A cell is cut from a chessboard $8\, x\, 8$, after which an open broken line was built, which vertices are the centers of the remaining cells. Each segment of the broken line has a length $\sqrt{17}$ or $\sqrt{65}$. When is the number of such broken lines bigger – when the cut cell is $(1,2)$ or $(3,6)$? (The rows and columns on the board are numerated consecutively from 1 to 8.)

There are $n$ ($n \ge 4$) straight lines on the plane. For two straight lines $a$ and $b$, if there are at least two straight lines among the remaining $n-2$ lines that intersect both straight lines $a$ and $b$, then $a$ and $b$ are called a [i]congruent [/i] pair of staight lines, otherwise it is called a [i]separated[/i] pair of straight lines. If the number of [i]congruent [/i] pairs of straight line among $n$ straight lines is $2012$ more than the number of [i]separated[/i] pairs of straight line , find the smallest possible value of $n$ (the order of the two straight lines in a pair is not counted).

$2000$ people are standing on a line. Each one of them is either a [i]liar[/i], who will always lie, or a [i]truth-teller[/i], who will always tell the truth. Each one of them says: "there are more liars to my left than truth-tellers to my right". Determine, if possible, how many people from each class are on the line.

We build a tower of $2\times 1$ dominoes in the following way. First, we place $55$ dominoes on the table such that they cover a $10\times 11$ rectangle; this is the first story of the tower. We then build every new level with $55$ domioes above the exact same $10\times 11$ rectangle. The tower is called [i]stable[/i] if for every non-lattice point of the $10\times 11$ rectangle, we can find a domino that has an inner point above it. How many stories is the lowest [i]stable[/i] tower?

(a) Prove that if 3n stars are placed in $3n$ cells of a $2n \times 2n$ array, then it is possible to remove $n$ rows and $n$ columns in such away that all stars will be removed . (b) Prove that it is possible to place $3n + 1$ stars in the cells of a $2n \times 2n$ array in such a way that after removing any $n$ rows and $n$ columns at least one star remains. (K . P. Kohas, Leningrad)

Determine with proof, the number of permutations $a_1,a_2,a_3,...,a_{2016}$ of $1,2,3,...,2016$ such that the value of $|a_i-i|$ is fixed for all $i=1,2,3,...,2016$, and its value is an integer multiple of $3$.

[b]p1.[/b] A [i]multimagic [/i] square is a $3 \times 3$ array of distinct positive integers with the property that the product of the $3$ numbers in each row, each column, and each of the two diagonals of the array is always the same. (a) Prove that the numbers $1, 2, 3, . . . , 9$ cannot be used to form a multimagic square. (b) Give an example of a multimagic square. [b]p2.[/b] A sequence $a_1, a_2, a_3, ... , a_n$ of real numbers is called an arithmetic progression if $$a_1 - a_2 = a_2 - a_3 = ... = a_{n-1} - a_n.$$ Prove that there exist distinct positive integers $n_1, n_2, n_3, ... , n_{2014}$ such that $$\frac{1}{n_1},\frac{1}{n_2}, ... ,\frac{1}{n_{2014}}$$ is an arithmetic progression. [b]p3.[/b] Let $\lfloor x \rfloor$ be the largest integer that is less than or equal to $x$. For example, $\lfloor 3.9 \rfloor = 3$ and $\lfloor 4\rfloor = 4$. Determine (with proof) all real solutions of the equation $$x^2 - 25 \lfloor x\rfloor + 100 = 0.$$ [b]p4.[/b] An army has $10$ cannons and $8$ carts. Each cart can carry at most one cannon. It takes one day for a cart to cross the desert. What is the least number of days that it takes to get the cannons across the desert? (Cannons can be left part way and picked up later during the procedure.) Prove that the amount of time that your solution requires to move the cannons across the desert is the smallest possible. [b]p5.[/b] Let $C$ be a convex polygon with $4031$ sides. Let $p$ be the length of its perimeter and let $d$ be the sum of the lengths of its diagonals. Show that $$\frac{d}{p}> 2014.$$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

