Given a polygon with $n$ sides, we assign the numbers $0,1,...,n-1$ to the vertices, and to each side is assigned the sum of the numbers assigned to its ends. The figure shows an example for $n = 5$. Notice that the numbers assigned to the sides are still in arithmetic progression. [img]https://cdn.artofproblemsolving.com/attachments/c/0/975969e29a7953dcb3e440884461169557f9a7.png[/img] $\bullet$ Make the respective assignment for a $9$-sided polygon, and generalize for odd $n$. $\bullet$ Prove that this is not possible if $n$ is even.

Let all distances between the vertices of a convex $n$-gon ($n > 3$) be different. a) A vertex is called uninteresting if the closest vertex is adjacent to it. What is the minimal possible number of uninteresting vertices (for a given $n$)? b) A vertex is called unusual if the farthest vertex is adjacent to it. What is the maximal possible number of unusual vertices (for a given $n$)? [i](Proposed by B.Frenkin)[/i]

There are $100$ piles of $400$ stones each. At every move, Pete chooses two piles, removes one stone from each of them, and is awarded the number of points, equal to the non- negative difference between the numbers of stones in two new piles. Pete has to remove all stones. What is the greatest total score Pete can get, if his initial score is $0$? (Maxim Didin)

There are $9$ point in the Cartezian plane with coordinates $(0,0),(0,1),(0,2),(1,0),(1,1),(1,2),(2,0),(2,1),(2,2).$ Some points are coloured in red and the others in blue. Prove that for any colouring of the points we can always find a right isosceles triangle whose vertexes have the same colour.

Let there be $n$ students, numbered $1$ through $n$. Let there be $n$ cards with numbers $1$ through $n$ written on them. Each student picks a card from the stack, and two students are called a pair if they pick each other's number. Let the probability that there are no pairs be $p_n$. Prove that $p_n - p_{n-1}=0$ if $n$ is odd, and prove that $p_n - p_{n-1}= \frac{1}{(-2)^kk^{1-k}}$ if $n = 2k$.

In a country, there are some two-way roads between the cities. There are $2010$ roads connected to the capital city. For all cities different from the capital city, there are less than $2010$ roads connected to that city. For two cities, if there are the same number of roads connected to these cities, then this number is even. $k$ roads connected to the capital city will be deleted. It is wanted that whatever the road network is, if we can reach from one city to another at the beginning, then we can reach after the deleting process also. Find the maximum value of $k.$

All segments joining $n$ points (no three of which are collinear) are coloured in one of $k$ colours. What is the smallest $k$ for which there always exists a closed polygonal line with the vertices at some of the $n$ points, whose sides are all of the same colour?

The four faces of a tetrahedral die are labelled $0, 1, 2,$ and $3,$ and the die has the property that, when it is rolled, the die promptly vanishes, and a number of copies of itself appear equal to the number on the face the die landed on. For example, if it lands on the face labelled $0,$ it disappears. If it lands on the face labelled $1,$ nothing happens. If it lands on the face labelled $2$ or $3,$ there will then be $2$ or $3$ copies of the die, respectively (including the original). Suppose the die and all its copies are continually rolled, and let $p$ be the probability that they will all eventually disappear. Find $\left\lfloor \frac{10}{p} \right\rfloor$.

