Find all positive integers $n \geqslant 2$ for which there exist $n$ real numbers $a_1<\cdots<a_n$ and a real number $r>0$ such that the $\tfrac{1}{2}n(n-1)$ differences $a_j-a_i$ for $1 \leqslant i<j \leqslant n$ are equal, in some order, to the numbers $r^1,r^2,\ldots,r^{\frac{1}{2}n(n-1)}$.

In an acute triangle $ ABC$ segments $ BE$ and $ CF$ are altitudes. Two circles passing through the point $ A$ and $ F$ and tangent to the line $ BC$ at the points $ P$ and $ Q$ so that $ B$ lies between $ C$ and $ Q$. Prove that lines $ PE$ and $ QF$ intersect on the circumcircle of triangle $ AEF$. [i]Proposed by Davood Vakili, Iran[/i]

For $n$ an odd positive integer, the unit squares of an $n\times n$ chessboard are coloured alternately black and white, with the four corners coloured black. A it tromino is an $L$-shape formed by three connected unit squares. For which values of $n$ is it possible to cover all the black squares with non-overlapping trominos? When it is possible, what is the minimum number of trominos needed?

Let $f(n)=n+\lfloor \sqrt{n}\rfloor$. Prove that, for every positive integer $m$, the sequence \[m, f(m), f(f(m)), f(f(f(m))), \cdots\] contains at least one square of an integer.

Let $ABC$ be an acute triangle with $ |AB | < |AC |$and orthocenter $H$. The circle with center A and radius$ |AC |$ intersects the circumcircle of $\triangle ABC$ at point $D$ and the circle with center $A$ and radius$ |AB |$ intersects the segment $\overline{AD}$ at point $K. $ The line through $K$ parallel to $CD $ intersects $BC$ at the point $ L.$ If $M$ is the midpoint of $\overline{BC}$ and N is the foot of the perpendicular from $H$ to $AL, $ prove that the line $ MN $ bisects the segment $\overline{AH}$.

Let $ABC$ be a triangle. An interior point $P$ of $ABC$ is said to be [i]good [/i]if we can find exactly $27$ rays emanating from $P$ intersecting the sides of the triangle $ABC$ such that the triangle is divided by these rays into $27$ [i]smaller triangles of equal area.[/i] Determine the number of good points for a given triangle $ABC$.

Let $AC$ be the greatest leg of a right triangle $ABC,$ and $CH$ be the altitude to its hypotenuse. The circle of radius $CH$ centered at $H$ intersects $AC$ in point $M.$ Let a point $B'$ be the reflection of $B$ with respect to the point $H.$ The perpendicular to $AB$ erected at $B'$ meets the circle in a point $K$. Prove that [b]a)[/b] $B'M \parallel BC$ [b]b)[/b] $AK$ is tangent to the circle.

Let $ABC$ be a triangle with $\angle C = 90^o$ and $D$ a point on the ray $CB$ such that $|AC| \cdot |CD| = |BC|^2$. A parallel line to $AB$ through $D$ intersects the ray $CA$ at $E$. Find $\angle BEC$.

Three integers are written on a blackboard. At every step one of them is erased and the sum of the other two decreased by $1$ is written instead. Is it possible to obtain the numbers $17,75,91$ if the three initial numbers were: $\textbf{(a)}~2,2,2$; $\textbf{(b)}~3,3,3$?

Let $ABCD$ be a cyclic quadrilateral and triangles $ACD, BCD$ are acute. Suppose that the lines $AB$ and $CD$ meet at $S$. Denote by $E$ the intersection of $AC, BD$. The circles $(ADE)$ and $(BC E)$ meet again at $F$. a) Prove that $SF \perp EF.$ b) The point $G$ is taken out side of the quadrilateral $ABCD$ such that triangle $GAB$ and $FDC$ are similar. Prove that $GA+ FB = GB + FA$

The sequence $(a_n)$ is defined as: $$a_1=1007$$ $$a_{i+1}\geq a_i+1$$ Prove the inequality: $$\frac{1}{2016}>\sum_{i=1}^{2016}\frac{1}{a_{i+1}^{2}+a_{i+2}^2}$$

In a quadrilateral $ABCD$, it is given that $AB$ is parallel to $CD$ and the diagonals $AC$ and $BD$ are perpendicular to each other. Show that (a) $AD \cdot BC \geq AB \cdot CD$ (b) $AD + BC \geq AB + CD.$

Prove that there exist a natural number $a$, for which 999 divides $2^{5n}+a.5^n$ for $\forall$ odd $n\in \mathbb{N}$ and find the smallest such $a$.

