At CMU, markers come in two colors: blue and orange. Zachary fills a hat randomly with three markers such that each color is chosen with equal probability, then Chase shuffles an additional orange marker into the hat. If Zachary chooses one of the markers in the hat at random and it turns out to be orange, the probability that there is a second orange marker in the hat can be expressed as simplified fraction $\tfrac{m}{n}$. Find $m+n$.

Yura put $2001$ coins of $1$, $2$ or $3$ kopeykas in a row. It turned out that between any two $1$-kopeyka coins there is at least one coin; between any two $2$-kopeykas coins there are at least two coins; and between any two $3$-kopeykas coins there are at least $3$ coins. How many $3$-koyepkas coins could Yura put?

In a volleyball tournament for the Euro-African cup, there were nine more teams from Europe than from Africa. Each pair of teams played exactly once and the Europeans teams won precisely nine times as many matches as the African teams, overall. What is the maximum number of matches that a single African team might have won?

Let $n\ge 2$ be a positive integer, and let $P_1P_2\dots P_{2n}$ be a polygon with $2n$ sides such that no two sides are parallel. Denote $P_{2n+1}=P_1$. For some point $P$ and integer $i\in\{1,2,\ldots,2n\}$, we say that $i$ is a $P$-good index if $PP_{i}>PP_{i+1}$, and that $i$ is a $P$-bad index if $PP_{i}<PP_{i+1}$. Prove that there's a point $P$ such that the number of $P$-good indices is the same as the number of $P$-bad indices.

There are two boxes containing balls. One of them contains $m$ balls, and the other contains $n$ balls, where $m, n > 0$. Two actions are permitted: (i) Remove an equal number of balls from both boxes. (ii) Increase the number of balls in one of the boxes by a factor $k$. Is it possible to remove all of the balls from both boxes with just these two actions, 1. if $k = 2$? 2. if $k = 3$?

Find the number of ordered quadruplets $(a_1,a_2,a_3,a_4)$ of integers, for which $a_1\geq 1$, $a_2\geq 2$, $a_3\geq 3$, and $-10\leq a_4\leq 10$ and $a_1+a_2+a_3+a_4=2011$ .

A maze is an $8 \times 8$ board with some adjacent squares separated by walls, so that any two squares can be connected by a path not meeting any wall. Given a command LEFT, RIGHT, UP, DOWN, a pawn makes a step in the corresponding direction unless it encounters a wall or an edge of the chessboard. God writes a program consisting of a finite sequence of commands and gives it to the Devil, who then constructs a maze and places the pawn on one of the squares. Can God write a program which guarantees the pawn will visit every square despite the Devil's efforts?

Let $n$ points, where there are not $3$ of them on a line, and consider the segments that are formed by connecting any $2$ of the points. There are enough colors available to paint the points and the segments, coloring them with the following two rules: a) All the segments that reach the same point are painted of different colors. b) Each point is painted a different color to all the segments that reach it. Find the minimum number of colors needed to make such a coloring.

A line $g$ is given in a plane. $n$ distinct points are chosen arbitrarily from $g$ and are named as $A_1, A_2, \ldots, A_n$. For each pair of points $A_i,A_j$, a semicircle is drawn with $A_i$ and $A_j$ as its endpoints. All semicircles lie on the same side of $g$. Determine the maximum number of points (which are not lying in $g$) of intersection of semicircles as a function of $n$.

The points $S(i, j)$ with integer Cartesian coordinates $0 < i \leq n, 0 < j \leq m, m \leq n$, form a lattice. Find the number of: [b](a)[/b] rectangles with vertices on the lattice and sides parallel to the coordinate axes; [b](b)[/b] squares with vertices on the lattice and sides parallel to the coordinate axes; [b](c)[/b] squares in total, with vertices on the lattice.

[b]p10.[/b] Three rectangles of dimension $X \times 2$ and four rectangles of dimension $Y \times 1$ are the pieces that form a rectangle of area $3XY$ where $X$ and $Y$ are positive, integer values. What is the sum of all possible values of $X$? [b]p11.[/b] Suppose we have a polynomial $p(x) = x^2 + ax + b$ with real coefficients $a + b = 1000$ and $b > 0$. Find the smallest possible value of $b$ such that $p(x)$ has two integer roots. [b]p12.[/b] Ten square slips of paper of the same size, numbered $0, 1, 2, ..., 9$, are placed into a bag. Four of these squares are then randomly chosen and placed into a two-by-two grid of squares. What is the probability that the numbers in every pair of blocks sharing a side have an absolute difference no greater than two? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Elmo has 2023 cookie jars, all initially empty. Every day, he chooses two distinct jars and places a cookie in each. Every night, Cookie Monster finds a jar with the most cookies and eats all of them. If this process continues indefinitely, what is the maximum possible number of cookies that the Cookie Monster could eat in one night? [i]Proposed by Espen Slettnes[/i]

