On this page we have an assortment of recreational math problems from Olympiad
math contests, SATs, university entrance exams, and the like. The point of this page
is to show how some of these problems can be attacked from the vantage of using the
unipodal algebra. The probems I choose to try often require some degree of cleverness
from the student just to solve them with traditional methods. The use of the unipodal
algebra allows us to fashion proofs that sometimes require less cleverness and at
other times, more cleverness. But that's where the fun comes from!!
Note 0: For help with the unipodal algebra, see My write-up on The Unipodal Algebra.
Note 1: Although I'm labeling the problems as 'olympiad' problems, they're actually of a
variety of sources. And, by the way, the solutions I have produced may not be the most
efficient solutions. (Time will tell.)
Note 2: From Problem #15 on, each(?) article contains a fairly good description
of the unipodal algebra that the earlier articles may not have.
Note 3: The type of 'olympiad' problem that is emenable to the methods of the unipodal
algebra, tend to be of a subclass of algebraic types, and not often of a statistical,
trigonmetric, or geometric nature.
Note 4: I have another page of 'olympiad'-style problems that you might be interested
in. It covers a wider scope of problems, some employing the unipodal algebra. Most
of the unipodal problems solved here are also on that other page. The root page for
it can be found at
Math Diversions.
Note 5: 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.]
Olympiad Problem 1: (I invented this problem on inspiration from one of the
olympian problems I tried from the Internet.)
Solve for the value of the expression (where $x$ is real) \begin{equation} \left(x + \frac{1}{x}\right)^{n} - \left(x - \frac{1}{x}\right)^{n}\quad (\mbox{$n$ a positive integer})\,,\label{eq:expression} \end{equation} given the two relations \begin{equation} \left(x + \frac{1}{x}\right)^{n} + \left(x - \frac{1}{x}\right)^{n} = x\,, \end{equation} and \begin{equation} x^2 - \frac{1}{x^2} = B\,, \end{equation} where $B$ is some nonzero real number.
Olympiad Problem 2:
This problem is by
Source: https://www.youtube.com/shorts/GkDVLurV3gYGiven the relation \begin{equation} a + \frac{1}{a} = 7\,, \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}
Titled: A Nice Algebraic Expansion Problem
Presenter: Numbers.Numbers
Olympiad Problem 3:
The problem is to solve for $y$ as a function of $x$, given the differential equation \begin{equation} \left(\frac{dy}{dx}\right)^2 - 1 = x^2\,. \end{equation}
Olympiad Problem 4:
Given the relations \begin{equation} x^2 - y^2 = 24\,, \end{equation} and \begin{equation} xy = 35\,, \end{equation} find the value of $x+y$ over the complex numbers.
Olympiad Problem 5:
Given the relation \begin{equation} a + \frac{1}{a} = 6 \,, \end{equation} find \begin{equation} a - \frac{1}{a}\,, \end{equation} over the complex numbers.
Olympiad Problem 6:
The YouTube video is found at:
Source: https://www.youtube.com/watch?v=-70c48Fl-x4
Titled: Evaluating x sqrt(x)+y sqrt(y)_Math is math
Presenter: Math is math
Given the relations \begin{align} x + y &= 19 \,,\label{eq:TheGiven1.6}\\ x y &= 9 \,,\label{eq:TheGiven2.6} \end{align} find \begin{equation} x \sqrt{x}+y \sqrt{y}\,,\label{eq:TheFind.6} \end{equation} where $x,y$ are positive real numbers.
Olympiad Problem 7:
The YouTube video is found at:
https://www.youtube.com/watch?v=fX-FrWzh-Vs
Titled:Can you Solve Harvard University System of Equations ?
Presenter: Super Academy
Given the relations
\begin{align}
10 ^2x + 10y^2 &= 29xy \,,\label{eq:TheGiven1.7}\\
x^2 - y^2 &= 21 \,,\label{eq:TheGiven2.7}
\end{align}
find
\begin{equation}
x + y\,,\label{eq:TheFind.7}
\end{equation}
where $x,y$ are positive real numbers.
