Mathematics Diversions 2

Patrick Reany

Abstract: This page presents mathematics diversions, which are basically just
math problems of varying difficulty, which one can try to solve for fun or for
testing one's math skills, or for learning new methods and concepts.

Disclaimer: Many of the following problems were taken from SATs or old
Olympiad tests. My alternative solutions may not follow Olympiad rules, however.
The techniques I use may not be fit for use in Olympiads or entrance exams or the like.
I believe that it's good to see alternative solutions to better fill out one's
mathematical abililties. Four decades ago, I quipped that one cannot truly
understand a theorem until one is able to prove it in essentially two different ways.
It's meant to be a thought-provoking claim, perhaps containing a morsel of truth.

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 Pascal's Triangle.

Link to my write-up on Geometric Series.

Link to my write-up on logarithms over the real numbers.

Link to my write-up on Hyperbolic trig functions over the real numbers.

Link to my write-up on Trigonometric functions.

Link to my write-up on the Lambert W function.

Link to my write-up on Word Problem solving.

To go to Diversions1 page Link to Diversions1

To go to Diversions3 page Link to Diversions3

To go to Diversions4 page Link to Diversions4

To go to Diversions5 page Link to Diversions5

To go to Diversions6 page Link to Diversions6

Problem 200 [Unipodal]:

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 $x$.

Solution to this problem.

Problem 201 [Not Unipodal: Complex Numbers]:

This problem is by

Source: https://www.youtube.com/watch?v=uzmtXoW9-W4
Title: I Solved An Equation in Three Ways | Problem 50
Presenter: aplusbi

Given the relation \begin{equation} z + i = \frac{z}{i}\,, \end{equation} find the values of $z$ over the complex numbers.

Solution to this problem.
Link to my write-up on Basic Complex Numbers.

Problem 202 [Not Unipodal: Lambert]:

This problem is by

Source: https://www.youtube.com/watch?v=rl7cEEA9VKg
Title: Requested Video: How To Solve This Classic Equation
Using Lambert W Product Log Function
Presenter: Learn Math By Doing

Given the relation \begin{equation} x^x = 2\,, \end{equation} find $x$.

Solution to this problem.
Link to my write-up on the Lambert W function.

Problem 203 [Not Unipodal]:

This problem is by

Source: https://www.youtube.com/watch?v=eDlv67bAdXA
Title: Germany | Can you solve this?
Presenter: Master T Maths Class

Given the relations \begin{align} x + y &= 10\,,\\ xy &= 50\,, \end{align} find the values of $x,y$.

Solution to this problem.

Problem 204 [Not Unipodal]:

This problem is by

Source: https://www.youtube.com/watch?v=MxdUD2riv4Q
Title: Cambridge University Admission Interview
Presenter: Super Academy

Given the relation \begin{equation} x = \sqrt{x}^{\sqrt{x}} \,, \end{equation} find the values of $x$.

Solution to this problem.

Problem 205 [Not Unipodal: Complex Numbers]:

This problem is by

Source: https://www.youtube.com/watch?v=LXCNrSiapMw
Title: I solved A System With Conjugates
| Problem 45
Presenter: aplusbi

Given the relations \begin{align} z+ \obz &= 2\,,\\ z\obz &= |z| + 2\,, \end{align}

Solution to this problem.
Link to my write-up on Basic Complex Numbers.

Problem 206 [Not Unipodal: Complex Numbers]:

This problem is by

Source: https://www.youtube.com/watch?v=suqfQuCzvd0
Title: I Checked A Complex Sum
| Problem 57
Presenter: aplusbi

Given the relation \begin{equation} w = (a+bi)^4 + (b+ai)^4\,, \end{equation} is $w$ a real number?

Solution to this problem.
Link to my write-up on Basic Complex Numbers.

Problem 207 [Unipodal: 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.

Solution to this problem.
Link to my write-up on Basic Complex Numbers.

Problem 208 [Unipodal]:

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 $x,y$.

Solution to this problem.

Problem 209 [Not Unipodal: Complex Numbers]:

This problem is by

Source: https://www.youtube.com/watch?v=CrQrFA5OWR4
Title: A Homemade Exponential Equation
| Problem 400
Presenter: aplusbi

Given the relation \begin{equation} z^{|z|} = \frac{1+\sqrt{3} i}{2}\,, \end{equation} find the values of $z$.

Solution to this problem.
Link to my write-up on Basic Complex Numbers.

Problem 210 [Not Unipodal: Complex Numbers]:

This problem is by

Source: https://www.youtube.com/watch?v=PXSVw-5GOnQ
Title: How I Solved An Equation With Absolute Value
| Problem 72
Presenter: aplusbi

Given the relation \begin{equation} z^2 = |z|\,, \end{equation} find the values of $z$.

Solution to this problem.
Link to my write-up on Basic Complex Numbers.

