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 abilities. 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.
Announcement: If I'm to continue posting on this page, I need to broaden
the scope of the mathematics I cover here. So, I intend to include topics on
groups, rings, matrices, abstract algebra, partial differentiation, word problems,
integration, chemistry, physics, and others. I also intend to include some
theory and proofs.
Note: 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: 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 FlowCharting Math Proofs.
Link to my write-up on Basic Geometric Algebra.
Link to my write-up on Geometric Series.
Link to my first write-up on Group Theory 1 (very basic).
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 logarithms over the real numbers.
Link to my write-up on A History of Scheme for solving word problems and stoichiometry problems.
Link to my write-up on Pascal's Triangle.
Link to my write-up on An introduction to stoichiometry (long version).
Link to my write-up on An introduction to stoichiometry (short version).
Link to my write-up on Trigonometric functions.
Link to my write-up on Word Problem solving.
Link to my write-up on Basic Matrix Algebra.
Link to my write-up on Mathematical Induction.
Link to my write-up on GCD & LCM.
Link to my write-up on Virtual Emplacement.
Link to my write-up on the Fibonacci sequence.
Link to my write-up on the Method of Partial Fractions.
Link to my write-up on Set Theory Basics.
Link to my write-up on Basic Ring Theory, 1.
Link to my write-up on Basic Gibbs Vector Calculus.
Link to my write-up on Basic Integration Techniques. (Short Paper)
Link to my write-up on Basic Probability.
Link to my write-up on The Unipodal Algebra.
To go to Diversions1 page Link to Diversions1
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
Problem 1000 [Unipodal]
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 $x$.
Solution to the problem.
Link to my write-up on The Unipodal Algebra.
Problem 1001 [Not Unipodal: Complex Numbers]
Source: https://www.youtube.com/watch?v=gJeDWfsgS9Y
Title: This Quadratic Leads to a Wild Power! l P590
Presenter: aplusbi
Given the relation \begin{equation} z^2 + 2z + 2 = 0\,, \end{equation} find the values of \begin{equation} \phi = z^{11} + z^{-11}\,. \end{equation} Look for solutions whose argument lies between 0 and $2\pi$.
Solution to the problem (or maybe not).
Link to my write-up on Basic Complex Numbers.
Problem 1002 [Not Unipodal: Algebra: Mixed-Rate Problem]
Source: http://www.theproblemsite.com/problems/high-school-math
/2008/mixture-problem
Title: A Mixed-Rate Problem
Presenter: Patrick
A man had a 10-gallon keg of wine and a jug. One day,
he drew off a jugful of wine and filled up the keg with
water. Later on, when the wine and water had got
thoroughly mixed, he drew off another jugful and again
filled up the keg with water. The keg then contained
equal quantities of wine and water. What was the
capacity of the jug?
Problem 1003 [Not Unipodal: Algebra]
Source: https://www.youtube.com/watch?v=DLmuvWL_RCg
Title: Can You Solve This Exponential Equation Without Guessing?
Presenter: Math Mastery with Amitesh
Given the relation \begin{equation} 16^{x^2+y} + 16^{x+y^2} = 1\,, \end{equation} find real solutions for $(x,y)$.
Problem 1004 [Not Unipodal: Lambert W function]
Source: https://www.youtube.com/shorts/xT2QV8LV2Z4
Title:Interesting Exponential Equation
Presenter: drpkmath
Given the relation \begin{equation} t^t = 49\,, \end{equation} find real solutions for $t$.
Solution to the problem .
Link to my write-up on the Lambert W function.
Problem 1005 [Unipodal]
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 $x$.
Solution to the problem.
Link to my write-up on The Unipodal Algebra.
Problem 1006 [Not Unipodal: Lambert W function: Logarithms]
Source: https://www.youtube.com/watch?v=iIAxAYeidIY
Title: This Log Equation Stumped Me
Presenter: NonsoMaths
Given the relation \begin{equation} x \log (x+1) +\log (x+1) = 2x\,, \end{equation} find real solutions for $x$.
Solution to the problem .
Link to my write-up on the Lambert W function.
Link to my write-up on logarithms over the real numbers.
Problem 1007 [Not Unipodal: Number Theory]
Source: https://www.youtube.com/watch?v=H1_yk1DSac8
Title: The last 2 digits of a perfect square cannot be both odd.
Presenter: Prime Newtons
Let $x$ be a positive integer. Show the rightmost 2 digits of $x^2$
cannot both be odd.
Problem 1008 [Not Unipodal: Algebra]
Source: https://www.youtube.com/watch?v=SELJwouvdSI
Title: A Radical Adventure
Presenter: SyberMath
Given the relations \begin{align} \sqrt{x} + \sqrt{y} &= 3 \,,\\ \sqrt{x+5} + \sqrt{y+3} &= 5 \,, \end{align} find the positive real solutions for $x,y$.
Problem 1009 [Not Unipodal: Algebra]
Source: https://www.youtube.com/watch?v=J6EgujRX0ss
Title: Can you Evaluate? | Calculator Not Allowed
Presenter: Math Beast
Given the relation \begin{equation} \phi = 2.25^{2.25}\,, \end{equation} re-express $\phi$ without decimals and with simple exponents.
Problem 1010 [Not Unipodal: Algebra: Mixed-Rate Problem]
Source: Source: http://iws.collin.edu/dkatz/Intermediate_Algebra
/Mixture_Problems.pdf
Title: A Mixed-Rate Problem
Presenter: Patrick
Your recipe calls for 2 cups of regular flour and one-half tablespoon of
baking powder. But your pantry only has a cup of regular flour and a
cup of self-rising flour, which is 4% baking powder. How much regular
flour should we add to the self-rising flour to get a mixture with the
desired concentration of baking powder?
Problem 1011 [Not Unipodal: Algebra]
Source: https://www.youtube.com/watch?v=j45XT7h7Yss
Title: Fun Math Problem
Presenter: Andy Math
Given the relation \begin{equation} 40^{x-1} = 2^{2x+1}\,, \end{equation} find real solutions for $x$.
