There are $n \ge 2$ coins, each with a different positive integer value. Call an integer $m$ [i]sticky [/i] if some subset of these $n$ coins have total value $m$. We call the entire set of coins a stick if all the sticky numbers form a consecutive range of integers. Compute the minimum total value of a stick across all sticks containing a coin of value $100$.

Alex and Bruno play the following game: each one, in your turn, the player writes, exactly one digit, in the right of the last number written. The game finishes if we have a number with $6$ digits( distincts ) and Alex starts the game. Bruno wins if the number with $6$ digits is a prime number, otherwise Alex wins. Which player has the winning strategy?

Let $K=(V, E)$ be a finite, simple, complete graph. Let $\phi: E \to \mathbb{R}^2$ be a map from the edge set to the plane, such that the preimage of any point in the range defines a connected graph on the entire vertex set $V$, and the points assigned to the edges of any triangle are collinear. Show that the range of $\phi$ is contained in a line.

Square $ABCD$ is divided into $n^2$ equal small squares by lines parallel to its sides.A spider starts from $A$ and moving only rightward or upwards,tries to reach $C$.Every "movement" of the spider consists of $k$ steps rightward and $m$ steps upwards or $m$ steps rightward and $k$ steps upwards(it can follow any possible order for the steps of each "movement").The spider completes $l$ "movements" and afterwards it moves without limitation (it still moves rightwards and upwards only).If $n=m\cdot l$,find the number of the possible paths the spider can follow to reach $C$.Note that $n,m,k,l\in \mathbb{N^{*}}$ with $k<m$.

There are $2008$ participants in a programming competition. In every round, all programmers are divided into two equal-sized teams. Find the minimal number of rounds after which there can be a situation in which every two programmers have been in different teams at least once.

Given an infinite sheet of square ruled paper. Some of the squares contain a piece. A move consists of a piece jumping over a piece on a neighbouring square (which shares a side) onto an empty square and removing the piece jumped over. Initially, there are no pieces except in an $m x n$ rectangle ($m, n > 1$) which has a piece on each square. What is the smallest number of pieces that can be left after a series of moves?

You are given $n$ not necessarily distinct real numbers $a_1, a_2, \ldots, a_n$. Let's consider all $2^n-1$ ways to select some nonempty subset of these numbers, and for each such subset calculate the sum of the selected numbers. What largest possible number of them could have been equal to $1$? For example, if $a = [-1, 2, 2]$, then we got $3$ once, $4$ once, $2$ twice, $-1$ once, $1$ twice, so the total number of ones here is $2$. [i](Proposed by Anton Trygub)[/i]

What is the greatest possible number of edges in a planar graph with $12$ vertices? A planar graph is one that can be drawn in a plane with none of the edges crossing (they intersect only at vertices).

Let $n$ be a positive integer. Initially, a bishop is placed in each square of the top row of a $2^n \times 2^n$ chessboard; those bishops are numbered from $1$ to $2^n$ from left to right. A [i]jump[/i] is a simultaneous move made by all bishops such that each bishop moves diagonally, in a straight line, some number of squares, and at the end of the jump, the bishops all stand in different squares of the same row. Find the total number of permutations $\sigma$ of the numbers $1, 2, \ldots, 2^n$ with the following property: There exists a sequence of jumps such that all bishops end up on the bottom row arranged in the order $\sigma(1), \sigma(2), \ldots, \sigma(2^n)$, from left to right. [i]Israel[/i]

All segments joining $n$ points (no three of which are collinear) are coloured in one of $k$ colours. What is the smallest $k$ for which there always exists a closed polygonal line with the vertices at some of the $n$ points, whose sides are all of the same colour?

