A triangle $ ABC$ is inscribed in a circle $ C(O,R)$ and has incenter $ I$. Lines $ AI,BI,CI$ meet the circumcircle $ (O)$ of triangle $ ABC$ at points $ D,E,F$ respectively. The circles with diameter $ ID,IE,IF$ meet the sides $ BC,CA, AB$ at pairs of points $ (A_1,A_2), (B_1, B_2), (C_1, C_2)$ respectively. Prove that the six points $ A_1,A_2, B_1, B_2, C_1, C_2$ are concyclic. Babis

On side $AB$ of triangle $ABC$ is marked point $K$ such that $AB = CK$. Points $N$ and $M$ are the midpoints of $AK$ and $BC$, respectively. The segments $NM$ and $CK$ intersect in point $P$. Prove that $KN = KP$.

Does there exist a quadratic trinomial $f(x)$ such that $f(1/2017)=1/2018$, $f(1/2018)=1/2017$, and two of its coefficients are integers? (A. Khrabrov)

The function $f(x) = ax^2 + bx + c$ satisfies the following conditions: $f(\sqrt2)=3$ and $ |f(x)| \le 1$ for all $x \in [-1, 1]$. Evaluate the value of $f(\sqrt{2013})$

Each coefficient of a polynomial is an integer with absolute value not exceeding $2015$. Prove that every positive root of this polynomial exceeds $\frac{1}{2016}$. [i]($6$ points)[/i]

We are going to colour the cells of a $2015 \times 2015$ board such that there are none of the following: $1)$ Three cells with the same colour where two of them are in the same column, and the third is in the same row and to the right of the upper cell, $2)$ Three cells with the same colour where two of them are in the same column, and the third is in the same row and to the left of the lower cell. What is the minimum number of colours $k$ required to make such a colouring possible?

The two sides of the triangle are equal to $1$ and $x$, and $ x \ge 1$. The values $a$ and $b$ are the largest and smallest angles of this triangle, respectively. Find the greatest value of $\cos a$ and the smallest value of $\cos b$.

$ABCD$ - convex quadrilateral. Bisectors of angles $A$ and $D$ intersect in $K$, Bisectors of angles $B$ and $C$ intersect in $L$. Prove $$2KL \geq |AB-BC+CD-DA|$$

How many handles has the Riemann surface of the function \[w=\sqrt{1+z^n}\]

The sequence $ (x_n)_{n \in \mathbb{N}}$ is defined by $ x_1\equal{}2, x_2\equal{}3,$ and $ x_{2m\plus{}1}\equal{}x_{2m}\plus{}x_{2m\minus{}1}$ for $ m \ge 1;$ $ x_{2m}\equal{}x_{2m\minus{}1}\plus{}2x_{2m\minus{}2}$ for $ m \ge 2.$ Determine $ x_n$ as a function of $ n$.

For how many integers $n$ with $1 \le n \le 2012$ is the product \[ \prod_{k=0}^{n-1} \left( \left( 1 + e^{2 \pi i k / n} \right)^n + 1 \right) \] equal to zero?

A line $l$ parallel to the side $BC$ of triangle $ABC$ touches its incircle and meets its circumcircle at points $D$ and $E$. Let $I$ be the incenter of $ABC$. Prove that $AI^2 = AD \cdot AE$.

Let triangle $ABC$ have altitudes $BE$ and $CF$ which meet at $H$. The reflection of $A$ over $BC$ is $A'$. Let $(ABC)$ meet $(AA'E)$ at $P$ and $(AA'F)$ at $Q$. Let $BC$ meet $PQ$ at $R$. Prove that $EF \parallel HR$. [i]Proposed by Daniel Hu[/i]

Define a sequence ${(n_k)_{k\ge 0}}$ by ${n_{0 }= n_{1} = 1}$, and ${n_{2k} = n_k + n_{k-1} }$ and ${n_{2k+1} = n_k}$ for ${k \ge 1}$. Let further ${q_k = n_k }$ / ${ n_{k-1} }$ for each ${k \ge 1}$. Show that every positive rational number is present exactly once in the sequence ${(q_k)_{k\ge 1}}$