Problem 1012 [Not Unipodal: Algebra]
Source: https://www.youtube.com/watch?v=VTxiSSLz7M4
Title: We Solved Using A Perfect Square Trinomial!
Presenter: Andy Math
Given the relations \begin{align} \frac{x^2}{y^2} + \frac{y^2}{x^2} = 3z+2\,,\\ \frac{x}{y} + \frac{y}{x} = z\,, \end{align} solve for $z$, where $x,y,z$ are real numbers.
Problem 1013 [Not Unipodal: Matrix Algebra]
Source: https://www.youtube.com/watch?v=pwkNnHVHVUc
Title: The SECRET Property of ALL 2 x 2 matrices!
Presenter: Math Mastery with Amitesh
Given two $2\times 2$ matrices $A,B$ with entries from a
commutative ring with unity $R$, show that
\begin{equation}
(AB - BA)^2= xI\,,
\end{equation}
where $x\in R$ and $I$ is the identity matrix for $2\times 2$ matrices.
Problem 1014 [Not Unipodal: Algebra: Geometry]
Source: https://www.youtube.com/watch?v=3VQ_oHa07oo
Title: Can YOU Find the Radius of This Circle?
Presenter: Math Queen
Find the radius $r$ of the circle inscribed in the right trapezoid:
Problem 1015 [Not Unipodal: Algebra: Mixed-Rate]
Source: http://www.beatthegmat.com/hard-solutions
-mixture-problem-t69704.html
Title: Mixed-Rate
Presenter: Patrick
Solution Y is 30 percent liquid X and 70 percent water. If 2 kilograms
of water evaporate from 8 kilograms of solutions Y and 2 kilograms of
solution Y are added to the remaining 6 kilograms of liquid, what percent
of this new liquid solution is liquid X?
Problem 1016 [Not Unipodal: Calculus: Algebra]
Source: https://www.youtube.com/watch?v=6JN-DJuSFUY
Title: Renormalization and envelopes
Presenter: Steven Strogatz
In the beginning of his lecture on "Renormalization and envelopes''
(Lecture 27), Steven Strogatz presents a figure similar to the following
one, which is to indicate a continuously sliding line segment of fixed
length (one unit for this case).
Figure 1. The continuously sliding line segment of fixed length (one unit).
We will show that there exists a curve in the $xy$-plane that each
of these line segments is tangent to. Such a curve is called an
"envelope'' to the set of tangent lines. In the next figure we pick a
typical line-segment tangent, showing a point $(x,y)$ on it, which it
shares with the envelop curve. The point of intersection $(x,y)$ is
parameterized by the angle $\tau$.
Figure 2. The continuously sliding line segment of fixed length
(one unit). Note that $\bar\tau = 90^\circ - \tau$ is the complement to $\tau$.
Show that the following relation holds between $x$ and $y$: \begin{equation} \frac{x}{\cos \tau} + \frac{y}{\sin \tau}= k + h = 1\,.\label{eq:k+h=1} \end{equation} After multiplying through by $\sin \tau \cos \tau$, we have that \begin{equation} x\sin \tau + y\cos \tau= \sin \tau \cos \tau\,.\label{eq:multiplying.through} \end{equation}
Next, we apply a standard trick. When a set of points is defined implicitly,
as in (\ref{eq:multiplying.through}), we can define a constant function
\begin{equation}
F(x,y,\tau) = x\sin \tau + y\cos \tau- \sin \tau \cos \tau = 0\,.\label{eq:F(x,y,tau)}
\end{equation}
Now, any differential change in coordinates $x,y$ along the envelope
defined by (\ref{eq:F(x,y,tau)}) will not change the value of $F$; hence
\begin{equation}
\frac{\partial F(x,y,\tau)}{\partial\tau} = 0\,,\label{eq:partial.F.partial.tau=0}
\end{equation}
which, when we work it out, becomes
\begin{equation}
x\cos \tau -y\sin \tau = \cos^2 \tau - \sin^2 \tau\,.\label{eq:differential.constraint}
\end{equation}
2) Use Eqs. (\ref{eq:multiplying.through}) and (\ref{eq:differential.constraint}) to show that the points $(x,y)$ on the envelope satisfy the equation \begin{equation} x^{2/3} + y^{2/3} = 1\,. \end{equation}
Problem 1017 [Not Unipodal: Algebra: Geometry]
Source: https://www.youtube.com/watch?v=nSTsMhedEKQ
Title: A Nice Geometry Problem
Presenter: Lines & Logic
Find the value of $x$ as it's depicted in the following right triangle:
Problem 1018 [Not Unipodal: Differential Equations]
Source: https://www.youtube.com/watch?v=6JN-DJuSFUY
Title: Renormalization and envelopes
Presenter: Steven Strogatz
In the beginning of his lecture on ``Renormalization and envelopes'' (Lecture 27),
Steven Strogatz presented a problem in the construction of an "envelope''
to the set of tangent lines. We solved that problem in Problem 1016.
This time, we move to the next problem, which is to find the solution to the
following boundary-value differential equation,
\begin{equation}
\epsilon y'' + (1+\epsilon)y'+y=0\,,
\end{equation}
where the boundary conditions are
\begin{equation}
y(0)=0\quad\mbox{and}\quad y(1)=1\,.
\end{equation}
Show that the solution is \begin{equation} y(x,\epsilon) = \frac{e^{-x}-e^{-x/\epsilon}}{e^{-1}-e^{-1/\epsilon}}\,. \end{equation}
Solution to the problem.
Link to my write-up on the Lambert W function.
Link to my write-up on logarithms over the real numbers.
Problem 1019 [Not Unipodal: Lambert W Function: Logarithms]
Source: https://www.youtube.com/watch?v=0q_933QGmWM
Title: Complex Analysis is Destroying this...
Presenter: Dr PK Math
Given the relation \begin{equation} z^{\tfrac{1+i}{\sqrt{2}}} = \left(\frac{1-i}{\sqrt{2}}\right)^z\,, \end{equation} solve for the complex number $z$.
