Let $D$ be the foot of perpendicular from $A$ to the Euler line (the line passing through the circumcentre and the orthocentre) of an acute scalene triangle $ABC$. A circle $\omega$ with centre $S$ passes through $A$ and $D$, and it intersects sides $AB$ and $AC$ at $X$ and $Y$ respectively. Let $P$ be the foot of altitude from $A$ to $BC$, and let $M$ be the midpoint of $BC$. Prove that the circumcentre of triangle $XSY$ is equidistant from $P$ and $M$.

Determine all pairs of integers $(a, b)$ such that \[\frac{a^{2}}{2ab^{2}-b^{3}+1}\] is a positive integer.

Let a hexagone with a diameter $D$ be given and let $d>\frac D 2.$ On each side of the hexagon one constructs a isosceles triangle with two equal sides of length $d$. Prove that the sum of the areas of those isoscele triangles is greater than the area of a rhombus with side lengths $d$ and a diagonal of length $D$. (The diameter of a polygon is the maximum of the lengths of all its sides and diagonals.)

Consider an 8x8 board divided in 64 unit squares. We call [i]diagonal[/i] in this board a set of 8 squares with the property that on each of the rows and the columns of the board there is exactly one square of the [i]diagonal[/i]. Some of the squares of this board are coloured such that in every [i]diagonal[/i] there are exactly two coloured squares. Prove that there exist two rows or two columns whose squares are all coloured.

Find the number of ordered triples of positive integers $(a,b,c),$ where $1 \leq a,b,c \leq 10,$ with the property that $\gcd(a,b), \gcd(a,c),$ and $\gcd(b,c)$ are all pairwise relatively prime. [i]Proposed by Kyle Lee[/i]

Let $n$ be a positive integer, and let $x=\frac{\sqrt{n+2}-\sqrt{n}}{\sqrt{n+2}+\sqrt{n}}$ and $y=\frac{\sqrt{n+2}+\sqrt{n}}{\sqrt{n+2}-\sqrt{n}}$. It is given that $14x^2+26xy+14y^2=2014$. Find the value of $n$.

Chloé chooses a real number uniformly at random from the interval $[0, 2017]$. Independently, Laurent chooses a real number uniformly at random from the interval $[0,4034]$. What is the probability that Laurent's number is greater than Chloé's number? $\textbf{(A)}~\frac12 \qquad \textbf{(B)}~\frac23 \qquad \textbf{(C)}~\frac34 \qquad \textbf{(D)}~\frac56\qquad \textbf{(E)}~\frac78$

Given a finite set $S$, $P(S)$ denotes the set of all the subsets of $S$. For any $f:P(S)\rightarrow \mathbb{R}$ ,prove the following inequality:$$\sum_{A\in P(S)}\sum_{B\in P(S)}f(A)f(B)2^{\left| A\cap B \right|}\geq 0.$$

Find the smallest natural number $n$ for which the equality $\sin n^o= \sin (1997n)^o$ holds.

Let $A$ be a finite ring. Show that there exists two natural numbers $m,p$ where $m> p\ge 1$, such that $a^m=a^p$ for all $a\in A$.

Each cell of a $29 \times 29$ table contains one of the integers $1, 2, 3, \ldots , 29$, and each of these integers appears $29$ times. The sum of all the numbers above the main diagonal is equal to three times the sum of all the numbers below this diagonal. Determine the number in the central cell of the table.

Square $EFGH$ has one vertex on each side of square $ABCD$. Point $E$ is on $\overline{AB}$ with $AE=7\cdot EB$. What is the ratio of the area of $EFGH$ to the area of $ABCD$? $\textbf{(A)}\,\frac{49}{64} \qquad\textbf{(B)}\,\frac{25}{32} \qquad\textbf{(C)}\,\frac78 \qquad\textbf{(D)}\,\frac{5\sqrt{2}}{8} \qquad\textbf{(E)}\,\frac{\sqrt{14}}{4} $

Rhombus $CKLN$ is inscribed into triangle $ABC$ in such way that point $L$ lies on side $AB$, point $N$ lies on side $AC$, point $K$ lies on side $BC$. $O_1, O_2$ and $O$ are the circumcenters of triangles $ACL, BCL$ and $ABC$ respectively. Let $P$ be the common point of circles $ANL$ and $BKL$, distinct from $L$. Prove that points $O_1, O_2, O$ and $P$ are concyclic. (D.Prokopenko)

Let $C_1$ and $C_2$ be circles in the same plane, $P_1$ and $P_2$ arbitrary points on $C_1$ and $C_2$ respectively, and $Q$ the midpoint of segment $P_1P_2.$ Find the locus of points $Q$ as $P_1$ and $P_2$ go through all possible positions. [i]Alternative version[/i]. Let $C_1, C_2, C_3$ be three circles in the same plane. Find the locus of the centroid of triangle $P_1P_2P_3$ as $P_1, P_2,$ and $P_3$ go through all possible positions on $C_1, C_2$, and $C_3$ respectively.