Olympiad Problem 8:
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 \,,\label{eq:TheGiven1.8}\\ x + y &= 2 \,,\label{eq:TheGiven2.8} \end{align} find all solutions for $x,y$ over the complex numbers.
Olympiad Problem 9:
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.
Olympiad Problem 10:
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.
Olympiad Problem 11:
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.
Olympiad Problem 12:
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.
Olympiad Problem 13:
The YouTube video is found at:
https://www.youtube.com/watch?v=v_QyUSa-_ME
Titled: Math Olympiad Problem | A Nice Algebra Challenge
Presenter: Math Booster
Given the relations \begin{align} a + b&= 1 \,,\\ a^2 + b^2 &= 2 \,, \end{align} find the value of $a^{11} + b^{11}$.
Olympiad Problem 14:
The YouTube video is found at:
https://www.youtube.com/watch?v=-B59K4zuyGo
Titled: Harvard University Aptitude Test Strategy You
Didn't Know Existed || Algebra Problem || 99% Failed
Presenter: Super Academy
Given the relations \begin{align} a^2&= b + 183 \,,\\ b^2&= a + 183 \,, \end{align} where $a\not= b$, find the values of $a$ and $b$.
Olympiad Problem 15:
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.
Olympiad Problem 16:
The YouTube video is found at:
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.
Olympiad Problem 17:
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.
Olympiad Problem 18:
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.
Olympiad Problem 19:
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, 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.)
Olympiad Problem 20:
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}
Olympiad Problem 21:
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}
Olympiad Problem 22:
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}
Olympiad Problem 23:
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}
Olympiad Problem 24:
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 $x$.
Olympiad Problem 24. This new solution is superior to the old one.
Olympiad Problem 25:
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 $a,b>0$.
Olympiad Problem 26:
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 $x$.
Olympiad Problem 26. This new solution is superior to the old one.
Olympiad Problem 27 [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 $x,y$
Olympiad Problem 28:
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 $x-y$.
Olympiad Problem 29:
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 $x$.
Olympiad Problem 30:
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.
Olympiad Problem 31:
This problem on YouTube at
Source: https://www.youtube.com/watch?v=yEW3O_ToYDE
Title: France | Can you solve this ? | Math Olympiad
Presenter: Learncommunolizer
Given the relation \begin{equation} \sqrt[4]{x} + \sqrt[4]{97 - x} = 5 \,, \end{equation} find the values of $x$.
Olympiad Problem 32:
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 $x$.
Olympiad Problem 32. This solution is distinctive
because I defined two distinct unipodes.
Olympiad Problem 33:
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 $z$.
Olympiad Problem 33. This solution is distinctive
because I summed on even terms of a unipode.
Olympiad Problem 34:
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 $x$.
Olympiad Problem 35:
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 $x$.
Olympiad Problem 36:
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 $x$.
Olympiad Problem 37: [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 $x$.
Olympiad Problem 38: [unipodal]
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 $x$.
Olympiad Problem 39: [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 $x$.
Olympiad Problem 40: [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 $x,y \in Reals$.
Olympiad Problem 41: [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.
Olympiad Problem 42:
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 $x$.
Olympiad Problem 43:
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 $x$.
Olympiad Problem 44:
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 $x$.
Olympiad Problem 45:
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 $x$.
Olympiad Problem 46:
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 $x,y$.
Olympiad Problem 47:
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}
*Unipodal solution in preparation*.
Olympiad Problem 48:
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 $x,y$.
Olympiad Problem 49:
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 $x$.
Olympiad Problem 50:
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 $x,y$.
Olympiad Problem 51:
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{x-1} + \sqrt{x+2} = 3\,, \end{equation} solve for $x$ values.
Olympiad Problem 52:
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 $xy$.
Olympiad Problem 53:
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 \definedas x + \frac{1}{x} \,, \end{equation} for $x$ over the positive reals.
Olympiad Problem 54:
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 $x,y$.
Olympiad Problem 55:
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.
Olympiad Problem 56:
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 $x$.
Olympiad Problem 57:
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.