Problem 1020 [Not Unipodal: Algebra]
Source: https://www.algebra.com/algebra/homework/
%word/age/Solving-Age-Problems.lesson
Title: An Age Problem
Presenter: Patrick
Cary is 9 years older than Dan. In 7 years, the sum of their ages will
equal 93. Find both of their ages now.
Problem 1021 [Not Unipodal: Lambert W Function]
Source: https://www.youtube.com/watch?v=tsZQ3X0IVzk
Title: Cambridge Maths Interview Question
Presenter: Math Beast
Given the relation \begin{equation} e^{2-2\sqrt{x}}=x\,,\label{eq:Given} \end{equation} solve for real values of $x$.
Problem 1022 [Not Unipodal: Calculus: Induction]
Source: https://tutorial.math.lamar.edu/Classes/CalcI/LogDiff.aspx
Title: Induction
Presenter: Patrick
Using induction, prove the relation \begin{equation} D_x x^n = nx^{n-1}\,. \end{equation}
Solution to the problem.
Link to my write-up on Mathematical Induction.
Problem 1023 [Not Unipodal: Calculus: Logarithmic Differentiation]
Source: https://tutorial.math.lamar.edu/Classes/CalcI/LogDiff.aspx
Title: Logarithmic Differentiation
Presenter: Patrick
Evaluate the derivative \begin{equation} D_x x^x =\ ?\,, \end{equation} where $x \ne 0$.
Problem 1024 [Not Unipodal: Algebra]
Source: https://www.youtube.com/watch?v=hMhds8tMZxk
Title: Maths Olympiad | Only 10%
Presenter: Rashel's Classroom
Given the relations \begin{align} xy &= 5\,,\\ yz &= 10\,,\\ zx &= 15\,, \end{align} find \begin{equation} \phi = x+y+z\,. \end{equation}
Problem 1025 [Not Unipodal: Algebra]
Source: To be determined (hopefully)
Title: ?
Presenter: Patrick
Starting with $x$ amount of pure liquid A in a beaker, we will $n$ times repeat the
following process: Draw out of the beaker $y$ amount of the liquid (by volume) and add $y$
amount of a different liquid Z to the beaker and let the contents come to a homogeneous
mixture before repeating the cycle. Show that, at the $n$th cycle, the amount of the original
liquid A is given by the formula
\begin{equation}
Q_n = x(1-y/x)^n\,.
\end{equation}
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Problem 1026 [Not Unipodal: Algebra]
Source: https://www.youtube.com/watch?v=FGZfZphuK_I
Title: Tricky Maths Question for Competitive Exams
Presenter: Math Beast
Given the relation \begin{equation} 7+10+13+\cdots+(x-3)+x = 282\,, \end{equation} solve for $x$ a positive integer.
Problem 1027 [Not Unipodal: Hyperbolic functions]
Source: https://www.youtube.com/watch?v=OYtXgvKQGwY
Title: Cambridge Math Entrance Exam
Presenter: Math Beast
Given the relation \begin{equation} \frac{e^x-e^{-x}}{3} = 1\,, \end{equation} solve for real values of $x$.
Problem 1028 [Not Unipodal: Algebra: Solved by a Table]
Source: https://www.youtube.com/watch?v=vGUfwLCHc4w
Title: Maths Olympiad Problem
Presenter: Spencer's Academy
\begin{equation} (x+500)^3 = 20 - x\,, \end{equation} solve for $x$.
Problem 1029 [Not Unipodal: Algebra: Word Problem]
Source: https://gmatclub.com/forum/the-ratio-by-
volume-of-soap-to-alcohol-to-water-in-a-68933.
Title: A ratios Problem
Presenter: Patrick
The ratio by volume of soap to alcohol to water in a certain solution is
$2:50:100$. After the solution is altered so that the ratio of soap to
alcohol is doubled, and the ratio of soap to water is halved, there is
100cc of alcohol. How many cc's of water does this new solution contain?
Problem 1030 [Not Unipodal: Algebra: Logrithms]
Source: https://www.youtube.com/watch?v=7Dby0KR7byc
Title: A Weird Log Equation | Can You Solve?
Presenter: SyberMath
Given the relation \begin{equation} \log_3 (x-2) = \log_5 x\,, \end{equation} solve for $x$.
Problem 1031 [Not Unipodal: Algebra: Logrithms]
Source: https://www.youtube.com/watch?v=psA6ZKKkPcQ
Title: Cambridge Maths Entrance Exam Question
Presenter: Math Beast
Given the relation \begin{equation} (-19)^z=19\,, \end{equation} solve for $z$.
Solution to the problem.
Link to my write-up on logarithms over the real numbers.
Problem 1032 [Not Unipodal: Algebra]
Source: https://gmatclub.com/forum/at-a-certain-company-40-of-the
-women-are-over-50-years -old-and-50-o-205994.html
Title: A mixed-Rate Problem
Presenter: Patrick
At a certain company, 40% of the women employees and 50% of the men
employees are 50 or older. If that amounts to 42% of all the company's
employees are 50 or older, what percentage of the company's employees
are men?
Problem 1033 [Not Unipodal: Algebra: Logrithms]
Source: https://www.youtube.com/watch?v=xDQC_O6dOiA
Title: How to Solve Logarithmic Equations
Presenter: Nuel's Academy
Given the relations \begin{align} 2^{2x-3y}&=32\,,\\ \log_y x &=2\,, \end{align} solve for $x,y$.
Solution to the problem.
Link to my write-up on logarithms over the real numbers.
Problem 1034 [Not Unipodal: Algebra: Geometry]
Source: https://www.youtube.com/watch?v=3N7qfm9cSow
Title: 90% FAILED to Solve this Math Problem
Presenter: Brain Station
The following figure shows a "large'' rectangle, subdivided into
four
smaller rectangles. Find the value of $x$.