[u]Round 1[/u] [b]p1.[/b] Ravi has a bag with $100$ slips of paper in it. Each slip has one of the numbers $3, 5$, or $7$ written on it. Given that half of the slips have the number $3$ written on them, and the average of the values on all the slips is $4.4$, how many slips have $7$ written on them? [b]p2.[/b] In triangle $ABC$, point $D$ lies on side $AB$ such that $AB \perp CD$. It is given that $\frac{CD}{BD}=\frac12$, $AC = 29$, and $AD = 20$. Find the area of triangle $BCD$. [b]p3.[/b] Compute $(123 + 4)(123 + 5) - 123\cdot 132$. [u]Round 2[/u] [b]p4. [/b] David is evaluating the terms in the sequence $a_n = (n + 1)^3 - n^3$ for $n = 1, 2, 3,....$ (that is, $a_1 = 2^3 - 1^3$ , $a_2 = 3^3 - 2^3$, $a_3 = 4^3 - 3^3$, and so on). Find the first composite number in the sequence. (An positive integer is composite if it has a divisor other than 1 and itself.) [b]p5.[/b] Find the sum of all positive integers strictly less than $100$ that are not divisible by $3$. [b]p6.[/b] In how many ways can Alex draw the diagram below without lifting his pencil or retracing a line? (Two drawings are different if the order in which he draws the edges is different, or the direction in which he draws an edge is different). [img]https://cdn.artofproblemsolving.com/attachments/9/6/9d29c23b3ca64e787e717ceff22d45851ae503.png[/img] [u]Round 3[/u] [b]p7.[/b] Fresh Mann is a $9$th grader at Euclid High School. Fresh Mann thinks that the word vertices is the plural of the word vertice. Indeed, vertices is the plural of the word vertex. Using all the letters in the word vertice, he can make $m$ $7$-letter sequences. Using all the letters in the word vertex, he can make $n$ $6$-letter sequences. Find $m - n$. [b]p8.[/b] Fresh Mann is given the following expression in his Algebra $1$ class: $101 - 102 = 1$. Fresh Mann is allowed to move some of the digits in this (incorrect) equation to make it into a correct equation. What is the minimal number of digits Fresh Mann needs to move? [b]p9.[/b] Fresh Mann said, “The function $f(x) = ax^2+bx+c$ passes through $6$ points. Their $x$-coordinates are consecutive positive integers, and their y-coordinates are $34$, $55$, $84$, $119$, $160$, and $207$, respectively.” Sophy Moore replied, “You’ve made an error in your list,” and replaced one of Fresh Mann’s numbers with the correct y-coordinate. Find the corrected value. [u]Round 4[/u] [b]p10.[/b] An assassin is trying to find his target’s hotel room number, which is a three-digit positive integer. He knows the following clues about the number: (a) The sum of any two digits of the number is divisible by the remaining digit. (b) The number is divisible by $3$, but if the first digit is removed, the remaining two-digit number is not. (c) The middle digit is the only digit that is a perfect square. Given these clues, what is a possible value for the room number? [b]p11.[/b] Find a positive real number $r$ that satisfies $$\frac{4 + r^3}{9 + r^6}=\frac{1}{5 - r^3}- \frac{1}{9 + r^6}.$$ [b]p12.[/b] Find the largest integer $n$ such that there exist integers $x$ and $y$ between $1$ and $20$ inclusive with $$\left|\frac{21}{19} -\frac{x}{y} \right|<\frac{1}{n}.$$ PS. You had better use hide for answers. Last rounds have been posted [url=https://artofproblemsolving.com/community/c4h2784267p24464980]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

We will say that two sets of distinct numbers are $\textit{linked}$ to each other if between any two numbers of each set lies at least one number of the other set. Is it possible to fill the cells of a $100 \times 200$ rectangle with distinct numbers so that any two rows of the rectangle are linked to one another, and any two columns of the rectangle are linked to one another?

Show for any positive integer $n$ that there exists a circle in the plane such that there are exactly $n$ grid points within the circle. (A grid point is a point having integer coordinates.)

Prove that from $x + y = 1 \ (x, y \in \mathbb R)$ it follows that \[x^{m+1} \sum_{j=0}^n \binom{m+j}{j} y^j + y^{n+1} \sum_{i=0}^m \binom{n+i}{i} x^i = 1 \qquad (m, n = 0, 1, 2, \ldots ).\]

Let $(x_1, y_1, z_1), (x_2, y_2, z_2), \cdots, (x_{19}, y_{19}, z_{19})$ be integers. Prove that there exist pairwise distinct subscripts $i, j, k$ such that $x_i+x_j+x_k$, $y_i+y_j+y_k$, $z_i+z_j+z_k$ are all multiples of $3$.

The first $1997$ natural numbers are written on the blackboard: $1,2,3,\ldots ,1997$. In front of each number, a "$+$" sign or a "$-$" sign will be written in order, from left to right. To decide each sign, a coin is tossed; If it comes up heads, you write "$+$", if it comes up tails, you write "$-$". Once the $1997$ signs are written, the algebraic sum of the expression on the blackboard is carried out and the result is $S$. What is the probability that $S$ is greater than $0$? Clarification: The probability of an event is equal to the number of favorable cases/number of possible cases.

Given any $22$ points in the plane, no three collinear. Show that the points can be divided into $11$ pairs, so that the $11$ line segments defined by the pairs have at least five different intersections

Let C be the number of ways to arrange the letters of the word CATALYSIS, T be the number of ways to arrange the letters of the word TRANSPORT, S be the number of ways to arrange the letters of the word STRUCTURE, and M be the number of ways to arrange the letters of the word MOTION. What is $\frac{C - T + S}{M}$ ?

Thirteen children are sitting at a round table, each holding two cards. Each card has one of the numbers $1, 2, ..., 13$ written on it, and each number is written on exactly two cards. On a signal, each child gives the card with the lower number to his neighbor on the right (and at the same time receives his card with the lower number from the neighbor on the left). Prove that after a finite number of such exchanges, a situation arises when at least one of the children will have two cards with the same number.

