On each cell of a $3\times 6$ the board lies one coin. It is known that some two coins lying on adjacent cells are fake. They have the same weigh, but are lighter than the real ones. All the other coins are real. How can one find both counterfeit coins in three weightings on a double-pan balance, without using weights? [i]Proposed by K. Knop[/i]

The tourist on an island can play the "getting treasure" game. He has to open a series of doors, each door is colored with one of n colors, according to the following rules: [i](i)[/i] The tourist has n keys, each key with a different color. [i](ii)[/i] Once a key is used, it is not permitted to change until it is destroyed. [i](iii)[/i] Each key can open any door, and keeps intact when it opens the door having different color with it, but is destroyed when it opens the door having the same color with it. Find the least number of doors to ensure that no tourist, no matter how he choose the order of the keys to use, can get the treasure.

On circle there are $2009$ positive integers which sum is $7036$. Show that it is possible to find two pairs of neighboring numbers such that sum of both pairs is greater or equal to $8$

In neverland, there are $n$ cities and $n$ airlines. Each airline serves an odd number of cities in a circular way, that is, if it serves cities $c_1,c_2,\dots,c_{2k+1}$, then they fly planes connecting $c_1c_2,c_2c_3,\dots,c_1c_{2k+1}$. Show that we can select an odd number of cities $d_1,d_2,\dots,d_{2m+1}$ such that we can fly $d_1\rightarrow d_2\rightarrow\dots\rightarrow d_{2m+1}\rightarrow d_1$ while using each airline at most once.

World Cup in America introduced a new point system. For a victory $3$ points are given, for a draw $1$ point and for defeat $0$ points. In the preliminary games, the teams are divided into groups of $4$ teams. In groups, teams play with each other, once, then in accordance with the points scored $a,b,c$ and $d$ ($a>b>c>d$) teams take the first, second, third and fourth place in their groups. Give all possible options for the distribution points $a,b,c$ and $d$

Two disjoint subsets of the set $\{1,2, ... ,m\}$ have the same sums of elements. Prove that each of the subsets $A,B$ has less than $m / \sqrt2$ elements.

[b]p1.[/b] How many real roots does the equation $2x^7 + x^5 + 4x^3 + x + 2 = 0$ have? [b]p2.[/b] Given that $f(n) = 1 +\sum^n_{j=1}(1 +\sum^j_{i=1}(2i + 1))$, find the value of $f(99)-\sum^{99}_{i=1} i^2$. [b]p3.[/b] A rectangular prism with dimensions $1\times a \times b$, where $1 < a < b < 2$, is bisected by a plane bisecting the longest edges of the prism. One of the smaller prisms is bisected in the same way. If all three resulting prisms are similar to each other and to the original box, compute $ab$. Note: Two rectangular prisms of dimensions $p \times q\times r$ and$ x\times y\times z$ are similar if $\frac{p}{x} = \frac{q}{y} = \frac{r}{z}$ . [b]p4.[/b] For fixed real values of $p$, $q$, $r$ and $s$, the polynomial $x^4 + px^3 + qx^2 + rx + s$ has four non real roots. The sum of two of these roots is $4 + 7i$, and the product of the other two roots is $3 - 4i$. Compute $q$. [b]p5.[/b] There are $10$ seats in a row in a theater. Say we have an infinite supply of indistinguishable good kids and bad kids. How many ways can we seat $10$ kids such that no two bad kids are allowed to sit next to each other? [b]p6.[/b] There are an infinite number of people playing a game. They each pick a different positive integer $k$, and they each win the amount they chose with probability $\frac{1}{k^3}$ . What is the expected amount that all of the people win in total? [b]p7.[/b] There are $100$ donuts to be split among $4$ teams. Your team gets to propose a solution about how the donuts are divided amongst the teams. (Donuts may not be split.) After seeing the proposition, every team either votes in favor or against the propisition. The proposition is adopted with a majority vote or a tie. If the proposition is rejected, your team is eliminated and will never receive any donuts. Another remaining team is chosen at random to make a proposition, and the process is repeated until a proposition is adopted, or only one team is left. No promises or deals need to be kept among teams besides official propositions and votes. Given that all teams play optimally to maximize the expected value of the number of donuts they receive, are completely indifferent as to the success of the other teams, but they would rather not eliminate a team than eliminate one (if the number of donuts they receive is the same either way), then how much should your team propose to keep? [b]p8.[/b] Dominic, Mitchell, and Sitharthan are having an argument. Each of them is either credible or not credible – if they are credible then they are telling the truth. Otherwise, it is not known whether they are telling the truth. At least one of Dominic, Mitchell, and Sitharthan is credible. Tim knows whether Dominic is credible, and Ethan knows whether Sitharthan is credible. The following conversation occurs, and Tim and Ethan overhear: Dominic: “Sitharthan is not credible.” Mitchell: “Dominic is not credible.” Sitharthan: “At least one of Dominic or Mitchell is credible.” Then, at the same time, Tim and Ethan both simultaneously exclaim: “I can’t tell exactly who is credible!” They each then think for a moment, and they realize that they can. If Tim and Ethan always tell the truth, then write on your answer sheet exactly which of the other three are credible. [b]p9.[/b] Pick an integer $n$ between $1$ and $10$. If no other team picks the same number, we’ll give you $\frac{n}{10}$ points. [b]p10.[/b] Many quantities in high-school mathematics are left undefined. Propose a definition or value for the following expressions and justify your response for each. We’ll give you $\frac15$ points for each reasonable argument. $$(i) \,\,\,(.5)! \,\,\, \,\,\,(ii) \,\,\,\infty \cdot 0 \,\,\, \,\,\,(iii) \,\,\,0^0 \,\,\, \,\,\,(iv)\,\,\, \prod_{x\in \emptyset}x \,\,\, \,\,\,(v)\,\,\, 1 - 1 + 1 - 1 + ...$$ [b]p11.[/b] On the back of your answer sheet, write the “coolest” math question you know, and include the solution. If the graders like your question the most, then you’ll get a point. (With your permission, we might include your question on the Mixer next year!) PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let us have an infinite grid of unit squares. We write in every unit square a real number, such that the absolute value of the sum of the numbers from any $n*n$ square is less or equal than $1$. Prove that the absolute value of the sum of the numbers from any $m*n$ rectangular is less or equal than $4$.

