Challenge Definition and 942 Threads

Let $x,\,y$ be positive integers such that $\dfrac{7}{10}<\dfrac{x}{y}<\dfrac{11}{15}$. Find the smallest possible value of $y$.

Prove that $(4\cos^2 9^{\circ}-3)(4\cos^2 27^{\circ}-3)=\tan 9^{\circ}$

Solve the equation $\sin a \cos b+ \sin b \cos c+ \sin c \cos a=\dfrac{3}{2}$

Show that $\dfrac{1}{2} \cdot \dfrac{3}{4} \cdot \dfrac{5}{6} \cdots \dfrac{1997}{1998} >\dfrac{1}{1999}$, where the use of induction method is not allowed.

Without the help of calculator, show that $\tan 50^{\circ}>1.18$

Evaluate: $$2^{2009}\frac{\displaystyle \int_0^1 x^{1004}(1-x)^{1004}\,dx}{\displaystyle \int_0^1x^{1004}(1-x^{2010})^{1004}\,dx}$$ ...of course without the use of beta or gamma functions. :p

Consider the sequences $(c_n)_n,\,(d_n)_n$ defined by $c_0=0$, $c_1=2$, $c_{n+1}=4c_n+c_{n-1}$, $n \ge 0$, $d_0=0$, $d_1=1$, $d_{n+1}=c_n-d_n+d_{n-1}$, $n \ge 0$. Prove that $(c_n)^3=d_{3n}$ for all $n$.

Let $k$ be an odd positive integer. Solve the equation $\cos kx=2^{k-1} \cos x$.

Let $p,\,q,\,r,\,s\,\in[0,\,\pi]$ and we are given that $2\cos p+6 \cos q+7 \cos r+9 \cos s=0$ and $2\sin p-6 \sin q+7 \sin r-9 \sin s=0$. Prove that $3 \cos (p+s)=7\cos(q+r)$.

Evaluate the following: $$\int_0^{\pi} e^{\cos x} \cos(\sin x)\,\,dx$$

Many of you don't know this, but as a young man, Greg once decided to worship the moon. He was so obsessed by the moon that he once decided to start following it. So at any given moment, he would check where the moon is and then walk in that direction. Greg has special powers so that he can see...

Let $f(x)=x^4+px^3+qx^2+rx+s$, where $p,\,q,\,r,\,s$ are real constants. Suppose $f(3)=2481$, $f(2)=1654$, $f(1)=827$. Determine the value of $\dfrac{f(-5)+f(9)}{4}$.

Show that the equation $8x^4-16x^3+16x^2-8x+p=0$ has at least one non-real root for every real number $p$ and find the sum of all the non-real roots of the equation.

Evaluate: $$\frac{1}{\cos^210^{\circ}}+\frac{1}{\sin^220^{ \circ }}+\frac{1}{\sin^240^{\circ}}-\frac{1}{\cos^245^{\circ}}$$

Evaluate: $$\Large \int_{\pi/2}^{5\pi/2} \frac{e^{\arctan(\sin x)}}{e^{\arctan(\sin x)}+e^{\arctan(\cos x)}}$$

Compute: $$\int_0^{\pi/2} \tan(x)\ln(\sin(x))\,dx$$

MIT Physics "Challenge Problems" (OCW) I had a question for anyone who is familiar with MIT and their OCW. I like to use their content to compliment my studies, but I feel some of their challenge problems are quite difficult. I am curious, are those challenge problems something they expect...

Homework Statement Show that if ##|\mathbb{Z}_p/(f)| = p^{deg(f(x))}## where the ##deg(f(x)) = n## and ##f(x) \in \mathbb{Z}_{p}## is irreducible over ##\mathbb{Z}_{p}[x]##, then ##\mathbb{Z}_{p}[x]/(f) \rightarrow_{\phi}^{\cong} \bigoplus_{i \in I} \mathbb{Z}_{p_{i}}## where ##|I| =...