Find the number of distinguishable $2\times 2\times 2$ cubes that can be formed by gluing together two blue, two green, two red, and two yellow $1\times 1\times 1$ cubes. Two cubes are indistinguishable if one can be rotated so that the two cubes have identical coloring patterns.

[b]p1.[/b] Find the sum of the seventeenth powers of the seventeen roots of the seventeeth degree polynomial equation $x^{17} - 17x + 17 = 0$. [b]p2.[/b] Four identical spherical cows, each of radius $17$ meters, are arranged in a tetrahedral pyramid (their centers are the vertices of a regular tetrahedron, and each one is tangent to the other three). The pyramid of cows is put on the ground, with three of them laying on it. What is the distance between the ground and the top of the topmost cow? [b]p3.[/b] If $a_n$ is the last digit of $\sum^{n}_{i=1} i$, what would the value of $\sum^{1000}_{i=1}a_i$ be? [b]p4.[/b] If there are $15$ teams to play in a tournament, $2$ teams per game, in how many ways can the tournament be organized if each team is to participate in exactly $5$ games against dierent opponents? [b]p5.[/b] For $n = 20$ and $k = 6$, calculate $$2^k {n \choose 0}{n \choose k}- 2^{k-1}{n \choose 1}{{n - 1} \choose {k - 1}} + 2^{k-2}{n \choose 2}{{n - 2} \choose {k - 2}} +...+ (-1)^k {n \choose k}{{n - k} \choose 0}$$ where ${n \choose k}$ is the number of ways to choose $k$ things from a set of $n$. [b]p6.[/b] Given a function $f(x) = ax^2 + b$, with a$, b$ real numbers such that $$f(f(f(x))) = -128x^8 + \frac{128}{3}x^6 - \frac{16}{22}x^2 +\frac{23}{102}$$ , find $b^a$. [b]p7.[/b] Simplify the following fraction $$\frac{(2^3-1)(3^3-1)...(100^3-1)}{(2^3+1)(3^3+1)...(100^3+1)}$$ [b]p8.[/b] Simplify the following expression $$\frac{\sqrt{3 + \sqrt5} + \sqrt{3 - \sqrt5}}{\sqrt{3 - \sqrt8}} -\frac{4}{ \sqrt{8 - 2\sqrt{15}}}$$ [b]p9.[/b] Suppose that $p(x)$ is a polynomial of degree $100$ such that $p(k) = k2^{k-1}$ , $k =1, 2, 3 ,... , 100$. What is the value of $p(101)$ ? [b]p10. [/b] Find all $17$ real solutions $(w, x, y, z)$ to the following system of equalities: $$ 2w + w^2x = x$$ $$ 2x + x^2y=y $$ $$ 2y + y^2z=z $$ $$ -2z+z^2w=w $$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

In a ballroom dance class $15$ boys and $15$ girls are lined up in parallel rows so that $15$ couples are formed. It so happens that the difference in height between the boy and the girl in each couple is not more than $10$ cm. Prove that if the boys and the girls were placed in each line in order of decreasing height, then the difference in height in each of the newly formed couples would still be at most $10$ cm. (AG Pechkovskiy, Moscow)

(a) Let $k,n\ge 1$.Find the number of sequences $\phi=S_0,S_1,\ldots,S_k$ of subsets of $[n]=\{1,2,3,\ldots,n\}$ if for all $1\le i\le k$ we have either (i)$S_{i-1}\subset S_i$ and $|S_i-S_{i-1}|$,or (ii)$S_i\subset S_{i-1}$ and $|S_{i-1}-S_i|=1$. (b) Suppose that we add the additional condition that $S_k=\phi$.Show that now the number $f_k(n)$ of sequences is given by$f_k(n)=\frac{1}{2^n}\sum_{i=0}^n\binom ni (n-2i)^k$. Note that $f_k(n)=0$ if $k$ is odd.

