Given $x,y,z\in \mathbb{R} ^+$ , that are the solutions to the system of equations : $$x^2+xy+y^2=57$$$$y^2+yz+z^2=84$$$$z^2+zx+x^2=111$$What is the value of $xy+3yz+5zx$? (maphybich)

Find all the functions $f:\mathbb{R}\to\mathbb{R}$ which satisfy the functional equation $$f(xy^2)+f(x^2y)=y^2f(x)+x^2f(y)$$for every $x,y\in\mathbb{R}$ and $f(2008) =f(-2008)$ (nooonuii)

Set $ M= \{1,2,\cdots,2550\} $ and $\min A ,\ \max A $ represents the minimum and maximum values of the elements in the set $A$. For $ k \in \{1,2,\cdots 2006\} $define $$ x_k = \frac{1}{2008} \bigg (\sum_{A \subset M : n(A)= k} (\ min A + \max A) \, \bigg) $$. Find remainder from division $\sum_{i=1}^{2006} x_i^2$ with $2551$. (passer-by)

Let $a,b$ and $c$ be positive integers that $$\frac{a\sqrt{3}+b}{b\sqrt3+c}$$is a rational number, show that $$\frac{a^2+b^2+c^2}{a+b+ c}$$is an integer. (Anonymous314)

There are $6$ irrational numbers. Prove that there are always three of them, suppose $a,b,c$ such that $a+b$,$b+c$,$c+a$ are irrational numbers. (Erken)

Find the total number of integer solutions of the equation $$x^5-y^2=4$$ (Erken)

$ABC$ is a triangle with an area of $1$ square meter. Given the point $D$ on $BC$, point $E$ on $CA$, point $F$ on $AB$, such that quadrilateral $AFDE$ is cyclic. Prove that the area of $DEF \le \frac{EF^2}{4 AD^2}$. (holmes)

Let $a,b,c,d \in R^+$ with $abcd=1$. Prove that $$\left(\frac{1+ab}{1+a}\right)^{2008}+\left(\frac{1+bc}{1+b}\right)^{2008}+\left(\frac{1+cd }{1+c}\right)^{2008}+\left(\frac{1+da}{1+d}\right)^{2008} \geq 4$$(dektep)

Let $x,y,z$ be a positive real numbers. Prove that $$\frac {x}{\sqrt {x + y}} + \frac {y}{\sqrt {y + z}} + \frac { z}{\sqrt {z + x}}\geq\sqrt [4]{\frac {27(yz + zx + xy)}{4}}$$ (dektep)

Find all polynomials $P(x)$ which have the properties: 1) $P(x)$ is not a constant polynomial and is a mononic polynomial. 2) $P(x)$ has all real roots and no duplicate roots. 3) If $P(a)=0$ then $P(a|a|)=0$ (nooonui)

Let ${a_1,a_2,\dots,a_n}$ be positive real numbers, ${n>1}$. Denote by $g_n$ their geometric mean, and by $A_1,A_2,\dots,A_n$ the sequence of arithmetic means defined by \[ A_k=\frac{a_1+a_2+\cdots+a_k}{k},\qquad k=1,2,\dots,n. \] Let $G_n$ be the geometric mean of $A_1,A_2,\dots,A_n$. Prove the inequality \[ n \root n\of{\frac{G_n}{A_n}}+ \frac{g_n}{G_n}\le n+1 \] and establish the cases of equality. Proposed by Finbarr Holland, Ireland

The trapezoid $ABCD$ has sides $AB$ and $CD$ that are parallel $\hat{DAB} = 6^{\circ}$ and $\hat{ABC} = 42^{\circ}$. Point $X$ lies on the side $AB$ , such that $\hat{AXD} = 78^{\circ}$ and $\hat{CXB} = 66^{\circ}$. The distance between $AB$ and $CD$ is $1$ unit . Prove that $AD + DX - (BC + CX) = 8$ units. (Heir of Ramanujan)

Let $P_1(x)=\frac{1}{x}$ and $P_n(x)=P_{n-1}(x)+P_{n-1}(x-1)$ for every natural $ n$ greater than $1$. Find the value of $P_{2008}(2008)$. (Mathophile)

Find all functions $f:\mathbb{R} \to \mathbb{R}$ satisfying the equation \[ f(x^2+y^2+2f(xy)) = (f(x+y))^2. \] for all $x,y \in \mathbb{R}$.