Problem 211 [Not Unipodal: Complex Numbers]:

This problem is by

Source: https://www.youtube.com/watch?v=MUpAYTkP_Zg
Title: I Solved An Equation with z and z bar
| Problem 77
Presenter: aplusbi

Given the relation \begin{equation} \obz + i = z i\,, \end{equation} find the values of $z$.

Solution to this problem.
Link to my write-up on Basic Complex Numbers.

Problem 212 [Not Unipodal: Complex Numbers]:

This problem is by

Source: https://www.youtube.com/watch?v=yq7FhX0QOtI
Title: Can You Solve An Equation With z(bar) And Modulus?
| Problem 91
Presenter: aplusbi

Given the relation \begin{equation} \frac{z}{|z|} = \obz + i \,, \end{equation} find the values of $z$.

Solution to this problem.
Link to my write-up on Basic Complex Numbers.

Problem 213 [Not Unipodal]:

This problem is by

Source: https://www.youtube.com/watch?v=kxP7mnHNaXY
Title: Cambridge University Admission Exam tricks
Presenter: Super Academy

Given the relation \begin{equation} x^{\log_5 3} = \sqrt{x} + 4 \,, \end{equation} find the values of $x$.

Solution to this problem.
Link to my write-up on logarithms over the real numbers.

Problem 214 [Not Unipodal: Alpha substitution]:

This problem is by

Source: ?
Title: ?
Presenter: ?

Given the relation \begin{equation} x^{2x^6} = 3 \,, \end{equation} find the values of $x$ over the real numbers.

Solution to this problem.

Problem 215 [Not Unipodal]:

This problem is by

Source: https://www.youtube.com/watch?v=kZY3T8fhF1k
Title: Japanese l can you solve??
Presenter: Math Master TV

Given the relation \begin{equation} 9^{4^m} = 4^{9^m} \,, \end{equation} find the values of $m$ over the real numbers.

Solution to this problem.

Problem 216 [Unipodal: Complex Numbers]:

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 $z,w$.

Solution to this problem.
Link to my write-up on Basic Complex Numbers.

Problem 217 [Not Unipodal: Complex Numbers]:

This problem is by

Source: https://www.youtube.com/watch?v=Q22yx5LUwwU
Title: A Curious Trigonometric Equation
| Problem 307
Presenter: aplusbi

Given the relation \begin{equation} \frac{\sin\theta}{1-\cos\theta} = i \,, \end{equation} find the values of $\theta$

Solution to this problem.
Link to my write-up on Basic Complex Numbers.

Problem 218 [Not Unipodal]:

This problem is by

Source: https://www.youtube.com/watch?v=QVB8nZRr2Ao
Title: A Nice Exponential System With Logs
Presenter: SyberMath Short5s

Given the relations \begin{align} x^{\log y}+ y^{\log x} &= 2\,,\\ x^{\log x}+ y^{\log y}&= 11\,, \end{align} find the values of $x,y$. (Note, $x,y>0$)

Solution to this problem.
Link to my write-up on logarithms over the real numbers.

Problem 219 [Not Unipodal]:

This problem is by

Source: https://www.youtube.com/watch?v=Q_ZGhFwT1w0
Title: You Probably Haven't Seen This Equation Before |
Problem 155
Presenter: aplusbi

Given the relation \begin{equation} z^z = i \,, \end{equation} find the values of $z$.

Solution to this problem.
Link to my write-up on Basic Complex Numbers.

Problem 220 [Not Unipodal: Alpha substitution]:

This problem is by

Source: https://www.youtube.com/watch?v=dDPelu7GCLs
Title: Exponential Equation Mathematics Problem
Presenter: New Track Mathematics

Given the relation \begin{equation} y^{y^4}= 64 \,, \end{equation} find the values of $y$.

Solution to this problem.

Problem 221 [Not Unipodal: Complex numbers]:

This problem is by

Source: https://www.youtube.com/watch?v=m_S9Yxx9x-4
Title: A Real Problem with Re(z) |
Problem 181
Presenter: aplusbi

Given the relation \begin{equation} \frac{z-1}{\Re(z)+1} = i\,, \end{equation} find the values of $z$.

Solution to this problem.
Link to my write-up on Basic Complex Numbers.

Problem 222 [Not Unipodal: Alpha substitution]:

This problem is by

Source: https://www.youtube.com/watch?v=3gYWZlYSlwE
Title: Germany - Math Olympiad Exponential Problem.
Presenter: KK Logic

Given the relation \begin{equation} 2^x = x^{32}\,, \end{equation} find the values of $x$.

Solution to this problem.

Problem 223 [Not Unipodal: Complex Numbers]:

This problem is by

Source: https://www.youtube.com/watch?v=MImKHu2qPOk
Title: Simplifying An Interesting Sum
Problem 305
Presenter: aplusbi

Given the relation \begin{equation} \phi= (1+i)^n + (1-i)^n\,, \end{equation} find a simpler form for $\phi$.

Solution to this problem.
Link to my write-up on Basic Complex Numbers.

