Given three positive real numbers $a,b,c$ such that following holds $a^2=b^2+bc$, $b^2=c^2+ac$ Prove that $\frac{1}{c}=\frac{1}{a}+\frac{1}{b}$.

Given a circle and a point $K$ inside it. An arbitrary circle equal to the given one and passing through the point $K$ has a common chord with the given circle. Find the geometric locus of the midpoints of these chords.

Given that $P(x)$ is the least degree polynomial with rational coefficients such that \[P(\sqrt{2} + \sqrt{3}) = \sqrt{2},\] find $P(10)$.

Consider a regular triangular pyramid with base $\vartriangle ABC$ and apex $D$. If we have $AB = BC =AC = 6$ and $AD = BD = CD = 4$, calculate the surface area of the circumsphere of the pyramid.

Find the number of integers $n$ with $n \ge 2$ such that the remainder when $2013$ is divided by $n$ is equal to the remainder when $n$ is divided by $3$. [i]Proposed by Michael Kural[/i]

Each valve $ A$, $ B$, and $ C$, when open, releases water into a tank at its own constant rate. With all three valves open, the tank fills in 1 hour, with only valves $ A$ and $ C$ open it takes 1.5 hours, and with only valves $ B$ and $ C$ open it takes 2 hours. The number of hours required with only valves $ A$ and $ B$ open is $ \textbf{(A)}\ 1.1 \qquad \textbf{(B)}\ 1.15 \qquad \textbf{(C)}\ 1.2 \qquad \textbf{(D)}\ 1.25 \qquad \textbf{(E)}\ 1.75$

Consider the following equation in $x$: $$ax (x^2 + ax + 1) = b (x^2 + b + 1).$$ It is known that $a, b$ are real such that $ab <0$ and furthermore the equation has exactly two integer roots positive. Prove that under these conditions $a^2 + b^2$ is not a prime number.

Semicircle $\Gamma$ has diameter $\overline{AB}$ of length $14$. Circle $\Omega$ lies tangent to $\overline{AB}$ at a point $P$ and intersects $\Gamma$ at points $Q$ and $R$. If $QR=3\sqrt3$ and $\angle QPR=60^\circ$, then the area of $\triangle PQR$ is $\frac{a\sqrt{b}}{c}$, where $a$ and $c$ are relatively prime positive integers, and $b$ is a positive integer not divisible by the square of any prime. What is $a+b+c$? $\textbf{(A) }110 \qquad \textbf{(B) }114 \qquad \textbf{(C) }118 \qquad \textbf{(D) }122\qquad \textbf{(E) }126$

Find the largest integer $n \ge 3$ for which there is a $n$-digit number $\overline{a_1a_2a_3...a_n}$ with non-zero digits $a_1, a_2$ and $a_n$, which is divisible by $\overline{a_2a_3...a_n}$.

Solve the system of equations in real numbers: \[ \frac{x}{y} + \frac{y}{z} + \frac{z}{x} = \frac{x}{z} + \frac{z}{y} + \frac{y}{x} \] \[ x^2 + y^2 + z^2 = 294 \] \[ x + y + z = 0 \]

Let $n \geq2$ be an integer number and $D_n$ the set of all the points $(x,y)$ in the plane such that its coordinates are integer numbers with: $-n \le x \le n$ and $-n \le y \le n$. (a) There are three possible colors in which the points of $D_n$ are painted with (each point has a unique color). Show that with any distribution of the colors, there are always two points of $D_n$ with the same color such that the line that contains them does not go through any other point of $D_n$. (b) Find a way to paint the points of $D_n$ with 4 colors such that if a line contains exactly two points of $D_n$, then, this points have different colors.

Let $ a$, $ b$ and $ k$ be three positive integers. We define two sequences $ \left( a_{n}\right)$ and $ \left( b_{n}\right)$ by the starting values $ a_{1}\equal{}a$ and $ b_{1}\equal{}b$ and the recurrent equations $ a_{n\plus{}1}\equal{}ka_{n}\plus{}b_{n}$ and $ b_{n\plus{}1}\equal{}kb_{n}\plus{}a_{n}$ for each positive integer $ n$. Prove that if $ a_{1}\perp b_{1}$, $ a_{2}\perp b_{2}$ and $ a_{3}\perp b_{3}$ hold, then $ a_{n}\perp b_{n}$ holds for every positive integer $ n$. Here, the abbreviation $ x\perp y$ stands for "the numbers $ x$ and $ y$ are coprime".

[color=darkred]Let $A,B\in\mathcal{M}_4(\mathbb{R})$ such that $AB=BA$ and $\det (A^2+AB+B^2)=0$ . Prove that: \[\det (A+B)+3\det (A-B)=6\det (A)+6\det (B)\ .\][/color]

Prove that for all positive real numbers $a,b,c$ we have \[\sqrt{a^2-ab+b^2}+\sqrt{b^2-bc+c^2}\ge\sqrt{a^2+ac+c^2} \]

Given a number $n\in\mathbb{Z}^+$ and let $S$ denotes the set $\{0,1,2,...,2n+1\}$. Consider the function $f:\mathbb{Z}\times S\to [0,1]$ satisfying two following conditions simultaneously: i) $f(x,0)=f(x,2n+1)=0\forall x\in\mathbb{Z}$; ii) $f(x-1,y)+f(x+1,y)+f(x,y-1)+f(x,y+1)=1$ for all $x\in\mathbb{Z}$ and $y\in\{1,2,3,...,2n\}$. Let $F$ be the set of such functions. For each $f\in F$, let $v(f)$ be the set of values of $f$. a) Proof that $|F|=\infty$. b) Proof that for each $f\in F$ then $|v(f)|<\infty$. c) Find the maximum value of $|v(f)|$ for $f\in F$.

