Determine the number of ordered pairs $(x,y)$ with $x$ and $y$ integers between $-5$ and $5,$ inclusive, such that $(x+y)(x+3y)=(x+2y)^2.$

There are $ n \plus{} 1$ cells in a row labeled from $ 0$ to $ n$ and $ n \plus{} 1$ cards labeled from $ 0$ to $ n$. The cards are arbitrarily placed in the cells, one per cell. The objective is to get card $ i$ into cell $ i$ for each $ i$. The allowed move is to find the smallest $ h$ such that cell $ h$ has a card with a label $ k > h$, pick up that card, slide the cards in cells $ h \plus{} 1$, $ h \plus{} 2$, ... , $ k$ one cell to the left and to place card $ k$ in cell $ k$. Show that at most $ 2^n \minus{} 1$ moves are required to get every card into the correct cell and that there is a unique starting position which requires $ 2^n \minus{} 1$ moves. [For example, if $ n \equal{} 2$ and the initial position is 210, then we get 102, then 012, a total of 2 moves.]

Let $S = \left\{ 1,2,\dots,n \right\}$, where $n \ge 1$. Each of the $2^n$ subsets of $S$ is to be colored red or blue. (The subset itself is assigned a color and not its individual elements.) For any set $T \subseteq S$, we then write $f(T)$ for the number of subsets of $T$ that are blue. Determine the number of colorings that satisfy the following condition: for any subsets $T_1$ and $T_2$ of $S$, \[ f(T_1)f(T_2) = f(T_1 \cup T_2)f(T_1 \cap T_2). \]

Compute the number of ways to color each cell of an $18 \times 18$ square grid either ruby or sapphire such that each contiguous $3 \times 3$ subgrid has exactly $1$ ruby cell.

Given a tetrahedron $ABCD$, in which $AB=CD= 13 , AC=BD=14$ and $AD=BC=15$. Show that the centers of the inscribed sphere and sphere around it coincide, and find the radii of these spheres.

Let $R$ be the rectangle on the cartesian plane with vertices $(0,0)$, $(5,0)$, $(5,7)$, and $(0,7)$. Find the number of squares with sides parallel to the axes and vertices that are lattice points that lie within the region bounded by $R$. [i]Proposed by Boyan Litchev[/i] [hide=Solution][i]Solution[/i]. $\boxed{85}$ We have $(6-n)(8-n)$ distinct squares with side length $n$, so the total number of squares is $5 \cdot 7+4 \cdot 6+3 \cdot 5+2 \cdot 4+1\cdot 3 = \boxed{85}$.[/hide]

Two circles $ C_1,C_2$ with different radii are given in the plane, they touch each other externally at $ T$. Consider any points $ A\in C_1$ and $ B\in C_2$, both different from $ T$, such that $ \angle ATB \equal{} 90^{\circ}$. (a) Show that all such lines $ AB$ are concurrent. (b) Find the locus of midpoints of all such segments $ AB$.

Let $d(n)$ denote the largest odd integer divides $n$. Calculate the sum $d(1)+d(2)+d(3)+\dots+d(2^{99})$.

For any real numbers $x, y, z$ with $xyz + x + y + z = 4, $show that $$(yz + 6)^2 + (zx + 6)^2 + (xy + 6)^2 \geq 8 (xyz + 5).$$

Let $ABC$ be an acute-angled triangle with $AC > AB$, let $O$ be its circumcentre, and let $D$ be a point on the segment $BC$. The line through $D$ perpendicular to $BC$ intersects the lines $AO, AC,$ and $AB$ at $W, X,$ and $Y,$ respectively. The circumcircles of triangles $AXY$ and $ABC$ intersect again at $Z \ne A$. Prove that if $W \ne D$ and $OW = OD,$ then $DZ$ is tangent to the circle $AXY.$

At a party there are eight guests, and each participant can't talk with at most three persons. Prove that we can group the persons in four pairs such that in every pair a conversation can take place.

Given two circles $k_{1}$ and $k_{2}$ with centers $O_{1}$ and $O_{2}$ that intersect at the points $A$ and $B$.Passes through A two lines that intersect the circle $k_{1}$ at the points $N_{1}$and $M_{1}$, and the circle $k_{2}$ at the points $N_{2}$ and $M_{2}$ (points $A, N_{1},M_{1}$ in colinear). Denote the midpoints of the segments $N_{1}N_{2}$ and $M_{1}M_{2]}$ , through $N$ and $M$.Prove that: $a)$ Points $M,N,A$ and $B$ lie on a circle $b)$The center of the circle passing through $M,N,A$ and $B$ lies in the middle of the segment $O_{1}O_{2}$

Let $I_a$ be the excenter of triangle $ABC$ with respect to $A$. The line $AI_a$ intersects the circumcircle of triangle ABC at $T$. Let $X$ be a point on segment $TI_a$ such that $X I_a^2 = XA \cdot X T$ The perpendicular line from $X$ to $BC$ intersects $BC$ at $A'$. Define $B'$ and $C'$ in the same way. Prove that $AA',BB'$ and $CC'$ are concurrent.

