_BOHCmRbo
Title: An Imaginarily Exponential Equation
| Problem 386
Presenter: aplusbi
Given the relation \begin{equation} z = i^z\,, \end{equation} find the complex values of $z$.
Solution to this problem.
Link to my write-up on Basic Complex Numbers.
Problem 179 [Not Unipodal, Complex Numbers]:
This problem is by
Source: https://www.youtube.com/watch?v=K0k4hdiiDBM
Title: A Ratio of Two Complex Numbers |
Problem 394
Presenter: aplusbi
Given the relation \begin{equation} \phi =\left( \frac{1+\sqrt{3} i }{1-\sqrt{3} i} \right)^{10}\,, \end{equation} simplify.
Solution to this problem.
Link to my write-up on Basic Complex Numbers.
Problem 180 [Not Unipodal, Complex Numbers]:
This problem is by
Source: https://www.youtube.com/watch?v=FhpAdhRMyk0
Title: A Nice Problem from A Nice Book |
Problem 377
Presenter: aplusbi
Given the relation \begin{equation} z + 2\obz = \frac{ 2-i }{1+3 i}\,, \end{equation} find the complex values of $z$.
Solution to this problem.
Link to my write-up on Basic Complex Numbers.
Problem 181 [Not Unipodal, Complex Numbers: Lambert]:
This problem is by
Source: https://www.youtube.com/watch?v=OMEu36Pth_8
Title: An Exponential Equation and A Special Function |
Problem 200
Presenter: aplusbi
Given the relation \begin{equation} z i^z = 4\,, \end{equation} find the complex values of $z$.
Solution to this problem.
Link to my write-up on the Lambert W function.
Link to my write-up on Basic Complex Numbers.
Problem 182 [Unipodal]:
This problem is by
Source: https://www.youtube.com/watch?v=cugNGIoe86o
Title: Interesting radical equation
Presenter: Chidex Online Math Class
Given the relation \begin{equation} \sqrt{9-x^2} + \sqrt{4-x^2} = 4\,, \end{equation} find the real values of $x$.
Problem 183 [Not Unipodal, Complex Numbers]:
This problem is by
Source: https://www.youtube.com/watch?v=BecmY7Gw85g
Title: A Problem With z And Its Conjugate |
Problem 218
Presenter: aplusbi
Given the relation \begin{equation} \frac{1}{z} + \frac{1}{\obz} = 4\,, \end{equation} find the complex values of $z$ (locus of points).
Solution to this problem.
Link to my write-up on Basic Complex Numbers.
Problem 184 [Not Unipodal, Complex Numbers]:
Pending completion.
Problem 185 [Not Unipodal, Complex Numbers]:
This problem is by
Source: https://www.youtube.com/watch?v=psS7fG_kFd8
Title: An Imaginary Exponential Equation
Problem 346
Presenter: aplusbi
Given the relation \begin{equation} i^{z+i} = 1\,,\ \end{equation} find the complex values of $z$.
Solution to this problem.
Link to my write-up on Basic Complex Numbers.
Problem 186 [Not Unipodal]:
This problem is by
Source: https://www.youtube.com/watch?v=v9Nsp67iNaM
Title: Harvard University Software Engineering Admission Exam
Presenter: Super Academy
Given the relation \begin{equation} x^{\log 64} + 4^{\log x} = 10\,, \end{equation} find the values of $x$.
Link to my write-up on logarithms over the real numbers.
Problem 187 [Not Unipodal]:
This problem is by
Source: https://www.youtube.com/watch?v=FXEFFSDwglE
Title: A Very Exponential Power Tower
| Problem 272
Presenter: aplusbi
Given the relation \begin{equation} z^{z^{z+1}} = 4\,, \end{equation} find the values of $z$
Problem 188 [Not Unipodal: Lambert]:
This problem is by
Source: https://www.youtube.com/watch?v=Pj7ju4ebtY0
Title: Can We Solve A Transcendental Equation
Presenter: SyberMath
Given the relation \begin{equation} e^x + x + 1=0\,, \end{equation} find the values of $x$.
Solution to this problem.
Link to my write-up on the Lambert W function.
Problem 189 [Not Unipodal, Complex Numbers]:
This problem is by
Source: https://www.youtube.com/watch?v=fE3i7kWdVYA
Title: A Short Locus Problem With Absolute Values
Presenter: aplusbi
Given the relation \begin{equation} |z| = |z-i|\,,\ \end{equation} find the values of $z$.