Lucy stands on the edge of a vertical cliff and throws a stone vertically upwards. The stone leaves her hand with speed 15m/s at the instant her hand is 80m above the surface of the sea. Air resistance is negligible and the acceleration of free fall is 10m/s/s. First part is to calculate the...

Let $a$ and $b$ be positive integers. Show that $\dfrac{(a+b)!}{(a+b)^{a+b}}\le \dfrac{a! \cdot b!}{a^ab^b}$.

Show that $\dfrac{1}{2}\cdot\dfrac{3}{4}\cdot\dfrac{5}{6} \cdots\dfrac{999999}{1000000}<\dfrac{1}{1000}$

If $a,\,b,\,c$ are positive numbers, show that $8(a^3+b^3+c^3)\ge (a+b)^3+(a+c)^3+(b+c)^3$.

Homework Statement F=-mg-m\alpha vF=m\frac{dv}{dt} = \int_0^v \frac{dv'}{g+\alpha v'} = -\int_0^t dt' This equals: \ln \frac{1+\alpha v}{g} = -\alpha t I don't understand why it does; I keep getting an incorrect answer. Can anyone tell me what I'm doing wrong? Homework Equations The Attempt...

A function f:\mathbb{R} \to \mathbb{R} is called "smooth" if its k-th derivative exists for all k. A function is called analytic at a if its Taylor series \sum_{n\geq 0} \frac{f^{(n)}(a)}{n!} (x-a)^n converges and is equal to f(x) in a small neighborhood around a. The challenge...

Show that for $ \displaystyle 0 \le a < \frac{\pi}{2}$, $$ \int_{0}^{\infty} e^{-x \cos a} \cos(x \sin a) \cos (bx) \ dx = \frac{(b^{2}+1) \cos a}{b^{4}+2b^{2} \cos (2a) + 1 }$$When I post integral challenge problems in the future, I'll just number them.

Prove that \sum_{k=0}^{n} \sin\left( \frac{k \pi}{n} \right) = \cot \left( \frac{\pi}{2n} \right)

Prove that $\sqrt[3]{\dfrac{2}{1}}+\sqrt[3]{\dfrac{3}{2}}+\cdots+\sqrt[3]{\dfrac{996}{995}}-\dfrac{1989}{2}<\dfrac{1}{3}+\dfrac{1}{6}+\cdots+ \dfrac{1}{8961}$

Suppose we have 2 disjoint cycles π and σ. How can one calculate π^σ? I know how to calculate σ^2 or σ^3 but I can't figure out how to solve that.

A man has a single coin in front of him that lands on heads with probability p and tails with probability 1-p. He begins flipping coins with his single coin. If a coin lands on heads it is removed from the game and n new coins are placed on his stack of coins, if it lands on tails it is...

A man begins flipping coins according to the following rules: He starts with a single coin that he flips. Every time he flips a heads, he removes the coin from the game and then puts out two more coins to add to his stack of coins to flip, every time he flips a tails he simply removes the coin...

Consider the following sequence: a1 = p, where p is a prime number. an+1 = 2an+1 Prove there is no value of p for which every an is a prime number, or make me look dumb and construct a counterexample.

Simplify $\large \dfrac{2}{\sqrt{4-3\sqrt[4]{5}+2\sqrt{5}-\sqrt[4]{125}}}$.

For the sake of this challenge, you can assume that the world behaves nicely (everything is continuous, differentiable etc, the world is a sphere blah blah blah) but figurative bonus points if you assume fewer things. If you're looking for an easier challenge check out Challenge 10a. The...

For the sake of this challenge, you can assume that the world behaves nicely (everything is continuous, differentiable etc, the world is a sphere blah blah blah) but figurative bonus points if you assume fewer things. If you're looking for a bigger challenge check out Challenge 10b. The...

Suppose $f(-x)=f(x)$, then compute the following definite integral: \int_{-a}^{a}\frac{1}{1+2^{f(x)}}\,dx where $0<a\in\mathbb{R}$.