On a table there are all 8 possible horizontal bars $1\times3$ such that each $1\times1$ square is either white or gray (see the figure). It is allowed to move them in any direction by any (not necessarily integer) distance. We may not rotate them or turn them over. Is it possible to move the bars so that they do not overlap, all the white points form a polygon bounded by a closed non-self-intersecting broken line and the same is true for all the gray points? [i]Mikhail Ilyinsky[/i]

What is the largest number of parts into which $n$ planes can divide space? We assume that the set of planes is non-degenerate in the sense that any three planes intersect in one point and no four planes have a common point (and for n=2 it is necessary to require that the planes are not parallel).

[hide=instructions]Time limit: 20 minutes. Fill in the crossword above with answers to the problems below. Notice that there are three directions instead of two. You are probably used to "down" and "across," but this crossword has "1," $e^{4\pi i/3}$, and $e^{5\pi i/3}$. You can think of these labels as complex numbers pointing in the direction to fill in the spaces. In other words "1" means "across", $e^{4\pi i/3}$ means "down and to the left," and $e^{5\pi i/3}$ means "down and to the right." To fill in the answer to, for example, $12$ across, start at the hexagon labeled $12$, and write the digits, proceeding to the right along the gray line. (Note: $12$ across has space for exactly $5$ digits.) Each hexagon is worth one point, and must be filled by something from the set $\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}$. Note that $\pi$ is not in the set, and neither is $i$, nor $\sqrt2$, nor $\heartsuit$,etc. None of the answers will begin with a $0$. "Concatenate $a$ and $b$" means to write the digits of $a$, followed by the digits of $b$. For example, concatenating $10$ and $3$ gives $103$. (It's not the same as concatenating $3$ and $10$.) Calculators are allowed! THIS SHEET IS PROVIDED FOR YOUR REFERENCE ONLY. DO NOT TURN IN THIS SHEET. TURN IN THE OFFICIAL ANSWER SHEET PROVIDED TO THE TEAM. OTHERWISE YOU WILL GET A SCORE OF ZERO! ZERO! ZERO! AND WHILE SOMETIMES "!" MEANS FACTORIAL, IN THIS CASE IT DOES NOT. Good luck, and have fun![/hide] [img]https://cdn.artofproblemsolving.com/attachments/b/f/f7445136e40bf4889a328da640f0935b2b8b82.png[/img] [u][b][i]Across[/i][/b][/u] (1) [b]A 3.[/b] (3 digits) Suppose you draw $5$ vertices of a convex pentagon (but not the sides!). Let $N$ be the number of ways you can draw at least $0$ straight line segments between the vertices so that no two line segments intersect in the interior of the pentagon. What is $N - 64$? (Note what the question is asking for! You have been warned!) [b]A 5.[/b] (3 digits) Among integers $\{1, 2,..., 10^{2012}\}$, let $n$ be the number of numbers for which the sum of the digits is divisible by $5$. What are the first three digits (from the left) of $n$? [b]A 6.[/b] (3 digits) Bob is punished by his math teacher and has to write all perfect squares, one after another. His teacher's blackboard has space for exactly $2012$ digits. He can stop when he cannot fit the next perfect square on the board. (At the end, there might be some space left on the board - he does not write only part of the next perfect square.) If $n^2$ is the largest perfect square he writes, what is $n$? [b]A 8. [/b](3 digits) How many positive integers $n$ are there such that $n \le 2012$, and the greatest common divisor of $n$ and $2012$ is a prime number? [b]A 9.