An $8 \times 8$ chessboard is covered completely and without overlaps by $32$ dominoes of size $1 \times 2$. Show that there are two dominoes forming a $2 \times 2$ square.

Suppose a $m \times n$ table. We write an integer in each cell of the table. In each move, we chose a column, a row, or a diagonal (diagonal is the set of cells which the difference between their row number and their column number is constant) and add either $+1$ or $-1$ to all of its cells. Prove that if for all arbitrary $3 \times 3$ table we can change all numbers to zero, then we can change all numbers of $m \times n$ table to zero. ([i]Hint[/i]: First of all think about it how we can change number of $ 3 \times 3$ table to zero.)

Let us call "[i]big[/i]" a triangle with all sides longer than $1$. Given a equilateral triangle with all the sides equal to $5$. Prove that: a) You can cut $100$ [i]big [/i] triangles out of given one. b) You can divide the given triangle onto $100$ [i]big [/i] nonintersecting ones fully covering the initial one. c) The same as b), but the triangles either do not have common points, or have one common side, or one common vertex. d) The same as c), but the initial triangle has the side $3$.

We attach to the vertices of a regular hexagon the numbers $1$, $0$, $0$, $0$, $0$, $0$. Now, we are allowed to transform the numbers by the following rules: (a) We can add an arbitrary integer to the numbers at two opposite vertices. (b) We can add an arbitrary integer to the numbers at three vertices forming an equilateral triangle. (c) We can subtract an integer $t$ from one of the six numbers and simultaneously add $t$ to the two neighbouring numbers. Can we, just by acting several times according to these rules, get a cyclic permutation of the initial numbers? (I. e., we started with $1$, $0$, $0$, $0$, $0$, $0$; can we now get $0$, $1$, $0$, $0$, $0$, $0$, or $0$, $0$, $1$, $0$, $0$, $0$, or $0$, $0$, $0$, $1$, $0$, $0$, or $0$, $0$, $0$, $0$, $1$, $0$, or $0$, $0$, $0$, $0$, $0$, $1$ ?)

Given a positive integer number $n$, determine the maximum number of edges a simple graph on $n$ vertices may have such that it contain no cycles of even length.

n \geq 6 is an integer. evaluate the minimum of f(n) s.t: any graph with n vertices and f(n) edge contains two cycle which are distinct( also they have no comon vertice)?