Olympiad Problem 58:
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 $ab$.
Olympiad Problem 59:
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 $x$.
Olympiad Problem 60:
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.
Olympiad Problem 61:
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 $x^{-1} - y^{-1}$.
Olympiad Problem 62:
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 $x,y$.
Olympiad Problem 63:
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 $x+y$.
Olympiad Problem 64:
This problem is by
?
Title: ?
Presenter: ?
Given the relation \begin{equation} x^2 = (5 - \sqrt{24})^x\,, \end{equation} find the values of $x \in$ Reals.
Olympiad Problem 65:
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.
Olympiad Problem 66:
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.
Olympiad Problem 67:
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 $x,y$.
Olympiad Problem 68:
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.
Olympiad Problem 69:
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 $x,y$.
Olympiad Problem 70:
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}
Olympiad Problem 71:
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 $x,y$.
Olympiad Problem 72:
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}
Olympiad Problem 73:
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}
Olympiad Problem 74:
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}
Olympiad Problem 75: (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 $a + b$.
Olympiad Problem 76:
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 $x,y$.
Olympiad Problem 77:
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 $x+y$.
Olympiad Problem 78:
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 $x$.
Olympiad Problem 79:
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 $x$
Olympiad Problem 80:
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 $x,y$.
Olympiad Problem 81:
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 $x$.
Olympiad Problem 82:
This problem is by
Source: https://www.youtube.com/watch?v=UMqOh1AoEt4
Title:Algebra Shortcut | Math Olympiad Problem |
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 \((0 < x < \pi)\).
Olympiad Problem 83:
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 $a,b$.
Olympiad Problem 84:
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 $x$
Olympiad Problem 85:
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 $x,y$.
Olympiad Problem 86:
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}
Olympiad Problem 87:
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 $x$.
Olympiad Problem 88:
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 $x$.
Olympiad Problem 89:
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 $\phi=x-y$.
Olympiad Problem 90:
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 $x,y$.
Olympiad Problem 91:
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}
Olympiad Problem 92:
This problem is by
Source: ?
Title: ?
Presenter: ?
Given the relation \begin{equation} \sqrt{x} + \sqrt{x-12} = 4\,, \end{equation} find the real values of $x$.
Olympiad Problem 93: [Complex Numbers]
This problem is by
Source: https://www.youtube.com/watch?v=OBtLU4KcNnk
Title: Finding The Square Root of 2+i
Problem 445
Presenter: aplusbi
Given the relation \begin{equation} z = \sqrt{2+i}\,, \end{equation} express $z$ without the square root sign.
Olympiad Problem 94:
This problem is by
Source: https://www.youtube.com/watch?v=sYL9gc5kvIk
Title: A Nice Algebra Equation |
Math Olympiad Questions
Presenter: Maths Black Board
Given the relations \begin{align} x+y&= 4\,,\\ x^5+y^5 &= 464\,, \end{align} find the values of $x,y$.
Olympiad Problem 95:
This problem is by
Source: https://www.youtube.com/watch?v=Bn5KBONOxDE
Title: I Solved A Complex Cubic System |
Problem 89
Presenter: aplusbi
Given the relations \begin{align} z + w&= 4\,,\\ z^3+ w^3 &= 4\,, \end{align} find the values of $z,w$.
Olympiad Problem 96:
This problem is by
Source: https://www.youtube.com/watch?v=qNEpmW3zyis
Title: How to solve this nice math Exponential
algebra problem
Presenter: Mathematics & Statistics Guru
Given the relations \begin{align} a+b &= -2\,,\\\ a^3+b^3 &= -56\,, \end{align} find the values of $a,b$.
Olympiad Problem 97:
This problem is by
Source: https://www.youtube.com/watch?v=8WqRzO5XjjM
Title: Harvard University Admission Interview tricks
Presenter: Super Academy
Given the relations \begin{align} x^6 + y^6 &= 793\,,\\ x^3+ y^3 &= 35\,, \end{align} find the values of $x,y$.