[/b] (4 digits) I have a random number machine generator that is very good at generating integers between $1$ and $256$, inclusive, with equal probability. However, right now, I want to produce a random number between $1$ and $n$, inclusive, so I do the following: $\bullet$ I use my machine to generate a number between $1$ and $256$. Call this $a$. $\bullet$ I take a and divide it by $n$ to get remainder $r$. If $r \ne 0$, then I record $r$ as the randomly generated number. If $r = 0$, then I record $n$ instead. Note that this process does not necessarily produce all numbers with equal probability, but that is okay. I apply this process twice to generate two numbers randomly between $1$ and $10$. Let $p$ be the probability that the two numbers are equal. What is $p \cdot 2^{16}$? [b]A 12.[/b] (5 digits) You and your friend play the following dangerous game. You two start off at some point $(x, y)$ on the plane, where $x$ and $y$ are nonnegative integers. When it is player $A$'s turn, A tells his opponent $B$ to move to another point on the plane. Then $A$ waits for a while. If $B$ is not eaten by a tiger, then $A$ moves to that point as well. From a point $(x, y)$ there are three places $A$ can tell $B$ to walk to: leftwards to $(x - 1, y)$, downwards to $(x, y-1)$, and simultaneously downwards and leftwards to $(x-1, y-1)$. However, you cannot move to a point with a negative coordinate. Now, what was this about being eaten by a tiger? There is a tiger at the origin, which will eat the first person that goes there! Needless to say, you lose if you are eaten. Consider all possible starting points $(x, y)$ with $0 \le x \le 346$ and $0 \le y \le 346$, and $x$ and $y$ are not both zero. Also suppose that you two play strategically, and you go first (i.e., by telling your friend where to go). For how many of the starting points do you win? [b][u][i]Down and to the left [/i][/u][/b] $e^{4\pi i/3}$ [b]DL 2.[/b] (2 digits) ABCDE is a pentagon with $AB = BC = CD = \sqrt2$, $\angle ABC = \angle BCD = 120$ degrees, and $\angle BAE = \angle CDE = 105$ degrees. Find the area of triangle $\vartriangle BDE$. Your answer in its simplest form can be written as $\frac{a+\sqrt{b}}{c}$ , where where $a, b, c$ are integers and $b$ is square-free. Find $abc$. [b]DL 3.[/b] (3 digits) Suppose $x$ and $y$ are integers which satisfy $$\frac{4x^2}{y^2} + \frac{25y^2}{x^2} = \frac{10055}{x^2} +\frac{4022}{y^2} +\frac{2012}{x^2y^2}- 20. $$ What is the maximum possible value of $xy -1$? [b]DL 5.[/b] (3 digits) Find the area of the set of all points in the plane such that there exists a square centered around the point and having the following properties: $\bullet$ The square has side length $7\sqrt2$. $\bullet$ The boundary of the square intersects the graph of $xy = 0$ at at least $3$ points. [b]DL 8.[/b] (3 digits) Princeton Tiger has a mom that likes yelling out math problems. One day, the following exchange between Princeton and his mom occurred: $\bullet$ Mom: Tell me the number of zeros at the end of $2012!$ $\bullet$ PT: Huh? $2012$ ends in $2$, so there aren't any zeros. $\bullet$ Mom: No, the exclamation point at the end was not to signify me yelling. I was not asking about $2012$, I was asking about $2012!$. What is the correct answer? [b]DL 9.