Let $ABCD$ be cyclic quadrilateral. Let $AC$ and $BD$ intersect at $R$, and let $AB$ and $CD$ intersect at $K$. Let $M$ and $N$ are points on $AB$ and $CD$ such that $\frac{AM}{MB}=\frac{CN}{ND}$. Let $P$ and $Q$ be the intersections of $MN$ with the diagonals of $ABCD$. Prove that circumcircles of triangles $KMN$ and $PQR$ are tangent at a fixed point.

Let $\mathcal{C}$ be the circumcircle of the square with vertices $(0, 0), (0, 1978), (1978, 0), (1978, 1978)$ in the Cartesian plane. Prove that $\mathcal{C}$ contains no other point for which both coordinates are integers.

Marisa has a collection of $2^8-1=255$ distinct nonempty subsets of $\{1, 2, 3, 4, 5, 6, 7, 8\}$. For each step she takes two subsets chosen uniformly at random from the collection, and replaces them with either their union or their intersection, chosen randomly with equal probability. (The collection is allowed to contain repeated sets.) She repeats this process $2^8-2=254$ times until there is only one set left in the collection. What is the expected size of this set?

Let $P$ be a fourth degree polynomial, with derivative $P^\prime$, such that $P(1)=P(3)=P(5)=P^\prime (7)=0$. Find the real number $x\neq 1,3,5$ such that $P(x)=0$.

Let $ABC$ be a triangle with angles $\angle A = 80^\circ, \angle B = 70^\circ, \angle C = 30^\circ$. Let $P$ be a point on the bisector of $\angle BAC$ satisfying $\angle BPC =130^\circ$. Let $PX, PY, PZ$ be the perpendiculars drawn from $P$ to the sides $BC, AC, AB$, respectively. Prove that the following equation with segment lengths is satisfied $$AY^3+BZ^3+CX^3=AZ^3+BX^3+CY^3.$$

Complex numbers $\alpha$ , $\beta$ , $\gamma$ have the property that $\alpha^k +\beta^k +\gamma^k$ is an integer for every natural number $k$. Prove that the polynomial \[(x-\alpha)(x-\beta )(x-\gamma )\] has integer coefficients.

Let $ABC$ be a right-angled triangle where ∠$ACB$=90°.Let $CD$ be an altitude of that triangle and points $M$ and $N$ be the midpoints of $CD$ and $BC$, respectively.If $S$ is the circumcenter of the triangle $AMN$, prove that $AS$ and $BC$ are paralel.

Which positive integers $m$ are such that $k^m - 1$ is divisible by $2^m$ for all odd numbers $k \ge 3$?

Compute $ \arctan\left(\tan65^\circ \minus{} 2\tan40^\circ\right)$. (Express your answer in degrees.)

Let $ABC$ be a triangle and $P$ be any point on the side $BC$. Let $I_1$,$I_2$ be the incenters of triangles $ABP$ and $ACP$, respectively. If $D$ is the point of tangency of the incircle of $ABC$ with the side $BC$, prove that $\angle I_1DI_2 = 90^o$.

The product $55\cdot60\cdot65$ is written as a product of 5 distinct numbers. Find the least possible value of the largest number, among these 5 numbers.

Let $n$ be a positive integer. We start with $n$ piles of pebbles, each initially containing a single pebble. One can perform moves of the following form: choose two piles, take an equal number of pebbles from each pile and form a new pile out of these pebbles. Find (in terms of $n$) the smallest number of nonempty piles that one can obtain by performing a finite sequence of moves of this form.

Find all $C\in \mathbb{R}$ such that every sequence of integers $\{a_n\}_{n=1}^{\infty}$ which is bounded from below and for all $n\geq 2$ satisfy $$0\leq a_{n-1}+Ca_n+a_{n+1}<1$$ is periodic. Proposed by Navid Safaei

Let $\triangle ABC$ be such that the midpoint of $BC$ is $D$. Let $E$ be the point on the opposite side of $AC$ as $B$ on the circumcircle of $\triangle ABC$ such that $\angle DEA = \angle DEC$ and let $\omega$ be the circumcircle of $\triangle CED$. If $\omega$ intersects $AE$ at $X$ and the tangent to $\omega$ at $D$ intersects $AB$ at $Y$, show that $XY$ is parallel to $BC$. [i]Proposed by Taco12[/i]

Given is triangle $ABC$ with centroid $T$, such that $\angle BAC + \angle BTC = 180^\circ$. Let $G$ and $H$ be the second points of intersection of lines $CT$ and $BT$ with the circumcircle of triangle $ABC$, respectively. Prove that the line $GH$ is tangent to the Euler circle of triangle $ABC$. [i]Proposed by Andrija Živadinović[/i]

