Let $ABC$ be a triangle, $P$ and $Q$ the intersections of the parallel line to $BC$ that passes through $A$ with the external angle bisectors of angles $B$ and $C$, respectively. The perpendicular to $BP$ at $P$ and the perpendicular to $CQ$ at $Q$ meet at $R$. Let $I$ be the incenter of $ABC$. Show that $AI = AR$.

Let $m,n\geq 2$ be positive integers and $S\subseteq [1,m]\times [1,n]$ be a set of lattice points. Prove that if \[|S|\geq m+n+\bigg\lfloor\frac{m+n}{4}-\frac{1}{2}\bigg\rfloor\]then there exists a circle which passes through at least four distinct points of $S.$

Two walls and the ceiling of a room meet at right angles at point $P$. A fly is in the air one meter from one wall, eight meters from the other wall, and $9$ meters from point $P$. How many meters is the fly from the ceiling? $\textbf{(A) }\sqrt{13}\qquad\textbf{(B) }\sqrt{14}\qquad\textbf{(C) }\sqrt{15}\qquad\textbf{(D) }4\qquad\textbf{(E) }\sqrt{17}$

Let be two quadrilaterals $ ABCD,A'B'C'D' $ with $ AB,BC,CD,AC,BD $ being perpendicular to $ A'B',B'C',C'D',A'C',B'D', $ respectively. Show that $ AD $ is perpendicular to $ A'D'. $

Show that $r = 2$ is the largest real number $r$ which satisfies the following condition: If a sequence $a_1$, $a_2$, $\ldots$ of positive integers fulfills the inequalities \[a_n \leq a_{n+2} \leq\sqrt{a_n^2+ra_{n+1}}\] for every positive integer $n$, then there exists a positive integer $M$ such that $a_{n+2} = a_n$ for every $n \geq M$.

Let$ p$ be an odd prime number. Determine the number of tuples $(a_1, a_2, . . . , a_p)$ of natural numbers with the following properties: 1) $1 \le ai \le p$ for all $i = 1, . . . , p$. 2) $a_1 + a_2 + · · · + a_p$ is not divisible by $p$. 3) $a_1a_2 + a_2a_3 + . . . +a_{p-1}a_p + a_pa_1$ is divisible by $p$.

Consider the following graph algorithm (where $V$ is the set of vertices and $E$ the set of edges in $G$). $\textbf{procedure }\textsc{s}(G)$ $\qquad \textbf{if } |V| = 0\textbf{ then return true}$ $\qquad \textbf{for }(u,v)\textbf{ in }E\textbf{ do}$ $\qquad\qquad H\gets G-u-v$ $\qquad\qquad\textbf{if } \textsc{s}(H)\textbf{ then return true}$ $\qquad\textbf{return false}$ Here $G - u - v$ means the subgraph of $G$ which does not contain vertices $u,v$ and all edges using them. How many graphs $G$ with vertex set $\{1,2,3,4,5,6\}$ and [i]exactly[/i] $6$ edges satisfy $s(G)$ being true?

A regular tetrahedron with $5$-inch edges weighs $2.5$ pounds. What is the weight in pounds of a similarly constructed regular tetrahedron that has $6$-inch edges? Express your answer as a decimal rounded to the nearest hundredth.

The area $T$ and an angle $\gamma$ of a triangle are given. Determine the lengths of the sides $a$ and $b$ so that the side $c$, opposite the angle $\gamma$, is as short as possible.

The vertices $X, Y , Z$ of an equilateral triangle $XYZ$ lie respectively on the sides $BC, CA, AB$ of an acute-angled triangle $ABC.$ Prove that the incenter of triangle $ABC$ lies inside triangle $XYZ.$ [i]Proposed by Nikolay Beluhov, Bulgaria[/i]

Find how many different solutions depending on $a$ has the system of equations : $$\begin{cases} x+z=2a \\ y+u+xz=a-3 \\ yz+xu=2a \\ yu=1 \end{cases}$$

The numbers $p,q>0$ are given. Construct a rectangle $ABCD$ with $AE=p,AF=q$ where $E,F$ are midpoints of $BC,CD,$ respectively. Discuss conditions of solvability.

A magician has $300$ cards with numbers from $1$ to $300$ written on them, each number on exactly one card. The magician then lays these cards on a $3 \times 100$ rectangle in the following way - one card in each unit square so that the number cannot be seen and cards with consecutive numbers are in neighbouring squares. Afterwards, the magician turns over $k$ cards of his choice. What is the smallest value of $k$ for which it can happen that the opened cards definitely determine the exact positions of all other cards?

Prove that every positive $x,y$ and real $a$ satisfy inequality $x^{\sin ^2a} y^{\cos^2a} < x + y$.

For $x, y$ positive integers, $x^2-4y+1$ is a multiple of $(x-2y)(1-2y)$. Prove that $|x-2y|$ is a square number.