[/b] (4 digits) Define the following: $\bullet$ $A = \sum^{\infty}_{n=1}\frac{1}{n^6}$ $\bullet$ $B = \sum^{\infty}_{n=1}\frac{1}{n^6+1}$ $\bullet$ $C = \sum^{\infty}_{n=1}\frac{1}{(n+1)^6}$ $\bullet$ $D = \sum^{\infty}_{n=1}\frac{1}{(2n-1)^6}$ $\bullet$ $E = \sum^{\infty}_{n=1}\frac{1}{(2n+1)^6}$ Consider the ratios $\frac{B}{A}, \frac{C}{A}, \frac{D}{A} , \frac{E}{A}$. Exactly one of the four is a rational number. Let that number be $r/s$, where $r$ and $s$ are nonnegative integers and $gcd \,(r, s) = 1$. Concatenate $r, s$. (It might be helpful to know that $A = \frac{\pi^6}{945}$ .) [b]DL 10.[/b] (3 digits) You have a sheet of paper, which you lay on the xy plane so that its vertices are at $(-1, 0)$, $(1, 0)$, $(1, 100)$, $(-1, 100)$. You remove a section of the bottom of the paper by cutting along the function $y = f(x)$, where $f$ satisfies $f(1) = f(-1) = 0$. (In other words, you keep the bottom two vertices.) You do this again with another sheet of paper. Then you roll both of them into identical cylinders, and you realize that you can attach them to form an $L$-shaped elbow tube. We can write $f\left( \frac13 \right)+f\left( \frac16 \right) = \frac{a+\sqrt{b}}{\pi c}$ , where $a, b, c$ are integers and $b$ is square-free. $Find a+b+c$. [b]DL 11.[/b] (3 digits) Let $$\Xi (x) = 2012(x - 2)^2 + 278(x - 2)\sqrt{2012 + e^{x^2-4x+4}} + 1392 + (x^2 - 4x + 4)e^{x^2-4x+4}$$ find the area of the region in the $xy$-plane satisfying: $$\{x \ge 0 \,\,\, and x \le 4 \,\,\, and \,\,\, y \ge 0 \,\,\, and \,\,\, y \le \sqrt{\Xi(x)}\}$$ [b]DL 13.[/b] (3 digits) Three cones have bases on the same plane, externally tangent to each other. The cones all face the same direction. Two of the cones have radii of $2$, and the other cone has a radius of $3$. The two cones with radii $2$ have height $4$, and the other cone has height $6$. Let $V$ be the volume of the tetrahedron with three of its vertices as the three vertices of the cones and the fourth vertex as the center of the base of the cone with height $6$. Find $V^2$. [b][u][i]Down and to the right[/i][/u][/b] $e^{5\pi i/3}$ [b]DR 1.[/b] (2 digits) For some reason, people in math problems like to paint houses. Alice can paint a house in one hour. Bob can paint a house in six hours. If they work together, it takes them seven hours to paint a house. You might be thinking "What? That's not right!" but I did not make a mistake. When Alice and Bob work together, they get distracted very easily and simultaneously send text messages to each other. When they are texting, they are not getting any work done. When they are not texting, they are painting at their normal speeds (as if they were working alone). Carl, the owner of the house decides to check up on their work. He randomly picks a time during the seven hours. The probability that they are texting during that time can be written as $r/s$, where r and s are integers and $gcd \,(r, s) = 1$. What is $r + s$? [b]DR 4.[/b] (3 digits) Let $a_1 = 2 +\sqrt2$ and $b_1 =\sqrt2$, and for $n \ge 1$, $a_{n+1} = |a_n - b_n|$ and $b_{n+1} = a_n + b_n$. The minimum value of $\frac{a^2_n+a_nb_n-6b^2_n}{6b^2_n-a^2_n}$ can be written in the form $a\sqrt{b} - c$, where $a, b, c$ are integers and $b$ is square-free. Concatenate $c, b, a$ (in that order!). [b]DR 7.[/b] (3 digits) How many solutions are there to $a^{503} + b^{1006} = c^{2012}$, where $a, b, c$ are integers and $|a|$,$|b|$, $|c|$ are all less than $2012$? PS. You should use hide for answers.