The four faces of a tetrahedral die are labelled $0, 1, 2,$ and $3,$ and the die has the property that, when it is rolled, the die promptly vanishes, and a number of copies of itself appear equal to the number on the face the die landed on. For example, if it lands on the face labelled $0,$ it disappears. If it lands on the face labelled $1,$ nothing happens. If it lands on the face labelled $2$ or $3,$ there will then be $2$ or $3$ copies of the die, respectively (including the original). Suppose the die and all its copies are continually rolled, and let $p$ be the probability that they will all eventually disappear. Find $\left\lfloor \frac{10}{p} \right\rfloor$.

Let $n$ be a positive integer and $f:[0,1] \to \mathbb{R}$ be an integrable function. Prove that there exists a point $c \in \left[0,1- \frac{1}{n} \right],$ such that [center] $ \int\limits_c^{c+\frac{1}{n}}f(x)\mathrm{d}x=0$ or $\int\limits_0^c f(x) \mathrm{d}x=\int\limits_{c+\frac{1}{n}}^1f(x)\mathrm{d}x.$ [/center]

Let $H$ be a regular hexagon of side length $x$. Call a hexagon in the same plane a "distortion" of $H$ if and only if it can be obtained from $H$ by translating each vertex of $H$ by a distance strictly less than $1$. Determine the smallest value of $x$ for which every distortion of $H$ is necessarily convex.

Determine the remainder when $(x^4-1)(x^2-1)$ is divided by $1+x+x^2$.

A particle starts at $(0,0,0)$ in three-dimensional space. Each second, it randomly selects one of the eight lattice points a distance of $\sqrt{3}$ from its current location and moves to that point. What is the probability that, after two seconds, the particle is a distance of $2\sqrt{2}$ from its original location? [i]Proposed by Connor Gordon[/i]

Consider a triangle $ABC$ and two congruent triangles $A_1B_1C_1$ and $A_2B_2C_2$ which are respectively similar to $ABC$ and inscribed in it: $A_i,B_i,C_i$ are located on the sides of $ABC$ in such a way that the points $A_i$ are on the side opposite to $A$, the points $B_i$ are on the side opposite to $B$, and the points $C_i$ are on the side opposite to $C$ (and the angle at A are equal to angles at $A_i$ etc.). The circumcircles of $A_1B_1C_1$ and $A_2B_2C_2$ intersect at points $P$ and $Q$. Prove that the line $PQ$ passes through the orthocenter of $ABC$.

Let $ a > b > 1$ be relatively prime positive integers. Define the weight of an integer $ c$, denoted by $ w(c)$ to be the minimal possible value of $ |x| \plus{} |y|$ taken over all pairs of integers $ x$ and $ y$ such that \[ax \plus{} by \equal{} c.\] An integer $ c$ is called a [i]local champion [/i]if $ w(c) \geq w(c \pm a)$ and $ w(c) \geq w(c \pm b)$. Find all local champions and determine their number. [i]Proposed by Zoran Sunic, USA[/i]

Let $p$ be a prime number greater than $2$ and let $m, n$ be integers such that: $$\frac{m}{n}=1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{p-1}.$$ Prove that $p$ divides $m$.

For a positive integer $n$, we dene $D_n$ as the largest integer that is a divisor of $a^n + (a + 1)^n + (a + 2)^n$ for all positive integers $a$. 1. Show that for all positive integers $n$, the number $D_n$ is of the form $3^k$ with $k \ge 0$ an integer. 2. Show that for all integers $k \ge 0$ there exists a positive integer n such that $D_n = 3^k$.

Let $A$ and $B$ be sets such that there are exactly $144$ sets which are subsets of either $A$ or $B$. Determine the number of elements $A \cup B$ has.

A regular heptagon $A_1A_2A_3A_4A_5A_6A_7$ is given. Straight $A_2A_3$ and $A_5A_6$ intersect at point $X$, and straight lines $A_3A_5$ and $A_1A_6$ intersect at point $Y$. Prove that lines $A_1A_2$ and $XY$ are parallel.

The sequence $a_n$ is defined for any $n\geq 10$ by the following inductive rule: [list] [*] $a_{10}=5778$ [*] If $a_n=0$ then $a_{n+1}=0$. [*] If $a_n\neq0$ then $a_{n+1}$ is the number whose base-$(n+1)$ representation equals the base $n$ representation of the number $a_n -1$. [/list] For example, $a_{11}=5\cdot11^3+7\cdot11^2+7\cdot11^1+7\cdot11^0=7586$ $a_{12}=5\cdot12^3+7\cdot12^2+7\cdot12^1+6\cdot12^0=9738$ [list=a] [*] Does there exist $n\geq10$ for which $a_n=0$? [*] Is $a_{1,000,000}=0$? [*] Is $a_{100^{100^{100}}}=0$? [/list]