Challenge Definition and 942 Threads

The CMS or cosmological background permates the whole universe. It's mentioned so often in study halls and on documentaries that NOTHING can peak further back into the universe evolution beyond the CMS. Just want to hear all of your ideas on how YOU would solve this problem if you were on this...

$n\in N,n\geq 2$ prove: $ \sum_{1}^{n}(\dfrac{1}{2n-1}-\dfrac{1}{2n})>\dfrac {2n}{3n+1}$

Factorize $\cos^2 x+\cos^2 2x+\cos^2 3x+\cos 2x+\cos 4x+\cos 6x$.

Find the number of distinct real solutions of the equation $(x − 1)(x − 3)(x − 5) · · · (x − 2017) = (x − 2)(x − 4)(x − 6) · · · (x − 2016)$.

$x$ and $y$ are two real numbers such that $x^3-y^3=2$ and $x^5-y^5\ge 4$. Prove that $x^2+y^2\gt 2$.

Let $a\,b$ and $c$ be the sides of a triangle. Prove that \frac{a}{b+c-a}+\frac{b}{a+c-b}+\frac{c}{a+b-c}\ge 3.

Prove \frac{ab}{a+b+ab}+\frac{bc}{b+c+bc}+\frac{ca}{c+a+ca}\le \frac{a^2+b^2+c^2+6}{9} for all positive real $a,\,b$ and $c$ and $a,\,b,\,c\ne 0$.

Let $1\lt a \lt 2$, $a$ is a root of the equation $x^5-x-2=0$. Prove that $\large a>\sqrt[9]{8}$.

Let $a,\,b,\,c$ and $d$ be positive real numbers such that $(a^2+b^2)^3=c^2+d^2$, prove that $\dfrac{a^3}{c}+\dfrac{b^3}{d}\ge 1$.

Let $a,\,b,\,c$ be real numbers such that $a\ge b\ge c>0$. Prove that \frac{a^2-b^2}{c}+\frac{c^2-b^2}{a}+\frac{a^2-c^2}{b}\ge 3a-4b+c.

A Youtube channel has just uploaded a video proposing a challenge. The question is pretty simple. It can be summed up to: A clock is moving towards you at 50% the speed of light, and eventually it passes you and continues its travel. When the clock is moving towards you, will it appear to tick...

Given the system of equation below: $a = zb + yc$ $b = xc + za$ $c = you + xb$ Prove that $\dfrac{a^2}{1-x^2}=\dfrac{b^2}{1-y^2}=\dfrac{c^2}{1-z^2}$.

If $a$ and $b$ are two positive real, and that $a^3+b^3=a-b$, prove that $2\left(\sqrt{2}-1\right)a^2-b^2\lt 1$.

Prove that $\sin 1+\sin 2+\sin 3+\cdots+\sin n<2$.

Prove that $\sqrt{8}^{\sqrt{7}}<\sqrt{7}^{\sqrt{8}}$.

Let I=\int_{2013}^{2014} \frac{\sin x}{x}\,dx. Determine with reason if $I<0,\,I=0$ or $I>0$?

Let $A,\,B$ and $C$ be three angles of a triangle $ABC$. Prove that $(1-\cos A)(1-\cos B)(1-\cos C)\ge \cos A\cos B \cos C$

Prove $\sqrt[3]{43}<\sqrt[3]{9}+\sqrt[3]{3}<\sqrt[3]{44}$

Prove \sum_{k=1}^{n}\left\lfloor{\left(\frac{k}{2}\right)^2}\right\rfloor=\left\lfloor{\dfrac{n(n+2)(2n-1)}{24}}\right\rfloor.

Hello fellow physicists, I teach first and second year physics at a community college. In particular, I teach two algebra based- and three calculus-based physics courses. I wanted to know if any of you have used any type of challenge exams to place out of the first year physics course. I am also...

Show that $\sqrt{99}-\sqrt{98}+\sqrt{97}-\sqrt{96}+\cdots-\sqrt{4}+\sqrt{3}-\sqrt{2}+\sqrt{1}> 5$.

Given that $S=1-\dfrac{1}{2}+\dfrac{1}{3}-\dfrac{1}{4} + ...+\dfrac{1}{99}-\dfrac{1}{100}$. Prove that $\dfrac{1}{2}<S<1$.

Marci is working a coin toss game at the fair. You get 4 coins to toss to try to get numbers that add up to 10. You can get a 0,1,2,3,4 or 5. How many different combinations of 10 are there? (Hint: 5+5+0+0 is different from 5+0+5+0)

$m,n,k\in N$, and $m>1,n>1$ prove : $(3^{m+1}-1)\times (5^{n+1}-1)\times(7^{k+1}-1)>98\times 3^m\times 5^n\times7^k$

Prove for all positive real $a,\,b,\,c,\,x,\,y,\,z$ that $\dfrac{a^3}{x}+\dfrac{b^3}{y}+\dfrac{c^3}{z}\ge \dfrac{(a+b+c)^3}{3(x+y+z)}$.

Let $a,\,b$ and $c$ be three distinct real numbers. Prove that $\sqrt[3]{a-b}+\sqrt[3]{b-c}+\sqrt[3]{c-a}\ne 0$.

Prove $\dfrac{1}{3}\cdot\dfrac{4}{6}\cdot\dfrac{7}{9}\cdots\dfrac{1000}{1002}\lt \dfrac{1}{79}$

Let $a\in \Bbb{Z^+}$, prove that $\dfrac{2}{2-\sqrt{2}}>\dfrac{1}{1\sqrt{1}}+\dfrac{1}{2\sqrt{2}}+\dfrac{1}{3\sqrt{3}}+\cdots+\dfrac{1}{a\sqrt{a}}$.

