Given an $ABC$ acute triangle with $O$ the center of the circumscribed circle. Suppose that $\omega$ is a circle that is tangent to the line $AO$ at point $A$ and also tangent to the line $BC$. Prove that $\omega$ is also tangent to the circumcircle of the triangle $BOC$.

Several pounamu stones weigh altogether $10$ tons and none of them weigh more than $1$ tonne. A truck can carry a load which weight is at most $3$ tons. What is the smallest number of trucks such that bringing all stones from the quarry will be guaranteed?

Let $f_{0}(x)=x$, and for each $n\geq 0$, let $f_{n+1}(x)=f_{n}(x^{2}(3-2x))$. Find the smallest real number that is at least as large as \[ \sum_{n=0}^{2017} f_{n}(a) + \sum_{n=0}^{2017} f_{n}(1-a)\] for all $a \in [0,1]$.

A $3\times3$ grid is to be painted with three colors (red, green, and blue) such that [list=i] [*] no two squares that share an edge are the same color and [*] no two corner squares on the same edge of the grid have the same color. [/list] As an example, the upper-left and bottom-left squares cannot both be red, as that would violate condition (ii). In how many ways can this be done? (Rotations and reflections are considered distinct colorings.)

Let $Q$ be a set of prime numbers, not necessarily finite. For a positive integer $n$ consider its prime factorization: define $p(n)$ to be the sum of all the exponents and $q(n)$ to be the sum of the exponents corresponding only to primes in $Q$. A positive integer $n$ is called [i]special[/i] if $p(n)+p(n+1)$ and $q(n)+q(n+1)$ are both even integers. Prove that there is a constant $c>0$ independent of the set $Q$ such that for any positive integer $N>100$, the number of special integers in $[1,N]$ is at least $cN$. (For example, if $Q=\{3,7\}$, then $p(42)=3$, $q(42)=2$, $p(63)=3$, $q(63)=3$, $p(2022)=3$, $q(2022)=1$.)

Let $r$ and $b$ be positive integers. The game of [i]Monis[/i], a variant of Tetris, consists of a single column of red and blue blocks. If two blocks of the same color ever touch each other, they both vanish immediately. A red block falls onto the top of the column exactly once every $r$ years, while a blue block falls exactly once every $b$ years. (a) Suppose that $r$ and $b$ are odd, and moreover the cycles are offset in such a way that no two blocks ever fall at exactly the same time. Consider a period of $rb$ years in which the column is initially empty. Determine, in terms of $r$ and $b$, the number of blocks in the column at the end. (b) Now suppose $r$ and $b$ are relatively prime and $r+b$ is odd. At time $t=0$, the column is initially empty. Suppose a red block falls at times $t = r, 2r, \dots, (b-1)r$ years, while a blue block falls at times $t = b, 2b, \dots, (r-1)b$ years. Prove that at time $t=rb$, the number of blocks in the column is $\left\lvert 1+2(r-1)(b+r)-8S \right\rvert$, where \[ S = \left\lfloor \frac{2r}{r+b} \right\rfloor + \left\lfloor \frac{4r}{r+b} \right\rfloor + ... + \left\lfloor \frac{(r+b-1)r}{r+b} \right\rfloor . \] [i]Proposed by Sammy Luo[/i]

Vitya wrote down $n{}$ different natural numbers in his notebook. For each pair of numbers from the notebook, he wrote out their smallest common multiple on the board. Could it happen for some $n>100$ that $n(n-1)/2$ numbers on the board are (in some order) consecutive terms of a non-constant arithmetic progression? [i]Proposed by S. Berlov[/i]

Prove that in all triangles $ABC$ with $\angle A = 2\angle B$ the distance from $C$ to $A$ and to the perpendicular bisector of $AB$ are in the same ratio.

Mojtaba and Hooman are playing a game. Initially Mojtaba draws $2018$ vectors with zero sum. Then in each turn, starting with Mojtaba, the player takes a vector and puts it on the plane. After the first move, the players must put their vector next to the previous vector (the beginning of the vector must lie on the end of the previous vector). At last, there will be a closed polygon. If this polygon is not self-intersecting, Mojtaba wins. Otherwise Hooman. Who has the winning strategy? [i]Proposed by Mahyar Sefidgaran, Jafar Namdar [/i]

Let $f: \mathbb {R} \to \mathbb {R}$ be a concave function and $g: \mathbb {R} \to \mathbb {R}$ be a continuous function . If $$ f (x + y) + f (x-y) -2f (x) = g (x) y^2 $$for all $x, y \in \mathbb {R}, $ prove that $f $ is a second degree polynomial.