Problem 1035 [Not Unipodal: Algebra: Proportions Problem]
Source: The Ether of Great Mathematical Ideas
Title: Proportions Problem
Presenter: Patrick
A, B, and C invested money in ratios $6:4:9$ and their interests from
the first cycle were in the ratios $3:2:5$. Assuming that their interests
were proportional to the times of their investments, show that their
investment times were in the ratios $9:9:10$.
Problem 1036 [Not Unipodal: Algebra: Lambert W Function]
Source: https://www.youtube.com/watch?v=dyicSIJAJ2s
Title: Why Most People Can't Find All 3 Solutions
Presenter: Mental Math
Given the relation \begin{equation} x^2=2^{-x} \,, \end{equation} find the real values of $x$.
Solution to the problem.
Link to my write-up on the Lambert W function.
Problem 1037 [Not Unipodal: Algebra]
Source: https://www.youtube.com/watch?v=C4oTVRdTlmo
Title: Nice Math Olympiad Algebra Simplification
Presenter: Math Beast
Given the relation \begin{equation} x\sqrt{x}= x+\sqrt{x} \,, \end{equation} find the real values of $x\ge 0$.
Problem 1038 [Not Unipodal: Logarithms, Lambert W Function]
Source: https://www.youtube.com/watch?v=PgEPrEbru5M
Title: solving a nested logarithmic equation
Presenter: blackpenredpen
Given the relation \begin{equation} \log\,(x + \ln x)=1 \,, \end{equation} find the real values of $x$.
Solution to the problem.
Link to my write-up on the Lambert W function.
Link to my write-up on logarithms over the real numbers.
Problem 1039 [Not Unipodal: Logarithms, Lambert W Function]
Source: https://www.youtube.com/watch?v=q0wnL9roqTI
Title: An Absolutely Complex Equation | P606
Presenter: aplusbi
Given the relation \begin{equation} |z|i - z = 1+2i \,, \end{equation} find the values of $z$.
Solution to the problem.
Link to my write-up on Basic Complex Numbers.
Problem 1040 [Not Unipodal: Algebra: A Ratios Problem]
Source: https://gmatclub.com/forum/the-ratio-by-weight-of-the
-four-ingredients-a-b-c-and-d-of-a-196635.html
Title: A Ratios Problem
Presenter: Patrick
The ratios, by weight, of four ingredients $A$, $B$, $C$, and $D$ of a certain
mixture is $4:7:8:12$. The mixture will be changed so that the ratio
of $A$ to $C$ is quadrupled and the ratio of $A$ to $D$ is decreased. The ratio
of $A$ to $B$ is held constant. If $B$ constitutes 20\% of the weight of the
new mixture, by approximately by what percent will the ratio of $A$ to
$D$ be decreased?
A. 15 $\qquad$ B. 25 $\qquad$ C. 35 $\qquad$ D. 45 $\qquad$ E. 55
Problem 1041 [Not Unipodal: Algebra: A Mixed-Rate Problem]
Source: https://gmatclub.com/forum/a-team-won-50-percent
-of-its-first-60-games-in-a-particular-season-an-205999.html
Title: A Mixed-Rate Problem
Presenter: Patrick
In a particular season, a team won 50% of its first 60 games and 80%
of the remaining games. If the team won 60% of all of its games that
season, how many games did the team play?
Problem 1042 [Not Unipodal: Algebra]
Source: https://www.youtube.com/watch?v=izGAhb2YgE8
Title: The Prettiest Radical Identity
Presenter: SyberMath
Given the expression \begin{equation} \phi = \sqrt{2+\sqrt{3}} - \sqrt{2-\sqrt{3}}\,, \end{equation} simplify $\phi$.
Problem 1043 [Not Unipodal: Algebra]
Source: https://www.youtube.com/watch?v=ptHnqpKn2s0
Title: Can You Find the Numbers in This Impossible-
Looking System?
Presenter: Math Queen
Let $a,b,c$ be elements of the nonnegative integers
(i.e., $a,b,c \in \Integers_{\ge0}$).
Find definitive values for $a,b,c$ consistent with
\begin{align}
a+bc &= 2024\,,\\
ab+c &= 2023\,.
\end{align}
Problem 1044 [Not Unipodal: A Geometry-Algebra Problem]
Source: https://www.youtube.com/watch?v=bGcA3Izipq4
Title: Many People Can't Solve This Math Problem
Presenter: Brain Station
Given the figure below, solve for the length $x$ of the hypotenuse
of right triangle ABC.
Problem 1045 [Not Unipodal: A Mixed-Rate Problem]
Source: https://www.algebra.com/algebra
Title: Question 250998
Presenter: Patrick
While hiking up and then down a trail, Rolf spent 60% of his time
hiking uphill and 40% hiking back down. If he averaged 2 mph
uphill, what was his average speed round trip?
Problem 1046 [Not Unipodal: A Functional Problem]
Source: https://www.youtube.com/watch?v=OIK7N5Odn_k
Title: 98% Failed To Solve This Math
Presenter: Math Beast
Given the relation \begin{equation} f(f( x)) = x^2-x+1 \,, \end{equation} find the value of $f(0)$.
Problem 1047 [Not Unipodal: Algebra]
Source: https://www.youtube.com/watch?v=_jxh0eTwsQA
Title: A Wonderful Math Olympiad Algebraic Question
Presenter: Mathematics & Statistics Guru
Given the relation \begin{equation} \sqrt{2}^{\sqrt{x}} - \sqrt{2}^{\sqrt{y}} = 112 \,, \end{equation} solve for $x,y$ where $x,y \in \Reals^+$.
Problem 1048 [Not Unipodal: Algebra]
Source: The Ether of Great Mathematical Ideas
Title: An alpha-transformation problem
Presenter: Patrick
Given the relation \begin{equation} t^t = 4^{t+16} \,, \end{equation} solve for $t$ where $t \in \Reals^+$.
Problem 1049 [Not Unipodal: Algebra]
Source: https://www.youtube.com/watch?v=ETzbGnYA5vI
Title: Cambridge Maths Exam Question
Presenter: Math Beast
Given the relation \begin{equation} x^{\log 2x} = 5 \quad\mbox{(the logarithm is base 10)}\,, \end{equation} solve for $x \in \Reals$.