[b]p1.[/b] What is the units digit of $1 + 9 + 9^2 +... + 9^{2015}$ ? [b]p2.[/b] In Fourtown, every person must have a car and therefore a license plate. Every license plate must be a $4$-digit number where each digit is a value between $0$ and $9$ inclusive. However $0000$ is not a valid license plate. What is the minimum population of Fourtown to guarantee that at least two people who have the same license plate? [b]p3.[/b] Two sides of an isosceles triangle $\vartriangle ABC$ have lengths $9$ and $4$. What is the area of $\vartriangle ABC$? [b]p4.[/b] Let $x$ be a real number such that $10^{\frac{1}{x}} = x$. Find $(x^3)^{2x}$. [b]p5.[/b] A Berkeley student and a Stanford student are going to visit each others campus and go back to their own campuses immediately after they arrive by riding bikes. Each of them rides at a constant speed. They first meet at a place $17.5$ miles away from Berkeley, and secondly $10$ miles away from Stanford. How far is Berkeley away from Stanford in miles? [b]p6.[/b] Let $ABCDEF$ be a regular hexagon. Find the number of subsets $S$ of $\{A,B,C,D,E, F\}$ such that every edge of the hexagon has at least one of its endpoints in $S$. [b]p7.[/b] A three digit number is a multiple of $35$ and the sum of its digits is $15$. Find this number. [b]p8.[/b] Thomas, Olga, Ken, and Edward are playing the card game SAND. Each draws a card from a $52$ card deck. What is the probability that each player gets a dierent rank and a different suit from the others? [b]p9.[/b] An isosceles triangle has two vertices at $(1, 4)$ and $(3, 6)$. Find the $x$-coordinate of the third vertex assuming it lies on the $x$-axis. [b]p10.[/b] Find the number of functions from the set $\{1, 2,..., 8\}$ to itself such that $f(f(x)) = x$ for all $1 \le x \le 8$. [b]p11.[/b] The circle has the property that, no matter how it's rotated, the distance between the highest and the lowest point is constant. However, surprisingly, the circle is not the only shape with that property. A Reuleaux Triangle, which also has this constant diameter property, is constructed as follows. First, start with an equilateral triangle. Then, between every pair of vertices of the triangle, draw a circular arc whose center is the $3$rd vertex of the triangle. Find the ratio between the areas of a Reuleaux Triangle and of a circle whose diameters are equal. [b]p12.[/b] Let $a$, $b$, $c$ be positive integers such that gcd $(a, b) = 2$, gcd $(b, c) = 3$, lcm $(a, c) = 42$, and lcm $(a, b) = 30$. Find $abc$. [b]p13.[/b] A point $P$ is inside the square $ABCD$. If $PA = 5$, $PB = 1$, $PD = 7$, then what is $PC$? [b]p14.[/b] Find all positive integers $n$ such that, for every positive integer $x$ relatively prime to $n$, we have that $n$ divides $x^2 - 1$. You may assume that if $n = 2^km$, where $m$ is odd, then $n$ has this property if and only if both $2^k$ and $m$ do. [b]p15.[/b] Given integers $a, b, c$ satisfying $$abc + a + c = 12$$ $$bc + ac = 8$$ $$b - ac = -2,$$ what is the value of $a$? [b]p16.[/b] Two sides of a triangle have lengths $20$ and $30$. The length of the altitude to the third side is the average of the lengths of the altitudes to the two given sides. How long is the third side? [b]p17.[/b] Find the number of non-negative integer solutions $(x, y, z)$ of the equation $$xyz + xy + yz + zx + x + y + z = 2014.$$ [b]p18.[/b] Assume that $A$, $B$, $C$, $D$, $E$, $F$ are equally spaced on a circle of radius $1$, as in the figure below. Find the area of the kite bounded by the lines $EA$, $AC$, $FC$, $BE$. [img]https://cdn.artofproblemsolving.com/attachments/7/7/57e6e1c4ef17f84a7a66a65e2aa2ab9c7fd05d.png[/img] [b]p19.[/b] A positive integer is called cyclic if it is not divisible by the square of any prime, and whenever $p < q$ are primes that divide it, $q$ does not leave a remainder of $1$ when divided by $p$. Compute the number of cyclic numbers less than or equal to $100$. [b]p20.[/b] On an $8\times 8$ chess board, a queen can move horizontally, vertically, and diagonally in any direction for as many squares as she wishes. Find the average (over all $64$ possible positions of the queen) of the number of squares the queen can reach from a particular square (do not count the square she stands on). PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Thirty pupils from the same class decided to exchange visits. Any pupil may make several visits during one evening, but must stay home if he is receiving guests that evening. Prove that in order that each pupil visit each of his classmates (a) four evenings are not enough (b) five evenings are not enough (c) ten evenings are enough (d) even seven evenings are enough

$2017$ voters vote by submitting a ranking of the integers $\{1, 2, ..., 38\}$ from favorite (a vote for that value in $1$st place) to least favorite (a vote for that value in $38$th/last place). Let $a_k$ be the integer that received the most $k$th place votes (the smallest such integer if there is a tie). Find the maximum possible value of $\Sigma_{k=1}^{38} a_k$.

Let $n\ge3$ be an integer. Each row in an $(n-2)\times n$ array consists of the numbers 1,2,...,$n$ in some order, and the numbers in each column are all different. Prove that this array can be expanded into an $n\times n$ array such that each row and each column consists of the numbers 1,2,...,$n$.