Mary Jane and Rachel are playing ping pong. Rachel has a $7/8$ chance of returning any shot, and Mary Jane has a $5/8$ chance. Mary Jane serves to Rachel (and doesn't mess up the serve). What is the average number of returns made?

Determine all positive integers $a$, $b$ and $c$ which satisfy the equation $$a^2+b^2+1=c!.$$ [i]Proposed by Nikola Velov[/i]

Let $ABCD$ be an isosceles trapezoid with bases $AB$ and $CD$. The incircle of the triangle $BCD$ touches $CD$ at $E$. Point $F$ is chosen on the bisector of the angle $DAC$ such that the lines $EF$ and $CD$ are perpendicular. The circumcircle of the triangle $ACF$ intersects the line $CD$ again at $G$. Prove that the triangle $AFG$ is isosceles.

Prove that all numbers of the sequence \[ \frac{107811}{3}, \quad \frac{110778111}{3}, \frac{111077781111}{3}, \quad \ldots \] are exact cubes.

Consider a set $S = \{a_1, \ldots, a_{2024}\}$ consisting of $2024$ distinct positive integers that satisfies the following property: [center] "For every positive integer $m < 2024,$ the sum of no $m$ distinct elements of $S$ is a multiple of $2024.$" [/center] Prove $a_1, \ldots, a_{2024}$ all leave the same remainder when divided by $2024.$ Justify your answer.

Suppose $a$, $b$, and $c$ are numbers satisfying the three equations: $$a + 2b = 20,$$ $$b + 2c = 2,$$ $$c + 2a = 3.$$ Find $9a + 9b + 9c$.

Nair has puzzle pieces shaped like an equilateral triangle. She has pieces of two sizes: large and small. [img]https://cdn.artofproblemsolving.com/attachments/a/1/aedfbfb2cb17bf816aa7daeb0d35f46a79b6e9.jpg[/img] Nair build triangular figures by following these rules: $\bullet$ Figure $1$ is made up of $4$ small pieces, Figure $2$ is made up of $2$ large pieces and $8$ small, Figure $3$ by $6$ large and $12$ small, and so on. $\bullet$ The central column must be made up exclusively of small parts. $\bullet$ Outside the central column, only large pieces can be placed. [img]https://cdn.artofproblemsolving.com/attachments/5/7/e7f6340de0e04d5b5979e72edd3f453f2ac8a5.jpg[/img] Following the pattern, how many pieces will Nair use to build Figure $20$?

Given triangle $ ABC$ with $ AB>AC$. $ l$ is tangent line of the circumcircle of triangle $ ABC$ at $ A$. A circle with center $ A$ and radius $ AC$, intersect $ AB$ at $ D$ and $ l$ at $ E$ and $ F$. Prove that the lines $ DE$ and $ DF$ pass through the incenter and excenter of triangle $ ABC$.

We have a rhombus $ABCD$ with $\angle BAD=60^\circ$. We take points $F,H,G$ on the sides $AD,DC$ and the diagonal $AC$, respectively, such that $DFGH$ is a parallelogram. Prove that $BFH$ is equilateral.

Compute $$\sum_{i=1}^{4} \sum_{t=1}^{4} \sum_{e=1}^{4} \left\lfloor \frac{ite}{5} \right\rfloor.$$

The map of a country is in the form of a convex polygon with $n (n\geq5)$ sides, such as any 3 diagonals do not concur inside the polygon. Some of the diagonals are roads, which allow movement in both directions and the other diagonals are not roads. There are cities on the intersection points of any two diagonals inside the polygon and at least one of the two diagonals is a road. Prove that you can move from any city to any other city using at most 3 roads.

Which of the following divides $3^{3n+1} + 5^{3n+2}+7^{3n+3}$ for every positive integer $n$? $ \textbf{(A)}\ 3 \qquad\textbf{(B)}\ 5 \qquad\textbf{(C)}\ 7 \qquad\textbf{(D)}\ 11 \qquad\textbf{(E)}\ 53 $