Olympiad Problem 98:
This problem is by
Source: https://www.youtube.com/watch?v=zscAAY-2hII
Title: A Nice Radical Equation With Parameters
Presenter: SyberMath Shorts
Given the relation \begin{equation} \sqrt{x+a} + \sqrt{x}= a\,, \end{equation} find the value of \begin{equation} \phi = \sqrt{x+a} - \sqrt{x}\,. \end{equation}
Olympiad Problem 99:
This problem is by
Source: https://www.youtube.com/watch?v=77Rf1Q0vsAc
Title: Harvard University Admission Interview Tricks
Presenter: Super Academy
Given the relation \begin{equation} x^2 = (5-\sqrt{24})^x\,, \end{equation} find the real values of $x$.
Olympiad Problem 100:
This problem is by
Source: https://www.youtube.com/watch?v=zqo4lTXwt5Y
Title: A awesome mathematics problem
Presenter: Mathematics & Statistics Guru
Given the relation \begin{equation} \left(\frac{n}{n-1}\right)^2 + \left(\frac{n}{n+1}\right)^2 = \frac{10}{9}\,, \end{equation} find the real values of $n$.
Olympiad Problem 101:
This problem is by
Source: https://www.youtube.com/watch?v=uEYkDU3HoDU
Title: Cambridge University Admission Interview Tricks
Presenter: Super Academy
Given the relations \begin{align} x^2 - y^2 &= 40\,,\\ xy &= 99\,, \end{align} find the values of $\phi = x+y$.
Olympiad Problem 102:
This problem is by
Source: ?
Title: ?
Presenter: ?
Given the relations \begin{align} a^3 + b^3 &= 16\,,\\ a+b &= 4\,, \end{align} find the values of $\phi = a^4 + b^4$.
Olympiad Problem 103:
This problem is by
Source: https://www.youtube.com/watch?v=5aicBGj7vYc
Title: De-nesting A Nice Radical
Presenter: SyberMath
Given the expression \begin{equation} \phi = \sqrt{16+\sqrt{255}}\,, \end{equation} express $\phi$ more simply.
Olympiad Problem 104: [Complex Numbers]
This problem is by
Source: https://www.youtube.com/watch?v=H342tw_GSTc
Title: A System of Equations
| Problem 316
Presenter: aplusbi
Given the relations \begin{align} zw &= 7 - i\,,\\ z + w &= 5\,, \end{align} find the values of $z,w$.
Olympiad Problem 105:
This problem is by
Source: https://www.youtube.com/watch?v=fgYn47lAHqg
Title: A Quick And Easy System
Presenter: SyberMath Shorts
Given the relations \begin{align} a^2 - b^2&= 24\,,\\ ab &= 35\,, \end{align} find the values of $a,b$.
Olympiad Problem 106:
This problem is by
Source: https://www.youtube.com/watch?v=8uHPC65T7Kc
Title: How to solve this nice math Exponential algebra problem
Presenter: Mathematics & Statistics Guru
Given the relations \begin{equation} (b-9)^4 = b^4\,, \end{equation} find the real values of $b$.
Olympiad Problem 107:
This problem is by
Source: https://www.youtube.com/watch?v=9z0ChCIXO58
Title: Harvard University Admission Test Tricks.
Presenter: Super Academy
Given the relations \begin{align} x^3 + y^3&= 98\,,\\ x - y &= 8\,, \end{align}
Olympiad Problem 108:
This problem is by
Source: https://www.youtube.com/watch?v=wDpPvH8WXdw
Title: Harvard University Admission Test Tricks!
Presenter: Super Academy
Given the relation \begin{equation} 5^x - 2^x = 2\sqrt{10^x}\,, \end{equation} find the real values of $x$.
Olympiad Problem 109:
This problem is by
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 $x,y$.
Olympiad Problem 110:
This problem is by
Source: https://www.youtube.com/watch?v=DZGz3n61Ph8
Title: Can you Pass Harvard University Admission Exam with Ease ?
Presenter: Super Academy
Given the relation \begin{equation} (4x+49)^{1/3} - (4x-49)^{1/3} = 2\,, \end{equation} find the real values of $x$.
