Find the value of $$\sum_{1\le a<b<c} \frac{1}{2^a3^b5^c}$$ (i.e. the sum of $\frac{1}{2^a3^b5^c}$ over all triples of positive integers $(a, b, c)$ satisfying $a<b<c$)

In a school there are $ 2008$ students. Students are members of certain committees. A committee has at most $ 1004$ members and every two students join a common committee. (i) Determine the smallest possible number of committees in the school. (ii) If it is further required that the union of any two committees consists of at most $ 1800$ students, will your answer in (i) still hold?

A game is played on a $n \times n$ chessboard. In the beginning Bars the cat occupies any cell according to his choice. The $d$ sparrows land on certain cells according to their choice (several sparrows may land in the same cell). Bars and the sparrows play in turns. In each turn of Bars, he moves to a cell adjacent by a side or a vertex (like a king in chess). In each turn of the sparrows, precisely one of the sparrows flies from its current cell to any other cell of his choice. The goal of Bars is to get to a cell containing a sparrow. Can Bars achieve his goal a) if $d=\lfloor \frac{3\cdot n^2}{25}\rfloor$, assuming $n$ is large enough? b) if $d=\lfloor \frac{3\cdot n^2}{19}\rfloor$, assuming $n$ is large enough? c) if $d=\lfloor \frac{3\cdot n^2}{14}\rfloor$, assuming $n$ is large enough?

Let $P$ be a polynomial with integer coefficients such that $P(0)=0$ and \[\gcd(P(0), P(1), P(2), \ldots ) = 1.\] Show there are infinitely many $n$ such that \[\gcd(P(n)- P(0), P(n+1)-P(1), P(n+2)-P(2), \ldots) = n.\]

Let $k$ be a positive integer. Prove that there exist integers $x$ and $y$, neither of which divisible by $7$, such that \begin{align*} x^2 + 6y^2 = 7^k. \end{align*}

In the acute triangle $ABC$, the midpoint of side $BC$ is called $M$. Point $X$ lies on the angle bisector of $\angle AMB$ such that $\angle BXM = 90^o$. Point $Y$ lies on the angle bisector of $\angle AMC$ such that $\angle CYM = 90^o$. Line segments $AM$ and $XY$ intersect in point $Z$. Prove that $Z$ is the midpoint of $XY$ . [asy] import geometry; unitsize (1.2 cm); pair A, B, C, M, X, Y, Z; A = (0,0); B = (4,1.5); C = (0.5,3); M = (B + C)/2; X = extension(M, incenter(A,B,M), B, B + rotate(90)*(incenter(A,B,M) - M)); Y = extension(M, incenter(A,C,M), C, C + rotate(90)*(incenter(A,C,M) - M)); Z = extension(A,M,X,Y); draw(A--B--C--cycle); draw(A--M); draw(M--interp(M,X,2)); draw(M--interp(M,Y,2)); draw(B--X, dotted); draw(C--Y, dotted); draw(X--Y); dot("$A$", A, SW); dot("$B$", B, E); dot("$C$", C, N); dot("$M$", M, NE); dot("$X$", X, NW); dot("$Y$", Y, NE); dot("$Z$", Z, S); [/asy]

Let $m$ and $n$ be positive integers greater than $1$. In each unit square of an $m\times n$ grid lies a coin with its tail side up. A [i]move[/i] consists of the following steps. [list=1] [*]select a $2\times 2$ square in the grid; [*]flip the coins in the top-left and bottom-right unit squares; [*]flip the coin in either the top-right or bottom-left unit square. [/list] Determine all pairs $(m,n)$ for which it is possible that every coin shows head-side up after a finite number of moves. [i]Thanasin Nampaisarn, Thailand[/i]

Given that the expression $\frac{20^{21}}{20^{20}} +\frac{20^{20}}{20^{21}}$ can be written in the form $m/n$ , where $m$ and $n$ are relatively prime positive integers, find $m +n$. [i]Proposed by Ada Tsui[/i]

Find the limit, when $n$ tends to the infinity, of $$\frac{\sum_{k=0}^{n} {{2n} \choose {2k}} 3^k} {\sum_{k=0}^{n-1} {{2n} \choose {2k+1}} 3^k}$$

$2014$ 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.

