Let $x_i,y_i$ be positive real numbers, $i = 1,2,...,n$. Prove that $$\left( \sum_{i=1}^n (x_i +y_i)^2\right)\left( \sum_{i=1}^n\frac{1}{x_iy_i}\right)\ge 4n^2$$

Let $k\ge2$ be an integer. Find the smallest integer $n \ge k+1$ with the property that there exists a set of $n$ distinct real numbers such that each of its elements can be written as a sum of $k$ other distinct elements of the set.

You are given $n$ different primes $p_1, p_2,..., p_n$. Consider the polynomial $$x^n + a_1x^{n -1} + a_2x^{n - 2} + ...+ a_{n - 1}x + a_n$$, where $a_i$ is the product of the first $i$ given prime numbers. For what $n$ can it have an integer root?

Let $f\in\mathbb{Z}[X]$ be an irreducible polynomial over the ring of integer polynomials, such that $|f(0)|$ is not a perfect square. Prove that if the leading coefficient of $f$ is 1 (the coefficient of the term having the highest degree in $f$) then $f(X^2)$ is also irreducible in the ring of integer polynomials. [i]Mihai Piticari[/i]

Solve the equation: $[\sqrt x+\sqrt{x+1}]+[\sqrt {4x+2}]=18$

Let $n$ be a positive integer. Show that there exist positive integers $a$ and $b$ such that \[ \frac{a^2 + a + 1}{b^2 + b + 1} = n^2 + n + 1. \]

Mr. Fat and Ms. Taf play a game. Mr. Fat chooses a sequence of positive integers $ k_1, k_2, \ldots , k_n$. Ms. Taf must guess this sequence of integers. She is allowed to give Mr. Fat a red card and a blue card, each with an integer written on it. Mr. Fat replaces the number on the red card with $ k_1$ times the number on the red card plus the number on the blue card, and replaces the number on the blue card with the number originally on the red card. He repeats this process with number $ k_2$. (That is, he replaces the number on the red card with $ k_2$ times the number now on the red card plus the number now on the blue card, and replaces the number on the blue card with the number that was just placed on the red card.) He then repeats this process with each of the numbers $ k_3, \ldots k_n$, in this order. After has has gone through the sequence of integers, Mr. Fat then gives the cards back to Ms. Taf. How many times must Ms. Taf submit the red and blue cards in order to be able to determine the sequence of integers $ k_1, k_2, \ldots k_n$?

If $a,b,c$ are positive real numbers, prove that $\frac{a^2b^2}{a+b}+\frac{b^2c^2}{b+c}+\frac{c^2a^2}{c+a}\le \frac{a^3+b^3+c^3}{2}$

Find the least possible cardinality of a set $A$ of natural numbers, the smallest and greatest of which are $1$ and $100$, and having the property that every element of $A$ except for $1$ equals the sum of two elements of $A$.

Find all functions $f: R \to R$ such that $f(x)f(yf(x)-1)=x^2f(y)-f(x)$ for all real $x ,y$

A polynomial $p$ with integer coefficients is such that $p(-n) < p(n) < n$ for some integer $n$. Prove that $p(-n) < -n$.

Find all functions $f:\mathbb{Z}_{>0}\rightarrow\mathbb{Z}_{>0}$ with the following properties: 1) For every natural number $n\geq 3$, $\gcd(f(n),n)\neq 1$. 2) For every natural number $n\geq 3$, there exists $i_n\in\mathbb{Z}_{>0}$, $1\leq i_n\leq n-1$, such that $f(n)=f(i_n)+f(n-i_n)$. [i]Proposed by Pavel Ciurea[/i]

Let $f : \mathbb R \to \mathbb R$ be a real-valued function defined on the set of real numbers that satisfies \[f(x + y) \leq yf(x) + f(f(x))\] for all real numbers $x$ and $y$. Prove that $f(x) = 0$ for all $x \leq 0$. [i]Proposed by Igor Voronovich, Belarus[/i]