Let $ABC$ be a triangle, $O$ its circumcenter, $S$ its centroid, and $H$ its orthocenter. Denote by $A_1, B_1$, and $C_1$ the centers of the circles circumscribed about the triangles $CHB, CHA$, and $AHB$, respectively. Prove that the triangle $ABC$ is congruent to the triangle $A_1B_1C_1$ and that the nine-point circle of $\triangle ABC$ is also the nine-point circle of $\triangle A_1B_1C_1$.

For how many ordered pairs of positive integers $(x, y)$ is $\frac{x^2+y^2}{x-y}$ an integer factor of $2310$?

Inside an acute angle is a circle. Investigate the possibility of constructing with only a compass and a ruler, a tangent to this circle that the point of contact will bisect the segment of the tangent that is cut off by the sides of the angle.

Find all pairs of prime numbers $p, q$ for which there exist positive integers $(m, n)$ such that $$(p+q)^m=(p-q)^n$$.

For a finite set $A$ of positive integers, a partition of $A$ into two disjoint nonempty subsets $A_1$ and $A_2$ is $\textit{good}$ if the least common multiple of the elements in $A_1$ is equal to the greatest common divisor of the elements in $A_2$. Determine the minimum value of $n$ such that there exists a set of $n$ positive integers with exactly $2015$ good partitions.

How many ways can $ 8$ mutually non-attacking rooks be placed on the $ 9\times9$ chessboard (shown here) so that all $ 8$ rooks are on squares of the same color? (Two rooks are said to be attacking each other if they are placed in the same row or column of the board.) [asy]unitsize(3mm); defaultpen(white); fill(scale(9)*unitsquare,black); fill(shift(1,0)*unitsquare); fill(shift(3,0)*unitsquare); fill(shift(5,0)*unitsquare); fill(shift(7,0)*unitsquare); fill(shift(0,1)*unitsquare); fill(shift(2,1)*unitsquare); fill(shift(4,1)*unitsquare); fill(shift(6,1)*unitsquare); fill(shift(8,1)*unitsquare); fill(shift(1,2)*unitsquare); fill(shift(3,2)*unitsquare); fill(shift(5,2)*unitsquare); fill(shift(7,2)*unitsquare); fill(shift(0,3)*unitsquare); fill(shift(2,3)*unitsquare); fill(shift(4,3)*unitsquare); fill(shift(6,3)*unitsquare); fill(shift(8,3)*unitsquare); fill(shift(1,4)*unitsquare); fill(shift(3,4)*unitsquare); fill(shift(5,4)*unitsquare); fill(shift(7,4)*unitsquare); fill(shift(0,5)*unitsquare); fill(shift(2,5)*unitsquare); fill(shift(4,5)*unitsquare); fill(shift(6,5)*unitsquare); fill(shift(8,5)*unitsquare); fill(shift(1,6)*unitsquare); fill(shift(3,6)*unitsquare); fill(shift(5,6)*unitsquare); fill(shift(7,6)*unitsquare); fill(shift(0,7)*unitsquare); fill(shift(2,7)*unitsquare); fill(shift(4,7)*unitsquare); fill(shift(6,7)*unitsquare); fill(shift(8,7)*unitsquare); fill(shift(1,8)*unitsquare); fill(shift(3,8)*unitsquare); fill(shift(5,8)*unitsquare); fill(shift(7,8)*unitsquare); draw(scale(9)*unitsquare,black);[/asy]

Let $ f(x)$and $ g(y)$ be two monic polynomials of degree=$ n$ having complex coefficients. We know that there exist complex numbers $ a_i,b_i,c_i \forall 1\le i \le n$, such that $ f(x)\minus{}g(y)\equal{}\prod_{i\equal{}1}^n{(a_ix\plus{}b_iy\plus{}c_i)}$. Prove that there exists $ a,b,c\in\mathbb{C}$ such that $ f(x)\equal{}(x\plus{}a)^n\plus{}c\text{ and }g(y)\equal{}(y\plus{}b)^n\plus{}c$.

A polynomial $f(x)$ with integer coefficients is called $\textit{splitty}$ if and only if for every prime $p$, there exists polynomials $g_p, h_p$ with integer coefficients with degrees strictly smaller than the the degree of $f$, such that all coefficients of $f-g_ph_p$ are divisible by $p$. Compute the sum of all positive integers $n \leq 100$ such that $x^4+16x^2+n$ is $\textit{splitty}$.

Among all sets of real numbers $\{ x_1 , x_2 , ... , x_{20} \}$ from the open interval $(0, 1 )$ such that $$x_1x_2...x_{20}= ( 1 - x_1 ) ( 1 -x_2 ) ... (1 - x_{20} )$$ find the one for which $x_1 x_2... x_{20}$ is maximal. (A Cherniatiev)

The positive integers are grouped as follows: $1, 2+3, 4+5+6, 7+8+9+10,\dots$. Find the value of the $n$-th sum.