Let $ABCD$ be a trapezoid such that $AD\parallel BC$. The interior angle bisectors of the corners $A$ and $B$ meet on $[DC]$. If $|BC|=9$ and $|AD|=4$, find $|AB|$.

How many ordered integer pairs $(x,y)$ ($0\leq x,y < 31$) are there satisfying $(x^2-18)^2\equiv y^2 (\mod 31)$? $ \textbf{(A)}\ 59 \qquad\textbf{(B)}\ 60 \qquad\textbf{(C)}\ 61 \qquad\textbf{(D)}\ 62 \qquad\textbf{(E)}\ \text{None} $

A grid consists of $n\times n$ points, with $n\in\mathbb{Z}^+$. In some of these points is a sentry. Every sentry chooses two directions, one perpendicular to the other (one vertical and the other horizontal) and watches over all the points that are found in the two chosen directions. Each sentry watches over her own point as well and the sentries on the edge of the grid can also watch the vacuum, depending on the directions they have chosen. For instance, in the figure below, representing a disposition of $5$ sentries in a $4\times 4$ grid, the sentries in $A,\,B,\,C,\,D,\,E$ watch over $1,\,3,\,4,\,5,\,7$ points, respectively; points $D$ and $E$ are watched by one sentry, point $C$ is watched by two sentries, points $A,\,B$ and $F$ are watched by three sentries. (a) Prove that we can place $12$ sentries in a $4\times 4$ grid in such a way that every point of the grid is watched by at most two sentries. (b) Let $S(n)$ be the maximum number of sentries we can place on an $n\times n$ grid in such a way that every point of the grid is watched by at most two sentries. Prove that $3n\leq S(n)\leq 4n$ for all $n\geq 3$.

A real number $x$ is chosen uniformly at random from the interval $[0, 1000].$ Find the probability that $$\left\lfloor\frac{\lfloor \tfrac{x}{2.5}\rfloor}{2.5}\right\rfloor=\left\lfloor\frac{x}{6.25}\right\rfloor.$$

The sequence of numbers $a_0,a_1,a_2,...$ is determined by $a_0 = 0$, and $$a_n= \begin{cases} 1+a_{n-1} \,\,\, when\,\,\, n \,\,\, is \,\,\, positive \,\,\, and \,\,\, odd \\ 3a_{n/2} \,\,\,when \,\,\,n \,\,\,is \,\,\,positive \,\,\,and \,\,\,even\end{cases}$$ How many of these numbers are less than $2007$ ?

Let $A$,$B$,$C$, and $D$ be points in the plane with $AB=AC=BC=BD=CD=36$ and such that $A \neq D$. Point $K$ lies on segment $AC$ such that $AK=2KC$. Point $M$ lies on segment $AB$, and point $N$ lies on line $AC$, such that $D$, $M$, and $N$ are collinear. Let lines $CM$ and $BN$ intersect at $P$. Then the maximum possible length of segment $KP$ can be expressed in the form $m+\sqrt{n}$ for positive integers $m$ and $n$. Compute $100m+n$. [i]Proposed by James Lin[/i]

Let $a$ and $ b$ be real numbers with $0\le a, b\le 1$. Prove that $$\frac{a}{b + 1}+\frac{b}{a + 1}\le 1$$ When does equality holds? (K. Czakler, GRG 21, Vienna)

Show that there are infinitely many positive integers which cannot be expressed as the sum of squares.

Given a acute triangle $PA_1B_1$ is inscribed in the circle $\Gamma$ with radius $1$. for all integers $n \ge 1$ are defined: $C_n$ the foot of the perpendicular from $P$ to $A_nB_n$ $O_n$ is the center of $\odot (PA_nB_n)$ $A_{n+1}$ is the foot of the perpendicular from $C_n$ to $PA_n$ $B_{n+1} \equiv PB_n \cap O_nA_{n+1}$ If $PC_1 =\sqrt{2}$, find the length of $PO_{2015}$ [hide=Source]Cono Sur Olympiad - 2015 - Day 1 - Problem 3[/hide]

A circular disk is partitioned into $ 2n$ equal sectors by $ n$ straight lines through its center. Then, these $ 2n$ sectors are colored in such a way that exactly $ n$ of the sectors are colored in blue, and the other $ n$ sectors are colored in red. We number the red sectors with numbers from $ 1$ to $ n$ in counter-clockwise direction (starting at some of these red sectors), and then we number the blue sectors with numbers from $ 1$ to $ n$ in clockwise direction (starting at some of these blue sectors). Prove that one can find a half-disk which contains sectors numbered with all the numbers from $ 1$ to $ n$ (in some order). (In other words, prove that one can find $ n$ consecutive sectors which are numbered by all numbers $ 1$, $ 2$, ..., $ n$ in some order.) [hide="Problem 8 from CWMO 2007"]$ n$ white and $ n$ black balls are placed at random on the circumference of a circle.Starting from a certain white ball,number all white balls in a clockwise direction by $ 1,2,\dots,n$. Likewise number all black balls by $ 1,2,\dots,n$ in anti-clockwise direction starting from a certain black ball.Prove that there exists a chain of $ n$ balls whose collection of numbering forms the set $ \{1,2,3\dots,n\}$.[/hide]

Let ${n}$ and $k$ be positive integers. There are given ${n}$ circles in the plane. Every two of them intersect at two distinct points, and all points of intersection they determine are pairwise distinct (i. e. no three circles have a common point). No three circles have a point in common. Each intersection point must be colored with one of $n$ distinct colors so that each color is used at least once and exactly $k$ distinct colors occur on each circle. Find all values of $n\geq 2$ and $k$ for which such a coloring is possible. [i]Proposed by Horst Sewerin, Germany[/i]

Determine all positive integers $n\geq 2$ for which there exist integers $x_1,x_2,\ldots ,x_{n-1}$ satisfying the condition that if $0<i<n,0<j<n, i\neq j$ and $n$ divides $2i+j$, then $x_i<x_j$.