Given $101$ distinct non-negative integers less than $5050$ show that one can choose four $a, b, c, d$ such that $a + b - c - d$ is a multiple of $5050$

The sum of six consecutive positive integers is 2013. What is the largest of these six integers? $\textbf{(A)}\ 335 \qquad \textbf{(B)}\ 338 \qquad \textbf{(C)}\ 340 \qquad \textbf{(D)}\ 345 \qquad \textbf{(E)}\ 350$

Determine all pairs of integer numbers $(x, y)$ such that: $$\frac{(x - y)^2}{x + y} = x - y + 6$$

Can every polynomial with integer coefficients be expressed as a sum of cubes of polynomials with integer coefficients? [hide]I found the following statement that can be linked to this problem: "It is easy to see that every polynomial in F[x] is sum of cubes if char (F)$\ne$3 and card (F)=2,4"[/hide]

Determine the locus of orthocenters of triangles, given the midpoint of a side and the feet of the altitudes drawn on two other sides.

A stack of $2000$ cards is labelled with the integers from $1$ to $2000,$ with different integers on different cards. The cards in the stack are not in numerical order. The top card is removed from the stack and placed on the table, and the next card is moved to the bottom of the stack. The new top card is removed from the stack and placed on the table, to the right of the card already there, and the next card in the stack is moved to the bottom of the stack. The process - placing the top card to the right of the cards already on the table and moving the next card in the stack to the bottom of the stack - is repeated until all cards are on the table. It is found that, reading from left to right, the labels on the cards are now in ascending order: $1,2,3,\ldots,1999,2000.$ In the original stack of cards, how many cards were above the card labelled 1999?

A chess king tours an entire $8\times 8$ chess board, visiting each square exactly once and returning at last to his starting position. Prove that he made an even number of diagonal moves. (V Proizvolov)

What's the smallest positive integer \( n > 3 \), for which there does [b]not[/b] exist a (not necessarily convex) \( n \)-gon such that all its diagonals have equal lengths? A diagonal of any polygon is defined as a segment connecting any two non-adjacent vertices of the polygon. [i]Proposed by Anton Trygub[/i]

For real numbers $w$ and $z$, \[ \frac{\frac{1}{w} + \frac{1}{z}}{\frac{1}{w} - \frac{1}{z}} = 2014. \] What is $\tfrac{w+z}{w-z}$ ? ${ \textbf{(A)}\ \ -2014\qquad\textbf{(B)}\ \frac{-1}{2014}\qquad\textbf{(C)}\ \frac{1}{2014}\qquad\textbf{(D)}}\ 1\qquad\textbf{(E)}\ 2014$

Let $O$ be an interior point in the equilateral triangle $ABC$, of side length $a$. The lines $AO, BO$, and $CO$ intersect the sides of the triangle in the points $A_1, B_1$, and $C_1$. Show that $OA_1 + OB_1 + OC_1 < a$.

Given 98 points in a circle. Mary and Joseph play alternatively in the next way: - Each one draw a segment joining two points that have not been joined before. The game ends when the 98 points have been used as end points of a segments at least once. The winner is the person that draw the last segment. If Joseph starts the game, who can assure that is going to win the game.

Let $ABC$ be a triangle with $AB\neq CB$. Let $C^{\prime}$ be a point on the ray $[AB$ such that $AC^{\prime}=CB$. Let $A^{\prime}$ be a point on the ray $[CB$ such that $CA^{\prime}=AB$. Let the circumcircles of triangles $ABA^{\prime}$ and $CBC^{\prime}$ intersect at a point $Q$ (apart from $B$). Prove that the line $BQ$ bisects the segment $CA$. Darij

Find the greatest prime factor of the sum of the two largest two-digit prime numbers.

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

Find the largest prime $p$ less than $210$ such that the number $210 - p$ is composite.