For every positive integer $n$, $\sigma(n)$ is equal to the sum of all the positive divisors of $n$ (for example, $\sigma(6)=1+2+3+6=12$) . Find the solution of the equation $$\sigma(p^2)=\sigma(q^b)$$where $p$ and $q$ are primes where $p>q$ and $b$ are positive integers. (gools)

Once upon a time, there was a tribe called the Goblin Tribe, and their regular game was ''The ATM Game (Level Giveaway)'' . The game stats with a number of Goblin standing in a circle. Then the Chieftain assigns a Level to each Goblin, which can be the same or different (Level is a number which is a non-negative integer). Start play by selecting a Goblin with Level $k$ ($k \not=). 0$) comes up. Let's assume Goblin $A$. Goblin $A$ explodes itself. Goblin A's Level becomes $0$. After that, Level of Goblin $k$ next to Goblin $A$ clockwise gets Level $1$. Prove that: 1.) If after that Goblin $k$ next to Goblin $A$ explodes itself and keep doing this, $k'$ next to that Goblin clockwise explodes itself. Prove that the level of each Goblin will be the same again. 2) 2.) If after that we can choose any Goblin whose level is not $0$ to explode itself. And keep doing this. Prove that no matter what the initial level is, we can make each level the way we want. But there is a condition that the sum of all Goblin's levels must be equal to the beginning. (gools)

In a triangle $ABC$, the angle bisector at $A,B,C$ meet the opposite sides at $A_1,B_1,C_1$, respectively. Prove that if the quadrilateral $BA_1B_1C_1$ is cyclic, then $$\frac{AC}{AB+BC}=\frac{AB}{AC+CB}+\frac{BC}{BA+AC}.$$

In triangle $ABC$ ($AB\not= AC$), the incircle is tangent to the sides of $BC$ ,$CA$ , $AB$ at $D$ ,$E$, $F$ respectively. Let $AD$ meet the incircle again at point $P$, let $EF$ and the line passing through the point $P$ and perpendicular to $AD$ intersect at $Q$. Let $AQ$ intersect $DE$ at $X$ and $DF$ at $Y$. Prove that $AX=AY$. (tatari/nightmare)

Let $ABC$ be a triangle whose side lengths are opposite the angle $A,B,C$ are $a,b,c$ respectively. Prove that $$\frac{ab\sin{2C}+bc\sin{ 2A}+ca\sin{2B}}{ab+bc+ca}\leq\frac{\sqrt{3}}{2}$$. (nooonuii)

Let $p,q,r \in \mathbb{R}^+$ and for every $n \in \mathbb{N}$ where $pqr=1$ , denote $$ \frac{1}{p^n+q^n+1} + \frac{1}{q^n+r^n+1} + \frac{1}{r^n+p^n+ 1} \leq 1$$ (Art-Ninja)

Let $a,b,c$ be positive real numbers where $ab+bc+ca = 3$. Prove that $$\dfrac{1}{a^2+1}+\dfrac{1}{b^2+1}+\dfrac{1}{c^2+1}\geq\dfrac{3} {2}.$$(dektep)

For even positive integers $a>1$. Prove that there are infinite positive integers $n$ that makes $n | a^n+1$. (tomoyo-jung)

Let $n,d$ be natural numbers. Prove that there is an arithmetic sequence of positive integers. $$a_1,a_2,...,a_n$$with common difference of $d$ and $a_i$ with prime factor greater than or equal to $i$ for all values $i=1,2,...,n$. (nooonuii)

Prove that there are different points $A_0 \,\, ,A_1 \,\, , \cdots A_{2550}$ on the $XY$ plane corresponding to the following properties simultaneously. (i) Any three points are not on the same line. (ii) If $ d(A_i,A_j)$ represents the distance between $A_i\,\, , A_j $ then $$ \sum_{0 \leq i < j \leq 2550}\{d(A_i,A_j)\} < 10^{-2008}$$Note : $ \{x \}$ represents the decimal part of x e.g. $ \{ 3.16\} = 0.16$. (passer-by)

Set $P$ as a polynomial function by $p_n(x)=\sum_{k=0}^{n-1} x^k$. a) Prove that for $m,n\in N$, when dividing $p_n(x)$ by $p_m(x)$, the remainder is $$p_i(x),\forall i=0,1,...,m-1.$$b) Find all the positive integers $i,j,k$ that make $$p_i(x)+p_j(x^2)+p_k(x^4)=p_{100}(x).$$(square1zoa)

One test is a multiple choice test with $5$ questions, each with $4$ options, $2000$ candidates, each choosing only one answer for each item.Find the smallest possible integer $n$ that gives a student's answer sheet the following properties: In the student's answer sheet $n$, there are four sheets in it. Any two of the four tiles have exactly the same three answers. (tatari/nightmare)