A three-digit number $x$ in base $10$ has a units-digit of $6$. When $x$ is written is base $9$, the second digit of the number is $4$, and the first and third digit are equal in value. Compute $x$ in base $10$.

Let $a_1, a_2, a_3, \ldots$ be a sequence of positive real numbers, and $s$ be a positive integer, such that \[a_n = \max \{ a_k + a_{n-k} \mid 1 \leq k \leq n-1 \} \ \textrm{ for all } \ n > s.\] Prove there exist positive integers $\ell \leq s$ and $N$, such that \[a_n = a_{\ell} + a_{n - \ell} \ \textrm{ for all } \ n \geq N.\] [i]Proposed by Morteza Saghafiyan, Iran[/i]

The lengths of the altitudes of a triangle are positive integers, and the length of the radius of the incircle is a prime number. Find the lengths of the sides of the triangle.

Let $x$ and $y$ be real numbers such that $x^2+y^2-22x-16y+113=0.$ Determine the smallest possible value of $x.$

How many triangles have area $ 10$ and vertices at $ (\minus{}5,0)$, $ (5,0)$, and $ (5\cos \theta, 5\sin \theta)$ for some angle $ \theta$? $ \textbf{(A)}\ 0\qquad \textbf{(B)}\ 2\qquad \textbf{(C)}\ 4\qquad \textbf{(D)}\ 6\qquad \textbf{(E)}\ 8$

Determine all positive integers $n$ for which the equation \[x^{n}+(2+x)^{n}+(2-x)^{n}= 0\] has an integer as a solution.

Rational numbers $a$ and $b$ satisfy the equality $$a^3b+ab^3+2a^2b^2+2a + 2b + 1 = 0. $$ Prove that the number $1-ab$ is the square of the rational numbers.

Let $\triangle ABC$ an acute triangle of incenter $I$ and incircle $\omega$. $\omega$ is tangent to $BC, CA$ and $AB$ at points $T_{A}, T_{B}$ and $T_{C}$, respectively. Let $l_{A}$ the line through $A$ and parallel to $BC$ and define $l_{B}$ and $l_{C}$ analogously. Let $L_{A}$ the second intersection point of $AI$ with the circumcircle of $\triangle ABC$ and define $L_{B}$ and $L_{C}$ analogously. Let $P_{A}=T_{B}T_{C}\cap l_{A}$ and define $P_{B}$ and $P_{C}$ analogously. Let $S_{A}=P_{B}T_{B}\cap P_{C}T_{C}$ and define $S_{B}$ and $S_{C}$ analogously. Prove that $S_{A}L_{A}, S_{B}L_{B}, S_{C}L_{C}$ are concurrent.

Let $x_n=2^{2^{n}}+1$ and let $m$ be the least common multiple of $x_2, x_3, \ldots, x_{1971}.$ Find the last digit of $m.$

Prove that for every positive integer $n$ there exists a polynomial $p(x)$ of degree $n$ with real coefficients, having $n$ distinct real roots and satisfying $$p(x)p(4-x)=p(x(4-x))$$

Find the number of positive integers $10 \le n \le 99$ with last digit at most $5$ such that the last two digits of $n^n$ are the same as $n.$

In how many consecutive zeros does the decimal expansion of $\frac{26!}{35^3}$ end? $\text{(A) }1\qquad\text{(B) }2\qquad\text{(C) }3\qquad\text{(D) }4\qquad\text{(E) }5$

You are initially given the number $n=1$. Each turn, you may choose any positive divisor $d\mid n$, and multiply $n$ by $d+1$. For instance, on the first turn, you must select $d=1$, giving $n=1\cdot(1+1)=2$ as your new value of $n$. On the next turn, you can select either $d=1$ or $2$, giving $n=2\cdot(1+1)=4$ or $n=2\cdot(2+1)=6$, respectively, and so on. Find an algorithm that, in at most $k$ steps, results in $n$ being divisible by the number $2021^{2021^{2021}} - 1$. An algorithm that completes in at most $k$ steps will be awarded: 1 pt for $k>2021^{2021^{2021}}$ 20 pts for $k=2021^{2021^{2021}}$ 50 pts for $k=10^{10^4}$ 75 pts for $k=10^{10}$ 90 pts for $k=10^5$ 95 pts for $k=6\cdot10^4$ 100 pts for $k=5\cdot10^4$

Twenty distinct points are marked on a circle and labeled $1$ through $20$ in clockwise order. A line segment is drawn between every pair of points whose labels differ by a prime number. Find the number of triangles formed whose vertices are among the original $20$ points.