[i]25 problems for 30 minutes.[/i] [b]p1.[/b] Chad, Ravi, Kevin, and Meena are four of the $551$ residents of Chadwick, Illinois. Expressing your answer to the nearest percent, how much of the population do they represent? [b]p2.[/b] Points $A$, $B$, and $C$ are on a line for which $AB = 625$ and $BC = 256$. What is the sum of all possible values of the length $AC$? [b]p3.[/b] An increasing arithmetic sequence has first term $2014$ and common difference $1337$. What is the least odd term of this sequence? [b]p4.[/b] How many non-congruent scalene triangles with integer side lengths have two sides with lengths $3$ and $4$? [b]p5.[/b] Let $a$ and $b$ be real numbers for which the function $f(x) = ax^2+bx+3$ satisfies $f(0)+2^0 = f(1)+2^1 = f(2) + 2^2$. What is $f(0)$? [b]p6.[/b] A pentomino is a set of five planar unit squares that are joined edge to edge. Two pentominoes are considered the same if and only if one can be rotated and translated to be identical to the other. We say that a pentomino is compact if it can fit within a $2$ by $3$ rectangle. How many distinct compact pentominoes exist? [b]p7.[/b] Consider a hexagon with interior angle measurements of $91$, $101$, $107$, $116$, $152$, and $153$ degrees. What is the average of the interior angles of this hexagon, in degrees? [b]p8.[/b] What is the smallest positive number that is either one larger than a perfect cube and one less than a perfect square, or vice versa? [b]p9.[/b] What is the first time after $4:56$ (a.m.) when the $24$-hour expression for the time has three consecutive digits that form an increasing arithmetic sequence with difference $1$? (For example, $23:41$ is one of those moments, while $23:12$ is not.) [b]p10.[/b] Chad has trouble counting. He wants to count from $1$ to $100$, but cannot pronounce the word "three," so he skips every number containing the digit three. If he tries to count up to $100$ anyway, how many numbers will he count? [b]p11.[/b] In square $ABCD$, point $E$ lies on side $BC$ and point $F$ lies on side $CD$ so that triangle $AEF$ is equilateral and inside the square. Point $M$ is the midpoint of segment $EF$, and $P$ is the point other than $E$ on $AE$ for which $PM = FM$. The extension of segment $PM$ meets segment $CD$ at $Q$. What is the measure of $\angle CQP$, in degrees? [b]p12.[/b] One apple is five cents cheaper than two bananas, and one banana is seven cents cheaper than three peaches. How much cheaper is one apple than six peaches, in cents? [b]p13.[/b] How many ordered pairs of integers $(a, b)$ exist for which |a| and |b| are at most $3$, and $a^3-a = b^3-b$? [b]p14.[/b] Five distinct boys and four distinct girls are going to have lunch together around a table. They decide to sit down one by one under the following conditions: no boy will sit down when more boys than girls are already seated, and no girl will sit down when more girls than boys are already seated. How many possible sequences of taking seats exist? [b]p15.[/b] Jordan is swimming laps in a pool. For each lap after the first, the time it takes her to complete is five seconds more than that of the previous lap. Given that she spends 10 minutes on the first six laps, how long does she spend on the next six laps, in minutes? [b]p16.[/b] Chad decides to go to trade school to ascertain his potential in carpentry. Chad is assigned to cut away all the vertices of a wooden regular tetrahedron with sides measuring four inches. Each vertex is cut away by a plane which passes through the three midpoints of the edges adjacent to that vertex. What is the surface area of the resultant solid, in square inches? Note: A tetrahedron is a solid with four triangular faces. In a regular tetrahedron, these faces are all equilateral triangles. [b]p17.[/b] Chad and Jordan independently choose two-digit positive integers. The two numbers are then multiplied together. What is the probability that the result has a units digit of zero? [b]p18.[/b] For art class, Jordan needs to cut a circle out of the coordinate grid. She would like to find a circle passing through at least $16$ lattice points so that her cut is accurate. What is the smallest possible radius of her circle? Note: A lattice point is defined as one whose coordinates are both integers. For example, $(5, 8)$ is a lattice point whereas $(3.5, 5)$ is not. [b]p19.[/b] Chad's ant Arctica is on one of the eight corners of Chad's toolbox, which measures two decimeters in width, three decimeters in length, and four decimeters in height. One day, Arctica wanted to go to the opposite corner of this box. Assuming she can only crawl on the surface of the toolbox, what is the shortest distance she has to crawl to accomplish this task, in decimeters? (You may assume that the toolbox is oating in the Exeter Space Station, so that Arctica can crawl on all six faces.) [b]p20.[/b] Jordan is counting numbers for fun. She starts with the number $1$, and then counts onward, skipping any number that is a divisor of the product of all previous numbers she has said. For example, she starts by counting $1$, $2$, $3$, $4$, $5$, but skips 6, a divisor of $1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 = 120$. What is the $20^{th}$ number she counts? [b]p21.[/b] Chad and Jordan are having a race in the lake shown below. The lake has a diameter of four kilometers and there is a circular island in the middle of the lake with a diameter of two kilometers. They start at one point on the edge of the lake and finish at the diametrically opposite point. Jordan makes the trip only by swimming in the water, while Chad swims to the island, runs across it, and then continues swimming. They both take the fastest possible route and, amazingly, they tie! Chad swims at two kilometers an hour and runs at five kilometers an hour. At what speed does Jordan swim? [img]https://cdn.artofproblemsolving.com/attachments/f/6/22b3b0bba97d25ab7aabc67d30821d0b12efc0.png[/img] [b]p22.[/b] Cameron has stolen Chad's barrel of oil and is driving it around on a truck on the coordinate grid on his truck. Cameron is a bad truck driver, so he can only move the truck forward one kilometer at a $4$ $EMC^2$ $2014$ Problems time along one of the gridlines. In fact, Cameron is so bad at driving the truck that between every two one-kilometer movements, he has to turn exactly $90$ degrees. After $50$ one-kilometer movements, given that Cameron's first one-kilometer movement was westward, how many points he could be on? [b]p23.[/b] Let $a$, $b$, and $c$ be distinct nonzero base ten digits. Assume there exist integers $x$ and $y$ for which $\overline{abc} \cdot \overline{cb} = 100x^2 + 1$ and $\overline{acb} \cdot \overline{bc} = 100y^2 + 1$. What is the minimum value of the number $\overline{abbc}$? Note: The notation $\overline{pqr}$ designates the number whose hundreds digit is $p$, tens digit is $q$, and units digit is $r$, not the product $p \cdot q \cdot r$. [b]p24.[/b] Let $r_1, r_2, r_3, r_4$ and $r_5$ be the five roots of the equation $x^5-4x^4+3x^2-2x+1 = 0$. What is the product of $(r_1 +r_2 +r_3 +r_4)$, $(r_1 +r_2 +r_3 +r_5)$, $(r_1 +r_2 +r_4 +r_5)$, $(r_1 +r_3 +r_4 +r_5)$, and $(r_2 +r_3 +r_4 +r_5)$? [b]p25.[/b] Chad needs seven apples to make an apple strudel for Jordan. He is currently at 0 on the metric number line. Every minute, he randomly moves one meter in either the positive or the negative direction with equal probability. Arctica's parents are located at $+4$ and $-4$ on the number line. They will bite Chad for kidnapping Arctica if he walks onto those numbers. Also, there is one apple located at each integer between $-3$ and $3$, inclusive. Whenever Chad lands on an integer with an unpicked apple, he picks it. What is the probability that Chad picks all the apples without getting bitten by Arctica's parents? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

