Let $x$, $y$, and $z$ be distinct real numbers that sum to $0$. Find the maximum possible value of \[ \dfrac {xy+yz+zx}{x^2+y^2+z^2}. \]

Determine all function $f$ from the set of real numbers to itself such that for every $x$ and $y$, \[f(x^2-y^2)=(x-y)(f(x)+f(y))\]

The greatest integer that will divide $13,511$, $13,903$, and $14,589$ and leave the same remainder is $\textbf{(A) }28\qquad\textbf{(B) }49\qquad\textbf{(C) }98\qquad$ $\textbf{(D) }\text{an odd multiple of }7\text{ greater than }49\qquad \textbf{(E) }\text{an even multiple of }7\text{ greater than }98$

Let $x, y,$ and $z$ be positive real numbers such that $xy + yz + zx = 3$. Prove that $$\frac{x + 3}{y + z} + \frac{y + 3}{z + x} + \frac{z + 3}{x + y} + 3 \ge 27 \cdot \frac{(\sqrt{x} + \sqrt{y} + \sqrt{z})^2}{(x + y + z)^3}.$$ Proposed by [i]Petar Filipovski, Macedonia[/i]

In a triangle $ABC$ points $A_1,B_1,C_1$ are the midpoints of $BC,CA,AB$ respectively,and $A_2,B_2,C_2$ are the midpoints of the altitudes from $A,B,C$ respectively. Show that the lines $A_1A_2,B_1B_2,C_1,C_2$ are concurrent.

Given are quadratic trinomials $f$ and $g$ with integral coefficients. For each positive integer $n$ there is an integer $k$ such that \[\frac{f(k)}{g(k)}=\frac{n + 1}{n}. \] Prove that $f$ and $g$ have a common root. [i] Proposed by A. Golovanov [/i]

Does there exist an integer $z$ that can be written in two different ways as $z = x! + y!$, where $x, y$ are natural numbers with $x \le y$ ?

Let $n$ be a given positive integer. We say that a positive integer $m$ is [i]$n$-good[/i] if and only if there are at most $2n$ distinct primes $p$ satisfying $p^2\mid m$. (a) Show that if two positive integers $a,b$ are coprime, then there exist positive integers $x,y$ so that $ax^n+by^n$ is $n$-good. (b) Show that for any $k$ positive integers $a_1,\ldots,a_k$ satisfying $\gcd(a_1,\ldots,a_k)=1$, there exist positive integers $x_1,\ldots,x_k$ so that $a_1x_1^n+a_2x_2^n+\cdots+a_kx_k^n$ is $n$-good. (Remark: $a_1,\ldots,a_k$ are not necessarily pairwise distinct) [i]Proposed by usjl.[/i]

Find all natural numbers $n$, $n < 10^7$, for which: If natural number $m$, $1 < m < n$, is not divisible by $n$, then $m$ is prime.

We are given $m,n \in \mathbb{Z}^+.$ Show the number of solution $4-$tuples $(a,b,c,d)$ of the system \begin{align*} ab + bc + cd - (ca + ad + db) &= m\\ 2 \left(a^2 + b^2 + c^2 + d^2 \right) - (ab + ac + ad + bc + bd + cd) &= n \end{align*} is divisible by 10.

In a test, $201$ students are trying to solve $6$ problems.We know that for each of $5$ first problems, there are at least $140$ students, who can solve it. Moreover, there is exactly $60$ students, who can solve $6^{th}$ problem. Show that there exist $2$ students, such that two of them combined are able to solve all $6$ question. (For example, number $1$ do $1,2,3,4$ and number $2$ do $3,5,6$)

Find all positive integers $ n$ for which the numbers in the set $ S \equal{} \{1,2, \ldots,n \}$ can be colored red and blue, with the following condition being satisfied: The set $ S \times S \times S$ contains exactly $ 2007$ ordered triples $ \left(x, y, z\right)$ such that: [b](i)[/b] the numbers $ x$, $ y$, $ z$ are of the same color, and [b](ii)[/b] the number $ x \plus{} y \plus{} z$ is divisible by $ n$. [i]Author: Gerhard Wöginger, Netherlands[/i]

The equation $z(z+i)(z+3i)=2002i$ has a zero of the form $a+bi$, where $a$ and $b$ are positive real numbers. Find $a$. $\textbf{(A) }\sqrt{118}\qquad\textbf{(B) }\sqrt{210}\qquad\textbf{(C) }2\sqrt{210}\qquad\textbf{(D) }\sqrt{2002}\qquad\textbf{(E) }100\sqrt2$

For each positive integer $n$ let $p(n)$ be the number of ordered pairs $(x,y)$ of positive integers such that$$\dfrac{1}{x}+\dfrac{1}{y} =\dfrac{1}{n}.$$For example, for $n=2$ the pairs are $(3,6),(4,4),(6,3)$. Therefore $p(2)=3$. a) Determine $p(n)$ for all $n$ and calculate $p(1995)$. b) Determine all pairs $n$ such that $p(n)=3$.