Problem 224 [Not Unipodal: Complex Numbers]:

This problem is by

Source: https://www.youtube.com/watch?v=8QvkcJvrNDI
Title: A Beautiful Homemade Equation
Problem 350
Presenter: aplusbi

Given the relation \begin{equation} z^{|z|}= 2+2\sqrt{3}\, i\,, \end{equation} find the values of $z$.

Solution to this problem.
Link to my write-up on Basic Complex Numbers.

Problem 225 [Unipodal]:

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 $a,b$.

Solution to this problem.

Problem 226 [Not Unipodal]:

This problem is by

Source: https://www.youtube.com/watch?v=FLLTAHlEqLk
Title: France | Junior Math Olympiad Exponent
Presenter: Super Academy

Given the relation \begin{equation} x^{\log 27} + 9^{\log x}=36\,, \end{equation} find the values of $x$.

Solution to this problem.
Link to my write-up on logarithms over the real numbers.

Problem 227 [Unipodal]:

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 $x,y$.

Solution to this problem.

Problem 228 [Not Unipodal: Complex Numbers]:

This problem is by

Source: https://www.youtube.com/watch?v=Q_ZGhFwT1w0
Title: You Probably Haven't Seen This Equation Before
Problem 155
Presenter: aplusbi

Given the relation \begin{equation} z^{z}= i\,, \end{equation} find the values of $z$.

Solution to this problem.
Link to my write-up on Basic Complex Numbers.

Problem 229 [Not Unipodal: Complex Numbers]:

This problem is by

Source: https://www.youtube.com/watch?v=KQXeTW9G1xg
Title: How To Solve z^4 = z^*
Problem 101
Presenter: aplusbi

Given the relation \begin{equation} z^4= \obz\,, \end{equation} find the values of $z$.

Solution to this problem.
Link to my write-up on Basic Complex Numbers.

Problem 230 [Not Unipodal: Complex Numbers]:

This problem is by

Source: https://www.youtube.com/watch?v=E0Z1hism9zU
Title: Let's Solve An Absolute Value Equation
Problem 113 (almost)
Presenter: aplusbi

Given the relation \begin{equation} |z^2 - 1| = |z|^2\,, \end{equation} find the values of $z$ (locus of points in the $x,y$ plane).

With apologies, I accidentally changed the problem a bit, as you can see.
However, since WolframAlpha agreed with my solution, I decided to
display the modified problem I solved.

Solution to this problem.
Link to my write-up on Basic Complex Numbers.

Problem 231 [Not Unipodal: Alpha substitution]:

This problem is by

Source: https://www.youtube.com/watch?v=et3fY76oy5Q
Title: Solving A Very Exponential Equation
Presenter: SyberMath Shorts

Given the relation \begin{equation} x^{x^2} = 2^{1024}\,, \end{equation} find the values of $x$.

Solution to this problem.

Problem 232 [Unipodal]:

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}

Solution to this problem.

Problem 233 [Not Unipodal]:

This problem is by

Source: https://www.youtube.com/watch?v=ls8t1LAGcBI
Title: Germany Math Olympiad Question
Presenter: Higher Mathematics

Given the relation \begin{equation} x^{ \sqrt{x}} = 10 \,, \end{equation} find the values of $x$ over the real numbers.

Solution to this problem.

Problem 234 [Not Unipodal: Solved by a table]:

This problem is by

Source: https://www.youtube.com/watch?v=dGLjIPkV9GY
Title: An Interesting Nonstandard Equation
Presenter: Syber Math

Given the relation \begin{equation} 256^{x} = \frac{1}{x} \,, \end{equation} find the values of $x$ over the real numbers.

Solution to this problem.

Problem 235 [Unipodal and Conventional]:

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 $x$.

Solution to this problem.

Problem 236 [Not Unipodal, and some useful Lambert $W$ function lemmas]:

This problem is by

Source: https://www.youtube.com/watch?v=LESu08vVvrQ
Title: International Mathematical Olympiad Problem
Presenter: Higher Mathematics

Given the relation \begin{equation} x^{5} = 9^x \,, \end{equation} find the values of $x$ over the real (or complex) numbers.

Solution to this problem.
Link to my write-up on the Lambert W function.

Problem 237 [Not Unipodal]:

This problem is by

Source: https://www.youtube.com/watch?v=bqFf5R2oofE
Title:How to solve this nice math Exponential algebra problem
Presenter: Mathematics & Statistics Guru

Given the relation \begin{equation} a^3 + b^3 + 3ab = 1 \,, \end{equation} find the values of $a+b$ over the real numbers.

Solution to this problem.

Problem 238 [Not Unipodal, Complex Numbers]:

This problem is by

Source: https://www.youtube.com/watch?v=6A5T1uhIozs
Title: An Interesting Homemade Equation
Problem 459
Presenter: aplusbi

Given the relation \begin{equation} z^{|z|^2}=2i\,, \end{equation} find the values of $z$.

Solution to this problem.
Link to my write-up on Basic Complex Numbers.