In a party among any four persons there are three people who are mutual acquaintances or mutual strangers. Prove that all the people can be separated into two groups $A$ and $B$ such that in $A$ everybody knows everybody else and in $B$ nobody knows anybody else.

Two guys are playing the game "Sea Battle-2000". On the board $ 1 \times 200 $, they take turns placing the letter "$ S $" or "$ O $" on the empty squares of the board. The winner is the one who gets the word "$ SOS $" first. Prove that the second player wins when played correctly.

Points $A$ and $B$ lie on a plane. A straight line passing through $A$ will divide the plane into 2 regions. A further straight line through $B$ will altogether divide the plane into 4 regions, and so on. If 1002 and 1000 straight lines are drawn passing through $A$ and $B$ respectively, what is the maximum number of regions formed?

Inside a square with side lengths $1$ a broken line of length $>1000$ without selfintersection is drawn. Show that there is a line parallel to a side of the square that intersects the broken line in at least $501$ points.

At a strange party, each person knew exactly $22$ others. For any pair of people $X$ and $Y$ who knew one another, there was no other person at the party that they both knew. For any pair of people $X$ and $Y$ who did not know each other, there were exactly six other people that they both knew. How many people were at the party?

Prove the identities: $$C_{n}^{1}+2C_{n}^{2}+3C_{n}^{3}+...+nC_{n}^{n}=n\cdot 2 ^{n-1}$$ $$C_{n}^{1}-2C_{n}^{2}+3C_{n}^{3}+...-(-1)^{n-1}nC_{n}^{n}=0$$

Given $5$ points in a plane, no three of them being collinear. Each two of these $5$ points are joined with a segment, and every of these segments is painted either red or blue; assume that there is no triangle whose sides are segments of equal color. [b]a.)[/b] Show that: [i](1)[/i] Among the four segments originating at any of the $5$ points, two are red and two are blue. [i](2)[/i] The red segments form a closed way passing through all $5$ given points. (Similarly for the blue segments.) [b]b.)[/b] Give a plan how to paint the segments either red or blue in order to have the condition (no triangle with equally colored sides) satisfied.

We have placed $n>3$ cards around a circle, facing downwards. In one step we may perform the following operation with three consecutive cards. Calling the one on the center $B$, the two on the ends $A$ and $C$, we put card $C$ in the place of $A$, then move $A$ and $B$ to the places originally occupied by $B$ and $C$, respectively. Meanwhile, we flip the cards $A$ and $B$. Using a number of these steps, is it possible to move each card to its original place, but facing upwards?

The plane is divided into a net of equilateral triangles of side length $1$, with disjoint interiors. A checker is placed initialy inside a triangle. The checker can be moved into another triangle sharing a common vertex (with the triangle hosting the checker) and having the opposite sides (with respect to this vertex) parallel. A path consists in a finite sequence of moves. Prove that there is no path between two triangles sharing a common side.

Let $n>1$ be an integer and $S_n$ the set of all permutations $\pi : \{1,2,\cdots,n \} \to \{1,2,\cdots,n \}$ where $\pi$ is bijective function. For every permutation $\pi \in S_n$ we define: \[ F(\pi)= \sum_{k=1}^n |k-\pi(k)| \ \ \text{and} \ \ M_{n}=\frac{1}{n!}\sum_{\pi \in S_n} F(\pi) \] where $M_n$ is taken with all permutations $\pi \in S_n$. Calculate the sum $M_n$.