The vertices of a regular $2005$-gon are colored red, white and blue. Whenever two vertices of different colors stand next to each other, we are allowed to recolor them into the third color. (a) Prove that there exists a finite sequence of allowed recolorings after which all the vertices are of the same color. (b) Is that color uniquely determined by the initial coloring?

Let $n$ be a positive integer and let $0\leq k\leq n$ be an integer. Show that there exist $n$ points in the plane with no three on a line such that the points can be divided into two groups satisfying the following properties. $\text{(i)}$ The first group has $k$ points and the distance between any two distinct points in this group is irrational. $\text{(ii)}$ The second group has $n-k$ points and the distance between any two distinct points in this group is an integer. $\text{(iii)}$ The distance between a point in the first group and a point in the second group is irrational.

[b](a)[/b]Prove that every pentagon with integral coordinates has at least two vertices , whose respective coordinates have the same parity. [b](b)[/b]What is the smallest area possible of pentagons with integral coordinates. Albanian National Mathematical Olympiad 2010---12 GRADE Question 3.

Jack plays a game in which he first rolls a fair six-sided die and gets some number $n$, then, he flips a coin until he flips $n$ heads in a row and wins, or he flips $n$ tails in a row in which case he rerolls the die and tries again. What is the expected number of times Jack must flip the coin before he wins the game.

Consider the set $A = \{1, 2, ..., n\}$, where $n \in N, n \ge 6$. Show that $A$ is the union of three pairwise disjoint sets, with the same cardinality and the same sum of their elements, if and only if $n$ is a multiple of $3$.

