Let $\mathcal{A}_n$ be the set of $n$-tuples $x = (x_1, ..., x_n)$ with $x_i \in \{0, 1, 2\}$. A triple $x, y, z$ of distinct elements of $\mathcal{A}_n$ is called [i]good[/i] if there is some $i$ such that $\{x_i, y_i, z_i\} = \{0, 1, 2\}$. A subset $A$ of $\mathcal{A}_n$ is called [i]good[/i] if every three distinct elements of $A$ form a good triple. Prove that every good subset of $\mathcal{A}_n$ has at most $2(\frac{3}{2})^n$ elements.

Determine how many $100$-positive integer sequences satisfy the two conditions following: - At least one term of the sequence is equal to $4$ or $5$. - Any two adjacent terms differ as a maximum in $2$.

A kangaroo jumps from lattice poin to lattice point in the coordinate plane. It can make only two kinds of jumps: $ (A)$ $ 1$ to right and $ 3$ up, and $ (B)$ $ 2$ to the left and $ 4$ down. $ (a)$ The start position of the kangaroo is $ (0,0)$. Show that it can jump to the point $ (19,95)$ and determine the number of jumps needed. $ (b)$ Show that if the start position is $ (1,0)$, then it cannot reach $ (19,95)$. $ (c)$ If the start position is $ (0,0)$, find all points $ (m,n)$ with $ m,n \ge 0$ which the kangaroo can reach.

A convex 2011-gon is drawn on the board. Peter keeps drawing its diagonals in such a way, that each newly drawn diagonal intersected no more than one of the already drawn diagonals. What is the greatest number of diagonals that Peter can draw?

Given positive integer $n,k$ such that $2 \le n <2^k$. Prove that there exist a subset $A$ of $\{0,1,\cdots,n\}$ such that for any $x \neq y \in A$, ${y\choose x}$ is even, and $$|A| \ge \frac{{k\choose \lfloor \frac{k}{2} \rfloor}}{2^k} \cdot (n+1)$$

Tarik wants to choose some distinct numbers from the set $S = \{2,...,111\}$ in such a way that each of the chosen numbers cannot be written as the product of two other distinct chosen numbers. What is the maximum number of numbers Tarik can choose ?

There are $2003$ pieces of candy on a table. Two players alternately make moves. A move consists of eating one candy or half of the candies on the table (the “lesser half” if there are an odd number of candies). At least one candy must be eaten at each move. The loser is the one who eats the last candy. Which player has a winning strategy?

A [i]permutation[/i] of the integers $1901, 1902, \cdots, 2000$ is a sequence $a_1, a_2, \cdots, a_{100}$ in which each of those integers appears exactly once. Given such a permutation, we form the sequence of partial sums \[s_1 = a_1,\;\;s_2 = a_1 + a_2,\;\;s_3 = a_1 + a_2 + a_3, \; \ldots\;, \; s_{100} = a_1 + a_2 + \cdots + a_{100}.\] How many of these permutations will have no terms of the sequence $s_1, \ldots, s_{100}$ divisible by three?

The cells of a square $100 \times 100$ table are colored in one of two colors, black or white. A coloring is called admissible if for any row or column, the number $b$ of black colored cells satisfies $50 \le b \le 60$. It is permitted to recolor a cell if by doing so the resulting configuration is still admissible. Prove that one can transition from any admissible coloring of the board to any other using a sequence of valid recoloring operations.

Joe and Penny play a game. Initially there are $5000$ stones in a pile, and the two players remove stones from the pile by making a sequence of moves. On the $k$-th move, any number of stones between $1$ and $k$ inclusive may be removed. Joe makes the odd-numbered moves and Penny makes the even-numbered moves. The player who removes the very last stone is the winner. Who wins if both players play perfectly?

How many five-card hands from a standard deck of $52$ cards are full houses? A full house consists of $3$ cards of one rank and $2$ cards of another rank.

