Point $K$ is marked on the diagonal $AC$ in rectangle $ABCD$ so that $CK = BC$. On the side $BC$, point $M$ is marked so that $KM = CM$. Prove that $AK + BM = CM$.

Let $n$ be a positive integer relatively prime to $6$. We paint the vertices of a regular $n$-gon with three colours so that there is an odd number of vertices of each colour. Show that there exists an isosceles triangle whose three vertices are of different colours.

Let $ABC$ be a triangle. Construct point $X$ such that $BX=BA$ and $X$ and $C$ lies on the same side of line $AB$. Construct point $Y$ such that $CY=CA$ and $Y$ and $B$ lies on different sides of line $AC$. Suppose that triangle $BAX$ and triangle $CAY$ are similar, prove that the circumcenter of triangle $AXY$ lies on the circumcircle of triangle $ABC$.

Let $H$ be the orthocenter of acute triangle $ABC$ and $D$ the midpoint of $BC$. A line through $H$ intersects $AB,AC$ at $F,E$ respectively, such that $AE=AF$. The ray $DH$ intersects the circumcircle of $\triangle ABC$ at $P$. Prove that $P,A,E,F$ are concyclic.

Twenty-one girls and twenty-one boys took part in a mathematical competition. It turned out that each contestant solved at most six problems, and for each pair of a girl and a boy, there was at least one problem that was solved by both the girl and the boy. Show that there is a problem that was solved by at least three girls and at least three boys.

Prove that if the numbers $a, b, c, d$ satisfy the system of equations $$\begin{cases} a^2+b^2=2cd \\ b^2+c^2=2da \\ c^2+d^2=2ab \end{cases}$$ then $a=b=c=d$.

Let \[I_n =\int_{0}^{n\pi} \frac{\sin x}{1+x} \, dx , \ \ \ \ n=1,2,3,4\] Arrange $I_1, I_2, I_3, I_4$ in increasing order of magnitude. Justify your answer.

Let $n \ge 2$ be a positive integer. Each square of an $n\times n$ board is coloured red or blue. We put dominoes on the board, each covering two squares of the board. A domino is called [i]even [/i] if it lies on two red or two blue squares and [i]colourful [/i] if it lies on a red and a blue square. Find the largest positive integer $k$ having the following property: regardless of how the red/blue-colouring of the board is done, it is always possible to put $k$ non-overlapping dominoes on the board that are either all [i]even [/i] or all [i]colourful[/i].

The real-valued function $f$ is defined for all positive integers. For any integers $a>1, b>1$ with $d=\gcd (a, b)$, we have \[f(ab)=f(d)\left(f\left(\frac{a}{d}\right)+f\left(\frac{b}{d}\right)\right) \] Determine all possible values of $f(2001)$.

Find all triples (p,a,m); p is a prime number, $a,m\in \mathbb{N}$, which satisfy: $a\leq 5p^2$ and $(p-1)!+a=p^m$.

In triangle $ABC$, $AB=13$, $BC=14$, and $CA=15$. Distinct points $D$, $E$, and $F$ lie on segments $\overline{BC}$, $\overline{CA}$, and $\overline{DE}$, respectively, such that $\overline{AD}\perp\overline{BC}$, $\overline{DE}\perp\overline{AC}$, and $\overline{AF}\perp\overline{BF}$. The length of segment $\overline{DF}$ can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$? ${ \textbf{(A)}\ 18\qquad\textbf{(B)}\ 21\qquad\textbf{(C)}\ 24\qquad\textbf{(D}}\ 27\qquad\textbf{(E)}\ 30 $

[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].

Compute the number of integers between $1$ and $100$, inclusive, that have an odd number of factors. Note that $1$ and $4$ are the first two such numbers.

A tamer wants to line up five lions and four tigers. We know that a tiger cannot go after another. How many ways can the beasts be distributed? The tamer cannot distinguish two animals of the same species.

Show that if \[ \frac{\cos x}{\cos y}+\frac{\sin x}{\sin y}=-1 \] then \[ \frac{\cos^3 y}{\cos x}+\frac{\sin^3 y}{\sin x}=1 \]