Find all integer solutions of $ p^3\equal{}p^2\plus{}q^2\plus{}r^2$ where $ p,q,r$ are primes.

Prove that for every given positive integer $n$, there exists a prime $p$ and an integer $m$ such that $(a)$ $p \equiv 5 \pmod 6$ $(b)$ $p \nmid n$ $(c)$ $n \equiv m^3 \pmod p$

Triangle $ABC$ is isosceles ($AB=AC$). From $A$, we draw a line $\ell$ parallel to $BC$. $P,Q$ are on perpendicular bisectors of $AB,AC$ such that $PQ\perp BC$. $M,N$ are points on $\ell$ such that angles $\angle APM$ and $\angle AQN$ are $\frac\pi2$. Prove that \[\frac{1}{AM}+\frac1{AN}\leq\frac2{AB}\]

Each of the $n$ students in a class sent a card to each of his $m$ colleagues. Prove that if $2m + 1 > n$, then at least two students sent cards to each other.

Let $G$ be the centroid of $\triangle ABC $ and let $D, E $ and $F$ be the midpoints of the line segments $BC, CA $ and $AB$ respectively. Suppose the circumcircle of $\triangle ABC $ meets $AD $ again at $X$, the circumcircle of $\triangle DEF $ meets $BE$ again at $Y$ and the circumcircle of $\triangle DEF $ meets $CF$ again at $Z$. Show that $G, X, Y $ and $Z$ are concyclic.

Let $x_0=a, x_1= b, x_2 = c$ be given real numbers and let $x_{n+2} = \frac{x_n + x_{n-1}}{2}$ for all $n\geq 1$. Show that the sequence $(x_n)_{n\geq 0}$ converges and find its limit.

Prove that there is no polynomial $P(x)$ with integer coefficients such that $$P(\sqrt[3]{5} + \sqrt[3]{25}) = 2\sqrt[3]{5}+3\sqrt[3]{25}$$

Let $a_n$ be the number of sets $S$ of positive integers for which \[ \sum_{k\in S}F_k=n,\] where the Fibonacci sequence $(F_k)_{k\ge 1}$ satisfies $F_{k+2}=F_{k+1}+F_k$ and begins $F_1=1$, $F_2=1$, $F_3=2$, $F_4=3$. Find the largest number $n$ such that $a_n=2020$.

There is a regular hexagon that is cut direct to $6n^2$ equilateral triangles (Fig.). There are arranged $2n$ rooks, neither of which beats each other (the rooks hit in directions parallel to sides of the hexagon). Prove that if we consider chess coloring all $6n^2$ equilateral triangles, then the number of rooks that stand on black triangles will be equal to the number of rooks standing on white triangles. [img]https://cdn.artofproblemsolving.com/attachments/d/0/43ce6c5c966f60a8ec893d5d8cd31e33c43fc0.png[/img] [hide=original wording] Є правильний шестикутник, що розрізаний прямими на 6n^2 правильних трикутників (рис. 2). У них розставлені 2n тур, ніякі дві з яких не б'ють одна одну (тура б'є в напрямках, що паралельні до сторін шестикутника). Доведіть, що якщо розглянути шахове розфарбування всіх 6n^2 правильних трикутників, то тоді кількість тур, що стоять на чорних трикутниках, буде рівна кількості тур, що стоять на білих трикутниках. [/hide]

Given two lines in a plane, intersecting at an acute angle. In the direction of one of the straight lines, compression is performed with a coefficient of 1/2. Prove that there is a point from which the distance to the point of intersection of the lines increases. Note: What is meant here is a transformation in which each point moves parallel to one straight line so that its distance to the second straight line is halved, while it remains the same side from the second straight line. [hide=original wording] На плоскости даны две прямые, пересекающиеся под острым углом. В направлении одной из прямых производится сжатие 1 с коэффициентом 1/2. Доказать, что найдется точка, расстояние от которой до точки пересечения прямых увеличится. Здесь имеется в виду преобразование, при котором каждая точка перемещается параллельно одной прямой так, что её расстояние до второй прямой уменьшается вдвое, причём она остаётся по ту же самую сторону от второй прямой[/hide]

The measures of the smallest angles of three different right triangles sum to $90^\circ$. All three triangles have side lengths that are primitive Pythagorean triples. Two of them are $3-4-5$ and $5-12-13$. What is the perimeter of the third triangle? $ \textbf{(A) }40 \qquad \textbf{(B) }126 \qquad \textbf{(C) }154 \qquad \textbf{(D) }176 \qquad \textbf{(E) }208 \qquad $