Weights of $1$ g, $2$ g,$ ...$ , $200$ g are placed on the two pans of a balance such that on each pan there are $100$ weights and the balance is in equilibrium. Prove that one can swap $50$ weights from one pan with $50$ weights from the other pan such that the balance remains in equilibrium. Kvant Magazine

A stromino is a $3 \times 1$ rectangle. Show that a $5 \times 5$ board divided into twenty-five $1 \times 1$ squares cannot be covered by $16$ strominos such that each stromino covers exactly three squares of the board, and every square is covered by one or two strominos. (A stromino can be placed either horizontally or vertically on the board.) [i]Proposed by Navilarekallu Tejaswi[/i]

Spot spotlight located at vertex $B$ of an equilateral triangle $ABC$, illuminates angle $\alpha$. Find all such values of $\alpha$, not exceeding $60^o$, which at any position of the spotlight, when the illuminated corner is entirely located inside the angle $ABC$, from the illuminated and two unlit segments of side $AC$ can be formed into a triangle.

$m,n,p$ are positive integers which do not simultaneously equal to zero. $3$D Cartesian space is divided into unit cubes by planes each perpendicular to one of $3$ axes and cutting corresponding axis at integer coordinates. Each unit cube is filled with an integer from $1$ to $60$. A filling of integers is called [i]Dien Bien[/i] if, for each rectangular box of size $\{2m+1,2n+1,2p+1\}$, the number in the unit cube which has common centre with the rectangular box is the average of the $8$ numbers of the $8$ unit cubes at the $8$ corners of that rectangular box. How many [i]Dien Bien[/i] fillings are there? Two fillings are the same if one filling can be transformed to the other filling via a translation. [hide]translation from [url=http://artofproblemsolving.com/community/c6h592875p3515526]here[/url][/hide]

Let $\,S\,$ be a finite set of points in three-dimensional space. Let $\,S_{x},\,S_{y},\,S_{z}\,$ be the sets consisting of the orthogonal projections of the points of $\,S\,$ onto the $yz$-plane, $zx$-plane, $xy$-plane, respectively. Prove that \[ \vert S\vert^{2}\leq \vert S_{x} \vert \cdot \vert S_{y} \vert \cdot \vert S_{z} \vert, \] where $\vert A \vert$ denotes the number of elements in the finite set $A$. [hide="Note"] Note: The orthogonal projection of a point onto a plane is the foot of the perpendicular from that point to the plane. [/hide]

Each edge of a graph with $15$ vertices is colored either red or blue in such a way that no three vertices are pairwise connected with edges of the same color. Determine the largest possible number of edges in the graph.

A cube with size $10\times 10\times 10$ consists of $1000$ unit cubes, all colored white. $A$ and $B$ play a game on this cube. $A$ chooses some pillars with size $1\times 10\times 10$ such that no two pillars share a vertex or side, and turns all chosen unit cubes to black. $B$ is allowed to choose some unit cubes and ask $A$ their colors. How many unit cubes, at least, that $B$ need to choose so that for any answer from $A$, $B$ can always determine the black unit cubes?

There are $2010$ red cards and $2010$ white cards. All of these $4020$ cards are shuffled and dealt in two randomly to each of the $2010$ round table players. The game consists of several rounds, each of which players simultaneously hand over cards to each other according to the following rules. If a player holds at least one red card, he passes one red card to the player sitting to his left, otherwise he transfers one white card to the left. The game ends after the round when each player has one red card and one white card. Determine as many rounds as possible.

Let $n$ be a positive integer. A sequence $(a, b, c)$ of $a, b, c \in \{1, 2, . . . , 2n\}$ is called [i]joke [/i] if its shortest term is odd and if only that smallest term, or no term, is repeated. For example, the sequences $(4, 5, 3)$ and $(3, 8, 3)$ are jokes, but $(3, 2, 7)$ and $(3, 8, 8)$ are not. Determine the number of joke sequences in terms of $n$.

Given a square $100\times 100$ on the sheet of cross-lined paper. There are several broken lines drawn inside the square. Their links consist of the small squares sides. They are neither pairwise- nor self-intersecting (have no common points). Their ends are on the big square boarder, and all the other vertices are in the big square interior. Prove that there exists (in addition to four big square angles) a node (corresponding to the cross-lining family, inside the big square or on its side) that does not belong to any broken line.

We have $100$ cards with two sides, the [i]even[/i] and the [i]odd[/i]. In each side there are written two succesive integers, in the [i]odd[/i] side and odd integer and at the back in the [i]even[/i] side the even number that follows the odd number of the [i]odd[/i] side, such that all intgers from $1$ to $200$ are used. Student $A$ randomly choses $21$ cards and sums all the numbers of boths sides and announces as their sum the number $913$. Student $B$ randomly choses from the remaining cards $20$ cards and sums all the numbers of boths sides and announces as their sum the number $2400$. a) Explain why student $A$ has done an error in the addition. b) If the correct result for student $A$ is $903$, explain why also student $B$ has done an error in the addition.

