Challenge Definition and 911 Threads

Show that $3^{2008}+4^{2009}$ can be written as product of two positive integers each of which is larger than $2009^{182}$

For $$m \in \mathbb{Z}^+$$, and $$a, \, z \in \mathbb{R} > 0$$, evaluate the definite integral:$$\int_0^z\frac{x^m}{(a+\log x)}\,dx$$[I'll be adding a few generalized forms like this in the logarithmic integrals thread, in Maths Notes, shortly... (Heidy) ]

A multiple zeta value is defined as \zeta(s_1,...,s_k) = \sum_{n_1 > n_2 ... > n_k > 0} \frac{1}{n_1^{s_1} n_2^{s_2}...n_k^{s_k}} . For example, \zeta(4) = \sum_{n = 1}^{\infty} \frac{1}{n^4} and \zeta(2,2) = \sum_{m =1}^{\infty} \sum_{n = 1}^{m-1} \frac{1}{ m^2 n^2} . Prove the...

What is this place? This forum is for people to come together and stretch their brains on math puzzles. Each week there will be a new challenge for the forum to try. What do I get for answering challenges? We're going to try a point system. The first person to post a solution will be awarded...

It's given that $p+m+n=12$ and that $p, m, n$ are non-negative integers. What is the maximal value of $pmn+pm+pn+mn$?

Find the four roots of the equation $(x-3)^4+(x-5)^4+8=0$.

Prove $$x^x \ge \left( \frac{x+1}{2} \right)^{x+1}$$ for $x>0$.

Evaluate $$\int_0^{\frac{\pi}{2}} \frac{dx}{1+(\tan x)^{\pi e}}$$.

Prove that $$\tan 20^{\circ}+4 \sin 20^{\circ}=\sqrt{3}$$.

Homework Statement The problem is attached as a picture. Homework Equations ... The Attempt at a Solution I have been trying a lot to prove this without any really fruitful approach. At first I thought that the statement was false, or that you could at least construct a sequence...

Find the real solution(s) to the equation $$\frac{36}{\sqrt{x}}+\frac{9}{\sqrt{y}}=42-9\sqrt{x}-\sqrt{y}$$.

Determine the smallest integer that is square and starts with the first four figure 3005. Calculator may be used but solution by computers will not be accepted.(Tongueout)

Hi all. I'm starting my college apps and started with MIT. Here's the prompt: "Tell us about the most significant challenge you've faced or something important that didn't go according to plan. How did you manage the situation?(*) (200-250 words)." Below is my response. What I want to know...

Prove $$2^{\frac{1}{3}}+2^{\frac{2}{3}}<3$$.

For $n=1,2,...,$ set $$S_n=\sum_{k=0}^{3n} {3n\choose k}$$ and $$T_n=\sum_{k=0}^{n} {3n\choose 3k}$$. Prove that $|S_n-3T_n|=2$.

Evaluate $$\int_0^{\pi} \frac{\cos 4x-\cos 4 \alpha}{\cos x-\cos \alpha} dx$$

I didn't even know this was going on. The FTC declared a winner in April for its FTC Robocall Challenge deal with the problem of illegal Robo calls. The winner will be going live soon and it will be free. They even have a easy to remember name - NoMoRobo. Although I am waiting to see how...

Prove that there are only two real numbers such that $(x-1)(x-2)(x-3)(x-4)(x-5)(x-6)=720$.

Solve $||||||| x^2 – x –1 |–3|–5|–7|–9| – 11|–13| = x^2 – 2x – 48$.

This challenge was suggested by jgens. The ##n##th harmonic number is defined by H_n = \sum_{k=1}^n \frac{1}{k} Show that ##H_n## is never an integer if ##n\geq 2##.

Find the following limit: \lim_{n\rightarrow +\infty} e^{-n} \sum_{k=0}^n \frac{n^k}{k!}

Find Residue at $z =0 $ of $$f(z) = \Gamma(z) \Gamma(z-1) x^{-z}$$ Try to find Residues for $ z=-n $

Part 1: Consider ##n## balls ##B_1##, ##B_2##, ..., ##B_n## having masses ##m_1##, ..., ##m_n##, such that ##m_1\ll m_2\ll ...\ll m_n##. The ##n## balls are stacked above each other. The bottom of ##B_1## is a height ##h## above the ground, and the bottom ##B_n## is a height ##\ell## above the...

This challenge was proposed by Boorglar. Many thanks to him! Let n be a natural number larger than 1, and a be a positive real number. Prove that if the sequence \{a\}, \{an\}, \{an^2\},... does not eventually become 0, then it will exceed 1/n infinitely many times. Here {x} means x -...

An open set in ##\mathbb{R}## is any set which can be written as the union of open intervals ##(a,b)## with ##a<b##. A subset of ##\mathbb{R}## is called a ##G_\delta## set if it is the countable intersection of open sets. Prove that if a set ##A\subseteq \mathbb{R}## is a ##G_\delta## set...

Assume that a cloud consists of tiny water droplets suspended (uniformly distributed, and at rest) in air, and consider a raindrop falling through them. What is the acceleration of the raindrop? Assume that the raindrop is initially of negligible size and that when it hits a water droplet, the...