In the square $ABCD$, points $E$ and $F$ were chosen on the sides $AB$ and $BC$ respectively, such that $BE=BF$. Let $L$ be midpoint of $EF$, $N$ be midpoint of $DF$, $O$ be the center of the square and $K=AL \cap DF$ (look at picture). Prove that points $C, K, L, O, N$ are lies on one circle. [i]A. Paleev[/i]

Determine all rational numbers $a$ for which the matrix $$\begin{pmatrix} a & -a & -1 & 0 \\ a & -a & 0 & -1 \\ 1 & 0 & a & -a\\ 0 & 1 & a & -a \end{pmatrix}$$ is the square of a matrix with all rational entries. [i]Proposed by Daniël Kroes, University of California, San Diego[/i]

Suppose that for a prime number $p$ and integers $a,b,c$ the following holds: \[6\mid p+1,\quad p\mid a+b+c,\quad p\mid a^4+b^4+c^4.\] Prove that $p\mid a,b,c$.

[b]p1.[/b] The length of the first side of a triangle is $1$, the length of the second side is $11$, and the length of the third side is an integer. Find that integer. [b]p2.[/b] Suppose $a, b$, and $c$ are positive numbers such that $a + b + c = 1$. Prove that $ab + ac + bc \le \frac13$. [b]p3.[/b] Prove that $1 +\frac12+\frac13+\frac14+ ... +\frac{1}{100}$ is not an integer. [b]p4.[/b] Find all of the four-digit numbers n such that the last four digits of $n^2$ coincide with the digits of $n$. [b]p5.[/b] (Bonus) Several ants are crawling along a circle with equal constant velocities (not necessarily in the same direction). If two ants collide, both immediately reverse direction and crawl with the same velocity. Prove that, no matter how many ants and what their initial positions are, they will, at some time, all simultaneously return to the initial positions. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Prove that for every positive integer $n$, there exist $n$ positive integers such that the sum of them is a perfect square and the product of them is a perfect cube.

If the tangents of all interior angles of a triangle are integers, what is the sum of these integers? $ \textbf{(A)}\ 4 \qquad\textbf{(B)}\ 5 \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}\ 9 \qquad\textbf{(E)}\ \text{None of above} $

Let ${ABC}$ be a triangle, ${k}$ its incircle and ${k_a,k_b,k_c}$ three circles orthogonal to ${k}$ passing through ${B}$ and ${C, A}$ and ${C}$ , and ${A}$ and ${B}$ respectively. The circles ${k_a,k_b}$ meet again in ${C'}$ ; in the same way we obtain the points ${B'}$ and ${A'}$ . Prove that the radius of the circumcircle of ${A'B'C'}$ is half the radius of ${k}$ .

Find all pairs $(x, y)$ of integers satisfying the following condition: each of the numbers $x^3 + y$ and $x + y^3$ is divisible by $x^2 + y^2$ . (S Zlobin)

Joe makes two cubes of sidelengths $9$ and $10$ from $1729$ randomly oriented and randomly arranged unit cubes, which are initially unpainted. These cubes are dipped into white paint. Then two cubes of sidelengths $1$ and $12$ are formed from the same unit cubes, again randomly oriented and randomly arranged, and these cubes are dipped into paint remover. Joe continues to alternately dip cubes of sides $9$ and $10$ into paint and cubes of sides $1$ and $12$ into paint remover ad nauseam. What is the limit of the expected number of painted unit cube faces immediately after dipping in paint remover?

For which $n{}$ is it possible that a product of $n{}$ consecutive positive integers is equal to a sum of $n{}$ consecutive (not necessarily the same) positive integers? [i]Boris Frenkin[/i]

Fix an odd integer $n > 1$. For a permutation $p$ of the set $\{1,2,...,n\}$, let S be the number of pairs of indices $(i, j)$, $1 \le i \le j \le n$, for which $p_i +p_{i+1} +...+p_j$ is divisible by $n$. Determine the maximum possible value of $S$. Croatia