Fifteen distinct points are designated on $\triangle ABC$: the 3 vertices $A$, $B$, and $C$; $3$ other points on side $\overline{AB}$; $4$ other points on side $\overline{BC}$; and $5$ other points on side $\overline{CA}$. Find the number of triangles with positive area whose vertices are among these $15$ points.

Are there positive integers $m,n$ such that there exist at least $2012$ positive integers $x$ such that both $m-x^2$ and $n-x^2$ are perfect squares? [i]David Yang.[/i]

Let $p$ be a prime number. Find all pairs of coprime positive integers $(m,n)$ such that $$ \frac{p+m}{p+n}=\frac{m}{n}+\frac{1}{p^2}.$$

A lattice point in the plane is a point whose coordinates are both integers. Given a set of 100 distinct lattice points in the plane, find the smallest number of line segments $ \overline{AB}$ for which $ A$ and $ B$ are distinct lattice points in this set and the midpoint of $ \overline{AB}$ is also a lattice point (not necessarily in the set).

Find all pairs of complex numbers $(z,w) \in \mathbb{C}^2$ such that the relation \[|z^{2n}+z^nw^n+w^{2n} | = 2^{2n}+2^n+1 \] holds for all positive integers $n$.

In acute triangle $ABC$, point $H$ is the intersection point of heights $CE$ on side $AB$ and $BD$ on side $AC$. A circle with diameter $DE$ intersects $AB$ and $AC$ at $F$ and $G$ respectively. $FG$ and $AH$ intersect at $K$. If $BC=25,BD=20, BE=7$, find the length of $AK$.

The three medians of a triangle has lengths $3, 4, 5$. What is the length of the shortest side of this triangle?

Solve the system equations \[\left\{\begin{array}{cc}x^{4}-y^{4}=240\\x^{3}-2y^{3}=3(x^{2}-4y^{2})-4(x-8y)\end{array}\right.\]

Determine all positive integers $x$, $y$ and $z$ such that $x^5 + 4^y = 2013^z$. ([i]Serbia[/i])

$Abdulqodir$ cut out $2024$ congruent regular $n-$gons from a sheet of paper and placed these $n-$gons on the table such that some parts of each of these $n-$gons may be covered by others. We say that a vertex of one of the afore-mentioned $n-$gons is $visible$ if it is not in the interior of another $n-$gon that is placed on top of it. For any $n>2$ determine the minimum possible number of visible vertices. \\ Proposed by David Hrushka, Slovakia

Three real numbers satisfy the following statements: (1) the square of their sum equals to the sum their squares. (2) the product of the first two numbers is equal to the square of the third number. Find these numbers.

Prove that, if there exist natural numbers $a_1,a_2…a_{2017}$ for which the product $(a_1^{2017}+a_2 )(a_2^{2017}+a_3 )…(a_{2016}^{2017}+a_{2017})(a_{2017}^{2017}+a_1)$ is a $k$-th power of a prime number, then $k=2017$ or $k\geq 2017.2018$.

There are a number of arcs on the edge of a circular disk. Each pair of arcs has the least one point in common. Show that on the circle you can choose two diametrical opposites points such that each arc contains at least one of these two points.

Determine whether the polynomial $P(x)=(x^2-2x+5)(x^2-4x+20)+1$ is irreducible over $\mathbb{Z}[X]$.

Each detail of the “Young Solderer” instructor is a bracket in the shape of the letter $\Pi$, consisting of three single segments. Is it possible from the parts of this constructor are soldered together, a complete wire frame of the cube $2 \times 2 \times 2$, divided into $1 \times 1 \times 1$ cubes? (The frame consists of 27 points, connected by single segments; any two adjacent points must be connected by exactly one piece of wire.) [hide]=original wording]Каждая деталько нструктора ''Юный паяльщик'' — это скобка в виде буквы П, остоящая из трех единичных отрезков. Можно ли издеталей этого конструктора спаятьполный роволочный каркас куба 2 × × 2 × 2, разбитого на кубики 1 × 1 × 1? (Каркас состоит из 27 точек,соединенных единичными отрезками; любые две соседние точки должны бытьсоединены ровно одним проволочным отрезком.)[/hide]

Suppose you started at the origin on the number line in a coin-flipping game. Every time you flip a heads, you move forward one step, otherwise you move back one step. However, there are walls at positions $8$ and $-8$; if you are at these positions and your coin flip dictates that you should move past them, instead you must stay. What is the expected number of coin flips needed to have visited both walls?