A segment of length $1$ is drawn such that its endpoints lie on a unit circle, dividing the circle into two parts. Compute the area of the larger region.

Consider an $8\times 8$ grid of squares. A rook is placed in the lower left corner, and every minute it moves to a square in the same row or column with equal probability (the rook must move; i.e. it cannot stay in the same square). What is the expected number of minutes until the rook reaches the upper right corner?

Let us consider one variable polynomials with the senior coefficient equal to one. We shall say that two polynomials $P(x)$ and $Q(x)$ commute, if $P(Q(x))=Q(P(x))$ (i.e. we obtain the same polynomial, having collected the similar terms). a) For every a find all $Q$ such that the $Q$ degree is not greater than three, and $Q$ commutes with $(x^2 - a)$. b) Let $P$ be a square polynomial, and $k$ is a natural number. Prove that there is not more than one commuting with $P$ $k$-degree polynomial. c) Find the $4$-degree and $8$-degree polynomials commuting with the given square polynomial $P$. d) $R$ and $Q$ commute with the same square polynomial $P$. Prove that $Q$ and $R$ commute. e) Prove that there exists a sequence $P_2, P_3, ... , P_n, ...$ ($P_k$ is $k$-degree polynomial), such that $P_2(x) = x^2 - 2$, and all the polynomials in this infinite sequence pairwise commute.

It is known that a sequence of positive real numbers \(\left(x_n\right)\) satisfies the relation: \[ x_{n+1} = x_n + \sqrt{x_n + \frac{1}{4}} + \sqrt{x_{n+1} + \frac{1}{4}}, \quad n \geq 1 \] Prove that the following inequality holds: \[ \frac{1}{x_2} + \frac{1}{x_3} + \cdots + \frac{1}{x_{2025}} < \frac{1}{\sqrt{x_1}} \] [i]Proposed by Oleksii Masalitin[/i]

Find all functions $f : R \to R$ which satisfy for all $x, y \in R$ the relation $f(f(f(x) + y) + y) = x + y + f(y)$

A $3000\times 3000$ square is tiled by dominoes (i. e. $1\times 2$ rectangles) in an arbitrary way. Show that one can color the dominoes in three colors such that the number of the dominoes of each color is the same, and each dominoe $d$ has at most two neighbours of the same color as $d$. (Two dominoes are said to be [i]neighbours[/i] if a cell of one domino has a common edge with a cell of the other one.)

Pam lists the four smallest positive prime numbers in increasing order. When she divides the positive integer $N$ by the first prime, the remainder is $1$. When she divides $N$ by the second prime, the remainder is $2$. When she divides $N$ by the third prime, the remainder is $3$. When she divides $N$ by the fourth prime, the remainder is $4$. Find the least possible value for $N$.

Let $a=-\sqrt{3}+\sqrt{5}+\sqrt{7}, b=\sqrt{3}-\sqrt{5}+\sqrt{7}, c=\sqrt{3}+\sqrt{5}-\sqrt{7}$. Evaluate \[\frac{a^4}{(a-b)(a-c)}+\frac{b^4}{(b-c)(b-a)}+\frac{c^4}{(c-a)(c-b)}.\]

Last academic year Yolanda and Zelda took different courses that did not necessarily administer the same number of quizzes during each of the two semesters. Yolanda's average on all the quizzes she took during the first semester was 3 points higher than Zelda's average on all the quizzes she took during the first semester. Yolanda's average on all the quizzes she took during the second semester was 18 points higher than her average for the first semester and was again 3 points higher than Zelda's average on all the quizzes Zelda took during her second semester. Which one of the following statements cannot possibly be true? (A) Yolanda's quiz average for the academic year was 22 points higher than Zelda's. (B) Zelda's quiz average for the academic year was higher than Yolanda's. (C) Yolanda's quiz average for the academic year was 3 points higher than Zelda's. (D) Zelda's quiz average for the academic year equaled Yolanda's. (E) If Zelda had scored 3 points higher on each quiz she took, then she would have had the same average for the academic year as Yolanda.

Let $a, b, c $ be the side lengths of a right angled triangle with c > a, b. Show that $$3<\frac{c^3-a^3-b^3}{c(c-a)(c-b)}\leq \sqrt{2}+2.$$

Four balls of radius $1$ are placed in space so that each of them touches the other three. What is the radius of the smallest sphere containing all of them?