Find all functions $ f: \mathbb{R} \rightarrow \mathbb{R}$ such that for all real numbers $ x,y$: $ x f(x)\minus{}yf(y)\equal{}(x\minus{}y)f(x\plus{}y)$.

We have $n$ points in the plane, no three on a line. We call $k$ of them good if they form a convex polygon and there is no other point in the convex polygon. Suppose that for a fixed $k$ the number of $k$ good points is $c_k$. Show that the following sum is independent of the structure of points and only depends on $n$ : \[ \sum_{i=3}^n (-1)^i c_i \]

Define $P(x)=((x-a_1)(x-a_2)...(x-a_n))^2 +1$, where $a_1,a_2...,a_n\in\mathbb{Z}$ and $n\in\mathbb{N^+}$. Prove that $P(x)$ couldn't be expressed as product of two non-constant polynomials with integer coefficients.

Which positive integers $n$ make the equation \[\sum_{i=1}^n \sum_{j=1}^n \left\lfloor \frac{ij}{n+1} \right\rfloor=\frac{n^2(n-1)}{4}\] true?

Let $m_1<m_2<\cdots<m_s$ be a sequence of $s\ge2$ positive integers, none of which can be written as the sum of (two or more) distinct other numbers in the sequence. For every integer $r$ with $1\le r<s$, prove that \[ r\cdot m_r+m_s ~\ge~ (r+1)(s-1). \] (Proposed by Gerhard Woeginger, Austria)

Let the sequence $(a_{n})_{n\geqslant 1}$ be defined as: $$a_{n}=\sqrt{A_{n+2}^{1}\sqrt[3]{A_{n+3}^{2}\sqrt[4]{A_{n+4}^{3}\sqrt[5]{A_{n+5}^{4}}}}},$$ where $A_{m}^{k}$ are defined by $$A_{m}^{k}=\binom{m}{k}\cdot k!.$$ Prove that $$a_{n}<\frac{119}{120}\cdot n+\frac{7}{3}.$$

Solve the system of equations $x+y+z=\pi$ $\tan x\tan z=2$ $\tan y\tan z=18$

Christian Reiher and Reid Barton want to open a security box, they already managed to discover the algorithm to generate the key codes and they obtained the following information: $i)$ In the screen of the box will appear a sequence of $n+1$ numbers, $C_0 = (a_{0,1},a_{0,2},...,a_{0,n+1})$ $ii)$ If the code $K = (k_1,k_2,...,k_n)$ opens the security box then the following must happen: a) A sequence $C_i = (a_{i,1},a_{i,2},...,a_{i,n+1})$ will be asigned to each $k_i$ defined as follows: $a_{i,1} = 1$ and $a_{i,j} = a_{i-1,j}-k_ia_{i,j-1}$, for $i,j \ge 1$ b) The sequence $(C_n)$ asigned to $k_n$ satisfies that $S_n = \sum_{i=1}^{n+1}|a_i|$ has its least possible value, considering all possible sequences $K$. The sequence $C_0$ that appears in the screen is the following: $a_{0,1} = 1$ and $a_0,i$ is the sum of the products of the elements of each of the subsets with $i-1$ elements of the set $A =$ {$1,2,3,...,n$}, $i\ge 2$, such that $a_{0, n+1} = n!$ Find a sequence $K = (k_1,k_2,...,k_n)$ that satisfies the conditions of the problem and show that there exists at least $n!$ of them.

An integer $m\ge 3$ and an infinite sequence of positive integers $(a_n)_{n\ge 1}$ satisfies the equation \[a_{n+2} = 2\sqrt[m]{a_{n+1}^{m-1} + a_n^{m-1}} - a_{n+1}. \] for all $n\ge 1$. Prove that $a_1 < 2^m$.

Product of square trinomials $ x ^ 2 + a_1x + b_1 $, $ x ^ 2 + a_2x + b_2 $, $ \dots $, $ x ^ 2 + a_n x + b_n $ equals polynomial $ P (x) = x ^ {2n} + c_1x ^ {2n-1} + c_2x ^ {2n-2} + \dots + c_ {2n-1} x + c_ {2n} $, where the coefficients $ c_1 $, $ c_2 $, $ \dots $, $ c_ {2n} $ are positive. Prove that for some $ k $ ($ 1 \leq k \leq n $) the coefficients $ a_k $ and $ b_k $ are positive.

