Challenge Definition and 911 Threads

As (almost) always: have a look on previous challenge threads, too. E.g. in https://www.physicsforums.com/threads/math-challenge-march-2019.967174/ are still problems to solve, and some of them easy, which I find, and in any case useful to know or at least useful to have seen. As a general...

An urn contains $n$ balls numbered $1, 2, . . . , n$. They are drawn one at a time at random until the urn is empty. Find the probability that throughout this process the numbers on the balls which have been drawn is an interval of integers. (That is, for $1 \leq k \leq n$, after the $k$th draw...

Questions 1.) (disclosed by @Demystifier ) Using the notion of double integrals prove that $$B(m,n) = \frac{\Gamma (m) \Gamma (n)}{\Gamma (m + n)}\; \;(m \gt 0\,,\, n\gt 0)$$ where ##B## and ##\Gamma## are the Beta and Gamma functions respectively. 2.) (solved by @Math_QED ) Show that the...

[FONT=Times New Roman]Prove that $$\tan18^\circ\ =\ \sqrt{1-\dfrac2{\sqrt5}}.$$ No calculator, computer program, Excel, Google, or any other kind of cheating tool allowed. (Smirk) Have fun!

In an equilateral triangle $ABC$, let $D$ be a point inside the triangle such that $\angle BAD=54^\circ$ and $\angle BCD=48^\circ$. Prove that $\angle DBA=42^\circ$.

This week I am at "General Relativity as a Challenge for Physics Education" 690. WE-Heraeus-Seminar https://www.we-heraeus-stiftung.de/veranstaltungen/seminare/2019/general-relativity-as-a-challenge-for-physics-education/ (...

Time for our new winter challenge! This time our challenge has also two Computer Science related questions and a separate section with five High School math problems. Enjoy! Rules: a) In order for a solution to count, a full derivation or proof must be given. Answers with no proof will be...

Merry Christmas to all who celebrate it today! Rules: a) In order for a solution to count, a full derivation or proof must be given. Answers with no proof will be ignored. Solutions will be posted around 15th of the following month. b) It is fine to use nontrivial results without proof as long...

I found this calendar with daily math puzzles. Based on the first three puzzles it seems to be much easier than the math challenges here, and require no advanced mathematics. The answer is always a three-digit number, and the answers to all 24 puzzles together create a larger puzzle.

It's December and we like to do a Special this month. The challenges will be posted like an Advent Calendar. We will add a new problem each day, from 12/1 to 12/25. They vary between relatively easy logical and numerical problems, calculations, to little proofs which hopefully add some...

Rules: a) In order for a solution to count, a full derivation or proof must be given. Answers with no proof will be ignored. Solutions will be posted around 15th of the following month. b) It is fine to use nontrivial results without proof as long as you cite them and as long as it is "common...

Summer is coming and brings ... Oops, time for a change! Fall (Spring) is here and what's better than to solve some tricky problems on a long dark evening (with the power of returning vitality all around). RULES: a) In order for a solution to count, a full derivation or proof must be given...

NASA is looking for a process to use CO2 as a Carbon source on Mars; ultimate goal is to use the Carbon in the synthesis of other products. $50,000 prize. Open to U.S. citizens, permanent residents, and U.S. business entities, work must be done in the U.S...

Summer is coming and brings a new basic math challenge! Enjoy! For more advanced problems you can check our other basic level math challenge thread! RULES: a) In order for a solution to count, a full derivation or proof must be given. Answers with no proof will be ignored. Solutions will be...

Summer is coming and brings a new basic math challenge! Enjoy! For more advanced problems you can check our other intermediate level math challenge thread! RULES: a) In order for a solution to count, a full derivation or proof must be given. Answers with no proof will be ignored. Solutions...

Given that $$a,\,b$$ and $$c$$ are real numbers that satisfy the system of equations below: $$(a+b)(b+c)=-1\\(a-b)^2+(a^2-b^2)^2=85\\(b-c)^2+(b^2-c^2)^2=75$$ Find $$(a-c)^2+(a^2-c^2)^2$$.

Summer is coming and brings a new intermediate math challenge! Enjoy! If you find the problems difficult to solve don't be disappointed! Just check our other basic level math challenge thread! RULES: 1) In order for a solution to count, a full derivation or proof must be given. Answers with no...