[b]p1.[/b] Twenty five horses participate in a competition. The competition consists of seven runs, five horse compete in each run. Each horse shows the same result in any run it takes part. No two horses will give the same result. After each run you can decide what horses participate in the next run. Could you determine the three fastest horses? (You don’t have stopwatch. You can only remember the order of the horses.) [b]p2.[/b] Prove that the equation $x^6-143x^5-917x^4+51x^3+77x^2+291x+1575=0$ does not have solutions in integer numbers. [b]p3.[/b] Show how we can cut the figure shown in the picture into two parts for us to be able to assemble a square out of these two parts. Show how we can assemble a square. [img]https://cdn.artofproblemsolving.com/attachments/7/b/b0b1bb2a5a99195688638425cf10fe4f7b065b.png[/img] [b]p4.[/b] The city of Vyatka in Russia produces local drink, called “Vyatka Cola”. «Vyatka Cola» is sold in $1$, $3/4$, and $1/2$-gallon bottles. Ivan and John bought $4$ gallons of “Vyatka Cola”. Can we say for sure, that they can split the Cola evenly between them without opening the bottles? [b]p5.[/b] Positive numbers a, b and c satisfy the condition $a + bc = (a + b)(a + c)$. Prove that $b + ac = (b + a)(b + c)$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

[u]Round 1[/u] [b]p1.[/b] Every angle of a regular polygon has degree measure $179.99$ degrees. How many sides does it have? [b]p2.[/b] What is $\frac{1}{20} + \frac{1}{1}+ \frac{1}{5}$ ? [b]p3.[/b] If the area bounded by the lines $y = 0$, $x = 0$, and $x = 3$ and the curve $y = f(x)$ is $10$ units, what is the area bounded by $y = 0$, $x = 0$, $x = 6$, and $y = f(x/2)$? [u]Round 2[/u] [b]p4.[/b] How many ways can $42$ be expressed as the sum of $2$ or more consecutive positive integers? [b]p5.[/b] How many integers less than or equal to $2015$ can be expressed as the sum of $2$ (not necessarily distinct) powers of two? [b]p6.[/b] $p,q$, and $q^2 - p^2$ are all prime. What is $pq$? [u]Round 3[/u] [b]p7.[/b] Let $p(x) = x^2 + ax + a$ be a polynomial with integer roots, where $a$ is an integer. What are all the possible values of $a$? [b]p8.[/b] In a given right triangle, the perimeter is $30$ and the sum of the squares of the sides is $338$. Find the lengths of the three sides. [b]p9.[/b] Each of the $6$ main diagonals of a regular hexagon is drawn, resulting in $6$ triangles. Each of those triangles is then split into $4$ equilateral triangles by connecting the midpoints of the $3$ sides. How many triangles are in the resulting figure? [u]Round 4[/u] [b]p10.[/b] Let $f = 5x+3y$, where $x$ and $y$ are positive real numbers such that $xy$ is $100$. Find the minimum possible value of $f$. [b]p11.[/b] An integer is called "Awesome" if its base $8$ expression contains the digit string $17$ at any point (i.e. if it ever has a $1$ followed immediately by a $7$). How many integers from $1$ to $500$ (base $10$) inclusive are Awesome? [b]p12.[/b] A certain pool table is a rectangle measuring $15 \times 24$ feet, with $4$ holes, one at each vertex. When playing pool, Joe decides that a ball has to hit at least $2$ sides before getting into a hole or else the shot does not count. What is the minimum distance a ball can travel after being hit on this table if it was hit at a vertex (assume it only stops after going into a hole) such that the shot counts? PS. You should use hide for answers. Rounds 5-8 have been posted [url=https://artofproblemsolving.com/community/c3h3157013p28696685]here [/url] and 9-12 [url=https://artofproblemsolving.com/community/c3h3158564p28715928]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

In a foreign island $5781$ sheep are sacrificed every year for the two deities of the island, Alice and Bob. Every deity wants as many sheep as he can to be sacrificed for him, and not for the other deity. Initially all $5781$ sheep are arranged around a circle with equal distances. At the first step, Alice puts one magic stone between several pairs of neighboring sheep, so that the total number of magic stones is odd. After that, Bob sacrifices one of the sheep for himself and replaces it by a food bucket. At the third step, Alice chooses a pair of neighboring sheep (not including the two which are separated by the bucket) and puts a border between them (the border may be at the same place as a magic stone). After that, all remaining sheep walk through an arc of the circle to the food bucket without crossing the border (so that there is only one possible route). Every sheep which walks on an odd number of magic stones is sacrificed for Alice, and every other sheep is for Bob. What is the maximal number of sheep which Alice can sacrifice for herself in a certain year, regardless of Bob's action?

There are $10$ girls in a class, all with different heights. They want to form a queue so that no girl stands directly between two girls shorter than her. How many ways are there to form the queue?

