On triangle $ABC$ let $D$ be the point on $AB$ so that $CD$ is an altitude of the triangle, and $E$ be the point on $BC$ so that $AE$ bisects angle $BAC.$ Let $G$ be the intersection of $AE$ and $CD,$ and let point $F$ be the intersection of side $AC$ and the ray $BG.$ If $AB$ has length $28,$ $AC$ has length $14,$ and $CD$ has length $10,$ then the length of $CF$ can be written as $\tfrac{k-m\sqrt{p}}{n}$ where $k, m, n,$ and $p$ are positive integers, $k$ and $n$ are relatively prime, and $p$ is not divisible by the square of any prime. Find $k - m + n + p.$

We number the columns of an $n\times n$-board from $1$ to $n$. In each cell, we place a number. This is done in such a way that each row precisely contains the numbers $1$ to $n$ (in some order), and also each column contains the numbers $1$ to $n$ (in some order). Next, each cell that contains a number greater than the cell's column number, is coloured grey. In the figure below you can see an example for the case $n = 3$. [asy] unitsize(0.6 cm); int i; fill((0,0)--(1,0)--(1,1)--(0,1)--cycle, gray(0.8)); fill(shift((1,0))*((0,0)--(1,0)--(1,1)--(0,1)--cycle), gray(0.8)); fill(shift((0,2))*((0,0)--(1,0)--(1,1)--(0,1)--cycle), gray(0.8)); for (i = 0; i <= 3; ++i) { draw((0,i)--(3,i)); draw((i,0)--(i,3)); } label("$1$", (0.5,3.5)); label("$2$", (1.5,3.5)); label("$3$", (2.5,3.5)); label("$3$", (0.5,2.5)); label("$1$", (1.5,2.5)); label("$2$", (2.5,2.5)); label("$1$", (0.5,1.5)); label("$2$", (1.5,1.5)); label("$3$", (2.5,1.5)); label("$2$", (0.5,0.5)); label("$3$", (1.5,0.5)); label("$1$", (2.5,0.5)); [/asy] (a) Suppose that $n = 5$. Can the numbers be placed in such a way that each row contains the same number of grey cells? (b) Suppose that $n = 10$. Can the numbers be placed in such a way that each row contains the same number of grey cells?

Prove that is $x,y$ are non negative integers then $5x\ge 7y$ if and only if there exist non-negative integers $(a,b,c,d)$ such that \[\left\{\begin{array}{l}x=a+2b+3c+7d\qquad\\ y=b+2c+5d\qquad\\ \end{array}\right.\]

Three points are chosen randomly and independently on a circle. The probability that all three pairwise distance between the points are less than the radius of the circle is $\frac{1}{K}$, $K\in\mathbb{N}$. Find $K$.

Let $S = \{(x, y) \in Z^2 | 0 \le x \le 11, 0\le y \le 9\}$. Compute the number of sequences $(s_0, s_1, . . . , s_n)$ of elements in $S$ (for any positive integer $n \ge 2$) that satisfy the following conditions: $\bullet$ $s_0 = (0, 0)$ and $s_1 = (1, 0)$, $\bullet$ $s_0, s_1, . . . , s_n$ are distinct, $\bullet$ for all integers $2 \le i \le n$, $s_i$ is obtained by rotating $s_{i-2}$ about $s_{i-1}$ by either $90^o$ or $180^o$ in the clockwise direction.

Let $A_1A_2A_3A_4$ and $B_1B_2B_3B_4$ be two squares such that the boundaries of $A_1A_2A_3A_4$ and $B_1B_2B_3B_4$ does not contain any line segment. Construct 16 line segments $A_iB_j$ for each possible $i,j \in \{1,2,3,4\}$. What is the maximum number of line segments that don't intersect the edges of $A_1A_2A_3A_4$ or $B_1B_2B_3B_4$? (intersection with a vertex is not counted). [i]Proposed by Allen Zheng[/i]

Color the six faces of a cube in six given colors. Each face is colored in exactly one color. Two faces with a common edge is in different colors. Then the number of ways to color the cube is________. Note: If we can make two cubes look the same by turning one of then, they are considered the same.