In the right triangle $ABC$ ( $\angle A = 90^o$), $D$ is the intersection of the bisector of the angle $A$ with the side $(BC)$, and $P$ and $Q$ are the projections of the point $D$ on the sides $(AB),(AC)$ respectively . If $BQ \cap DP=\{M\}$, $CP \cap DQ=\{N\}$, $BQ\cap CP=\{H\}$, show that: a) $PM = DN$ b) $MN \parallel BC$ c) $AH \perp BC$.

The areas of the faces $ABD,ACD,BCD,BCA$ of a tetrahedron $ABCD$ are $S_1,S_2,Q_1,Q_2$, respectively. The angle between the faces $ABD$ and $ACD$ equals $\alpha$, and the angle between $BCD$ and $BCA$ is $\beta$. Prove that $$S_1^2+S_2^2-2S_1S_2\cos\alpha=Q_1^2+Q_2^2-2Q_1Q_2\cos\beta.$$

Let $m$ be a positive integer. The positive integer $a$ is called a [i]golden residue[/i] modulo $m$ if $\gcd(a,m)=1$ and $x^x \equiv a \pmod m$ has a solution for $x$. Given a positive integer $n$, suppose that $a$ is a golden residue modulo $n^n$. Show that $a$ is also a golden residue modulo $n^{n^n}$. [i]Proposed by Mahyar Sefidgaran[/i]

Let $\Gamma$ be a circle and let $d$ be a line such that $\Gamma$ and $d$ have no common points. Further, let $AB$ be a diameter of the circle $\Gamma$; assume that this diameter $AB$ is perpendicular to the line $d$, and the point $B$ is nearer to the line $d$ than the point $A$. Let $C$ be an arbitrary point on the circle $\Gamma$, different from the points $A$ and $B$. Let $D$ be the point of intersection of the lines $AC$ and $d$. One of the two tangents from the point $D$ to the circle $\Gamma$ touches this circle $\Gamma$ at a point $E$; hereby, we assume that the points $B$ and $E$ lie in the same halfplane with respect to the line $AC$. Denote by $F$ the point of intersection of the lines $BE$ and $d$. Let the line $AF$ intersect the circle $\Gamma$ at a point $G$, different from $A$. Prove that the reflection of the point $G$ in the line $AB$ lies on the line $CF$.

Determine the greatest natural number $n$, such that for each set $S$ of 2015 different integers there exist 2 subsets of $S$ (possible to be with 1 element and not necessarily non-intersecting) each of which has a sum of its elements divisible by $n$.

Let $n>2$ be a given positive integer. There are $n$ guests at Georg's bachelor party and each guest is friends with at least one other guest. Georg organizes a party game among the guests. Each guest receives a jug of water such that there are no two guests with the same amount of water in their jugs. All guests now proceed simultaneously as follows. Every guest takes one cup for each of his friends at the party and distributes all the water from his jug evenly in the cups. He then passes a cup to each of his friends. Each guest having received a cup of water from each of his friends pours the water he has received into his jug. What is the smallest possible number of guests that do not have the same amount of water as they started with?

$(x_1, y_1,z_1)$ and $(x_2, y_2, z_2)$ are triples of real numbers such that for every pair of integers $(m,n)$ at least one of $x_{1m} + y_{1n} + z_1$, $x_{2m} + y_{2n} + z_2$ is an even integer. Prove that one of the triples consists of three integers.