Problem 1050 [Not Unipodal: Algebra]
Source: https://www.algebra.com/algebra
Title: Solution by tree (Question 10300)
Presenter: Patrick
Mary bought some donuts. She gave 1/2 her donuts and 1/2 a donut
to her mom. Then she gave away 1/2 her remaining donuts and 1/2
a donut to her aunt. Then she gave 1/2 of her remaining donuts and
1/2 a donut to her sister, Kathy. This left her with 1/4 of a dozen donuts.
How many doughnuts had she bought? [Ans. 31]
Problem 1051 [Not Unipodal: Algebra/Lambert]
Source: The Ether of Great Mathematical Ideas
Title: To Lambert or not to Lambert?
Presenter: Patrick
The problem is to find a real solution to \begin{equation} x 36^x = 3. \end{equation}
Solution to the problem.
Link to my write-up on the Lambert W function.
Problem 1052 [Unipodal]
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}
Solution to the problem.
Link to my write-up on The Unipodal Algebra.
Problem 1053 [Not Unipodal: Logarithms]
Source: https://www.youtube.com/watch?v=1JGOZ2in2IQ
Title: Logarithmic Equation
Presenter: Maths Simplified Solutions
Given the relation \begin{equation} 16^{\log_4 x} = x^x\,, \end{equation} find the real values of $x$.
Solution to the problem.
Link to my write-up on logarithms over the real numbers.
Problem 1054 [Not Unipodal: Alpha Transformation]
Source: https://www.youtube.com/watch?v=xE1cDs85EcA
Title: 99% of Students Can't Solve This
Presenter: MindYourDecisions
Given the relation \begin{equation} x^{x^3} = 3\,, \end{equation} find the real values of $x$.
Problem 1055 [Not Unipodal: Calculus]
Source: https://www.youtube.com/watch?v=ooanJuau0Ko
Title:How to Integrate - Easy Substitution Method
Presenter: PreMath
Find the value of the integral \begin{equation} \int \!x \sqrt{x^2+1}\, dx\,. \end{equation}
Problem 1056 [Not Unipodal: Lambert]
Source: https://www.youtube.com/watch?v=yCmtrraOqZI
Title: Solve for X
Presenter: PreMath
Given the relation \begin{equation} 3125^x = x^{-1}\,, \end{equation} solve for real values of $x$.
Solution to the problem.
Link to my write-up on the Lambert W function.
Problem 1057 [Not Unipodal: Complex Numbers]
Source: https://www.youtube.com/watch?v=YRi6NTW4TZM
Title: Cambridge Maths Entrance Exam
Presenter: Math Beast
Given the relation \begin{equation} \phi =i^{1/i}\,, \end{equation} solve for real values of $\phi$.
Solution to the problem.
Link to my write-up on Basic Complex Numbers.
Problem 1058 [Not Unipodal: Lambert: Alpha Transformation]
Source: https://www.youtube.com/watch?v=rfArXK422FM
Title: Russian Math Olympiad Questions
Presenter: Math Beast
Given the relation \begin{equation} a^a = 2^{\sqrt{200}}\,, \end{equation} solve for real values of $a$.
Solution to the problem.
Link to my write-up on the Lambert W function.
Problem 1059 [Not Unipodal: A Mixed-Rate Problem]
Source: https://www.algebra.com/algebra
Title: Question 6293
Presenter: Patrick
In printing an article of 48,000 words, a printer decides to use two
sizes of type. Using the larger type, a printed page contains 1,800
words. Using smaller type, a page contains 2,400 words. The article
is allotted 21 full pages in a magazine. How many pages must be in
smaller type?
Problem 1060 [Not Unipodal: Logarithms]
Source: https://www.youtube.com/watch?v=60G2F3I9zM8
Title: Power Tower | Can you solve for X?
Presenter: PreMath
Given the relation \begin{equation} x^{x^x} = (0.5)^{\sqrt{2}}\,, \end{equation} solve for real values of $x$.
Solution to the problem.
Link to my write-up on logarithms over the real numbers.
Problem 1061 [Not Unipodal: Polynomials]
Source: https://www.youtube.com/watch?v=1b1G2hARThc
Title: Oxford Entrance Question
Presenter: PreMath
Given the relation \begin{equation} 2^{x+2} + 4^x =4 + 8^{x}\,, \end{equation} solve for real values of $x$.
Problem 1062 [Not Unipodal: Mixed Rates]
Source: https://www.algebra.com/algebra
Title: Question 2508
Presenter: Patrick
Gold is 19 times heavier than water. Copper is 9 times heavier
than water. In what ratio should they be mixed so that the alloy
is 15 times heavier than water?
Problem 1063 [Not Unipodal: Lambert]
Source: The Ether of Great Mathematical Ideas
Title: DIY Problem
Presenter: Patrick
Given the relation \begin{equation} 6^x - 9x = 2\,, \end{equation} solve for all values of $x$.
Solution to the problem.
Link to my write-up on the Lambert W function.
Problem 1064 [Unipodal]
Source: https://www.youtube.com/watch?v=gIYT9Vp_gdk
Title: This Radical Equation Has a Hidden Trick
Presenter: SyberMath
Given the relation \begin{equation} \left(\sqrt{2+\sqrt{3}}\right)^x + \left(\sqrt{2-\sqrt{3}}\right)^x = 4\,, \end{equation} solve for all real values of $x$.
Solution to the problem.
Link to my write-up on The Unipodal Algebra.
Problem 1065 [Not Unipodal: Mixed Rates]
Source: Intermediate Algebra for College Students,
3rd Ed. Prentice-Hall (2002), p. 181 #21.
Title: Mixed Rate
Presenter: R. Blitzer
At the north campus of a small liberal arts college, 10%
of the students are women. At the south campus, 50% of
the students are women. The campuses are merged into
one east campus, of which 40% of the 1200 students are
women. How many students were in the north and south
campuses before the merger?