An $8$ by $8$ square is divided into $64$ $1$ by $1$ squares, and must be covered by $64$ black and $64$ white, isosceles, right-angled triangles (each square must be covered by two triangles). A covering is said to be “fine” if any two neighbouring triangles (i.e. having a common side) are of different colours. How many different fine coverings are there? (NB Vassiliev)

In each 1x1 square of a nxn board we write $n^2$ numbers with sum S.A move is choosing a 2x2 square and adding 1 to three numbers(from three different 1x1 squares).We say that a number n is good if we can make all the numbers on the board equal by applying a successive number of moves and it not depends of S. a)Show that 6 is not good b)Show that 4 and 1024 are good

The sides of a convex regular polygon of $L + M + N$ sides are to be given draw in three colors: $L$ of them with a red stroke, $M$ with a yellow stroke, and $N$ with a blue. Express, through inequalities, the necessary and sufficient conditions so that there is a solution (several, in general) to the problem of doing it without leaving two adjacent sides drawn with the same color.

10 vertices of a regular 100-gon are coloured red and ten other (distinct) vertices are coloured blue. Prove that there is at least one connection edge (segment) of two red which is as long as the connection edge of two blue points. [hide="Hint"]Possible approaches are pigeon hole principle, proof by contradiction, consider turns (bijective congruent mappings) which maps red in blue points. [/hide]

There is a tiger (which is treated as a point) in the plane that is trying to escape. It starts at the origin at time $t = 0$, and moves continuously at some speed $k$. At every positive integer time $t$, you can place one closed unit disk anywhere in the plane, so long as the disk does not intersect the tiger's current position. The tiger is not allowed to move into any previously placed disks (i.e. the disks block the tiger from moving). Note that when you place the disks, you can "see" the tiger (i.e. know where the tiger currently is). Your goal is to prevent the tiger from escaping to infinity. In other words, you must show there is some radius $R(k)$ such that, using your algorithm, it is impossible for a tiger with speed $k$ to reach a distance larger than $R(k)$ from the origin (where it started). Find an algorithm for placing disks such that there exists some finite real $R(k)$ such that the tiger will never be a distance more than $R(k)$ away from the origin. An algorithm that can trap a tiger of speed $k$ will be awarded: 1 pt for $k<0.05$ 10 pts for $k=0.05$ 20 pts for $k=0.2$ 30 pts for $k=0.3$ 50 pts for $k=1$ 70 pts for $k=2$ 80 pts for $k=2.3$ 85 pts for $k=2.6$ 90 pts for $k=2.9$ 100 pts for $k=3.9$

A regular $1985$-gon is given in the plane. Let's pass a straight line through each side of it. Determine the number of parts into which these lines divide the plane.

300 bureaucrats are split into three comissions of 100 people. Each two bureaucrats are either familiar to each other or non familiar to each other. Prove that there exists two bureaucrats from two distinct commissions such that the third commission contains either 17 bureaucrats familiar to both of them, or 17 bureaucrats familiar to none of them. _________________________________________ This problem is taken from Russian Olympiad 2007-2008 district round 9.8 $ Tipe$

Let $a_n$ be the number of ways to express $n$ as an ordered sum of powers of $3.$ For example $a_4=3,$ since $$4=1+1+1+1=1+3=3+1.$$ Let $b_n$ denote the remainder upon dividing $a_n$ by $3.$ Evaluate $$\sum_{n=1}^{3^{2025}} b_n.$$