Let $f(t)=t^3+t$. Decide if there exist rational numbers $x, y$ and positive integers $m, n$ such that $xy=3$ and: \begin{align*} \underbrace{f(f(\ldots f(f}_{m \ times}(x))\ldots)) = \underbrace{f(f(\ldots f(f}_{n \ times}(y))\ldots)). \end{align*}

Find all second degree polynomial $d(x)=x^{2}+ax+b$ with integer coefficients, so that there exists an integer coefficient polynomial $p(x)$ and a non-zero integer coefficient polynomial $q(x)$ that satisfy: \[\left( p(x) \right)^{2}-d(x) \left( q(x) \right)^{2}=1, \quad \forall x \in \mathbb R.\]

Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that \[f(x + y) + y \le f(f(f(x)))\] holds for all $x, y \in \mathbb{R}$.

For a real number $x$, the minimum value of the expression $$\frac{2x^2 + x - 3}{x^2 - 2x + 3}$$ can be written in the form $\frac{a-\sqrt{b}}{c}$, where $a, b$, and $c$ are positive integers such that $a$ and $c$ are relatively prime. Find $a + b + c$

Determine all triples of real numbers that satisfy the following system of equations: \[x+y-z=-1\\ x^2-y^2+z^2=1\\ -x^3+y^3+z^3=-1\]

Find all $a\in\mathbb{R}$ for which there exists a non-constant function $f: (0,1]\rightarrow\mathbb{R}$ such that \[a+f(x+y-xy)+f(x)f(y)\leq f(x)+f(y)\] for all $x,y\in(0,1].$

Let $q$ be the sum of the expressions $a_1^{-a_2^{a_3^{a_4}}}$ over all permutations $(a_1, a_2, a_3, a_4)$ of $(1,2,3,4).$ Determine $\lfloor q \rfloor.$

Let $n$ a positive integer and let $f\colon [0,1] \to \mathbb{R}$ an increasing function. Find the value of : \[ \max_{0\leq x_1\leq\cdots\leq x_n\leq 1}\sum_{k=1}^{n}f\left ( \left | x_k-\frac{2k-1}{2n} \right | \right )\]

For a polynomial $P$ with integer coefficients, $P(5)$ is divisible by $2$ and $P(2)$ is divisible by $5$. Prove that $P(7)$ is divisible by $10$.

It is given the equation $x^4-2ax^3+a(a+1)x^2-2ax+a^2=0$. a) Find the greatest value of $a$, such that this equation has at least one real root. b) Find all the values of $a$, such that the equation has at least one real root.

$f : [-\sqrt{1995},\sqrt{1995}] \to\mathbb{R}$ is defined by $f(x) = x(1993+\sqrt{1995-x^{2}})$. Find its maximum and minimum values.

Let $n$ be a positive integer that is not a perfect cube. Define real numbers $a,b,c$ by \[a=\root3\of n\kern1.5pt,\qquad b={1\over a-[a]}\kern1pt,\qquad c={1\over b-[b]}\kern1.5pt,\] where $[x]$ denotes the integer part of $x$. Prove that there are infinitely many such integers $n$ with the property that there exist integers $r,s,t$, not all zero, such that $ra+sb+tc=0$.

Put $x = \log_{10} 2$, $y = \log_{10} 3$. Then $15 < 16$ implies $1 - x + y < 4x$, so $1 + y < 5x$. Derive similar inequalities from $80 < 81$ and $243 < 250$. Hence show that \[ 0.47 < \log_{10} 3 < 0.482. \]

For any $n\in\mathbb N$, denote by $a_n$ the sum $2+22+222+\cdots+22\ldots2$, where the last summand consists of $n$ digits of $2$. Determine the greatest $n$ for which $a_n$ contains exactly $222$ digits of $2$.

(a) A polynomial $P(x)$ with integer coefficients takes the value $-2$ for at least seven distinct integers $x$. Prove that it cannot take the value $1996$. (b) Prove that there are irrational numbers $x,y$ such that $x^y$ is rational.