Let $\dfrac{\cos^4 a}{x}+\dfrac{\sin^4 a}{y}=\dfrac{1}{x+y}$ for all real $a,\,b,\,x,\,y$. Prove that $\dfrac{\cos^8 a}{x^3}+\dfrac{\sin^8 a}{y^3}=\dfrac{1}{(x+y)^3}$

For reals $x,\,y,\,z$ and $a,\,b$ and $c$ that satisfy $a + b + c = ax + by + cz = x^2a + y^2b + z^2c = 1$, prove that $x^3a + y^3b + cz^3c = 1 − (1 − x)(1 − y)(1 − z)$

I found this question in my study guide triangle ABC is similar to triangle DEF. Triangle ABC has sides 4,6,8. Wich could be the corresponding sides of a triangle DEF? Indicate all that apply A) 1, 1.5, 2 B)1.5, 2.25, 3 C)6, 9, 12 D) 8, 12, 16 E)10, 15, 20 What I did was add the...

So I was checking the How to self-study math thread and saw that someone suggested that It would be helpfull to create this kind of thread. And because we are writting a test on thursday on Probability I though it would be nice to find out which parts I still need to double-check. So these...

Ok, I got an experimental idea for a game. I'm going to post a picture from a movie and the first person that guesses it can then post a new picture. Pretty inventive, huh? Oh, and the rules are you can't upload the picture to that google image recognition thing (of course). Be honest. Name...

In triangle ABC, $\angle ACB=\angle ABC=80^\circ$ and $P$ is on the line segment $AB$ such that $BC=AP$. Find $\angle BPC$.

Hey! :o A cycloid is a flat curve that is traced by point of the rim of a circle while the circle rolls without slippage on the line. Show that if the line is the axis $x$ and the circle has radius $a>0$, then the cycloid can be parametrized by $$\gamma (t)=a(t-\sin t, 1-\cos t)$$ Could you...

Let $s,\,t,\,u,\,v$ be real numbers such that $s+t+u+v=0$. Prove that $(s^3+t^3+u^3+v^3)^2=9(st-uv)(tu-sv)(us-tv)$.

Homework Statement A solid cube of side ##l = r*pi/2## and of uniform density is placed on the highest point of a cylinder of radius ##r## as shown in the attached figure. If the cylinder is sufficiently rough that no sliding occurs, calculate the full range of the angle through which the block...

Hello I'm relatively new to teaching physics and was wondering if you anyone had advices as to the parameters to put on this challenge. How high should I drop their containers from? 2nd or 3rd story? When I was in school, I think we could only use glue and toothpicks, but I don't remember...

Solve the equation $\left\lfloor{\dfrac{2a + 1}{a^2+ 1}}\right\rfloor\left\{\dfrac{a^2+ 2a +2}{a^2+ 1}\right\}=\dfrac{2a -a^2}{a^2 + 1}$, where $\{a\}$ denotes the fractional part of $a$.

Prove $\tan 6^{\circ}+\tan 24^{\circ}+\tan 42^{\circ}+\tan 86^{\circ}\gt 4$.

Prove $\sqrt{2}+\sqrt{3}\gt \pi$.

Prove $\dfrac{(x+y)^2}{2}+\dfrac{x+y}{4}\ge x\sqrt{y}+y\sqrt{x}$.

Prove $35\sqrt{55}+55\sqrt{77}+77\sqrt{35}+35\sqrt{77}+55\sqrt{35}+77\sqrt{55}\gt 2310$.

$a,b,c \in N$,prove : $\dfrac{a^3}{a^2+ab+b^2}+\dfrac{b^3}{b^2+bc+c^2}+\dfrac{c^3}{c^2+ca+a^2}\geq\dfrac{a+b+c}{3}$

Suppose $k>0$. Show that $\dfrac{2015}{(k+1)(k+4030)}<\dfrac{1}{k+1}-\dfrac{1}{k+2}+\dfrac{1}{k+3}-\dfrac{1}{k+4}+\cdots+\dfrac{1}{k+4029}-\dfrac{1}{k+4030}$.

I must be in the 'optional for experts' section. I can get a series for coth using mathmatica, but would like to do it 'manually'. The power series for cosh & sinh leaves a division I don't want to attempt. I tried $ \frac{cosh}{sinh} = \frac{{e}^{2x+1}+1}{{e}^{2x-1}+1} $ and using...

Prove \cos20^\circ\cdot\cos40^\circ\cdot\cos80^\circ=\frac18

Find the minimum of $\dfrac{1}{a-b}+\dfrac{1}{b-c}+\dfrac{1}{a-c}$ for real $a>b>c$ given $(a-b)(b-c)(a-c)=17$.

Real numbers $u,\,v,\,x,\,y$ satisfy the following conditions: $|u|>1$, $|v|>1$, $|x|>1$, $|y|>1$, and $u+v+x+y+uv(x+y)+xy(u+v)=0$ Prove that $\dfrac{1}{u-1}+\dfrac{1}{v-1}+\dfrac{1}{x-1}+\dfrac{1}{y-1}>0$.

It occurred to me that some of the PF community might enjoy learning the game of Microshogi ( otherwise known as Five-minute Poppy Shogi, invented by Yasuharu Ōyama in 1981 ), so I composed this Microshogi tsume problem as a challenge. Does anyone play Microshogi? Then solve the tsume. Have fun...