Let $P_0P_5Q_5Q_0$ be a rectangular chocolate bar, one half dark chocolate and one half white chocolate, as shown in the diagram below. We randomly select $4$ points on the segment $P_0P_5$, and immediately after selecting those points, we label those $4$ selected points $P_1, P_2, P_3, P_4$ from left to right. Similarly, we randomly select $4$ points on the segment $Q_0Q_5$, and immediately after selecting those points, we label those $4$ points $Q_1, Q_2, Q_3, Q_4$ from left to right. The segments $P_1Q_1, P_2Q_2, P_3Q_3, P_4Q_4$ divide the rectangular chocolate bar into $5$ smaller trapezoidal pieces of chocolate. The probability that exactly $3$ pieces of chocolate contain both dark and white chocolate can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$. [Diagram in the individuals file for this exam on the Chmmc website]

Let $ P$ be a convex $ n$ polygon each of which sides and diagnoals is colored with one of $ n$ distinct colors. For which $ n$ does: there exists a coloring method such that for any three of $ n$ colors, we can always find one triangle whose vertices is of $ P$' and whose sides is colored by the three colors respectively.

Let $G$ be a simple $k$ edge-connected graph on $n$ vertices and let $u$ and $v$ be different vertices of $G$. Prove that there are $k$ edge-disjoint paths from $u$ to $v$ each having at most $\frac{20n}{k}$ edges.

In a party of eight people, each pair of people either know each other or do not know each other. Each person knows exactly three of the others. Determine whether the following two conditions can be satisfied simultaneously: [list] – for any three people, at least two do not know each other; – for any four people there are at least two who know each other. [/list]

In the cells of a $1 \times 100$ board, Julián writes all the integers from $1$ to $100$ (inclusive) in any order of his choice, without repeating numbers. For every three consecutive cells on the board, the cell containing the middle value of the three numbers in those cells is marked. For example, if the three numbers are $7$, $99$ and $22$, then the cell with $22$ is marked. Let $S$ be the sum of all the numbers in the marked cells. Determine the minimum value that $S$ can take. [b]Note:[/b] Each marked number contributes to the sum $S$ exactly once, but it can be marked multiple times.

Let $n \geq 1$ be an integer. What is the maximum number of disjoint pairs of elements of the set $\{ 1,2,\ldots , n \}$ such that the sums of the different pairs are different integers not exceeding $n$?

Sally the snail sits on the $3 \times 24$ lattice of points $(i, j)$ for all $1 \le i \le 3$ and $1 \le j \le 24$. She wants to visit every point in the lattice exactly once. In a move, Sally can move to a point in the lattice exactly one unit away. Given that Sally starts at $(2, 1)$, compute the number of possible paths Sally can take.

In a group of $6$ people playing the card game Tractor, all $54$ cards from $3$ decks are dealt evenly to all the players at random. Each deck is dealt individually. Let the probability that no one has at least two of the same card be $X$. Find the largest integer $n$ such that the $n$th root of $X$ is rational. [i]Proposed by Sammy Charney[/i] [b]Due to the problem having infinitely many solutions, all teams who inputted answers received points.[/b]