[b]p1.[/b] Compute the degree of the least common multiple of the polynomials $x - 1$, $x^2 - 1$, $x^3 - 1$,$...$, $x^{10} -1$. [b]p2.[/b] A line in the $xy$ plane is called wholesome if its equation is $y = mx+b$ where $m$ is rational and $b$ is an integer. Given a point with integer coordinates $(x,y)$ on a wholesome line $\ell$, let $r$ be the remainder when $x$ is divided by $7$, and let $s$ be the remainder when y is divided by $7$. The pair $(r, s)$ is called an [i]ingredient[/i] of the line $\ell$. The (unordered) set of all possible ingredients of a wholesome line $\ell$ is called the [i]recipe [/i] of $\ell$. Compute the number of possible recipes of wholesome lines. [b]p3.[/b] Let $\tau (n)$ be the number of distinct positive divisors of $n$. Compute $\sum_{d|15015} \tau (d)$, that is, the sum of $\tau (d)$ for all $d$ such that $d$ divides $15015$. [b]p4.[/b] Suppose $2202010_b - 2202010_3 = 71813265_{10}$. Compute $b$. ($n_b$ denotes the number $n$ written in base $b$.) [b]p5.[/b] Let $x = (3 -\sqrt5)/2$. Compute the exact value of $x^8 + 1/x^8$. [b]p6.[/b] Compute the largest integer that has the same number of digits when written in base $5$ and when written in base $7$. Express your answer in base $10$. [b]p7.[/b] Three circles with integer radii $a$, $b$, $c$ are mutually externally tangent, with $a \le b \le c$ and $a < 10$. The centers of the three circles form a right triangle. Compute the number of possible ordered triples $(a, b, c)$. [b]p8.[/b] Six friends are playing informal games of soccer. For each game, they split themselves up into two teams of three. They want to arrange the teams so that, at the end of the day, each pair of players has played at least one game on the same team. Compute the smallest number of games they need to play in order to achieve this. [b]p9.[/b] Let $A$ and $B$ be points in the plane such that $AB = 30$. A circle with integer radius passes through $A$ and $B$. A point $C$ is constructed on the circle such that $AC$ is a diameter of the circle. Compute all possible radii of the circle such that $BC$ is a positive integer. [b]p10.[/b] Each square of a $3\times 3$ grid can be colored black or white. Two colorings are the same if you can rotate or reflect one to get the other. Compute the total number of unique colorings. [b]p11.[/b] Compute all positive integers $n$ such that the sum of all positive integers that are less than $n$ and relatively prime to $n$ is equal to $2n$. [b]p12.[/b] The distance between a point and a line is defined to be the smallest possible distance between the point and any point on the line. Triangle $ABC$ has $AB = 10$, $BC = 21$, and $CA = 17$. Let $P$ be a point inside the triangle. Let $x$ be the distance between $P$ and $\overleftrightarrow{BC}$, let $y$ be the distance between $P$ and $\overleftrightarrow{CA}$, and let $z$ be the distance between $P$ and $\overleftrightarrow{AB}$. Compute the largest possible value of the product $xyz$. [b]p13.[/b] Alice, Bob, David, and Eve are sitting in a row on a couch and are passing back and forth a bag of chips. Whenever Bob gets the bag of chips, he passes the bag back to the person who gave it to him with probability $\frac13$ , and he passes it on in the same direction with probability $\frac23$ . Whenever David gets the bag of chips, he passes the bag back to the person who gave it to him with probability $\frac14$ , and he passes it on with probability $\frac34$ . Currently, Alice has the bag of chips, and she is about to pass it to Bob when Cathy sits between Bob and David. Whenever Cathy gets the bag of chips, she passes the bag back to the person who gave it to her with probability $p$, and passes it on with probability $1-p$. Alice realizes that because Cathy joined them on the couch, the probability that Alice gets the bag of chips back before Eve gets it has doubled. Compute $p$. [b]p14.[/b] Circle $O$ is in the plane. Circles $A$, $B$, and $C$ are congruent, and are each internally tangent to circle $O$ and externally tangent to each other. Circle $X$ is internally tangent to circle $O$ and externally tangent to circles $A$ and $B$. Circle $X$ has radius $1$. Compute the radius of circle $O$. [img]https://cdn.artofproblemsolving.com/attachments/f/d/8ddab540dca0051f660c840c0432f9aa5fe6b0.png[/img] [b]p15.[/b] Compute the number of primes $p$ less than 100 such that $p$ divides $n^2 +n+1$ for some integer $n$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

There are $76$ cards with different numbers written on them. These cards are laid out on the table in a circle, number down. Try to find some three cards in a row such that the number written on the middle of these three cards is greater than on each of the two neighboring ones. You can turn over no more than $10$ cards in succession. How should one proceed to be sure to find three cardboard boxes for which the specified condition is met?