Find all real numbers $x$ that can be written as $$x= \frac{a_0}{a_1a_2..a_n}+\frac{a_1}{a_2a_3..a_n}+\frac{a_2}{a_3a_4..a_n}+...+\frac{a_{n-2}}{a_{n-1}a_n}+\frac{a_{n-1}}{a_n}$$ where $n, a_1,a_2,...,a_n$ are positive integers and $1 = a_0 \le a_1 <... < a_n$

Find all functions $f:\mathbb Z\rightarrow \mathbb Z$ such that, for all integers $a,b,c$ that satisfy $a+b+c=0$, the following equality holds: \[f(a)^2+f(b)^2+f(c)^2=2f(a)f(b)+2f(b)f(c)+2f(c)f(a).\] (Here $\mathbb{Z}$ denotes the set of integers.) [i]Proposed by Liam Baker, South Africa[/i]

Let be two natural numbers $ b>a>0 $ and a function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ having the following property. $$ f\left( x^2+ay\right)\ge f\left( x^2+by\right) ,\quad\forall x,y\in\mathbb{R} $$ [b]a)[/b] Show that $ f(s)\le f(0)\le f(t) , $ for any real numbers $ s<0<t. $ [b]b)[/b] Prove that $ f $ is constant on the interval $ (0,\infty ) . $ [b]c)[/b] Give an example of a non-monotone such function.

Does there exist any continuous function $ f$ such that $ f(f(x))=-x^{2019}\ \forall\ x \in \mathbb{R}$

Find all functions $f: \mathbb{R}\rightarrow \mathbb{R}$ such that \[ f(xy-1) + f(x)f(y) = 2xy-1 \] for all x and y

If $x$ and $y$ are real numbers, compute the minimum possible value of \[\frac{4xy(3x^2+10xy+6y^2)}{x^4+4y^4}.\]

Polynomial $P(x)$ with integer coefficients is given. For some positive integer $n$ numbers $P(0),P(1),\dots,P(2^n+1)$ are all divisible by $2^{2^n}$. Prove that values of $P(x)$ in all integer points are divisible by $2^{2^n}$.

Let $k$ be a positive integer. Find all collection of integers $(a_1, a_2,\cdots, a_k)$ such that there exist a non-linear polynomial $P$ with integer coefficients, so that for all positive integers $n$ there exist a positive integer $m$ satisfying: $$P(n+a_1)+P(n+a_2)+...+P(n+a_k)=P(m)$$ [i]Proposed by Ivan Chan Kai Chin[/i]

Solve the system of equations $$ \begin{array}{l}<br /> x^2y^2 + x^2z^2 = axyz\\<br /> y^2z^2 + y^2x^2 = bxyz\\<br /> z^2x^2 + z^2y^2 = cxyz.<br /> \end{array}$$

For positive reals $a,b,c$ with $\sqrt{a}+\sqrt{b}+\sqrt{c}\ge 3$ prove that $$\frac{a^3}{a^2+b}+\frac{b^3}{b^2+c}+\frac{c^3}{c^2+a}\ge \frac{3}{2}$$

Let $f(x)$ be a monic cubic polynomial with $f(0)=-64$, and all roots of $f(x)$ are non-negative real numbers. What is the largest possible value of $f(-1)$? (A polynomial is monic if its leading coefficient is 1.)

Let $ f$ be a function from the set of real numbers $ \mathbb{R}$ into itself such for all $ x \in \mathbb{R},$ we have $ |f(x)| \leq 1$ and \[ f \left( x \plus{} \frac{13}{42} \right) \plus{} f(x) \equal{} f \left( x \plus{} \frac{1}{6} \right) \plus{} f \left( x \plus{} \frac{1}{7} \right).\] Prove that $ f$ is a periodic function (that is, there exists a non-zero real number $ c$ such $ f(x\plus{}c) \equal{} f(x)$ for all $ x \in \mathbb{R}$).