For some positive numbers $a$, $b$, $c$ and $d$, we know that $$ \frac{1}{a^3 + 1}+ \frac{1}{b^3 + 1}+ \frac{1}{c^3 + 1} + \frac{1}{d^3 + 1} = 2. $$ Prove that $$ \frac{1 - a}{a^2 - a + 1} + \frac{1-b}{b^2 - b + 1} + \frac{1-c}{c^2 - c + 1} +\frac{1-d}{d^2 - d + 1} \geq 0. $$

The square of $ 5\minus{}\sqrt{y^2\minus{}25}$ is: $ \textbf{(A)}\ y^2\minus{}5\sqrt{y^2\minus{}25} \qquad \textbf{(B)}\ \minus{}y^2 \qquad \textbf{(C)}\ y^2 \\ \textbf{(D)}\ (5\minus{}y)^2 \qquad \textbf{(E)}\ y^2\minus{}10\sqrt{y^2\minus{}25}$

A line $MN$ is given in the plane. Consider circles $k_1$, $k_2$ tangent to the line at points $M$, $N$, respectively, while touching each other externally. Let $X$ be the midpoint of the segment $PQ$, where $P$, $Q$ are in this order tangent points of the second common external tangent of the circles $k_1$, $k_2$. Find the locus of the points $X$ for all pairs of circles of the specified properties.

Triangles $ABC$, $ADE$, and $EFG$ are all equilateral. Points $D$ and $G$ are midpoints of $\overline{AC}$ and $\overline{AE}$, respectively. If $AB = 4$, what is the perimeter of figure $ABCDEFG$? [asy] pair A,B,C,D,EE,F,G; A = (4,0); B = (0,0); C = (2,2*sqrt(3)); D = (3,sqrt(3)); EE = (5,sqrt(3)); F = (5.5,sqrt(3)/2); G = (4.5,sqrt(3)/2); draw(A--B--C--cycle); draw(D--EE--A); draw(EE--F--G); label("$A$",A,S); label("$B$",B,SW); label("$C$",C,N); label("$D$",D,NE); label("$E$",EE,NE); label("$F$",F,SE); label("$G$",G,SE); [/asy] $\text{(A)}\ 12 \qquad \text{(B)}\ 13 \qquad \text{(C)}\ 15 \qquad \text{(D)}\ 18 \qquad \text{(E)}\ 21$

Let $ABC$ be an arbitrary triangle and $O$ is the circumcenter of $\triangle {ABC}$.Points $X,Y$ lie on $AB,AC$,respectively such that the reflection of $BC$ WRT $XY$ is tangent to circumcircle of $\triangle {AXY}$.Prove that the circumcircle of triangle $AXY$ is tangent to circumcircle of triangle $BOC$.

Let $ABCD$ be a square and let $P$ be a point on side $AB$. The point $Q$ lies outside the square such that $\angle ABQ = \angle ADP$ and $\angle AQB = 90^{\circ}$. The point $R$ lies on the side $BC$ such that $\angle BAR = \angle ADQ$. Prove that the lines $AR, CQ$ and $DP$ pass through a common point.

Let $a,b,c$ are positive real numbers and $\alpha$ be any real number. Denote $f(\alpha)=abc(a^{\alpha}+b^{\alpha}+c^{\alpha}), g(\alpha)=a^{2+\alpha}(b+c-a)+b^{2+\alpha}(-b+c+a)+c^{2+\alpha}(b-c+a)$. Determine $\min{|f(\alpha)-g(\alpha)|}$ and $\max{|f(\alpha)-g(\alpha)|}$, if they are exists.

Let $H$ be the orthocenter of an obtuse triangle $ABC$ and $A_1B_1C_1$ arbitrary points on the sides $BC,AC,AB$ respectively.Prove that the tangents drawn from $H$ to the circles with diametrs $AA_1,BB_1,CC_1$ are equal.

Professor David proposed to his wife to calculate the steps of an escalator that worked in a shopping mall, asking him to walk up counting the steps that rise from the bottom to the end. The teacher, in turn, left with his wife, but walking the twice as fast, so that the woman advanced one step each time her husband advanced $2$. When The lady arrived at the top reported that she had counted $21$ steps, while the teacher counted $28$ of them. How many steps are there in sight on the ladder at any given time? [hide=original wording]El profesor David propuso a su senora calcular los escalones de una escalera mecanica que funcionaba en un centro comercial, pidiendole que caminara hacia arriba contando los escalones que subiera desde la base hasta el final. El profesor a su vez, partio junto a su senora, pero caminando el doble de rapido, de modo que la senora avanzaba un escalon cada vez que su marido avanzaba 2. Cuando la senora llego arriba informo que habıa contado 21 escalones, mientras que el profesor conto 28 de ellos, ¿Cuantos escalones hay a la vista en la escalera en un instante cualquiera?[/hide]

S is the set of all ($a$, $b$, $c$, $d$, $e$, $f$) where $a$, $b$, $c$, $d$, $e$, $f$ are integers such that $a^2 + b^2 + c^2 + d^2 + e^2 = f^2$. Find the largest $k$ which divides abcdef for all members of $S$.