Does there exist a positive integer that is a power of $2$ and we get another power of $2$ by swapping its digits? Justify your answer.

In a certain year the price of gasoline rose by $ 20\%$ during January, fell by $ 20\%$ during February, rose by $ 25\%$ during March, and fell by $ x\%$ during April. The price of gasoline at the end of April was the same as it had been at the beginning of January. To the nearest integer, what is $ x$? $ \textbf{(A)}\ 12\qquad \textbf{(B)}\ 17\qquad \textbf{(C)}\ 20\qquad \textbf{(D)}\ 25\qquad \textbf{(E)}\ 35$

On the circumcircle of triangle $ABC$, point $P$ is taken in such a way that the perpendicular drawn by the point $P$ to the line $AC$ cuts the circle also at the point $Q$, the perpendicular drawn by the point $Q$ to the line $AB$ cuts the circle also at point R and the perpendicular drawn by point $R$ to the line BC cuts the circle also at the point $P$. Let $O$ be the center of this circle. Prove that $\angle POC = 90^o$ .

Given a convex polygon with vertices at lattice points on a plane containing origin $O$. Let $V_1$ be the set of vectors going from $O$ to the vertices of the polygon, and $V_2$ be the set of vectors going from $O$ to the lattice points that lie inside or on the boundary of the polygon (thus, $V_1$ is contained in $V_2$.) Two grasshoppers jump on the whole plane: each jump of the first grasshopper shift its position by a vector from the set $V_1$, and the second by the set $V_2$. Prove that there exists positive integer $c$ that the following statement is true: if both grasshoppers can jump from $O$ to some point $A$ and the second grasshopper needs $n$ jumps to do it, then the first grasshopper can use at most $n+c$ jumps to do so.

The center of a circle and nine randomly selected points on this circle are colored in red. Every pair of those points is connected by a line segment, and every point of intersection of two line segments inside the circle is colored in red. What is the largest possible number of red points? A. $235$ B. $245$ C. $250$ D. $220$ E. $265$

Let $f: \mathbb{R}^2 \to \mathbb{R}^2$ be a function having the following property: For any two points $A$ and $B$ in $\mathbb{R}^2$, the distance between $A$ and $B$ is the same as the distance between the points $f(A)$ and $f(B)$. Denote the unique straight line passing through $A$ and $B$ by $l(A,B)$ (a) Suppose that $C,D$ are two fixed points in $\mathbb{R}^2$. If $X$ is a point on the line $l(C,D)$, then show that $f(X)$ is a point on the line $l(f(C),f(D))$. (b) Consider two more point $E$ and $F$ in $\mathbb{R}^2$ and suppose that $l(E,F)$ intersects $l(C,D)$ at an angle $\alpha$. Show that $l(f(C),f(D))$ intersects $l(f(E),f(F))$ at an angle $\alpha$. What happens if the two lines $l(C,D)$ and $l(E,F)$ do not intersect? Justify your answer.

Let $a, b, c >1$ be positive real numbers such that $a^{\log_b c}=27, b^{\log_c a}=81,$ and $c^{\log_a b}=243$. Then the value of $\log_3{abc}$ can be written as $\sqrt{x}+\sqrt{y}+\sqrt{z}$ for positive integers $x,y,$ and $z$. Find $x+y+z$. [i]Proposed by [b]AOPS12142015[/b][/i]

Find all functions $f : \mathbb{Q}[x] \to \mathbb{Q}[x]$ such that two following conditions holds : $$\forall P , Q \in \mathbb{Q}[x] : f(P+Q)=f(P)+f(Q)$$ $$\forall P \in \mathbb{Q}[x] : gcd(P , f(P))=1 \iff$$ $P$ is square-free. Which a square-free polynomial with rational coefficients is a polynomial such that there doesn't exist square of a non-constant polynomial with rational coefficients that divides it. [i]Proposed by Sina Azizedin[/i]

Prove that for any positive integers $a$ and $b$ we have $$a+(-1)^b \sum^a_{m=0} (-1)^{\lfloor{\frac{bm}{a}\rfloor}} \equiv b+(-1)^a \sum^b_{n=0} (-1)^{\lfloor{\frac{an}{b}\rfloor}} \pmod{4}.$$