Find the number of arithmetic sequences $a_1,a_2,a_3$ of three nonzero integers such that the sum of the terms in the sequence is equal to the product of the terms in the sequence. [i]Proposed by Sammy Charney[/i]

Let $c$ be a positive real number. Prove that $c$ can be expressed in infinitely many ways as a sum of infinitely many distinct terms selected from the sequence $\left( \frac{1}{10n} \right)_{n\in \mathbb{N}}$

If $0 \leq a \leq b \leq c \leq d,$ prove that \[a^bb^cc^dd^a \geq b^ac^bd^ca^d.\]

A museum has the shape of a $n \times n$ square divided into $n^2$ rooms of the shape of a unit square $(n>1)$. Between every two adjacent rooms (i.e. sharing a wall) there is a door. A night guardian wants to organize an inspection journey through the museum according to the following rules. He starts from some room and, whenever he enters a room, he stays there for exactly one minute and then proceeds to another room. He is allowed to enter a room more than once, but at the end of his journey he must have spent exactly $k$ minutes in every room. Find all $n$ and $k$ for which it is possible to organize such a journey.

Function $f(x)=\frac{x}{1-2^x}-\frac{x}{2}$ is $\text{(A)}$ an even function, not an odd function. $\text{(B)}$ an odd function, not an even function. $\text{(C)}$ an even function, also an odd function. $\text{(D)}$ neither an even function, nor an odd function.

Let $m > n$ be positive integers such that $3(3mn - 2)^2 - 2(3m -3n)^2 = 2019$. Find $3m + n$.

Let $p$ be a prime number. Call a positive integer $n$ interesting if \[x^n-1=(x^p-x+1)f(x)+pg(x)\] for some polynomials $f$ and $g$ with integer coefficients. a) Prove that the number $p^p-1$ is interesting. b) For which $p$ is $p^p-1$ the minimal interesting number?

In triangle $ABC$ let $N, M, P$ be the midpoints of the sides $BC, CA, AB$ and $G$ be the centroid of this triangle. Let the circle circumscribed to $BGP$ intersect the line $MP$ in point $K$, $P \neq K$, and the circle circumscribed to $CGN$ intersect the line $MN$ in point $L$, $N \neq L$. Prove that $ \angle BAK = \angle CAL $.

As Emily is riding her bike on a long straight road, she spots Ermenson skating in the same direction $1/2$ mile in front of her. After she passes him, she can see him in her rear mirror until he is $1/2$ mile behind her. Emily rides at a constant rate of $12$ miles per hour. Ermenson skates at a constant rate of $8$ miles per hour. For how many minutes can Emily see Ermenson? $ \textbf{(A)}\ 6 \qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ 12\qquad\textbf{(D)}\ 15\qquad\textbf{(E)}\ 16 $

In a sports league, each team uses a set of at most $t$ signature colors. A set $S$ of teams is[i] color-identifiable[/i] if one can assign each team in $S$ one of their signature colors, such that no team in $S$ is assigned any signature color of a different team in $S$. For all positive integers $n$ and $t$, determine the maximum integer $g(n, t)$ such that: In any sports league with exactly $n$ distinct colors present over all teams, one can always find a color-identifiable set of size at least $g(n, t)$.

A quadrilateral $PQRS$ has an inscribed circle, the points of tangencies with sides $PQ$, $QR$, $RS$, $SP$ being $A$, $B$, $C$, $D$, respectively. Let the midpoints of $AB$, $BC$, $CD$, $DA$ be $E$, $F$, $G$, $H$, respectively. Prove that the angle between segments $PR$ and $QS$ is equal to the angle between segments $EG$ and $FH$.

Let $ABC$ be a triangle with $90^\circ \ne \angle A \ne 135^\circ$. Let $D$ and $E$ be external points to the triangle $ABC$ such that $DAB$ and $EAC$ are isoscele triangles with right angles at $D$ and $E$. Let $F = BE \cap CD$, and let $M$ and $N$ be the midpoints of $BC$ and $DE$, respectively. Prove that, if three of the points $A$, $F$, $M$, $N$ are collinear, then all four are collinear.

Consider a $2018 \times 2019$ board with integers in each unit square. Two unit squares are said to be neighbours if they share a common edge. In each turn, you choose some unit squares. Then for each chosen unit square the average of all its neighbours is calculated. Finally, after these calculations are done, the number in each chosen unit square is replaced by the corresponding average. Is it always possible to make the numbers in all squares become the same after finitely many turns?