Let $(x,y)$ be the coordinates of a point chosen uniformly at random within the unit square with vertices at $(0,0), (0,1), (1,0),$ and $(1,1).$ The probability that $|x - \tfrac{1}{2}| + |y - \tfrac{1}{2}| < \tfrac{1}{2}$ is $\tfrac{p}{q},$ where $p$ and $q$ are relatively prime integers. Find $p + q.$

The numbers $1$ to $15$ are each coloured blue or red. Determine all possible colourings that satisfy the following rules: • The number $15$ is red. • If numbers $x$ and $y$ have different colours and $x + y \le 15$, then $x + y$ is blue. • If numbers $x$ and $y$ have different colours and $x \cdot y \le 15$, then $x \cdot y$ is red.

Which one below cannot be expressed in the form $x^2+y^5$, where $x$ and $y$ are integers? $ \textbf{(A)}\ 59170 \qquad\textbf{(B)}\ 59149 \qquad\textbf{(C)}\ 59130 \qquad\textbf{(D)}\ 59121 \qquad\textbf{(E)}\ 59012 $

In the equation below, $A$ and $B$ are consecutive positive integers, and $A$, $B$, and $A+B$ represent number bases: \[132_A + 43_B = 69_{A+B.}\] What is $A + B$? $ \textbf{(A)}\ 9\qquad\textbf{(B)}\ 11\qquad\textbf{(C)}\ 13\qquad\textbf{(D)}\ 15\qquad\textbf{(E)}\ 17 $

Let the diagonals of cyclic quadrilateral $ABCD$ meet at point $P$. The line passing through $P$ and perpendicular to $PD$ meets $AD$ at point $D_1$, a point $A_1$ is defined similarly. Prove that the tangent at $P$ to the circumcircle of triangle $D_1PA_1$ is parallel to $BC$.

At a strange party, each person knew exactly $22$ others. For any pair of people $X$ and $Y$ who knew each other, there was no other person at the party that they both knew. For any pair of people $X$ and $Y$ who did not know one another, there were exactly 6 other people that they both knew. How many people were at the party?

I have a set $A$ containing $n$ distinct integers. This set has the property that if $a,b \in A$, then $12 \nmid |a+b|$ and $12 \nmid |a-b|$. What is the largest possible value of $n$?

Alice and Bob are playing a game, starting with a binary string$ b$ of length $2022$. In each step, the rightmost digit of the string is deleted. If the deleted digit was $1$, Alice gets to choose which digit she wants to append on the left. Otherwise, Bob gets to choose the digit to append on the left of the string. Alice would like to turn the string $b$ into the all-zero string $\underbrace{00 . . . 0}_{2022}$, in the least number of steps possible, while Bob would like to maximize the number of steps necessary, or prevent Alice from doing this at all. a) Is there a string $b$ for which Bob can prevent Alice in her goal, if both players play optimally? b) If the answer to part a is yes, find all such strings $b$. If the answer is no, find the maximal game time and find the set of strings $b$ for which the game time is maximal.

Let $n =123456789101112 ... 998999$ be the natural number where is obtained by writing the natural numbers from $1$ to $999$ one after the other. What is the $1997$-th digit number in $n$?

Two rays emanate from the origin $O$ and form a $45^\circ$ angle in the first quadrant of the Cartesian coordinate plane. For some positive numbers $X$, $Y$, and $S$, the ray with the larger slope passes through point $A = (X, S)$, and the ray with the smaller slope passes through point $B = (S, Y)$. If $6X + 6Y + 5S = 600$, then determine the maximum possible area of $\triangle OAB$.

Given a sequence of natural numbers $ (a_i) $, for each natural number $ n $ the sum of the terms of the sequence that are not greater than $ n $ is a number not less than $ n $. Prove that for every natural number $ k $ it is possible to choose from the sequence $ (a_i) $ a finite sequence with the sum of terms equal to $ k $.

Omar has four fair standard six-sided dice. Omar invented a game where he rolls his four dice over and over again until the number 1 does not appear on the top face of any of the dice. Omar wins the game if on that roll the top faces of his dice show at least one 2 and at least one 5. The probability that Omar wins the game is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

For $H\subset \mathbb Z$ and $n\in\mathbb Z$ let $h_n$ denote the number of finite subsets of $H$ in which the sum of the elements is $n$. Determine whether there exists $H\subset \mathbb Z$ for which $0\notin H$ and $h_n$ is a finite even number for every $n\in\mathbb{Z}$. (The sum of the elements of the empty set is $0$.) [i]Proposed by Csongor Beke, Cambridge[/i]

Zscoder has an simple undirected graph $G$ with $n\ge 3$ vertices. Navi labels a positive integer to each vertex, and places a token at one of the vertex. This vertex is now marked red. In each turn, Zscoder plays with following rule: $\bullet$ If the token is currently at vertex $v$ with label $t$, then he can move the token along the edges in $G$ (possibly repeating some edges) exactly $t$ times. After these $t$ moves, he marks the current vertex red where the token is at if it is unmarked, or does nothing otherwise, then finishes the turn. Zscoder claims that he can mark all vertices in $G$ red after finite number of turns, regardless of Navi's labels and starting vertex. What is the minimum number of edges must $G$ have, in terms of $n$? [i]Proposed by Yeoh Zi Song[/i]

$ABCD$ is a rectangle. Segment $BA$ is extended through $A$ to a point $E$. Let the intersection of $EC$ and $AD$ be point $F$. Suppose that [the] measure of $\angle{ACD}$ is $60$ degrees, and that the length of segment $EF$ is twice the length of diagonal $AC$. What is the measure of $\angle{ECD}$?

