Prove that in any group of people, the number of those who know an odd number of the others in the group is even. Assume that knowing is a symmetric relation.

Let $m$ and $n$ two given integers. Ana thinks of a pair of real numbers $x$, $y$ and then she tells Beto the values of $x^m+y^m$ and $x^n+y^n$, in this order. Beto's goal is to determine the value of $xy$ using that information. Find all values of $m$ and $n$ for which it is possible for Beto to fulfill his wish, whatever numbers that Ana had chosen.

A set of points on the plane is [i]antiparallelogram [/i] if any four points of the set are [b]not[/b] vertices of a parallelogram. Prove that for any set of $2023$ points on the plane, [b]no[/b] three of them are collinears, there exists a subset of $17$ points, such that this subset is antiparallelogram.

The total number of solutions of $xyz=2520$ (A)$2520$ (B)$2160$ (C)$540$ (D)None of these

[b]p1.[/b] Triangle $ABC$ has side lengths $AB = 3^2$ and $BC = 4^2$. Given that $\angle ABC$ is a right angle, determine the length of $AC$. [b]p2.[/b] Suppose $m$ and $n$ are integers such that $m^2+n^2 = 65$. Find the largest possible value of $m-n$. [b]p3.[/b] Six middle school students are sitting in a circle, facing inwards, and doing math problems. There is a stack of nine math problems. A random student picks up the stack and, beginning with himself and proceeding clockwise around the circle, gives one problem to each student in order until the pile is exhausted. Aditya falls asleep and is therefore not the student who picks up the pile, although he still receives problem(s) in turn. If every other student is equally likely to have picked up the stack of problems and Vishwesh is sitting directly to Aditya’s left, what is the probability that Vishwesh receives exactly two problems? [b]p4.[/b] Paul bakes a pizza in $15$ minutes if he places it $2$ feet from the fire. The time the pizza takes to bake is directly proportional to the distance it is from the fire and the rate at which the pizza bakes is constant whenever the distance isn’t changed. Paul puts a pizza $2$ feet from the fire at $10:30$. Later, he makes another pizza, puts it $2$ feet away from the fire, and moves the first pizza to a distance of $3$ feet away from the fire instantly. If both pizzas finish baking at the same time, at what time are they both done? [b]p5.[/b] You have $n$ coins that are each worth a distinct, positive integer amount of cents. To hitch a ride with Charon, you must pay some unspecified integer amount between $10$ and $20$ cents inclusive, and Charon wants exact change paid with exactly two coins. What is the least possible value of $n$ such that you can be certain of appeasing Charon? [b]p6.[/b] Let $a, b$, and $c$ be positive integers such that $gcd(a, b)$, $gcd(b, c)$ and $gcd(c, a)$ are all greater than $1$, but $gcd(a, b, c) = 1$. Find the minimum possible value of $a + b + c$. [b]p7.[/b] Let $ABC$ be a triangle inscribed in a circle with $AB = 7$, $AC = 9$, and $BC = 8$. Suppose $D$ is the midpoint of minor arc $BC$ and that $X$ is the intersection of $\overline{AD}$ and $\overline{BC}$. Find the length of $\overline{BX}$. [b]p8.[/b] What are the last two digits of the simplified value of $1! + 3! + 5! + · · · + 2009! + 2011!$ ? [b]p9.[/b] How many terms are in the simplified expansion of $(L + M + T)^{10}$ ? [b]p10.[/b] Ben draws a circle of radius five at the origin, and draws a circle with radius $5$ centered at $(15, 0)$. What are all possible slopes for a line tangent to both of the circles? PS. You had better use hide for answers.

Let $n \ge 3$ be a positive integer. We call a $3 \times 3$ grid [i]beautiful[/i] if the cell located at the center is colored white and all other cells are colored black, or if it is colored black and all other cells are colored white. Determine the minimum value of $a+b$ such that there exist positive integers $a$, $b$ and a coloring of an $a \times b$ grid with black and white, so that it contains $n^2 - n$ [i]beautiful[/i] subgrids.

A farmer noticed that, during the last year, there were exactly as many calves born as during the two preceding years together. Even better, the number of pigs born during the last year was one larger than the number of pigs born during the two preceding years together. The farmer promised that if such a trend will continue then, after some years, at least twice as many pigs as calves will be born in his cattle, even though this far this target has not yet ever been reached. Will the farmer be able to keep his promise?

Let $m$ be a positive integer, and consider a $m\times m$ checkerboard consisting of unit squares. At the centre of some of these unit squares there is an ant. At time $0$, each ant starts moving with speed $1$ parallel to some edge of the checkerboard. When two ants moving in the opposite directions meet, they both turn $90^{\circ}$ clockwise and continue moving with speed $1$. When more than $2$ ants meet, or when two ants moving in perpendicular directions meet, the ants continue moving in the same direction as before they met. When an ant reaches one of the edges of the checkerboard, it falls off and will not re-appear. Considering all possible starting positions, determine the latest possible moment at which the last ant falls off the checkerboard, or prove that such a moment does not necessarily exist. [i]Proposed by Toomas Krips, Estonia[/i]

Consider the sequences with 2016 terms formed by the digits 1, 2, 3, and 4. Find the number of those sequences containing an even number of the digit 1.