Solution to this problem.
Link to my write-up on Basic Complex Numbers.
Problem 190 [Not Unipodal, Complex Numbers]:
This problem is by
Source: https://www.youtube.com/watch?v=G66wNWWpjDM
Title: An Infinite Complex Fraction
Presenter: aplusbi
Given the relation
Solution to this problem.
Link to my write-up on Basic Complex Numbers.
Problem 191 [Not Unipodal]:
This problem is by
Source: https://www.youtube.com/watch?v=elkUO6eRMvM
Title: Solving 8 Equations w/ Lambert W function
Presenter: blackpenredpen
Given the relation \begin{equation} W(e^{e^2+1+x^x}) = x^x\,, \end{equation} find the values of $x$.
Solution to this problem.
Link to my write-up on the Lambert W function.
Problem 192[Unipodal]:
This problem is by
Source: https://www.youtube.com/watch?v=_lJ1NsXB-gE
Title: Can you Solve Admission Question from
Cambridge University?
Presenter: Super Academy
Given the relations \begin{align} 4^x - 4^y&= 24\,,\\ 2^{x + y} &= 35\,, \end{align} find the real values of $\phi=x-y$.
Problem 193 [Unipodal]:
This problem is by
Source: https://www.youtube.com/watch?v=NQVQyttRC7o
Title: Cambridge University Entrance Exam
Secrets EXPOSED
Presenter: Super Academy
Given the relations \begin{align} x+y&= 7\,,\\ x y &= 7\,, \end{align} find the values of $x,y$.
Problem 194 [Not Unipodal]:
This problem is by
Source: https://www.youtube.com/watch?v=1zLfC_TVP1Y
Title: China | A nice Math Olympiad Exponential Simplification
Presenter: Super Academy
Given the relations \begin{equation} x^x = 2^{8+2x}\,, \end{equation} find the values of $x$.
Problem 195 [Not Unipodal]:
This problem is by
Source: https://www.youtube.com/watch?v=evGZ4AXFILE
Title: High School Mathematics Tournament | Diophantine Equation
Presenter: Super Academy
Given the relations \begin{align} \sqrt{x^n} + \sqrt{y^n} &= 13\,,\\ {x^n} - {y^n} &= 13\,, \end{align} find the positive integer values of $n,x,y$.
Problem 196 [Not Unipodal: the first word problem in the set]:
This problem is by
Source: https://www.youtube.com/watch?v=rqDnyMZ9oiU
Title: Can You Solve? | America's Hard Maths Word Problem
Presenter: Daily Mathematics
Solve the word problem:
A man invested a sum of \$280, partly at
5% and partly at 4%. If the total
interest is \$12.90 per annum,
find the amount invested at 5%.
If you're interested, see my many articles on solving algebra word problems at:
Solution to this problem.
Link to my write-up on Word Problem solving.
Problem 197 [Unipodal]:
This problem is by
Source: https://www.youtube.com/watch?v=yl_0f9wJQeQ
Title: How to solve this nice math rational
Exponential algebra problem
Presenter: Mathematics and Statistics Guru
Given the relation \begin{equation} \frac{x}{y} + \frac{ y}{ x} = 1\,, \end{equation} find the value of \begin{equation} \phi = \left( \frac{x}{y}\right)^{153}+ \left(\frac{ y}{ x}\right)^{153}\,. \end{equation}
Problem 198 [Not Unipodal, Complex Numbers]:
This problem is by
Source: https://www.youtube.com/watch?v=RughThc09xU
Title: Solving A Nice Problem With z-bar | Problem 139
Presenter: aplusbi
Given the relation \begin{equation} z = \obz^2 + 1\,, \end{equation} find the values of $z$.
Solution to this problem.
Link to my write-up on Basic Complex Numbers.
Problem 199 [Not Unipodal, Complex Numbers]:
This problem is by
Source: https://www.youtube.com/watch?v=H2GbSHIkybo
Title:Solving An Absolute Value Equation
| Problem 127
Presenter: aplusbi
Given the relation \begin{equation} |z+w| = |z-w|\,, \end{equation} find the values of \begin{equation} \phi = \arg{z} - \arg{w} \,. \end{equation}
Solution to this problem.
Link to my write-up on Basic Complex Numbers.
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