Each point on the plane is colored either red, green, or blue. Prove that there exists an isosceles triangle whose vertices all have the same color.

A rectangle with sides of lengths $a$ and $b$ ($a > b$) is cut into rightangled triangles so that any two of these triangles either have a common side, a common vertex or no common points. Moreover, any common side of two triangles is a leg of one of them and the hypotenuse of the other. Prove that $a > 2b$. (A Shapovalov)

There are $n$ boxes which is numbere from $1$ to $n$. The box with number $1$ is open, and the others are closed. There are $m$ identical balls ($m\geq n$). One of the balls is put into the open box, then we open the box with number $2$. Now, we put another ball to one of two open boxes, then we open the box with number $3$. Go on until the last box will be open. After that the remaining balls will be randomly put into the boxes. In how many ways this arrangement can be done?

Let $m,n$ be positive integers with $m \le n$, and let $F$ be a family of $m$-element subsets of $\{1,2,...,n\}$ satisfying $A \cap B \ne \varnothing$ for all $A,B \in F$. Determine the maximum possible number of elements in $F$.

Is it possible to partition $3$-dimensional Euclidean space into $1979$ mutually isometric subsets?

Let $n > 2$ be an integer. The numbers $1, 2, . . . , n$ are written at the vertices of an $n$-gon in that order. A move consists of choosing two adjacent vertices and adding $1$ to the numbers written there. Determine all n for which it is possible to achieve that all numbers are identical after a finite number of moves.

A group of aliens from Gliese $667$ Cc come to Earth to test the hypothesis that mathematics is indeed a universal language. To do this, they give you the following information about their mathematical system: $\bullet$ For the purposes of this experiment, the Gliesians have decided to write their equations in the same syntactic format as in Western math. For example, in Western math, the expression “$5+4$” is interpreted as running the “$+$” operation on numbers $5$ and $4$. Similarly, in Gliesian math, the expression $\alpha \gamma \beta$ is interpreted as running the “$\gamma $” operation on numbers $\alpha$ and $ \beta$. $\bullet$ You know that $\gamma $ and $\eta$ are the symbols for addition and multiplication (which works the same in Gliesian math as in Western math), but you don’t know which is which. By some bizarre coincidence, the symbol for equality is the same in Gliesian math as it is in Western math; equality is denoted with an “$=$” symbol between the two equal values. $\bullet$ Two symbols that look exactly the same have the same meaning. Two symbols that are different have different meanings and, therefore, are not equal. They then provide you with the following equations, written in Gliesian, which are known to be true: [img]https://cdn.artofproblemsolving.com/attachments/b/e/e2e44c257830ce8eee7c05535046c17ae3b7e6.png[/img]

A subset $S$ of size $n$ of a plane consisting of points with both coordinates integer is given, where $n$ is an odd number. The injective function $f\colon S\rightarrow S$ satisfies the following: for each pair of points $A, B\in S$, the distance between points $f(A)$ and $f(B)$ is not smaller than the distance between points $A$ and $B$. Prove there exists a point $X$ such that $f(X)=X$.

For each positive integer $ n$, define $ f(n)$ as the number of digits $ 0$ in its decimal representation. For example, $ f(2)\equal{}0$, $ f(2009)\equal{}2$, etc. Please, calculate \[ S\equal{}\sum_{k\equal{}1}^{n}2^{f(k)},\] for $ n\equal{}9,999,999,999$. [i]Yudi Satria, Jakarta[/i]

How many ways are there to represent $10^6$ as the product of three factors? Factorizations which only differ in the order of the factors are considered to be distinct.