[b]p1. [/b] What is the maximum number of $T$-shaped polyominos (shown below) that we can put into a $6 \times 6$ grid without any overlaps. The blocks can be rotated. [img]https://cdn.artofproblemsolving.com/attachments/7/6/468fd9b81e9115a4a98e4cbf6dedf47ce8349e.png[/img] [b]p2.[/b] In triangle $\vartriangle ABC$, $\angle A = 30^o$. $D$ is a point on $AB$ such that $CD \perp AB$. $E$ is a point on $AC$ such that $BE \perp AC$. What is the value of $\frac{DE}{BC}$ ? [b]p3.[/b] Given that f(x) is a polynomial such that $2f(x) + f(1 - x) = x^2$. Find the sum of squares of the coefficients of $f(x)$. [b]p4. [/b] For each positive integer $n$, there exists a unique positive integer an such that $a^2_n \le n < (a_n + 1)^2$. Given that $n = 15m^2$ , where $m$ is a positive integer greater than $1$. Find the minimum possible value of $n - a^2_n$. [b]p5.[/b] What are the last two digits of $\lfloor (\sqrt5 + 2)^{2016}\rfloor$ ? Note $\lfloor x \rfloor$ is the largest integer less or equal to x. [b]p6.[/b] Let $f$ be a function that satisfies $f(2^a3^b)) = 3a+ 5b$. What is the largest value of f over all numbers of the form $n = 2^a3^b$ where $n \le 10000$ and $a, b$ are nonnegative integers. [b]p7.[/b] Find a multiple of $21$ such that it has six more divisors of the form $4m + 1$ than divisors of the form $4n + 3$ where m, n are integers. You can keep the number in its prime factorization form. [b]p8.[/b] Find $$\sum^{100}_{i=0} \lfloor i^{3/2} \rfloor +\sum^{1000}_{j=0} \lfloor j^{2/3} \rfloor$$ where $\lfloor x \rfloor$ is the largest integer less or equal to x. [b]p9. [/b] Let $A, B$ be two randomly chosen subsets of $\{1, 2, . . . 10\}$. What is the probability that one of the two subsets contains the other? [b]p10.[/b] We want to pick $5$-person teams from a total of $m$ people such that: 1. Any two teams must share exactly one member. 2. For every pair of people, there is a team in which they are teammates. How many teams are there? (Hint: $m$ is determined by these conditions). PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

All positive integers from \( 1 \) to \( 2025 \) are written on a board. Mykhailo and Oleksii play the following game. They take turns, starting with Mykhailo, erasing one of the numbers written on the board. The game ends when exactly two numbers remain on the board. If their sum is a perfect square of an integer, Mykhailo wins; otherwise, Oleksii wins. Who wins if both players play optimally? [i]Proposed by Fedir Yudin[/i]

Two grade seven students were allowed to enter a chess tournament otherwise composed of grade eight students. Each contestant played once with each other contestant and received one point for a win, one half point for a tie and zero for a loss. The two grade seven students together gained a total of eight points and each grade eight student scored the same number of points as his classmates. How many students for grade eight participated in the chess tournament? Is the solution unique?

We say that a subset \( T \) of \(\{1, 2, \dots, 2024\}\) is [b]kawaii[/b] if \( T \) has the following properties: 1. \( T \) has at least two distinct elements; 2. For any two distinct elements \( x \) and \( y \) of \( T \), \( x - y \) does not divide \( x + y \). For example, the subset \( T = \{31, 71, 2024\} \) is [b]kawaii[/b], but \( T = \{5, 15, 75\} \) is not [b]kawaii[/b] because \( 15 - 5 = 10 \) divides \( 15 + 5 = 20 \). What is the largest possible number of elements that a [b]kawaii [/b]subset can have?

Fix an integer $n \geq 2$. An $n\times n$ sieve is an $n\times n$ array with $n$ cells removed so that exactly one cell is removed from every row and every column. A stick is a $1\times k$ or $k\times 1$ array for any positive integer $k$. For any sieve $A$, let $m(A)$ be the minimal number of sticks required to partition $A$. Find all possible values of $m(A)$, as $A$ varies over all possible $n\times n$ sieves. [i]Palmer Mebane[/i]

The diagram below shows four congruent squares and some of their diagonals. Let $T$ be the number of triangles and $R$ be the number of rectangles that appear in the diagram. Find $T + R$. [img]https://cdn.artofproblemsolving.com/attachments/1/5/f756bbe67c09c19e811011cb6b18d0ff44be8b.png[/img]

There are 2014 houses in a circle. Let $A$ be one of these houses. Santa Claus enters house $A$ and leaves a gift. Then with probability $1/2$ he visits $A$'s left neighbor and with probability $1/2$ he visits $A$'s right neighbor. He leaves a gift also in that second house, and then repeats the procedure (visits with probability $1/2$ either of the neighbors, leaves a gift, etc). Santa finishes as soon as every house has received at least one gift. Prove that any house $B$ different from $A$ has a probability of $1/2013$ of being the last house receiving a gift.