Olympiad Problem 111:
This problem is by
Source:https://www.youtube.com/watch?v=Btzs5Pq_HyI
Title: 60 years ago this question was on the International
Mathematical Olympiad
Presenter: MindYourDecisions
Given the relation \begin{equation} \sqrt{x+\sqrt{2x-1}}+ \sqrt{x-\sqrt{2x-1}} = A\,, \end{equation} find the real values of $x$ in the following three cases:
(a) $A=\sqrt{2}$,
(b) $A=1$,
(c) $A= 2$.
We also have the requirement that the radicand under the square
root sign is never negative.
Olympiad Problem 112: (Also see Problem 59.)
This problem is by
Source: https://www.youtube.com/watch?v=Rj8AUADp8hw
Title: High School Mathematics Tournament Algebra
Presenter: Super Academy
Given the relation \begin{equation} \sqrt{x+7}- \sqrt{x-9} = 2\,, \end{equation} find the values for $x$.
Olympiad Problem 113:
This problem is by
Source: https://www.youtube.com/watch?v=nBu-7ughFE8
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 = 1/5\,, \end{equation} find the values of \begin{equation} \phi = \tan x \,, \end{equation} where $0< x <\pi$.
Olympiad Problem 114:
This problem is by
Source:https://www.youtube.com/watch?v=vc2dev1THoA
Title: Canada | A Nice Algebra Problem
Presenter: Clear Math Concepts
Given the relation \begin{equation} \sqrt{x+35} = \sqrt{x} + \sqrt{5}\,, \end{equation} find the values for $x$.
Olympiad Problem 115:
This problem is by
Source: https://www.youtube.com/watch?v=XK0kkWVQ-Z4
Title: Can You Simplify Another Radical?
Presenter: SyberMath
Given the relation \begin{equation} \phi = \sqrt{a-\sqrt{a^2-9}} \,, \end{equation} find a representation of $\phi$ that has one less level of nesting of square roots.
Olympiad Problem 116:
This problem is by
Source: https://www.youtube.com/watch?v=VO6BqO-0sbk
Title: Radical sum - Viewer Submission
Presenter: Math Out Loud
Given the relation \begin{equation} \sqrt{2016}+\sqrt{56} = 14^k \,, \end{equation} determine the value of $k$:
(A) $ \frac{1}{2}$, (B) $\frac{3}{4}$, (C) $\frac{5}{4}$, (D) $\frac{3}{2}$, (E) $\frac{5}{2}$.
Olympiad Problem 117:
This problem is by
Source: https://www.youtube.com/watch?v=ejoQngmazms
Title: A Nice Quartic Equation
Presenter: SyberMath
Given the relation \begin{equation} x^4+(x+1)^4 = 1\,, \end{equation} find the values for $x$.
Olympiad Problem 118:
This problem is by
Source: https://www.youtube.com/watch?v=4MPaTM2k-8I
Title: An Interesting Exponential Equation
Presenter: SyberMath
Given the relation \begin{equation} (\sqrt{5}+2)^x+ (\sqrt{5}-2)^x= 18\,, \end{equation} find the (real) values for $x$.
Olympiad Problem 119:
This problem is by
Source: https://www.youtube.com/watch?v=ETQBNy7qKWw
Title: Can you Pass Oxford University Admission Test ?
Presenter: Super Academy
Given the relation \begin{equation} \sqrt{x\fifthroot{x}\,} - \fifthroot{x\,\sqrt{x}\,}= 702\,, \end{equation} find the (real) values for $x$.
Olympiad Problem 120:
This problem is by
Source: https://www.youtube.com/watch?v=FFC2Ts1GamQ
Title: Solving A Cubic System | Problem 335
Presenter: aplusbi
Given the relation \begin{align} z^2+w^2&= 0\,,\\ z^3+w^3 &= -4\,, \end{align} find the values of $z,w$.
