Given a convex quadrilateral $ABCD$ such that $m(\widehat{ABC})=m(\widehat{BCD})$. The lines $AD$ and $BC$ intersect at a point $P$ and the line passing through $P$ which is parallel to $AB$, intersects $BD$ at $T$. Prove that $$m(\widehat{ACB})=m(\widehat{PCT})$$

Euclid has a tool called [i]cyclos[/i] which allows him to do the following: [list] [*] Given three non-collinear marked points, draw the circle passing through them. [*] Given two marked points, draw the circle with them as endpoints of a diameter. [*] Mark any intersection points of two drawn circles or mark a new point on a drawn circle. [/list] Show that given two marked points, Euclid can draw a circle centered at one of them and passing through the other, using only the cyclos. [i]Proposed by Rohan Goyal, Anant Mudgal, and Daniel Hu[/i]

If $f(x)=e^xg(x)$, where $g(2)=1$ and $g'(2)=2$, find $f'(2)$.

Four distinct points are marked in a line. For each point, the sum of the distances from said point to the other three is calculated; getting in total 4 numbers. Decide whether these 4 numbers can be, in some order: a) $29,29,35,37$ b) $28,29,35,37$ c) $28,34,34,37$

Given is a board $n \times n$ and in every square there is a checker. In one move, every checker simultaneously goes to an adjacent square (two squares are adjacent if they share a common side). In one square there can be multiple checkers. Find the minimum and the maximum number of covered cells for $n=5, 6, 7$.

Find the best approximation of $\sqrt{3}$ by a rational number with denominator less than or equal to $15$

Let $f : \mathbb{R} \to \mathbb{R}$ be a function whose second derivative is continuous. Suppose that $f$ and $f''$ are bounded. Show that $f'$ is also bounded.

Let $ABC$ be a triangle with $ \angle ACB = 90^o + \frac12 \angle ABC$ . The point $M$ is the midpoint of the side $BC$ . A circle with center at vertex $A$ intersects the line $BC$ at points $M$ and $D$. Prove that $MD = AB$.

An ordinary clock in a factory is running slow so that the minute hand passes the hour hand at the usual dial position($12$ o'clock, etc.) but only every $69$ minutes. At time and one-half for overtime, the extra pay to which a $\textdollar 4.00$ per hour worker should be entitled after working a normal $8$ hour day by that slow running clock, is $\textbf{(A) }\textdollar 2.30\qquad\textbf{(B) }\textdollar 2.60\qquad\textbf{(C) }\textdollar 2.80\qquad\textbf{(D) }\textdollar 3.00\qquad \textbf{(E) }\textdollar 3.30$

Find all functions $R-->R$ such that: $f(x^2) - f(y^2) + 2x + 1 = f(x + y)f(x - y)$

Two teams, $ A$ and $ B$, fight for a territory limited by a circumference. $ A$ has $ n$ blue flags and $ B$ has $ n$ white flags ($ n\geq 2$, fixed). They play alternatively and $ A$ begins the game. Each team, in its turn, places one of his flags in a point of the circumference that has not been used in a previous play. Each flag, once placed, cannot be moved. Once all $ 2n$ flags have been placed, territory is divided between the two teams. A point of the territory belongs to $ A$ if the closest flag to it is blue, and it belongs to $ B$ if the closest flag to it is white. If the closest blue flag to a point is at the same distance than the closest white flag to that point, the point is neutral (not from $ A$ nor from $ B$). A team wins the game is their points cover a greater area that that covered by the points of the other team. There is a draw if both cover equal areas. Prove that, for every $ n$, team $ B$ has a winning strategy.

Let $ABC$ be an acute triangle inscribed in a circle of center $O$. If the altitudes $BD,CE$ intersect at $H$ and the circumcenter of $\triangle BHC$ is $O_1$, prove that $AHO_1O$ is a parallelogram.

Let $n\geqslant 2$ be a positive integer and $a_1,a_2, \ldots ,a_n$ be real numbers such that \[a_1+a_2+\dots+a_n=0.\] Define the set $A$ by \[A=\left\{(i, j)\,|\,1 \leqslant i<j \leqslant n,\left|a_{i}-a_{j}\right| \geqslant 1\right\}\] Prove that, if $A$ is not empty, then \[\sum_{(i, j) \in A} a_{i} a_{j}<0.\]

Find the maximum number of cells that can be coloured from a $4\times 3000$ board such that no tetromino is formed. [i]Proposed by Arian Zamani, Matin Yousefi[/i] [b]Rated 5[/b]

Let $a_1, a_2, \ldots , a_n, \ldots $ be a sequence of real numbers such that $0 \leq a_n \leq 1$ and $a_n - 2a_{n+1} + a_{n+2} \geq 0$ for $n = 1, 2, 3, \ldots$. Prove that \[0 \leq (n + 1)(a_n - a_{n+1}) \leq 2 \qquad \text{ for } n = 1, 2, 3, \ldots\]

Find the smallest positive real constant $a$, such that for any three points $A,B,C$ on the unit circle, there exists an equilateral triangle $PQR$ with side length $a$ such that all of $A,B,C$ lie on the interior or boundary of $\triangle PQR$.

