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]

Let $ABCD$ be a square. Let $P$ be a point on the semicircle of diameter $AB$ outside the square. Let $M$ and $N$ be the intersections of $PD$ and $PC$ with $AB$, respectively. Prove that $MN^2 = AM \cdot BN$.

Find all finite sets $S$ of points in the plane with the following property: for any three distinct points $A,B,$ and $C$ in $S,$ there is a fourth point $D$ in $S$ such that $A,B,C,$ and $D$ are the vertices of a parallelogram (in some order).

Each cell of an $n \times n$ board is colored either black or white. A coloring is called [i]good[/i] if every $2 \times 2$ square contains an even number of black cells, and every cross contains an odd number of black cells. Determine all $n \geqslant 3$ such that, in every good coloring, the four corner cells of the board are the same color. [b]Note:[/b] Each $2 \times 2$ square contains exactly $4$ cells of the board. Each cross contains exactly $5$ cells of the board. [asy] size(5cm); // Function to draw a filled square centered at a given position void drawFilledSquare(pair center, real sideLength) { real halfSide = sideLength / 2; fill(shift(center) * box((-halfSide, -halfSide), (halfSide, halfSide)), lightgray); draw(shift(center) * box((-halfSide, -halfSide), (halfSide, halfSide))); } // Side length of each square real sideLength = 1; // Coordinates for the cross (left shape) pair[] crossPositions = { (0, 0), (-1, 0), (1, 0), (0, -1), (0, 1) }; // Coordinates for the square (right shape) pair[] squarePositions = { (3, -0.5), (3, 0.5), (4, -0.5), (4, 0.5) }; // Draw the cross for (pair pos : crossPositions) { drawFilledSquare(pos, sideLength); } // Draw the square for (pair pos : squarePositions) { drawFilledSquare(pos, sideLength); } [/asy]

Square $ABCD$ is shown in the diagram below. Points $E$, $F$, and $G$ are on sides $\overline{AB}$, $\overline{BC}$ and $\overline{DA}$, respectively, so that lengths $\overline{BE}$, $\overline{BF}$, and $\overline{DG}$ are equal. Points $H$ and $I$ are the midpoints of segments $\overline{EF}$ and $\overline{CG}$, respectively. Segment $\overline{GJ}$ is the perpendicular bisector of segment $\overline{HI}$. The ratio of the areas of pentagon $AEHJG$ and quadrilateral $CIHF$ can be written as $\dfrac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$. [asy] draw((0,0)--(50,0)--(50,50)--(0,50)--cycle); label("$A$",(0,50),NW); label("$B$",(50,50),NE); label("$C$",(50,0),SE); label("$D$",(0,0),SW); label("$E$",(0,100/3-1),W); label("$F$",(100/3-1,0),S); label("$G$",(20,50),N); label("$H$",((100/3-1)/2,(100/3-1)/2),SW); label("$I$",(35,25),NE); label("$J$",(((100/3-1)/2+35)/2,((100/3-1)/2+25)/2),S); draw((0,100/3-1)--(100/3-1,0)); draw((20,50)--(50,0)); draw((100/6-1/2,100/6-1/2)--(35,25)); draw((((100/3-1)/2+35)/2,((100/3-1)/2+25)/2)--(20,50)); [/asy]

In the cyclic quadrilateral $ABCD$, the diagonals $AC$ and $BD$ intersect at $P$. Let $O$ be the center of the circumcircle $ABCD$, and $E$ a point of the extension of $OC$ beyond $C$. A parallel line to $CD$ is drawn through $E$ that cuts the extension of $OD$ beyonf $D$ at $F$. Let $Q$ be a point interior to $ABCD$, such that $\angle AFQ = \angle BEQ$ and $\angle FAQ = \angle EBQ$. Prove that $PQ \perp CD$.

Let $G$ be a simple graph on $n$ vertices with at least one edge, and let us consider those $S:V(G)\to\mathbb R^{\ge 0}$ weighings of the vertices of the graph for which $\sum_{v\in V(G)} S(v)=1$. Furthermore define \[f(G)=\max_S\min_{(v,w)\in E(G)}S(v)S(w),\] where $S$ runs through all possible weighings. Prove that $f(G)=\frac1{n^2}$ if and only if the vertices of $G$ can be covered with a disjoint union of edges and odd cycles. ($V(G)$ denotes the vertices of graph $G$, $E(G)$ denotes the edges of graph $G$.)

If $a$, $b$, $c$ are positive reals such that $a+b+c=1$, prove that \[\frac{1+a}{1-a}+\frac{1+b}{1-b}+\frac{1+c}{1-c}\leq 2\left(\frac{b}{a}+\frac{c}{b}+\frac{a}{c}\right).\]

Given a triangle $ABC$, $BC = 2 \cdot AC$. Point $M$ is the midpoint of side $ BC$ and point $D$ lies on $AB$, with $AD = 2 \cdot BD$. Prove that the lines $AM$ and $MD$ are perpendicular.

Famous French mathematician Pierre Fermat believed that all numbers of the form $F_n = 2^{2^n} + 1$ are prime for all non-negative integers $n$. Indeed, one can check that $F_0 = 3$, $F_1 = 5$, $F_2 = 17$, $F_3 = 257$ are all prime. a) Prove that $F_5$ is divisible by $641$. (Hence Fermat was wrong.) b) Prove that if $k \ne n$ then $F_k$ and $F_n$ are relatively prime (i.e. they do not have any common divisor except $1$) (Notice: using b) one can prove that there are infinitely many prime numbers)

Let $n \geq 3$ be a positive integer. Find the maximum number of diagonals in a regular $n$-gon one can select, so that any two of them do not intersect in the interior or they are perpendicular to each other.

Find the smallest constant $C > 0$ for which the following statement holds: among any five positive real numbers $a_1,a_2,a_3,a_4,a_5$ (not necessarily distinct), one can always choose distinct subscripts $i,j,k,l$ such that \[ \left| \frac{a_i}{a_j} - \frac {a_k}{a_l} \right| \le C. \]

Let $Q(x)$ be a cubic polynomial with integer coefficients. Suppose that a prime $p$ divides $Q(x_j)$ for $j = 1$ ,$2$ ,$3$ ,$4$ , where $x_1 , x_2 , x_3 , x_4$ are distinct integers from the set $\{0,1,\cdots, p-1\}$. Prove that $p$ divides all the coefficients of $Q(x)$.