Olympiad Problem 121:
This problem is by
This time, I take a problem from the textbook New FoundationsGiven the relation \begin{equation} L^2 = \gamma(1 + \bv/c)\,, \end{equation} show that \begin{equation} \pm L = \left(\frac{\gamma+1}{2}\right)^{1/2}\! +\,\, \hat\bv\left(\frac{\gamma-1}{2}\right)^{1/2} \,. \end{equation}
for Classical Mechanics by David Hestenes, p. 588.
Olympiad Problem 122:
This problem is by
Source: https://www.youtube.com/watch?v=cmPno8PC1RQ
Title: A Nice Polynomial Equation From A Nice Book
Presenter: SyberMath
Given the relation \begin{equation} (x+2)^5 = x^5 + 242\,, \end{equation} find the values for $x$.
Olympiad Problem 123:
This problem is by
Source: ---
Title: The Inverse Sinh-to-Natural-Log Identity Proof
Presenter: Patrick
Prove that \begin{equation} \sinh^{-1} y = \ln\,\big[\,y + \sqrt{y^2+1}\,\big]\,. \end{equation}
Olympiad Problem 124:
This problem is by
Source: https://www.youtube.com/watch?v=rEyGlcSdEfk
Title: Brazil Olympiad Simplification Challenge
Presenter: Smart math tricks
Given the relation \begin{equation} \sqrt{x}+\sqrt{2x} = x \,, \end{equation} determine the values of $x$:
Olympiad Problem 125:
This problem is by
Source: https://www.advancedmath.org
Title: An Instructive Unipodal Integral
Presenter: Patrick
perform the integral \begin{equation} I= \int (a\cosh x - \sinh x)\cosh bx\,dx\,. \end{equation}
Olympiad Problem 126:
This problem is by
Source: https://www.youtube.com/watch?v=16p5Lxh-H_8
Title: Oxford University Entrance Exam Tricks
Presenter: Super Academy
Given the relation \begin{equation} (\sqrt{10}+3)^x+ (\sqrt{10}-3)^x= 38\,, \end{equation} find the (real) values for $x$.
Olympiad Problem 127:
This problem is by
Source: https://www.youtube.com/watch?v=gz-j_IiNaX0
Title: Wow! Derivative + Hyperbolic Function + Math Induction!
Presenter: bprp calculus basics
Given the relation \begin{equation} y = e^{2x}\sinh x\,, \end{equation} use induction to show that \begin{equation} \frac{d^ny}{dx^n} = e^{2x}\left[\,\frac{3^n+1}{2}\sinh x + \frac{3^n-1}{2}\cosh x\,\right]\,. \end{equation}
Olympiad Problem 128:
This problem is by
Source: https://www.youtube.com/watch?v=kVQqaYo9DEA
Title: Sample JEE main question from India
Presenter: Prime Newtons
Given the relation \begin{equation} (\sqrt{3}+\sqrt{2})^x+ (\sqrt{3}-\sqrt{2})^x= 10\,, \end{equation} find the (real) values for $x$.
Olympiad Problem 129:
This problem is by
Source: Patrick
Title: A made up problem
Given the relation \begin{equation} \fourthroot{x+\sqrt{x^2+1}} - \fourthroot{x-\sqrt{x^2+1}}= 2\,, \end{equation} find the values for $x$.
Olympiad Problem 130: The 'Babylonian Quadratic' Problem
This problem is by
Source: https://www.youtube.com/watch?v=e1mLmkSFQa8&list
=PLd8BS_A4wDvEilFB9VQNo27V8iSaCzjkQ&index=21
Title: 21 Babylonian Quadratics With a=1
Presenter: Gary Rubinstein
Given the relations \begin{align} a + b&= 18\,,\\ ab &= 77\,, \end{align} find the values of $a,b$.
Olympiad Problem 131:
This problem is by
Source: https://www.youtube.com/watch?v=_quVl1cobqU
Title: How to solve System of Equations - Did you know this?
Presenter: Maths & Olympiad
Given the relations \begin{align} 3^x + 9^y&= 30\,,\\ x + 2y &= 4\,, \end{align} find the solutions for $x,y$.