This new challenge was suggested by jostpuur. It is rather number theoretic. Assume that q\in \mathbb{Q} is an arbitrary positive rational number. Does there exist a natural number L\in \mathbb{N} such that Lq=99…9900…00 with some amounts of nines and zeros? Prove or find a counterexample.

Written by micromass: The newest challenge was the following: This was solved by HS-Scientist. Here's his solution: This is a very beautiful construction. Here's yet another way of showing it. Definition: Let ##X## be a countable set. Let ##A,B\subseteq X##, we say that ##A## and...

Written by micromass: I have recently posted a challenge in my signature. The challenge read as follows: The first answer I got was from Millenial. He gave the following correct solution: This solution is very elegant. But there are other solutions. For example, we can prove the...

NEW CHALLENGE: This challenge was a suggestion by jgens. I am very thankful that he provided me with this neat problem. A 15-puzzle has the following form: The puzzle above is solved. The object of the game is to take an unsolved puzzle, such as and to make a combination of...

Let's put up a new challenge: This is called the Fano plane: This is a geometric figure consisting of 7 points and 7 lines. However, it is a so-called projective plane. This means that it satisfies the following axioms: 1) Through any two points, there is exactly one line 2) Any two...

This is a well-known result in complex analysis. But let's see what people come up with anyway: Challenge: Prove that there is no continuous function ##f:\mathbb{C}\rightarrow \mathbb{C}## such that ##(f(x))^2 = x## for each ##x\in \mathbb{C}##.

The newest challenge is the following: As an example, we can easily go from ##0## to ##-1/3##. Indeed, we can apply ##T## to ##0## to go to ##1##, we apply ##T## to go to ##2##, we apply ##T## to go to ##3##, and then we apply ##R## to go to ##-1/3##.

Evaluate $$\int_2^4 \frac{\sqrt{\ln (9-x)}\,dx}{\sqrt{\ln (9-x)}+\sqrt{\ln (x+3)}}$$

Consider the sequence of positive integers which satisfies $$a_n=a_{n-1}^2+a_{n-2}^2+a_{n-3}^2$$ for all $n \ge 3$. Prove that if $a_k=1997$, then $k \le 3$.

Suppose you have two poles separated by the distance $w$, the first of height $h_1$ and the second of $h_2$, where $0<h_1<h_2$. You wish to attach two wires to the ground in between the poles, one to the top of each pole, such that the angle subtended by the two wires is a maximum. What portion...

Find all pairs $(p, q)$ of integers such that $1+1996p+1998q=pq$.

Evaluate $$\tan\frac{\pi}{13}\tan\frac{2\pi}{13}\tan\frac{3 \pi}{13}\tan\frac{4\pi}{13}\tan\frac{5\pi}{13} \tan \frac{6\pi}{13}$$.

Solve $$2\sin^4 (x)(\sin((2x)-3)-2\sin^2 (x)(\sin((2x)-3)-1=0$$.

Find distinct positive integers $$a,\;b,$$ and $$c$$ such that $$a+b+c,\;ab+bc+ac,\;abc$$ forms an arithmetic progression.

Prove the following $$_2F_1 \left(a,1-a;c; \frac{1}{2} \right) = \frac{\Gamma \left(\frac{c}{2} \right)\Gamma \left(\frac{1+c}{2} \right) } {\Gamma \left(\frac{c+a}{2}\right)\Gamma \left(\frac{1+c-a}{2}\right)}.$$

Consider a pyramid whose base is an $n$-gon with side length $s$, and whose height is $h$. What is the radius of the largest sphere that will fit entirely within the pyramid?

Prove the following $$ {}_2 F_1 \left( a,b; c ; x \right) = \frac{\Gamma(c)}{\Gamma(b)\Gamma(c-b)}\int^1_0 t^{b-1}(1-t)^{c-b-1} (1-xt)^{-a} \, dt$$ Hypergeometric function .

d is deposited in a savings account for 3 years, then 2d for 3years, then 3d for 3 years, and so on similarly; here's an example for 9 years, 1st deposit = $100, rate = 10% annual: YEAR DEPOSIT INTEREST BALANCE 0 .00 1 100.00 .00...

Solve the following system in real numbers: $$a^2+b^2=2c$$ $$1+a^2=2ac$$ $$c^2=ab$$

Show that $$\frac{1}{44}>\left(\frac{1}{2}\right)\left(\frac{3}{4}\right)\left(\frac{5}{6}\right)\cdots\left( \frac{1997}{1998}\right)>\frac{1}{1999}$$

Solve $$\sqrt{x^2-1}+\sqrt{x-1}=x\sqrt{x}$$.

How many five digit numbers are divisible by 3 and contain the digit 6?

Find the constants $$a,\;b, \;c,\; d$$ such that $$4x^3-3x+\frac{\sqrt{3}}{2}=a(x-b)(x-c)(x-d)$$.

Find all $$x$$ in the interval $$[0, 2\pi] $$ which satisfies $$2\cos(x) \le \left|\sqrt{1+\sin (2x)}-\sqrt{1-\sin (2x)} \right|\le \sqrt{2}$$