Denote the set of all positive integers by $\mathbb{N}$, and the set of all ordered positive integers by $\mathbb{N}^2$. For all non-negative integers $k$, define [i]good functions of order k[/i] recursively for all non-negative integers $k$, among all functions from $\mathbb{N}^2$ to $\mathbb{N}$ as follows: (i) The functions $f(a,b)=a$ and $f(a,b)=b$ are both good functions of order $0$. (ii) If $f(a,b)$ and $g(a,b)$ are good functions of orders $p$ and $q$, respectively, then $\gcd(f(a,b),g(a,b))$ is a good function of order $p+q$, while $f(a,b)g(a,b)$ is a good function of order $p+q+1$. Prove that, if $f(a,b)$ is a good function of order $k\leq \binom{n}{3}$ for some positive integer $n\geq 3$, then there exist a positive integer $t\leq \binom{n}{2}$ and $t$ pairs of non-negative integers $(x_1,y_1),\ldots,(x_n,y_n)$ such that $$f(a,b)=\gcd(a^{x_1}b^{y_1},\ldots,a^{x_t}b^{y_t})$$ holds for all positive integers $a$ and $b$. [i]Proposed by usjl[/i]

Find all natural numbers $n$ for which there is a permutation $\sigma$ of $\{1,2,\ldots, n\}$ that satisfies: \[ \sum_{i=1}^n \sigma(i)(-2)^{i-1}=0 \]

Let $$\frac{r}{s}= 0.k_1k_2k_3 ...$$ be the decimal expansion of a rational number (If this is a terminating decimal, all $k_i$ from a certain one on are $0$). Prove that at least two of the numbers $$\sigma_1 = 10\frac{r}{s} - k_i, \sigma_2 = 10^2- (10k_1 + k_2),$$ $$\sigma_3 = 10^2 - (10^2k_1 + 10k_2 + k_3), ...$$ are equal.

The two circles ${{w} _ {1}}, \, \, {{w} _ {2}}$ touch externally at the point $Q$. The common external tangent of these circles is tangent to ${{w} _ {1}}$ at the point $B$, $BA$ is the diameter of this circle. A tangent to the circle ${{w} _ {2}} $ is drawn through the point $A$, which touches this circle at the point $C$, such that the points $B$ and $C$ lie in one half-plane relative to the line $AQ$. Prove that the circle ${{w} _ {1}}$ bisects the segment $C $. (Igor Nagel)

Let $p$ and $a$ be positive integer numbers having no common divisors except of $1$. Prove that $p$ is prime if and only if all the coefficients of the polynomial \[ F(x) = (x-a)^p - (x^p - a) \] are divisible by $p$.

Esmeralda writes $2n$ real numbers $x_1, x_2, \dots , x_{2n}$, all belonging to the interval $[0, 1]$, around a circle and multiplies all the pairs of numbers neighboring to each other, obtaining, in the counterclockwise direction, the products $p_1 = x_1x_2$, $p_2 = x_2x_3$, $\dots$ , $p_{2n} = x_{2n}x_1$. She adds the products with even indices and subtracts the products with odd indices. What is the maximum possible number Esmeralda can get?

Two non-decreasing sequences of nonnegative integers have different first terms. Each sequence has the property that each term beginning with the third is the sum of the previous two terms, and the seventh term of each sequence is $N$. What is the smallest possible value of $N$? ${ \textbf{(A)}\ 55\qquad\textbf{(B)}\ 89\qquad\textbf{(C)}\ 104\qquad\textbf{(D}}\ 144\qquad\textbf{(E)}\ 273 $

Kimothy starts in the bottom-left square of a $4$ by $4$ chessboard. In one step, he can move up, down, left, or right to an adjacent square. Kimothy takes $16$ steps and ends up where he started, visiting each square exactly once (except for his starting/ending square). How many paths could he have taken?

Determine all polynomials $P{}$ with integer coefficients, satisfying $0 \leqslant P (n) \leqslant n!$ for all non-negative integers $n$. [i]Andrei Chirita[/i]