A teacher and her 30 students play a game on an infinite cell grid. The teacher starts first, then each of the 30 students makes a move, then the teacher and so on. On one move the person can color one unit segment on the grid. A segment cannot be colored twice. The teacher wins if, after the move of one of the 31 players, there is a $1\times 2$ or $2\times 1$ rectangle , such that each segment from it's border is colored, but the segment between the two adjacent squares isn't colored. Prove that the teacher can win.

$5000$ movie fans gathered at a convention. Each participant had watched at least one movie. The participants should be split into discussion groups of two kinds. In each group of the first kind, the members would discuss a movie they all watched. In each group of the second kind, each member would tell about the movie that no one else in this group had watched. Prove that the chairman can always split the participants into exactly 100 groups. (A group consisting of one person is allowed; in this case this person submits a report).

Max flips $2020$ fair coins. Let the probability that there are at most $505$ heads be $p$. Estimate $-\log_2(p)$ to 5 decimal places, in the form $x.abcde$ where $x$ is a positive integer and $a, b, c, d, e$ are decimal digits.

Can $29$ boys and $31$ girls be lined up holding hands so no one is holding hands with two girls?

Let $S = \{1,2,3,4,5,6,7\}$. Compute the number of sets of subsets $T = \{A, B, C\}$ with $A, B, C \in S$ such that $A \cup B \cup C = S$, $(A \cap C) \cup (B \cap C) = \emptyset$, and no subset contains two consecutive integers.

Danny likes seven-digit numbers with the following property: the 1's digit is divisible by the 10's digit, the 10's digit is divisible by the 100's digit, and so on. For example, Danny likes the number $1133366$ but doesn't like $9999993$. Is the amount of numbers Danny likes divisible by $7$?

We have $2015$ prisinoers.The king gives everyone a hat coloured in one of $5$ colors.Everyone sees all hats expect his own.Now,the King orders them in a line(a prisioner can see all guys behind and in front of him).The king asks the prisinoers one by one does he know the color of his hat.If he answers [b]NO[/b],then he is killed.If he answers [b]YES[/b],then answers which color is his hat,if his answers is true,he goes to freedom,if not,he is killed.All the prisinors can hear did he answer [b]YES[/b] or [b]NO[/b],but if he answered [b]YES[/b],they don't know what did he answered(he is killed in public).They can think of a strategy before the King comes,but after that they can't comunicate.What is the largest number of prisinors we can guarentee that can survive?

The numbers $\frac 32$, $\frac 43$ and $\frac 65$ are intially written on the blackboard. A move consists of erasing one of the numbers from the blackboard, call it $a$, and replacing it with $bc-b-c+2$, where $b,c$ are the other two numbers currently written on the blackboard. Is it possible that $\frac{1000}{999}$ would eventually appear on the blackboard? What about $\frac{113}{108}$? [i] (Andrei Bâra)[/i]

There are $1000$ cities in the country of Euleria, and some pairs of cities are linked by dirt roads. It is possible to get from any city to any other city by traveling along these roads. Prove that the government of Euleria may pave some of the roads so that every city will have an odd number of paved roads leading out of it.

There are $1,000,000$ piles of $1996$ coins in each of them, and in one pile there are only fake coins, and in all the others - only real ones. What is the smallest weighing number that can be used to determine a heap containing counterfeit coins if the scales used have one bowl and allow weighing as much weight as desired with an accuracy of one gram, and it is also known that each counterfeit coin weighs $9$ grams, and each real coin weighs $10$ grams?

Some squares of an $n \times n$ chessboard have been marked ($n \in N^*$). Prove that if the number of marked squares is at least $n\left(\sqrt{n} + \frac12\right)$, then there exists a rectangle whose vertices are centers of marked squares.

There are $n$ ($n \ge 10$) cards with numbers $1, 2, \ldots, n$ lying in a row on a table, face down, so that the numbers on any adjacent cards differ by at least $5.$ Is it always enough to turn at most $n-5$ cards to determine which of the cards has number $n$? (It is not necessary to turn the card with number $n$.)

There are $100$ books in a row, numbered from $1$ to $100$ in some order. An operation is choose three books and reorder in any order between them(the others $97$ books stay at the same place). Denote that a book is in [i]correct position[/i] if the book $i$ is in the position $i$. Determine the least integer $m$ such that, for any initial configuration, we can realize $m$ operations and all the books will be in the correct position.

Let $m,n$ be naturals satisfying $n \geq m \geq 2$ and let $S$ be a set consisting of $n$ naturals. Prove that $S$ has at least $2^{n-m+1}$ distinct subsets, each whose sum is divisible by $m$. (The zero set counts as a subset).

Does there exist a convex polyhedron having equal number of edges and diagonals? [i](A diagonal of a polyhedron is a segment through two vertices not lying on the same face) [/i]

Alice and Bob play a game on a 6 by 6 grid. On his or her turn, a player chooses a rational number not yet appearing in the grid and writes it in an empty square of the grid. Alice goes first and then the players alternate. When all squares have numbers written in them, in each row, the square with the greatest number in that row is colored black. Alice wins if she can then draw a line from the top of the grid to the bottom of the grid that stays in black squares, and Bob wins if she can't. (If two squares share a vertex, Alice can draw a line from one to the other that stays in those two squares.) Find, with proof, a winning strategy for one of the players.