Let $f : R \to R$ and $g : R \to R$ be strictly increasing linear functions such that $f(x)$ is an integer if and only if $g(x)$ is an integer. Prove that $f(x) - g(x)$ is an integer for any $x \in R$.

Two circles $ \Omega_{1}$ and $ \Omega_{2}$ are externally tangent to each other at a point $ I$, and both of these circles are tangent to a third circle $ \Omega$ which encloses the two circles $ \Omega_{1}$ and $ \Omega_{2}$. The common tangent to the two circles $ \Omega_{1}$ and $ \Omega_{2}$ at the point $ I$ meets the circle $ \Omega$ at a point $ A$. One common tangent to the circles $ \Omega_{1}$ and $ \Omega_{2}$ which doesn't pass through $ I$ meets the circle $ \Omega$ at the points $ B$ and $ C$ such that the points $ A$ and $ I$ lie on the same side of the line $ BC$. Prove that the point $ I$ is the incenter of triangle $ ABC$. [i]Alternative formulation.[/i] Two circles touch externally at a point $ I$. The two circles lie inside a large circle and both touch it. The chord $ BC$ of the large circle touches both smaller circles (not at $ I$). The common tangent to the two smaller circles at the point $ I$ meets the large circle at a point $ A$, where the points $ A$ and $ I$ are on the same side of the chord $ BC$. Show that the point $ I$ is the incenter of triangle $ ABC$.

The three roots of the cubic $ 30 x^3 \minus{} 50x^2 \plus{} 22x \minus{} 1$ are distinct real numbers between $ 0$ and $ 1$. For every nonnegative integer $ n$, let $ s_n$ be the sum of the $ n$th powers of these three roots. What is the value of the infinite series \[ s_0 \plus{} s_1 \plus{} s_2 \plus{} s_3 \plus{} \dots \, ?\]

Given two acute-angled triangles $ABC$ and $A'B'C'$ with the points $O$ and $O'$ inside. Three pairs of the perpendiculars are drawn: $[OA_1]$ to the side $[BC]$, $[O'A'_1]$ to the side $[B'C']$, $[OB_1]$ to the side $[AC]$, $[O'B'_1]$ to the side $[A'C']$, $[OC_1] $ to the side $[AB]$, $[O'C'_1]$ to the side $[A'B']$; It is known that $$[OA_1] \parallel [O'A'], [OB_1] \parallel [O'B'], [OC_1] \parallel [O'C'] $$ and $$|OA_1|\cdot|O'A'| = |OB_1|\cdot |O'B'| = |OC_1|\cdot |O'C'|$$ Prove that $$[O'A'_1] \parallel [OA], [O'B'_1] \parallel[OB], [O'C'_1] \parallel[OC]$$ and $$|O'A'_1|\cdot|OA| = |O'B'_1|\cdot|OB| = |O'C'_1|\cdot|OC|$$

Find all pairs of positive integers $ m$ and $ n$ that satisfy (both) following conditions: (i) $ m^{2}\minus{}n$ divides $ m\plus{}n^{2}$ (ii) $ n^{2}\minus{}m$ divides $ n\plus{}m^{2}$

Adrian has $2n$ cards numbered from $ 1$ to $2n$. He gets rid of $n$ cards that are consecutively numbered. The sum of the numbers of the remaining papers is $1615$. Find all the possible values of $n$.

Determine the largest $k\ge0$ such that the inequality \[\left(\sum_{j=1}^n x_j\right)^2\left(\sum_{j=1}^n x_jx_{j+1}\right)\ge k\sum_{j=1}^n x_j^2x_{j+1}^2\] holds for every $n\ge2$ and any $n$-tuple $x_1,\ldots,x_n$ of non-negative numbers (given that $x_{n+1}=x_1$)

Emily draws six dots on a piece of paper such that no three lie on a straight line, then draws a line segment connecting each pair of dots. She then colors five of these segments red. Her coloring is said to be $\emph{red-triangle-free}$ if for every set of three points from her six drawn points there exists an uncolored segment connecting two of the three points. In how many ways can Emily color her drawing such that it is red-triangle-free?

Suppose we have $10$ balls and $10$ colors. For each ball, we (independently) color it one of the $10$ colors, then group the balls together by color at the end. If $S$ is the expected value of the square of the number of distinct colors used on the balls, find the sum of the digits of $S$ written as a decimal. [i]Proposed by Michael Kural[/i]

Let $ABC$ be a triangle with $AB = 9$, $BC = 10$, and $CA = 17$. Let $B'$ be the reflection of the point $B$ over the line $CA$. Let $G$ be the centroid of triangle $ABC$, and let $G'$ be the centroid of triangle $AB'C$. Determine the length of segment $GG'$.