Problem 1066 [Not Unipodal: Calculus: Logarithms]
Source: https://www.youtube.com/watch?v=Qc2u8licPD4
Title: Derivative of an Infinite Power of x
Presenter: Brain Station Advanced
Given the relation \begin{equation} \phi(x) = x^{x^{x^{.^{\cdot^{\cdot}}}}}\,, \end{equation} solve for all real values of $x$.
Problem 1067 [Not Unipodal: Lambert]
Source: https://www.youtube.com/watch?v=it6kW89PpI0
Title: This Broke My Brain
Presenter: SyberMath
Given the relation \begin{equation} 7^{6-x}=x+2\,, \end{equation} solve for all values of $x$.
Solution to the problem.
Link to my write-up on the Lambert W function.
Problem 1068 [Not Unipodal: Algebra]
Source: https://www.youtube.com/watch?v=AFU4Q7TIXU4Given the relation \begin{equation} \frac{x}{y}+ \frac{y}{x}=x+y\quad \mbox{where}\quad x,y \in \Integers^+\,, \end{equation} solve for all allowed values of $x,y$.
Title: Tricky Maths Question for Competitive Exams
Presenter: Math Beast
Problem 1069 [Not Unipodal: Complex: Logarithms]
Source: https://www.youtube.com/watch?v=WgvrsAF3pEwGiven the relation \begin{equation} (-2)^c=2\,, \end{equation} solve for all allowed complex values of $c$.
Title: Cambridge Maths Interview Question | 95% got it wrong
Presenter: Math Beast
Solution to the problem.
Link to my write-up on logarithms over the real numbers.
Link to my write-up on Basic Complex Numbers.
Problem 1070 [Not Unipodal: Logarithms]
Source: ?Given the relation \begin{equation} \frac{80}{8^x} = 8^x\,, \end{equation} solve for real values of $x$.
Title: More Logarithms
Presenter: Interesting Math
Problem 1071 [Not Unipodal: Word Problem]
Source: https://www.algebra.com/algebraIf a hen and a half can lay an egg and a half in a day and a half, how many
Title: Question 177914
Presenter: Patrick
eggs will six hens lay in seven days?
Problem 1072 [Not Unipodal: transcendental numbers]
Source: The Ether of Great Mathematical Ideas
Title: Is phi = e^e transcendental?
Presenter: Patrick
Is $\phi = e^e$ transcendental? (Actually, I don't answer this question.)
Problem 1073 [Not Unipodal: Word Problem: Ratios]
Source: http://advancedmathtutoring.com/
a-hard-sat-math-mixture-word-problem
Title: Ratios
A cylindrical glass is filled with three different juice mixtures.
First, 1/4 of the glass is filled with a mixture that is half apple
juice and half orange juice. Then, the glass is filled to the 80%
mark with a mixture that has twice as much orange juice as
apple juice and twice as much apple juice as pineapple juice.
The remainder of the glass is filled with pineapple juice. What
percent of the final mixture is apple juice? [Ans. 28.8%]
Problem 1074 [Not Unipodal: Geometric Calculus]
Source: The Ether of Great Mathematical Ideas
Title: The Adjoint of the Pseudoscalar.
Presenter: Patrick
Preliminaries
Let $\calV$ be a vector space of dimension $n$. Let $\calG(\calV)$ be the geometric algebra over $\calV$. Let $f$ be a linear function on $\calV$.
The so-called adjoint of $f$, $\obf$, is defined implicitly by the relation \begin{equation} x \cdot f(y) = \obf(x)\cdot y\quad \mbox{ for all}\ x,y \in \calV\,. \end{equation}
Let $\ubf$ (the differential of $f$) be the extension of $f$ to apply to all multivectors of the geometric algebra $\calG(\calV)$. We demand that $\ubf$ be a linear function over its space of multivectors. Let vector $a$ be in $\calV$. Then $\ubf(a) = f(a)$. That is, the differential of $f$ treats the vectors of $\calV$ the same as does $f$.
And to bivectors, such as $a\wedge b $, we get \begin{equation} \ubf(a\wedge b) =\ubf(a) \wedge \ubf(b)\,, \end{equation} and similar expansions are to be made for trivectors and above. David Hestenes refers to this process of "distributing" $\ubf$ over a serial wedge product the ``outermorphism." The trivector example is: \begin{equation} \ubf(a\wedge b\wedge c) =\ubf(a) \wedge \ubf(b)\wedge \ubf(c)\,, \end{equation} and so on.
That leaves the question of how we should define the differential of a scalar, say $\lambda$. Scalars of a geometric algebra are real numbers, and they pass through, so to speak, unaffected, that is: \begin{equation} \ubf(\lambda) = \lambda. \end{equation}
So, what happens when we apply this differential to the pseudoscalar $I$ of $\calG(\calV)$? (The pseudoscalar $I$ of $\calG(\calV)$ can be formed by taking the wedge product of all the vectors of an orthonormal basis for $\calV$.) We'll solve this by a definition: \begin{equation} \ubf(I) = (\det f) I\,, \end{equation} where $\det f$ is the determinant of the linear transformation $f$, and it's a scalar.
Problem
Assuming that $\obf$ is itself a linear, outermorphism, (For scalar $\lambda$, $\obf(\lambda) = \lambda$.) show that \begin{equation} \obf(I) = (\det f) I\,, \end{equation} which might seem counterintuitive.
Solution to the problem.
Link to my write-up on Basic Geometric Algebra.
Problem 1075 [Not Unipodal: Geometric Algebra-Calculus]
Source: The Ether of Great Mathematical Ideas
Title: Two-Sided Inverse of a Linear Function.