[i]20 problems for 25 minutes.[/i] [b]p1.[/b] Compute the value of $2 + 20 + 201 + 2016$. [b]p2.[/b] Gleb is making a doll, whose prototype is a cube with side length $5$ centimeters. If the density of the toy is $4$ grams per cubic centimeter, compute its mass in grams. [b]p3.[/b] Find the sum of $20\%$ of $16$ and $16\%$ of $20$. [b]p4.[/b] How many times does Akmal need to roll a standard six-sided die in order to guarantee that two of the rolled values sum to an even number? [b]p5.[/b] During a period of one month, there are ten days without rain and twenty days without snow. What is the positive difference between the number of rainy days and the number of snowy days? [b]p6.[/b] Joanna has a fully charged phone. After using it for $30$ minutes, she notices that $20$ percent of the battery has been consumed. Assuming a constant battery consumption rate, for how many additional minutes can she use the phone until $20$ percent of the battery remains? [b]p7.[/b] In a square $ABCD$, points $P$, $Q$, $R$, and $S$ are chosen on sides $AB$, $BC$, $CD$, and $DA$ respectively, such that $AP = 2PB$, $BQ = 2QC$, $CR = 2RD$, and $DS = 2SA$. What fraction of square $ABCD$ is contained within square $PQRS$? [b]p8.[/b] The sum of the reciprocals of two not necessarily distinct positive integers is $1$. Compute the sum of these two positive integers. [b]p9.[/b] In a room of government officials, two-thirds of the men are standing and $8$ women are standing. There are twice as many standing men as standing women and twice as many women in total as men in total. Find the total number of government ocials in the room. [b]p10.[/b] A string of lowercase English letters is called pseudo-Japanese if it begins with a consonant and alternates between consonants and vowels. (Here the letter "y" is considered neither a consonant nor vowel.) How many $4$-letter pseudo-Japanese strings are there? [b]p11.[/b] In a wooden box, there are $2$ identical black balls, $2$ identical grey balls, and $1$ white ball. Yuka randomly draws two balls in succession without replacement. What is the probability that the first ball is strictly darker than the second one? [b]p12.[/b] Compute the real number $x$ for which $(x + 1)^2 + (x + 2)^2 + (x + 3)^2 = (x + 4)^2 + (x + 5)^2 + (x + 6)^2$. [b]p13.[/b] Let $ABC$ be an isosceles right triangle with $\angle C = 90^o$ and $AB = 2$. Let $D$, $E$, and $F$ be points outside $ABC$ in the same plane such that the triangles $DBC$, $AEC$, and $ABF$ are isosceles right triangles with hypotenuses $BC$, $AC$, and $AB$, respectively. Find the area of triangle $DEF$. [b]p14.[/b] Salma is thinking of a six-digit positive integer $n$ divisible by $90$. If the sum of the digits of n is divisible by $5$, find $n$. [b]p15.[/b] Kiady ate a total of $100$ bananas over five days. On the ($i + 1$)-th day ($1 \le i \le 4$), he ate i more bananas than he did on the $i$-th day. How many bananas did he eat on the fifth day? [b]p16.[/b] In a unit equilateral triangle $ABC$; points $D$,$E$, and $F$ are chosen on sides $BC$, $CA$, and $AB$, respectively. If lines $DE$, $EF$, and $FD$ are perpendicular to $CA$, $AB$ and $BC$, respectively, compute the area of triangle $DEF$. [b]p17.[/b] Carlos rolls three standard six-sided dice. What is the probability that the product of the three numbers on the top faces has units digit 5? [b]p18.[/b] Find the positive integer $n$ for which $n^{n^n}= 3^{3^{82}}$. [b]p19.[/b] John folds a rope in half five times then cuts the folded rope with four knife cuts, leaving five stacks of rope segments. How many pieces of rope does he now have? [b]p20.[/b] An integer $n > 1$ is conglomerate if all positive integers less than n and relatively prime to $n$ are not composite. For example, $3$ is conglomerate since $1$ and $2$ are not composite. Find the sum of all conglomerate integers less than or equal to $200$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $m$, $n$ be positive integers. On an $m\times{n}$ checkerboard, divided into $1\times1$ squares, we consider all paths that go from upper right vertex to the lower left vertex, travelling exclusively on the grid lines by going down or to the left. We define the area of a path as the number of squares on the checkerboard that are below this path. Let $p$ be a prime such that $r_{p}(m)+r_{p}(n)\geq{p}$, where $r_{p}(m)$ denotes the remainder when $m$ is divided by $p$ and $r_{p}(n)$ denotes the remainder when $n$ is divided by $p$. How many paths have an area that is a multiple of $p$?

Alice is drawing a shape on a piece of paper. She starts by placing her pencil at the origin, and then draws line segments of length one, alternating between vertical and horizontal segments. Eventually, her pencil returns to the origin, forming a closed, non-self-intersecting shape. Show that the area of this shape is even if and only if its perimeter is a multiple of eight.