Problem 239 [Not Unipodal, Complex Numbers]:

This problem is by

Source: https://www.youtube.com/watch?v=XgUcfL7Vj9g
Title: A Nice Absolute Value Equation
Problem 431
Presenter: aplusbi

Given the relation \begin{equation} z|z-1|=20+20i\,, \end{equation} find the values of $z$.

Solution to this problem.
Link to my write-up on Basic Complex Numbers.

Problem 240 [Not Unipodal, Complex Numbers]:

This problem is by

Source: https://www.youtube.com/watch?v=Xmquv6CpTco
Title: Can We Solve A Tangential Equation
Problem 435
Presenter: aplusbi

Given the relation \begin{equation} (1+i\tan\theta)^5 = 32\,, \end{equation} find the values of $\theta$.

Solution to this problem.
Link to my write-up on Trigonometric functions.

Problem 241 [Not Unipodal, Lambert]:

This problem is by

Source: https://www.youtube.com/watch?v=wZSxh_WB_JY
Title: An Interesting Nonstandard Equation
Presenter: SyberMath

Given the relation \begin{equation} e^{x^2-1} = x\,, \end{equation} find the values of $x$.

Solution to this problem.
Link to my write-up on the Lambert W function.

Problem 242 [Not Unipodal]:

This problem is by

Source: https://www.youtube.com/watch?v=EMu-kYY5rdE
Title: Is e^x=ln(x) solvable?
Presenter: blackpenredpen

Given the relation \begin{equation} e^{x} = \ln x\,, \end{equation} find the values of $x$.

Solution to this problem.
Link to my write-up on logarithms over the real numbers.

Problem 243 [Not Unipodal: Alpha substitution]:

This problem is by

Source: https://www.youtube.com/watch?v=nWWocG37KzI
Title: Oxford University Pure Mathematics Admission Exam
Presenter: Super Academy

Given the relation \begin{equation} \sqrt{x}^x = 2^{ x+16}\,, \end{equation} find the values of $x$.

Solution to this problem.

Problem 244 [Unipodal]:

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 $n$.

Solution to this problem.

Problem 245 [Unipodal]:

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 $\phi = x+y$.

Solution to this problem.

Problem 246 [Not Unipodal: Lambert]:

This problem is by

Source: https://www.youtube.com/watch?v=bCG6L0h5R4A
Title: The Hardest SAT Problem
Presenter: Higher Mathematics

Given the relation \begin{equation} x^4 = 3^{ x}\,,\ \end{equation} find the values of $x$

Solution to this problem.
Link to my write-up on the Lambert W function.

Problem 247 [Not Unipodal: Complex Numbers]:

This problem is by

Source: https://www.youtube.com/watch?v=8n1J4goP6vM
Title: A Nice Equation | Problem 433
Presenter: aplusbi

Given the relation \begin{equation} \frac{\ln z}{z} = \frac{\pi}{2} \,, \end{equation}

Solution to this problem.
Link to my write-up on logarithms over the real numbers.

Problem 248 [Unipodal]:

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 $\phi = a^4 + b^4$.

Solution to this problem.

Problem 249 [Not Unipodal: Lambert]:

This problem is by

Source: https://www.youtube.com/watch?v=QkiWK7mk3bc
Title: Nice Exponent Math Simplification
Presenter: Master T Maths Class

Given the relation \begin{equation} 5^x\cdot 2^x \cdot x^x = \sqrt{5} \,, \end{equation} find the values of $x$

Solution to this problem.
Link to my write-up on the Lambert W function.

Problem 250 [Not Unipodal]:

This problem is by

Source: https://www.youtube.com/watch?v=Pj7ju4ebtY0
Title: Can We Solve A Transcendental Equation
Presenter: SyberMath

Given the relation \begin{equation} e^x+x+1=0 \,, \end{equation} find the values of $x$.

Solution to this problem.
Link to my write-up on the Lambert W function.

Problem 251 [Not Unipodal: Lambert]:

This problem is by

Source: https://www.youtube.com/watch?v=YknH4GYKdBo
Title: An Interesting Nonstandard Equation
Presenter: SyberMath

Given the relation \begin{equation} \sqrt{x}=\left( \frac{1}{2}\right)^x \,, \end{equation} find the real values of $x$.

Solution to this problem.
Link to my write-up on the Lambert W function.

Problem 252 [Not Unipodal: Lambert]:

This problem is by

Source: https://www.youtube.com/watch?v=YwElQV393M4
Title: An Interesting Nonstandard Equation
Presenter: SyberMath

Given the relation \begin{equation} x e^{(x-1)/x} = 1 \,, \end{equation} find the values of $x$.

Solution to this problem.
Link to my write-up on the Lambert W function.

Problem 253 [Unipodal]:

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.

Solution to this problem.

Problem 254 [Not Unipodal: Complex Numbers]:

This problem is by

Source: https://www.youtube.com/watch?v=z-aTQRqvWlM
Title: Solving A Locus Problem With Im
| Problem 143
Presenter: aplusbi

Given the relation \begin{equation} \Im(z^2) = 4 \,, \end{equation} find the values of $z$ (locus of points).