[u]Round 9[/u] [b]p25.[/b] Let $S$ be the region bounded by the lines $y = x/2$, $y = -x/2$, and $x = 6$. Pick a random point $P = (x, y)$ in $S$ and translate it $3$ units right to $P' = (x + 3, y)$. What is the probability that $P'$ is in $S$? [b]p26.[/b] A triangle with side lengths $17$, $25$, and $28$ has a circle centered at each of its three vertices such that the three circles are mutually externally tangent to each other. What is the combined area of the circles? [b]p27.[/b] Find all ordered pairs $(x, y)$ of integers such that $x^2 - 2x + y^2 - 6y = -9$. [u]Round 10[/u] [b]p28.[/b] In how many ways can the letters in the word $SCHAFKOPF$ be arranged if the two $F$’s cannot be next to each other and the $A$ and the $O$ must be next to each other? [b]p29.[/b] Let a sequence $a_0, a_1, a_2, ...$ be defined by $a_0 = 20$, $a_1 = 11$, $a_2 = 0$, and for all integers $n \ge 3$, $$a_n + a_{n-1 }= a_{n-2} + a_{n-3}.$$ Find the sum $a_0 + a_1 + a_2 + · · · + a_{2010} + a_{2011}$. [b]p30.[/b] Find the sum of all positive integers b such that the base $b$ number $190_b$ is a perfect square. [u]Round 11[/u] [b]p31.[/b] Find all real values of x such that $\sqrt[3]{4x -1} + \sqrt[3]{4x + 1 }= \sqrt[3]{8x}$. [b]p32.[/b] Right triangle $ABC$ has a right angle at B. The angle bisector of $\angle ABC$ is drawn and extended to a point E such that $\angle ECA = \angle ACB$. Let $F$ be the foot of the perpendicular from $E$ to ray $\overrightarrow{BC}$. Given that $AB = 4$, $BC = 2$, and $EF = 8$, find the area of triangle $ACE$. [b]p33.[/b] You are the soul in the southwest corner of a four by four grid of distinct souls in the Fields of Asphodel. You move one square east and at the same time all the other souls move one square north, south, east, or west so that each square is now reoccupied and no two souls switched places directly. How many end results are possible from this move? [u]Round 12[/u] [b]p34.[/b] A [i]Pythagorean [/i] triple is an ordered triple of positive integers $(a, b, c)$ with $a < b < c $and $a^2 + b^2 = c^2$ . A [i]primitive [/i] Pythagorean triple is a Pythagorean triple where all three numbers are relatively prime to each other. Find the number of primitive Pythagorean triples in which all three members are less than $100,000$. If $P$ is the true answer and $A$ is your team’s answer to this problem, your score will be $max \left\{15 -\frac{|A -P|}{500} , 0 \right\}$ , rounded to the nearest integer. [b]p35.[/b] According to the Enable2k North American word list, how many words in the English language contain the letters $L, M, T$ in order but not necessarily together? If $A$ is your team’s answer to this problem and $W$ is the true answer, the score you will receive is $max \left\{15 -100\left| \frac{A}{W}-1\right| , 0 \right\}$, rounded to the nearest integer. [b]p36.[/b] Write down $5$ positive integers less than or equal to $42$. For each of the numbers written, if no other teams put down that number, your team gets $3$ points. Otherwise, you get $0$ points. Any number written that does not satisfy the given requirement automatically gets $0$ points. PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h2952214p26434209]here[/url] and 5-8 [url=https://artofproblemsolving.com/community/c3h3133709p28395558]here[/url]. Rest Rounds soon. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Eight football teams participate in a tournament of one round (each team plays each other team once) . There are no draws. Prove that it is possible at the conclusion of the tournament to be able to find $4$ teams , say $A, B, C$ and $D$ so that $A$ defeated $B, C$ and $D, B$ defeated $C$ and $D$ , and $C$ defeated $D$ .

Find maximal numbers of planes, such there are $6$ points and 1) $4$ or more points lies on every plane. 2) No one line passes through $4$ points.

A baseball league has $64$ people, each with a different $6$-digit binary number whose base-$10$ value ranges from $0$ to $63$. When any player bats, they do the following: for each pitch, they swing if their corresponding bit number is a $1$, otherwise, they decide to wait and let the ball pass. For example, the player with the number $11$ has binary number $001011$. For the first and second pitch, they wait; for the third, they swing, and so on. Pitchers follow a similar rule to decide whether to throw a splitter or a fastball, if the bit is $0$, they will throw a splitter, and if the bit is $1$, they will throw a fastball. If a batter swings at a fastball, then they will score a hit; if they swing on a splitter, they will miss and get a “strike.” If a batter waits on a fastball, then they will also get a strike. If a batter waits on a splitter, then they get a “ball.” If a batter gets $3$ strikes, then they are out; if a batter gets $4$ balls, then they automatically get a hit. For example, if player $11$ pitched against player $6$ (binary is $000110$), the batter would get a ball for the first pitch, a ball for the second pitch, a strike for the third pitch, a strike for the fourth pitch, and a hit for the fifth pitch; as a result, they will count that as a “hit.” If player $11$ pitched against player $5$ (binary is $000101$), however, then the fifth pitch would be the batter’s third strike, so the batter would be “out.” Each player in the league plays against every other player exactly twice; once as batter, and once as pitcher. They are then given a score equal to the number of outs they throw as a pitcher plus the number of hits they get as a batter. What is the highest score received?