Given triangle $ ABC$ and its circumcircle $ \Gamma$, let $ M$ and $ N$ be the midpoints of arcs $ BC$ (that does not contain $ A$) and $ CA$ (that does not contain $ B$), repsectively. Let $ X$ be a variable point on arc $ AB$ that does not contain $ C$. Let $ O_1$ and $ O_2$ be the incenter of triangle $ XAC$ and $ XBC$, respectively. Let the circumcircle of triangle $ XO_1O_2$ meets $ \Gamma$ at $ Q$. (a) Prove that $ QNO_1$ and $ QMO_2$ are similar. (b) Find the locus of $ Q$ as $ X$ varies.

Let $n \in \mathbb{N}$ and $z \in \mathbb{C}^{*}$. Prove that $\left | n\textrm{e}^{z} - \sum_{j=1}^{n}\left (1+\frac{z}{j^2}\right )^{j^2}\right | < \frac{1}{3}\textrm{e}^{|z|}\left (\frac{\pi|z|}{2}\right)^2$.

Let $A, B, C$ be points on a unit circle such that $|AB|=|BC|$ and $m(\widehat{ABC})=72^\circ$. Let $D$ be a point such that $\triangle BCD$ is equilateral. If $AD$ meets the circle at $D$, what is $|DE|$? $ \textbf{(A)}\ \dfrac 12 \qquad\textbf{(B)}\ \dfrac {\sqrt 3}2 \qquad\textbf{(C)}\ \dfrac {\sqrt 2}2 \qquad\textbf{(D)}\ \sqrt 3 -1 \qquad\textbf{(E)}\ \text{None of the above} $

Inside a square of sidelength $ 1$ there are finitely many disks that are allowed to overlap. The sum of all circumferences is $ 10$. Show that there is a line intersecting or touching at least $ 4$ disks.

Find the volume $V$ of the solid formed by a rotation of the region enclosed by the curve $y=2^{x}-1$ and two lines $x=0,\ y=1$ around the $y$ axis.

The circumference of the circle with center $O$ is divided into 12 equal arcs, marked the letters $A$ through $L$ as seen below. What is the number of degrees in the sum of the angles $x$ and $y$? [asy] size(230); defaultpen(linewidth(0.65)); pair O=origin; pair[] circum = new pair[12]; string[] let = {"$A$","$B$","$C$","$D$","$E$","$F$","$G$","$H$","$I$","$J$","$K$","$L$"}; draw(unitcircle); for(int i=0;i<=11;i=i+1) { circum[i]=dir(120-30*i); dot(circum[i],linewidth(2.5)); label(let[i],circum[i],2*dir(circum[i])); } draw(O--circum[4]--circum[0]--circum[6]--circum[8]--cycle); label("$x$",circum[0],2.75*(dir(circum[0]--circum[4])+dir(circum[0]--circum[6]))); label("$y$",circum[6],1.75*(dir(circum[6]--circum[0])+dir(circum[6]--circum[8]))); label("$O$",O,dir(60)); [/asy] $\textbf{(A) }75\qquad\textbf{(B) }80\qquad\textbf{(C) }90\qquad\textbf{(D) }120\qquad \textbf{(E) }150$

A square with side $2008$ is broken into regions that are all squares with side $1$. In every region, either $0$ or $1$ is written, and the number of $1$'s and $0$'s is the same. The border between two of the regions is removed, and the numbers in each of them are also removed, while in the new region, their arithmetic mean is recorded. After several of those operations, there is only one square left, which is the big square itself. Prove that it is possible to perform these operations in such a way, that the final number in the big square is less than $\frac{1}{2^{10^6}}$.

Let $ a_1 \geq a_2 \geq a_3 \in \mathbb{Z}^\plus{}$ be given and let N$ (a_1, a_2, a_3)$ be the number of solutions $ (x_1, x_2, x_3)$ of the equation \[ \sum^3_{k\equal{}1} \frac{a_k}{x_k} \equal{} 1.\] where $ x_1, x_2,$ and $ x_3$ are positive integers. Prove that \[ N(a_1, a_2, a_3) \leq 6 a_1 a_2 (3 \plus{} ln(2 a_1)).\]

Let $A$ and $B$ be two spheres of different radii, both inscribed in a cone $K$. There are $m$ other, congruent spheres arranged in a ring such that each of them touches $A, B, K$ and two of the other spheres. Prove that this is possible for at most three values of $m.$

Real numbers $a_1,a_2,...,a_n$ ($n \ge 3$) satisfy the conditions $a_1 = a_n = 0$ and $$a_{k-1}+a_{k+1} \ge 2a_k$$ for $k = 2$,$3$$,...,$$n -1$. Prove that none of the numbers $a_1$,$...$,$a_n$ is positive.