If $a=\sqrt[3]{9}-\sqrt[3]{3}+1$, what is $(\frac {4-a}a)^6$? $ \textbf{(A)}\ 3 \qquad\textbf{(B)}\ 6 \qquad\textbf{(C)}\ 8 \qquad\textbf{(D)}\ 9 \qquad\textbf{(E)}\ 12 $

A given finite number of lines in the plane, no two of which are parallel and no three of which are concurrent, divide the plane into finite and infinite regions. In each finite region we write $1$ or $-1$. In one operation, we can choose any triangle made of three of the lines (which may be cut by other lines in the collection) and multiply by $-1$ each of the numbers in the triangle. Determine if it is always possible to obtain $1$ in all the finite regions by successively applying this operation, regardless of the initial distribution of $1$s and $-1$s.

Two players in turns color the squares of a $4 \times 4$ grid, one square at the time. Player loses if after his move a square of $2\times2$ is colored completely. Which of the players has the winning strategy, First or Second? [i](3 points)[/i]

If $a$ is a root of $x^3-x-1 = 0$, compute the value of $$a^{10 }+ 2a^8 -a^7 - 3a^6 - 3a^5 + 4a^4 + 2a^3 - 4a^4 - 6a - 17.$$

$S$ be a set of ordered triples $(a,b,c)$ of distinct elements of a finite set $A$. Suppose that [list=1] [*] $(a,b,c)\in S\iff (b,c,a)\in S$ [*] $(a,b,c)\in S\iff (c,b,a)\not\in S$ [*] $(a,b,c),(c,d,a)\text{ both }\in S\iff (b,c,d),(d,a,b)\text{ both }\in S$[/list] Prove there exists $g: A\to \mathbb{R}$, such that $g$ is one-one and $g(a)<g(b)<g(c)\implies (a,b,c)\in S$