The parabolas $y=x^2+15x+32$ and $x = y^2+49y+593$ meet at one point $(x_0,y_0)$. Find $x_0+y_0$.

Let $ABC$ be an acute-angled triangle and let $P$ be the intersection of the tangents at $B$ and $C$ of the circumscribed circle of $\vartriangle ABC$. The line through $A$ perpendicular on $AB$ and cuts the line perpendicular on $AC$ through $C$ at $X$. The line through $A$ perpendicular on $AC$ cuts the line perpendicular on $AB$ through $B$ at $Y$. Show that $AP \perp XY$.

Prove that for all real numbers $ a,b$ with $ ab>0$ we have: $ \sqrt[3]{\frac{a^2 b^2 (a\plus{}b)^2}{4}} \le \frac{a^2\plus{}10ab\plus{}b^2}{12}$ and find the cases of equality. Hence, or otherwise, prove that for all real numbers $ a,b$ $ \sqrt[3]{\frac{a^2 b^2 (a\plus{}b)^2}{4}} \le \frac{a^2\plus{}ab\plus{}b^2}{3}$ and find the cases of equality.

Given real numbers $(a_i)$ and $(b_i)$ (for $i=1,2,3,4$) such that $a_1 b _2 \ne a_2 b_1 .$ Consider the set of all solutions $(x_1 ,x_2 ,x_3 , x_4)$ of the simultaneous equations $$ a_1 x_1 +a _2 x_2 +a_3 x_3 +a_4 x_4 =0 \;\; \text{and}\;\; b_1 x_1 +b_2 x_2 +b_3 x_3 +b_4 x_4 =0 $$ for which no $x_i$ is zero. Each such solution generates a $4$-tuple of plus and minus signs (by considering the sign of $x_i$). [list=a] [*] Determine, with proof, the maximum number of distinct $4$-tuples possible. [*] Investigate necessary and sufficient conditions on $(a_i)$ and $(b_i)$ such that the above maximum of distinct $4$-tuples is attained.

Point $C$ is a midpoint of $AB$. Circle $o_1$ which passes through $A$ and $C$ intersect circle $o_2$ which passes through $B$ and $C$ in two different points $C$ and $D$. Point $P$ is a midpoint of arc $AD$ of circle $o_1$ which doesn't contain $C$. Point $Q$ is a midpoint of arc $BD$ of circle $o_2$ which doesn't contain $C$. Prove that $PQ \perp CD$.

Find the real values $ m\in\mathbb{R} $ such that all solutions of the equation $$ 1=2mx(2x-1)(2x-2)(2x-3) $$ are real.

Find all functions $f:\mathbb{R}\to \mathbb{R}$ such that for all $x, y\in \mathbb{R}$ the following holds: \[f(x)+f(y)=f(xy)+f(f(x)+f(y))\]

Let $b = \tfrac 12 (-1 + 3\sqrt{5})$. Determine the number of rational numbers which can be written in the form \[ a_{2014}b^{2014} + a_{2013}b^{2013} + \dots + a_1b + a_0 \] where $a_0, a_1, \dots, a_{2014}$ are nonnegative integers less than $b$. [i]Proposed by Michael Kural and Evan Chen[/i]

Find all ordered triples $(a,b, c)$ of real numbers such that $$(a-b)(b-c) + (b-c)(c-a) + (c-a)(a-b) = 0.$$

Let $ABC$ be a fixed triangle. Point $D$ is an arbitrary point on the side $BC$. Point $P$ is fixed on $AD$. The circumcircle of triangle $BPD$ meets $AB$ at $E$ distinct from $B$. Point $Q$ varies on $AP$. Let $BQ$ and $CQ$ meet the circumcircles of triangles $BPD, CPD$ respectively at $F,Z$ distinct from $B,C$. Prove that the circumcircle $EFZ$ is through a fixed point distinct from $E$ and this fixed point is on the circumcircle of triangle $CPD$. Kostas Vittas

In an $n\times{n}$ grid, there are some cats living in each cell (the number of cats in a cell must be a non-negative integer). Every midnight, the manager chooses one cell: (a) The number of cats living in the chosen cell must be greater than or equal to the number of neighboring cells of the chosen cell. (b) For every neighboring cell of the chosen cell, the manager moves one cat from the chosen cell to the neighboring cell. (Two cells are called "neighboring" if they share a common side, e.g. there are only $2$ neighboring cells for a cell in the corner of the grid) Find the minimum number of cats living in the whole grid, such that the manager is able to do infinitely many times of this process.