Given unequal integers $a, b, c$ prove that $(a-b)^5+(b-c)^5+(c-a)^5$ is divisible by $5(a-b)(b-c)(c-a)$.

For any triangle $ABC$, prove that $\cos \dfrac{A}{2} \cot \dfrac{A}{2}+\cos \dfrac{B}{2} \cot \dfrac{B}{2}+\cos \dfrac{C}{2} \cot \dfrac{C}{2} \ge \dfrac{\sqrt{3}}{2} \left( \cot \dfrac{A}{2}+\cot \dfrac{B}{2}+\cot \dfrac{C}{2} \right)$

I have to design a space rover for Mars that has to have quite a few operating gadgets on it. Without giving a lot of details I am using sketch up and needless to say it sucks. I can't even figure out how to make a wheel. and I can't make objects at angles, like shafts or motors or anything...

Countably infinite prisoners, numbered 1,2,3 etc. are told that tomorrow they will be given hats, colored either black or white, and lined up so that each prisoner can only see the prisoners whose hats are labeled with a greater number. The prisoners then get to guess which color hat they wear...

Homework Statement If a 0.505m long wire is excited into its lowest electrical resonance by a 2.2E7 Hz electrical oscillator, what is the ratio of the speed of the electrical current to that of light? Assume that the wire is like a tube with both ends closed. f=2.2E7 L=0.505 v=...

Let $x, y$ be real numbers such that $9x^2+8xy+7y^2 \le 6$. Show that $7x+12xy+5y \le 9$.

Using any of \{+,\;-,\;\times,\;\div,\;x^y,\;\sqrt{x},\;x!\} . . create 64 with two 4's.

Express $\cos^7 x+\cos^7 \left( x+\dfrac{2 \pi}{3} \right)+\cos^7 \left( x+\dfrac{4 \pi}{3} \right)$ in terms of $\cos 3x$.

Show that $$\int_0^{\frac{\pi}{2}}\frac{\log \tan \theta}{\sqrt{1+\cos^2 \theta}}d\theta = \frac{\log 2}{16 \Gamma \left(\frac{3}{4} \right)^2}\sqrt{2\pi^3}$$ This integral is harder than the http://mathhelpboards.com/challenge-questions-puzzles-28/integration-challenge-7720.html. :D

i wrote a script that codes phrases for me in c++ that should be impossible to solve if anybody can solve the code i'll respond with the key and explain how it works so here it is 05 08 13 52 19 55 89 44 33 77 10 82 88 84 23 65 46 52 57 95 37 93 18 11 29 40 69 34 97 87 21 52 17 33 86 20 37 96...

A random variable X is called sub-gaussian if P( |X| > t) \leq 2e^{-K t^2} for some constant K. Equivalently, the probability density p(x) is sub-gaussian if \int_{-t}^{t} p(x) dx \geq 1 - 2 e^{-Kt^2}. The challenge: Prove that the standard normal distribution (with mean 0 and...

This question is courtesy of mfb: An immortal snail is at one end of a perfect rubber band with a length of 1km. Every day, it travels 10cm in a random direction, forwards or backwards on the rubber band. Every night, the rubber band gets stretched uniformly by 1km. As an example, during the...

Similar to the previous two-part question, if you find part b to be an appropriate challenge please leave part a to those who are appropriately challenged by it. This question is courtesy of mfb An immortal snail is at one end of a perfect rubber band with a length of 1km. Every day, it...

Here is an interesting integral, which I would like to share with you: Show that $$ \begin{align*} \int_0^{\frac{\pi}{2}}\sin^{-1}\left( \frac{\sin x}{\phi}\right) dx&= \frac{\pi^2}{12}-\frac{3}{4}\log^2 \phi \end{align*} $$ where $\phi$ is the Golden Ratio.

A tester of basic quantum mechanics: 1) Let the state of a quantum particle be represented by \phi. Show that if \phi satisfies Schrodinger's equation, then its norm is constant. 2) Now consider a quantum particle with state \phi_{t} defined on [-a,a] subject to potential V=0. State the...