[u]Set 7[/u] [b]p19.[/b] Let circles $\omega_1$ and $\omega_2$, with centers $O_1$ and $O_2$, respectively, intersect at $X$ and $Y$ . A lies on $\omega_1$ and $B$ lies on $\omega_2$ such that $AO_1$ and $BO_2$ are both parallel to $XY$, and $A$ and $B$ lie on the same side of $O_1O_2$. If $XY = 60$, $\angle XAY = 45^o$, and $\angle XBY = 30^o$, then the length of $AB$ can be expressed in the form $\sqrt{a - b\sqrt2 + c\sqrt3}$, where $a, b, c$ are positive integers. Determine $a + b + c$. [b]p20.[/b] If $x$ is a positive real number such that $x^{x^2}= 2^{80}$, find the largest integer not greater than $x^3$. [b]p21.[/b] Justin has a bag containing $750$ balls, each colored red or blue. Sneaky Sam takes out a random number of balls and replaces them all with green balls. Sam notices that of the balls left in the bag, there are $15$ more red balls than blue balls. Justin then takes out $500$ of the balls chosen randomly. If $E$ is the expected number of green balls that Justin takes out, determine the greatest integer less than or equal to $E$. [u]Set 8[/u] These three problems are interdependent; each problem statement in this set will use the answers to the other two problems in this set. As such, let the positive integers $A, B, C$ be the answers to problems $22$, $23$, and $24$, respectively, for this set. [b]p22.[/b] Let $WXYZ$ be a rectangle with $WX =\sqrt{5B}$ and $XY =\sqrt{5C}$. Let the midpoint of $XY$ be $M$ and the midpoint of $YZ$ be $N$. If $XN$ and $W Y$ intersect at $P$, determine the area of $MPNY$ . [b]p23.[/b] Positive integers $x, y, z$ satisfy $$xy \equiv A \,\, (mod 5)$$ $$yz \equiv 2A + C\,\, (mod 7)$$ $$zx \equiv C + 3 \,\, (mod 9).$$ (Here, writing $a \equiv b \,\, (mod m)$ is equivalent to writing $m | a - b$.) Given that $3 \nmid x$, $3 \nmid z$, and $9 | y$, find the minimum possible value of the product $xyz$. [b]p24.[/b] Suppose $x$ and $y$ are real numbers such that $$x + y = A$$ $$xy =\frac{1}{36}B^2.$$ Determine $|x - y|$. [u]Set 9[/u] [b]p25. [/b]The integer $2017$ is a prime which can be uniquely represented as the sum of the squares of two positive integers: $$9^2 + 44^2 = 2017.$$ If $N = 2017 \cdot 128$ can be uniquely represented as the sum of the squares of two positive integers $a^2 +b^2$, determine $a + b$. [b]p26.[/b] Chef Celia is planning to unveil her newest creation: a whole-wheat square pyramid filled with maple syrup. She will use a square flatbread with a one meter diagonal and cut out each of the five polygonal faces of the pyramid individually. If each of the triangular faces of the pyramid are to be equilateral triangles, the largest volume of syrup, in cubic meters, that Celia can enclose in her pyramid can be expressed as $\frac{a-\sqrt{b}}{c}$ where $a, b$ and $c$ are the smallest possible possible positive integers. What is $a + b + c$? [b]p27.[/b] In the Cartesian plane, let $\omega$ be the circle centered at $(24, 7)$ with radius $6$. Points $P, Q$, and $R$ are chosen in the plane such that $P$ lies on $\omega$, $Q$ lies on the line $y = x$, and $R$ lies on the $x$-axis. The minimum possible value of $PQ+QR+RP$ can be expressed in the form $\sqrt{m}$ for some integer $m$. Find m. [u]Set 10[/u] [i]Deja vu?[/i] [b]p28. [/b] Let $ABC$ be a triangle with incircle $\omega$. Let $\omega$ intersect sides $BC$, $CA$, $AB$ at $D, E, F$, respectively. Suppose $AB = 7$, $BC = 12$, and $CA = 13$. If the area of $ABC$ is $K$ and the area of $DEF$ is $\frac{m}{n}\cdot K$, where $m$ and $n$ are relatively prime positive integers, then compute $m + n$. [b]p29.[/b] Sebastian is playing the game Split! again, but this time in a three dimensional coordinate system. He begins the game with one token at $(0, 0, 0)$. For each move, he is allowed to select a token on any point $(x, y, z)$ and take it off, replacing it with three tokens, one at $(x + 1, y, z)$, one at $(x, y + 1, z)$, and one at $(x, y, z + 1)$ At the end of the game, for a token on $(a, b, c)$, it is assigned a score $\frac{1}{2^{a+b+c}}$ . These scores are summed for his total score. If the highest total score Sebastian can get in $100$ moves is $m/n$, then determine $m + n$. [b]p30.[/b] Determine the number of positive $6$ digit integers that satisfy the following properties: $\bullet$ All six of their digits are $1, 5, 7$, or $8$, $\bullet$ The sum of all the digits is a multiple of $5$. [u]Set 11[/u] [b]p31.