Let $T$ be a non-empty finite subset of positive integers $\ge 1$. A subset $S$ of $T$ is called [b]good [/b] if for every integer $t\in T$ there exists an $s$ in $S$ such that $gcd(t,s) >1$. Let \[A={(X,Y)\mid X\subseteq T,Y\subseteq T,gcd(x,y)=1 \text{for all} x\in X, y\in Y}\] Prove that : $a)$ If $X_0$ is not [b]good[/b] then the number of pairs $(X_0,Y)$ in $A$ is [b]even[/b]. $b)$ the number of good subsets of $T$ is [b]odd[/b].

[b]p1.[/b] How many nonzero digits are in the number $(5^{94} + 5^{92})(2^{94} + 2^{92})$? [b]p2.[/b] Suppose $A$ is a set of $2013$ distinct positive integers such that the arithmetic mean of any subset of $A$ is also an integer. Find an example of $A$. [b]p3.[/b] How many minutes until the smaller angle formed by the minute and hour hands on the face of a clock is congruent to the smaller angle between the hands at $5:15$ pm? Round your answer to the nearest minute. [b]p4.[/b] Suppose $a$ and $b$ are positive real numbers, $a + b = 1$, and $$1 +\frac{a^2 + 3b^2}{2ab}=\sqrt{4 +\frac{a}{b}+\frac{3b}{a}}.$$ Find $a$. [b]p5.[/b] Suppose $f(x) = \frac{e^x- 12e^{-x}}{ 2}$ . Find all $x$ such that $f(x) = 2$. [b]p6.[/b] Let $P_1$, $P_2$,$...$,$P_n$ be points equally spaced on a unit circle. For how many integer $n \in \{2, 3, ... , 2013\}$ is the product of all pairwise distances: $\prod_{1\le i<j\le n} P_iP_j$ a rational number? Note that $\prod$ means the product. For example, $\prod_{1\le i\le 3} i = 1\cdot 2 \cdot 3 = 6$. [b]p7.[/b] Determine the value $a$ such that the following sum converges if and only if $r \in (-\infty, a)$ : $$\sum^{\infty}_{n=1}(\sqrt{n^4 + n^r} - n^2).$$ Note that $\sum^{\infty}_{n=1}\frac{1}{n^s}$ converges if and only if $s > 1$. [b]p8.[/b] Find two pairs of positive integers $(a, b)$ with $a > b$ such that $a^2 + b^2 = 40501$. [b]p9.[/b] Consider a simplified memory-knowledge model. Suppose your total knowledge level the night before you went to a college was $100$ units. Each day, when you woke up in the morning you forgot $1\%$ of what you had learned. Then, by going to lectures, working on the homework, preparing for presentations, you had learned more and so your knowledge level went up by $10$ units at the end of the day. According to this model, how long do you need to stay in college until you reach the knowledge level of exactly $1000$? [b]p10.[/b] Suppose $P(x) = 2x^8 + x^6 - x^4 +1$, and that $P$ has roots $a_1$, $a_2$, $...$ , $a_8$ (a complex number $z$ is a root of the polynomial $P(x)$ if $P(z) = 0$). Find the value of $$(a^2_1-2)(a^2_2-2)(a^2_3-2)...(a^2_8-2).$$ [b]p11.[/b] Find all values of $x$ satisfying $(x^2 + 2x-5)^2 = -2x^2 - 3x + 15$. [b]p12.[/b] Suppose $x, y$ and $z$ are positive real numbers such that $$x^2 + y^2 + xy = 9,$$ $$y^2 + z^2 + yz = 16,$$ $$x^2 + z^2 + xz = 25.$$ Find $xy + yz + xz$ (the answer is unique). [b]p13.[/b] Suppose that $P(x)$ is a monic polynomial (i.e, the leading coefficient is $1$) with $20$ roots, each distinct and of the form $\frac{1}{3^k}$ for $k = 0,1,2,..., 19$. Find the coefficient of $x^{18}$ in $P(x)$. [b]p14.[/b] Find the sum of the reciprocals of all perfect squares whose prime factorization contains only powers of $3$, $5$, $7$ (i.e. $\frac{1}{1} + \frac{1}{9} + \frac{1}{25} + \frac{1}{419} + \frac{1}{811} + \frac{1}{215} + \frac{1}{441} + \frac{1}{625} + ...$). [b]p15.[/b] Find the number of integer quadruples $(a, b, c, d)$ which also satisfy the following system of equations: $$1+b + c^2 + d^3 =0,$$ $$a + b^2 + c^3 + d^4 =0,$$ $$a^2 + b^3 + c^4 + d^5 =0,$$ $$a^3+b^4+c^5+d^6 =0.$$ PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].