We are having  99 equal circles in a row and in the interior, we write inside them all the numbers from 1 up to 99 (one number in each circle).We color each of the circles with one of the two colors available: red and green. A coloring is good if it has the ability: Red circles lying in the interval of the numbers from 1 up to 50  are more than the red circles lying in the interval of the numbers from 51  up to 99 . a) Find how many different colorings can be constructed. b) Find how many different good colorings can be constructed. (Note: Two colorings are different, if they have different color in at least one of their circles.)

Druv has a $33 \times 33$ grid of unit squares, and he wants to color each unit square with exactly one of three distinct colors such that he uses all three colors and the number of unit squares with each color is the same. However, he realizes that there are internal sides, or unit line segments that have exactly one unit square on each side, with these two unit squares having different colors. What is the minimum possible number of such internal sides?

Let $G$ be a weighted bipartite graph $A \cup B$, with $|A| = |B| = n$. In other words, each edge in the graph is assigned a positive integer value, called its [i]weight.[/i] Also, define the weight of a perfect matching in $G$ to be the sum of the weights of the edges in the matching. Let $G'$ be the graph with vertex set $A \cup B$, and (which) contains the edge $e$ if and only if $e$ is part of some minimum weight perfect matching in $G$. Show that all perfect matchings in $G'$ have the same weight.

The cells of some rectangular $M \times n$ array are colored, each by one of two colors, so that for any two columns the number of pairs of cells situated on a same row and bearing the same color is less than the number of pairs of cells situated on a same row and bearing different colors. i) Prove that if $M=2011$ then $n \leq 2012$ (a model for the extremal case $n=2012$ does indeed exist, but you are not asked to exhibit one). ii) Prove that if $M=2011=n$, each of the colors appears at most $1006\cdot 2011$ times, and at least $1005\cdot 2011$ times. iii) Prove that if however $M=2012$ then $n \leq 1007$. ([i]Dan Schwarz[/i])

Finite set $\{a_1, a_2, ..., a_n, b_1, b_2, ..., b_n\}=\{1, 2, …, 2n\}$ is given. If $a_1<a_2<...<a_n$ and $b_1>b_2>...>b_n$, show that $$\sum_{i=1}^n |a_i-b_i|=n^2$$

Consider $n$ points $A_1,A_2,A_3,\ldots, A_n$ ($n\geq 4$) in the plane, such that any three are not collinear. Some pairs of distinct points among $A_1,A_2,A_3,\ldots, A_n$ are connected by segments, such that every point is connected with at least three different points. Prove that there exists $k>1$ and the distinct points $X_1,X_2,\ldots, X_{2k}$ in the set $\{A_1,A_2,A_3,\ldots, A_n\}$, such that for every $i\in \overline{1,2k-1}$ the point $X_i$ is connected with $X_{i+1}$, and $X_{2k}$ is connected with $X_1$.

Two players, Aurelio and Bernardo, play the following game. Aurelio begins by writing the number $1$. Next it is Bernardo's turn, who writes number $2$. From then on, each player chooses whether to add $1$ to the number just written by the previous player, or whether multiply that number by $2$. Then write the result and it's the other player's turn. The first player to write a number greater than $ 2007$ loses the game. Determine if one of the players can ensure victory no matter what the other does.

Ana and Carlos entertain themselves with the next game. At the beginning of game in each vertex of the square there is an empty box. In each step, the corresponding player has two possibilities: either he adds a stone to an arbitrary box, or move each box clockwise to the next vertex of the square. Carlos starts and they take 2012 steps in turn (each player 1006). So Carlos marks one of the vertices of the square and allows Ana to make a more play. Carlos wins if after this last step the number ofstones in some box is greater than the number of stones in the box which is at the vertex marked by Carlos; otherwise Ana wins. Which of the two players has a winning strategy?

[b]p1.[/b] Show that $9x + 5y$ is a multiple of$ 17$ whenever $2x + 3y$ is a multiple of $17$. [b]p2.[/b] Express the five distinct fifth roots of $1$ in terms of radicals. [b]p3.[/b] Prove that the three perpendiculars dropped to the three sides of an equilateral triangle from any point inside the triangle have a constant sum. [b]p4.[/b] Find the volume of a sphere which circumscribes a regular tetrahedron of edge $a$. [b]p5.[/b] For any integer $n$ greater than $1$, show that $n^2-2n + 1$ is a factor at $n^{n-1}-1$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find the least natural number $ n$ such that, if the set $ \{1,2, \ldots, n\}$ is arbitrarily divided into two non-intersecting subsets, then one of the subsets contains 3 distinct numbers such that the product of two of them equals the third.