Solution to this problem.
Link to my write-up on Basic Complex Numbers.

Problem 255 [Not Unipodal: Complex Numbers]:

This problem is by

Source: https://www.youtube.com/watch?v=pJqyXQ-UUbI
Title: Finding The Absolute Value of z + 1
| Problem 255
Presenter: aplusbi

Given the relation \begin{equation} z = e^{i\theta}\,, \end{equation} find the values of \begin{equation} \phi = |z+1|\,. \end{equation}

Solution to this problem.
Link to my write-up on Basic Complex Numbers.

Problem 256 [Not Unipodal: Complex Numbers]:

This problem is by

Source: https://www.youtube.com/watch?v=inTOdWElDLQ
Title: A Nice Exponential Equation
| Problem 358
Presenter: aplusbi

Given the relation \begin{equation} z^{-z} = e^{\pi/2}\,, \end{equation} find the values of $z$

Solution to this problem.
Link to my write-up on Basic Complex Numbers.

Problem 257 [Not Unipodal: 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 $z,w$.

Solution to this problem.
Link to my write-up on Basic Complex Numbers.

Problem 258 [Unipodal: Complex Numbers]:

Source: https://www.youtube.com/watch?v=H342tw_GSTc
Title: A System of Equations
| Problem 316
Presenter: aplusbi

Note: This is the same problem as the last one, but using the unipodal
algebra this time.

Given the relations \begin{align} zw &= 7 - i\,,\\ z + w &= 5\,, \end{align} find the values of $z,w$.

Solution to this problem.
Link to my write-up on Basic Complex Numbers.

Problem 259 [Not Unipodal: Lambert]:

This problem is by

Source: https://www.youtube.com/watch?v=_pv3HYNOnFY
Title:Solving An Exponential Equation With A Parameter
Presenter: SyberMath

Given the relation \begin{equation} x^{(\ln x)/x} = a\,, \end{equation} find the values of $x$.

Solution to this problem.
Link to my write-up on the Lambert W function.
Link to my write-up on logarithms over the real numbers.

Problem 260 [Not Unipodal: Complex Numbers]:

This problem is by

Source: https://www.youtube.com/watch?v=LLN6dZgj65U
Title: An Exponent That Doubles |
Problem 353
Presenter: aplusbi

Given the relation \begin{equation} i^z = 2i\,, \end{equation} find the values of $z$

Solution to this problem.
Link to my write-up on Basic Complex Numbers.

Problem 261 [Not Unipodal: Complex Numbers]:

This problem is by

Source: https://www.youtube.com/watch?v=xTc9xaO6kHo
Title: A Quick And EZ Rational Equation |
Problem 315
Presenter: aplusbi

Given the relation \begin{equation} \frac{1+ e^{i\theta}}{1+ e^{-i\theta}} =i\,, \end{equation} find the values of $\theta$.

Solution to this problem.
Link to my write-up on Basic Complex Numbers.

Problem 262 [Not Unipodal: Complex Numbers: Lambert]:

This problem is by

Source: https://www.youtube.com/watch?v=Fcp92u28eUY
Title: You Probably Haven't Seen This Before |
Problem 212
Presenter: aplusbi

Given the relation \begin{equation} \ln(\ln z) = z\,, \end{equation} find the values of $z$.

Solution to this problem.
Link to my write-up on the Lambert W function.

Link to my write-up on logarithms over the real numbers.

Problem 263 [Not Unipodal: Complex Numbers: Lambert]:

This problem is by

Source: https://www.youtube.com/watch?v=OGmWI_7qRuA
Title: Let's Solve A Homemade Exponential Equation |
Problem 239
Presenter: aplusbi

Given the relation \begin{equation} z^z= e^{-\pi +2i\ln 2}\,, \end{equation} find the values of $z$

Solution to this problem.
Link to my write-up on the Lambert W function.
Link to my write-up on Basic Complex Numbers.

Problem 264 [Not Unipodal: Complex Numbers]:

This problem is by

Source: https://www.youtube.com/watch?v=9s9D5NZjVMc
Title: A Highly Complex Exponential |
Problem 257
Presenter: aplusbi

Given the relation \begin{equation} e^{e^i}= a+bi = z\,, \end{equation} find the values of \begin{equation} \phi = a^2+b^2 = r^2=z\obz\,. \end{equation}

Solution to this problem.
Link to my write-up on Basic Complex Numbers.

Problem 265 [Unipodal: Complex Numbers]:

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 $a,b$.

Solution to this problem.

Problem 266 [Not Unipodal]:

This problem is by

Source: https://www.youtube.com/watch?v=S0NfTf1BjCY
Title: New Zealand exam leaves students in tears
Presenter: MindYourDecisions

Given the relation \begin{equation} \frac{x^2-2x}{4x-1} = \frac{k-1}{k+1}\,, \end{equation} and we are told that the roots to this equation come in a pair
of equal magnitude but opposite signs, such as $a$ and $-a$,
we are to find the values of $k$.

