Find all functions $f\colon R \to R$ such that \[f\left(x^{2}+yf(x)\right) = f(x)^{2}+xf(y)\] for all reals $x,y$.

Find all real numbers $x$ which satisfy $\frac{n}{3n+1}\leq x\leq \frac{4n+1}{2n-1}$, for all $n\in\mathbb{N}^*$. [i]Gheorghe Boroica[/i]

Let $a,b,c$ be positive real numbers such that $a + b + c = 1$. Show that $$\left(\frac{1}{a} + \frac{1}{bc}\right)\left(\frac{1}{b} + \frac{1}{ca}\right)\left(\frac{1}{c} + \frac{1}{ab}\right) \geq 1728$$

Let $n \geq 1$ be an integer. What is the maximum number of disjoint pairs of elements of the set $\{ 1,2,\ldots , n \}$ such that the sums of the different pairs are different integers not exceeding $n$?

Let $ABCD$ be a parallelogram. A variable line $g$ through the vertex $A$ intersects the rays $BC$ and $DC$ at the points $X$ and $Y$, respectively. Let $K$ and $L$ be the $A$-excenters of the triangles $ABX$ and $ADY$. Show that the angle $\measuredangle KCL$ is independent of the line $g$. [i]Proposed by Vyacheslev Yasinskiy, Ukraine[/i]

The sequence $(a_{n})$ is defined by $a_1=1$ and $a_n=n(a_1+a_2+\cdots+a_{n-1})$ , $\forall n>1$. [b](a)[/b] Prove that for every even $n$, $a_{n}$ is divisible by $n!$. [b](b)[/b] Find all odd numbers $n$ for the which $a_{n}$ is divisible by $n!$.

Three lines are drawn in a plane. Which of the following could NOT be the total number of points of intersections? (A) $0$ (B) $1$ (C) $2$ (D) $3$ (E) They all could.

Sides $\overline{AB}$ and $\overline{AC}$ of equilateral triangle $ABC$ are tangent to a circle at points $B$ and $C$ respectively. What fraction of the area of $\triangle ABC$ lies outside the circle? $ \textbf{(A) }\dfrac{4\sqrt{3}\pi}{27}-\frac{1}{3}\qquad \textbf{(B) } \frac{\sqrt{3}}{2}-\frac{\pi}{8}\qquad \textbf{(C) } \frac{1}{2} \qquad \textbf{(D) }\sqrt{3}-\frac{2\sqrt{3}\pi}{9}\qquad \textbf{(E) } \frac{4}{3}-\dfrac{4\sqrt{3}\pi}{27}$

Determine all functions $f:\mathbb{Z}\rightarrow\mathbb{Z}$ with the property that \[f(x-f(y))=f(f(x))-f(y)-1\] holds for all $x,y\in\mathbb{Z}$.

Let $x$ be an integer of $n$ digits, all equal to $ 1$. Show that if $x$ is prime, then $n$ is also prime.

For positive numbers $ a $, $ b $ and $ c $ satisfying the equality $ a + b + c = 1 $, prove the inequality $$ \frac {a ^ 7 + b ^ 7} {a ^ 5 + b ^ 5} + \frac {b ^ 7 + c ^ 7} {b ^ 5 + c ^ 5} + \frac {c ^ 7 + a ^ 7} {c ^ 5 + a ^ 5} \geq \frac {1} {3}. $$

$P(x,y)$ is polynomial with real coefficients and $P(x+2y,x+y)=P(x,y)$. Prove that exists polynomial $Q(t)$ such that $P(x,y)=Q((x^2-2y^2)^2)$ [i]A. Golovanov[/i]

What is the last two digits of base-$3$ representation of $2005^{2003^{2004}+3}$? $ \textbf{(A)}\ 21 \qquad\textbf{(B)}\ 01 \qquad\textbf{(C)}\ 11 \qquad\textbf{(D)}\ 02 \qquad\textbf{(E)}\ 22 $

Let $Z_1,Z_2,\cdots ,Z_n$ be complex numbers satisfying $|Z_1|+|Z_2|+\cdots +|Z_n|=1$. Show that there exist some among the $n$ complex numbers such that the modulus of the sum of these complex numbers is not less than $1/6$.