The leader of an IMO team chooses positive integers $n$ and $k$ with $n > k$, and announces them to the deputy leader and a contestant. The leader then secretly tells the deputy leader an $n$-digit binary string, and the deputy leader writes down all $n$-digit binary strings which differ from the leader’s in exactly $k$ positions. (For example, if $n = 3$ and $k = 1$, and if the leader chooses $101$, the deputy leader would write down $001, 111$ and $100$.) The contestant is allowed to look at the strings written by the deputy leader and guess the leader’s string. What is the minimum number of guesses (in terms of $n$ and $k$) needed to guarantee the correct answer?

Nine distinct light bulbs are placed in a circle, each of which is off. Determine the number of ways to turn on some of the light bulbs in the circle such that no four consecutive bulbs are all off.

Let $p$ be a prime number and $a_1, \ldots, a_{p-1}$ be not necessarily distinct nonzero elements of the $p$-element $\mathbb Z_p \pmod{p}$ group. Prove that each element of $\mathbb Z_p$ equals a sum of some of the $a_i$'s (the empty sum is $0$). (translated by Miklós Maróti)

[u]Round 1[/u] [b]1.1.[/b] There are $99$ dogs sitting in a long line. Starting with the third dog in the line, if every third dog barks three times, and all the other dogs each bark once, how many barks are there in total? [b]1.2.[/b] Indigo notices that when she uses her lucky pencil, her test scores are always $66 \frac23 \%$ higher than when she uses normal pencils. What percent lower is her test score when using a normal pencil than her test score when using her lucky pencil? [b]1.3.[/b] Bill has a farm with deer, sheep, and apple trees. He mostly enjoys looking after his apple trees, but somehow, the deer and sheep always want to eat the trees' leaves, so Bill decides to build a fence around his trees. The $60$ trees are arranged in a $5\times 12$ rectangular array with $5$ feet between each pair of adjacent trees. If the rectangular fence is constructed $6$ feet away from the array of trees, what is the area the fence encompasses in feet squared? (Ignore the width of the trees.) [u]Round 2[/u] [b]2.1.[/b] If $x + 3y = 2$, then what is the value of the expression $9^x * 729^y$? [b]2.2.[/b] Lazy Sheep loves sleeping in, but unfortunately, he has school two days a week. If Lazy Sheep wakes up each day before school's starting time with probability $1/8$ independent of previous days, then the probability that Lazy Sheep wakes up late on at least one school day over a given week is $p/q$ for relatively prime positive integers $p, q$. Find $p + q$. [b]2.3.[/b] An integer $n$ leaves remainder $1$ when divided by $4$. Find the sum of the possible remainders $n$ leaves when divided by $20$. [u]Round 3[/u] [b]3.1. [/b]Jake has a circular knob with three settings that can freely rotate. Each minute, he rotates the knob $120^o$ clockwise or counterclockwise at random. The probability that the knob is back in its original state after $4$ minutes is $p/q$ for relatively prime positive integers $p, q$. Find $p + q$. [b]3.2.[/b] Given that $3$ not necessarily distinct primes $p, q, r$ satisfy $p+6q +2r = 60$, find the sum of all possible values of $p + q + r$. [b]3.3.[/b] Dexter's favorite number is the positive integer $x$, If $15x$ has an even number of proper divisors, what is the smallest possible value of $x$? (Note: A proper divisor of a positive integer is a divisor other than itself.) [u]Round 4[/u] [b]4.1.[/b] Three circles of radius $1$ are each tangent to the other two circles. A fourth circle is externally tangent to all three circles. The radius of the fourth circle can be expressed as $\frac{a\sqrt{b}-\sqrt{c}}{d}$ for positive integers $a, b, c, d$ where $b$ is not divisible by the square of any prime and $a$ and $d$ are relatively prime. Find $a + b + c + d$. [b]4.2. [/b]Evaluate $$\frac{\sqrt{15}}{3} \cdot \frac{\sqrt{35}}{5} \cdot \frac{\sqrt{63}}{7}... \cdot \frac{\sqrt{5475}}{73}$$ [b]4.3.[/b] For any positive integer $n$, let $f(n)$ denote the number of digits in its base $10$ representation, and let $g(n)$ denote the number of digits in its base $4$ representation. For how many $n$ is $g(n)$ an integer multiple of $f(n)$? PS. You should use hide for answers. Rounds 5-8 have been posted [url=https://artofproblemsolving.com/community/c3h2784571p24468619]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

A convex polyhedron has $m$ triangular faces (there can be faces of other kind too). From each vertex there are exactly 4 edges. Find the least possible value of $m$.

Find the number of sets with $10$ distinct positive integers such that the average of its elements is less than 6.

Two players $A$ and $B$ play the following game. An even number of cells are placed on a circle. $A$ begins and $A$ and $B$ play alternately, where each move consists of choosing a free cell and writing either $O$ or $M$ in it. The player after whose move the word $OMO$ (OMO = [i]Osterreichische Mathematik Olympiade[/i]) occurs for the first time in three successive cells wins the game. If no such word occurs, then the game is a draw. Prove that if player $B$ plays correctly, then player $A$ cannot win.