Solution to this problem.

Problem 267 [Not Unipodal: Complex Numbers]:

This problem is by

Source: https://www.youtube.com/watch?v=kVKFUMlusAc
Title: Can We Solve e^z = ln(z)? |
Problem 147
Presenter: aplusbi

Given the relation \begin{equation} e^z= \ln z\,, \end{equation} find the values of $z$.

Solution to this problem.
Link to my write-up on Basic Complex Numbers.
Link to my write-up on logarithms over the real numbers.

Problem 268 [Not Unipodal: Complex Numbers]:

This problem is by

Source: https://www.youtube.com/watch?v=ebvPDeSLUHM
Title:A Somewhat Exponential Equation |
Problem 178
Presenter: aplusbi

Given the relation \begin{equation} z^i= i^z\,, \end{equation} find the values of $z$.

Solution to this problem.
Link to my write-up on Basic Complex Numbers.

Problem 269 [Not Unipodal: Complex Numbers]:

This problem is by

Source: https://www.youtube.com/watch?v=hhQ1nBeU-Mg
Title: Can We Solve i^z = 1 + i |
Problem 203
Presenter: aplusbi

Given the relation \begin{equation} i^z = 1 + i \,, \end{equation} find the values of $z$.

Solution to this problem.
Link to my write-up on Basic Complex Numbers.

Problem 270 [Not Unipodal: Complex Numbers]:

This problem is by

Source: https://www.youtube.com/watch?v=OMEu36Pth_8
Title: An Exponential Equation and A Special Function
Problem 200
Presenter: aplusbi

Given the relation \begin{equation} z i^z = 4 \,, \end{equation} find the values of $z$.

Solution to this problem.
Link to my write-up on Basic Complex Numbers.

Problem 271 [Not Unipodal: Complex Numbers]:

This problem is by

Source: https://www.youtube.com/watch?v=hUUrZdeCUx0
Title: A Problem With Conjugates |
Problem 142
Presenter: aplusbi

Given the relation \begin{equation} z(\obz+2) = 2i \,, \end{equation} find the values of $z$.

Solution to this problem.

Problem 272 [Not Unipodal]:

This problem is by

Source: https://www.youtube.com/watch?v=vrudNt_bPK8
Title: Brazil | A Nice Olympiad Math Problem |
Presenter: MathMinds

Given the relation \begin{equation} x^x = x \,, \end{equation} find the values of $x$.

Solution to this problem.

Problem 273 [Not Unipodal: Alpha substitution]:

This problem is by

Source: https://www.youtube.com/watch?v=g_fNga4O4i0
Title: An Interesting Equation With Exponentials
Presenter: SyberMath Shorts

Given the relation \begin{equation} x^x = 5^{x+25} \,, \end{equation} find the values of $x$.

Solution to this problem.

Problem 274 [Not Unipodal: Lambert]:

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 values of $x$.

Solution to this problem.
Link to my write-up on the Lambert W function.

Problem 275 [Not Unipodal: Complex numbers]:

This problem is by

Source: https://www.youtube.com/watch?v=LHyg9-fBv_U
Title: A Trigonometric Exponential Equation |
Problem 103
Presenter: aplusbi

Given the relation \begin{equation} \cos e^z = i \,, \end{equation} find the values of $z$.

Solution to this problem.
Link to my write-up on Basic Complex Numbers.

Problem 276 [Not Unipodal: Complex numbers]:

This problem is by

Source: https://www.youtube.com/watch?v=dlCHixjJr4I
Title:Cambridge University Admission Interview Tricks
Presenter: Super Academy

Given the relation \begin{equation} 4^{x^2} = x^{128} \,, \end{equation} find the values of $x$.

Solution to this problem.
Link to my write-up on Basic Complex Numbers.

Problem 277 [Not Unipodal: Complex numbers]:

This problem is by

Source: https://www.youtube.com/watch?v=hRhwrN88IWo
Title: Let's Solve A Problem With Absolute Values |
Problem 122
Presenter: aplusbi

Given the relation \begin{equation} |z + w| = |z| + |w| \,, \end{equation} find the values of \begin{equation} \phi = \Im \left( \frac{z}{w} \right)\,. \end{equation}

Solution to this problem.
Link to my write-up on Basic Complex Numbers.

Problem 278 [Not Unipodal: Lambert]:

This problem is by

Source: https://www.youtube.com/watch?v=qSlHrGHWWQE
Title: A Nonstandard Equation
Presenter: SyberMath Shorts

Given the relation \begin{equation} a^{a^5} = 4^{1/5} \,, \end{equation} find the values of $a$.

Solution to this problem.
Link to my write-up on the Lambert W function.

Problem 279 [Not Unipodal]:

This problem is by

Source: https://www.youtube.com/watch?v=ET_O1idb0No
Title: A Radical Exponential
Presenter: SyberMath Shorts

Given the relation \begin{equation} a^{x} = x^{1/2} \,, \end{equation} find the values of $x$.

Solution to this problem.