[b]p1.[/b] Nine coins are placed in a row, alternating between heads and tails as follows: $H T H T H T H T H$. A legal move consists of turning over any two adjacent coins. (a) Give a sequence of legal moves that changes the configuration into $H H H H H H H H H$. (b) Prove that there is no sequence of legal moves that changes the original configuration into $T T T T T T T T T$. [b]p2.[/b] Find (with proof) all integers $k $that satisfy the equation $$\frac{k - 15}{2000}+\frac{k - 12}{2003}+\frac{k - 9}{2006}+\frac{k - 6}{2009}+\frac{k - 3}{2012} = \frac{k - 2000}{15}+\frac{k - 2003}{12}+\frac{k - 2006}{9}+\frac{k - 2009}{6}+\frac{k - 2012}{3}.$$ [b]p3.[/b] Some (not necessarily distinct) natural numbers from $1$ to $2015$ are written on $2015$ lottery tickets, with exactly one number written on each ticket. It is known that the sum of the numbers on any nonempty subset of tickets (including the set of all tickets) is not divisible by $2016$. Prove that the same number is written on all of the tickets. [b]p4.[/b] A set of points $A$ is called distance-distinct if every pair of points in $A$ has a different distance. (a) Show that for all infinite sets of points $B$ on the real line, there exists an infinite distance-distinct set A contained in $B$. (b) Show that for all infinite sets of points $B$ on the real plane, there exists an infinite distance-distinct set A contained in $B$. [b]p5.[/b] Let $ABCD$ be a (not necessarily regular) tetrahedron and consider six points $E, F, G, H, I, J$ on its edges $AB$, $BC$, $AC$, $AD$, $BD$, $CD$, respectively, such that $$|AE| \cdot |EB| = |BF| \cdot |FC| = |AG| \cdot |GC| = |AH| \cdot |HD| = |BI| \cdot |ID| = |CJ| \cdot |JD|.$$ Prove that the points $E, F, G, H, I$, and $J$ lie on the surface of a sphere. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

The number of revolutions of a wheel, with fixed center and with an outside diameter of $ 6$ feet, required to cause a point on the rim to go one mile is: $ \textbf{(A)}\ 880 \qquad\textbf{(B)}\ \frac{440}{\pi} \qquad\textbf{(C)}\ \frac{880}{\pi} \qquad\textbf{(D)}\ 440\pi \qquad\textbf{(E)}\ \text{none of these}$

Find all ordered pairs of positive integers $(m,n)$ for which there exists a set $C=\{c_1,\ldots,c_k\}$ ($k\ge1$) of colors and an assignment of colors to each of the $mn$ unit squares of a $m\times n$ grid such that for every color $c_i\in C$ and unit square $S$ of color $c_i$, exactly two direct (non-diagonal) neighbors of $S$ have color $c_i$. [i]David Yang.[/i]

Consider a parallelogram $ABCD$ such that the midpoint $M$ of the side $CD$ lies on the angle bisector of $\angle BAD$. Show that $\angle AMB$ is a right angle.

Let $a_n$ be the local maximum of $f_n(x)=\frac{x^ne^{-x+n\pi}}{n!}\ (n=1,\ 2,\ \cdots)$ for $x>0$. Find $\lim_{n\to\infty} \ln \left(\frac{a_{2n}}{a_n}\right)^{\frac{1}{n}}$.

Grandma has just finished baking a large rectangular pan of brownies. She is planning to make rectangular pieces of equal size and shape, with straight cuts parallel to the sides of the pan. Each cut must be made entirely across the pan. Grandma wants to make the same number of interior pieces as pieces along the perimeter of the pan. What is the greatest possible number of brownies she can produce? $(\textbf{A}) \: 24 \qquad (\textbf{B}) \: 30 \qquad (\textbf{C}) \: 48 \qquad (\textbf{D}) \: 60 \qquad (\textbf{E}) \: 64$

There are $n$ pawns on $n$ distinct squares of a $19\times 19$ chessboard. In each move, all the pawns are simultaneously moved to a neighboring square (horizontally or vertically) so that no two are moved onto the same square. No pawn can be moved along the same line in two successive moves. What is largest number of pawns can a player place on the board (being able to arrange them freely) so as to be able to continue the game indefinitely?

Let $Q_n$ be the product of the squares of even numbers less than or equal to $n$ and $K_n$ equal to the product of cubes of odd numbers less than or equal to $n$. What is the highest power of $98$, that [b]a)[/b]$Q_n$, [b]b)[/b] $K_n$ or [b]c)[/b] $Q_nK_n$ divides? If one divides $Q_{98}K_{98}$ by the highest power of $98$, then one get a number $N$. By which power-of-two number is $N$ still divisible?

An arbitrary number of lines divide the plane into regions. Show that the regions can be colored red and blue so that neighboring regions have different colors.

A positive integer will be called [i]typical[/i] if the sum of its decimal digits is a multiple of $2011$. a) Show that there are infinitely many [i]typical[/i] numbers, each having at least $2011$ multiples which are also typical numbers. b) Does there exist a positive integer such that each of its multiples is typical?