Olympiad Problem 132:
This problem is by
Source:https://www.youtube.com/watch?v=PLakuVTrLWM
Title: The Sum Of Two Cube Roots | Problem 511
Presenter: aplusbi
Simplify the expression \begin{equation} \phi = (2+11i)^{1/3} + (2-11i)^{1/3}\,. \end{equation}
Olympiad Problem 133:
This problem is by
Source: ?
Title: Circle and hyperbola
Presenter: Patrick
Given the relations \begin{align} x^2 + y^2&= r^2\,,\\ x y &= \lambda\,, \end{align} find the values of $x,y$, where $r,\lambda$ are arbitrary positive real numbers.
Olympiad Problem 134:
This problem is by
Source: https://www.youtube.com/watch?v=sYL9gc5kvIk
Title: A Nice Algebra Equation |
Math Olympiad Questions
Presenter: Maths Black Board (redone)
Given the relations \begin{align} x+y&= 4\,,\\ x^5+y^5 &= 464\,, \end{align} find the values of $x,y$.
Olympiad Problem 135:
This problem is by
Source: https://www.youtube.com/watch?v=aPdJpfEx_oo
Title: Italy | a nice Math Olympiad Question
Presenter: Math-X
Given the relation \begin{equation} \sqrt{1+\sqrt{1+x}} = x^{1/3}\,, \end{equation} find the real values of $x>0$.
Olympiad Problem 136:
This problem is by
Source: https://www.youtube.com/watch?v=WVWePTHo8HI
Title: German Olympiad Question
Presenter: Higher Mathematics
Given the relation \begin{equation} 3^a + 2^a = 35\,, \end{equation} find the integer values of $a$.
After that, do similarly for the relation \begin{equation} 3^a + 2^a = 36\,, \end{equation} to find its real value solutions, which WolframAlpha claims is \begin{equation} a \approx 3.02799\,. \end{equation}
Olympiad Problem 137:
This problem is by
Source: https://www.youtube.com/watch?v=R_ouo5iekqw
Title: A Nice Algebra Problem | Math Olympiad
Presenter: SALogic
Given the relation \begin{equation} (x+2)^4+x^4 = 80\,, \end{equation} find the real values for $x$.
Note: I inadvertantly solved a slightly different problem.
Olympiad Problem 138:
This problem is by
Source: https://www.youtube.com/watch?v=U0F7CqXZ0IA
Title: A very interesting algebra math simplification
Presenter: Math Beast
Given the relations \begin{align} m + n&= 2\,,\\ m^4 + n^4 &= 272\,, \end{align} find the values of $\phi = mn$.
Olympiad Problem 139:
This problem is by
Source: https://www.youtube.com/watch?v=sV3u3F7ABzw
Title: Cambridge University Interview Trick
Presenter: Higher Mathematics
Given the relations \begin{align} a + b&= 1\,,\\ a^2 + b^2 &= 2\,, \end{align} find the values of \begin{equation} \phi = a^8 + b^8\,. \end{equation}
Olympiad Problem 140:
This problem is by
Source: The Ether of Mathematical Ideas
Title: Logarithm and square root of a vector
Presenter: Patrick
Using the unipodal algebra, find the Logarithm of the unit
vector $u$ to be
\begin{equation}
\Log u = i\pi u_- \,.
\end{equation}
And then find the square root of $u$.
Olympiad Problem 141:
This problem is by
Source: https://www.youtube.com/watch?v=GgORRgXLoiY
Title: Can you Solve Cambridge University Admission Interview Exam?
Presenter: Super Academy
Given the relation \begin{equation} \Big (\cuberoot{5+2\sqrt{6}}\,\Big)^x + \Big (\cuberoot{5-2\sqrt{6}}\,\Big)^x= 10\,, \end{equation} find the real values for $x$.
Olympiad Problem 142:
This problem is by
Source: https://www.youtube.com/watch?v=OmsaCaIgoeo
Title:Harvard University Admission Interview Tricks
Presenter: Super Academy
Given the relations \begin{align} x^2 - y^2&= 119\,,\\ xy &= 60\,, \end{align} find the values of $x,y$.