Summer is coming and brings a new basic math challenge! Enjoy! For more advanced problems you can check our other intermediate level math challenge thread! RULES: 1) In order for a solution to count, a full derivation or proof must be given. Answers with no proof will be ignored. Solutions...

Jack, John and CWS's ============== Canadian Wild Strawberries (CWS) are tiny but tasty. A and B each have a jar containing 400 CWS; they decide to have a CWS eating race; A wins, swallowing his last CWS when B still has 23 left. Took A 13.2 seconds; burp! Next, B takes on C, each with a jar...

Summer is coming and brings a new intermediate math challenge! Enjoy! If you find the problems difficult to solve don't be disappointed! Just check our other basic level math challenge thread! RULES: 1) In order for a solution to count, a full derivation or proof must be given. Answers with no...

Summer is coming and brings a new basic math challenge! Enjoy! For more advanced problems you can check our other intermediate level math challenge thread! RULES: 1) In order for a solution to count, a full derivation or proof must be given. Answers with no proof will be ignored. Solutions will...

[FONT=Times New Roman]If $A$ and $B$ are nonempty sets of complex numbers, define $$A\circ B\ =\ \{z_1z_2:z_1\in A,\,z_2\in B\}.$$ Further define $A^{[1]}=A$ and recursively $A^{[n]}=A^{[n-1]}\circ A$ for $n>1$. Let $\zeta_n=\{z\in\mathbb C:z^n=1\}$. Given a fixed integer $n\geqslant2$ and any...

Summer is coming and brings a new intermediate math challenge! Enjoy! If you find the problems difficult to solve don't be disappointed! Just check our other basic level math challenge thread! RULES: 1) In order for a solution to count, a full derivation or proof must be given. Answers with no...

Summer is coming and brings a new basic math challenge! Enjoy! For more advanced problems you can check our other intermediate level math challenge thread! RULES: 1) In order for a solution to count, a full derivation or proof must be given. Answers with no proof will be ignored. 2) It is fine...

[FONT=Times New Roman]$n$ lights are arranged in a circle, each operated by exactly one of $n$ switches (with each switch operating exactly one light). Flicking a switch turns the light it is operating on if it is off, and off if it is on. Initially all the lights are off. The first person comes...

[FONT=Times New Roman]If the line $x=k$ is tangent to the curve $$\large y\:=\:x+\sqrt2\,e^{\frac{x+y}{\sqrt2}}$$ what is the value of $k$?

Given that √5 tanA=-2 and CosB=8/17 in ∆ABC State why we may assume that angle C is acute and determine the value of Sin CAttempt made: tanA=-2/√5 CosB=8/17 A is obtuse angle of 138° or reflex angle 318.19° B is an acute angle of61.9° or reflex angle298.1 °.Since it is a right...

It's time for an intermediate math challenge! If you find the problems difficult to solve don't be disappointed! Just check our other basic level math challenge thread! RULES: 1) In order for a solution to count, a full derivation or proof must be given. Answers with no proof will be ignored...

It's time for a basic math challenge! For more advanced problems you can check our other intermediate level math challenge thread! RULES: 1) In order for a solution to count, a full derivation or proof must be given. Answers with no proof will be ignored. 2) It is fine to use nontrivial results...

We, a small group of currently four members, want to try a new version of the math-challenges-threads once a month. It turned out to be not as easy as we thought, to find good problems. So what we've gathered are ten questions on "B" level and ten on "I" level for May, and plan to do the same...

[FONT=Times New Roman]Define a Fibonacci sequence by $$\varphi_0=0,\,\varphi_1=1;\ \varphi_{n+2}=\varphi_{n+1}+\varphi_n\ \forall \,n\in\mathbb Z^+\cup\{0\}.$$ Show that $$5\varphi_n^2+4(-1)^n$$ is a perfect square for all non-negative integers $n$.

Hi all. I would like to post some challenge problems from time to time. I’ll start with a simple one. :) Find all real numbers $x,y$ satisfying the following equation: $$2x^2y^2-2xy+x^2+y^2-2x-2y+3=0.$$

Suppose a pond contains $x(t)$ fish at time $t$, and $x(t)$ changes according to the DE: \[\frac{\mathrm{d} x}{\mathrm{d} t} = x\left ( 1-\frac{x}{x_0} \right )-R_f\] where $x_0$ is the equilibrium amount with no fishing and $R_f > 0$ is the constant rate of removal due to fishing. Assume $x(0)...