Let $a, b, c$ be positive real numbers such that $abc = 1$. Prove that $$\sqrt{\frac{a+b}{b+1}}+\sqrt{\frac{b+c}{c+1}}+\sqrt{\frac{c+a}{a+1}} \ge 3$$

What is the remainder when $7^{8^9}$ is divided by $1000?$

In triangle $ABC$, $\angle C = 30^{\circ}$, $O$ and $I$ are the circumcenter and incenter respectively, Points $D \in AC$ and $E \in BC$, such that $AD = BE = AB$. Prove that $OI = DE$ and $OI \bot DE$.

A semicircle with diameter $AB = 2r$ is divided into two sectors by an arbitrary radius. To each of the sectors a circle is inscribed. These two circles touch A$B$ at $S$ and $T$. Show that $ST \ge 2r(\sqrt{2}-1)$.

The integers from $1$ to $n$ are written, one on each of $n$ cards. The first player removes one card. Then the second player removes two cards with consecutive integers. After that the first player removes three cards with consecutive integers. Finally, the second player removes four cards with consecutive integers. What is th smallest value of $n$ for which the second player can ensure that he competes both his moves?

Let $F=\{1,2,...,100\}$ and let $G$ be any $10$-element subset of $F$. Prove that there exist two disjoint nonempty subsets $S$ and $T$ of $G$ with the same sum of elements.

Consider an acute triangle $ABC$. Let $D$, $E$ and $F$ be the feet of the altitudes through vertices $A$, $B$ and $C$. Denote by $A'$, $B'$, $C'$ the projections of $A$, $B$, $C$ onto lines $EF$, $FD$, $DE$, respectively. Further, let $H_D$, $H_E$, $H_F$ be the orthocenters of triangles $DB'C'$, $EC'A'$, $FA'B'$. Show that $$H_DB^2 + H_EC^2 + H_FA^2 = H_DC^2 + H_EA^2 + H_FB^2.$$

Let $F(x_1,..., x_n)$ be a polynomial with real coefficients in $ n > 1$ “indeterminate” variables $x_1,..., x_n$. We say that $F$ is $n$-[i]alternating [/i]if for all integers $1 \le i < j \le n$, $$F(x_1,..., x_i,..., x_j,..., x_n) = - F(x_1,..., x_j,..., x_i,..., x_n),$$ i.e. swapping the order of indeterminates $x_i, x_j$ flips the sign of the polynomial. For example, $x^2_1x_2 - x^2_2x_1$ is $2$-alternating, whereas $x_1x_2x_3 +2x_2x_3$ is not $3$-alternating. [i]Note: two polynomials $P(x_1,..., x_n)$ and $Q(x_1,..., x_n)$ are considered equal if and only if each monomial constituent $ax^{k_1}_1... x^{k_n}_n$ of $P$ appears in $Q$ with the same coefficient $a$, and vice versa. This is equivalent to saying that $P(x_1,..., x_n) = 0$ if and only if every possible monomial constituent of $P$ has coefficient $0$. [/i] (1) Compute a $3$-alternating polynomial of degree $3$. (2) Prove that the degree of any nonzero $n$-alternating polynomial is at least ${n \choose 2}$.

Prove that there do not exist polynomials $ P$ and $ Q$ such that \[ \pi(x)\equal{}\frac{P(x)}{Q(x)}\] for all $ x\in\mathbb{N}$.

Let $S = \{(x,y) | x = 1, 2, \ldots, 1993, y = 1, 2, 3, 4\}$. If $T \subset S$ and there aren't any squares in $T.$ Find the maximum possible value of $|T|.$ The squares in T use points in S as vertices.

For each natural number $x$, let $S(x)$ be the sum of its digits. Find the smallest natural number $n$ such that $9S(n) = 16S(2n)$.