How many complex numbers $z$ such that $\left| z \right| < 30$ satisfy the equation \[ e^z = \frac{z - 1}{z + 1} \, ? \]

Let $a, n \in \mathbb{Z}^{*}_{+}$. $a$ is defined inductively in the base $n$-[i]recursive[/i]. We first write $a$ in the base $n$, e.g., as a sum of terms of the form $k_tn^t$, with $0 \le k_t < n$. For each exponent $t$, we write $t$ in the base $n$-[i]recursive[/i], until all the numbers in the representation are less than $n$. For instance, $1309 = 3^6 + 2.3^5 + 1.3^4 + 1.3^2 + 1.3 + 1$ $ = 3^{2.3} + 2.3^{3+2} + 1.3^{3+1} + 1.3^2 + 1$ Let $x_1 \in \mathbb{Z}$ arbitrary. We define $x_n$ recursively, as following: if $x_{n-1} > 0$, we write $x_{n-1}$ in the base $n$-[i]recursive[/i] and we replace all the numbers $n$ for $n+1$ (even the exponents!), so we obtain the successor of $x_n$. If $x_{n-1} = 0$, then $x_n = 0$. Example: $x_1 = 2^{2^{2} + 2 + 1} + 2^{2+1} + 2 + 1$ $\Rightarrow x_2 = 3^{3^{3} + 3 + 1} + 3^{3+1} + 3$ $\Rightarrow x_3 = 4^{4^{4} + 4 + 1} + 4^{4+1} + 3$ $\Rightarrow x_4 = 5^{5^{5} + 5 + 1} + 5^{5+1} + 2$ $\Rightarrow x_5 = 6^{6^{6} + 6 + 1} + 6^{6+1} + 1$ $\Rightarrow x_6 = 7^{7^{7} + 7 + 1} + 7^{7+1}$ $\Rightarrow x_7 = 8^{8^{8} + 8 + 1} + 7.8^8 + 7.8^7 + 7.8^6 + ... + 7$ $.$ $.$ $.$ Prove that $\exists N : x_N = 0$.

For which natural number $k$ does $\frac{k^2}{1.001^k}$ attain its maximum value?

Ali and Veli goes to hunting. The probability that each will successfully hit a duck is $1/2$ on any given shot. During the hunt, Ali shoots $12$ times, and Veli shoots $13$ times. What is the probability that Veli hits more ducks than Ali? $ \textbf{(A)}\ \dfrac 12 \qquad\textbf{(B)}\ \dfrac{13}{25} \qquad\textbf{(C)}\ \dfrac{13}{24} \qquad\textbf{(D)}\ \dfrac{7}{13} \qquad\textbf{(E)}\ \dfrac{3}{4} $

For two real numbers $x$ and $y$, let $x\circ y=\frac{xy}{x+y}$. The value of \[1 \circ (2 \circ (3 \circ (4 \circ 5)))\] can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m+n$. [i]Proposed by Nathan Xiong[/i]

Let $*$ be a commutative and associative binary operation on a set $S.$ Assume that for every $x$ and $y$ in $S,$ there exists $z$ in $S$ such that $x*z=y.$ (This $z$ may depend on $x$ and $y.$) Show that if $a,b,c$ are in $S$ and $a*c=b*c,$ then $a=b.$

a) Let $a, b, c$ be positive integers such that $ab + b + 1$, $bc + c + 1$ and $ca + a + 1$ are divisors of the number $abc - 1$, prove that $a = b = c$. b) Find all triples $(a, b, c)$ of positive integers such that the product $$(ab - b + 1)(bc - c + 1)(ca - a + 1)$$ is a divisor of the number $(abc + 1)^2$.

Show that the triangle whose angles satisfy the equality \[\frac{\sin^2A+\sin^2B+\sin^2C}{\cos^2A+\cos^2B+\cos^2C} = 2\] is right angled.

Two circles, one inside the other, are given in the plane. Construct a point $O$, inside the inner circle, such that if a ray from $O$ cuts the circles at $A$ and $B$ respectively, then the ratio $OA/OB$ is constant. (Folklore)

Altitudes $AH1$ and $BH2$ of acute triangle $ABC$ intersect at $H$. Let $w1$ be the circle that goes through $H2$ and touches the line $BC$ at $H1$, and let $w2$ be the circle that goes through $H1$ and touches the line $AC$ at $H2$. Prove, that the intersection point of two other tangent lines $BX$ and $AY$( $X$ and $Y$ are different from $H1$ and $H2$) to circles $w1$ and $w2$ respectively, lies on the circumcircle of triangle $HXY$. Proposed by [i]Danilo Khilko[/i]