Olympiad Problem 143:
This problem is by
Source: https://www.youtube.com/watch?v=OkJc1pyowP0
Title: How Fast Can You Crack This Math Challenge?
Presenter: Khem math
Given the relation \begin{equation} (\sqrt{5}-1)^x - (\sqrt{5}+1)^x= 2^x\,, \end{equation} find the values of $x$ over the complex numbers.
Olympiad Problem 144:
Source: https://www.youtube.com/watch?v=aX4IHM0VcXk
Title: A fancier way to solve this radical equation
Presenter: blackpenredpen
Given the relation \begin{equation} (x-40)^{1/3} + (-x +3)^{1/3} = -1\,, \end{equation} find the real values of $x$.
Olympiad Problem 145:
Source: https://www.youtube.com/watch?v=wgd6Gp-Kfbk
Title: This Question Tricked Thousands of Students
Presenter: J Educational Tutorials
Given the relation \begin{equation} x^2 - 5x + 7 = 0\,, \end{equation} find the value of $\phi$ given by \begin{equation} \phi \definedas (x-2)^{90} + (3-x)^{90}\,. \end{equation}
Olympiad Problem 146:
Source: The Ether of Great Mathematical Ideas
Title: Yet another example
Presenter: Patrick
Given the relation \begin{equation} \sqrt{x+1} + \sqrt{x-1} = 2\,, \end{equation} find the value of $x$.
Olympiad Problem 147:
Source: https://www.youtube.com/watch?v=aX4IHM0VcXk
Title: A fancier way to solve this radical equation
Presenter: blackpenredpen
Given the relation \begin{equation} \cuberoot{x-40} + \cuberoot{-x+3} = -1\,, \end{equation} find the real values of $x$.
Olympiad Problem 147.
Link to my write-up on The Unipodal Algebra.
Olympiad Problem 148:
Source: https://www.youtube.com/watch?v=euBnlvAanxY
Title: France | Can you solve?
Presenter: Math Master TV
The following relation \begin{equation}\ \phi = (\sqrt{11}+\sqrt{5}\,)^8 +(\sqrt{11}-\sqrt{5}\,)^8\,, \end{equation} has a simple (though large) integer value. Find it.
Olympiad Problem 148.
Link to my write-up on The Unipodal Algebra.
Olympiad Problem 149:
Source: https://www.youtube.com/watch?v=j1zdTAsfbbI
Title: This Radical Equation is EASIER Than it Looks
Presenter: NonsoMaths
Given the relation \begin{equation} \sqrt{2x^2-7x+1} - \sqrt{2x^2-9x+4} = 1\,, \end{equation} solve for real values of $x$.
Olympiad Problem 149.
Link to my write-up on The Unipodal Algebra.
Olympiad Problem 150:
Source: https://www.youtube.com/watch?v=QvW1zHap2D4
Title: A nice radical maths olympiad question
Presenter: Rashel's Classroom
Given the relation \begin{equation} \sqrt{39-x} + \sqrt{7-x} = 8\,, \end{equation} solve for real values of $x$.
Olympiad Problem 150.
Link to my write-up on The Unipodal Algebra.
Olympiad Problem 151:
Source: https://www.youtube.com/watch?v=k0BOCf7CO_g
Title: The Square Root Trap!
Presenter: SyberMath
Given the relation \begin{equation} x - \sqrt{a-x^2} = 1\,, \end{equation} solve for real values of $x$.
Olympiad Problem 151.
Link to my write-up on The Unipodal Algebra.
Olympiad Problem 152
Source: https://www.youtube.com/watch?v=kDLP3lZh2TI
Title: ALGEBRA CHALLENGE
Presenter: Maths Simplified Solutions
Given the relation \begin{equation} x^{332} + x^{-332} = 963\,, \end{equation} find the real values of \begin{equation} \phi = x^{166} - x^{-166} \,. \end{equation}
Olympiad Problem 152.
Link to my write-up on The Unipodal Algebra.
| Olympiad Problems with Unipodal Algebra |