Problem 280 [Not Unipodal: Lambert]:

This problem is by

Source: https://www.youtube.com/watch?v=qNOrL5fBVVM
Title: High School Math Tournament
Presenter: Super Academy

Given the relation \begin{equation} (x-3)^{\sqrt{x-3}} = 3\,, \end{equation} find the real values of $x$.

Solution to this problem.
Link to my write-up on the Lambert W function.

Problem 281 [Not Unipodal: Complex Numbers]:

This problem is by

Source: https://www.youtube.com/watch?v=buI2Z5JGr24
Title:I Solved An Equation With z and z(bar)
Presenter: aplusbi

Given the relation \begin{equation} z + zi + 1= \obz\,, \end{equation} find the real values of $z$.

Solution to this problem.
Link to my write-up on Basic Complex Numbers.

Problem 282 [Not Unipodal: Complex Numbers]:

This problem is by

Source: https://www.youtube.com/watch?v=suqfQuCzvd0
Title: I Checked A Complex Sum |
Problem 57
Presenter: aplusbi

Given the relation \begin{equation} \phi = (a+ib)^4 + (b+ia)^4 \,, \end{equation} determine if $\phi$ is real or not.

Solution to this problem.
Link to my write-up on Basic Complex Numbers.

Problem 283 [Not Unipodal: Complex Numbers]:

This problem is by

Source: https://www.youtube.com/watch?v=jFO4OTGCHJ0
Title: Solving an Equation with Absolute Value |
Problem 16
Presenter: aplusbi

Given the relation \begin{equation} z|z|=1-i\sqrt{3}\,, \end{equation} find the values of $z$.

Solution to this problem.
Link to my write-up on Basic Complex Numbers.

Problem 284 [Not Unipodal: Complex Numbers]:

This problem is by

Source: https://www.youtube.com/watch?v=1m2Ye6Izeig
Title:How To Solve A System with Absolute Values |
Problem 42
Presenter: aplusbi

Given the relation \begin{align} |z| &= |z-2| \,,\\ |z+i| &= \sqrt{5}\,, \end{align} find the values of $z$.

Solution to this problem.
Link to my write-up on Basic Complex Numbers.

Problem 285 [Not Unipodal]:

This problem is by

Source: https://www.youtube.com/watch?v=qXk9460YCow
Title: Stanford University Admission Exam Tricks Probably Never knew
Presenter: Super Academy

Given the relation \begin{align} x^2+xy &= 35 \,,\\ y^2+xy &=14\,, \end{align} find the values of $xy$.

Solution to this problem.

Problem 286 [Not Unipodal]:

This problem is by

Source: https://www.youtube.com/watch?v=_pv3HYNOnFY
Title: Solving An Exponential Equation With A Parameter
Presenter: SyberMath

Given the relation \begin{equation} x^{(\ln x)/x} = a\,, \end{equation} find the values of $x$.

Solution to this problem.

Link to my write-up on logarithms over the real numbers.

Problem 287 [Not Unipodal]:

This problem is by

Source: https://www.youtube.com/watch?v=FP-CreHbJ0w
Title: Stanford University Admission Interview Tricks
Presenter: Super Academy

Given the relation \begin{equation} a^4+b^4=10a^2b^2 \,, \end{equation} find the values of \begin{equation} \phi= \frac{a+b}{a-b}\,. \end{equation}

Solution to this problem.

Problem 288 [Not Unipodal: Complex Numbers]:

This problem is by

Source: https://www.youtube.com/watch?v=5ytLQ4rV3fQ
Title: An Ln Equation |
Problem 439
Presenter: aplusbi

Given the relation \begin{equation} z \ln z = -\sqrt{2}\pi/8+ i\sqrt{2}\pi/8 \,, \end{equation} find the values of $z$.

Solution to this problem.
Link to my write-up on Basic Complex Numbers.
Link to my write-up on logarithms over the real numbers.

Problem 289 [Not Unipodal: Complex Numbers]:

This problem is by

Source: https://www.youtube.com/watch?v=heY8pAsQ6uU
Title: A System With Conjugates |
Problem 341
Presenter: aplusbi

Given the relation \begin{align} z + \overline{w} &= 3 + 2i \,,\\ w - \obz &= 5-4i \,, \end{align} find the values of $z,w$.

Solution to this problem.
Link to my write-up on Basic Complex Numbers.

Problem 290 [Unipodal]:

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 $b$.

Solution to this problem.

Problem 291 [Not Unipodal: Complex Numbers]:

This problem is by

Source: https://www.youtube.com/watch?v=FpQN3LmTpy8
Title: A Homemade Exponential Equation |
Problem 427
Presenter: aplusbi

Given the relations \begin{equation} z^{e^z} = \frac{-i}{\pi} \,, \end{equation} find the values of $z$.

Solution to this problem.
Link to my write-up on Basic Complex Numbers.