[/b] The triangular numbers are defined as $T_n =\frac{n(n+1)}{2}$. We also define $S_n =\frac{n(n+2)}{3}$. If the sum $$\sum_{i=16}^{32} \left(\frac{1}{T_i}+\frac{1}{S_i}\right)= \left(\frac{1}{T_{16}}+\frac{1}{S_{16}}\right)+\left(\frac{1}{T_{17}}+\frac{1}{S_{17}}\right)+...+\left(\frac{1}{T_{32}}+\frac{1}{S_{32}}\right)$$ can be written in the form $a/b$ , where $a$ and $b$ are positive integers with $gcd(a, b) = 1$, then find $a + b$. [b]p32.[/b] Farmer Will is considering where to build his house in the Cartesian coordinate plane. He wants to build his house on the line $y = x$, but he also has to minimize his travel time for his daily trip to his barnhouse at $(24, 15)$ and back. From his house, he must first travel to the river at $y = 2$ to fetch water for his animals. Then, he heads for his barnhouse, and promptly leaves for the long strip mall at the line $y =\sqrt3 x$ for groceries, before heading home. If he decides to build his house at $(x_0, y_0)$ such that the distance he must travel is minimized, $x_0$ can be written in the form $\frac{a\sqrt{b}-c}{d}$ , where $a, b, c, d$ are positive integers, $b$ is not divisible by the square of a prime, and $gcd(a, c, d) = 1$. Compute $a+b+c+d$. [b]p33.[/b] Determine the greatest positive integer $n$ such that the following two conditions hold: $\bullet$ $n^2$ is the difference of consecutive perfect cubes; $\bullet$ $2n + 287$ is the square of an integer. [u]Set 12[/u] The answers to these problems are nonnegative integers that may exceed $1000000$. You will be awarded points as described in the problems. [b]p34.[/b] The “Collatz sequence” of a positive integer n is the longest sequence of distinct integers $(x_i)_{i\ge 0}$ with $x_0 = n$ and $$x_{n+1} =\begin{cases} \frac{x_n}{2} & if \,\, x_n \,\, is \,\, even \\ 3x_n + 1 & if \,\, x_n \,\, is \,\, odd \end{cases}.$$ It is conjectured that all Collatz sequences have a finite number of elements, terminating at $1$. This has been confirmed via computer program for all numbers up to $2^{64}$. There is a unique positive integer $n < 10^9$ such that its Collatz sequence is longer than the Collatz sequence of any other positive integer less than $10^9$. What is this integer $n$? An estimate of $e$ gives $\max\{\lfloor 32 - \frac{11}{3}\log_{10}(|n - e| + 1)\rfloor, 0\}$ points. [b]p35.[/b] We define a graph $G$ as a set $V (G)$ of vertices and a set $E(G)$ of distinct edges connecting those vertices. A graph $H$ is a subgraph of $G$ if the vertex set $V (H)$ is a subset of $V (G)$ and the edge set $E(H)$ is a subset of $E(G)$. Let $ex(k, H)$ denote the maximum number of edges in a graph with $k$ vertices without a subgraph of $H$. If $K_i$ denotes a complete graph on $i$ vertices, that is, a graph with $i$ vertices and all ${i \choose 2}$ edges between them present, determine $$n =\sum_{i=2}^{2018} ex(2018, K_i).$$ An estimate of $e$ gives $\max\{\lfloor 32 - 3\log_{10}(|n - e| + 1)\rfloor, 0\}$ points. [b]p36.[/b] Write down an integer between $1$ and $100$, inclusive. This number will be denoted as $n_i$ , where your Team ID is $i$. Let $S$ be the set of Team ID’s for all teams that submitted an answer to this problem. For every ordered triple of distinct Team ID’s $(a, b, c)$ such that a, b, c ∈ S, if all roots of the polynomial $x^3 + n_ax^2 + n_bx + n_c$ are real, then the teams with ID’s $a, b, c$ will each receive one virtual banana. If you receive $v_b$ virtual bananas in total and $|S| \ge 3$ teams submit an answer to this problem, you will be awarded $$\left\lfloor \frac{32v_b}{3(|S| - 1)(|S| - 2)}\right\rfloor$$ points for this problem. If $|S| \le 2$, the team(s) that submitted an answer to this problem will receive $32$ points for this problem. PS. You had better use hide for answers. First sets have been posted [url=https://artofproblemsolving.com/community/c4h2777264p24369138]here[/url].Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $x,y$ and $z$ be real numbers such that $0 \leq x,y,z \leq 1.$ Prove that \[xyz+(1-x)(1-y)(1-z) \leq 1.\] When does equality hold?

Points $A$, $B$, and $C$ lie on a line, in that order, with $AB=8$ and $BC=2$. $B$ is rotated $20^\circ$ counter-clockwise about $A$ to a point $B'$, tracing out an arc $R_1$. $C$ is then rotated $20^\circ$ clockwise about $A$ to a point $C'$, tracing out an arc $R_2$. What is the area of the region bounded by arc $R_1$, segment $B'C$, arc $R_2$, and segment $C'B$? [i]Proposed by Thomas Lam[/i]