Four teams $A$, $B$, $C$ and $D$ play a football tournament in which each team plays exactly two times against each of the remaining three teams (there are $12$ matches). In each matchif it's a tie each team gets $1$ point and if it isn't a tie then the winner gets $3$ points and the loser gets $0$ points. At the end of the tournament the teams $A$, $B$ and $C$ have $8$ points each. Determine all possible points of team $D$.

Sofia and Viktor are playing the following game on a $2022 \times 2022$ board: - Firstly, Sofia covers the table completely by dominoes, no two are overlapping and all are inside the table; - Then Viktor without seeing the table, chooses a positive integer $n$; - After that Viktor looks at the table covered with dominoes, chooses and fixes $n$ of them; - Finally, Sofia removes the remaining dominoes that aren't fixed and tries to recover the table with dominoes differently from before. If she achieves that, she wins, otherwise Viktor wins. What is the minimum number $n$ for which Viktor can always win, no matter the starting covering of dominoes. [i]Proposed by Viktor Simjanoski[/i]

Is it possible to paint all natural numbers in $6$ colors, for each one color to be used and the sum of any five numbers of different color to be painted in the sixth color?

Julia is placing identical $1$-by-$1$ tiles on the $2$-by-$2$ grid pictured, one piece at a time, so that every piece she places after the first is adjacent to, but not on top of, some piece she’s already placed. Determine the number of ways that Julia can complete the grid. [img]https://cdn.artofproblemsolving.com/attachments/4/6/4a585593b9301ddb0e4ac3ceced212c378c9f8.png[/img]

Prove that for any given number $a_k, 1 \le k \le 5$, there are $\lambda_k \in \{-1, 0, 1\}, 1 \le k \le 5$, which are not all equal zero, such that $11 | \lambda_1a_1^2+\lambda_2a_2^2+\lambda_3a_3^2+\lambda_4a_4^2+\lambda_5a_5^2$

In a tournament with $55$ participants, one match is played at a time, with the loser dropping out. In each match, the numbers of wins so far of the two participants differ by not more than $1$. What is the maximal number of matches for the winner of the tournament?

Given are sheets and the numbers $00, 01, \ldots, 99$ are written on them. We must put them in boxes $000, 001, \ldots, 999$ so that the number on the sheet is the number on the box with one digit erased. What is the minimum number of boxes we need in order to put all the sheets?

In a certain forest there are at least three crossroads, and for any three crossroads of roads A, B and C there is a road from A to B without passing through C. A deer and a hunter are standing at crossroads of different paths. Is it possible that they can exchange positions without their paths crossing at other points, that are not their initial positions?

Consider a random string $s$ of $10^{2020}$ base-ten digits (there can be leading zeroes). We say a substring $s' $ (which has no leading zeroes) is self-locating if $s' $ appears in $s$ at index $s' $ where the string is indexed at $ 1$. For example the substring $11$ in the string “$122352242411$” is selflocating since the $11$th digit is $ 1$ and the $12$th digit is $ 1$. Let the expected number of self-locating substrings in s be $G$. Compute $\lfloor G \rfloor$.

$2014$ points are placed on a circumference. On each of the segments with end points on two of the $2014$ points is written a non-negative real number. For any convex polygon with vertices on some of the $2014$ points, the sum of the numbers written on their sides is less or equal than $1$. Find the maximum possible value for the sum of all the written numbers.

Is it true that for all natural $n$, it is always possible to give each of the $n$ children a hat painted in one of $100$ colors so that if a girl is known to more than $2015$ boys, then not all of these boys have hats of the same color, and, if a boy is acquainted with more than $2015$ girls, don't all these girls have hats of the same color? [hide=original wording]Vai tiesa, ka visiem naturāliem n vienmēr iespējams katram no n bērniem iedot pa cepurei, kas nokrāsota vienā no 100 krāsām tā, ka, ja kāda meitene ir pazīstama ar vairāk nekā 2015 zēniem, tad ne visiem šiem zēniem cepures ir vienā krāsā, un, ja kāds zēns ir pazīštams ar vairāk nekā 2015 meitenēm, tad ne visām šīm meitenēm cepures ir vienā krāsā?[/hide]