Problem 292 [Not Unipodal: Complex Numbers: Lambert]:

This problem is by

Source: https://www.youtube.com/watch?v=5MxFlElJEaQ
Title: Let's Solve An Exponential Equation |
Problem 206
Presenter: aplusbi

Given the relations \begin{equation} z^{z} = 1 \,, \end{equation} find the values of $z$.

Solution to this problem.
Link to my write-up on the Lambert W function.
Link to my write-up on Basic Complex Numbers.

Problem 293 [Not Unipodal: Complex Numbers: Lambert]:

This problem is by

Source: https://www.youtube.com/watch?v=Q4MV_jqpH_k
Title: Cambridge University Admission Test Tricks !
Presenter: Super Academy

Given the relations \begin{equation} 5^{10x} = x^2 \,, \end{equation} find the real values of $x$.

Solution to this problem.
Link to my write-up on the Lambert W function.

Problem 294 [Unipodal]:

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}

Solution to this problem.

Problem 295 [Not Unipodal: Complex Numbers: Lambert]:

This problem is by

Source: https://www.youtube.com/watch?v=iwj7lgRWu5U
Title: Harvard University Admission Interview Tricks
Presenter: Super Academy

Given the relations \begin{equation} 8^{x} = 6x \,, \end{equation} find the real values of $x$.

Solution to this problem.
Link to my write-up on the Lambert W function.

Problem 296 [Not Unipodal: Lambert]:

This problem is by

The website is found at: Source: Wikipedia page on the Lambert W function

Problem # 1:

Given the relation \begin{equation} 3^{x}=2x+2 \,, \end{equation} find the real values of $x$.

Problem # 2:

Given the relation \begin{equation} x=a+b\, e^{cx}\,, \end{equation} find the values of $x$ (where $a,b,c$ are complex numbers).

Solution to both problems.
Link to my write-up on the Lambert W function.

Problem 297 [Not Unipodal: Alpha substitution]:

This problem is by

Source: https://www.youtube.com/watch?v=c8WG50X6ZpE
Title: Stanford University Entrance Exam
Presenter: Super Academy

Given the relation \begin{equation} 243^{x^2} = x^{45} \,, \end{equation} find the real values of $x$.

Solution to this problem.

Problem 298 [Not Unipodal]:

This problem is by

Source: https://www.youtube.com/watch?v=tPpte7ivc2s
Title: Harvard University Admission Interview Tricks
Presenter: Super Academy

Given the relation \begin{equation} \log_{81}x + \log_{9}x= 6\,, \end{equation} find the real values of $x$.

Solution to this problem.
Link to my write-up on logarithms over the real numbers.

Problem 299 [Not Unipodal: Lambert]:

This problem is by

Source: https://www.youtube.com/watch?v=glE7fMKJO1Q
Title: Oxford University Pure Mathematics Course Admission Exam
Presenter: Super Academy

Given the relation \begin{equation} \log x + 64^{\log {x}}= \third\,, \end{equation} find the real values of $x$.

Solution to this problem.
Link to my write-up on the Lambert W function.

Link to my write-up on logarithms over the real numbers.

Problem 300 [Not Unipodal]:

This problem is by

Source: https://www.youtube.com/watch?v=Jelc62QcZMs
Title: Entrance Exam Tricks from Stanford Universty Interview
Presenter: Super Academy

Given the relation \begin{equation} x^{\log_5 x} = 625 \,, \end{equation} find the values of $x$.

Solution to this problem.

Link to my write-up on logarithms over the real numbers.

Problem 301 [Not Unipodal: Complex numbers]:

This problem is by

Source: https://www.youtube.com/watch?v=NLT1vLIJjI0
Title: Can you Pass Harvard University Admission Interview ?
Presenter: Super Academy

Given the relation \begin{equation} \sqrt{x } + \sqrt{-x } = 36 \,, \end{equation} find the values of $x$.

Solution to this problem.

Problem 302 [Not Unipodal: Lambert]:

This problem is by

Source: https://www.youtube.com/watch?v=dCQgK40HOR8
Title: Stanford University Admission Test Tricks !
Presenter: Super Academy

Given the relation \begin{equation} 6^x = 6x + 24\,, \end{equation} find the real values of $x$.

Solution to this problem.
Link to my write-up on the Lambert W function.

Problem 303 [Not Unipodal: Lambert]:

This problem is by

The website is found at: Source: Wikipedia page on the Lambert W function, under
`Thermodynamic equilibrium'.

Given the relation \begin{equation} \ln K= \frac {a}{T}+b+c\ln T\,,\label{eq:TheGiven1} \end{equation} find the real values of $T$.

Solution to the problem.
Link to my write-up on logarithms.
Link to my write-up on the Lambert W function.

Problem 304 [Not Unipodal]:

This problem is by

Source: https://www.youtube.com/watch?v=6KJgiNW2pDg
Title: Can you Pass Harvard University Admission Interview ?
Presenter: Super Academy

Given the relation \begin{align} x^2 + xy + y^2 &= 96 \,,\\ x + \sqrt{xy} + y &=16\,, \end{align} find the values of $x,y$.