Presenter: Patrick
Preliminaries
Let $\calV$ be a vector space of dimension $n$. Let $\calG(\calV)$ be the geometric algebra over $\calV$. Let $f$ be a linear function on $\calV$. The so-called adjoint of $f$, $\obf$, is defined implicitly by the relation \begin{equation} x \cdot f(y) = \obf(x)\cdot y\quad \mbox{ for all}\ x,y \in \calV\,.\label{eq:implicitly.obf} \end{equation} Let $\ubf$ (the differential of $f$) be the extension of $f$ to apply to all multivectors of the geometric algebra $\calG(\calV)$. We demand that $\ubf$ be a linear function over its space of multivectors. Let vector $a$ be in $\calV$. Then $\ubf(a) = f(a)$. That is, the differential of $f$ treats the vectors of $\calV$ the same as does $f$.
Problem
Establish that the general rule \begin{equation} \ubf^{-1}A = \frac{\obf( AI)I^{-1}}{\det f} \end{equation} is a genuine two-sided inverse for $f$ when $A=x$ where $x$ a vector.
Solution to the problem.
Link to my write-up on Basic Geometric Algebra.
Problem 1076 [Not Unipodal: Word Problem: Mixtures]
Source: http://johnrdixonbooks.com/images/Word.pdf, p. 6.
Title: Mixture Problem
Presenter: Patrick
A man has 10 gallons of a 50% sulfuric acid solution, 20 gallons
of a 20% solution, and 5 gallons of a 10% solution. He wants
to use up all the 10% solution and make 15 gallons of 30% solution.
How much of each solution should he use?
Problem 1077 [Not Unipodal: Logarithms: alpha transformation]
Source: https://www.youtube.com/watch?v=yoilUnHtr_M
Title: This Equation Shouldn’t Exist…
Presenter: Patrick
Given the relation \begin{equation} x^{\log x} =10^{10^{10}}\,, \end{equation} solve for positive real values of $x$, where the logarithm is base 10.
Problem 1078 [Not Unipodal: Logarithms: alpha transformation]
Source: https://answers.yahoo.com/question/
index?qid=20080214222603AAcQyvt
Title: A Mixed-Rate Problem
Presenter: Patrick
An employer has a daily payroll of \(\$\)1950 when employing some workers
at \(\$\)120 per day [type B worker] and others at $150 per day [type A worker].
When the number of \(\$\)120 workers is increased by 50% and the number of
\(\$\)150 workers is decreased by 1/5, the new daily payroll is \(\$\)2,400. Find
how many workers were originally employed at each rate.
Problem 1079 [Not Unipodal: Lambert]
Source: https://www.youtube.com/watch?v=0sGS1yVAoHE
Title: A Clean Trick to Solve This
Presenter: Sybermath
Given the relation \begin{equation} x = 2^{x-2}\,, \end{equation} find the two real values of $x$.
Note: See my write-up on the Lambert $W$ function for the
Lambert $W$ function base $B$.
Solution to the problem.
Link to my write-up on the Lambert W function.
Problem 1080 [Not Unipodal: Mixed-Rate]
Source: http://www.analyzemath.com/high_school
_math/grade_12/problems.html
Title: A Mixed-Rate Problem
Presenter: Patrick
Two large and 1 small pumps can fill a swimming pool in 4 hours.
One large and 3 small pumps can also fill the same swimming pool
in 4 hours. How many hours will it take 4 large and 4 small pumps
to fill the swimming pool. (We assume that all large pumps are
similar and all small pumps are also similar.)
Problem 1081 [Not Unipodal: Logarithm]
Source: https://www.youtube.com/watch?v=1HVUZP69W0c
Title: Solving Logarithmic Equations with Different
Bases Made Easy!
Presenter: L.JARED2
Given the relation \begin{equation} \log_4(6-x) = \log_2 x \,, \end{equation} find the real solutions of $x$.
Solution to the problem.
Link to my write-up on logarithms over the real numbers.
Problem 1082 [Not Unipodal: Logarithm]
Source: https://www.youtube.com/watch?v=xBpZRWCGw30
Title: Is this equation solvable?
Presenter: blackpenredpen
Given the relation \begin{equation} x^{\ln 4}+ x^{\ln 10} = x^{\ln 25} \,, \end{equation} find the positive real solutions for $x$.
Solution to the problem.
Link to my write-up on logarithms over the real numbers.
Problem 1083 [Unipodal]
Source: https://www.youtube.com/watch?v=P66nkc3bKrE
Title: find the value of x ll maths olympiad
Presenter: maths curiosity
Given the relation \begin{equation} \sqrt{x+5}+ \sqrt{x-5} = 5 \,, \end{equation} find the real solutions for $x$.
Solution to the problem.
Link to my write-up on The Unipodal Algebra.
Problem 1084 [Not Unipodal: Mixed-Rate]
Source: http://www.beatthegmat.com/650-800
-question-t68610.html
Title: A percentage problem
Presenter: Patrick
At a technology consulting firm with x computers, all of which are
laptops or desktops, 30% are laptops; if 80% of the total number of
computers have more than 1GB of RAM and 10% of the computers
with less than 1 GB of RAM are laptops (and no computers have
exactly 1GB of RAM), approximately what percent of desktops have
more than 1GB of RAM?
Problem 1085 [Not Unipodal: Algebra: Complex Numbers]
Source: https://www.youtube.com/watch?v=z2Yf5nQY5Kc
Title: Can You Solve This Harvard University Interview Question
Presenter: HkLogics
Given the relation \begin{equation} (x/2)^6=6^6\,, \end{equation} solve for $x$ in the complex numbers.
Solution to the problem.
Link to my write-up on Basic Complex Numbers.
Problem 1086 [Not Unipodal: Algebra: Alpha Transform]
Source: https://www.youtube.com/shorts/sHuzHybS5MA
Title: Square Root Maths
Presenter: Rachel Shorts
Given the relation \begin{equation} (x-1)^x=81\,, \end{equation} solve for $x$ in the real numbers.
Problem 1087 [Not Unipodal: Complex Functions]
Source: https://www.youtube.com/watch?v=m9N5S3Hq30o
Title: Simplify If You Can!
Presenter: aplusbi
Given the relation \begin{equation} \phi = \frac{e^{i\theta/2}}{e^{i\theta} + 1}\,, \end{equation} simplify $\phi$.
Solution to the problem.
Link to my write-up on Basic Complex Numbers.