[u]Round 1[/u] [b]p1.[/b] The fractal T-shirt for this year's Duke Math Meet is so complicated that the printer broke trying to print it. Thus, we devised a method for manually assembling each shirt - starting with the full-size 'base' shirt, we paste a smaller shirt on top of it. And then we paste an even smaller shirt on top of that one. And so on, infinitely many times. (As you can imagine, it took a while to make all the shirts.) The completed T-shirt consists of the original 'base' shirt along with all of the shirts we pasted onto it. Now suppose the base shirt requires $2011$ $cm^2$ of fabric to make, and that each pasted-on shirt requires $4/5$ as much fabric as the previous one did. How many $cm^2$ of fabric in total are required to make one complete shirt? [b]p2.[/b] A dog is allowed to roam a yard while attached to a $60$-meter leash. The leash is anchored to a $40$-meter by $20$-meter rectangular house at the midpoint of one of the long sides of the house. What is the total area of the yard that the dog can roam? [b]p3.[/b] $10$ birds are chirping on a telephone wire. Bird $1$ chirps once per second, bird $2$ chirps once every $2$ seconds, and so on through bird $10$, which chirps every $10$ seconds. At time $t = 0$, each bird chirps. Define $f(t)$ to be the number of birds that chirp during the $t^{th}$ second. What is the smallest $t > 0$ such that $f(t)$ and $f(t + 1)$ are both at least $4$? [u]Round 2[/u] [b]p4.[/b] The answer to this problem is $3$ times the answer to problem 5 minus $4$ times the answer to problem 6 plus $1$. [b]p5.[/b] The answer to this problem is the answer to problem 4 minus $4$ times the answer to problem 6 minus $1$. [b]p6.[/b] The answer to this problem is the answer to problem 4 minus $2$ times the answer to problem 5. [u]Round 3[/u] [b]p7.[/b] Vivek and Daniel are playing a game. The game ends when one person wins $5$ rounds. The probability that either wins the first round is $1/2$. In each subsequent round the players have a probability of winning equal to the fraction of games that the player has lost. What is the probability that Vivek wins in six rounds? [b]p8.[/b] What is the coefficient of $x^8y^7$ in $(1 + x^2 - 3xy + y^2)^{17}$? [b]p9.[/b] Let $U(k)$ be the set of complex numbers $z$ such that $z^k = 1$. How many distinct elements are in the union of $U(1),U(2),...,U(10)$? [u]Round 4[/u] [b]p10.[/b] Evaluate $29 {30 \choose 0}+28{30 \choose 1}+27{30 \choose 2}+...+0{30 \choose 29}-{30\choose 30}$. You may leave your answer in exponential format. [b]p11.[/b] What is the number of strings consisting of $2a$s, $3b$s and $4c$s such that $a$ is not immediately followed by $b$, $b$ is not immediately followed by $c$ and $c$ is not immediately followed by $a$? [b]p12.[/b] Compute $\left(\sqrt3 + \tan (1^o)\right)\left(\sqrt3 + \tan (2^o)\right)...\left(\sqrt3 + \tan (29^o)\right)$. [u]Round 5[/u] [b]p13.[/b] Three massless legs are randomly nailed to the perimeter of a massive circular wooden table with uniform density. What is the probability that the table will not fall over when it is set on its legs? [b]p14.[/b] Compute $$\sum^{2011}_{n=1}\frac{n + 4}{n(n + 1)(n + 2)(n + 3)}$$ [b]p15.[/b] Find a polynomial in two variables with integer coefficients whose range is the positive real numbers. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Some of the $n$ faces of a polyhedron are colored in black such that any two black-colored faces have no common vertex. The rest of the faces of the polyhedron are colored in white. Prove that the number of common sides of two white-colored faces of the polyhedron is at least $n-2$.

There are $n$ sticks which have distinct integer length. Suppose that it's possible to form a non-degenerate triangle from any $3$ distinct sticks among them. It's also known that there are sticks of lengths $5$ and $12$ among them. What's the largest possible value of $n$ under such conditions? [i](Proposed by Bogdan Rublov)[/i]

5. Tom has 13 weight pieces that look equal, however 12 of them weigh the same and the 13th piece is fake and weighs more than the others. He also has two balances: one shows correctly which pan is heavier or that their weights are equal, the other one gives the correct result when the weights on the pans differ, and gives a random result when the weights are equal. (Tom does not know which balance is which). Tom can choose the balance before each weighting. Prove that he can surely determine the fake weight piece in three weighings. Andrey Arzhantsev