A convex hexagon $ABCDEF$ is given where $ \measuredangle FAB + \measuredangle BCD + \measuredangle DEF = 360^{\circ}$ and $ \measuredangle AEB = \measuredangle ADB$. Suppose the lines $AB$ and $DE$ are not parallel. Prove that the circumcenters of the triangles $ \triangle AFE, \triangle BCD$ and the intersection of the lines $AB$ and $DE$ are collinear.

Square trinomials $P(x) = x^2 + ax + b$ and $Q(x) = x^2 + cx + d$ are such that the equation $P(Q(x)) = Q(P(x))$ has no real roots. Prove that $b \ne d$.

Let $P$ be a variable point on arc $BC$ of the circumcircle of triangle $ABC$ not containing $A$. Let $I_1$ and $I_2$ be the incenters of the triangles $PAB$ and $PAC$, respectively. Prove that: [b](a)[/b] The circumcircle of $?PI_1I_2$ passes through a fixed point. [b](b)[/b] The circle with diameter $I_1I_2$ passes through a fixed point. [b](c)[/b] The midpoint of $I_1I_2$ lies on a fixed circle.

Find all the pairs of positive integers $(x,y)$ such that $x\leq y$ and \[\frac{(x+y)(xy-1)}{xy+1}=p,\]where $p$ is a prime number.

In a certain country there are $10$ cities connected by a network of one-way nonstop flights so that it is possible to fly (using one or more flights) from any city to any other. Let $n$ be the least number of flights needed to complete a trip starting from one of the cities, visiting all others and returning to the starting point. Find the greatest possible value of $n$.

Students guess that Norb's age is $24, 28, 30, 32, 36, 38, 41, 44, 47$, and $49$. Norb says, "At least half of you guessed too low, two of you are off by one, and my age is a prime number." How old is Norb? $ \textbf{(A)}29\qquad\textbf{(B)}31\qquad\textbf{(C)}37\qquad\textbf{(D)}43\qquad\textbf{(E)}48 $

A number $n$ is $\textit{almost prime}$ if any of $n-2$, $n-1$, $n$, $n+1$, or $n+2$ is prime. Compute the smallest positive integer that is not $\textit{almost prime}$.

Five equally skilled tennis players named Allen, Bob, Catheryn, David, and Evan play in a round robin tournament, such that each pair of people play exactly once, and there are no ties. In each of the ten games, the two players both have a 50% chance of winning, and the results of the games are independent. Compute the probability that there exist four distinct players $P_1$, $P_2$, $P_3$, $P_4$ such that $P_i$ beats $P_{i+1}$ for $i=1, 2, 3, 4$. (We denote $P_5=P_1$).

An $ n \times n, n \geq 2$ chessboard is numbered by the numbers $ 1, 2, \ldots, n^2$ (and every number occurs). Prove that there exist two neighbouring (with common edge) squares such that their numbers differ by at least $ n.$

We are given $n$ coins of different weights and $n$ balances, $n>2$. On each turn one can choose one balance, put one coin on the right pan and one on the left pan, and then delete these coins out of the balance. It's known that one balance is wrong (but it's not known ehich exactly), and it shows an arbitrary result on every turn. What is the smallest number of turns required to find the heaviest coin? [hide=Thanks]Thanks to the user Vlados021 for translating the problem.[/hide]

Let $\mathbb{Q}^+$ denote the set of all positive rational numbers. Determine all functions $f:\mathbb{Q}^+\to \mathbb{Q}^+$ satisfying$$f(x^2f(y)^2)=f(x^2)f(y)$$for all $x,y\in\mathbb{Q}^+$

Maria have $14$ days to train for an olympiad. The only conditions are that she cannot train by $3$ consecutive days and she cannot rest by $3$ consecutive days. Determine how many configurations of days(in training) she can reach her goal.

Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that: $$f(xy+f(x))=xf(y)$$ for all $x,y\in\mathbb{R}$.

A natural number is called [i]good[/i] if it can be represented as sum of two coprime natural numbers, the first of which decomposes into odd number of primes (not necceserily distinct) and the second to even. Prove that there exist infinity many $n$ with $n^4$ being good.

Let be three natural numbers $ n,p,q , $ a field $ \mathbb{F} , $ and two matrices $ A,B\in\mathcal{M}_n\left( \mathbb{F} \right) $ such that $$ A^pB=0=(A+I)^qB. $$ Prove that $ B=0. $ [i]D.M. Bătinețu[/i]