Problem 1088 [Not Unipodal: Algebra]
Source: https://www.youtube.com/watch?v=-fr70Fsq2iY
Title: Canada | Can You Solve This?
Presenter: HkLogics
Given the relation \begin{equation} \sqrt{2}^{\sqrt{x}} - \sqrt{2}^{\sqrt{y}} = 48 \,, \end{equation} find integer solutions.
Problem 1089 [Not Unipodal: Algebra: Kinematics]
Source: https://www.algebra.com/algebra
Title: Question 12377
Presenter: Patrick
A rock is dropped from a cliff into the ocean. It travels $16t^2$ feet in $t$
seconds. If the splash is heard 1.5 second later, how high is the cliff?
[Note: Assume the speed of sound at sea level is 1100 feet per second.]
Problem 1090 [Not Unipodal: Algebra: Complex Numbers]
Source: https://www.youtube.com/watch?v=BgtvGKsTOtM
Title: Harvard Entrance Exam Maths Question
Presenter: Math Beast
Given the relation \begin{equation} 1+z+z^2+z^3+z^4+z^5+z^6 = 0\quad\mbox{where}\quad z \ne 1\,, \end{equation} find the value of \begin{equation} \phi = z+z^2+z^4\,.\label{eq:phi} \end{equation}
Solution to the problem.
Link to my write-up on Basic Complex Numbers.
Problem 1091 [Not Unipodal: Algebra]
Source: Khan Academy YouTube video
Title: Advanced Ratio Problem
Presenter: Patrick
In a group of 57 children the ratio of girls to boys is 4 : 15. How
many boys must leave the group so that the resulting ratio
of girls to boys is 4 : 11?
Problem 1092 [Not Unipodal: Logarithms]
Source: https://www.youtube.com/watch?v=dX7KR1CrXwU
Title: Solving a 'Harvard' University Entrance Exam Question
Presenter: Maths Explorer
Given the relation \begin{equation} 4^x \cdot 8^x = 30\,, \end{equation} solve for the real values of $x$
Solution to the problem.
Link to my write-up on logarithms over the real numbers.
Problem 1093 [Not Unipodal: Mixed Rate]
Source: Nursing School Entrance Exam, 2005,
LearningExpress, p. 52.
Title: Mixed Rate
Presenter: Patrick
If jogging for 1 mile uses 150 calories and fast walking for 1 mile
uses 100 calories, a jogger has to go how many times as far as a
walker to use the same number of calories?
Problem 1094 [Not Unipodal: Logarithms]
Source: https://www.youtube.com/watch?v=d4yF_duiq-c
Title: Can You Pass Stanford University Entrance Exam?
Presenter: Sagans Online Maths
Given the relation \begin{equation} 8^{\log x} - 2^{\log x} = 5!\,, \end{equation} solve for all real values of $x$, where the logarithm is base 10.
Solution to the problem.
Link to my write-up on logarithms over the real numbers.
Problem 1095 [Not Unipodal: Algebra]
Source: https://www.algebra.com/algebra
Title: Question 123779
Presenter: Patrick
A rock is dropped from a cliff into the ocean. It travels $16t^2$ feet in
$t$ seconds. If the splash is heard 1.5 second later, how high is the
cliff? [Note: Assume the speed of sound at sea level is 1100 feet
per second.]
Problem 1096 [Not Unipodal: Integral Calculus]
Source: https://www.youtube.com/watch?v=6ALDtCgeXhg
Title: This AP Calculus BC Integral Looks Impossible
Presenter: Dr PK Math
Given the relation \begin{equation} I =\int_0^1 \frac{5x+8}{3x^2+2x+2}\,dx\,, \end{equation} solve for $I$.
Problem 1097 [Not Unipodal: Algebra: Logarithms]
Source: https://www.youtube.com/watch?v=fMm3I6ctXK0
Title: Japanese| A Mind Blowing & Brain Buster
Olympiad Math Problem
Presenter: Shorif Sir
Given the relation \begin{equation} 8^{x+1}+ 8^{x-1}= 100\,, \end{equation} solve for all real values of $x$.
Solution to the problem.
Link to my write-up on logarithms over the real numbers.
Problem 1098 [Not Unipodal: Algebra]
Source: https://www.youtube.com/watch?v=V1_0AQSSU6k
Title: x/4 + 8/x = 3 This Algebra Equation is NOT so simple!
Presenter: TabletClass Math
Given the relation \begin{equation} \frac{x}{4}+ \frac{8}{x}= 3\,, \end{equation} solve for all values of $x$.
Problem 1099 [Not Unipodal: Algebra: Percentages]
Source: https://www.youtube.com/shorts/3hsydDnzOKc
Title: Percentages
Presenter: Guinness and Math Guy
40% of 30 equals what percent of 48?
Problem 1100 [Not Unipodal: Logarithms]
Source: https://www.youtube.com/shorts/z2H_JxmVeog
Title: Logarithms
Presenter: MindSphereYT
Simplify \begin{equation} \phi = \log_{\sqrt{8}}4^{1/3}\,. \end{equation}
Solution to the problem.
Link to my write-up on logarithms over the real numbers.
Problem 1101 [Not Unipodal: Logarithms]
Source: https://www.basic-mathematics.com/hard-word-problems-in-algebra.html
Title: Word Problem 8.
Presenter: Patrick
One ounce of solution X contains only ingredients $a$ and $b$ in a ratio of 2:3.
One ounce of solution Y contains only ingredients $a$ and $b$ in a ratio of 1:2.
If solution Z is created by mixing solutions X and Y in a ratio of 3:11, then
2520 ounces of solution Z contains how many ounces of a? [Also, find the
ratio of $a$ to $b$ in Z.]
Problem 1102 [Not Unipodal: Calculus: Integration]
Source: The Ether of Great Mathematical Ideas
Title: An Integral
Presenter: Patrick
Perform the indefinite integration: \begin{equation} I = \int \sin x \cos x\, dx\,. \end{equation}
To go to Diversions1 page Link to Diversions1
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