Consider a round-robin tournament with $2n+1$ teams, where each team plays each other team exactly one. We say that three teams $X,Y$ and $Z$, form a [i]cycle triplet [/i] if $X$ beats $Y$, $Y$ beats $Z$ and $Z$ beats $X$. There are no ties. a)Determine the minimum number of cycle triplets possible. b)Determine the maximum number of cycle triplets possible.

Let $n$ be a positive integer. Every square in a $n \times n$-square grid is either white or black. How many such colourings exist, if every $2 \times 2$-square consists of exactly two white and two black squares? The squares in the grid are identified as e.g. in a chessboard, so in general colourings obtained from each other by rotation are different.

The positive integer $n \geqslant 2$ is given. How many ways can the cells of the $n\times n$ square be colored in four colors so that any two cells with a common side or vertex are colored in different colors? [i] I. Efremov [/i]

Construct a tetromino by attaching two $2 \times 1$ dominoes along their longer sides such that the midpoint of the longer side of one domino is a corner of the other domino. This construction yields two kinds of tetrominoes with opposite orientations. Let us call them $S$- and $Z$-tetrominoes, respectively. Assume that a lattice polygon $P$ can be tiled with $S$-tetrominoes. Prove that no matter how we tile $P$ using only $S$- and $Z$-tetrominoes, we always use an even number of $Z$-tetrominoes. [i]Proposed by Tamas Fleiner and Peter Pal Pach, Hungary[/i]

The integers from $1$ to $n$ are written, one on each of $n$ cards. The first player removes one card. Then the second player removes two cards with consecutive integers. After that the first player removes three cards with consecutive integers. Finally, the second player removes four cards with consecutive integers. What is th smallest value of $n$ for which the second player can ensure that he competes both his moves?

A table $10 \times 10$ was filled according to the rules of the game “Bomb Squad”: several cells contain bombs (one bomb per cell) while each of the remaining cells contains a number, equal to the number of bombs in all cells adjacent to it by side or by vertex. Then the table is rearranged in the “reverse” order: bombs are placed in all cells previously occupied with numbers and the remaining cells are filled with numbers according to the same rule. Can it happen that the total sum of the numbers in the table will increase in a result?

Suppose that integers are given $m <n $. Consider a spreadsheet of size $n \times n $, whose cells arbitrarily record all integers from $1 $ to ${{n} ^ {2}} $. Each row of the table is colored in yellow $m$ the largest elements. Similarly, the blue colors the $m$ of the largest elements in each column. Find the smallest number of cells that are colored yellow and blue at a time

There is a $2n \times 2n$ array (matrix) consisting of $0's$ and $1's$ and there are exactly $3n$ zeroes. Show that it is possible to remove all the zeroes by deleting some $n$ rows and some $n$ columns.

Each of the squares in a $50 \times 50$ square board is filled with a number from $1$ to $50$ so that each of the numbers $1,2, ..., 50$ appears exactly $50$ times. Prove that there is a row or a column containing at least $8$ distinct numbers.

We have $2^m$ sheets of paper, with the number $1$ written on each of them. We perform the following operation. In every step we choose two distinct sheets; if the numbers on the two sheets are $a$ and $b$, then we erase these numbers and write the number $a + b$ on both sheets. Prove that after $m2^{m -1}$ steps, the sum of the numbers on all the sheets is at least $4^m$ . [i]Proposed by Abbas Mehrabian, Iran[/i]

The eight points $A, B,. . ., G$ and $H$ lie on five circles as shown. Each of these letters are represented by one of the eight numbers $1, 2,. . ., 7$ and $ 8$ replaced so that the following conditions are met: (i) Each of the eight numbers is used exactly once. (ii) The sum of the numbers on each of the five circles is the same. How many ways are there to get the letters substituted through the numbers in this way? (Walther Janous) [img]https://cdn.artofproblemsolving.com/attachments/5/e/511cdd2fc31e8067f400369c4fe9cf964ef54c.png[/img]