Prove, that $$\prod_{j = 1}^{n}\left(1+2\cos \left(\frac{3^j}{3^n+1}2\pi\right)\right) = 1.$$

Evaluate the definite integral $$\int_{0}^{1} \ln^2(1+x^{-1}) \,dx$$

$f(x)$ is a degree 10 polynomial such that $f(p)=q$, $f(q)=r$, $f(r)=p$, where $p$, $q$, $r$ are integers with $p<q<r$. Show that not all the coefficients of $f(x)$ are integers.

Prove, that the definite integral $$\int_{0}^{\pi}\ln (2-2\cos x)dx = 0.$$

Let $r_1,r_2, …,r_7$ be the distinct roots (one real and six complex) of the equation $x^7-7= 0$. Let \[p = (r_1+r_2)(r_1+r_3)…(r_1+r_7)(r_2+r_3)(r_2+r_4)…(r_2+r_7)…(r_6+r_7) = \prod_{1\leq i<j\leq 7}(r_i+r_j).\] Evaluate $p^2$.

Let $f$ be a positive and continuous function on the real line which satisfies $f(x + 1) = f(x)$ for all numbers $x$. Prove \[\int_{0}^{1}\frac{f(x)}{f(x+\frac{1}{2})}dx \geq 1.\]

Let the sequence $\left\{x_n\right\}$ of integers (modulo $11$) be defined by the recurrence relation: $x_{n+3} \equiv \frac{1}{3}(x_{n+2}+x_{n+1}+x_n)$ (mod $11$), for $n=1,2,..$ Show, that every such sequence $\left\{x_n\right\}$ is either constant or periodic with period $10$.

Solve the definite integral \[I(a) = \int_{0}^{2\pi}\frac{\sin^2 \theta }{1-2a\cos \theta + a^2}\: \: d\theta,\;\;\; 0<a<1.\]

Determine the sum: \[\frac{1}{4!}+\frac{4!}{8!}+\frac{8!}{12!}+\frac{12!}{16!}+...\]

I'm a little surprised this has not been posted already, also because of constraints of the rules I'm putting it in mechanical engineering (mods are welcome to change it as they see fit) :smile: https://herox.com/GoFly/guidelines VISION Remember when you were a child and wanted to fly? We...

Who can make the most impressive, interesting, or pretty TikZ picture? This third challenge is to create a vector diagram. Such as used in geometric figures, or in physical diagrams with forces and velocities, or in state diagrams. For more impressive arrows, we might use the arrows tikz...

We have an $n \times n$ square grid of dots ($n \ge 2$). Let $s_n$ denote the number of squares that can be constructed from the grid points. (a). Show, that $$s_n = \frac{n^4-n^2}{12}.$$ Note, that squares with "diagonal sides" also count. (b). Evaluate the sum: \[S = \sum_{k = 2}^{\infty...

Evaluate the double integral: \[I = \int \int _R\frac{1}{(1+x^2)y}dxdy\] - where $R$ is the region in the upper half plane between the two curves: $2x^4+y^4+ y = 2$ and $x^4 + 8y^4+y = 1$.

RULES: 1) In order for a solution to count, a full derivation or proof must be given. Answers with no proof will be ignored. 2) It is fine to use nontrivial results without proof as long as you cite them and as long as it is "common knowledge to all mathematicians". Whether the latter is...

Who can make the most impressive, interesting, or pretty TikZ picture? This second challenge is to create a function graph. We can use vanilla TikZ, or the pgfplots package, or... well... that's up to you! If it's not immediately obvious, please mention what makes your picture special. Please...

Hey all, 2 weeks ago I created a challenge to create a geometrical diagram, like a triangle, that is somehow interesting or impressive. Now the moment of truth is here. Please everyone, give your vote! Voting will close in 2 weeks time. Let me recap the submissions.I like Serena...

Who can make the most impressive, interesting, or pretty TikZ picture? This first challenge is to create a geometrical diagram, like a triangle, that is somehow interesting or impressive. We might make it a very complicated figure, or an 'impossible' figure, or use pretty TikZ embellishments...