Let $f(x)=a_0+a_1|x|+a_2|x|^2+a_3|x|^3$, where $a_0,a_1,a_2,a_3$ are constant. Then $\textbf{(A)}~f(x)\text{ is differentiable at }x=0\text{ if whatever be }a_0,a_1,a_2,a_3$ $\textbf{(B)}~f(x)\text{ is not differentiable at }x=0\text{ if whatever be }a_0,a_1,a_2,a_3$ $\textbf{(C)}~f(x)\text{ is differentiable at }x=0\text{ only if }a_1=0$ $\textbf{(D)}~f(x)\text{ is differentiable at }x=0\text{ only if }a_1=0,a_3=0$

Given a set $A$ and a function $f: A\rightarrow A$, denote by $f_{1}(A)=f(A)$, $f_{2}(A)=f(f_{1}(A))$, $f_{3}(A)=f(f_{2}(A))$, and so on, ($f_{n}(A)=f(f_{n-1}(A))$, where the notation $f(B)$ means the set $\{ f(x) \ : \ x\in B\}$ of images of points from $B$). Denote also by $f_{\infty}(A)=f_{1}(A)\cap f_{2}(A)\cap \ldots = \bigcap_{n\geq 1}f_{n}(A)$. a) Show that if $A$ is finite, then $f(f_{\infty}(A))=f_{\infty}(A)$. b) Determine if the above is true for $A=\mathbb{N}\times \mathbb{N}$ and the function \[f\big((m,n)\big)=\begin{cases}(m+1,n) & \mbox{if }n\geq m\geq 1 \\ (0,0) & \mbox{if }m>n \\ (0,n+1) & \mbox{if }n=0. \end{cases}\]

Let $ABC$ be an equilateral triangle of side length $2$. Point $A'$ is chosen on side $BC$ such that the length of $A'B$ is $k<1$. Likewise points $B'$ and $C'$ are chosen on sides $CA$ and $AB$. with $CB'=AC'=k$. Line segments are drawn from points $A',B',C'$ to their corresponding opposite vertices. The intersections of these line segments form a triangle, labeled $PQR$. Prove that $\Delta PQR$ is an equilateral triangle with side length ${4(1-k) \over \sqrt{k^2-2k+4}}$.

We consider two sequences of real numbers $x_{1} \geq x_{2} \geq \ldots \geq x_{n}$ and $\ y_{1} \geq y_{2} \geq \ldots \geq y_{n}.$ Let $z_{1}, z_{2}, .\ldots, z_{n}$ be a permutation of the numbers $y_{1}, y_{2}, \ldots, y_{n}.$ Prove that $\sum \limits_{i=1}^{n} ( x_{i} -\ y_{i} )^{2} \leq \sum \limits_{i=1}^{n}$ $( x_{i} - z_{i})^{2}.$

For every non-negative integer $i$, define the number $M(i)$ as follows: write $i$ down as a binary number, so that we have a string of zeroes and ones, if the number of ones in this string is even, then set $M(i) = 0$, otherwise set $M(i) = 1$. (The first terms of the sequence $M(i)$, $i = 0, 1, 2, ...$ are $0, 1, 1, 0, 1, 0, 0, 1,...$ ) (a) Consider the finite sequence $M(O), M(1), . . . , M(1000) $. Prove that there are at least $320$ terms in this sequence which are equal to their neighbour on the right : $M(i) = M(i + 1 )$ . (b) Consider the finite sequence $M(O), M(1), . . . , M(1000000)$ . Prove that the number of terms $M(i)$ such that $M(i) = M(i +7)$ is at least $450000$. (A Kanel)

Let $n$ be a positive integer. Ana and Banana play a game with $2n$ lamps numbered $1$ to $2n$ from left to right. Initially, all lamps numbered $1$ through $n$ are on, and all lamps numbered $n+1$ through $2n$ are off. They play with the following rules, where they alternate turns with Ana going first: [list] [*] On Ana's turn, she can choose two adjacent lamps $i$ and $i+1$, where lamp $i$ is on and lamp $i+1$ is off, and toggle both. [*] On Banana's turn, she can choose two adjacent lamps which are either both on or both off, and toggle both. [/list] Players must move on their turn if they are able to, and if at any point a player is not able to move on her turn, then the game ends. Determine all $n$ for which Banana can turn off all the lamps before the game ends, regardless of the moves that Ana makes. [i]Andrew Wen[/i]

All vertices of a regular 100-gon are colored in 10 colors. Prove that there exist 4 vertices of the given 100-gon which are the vertices of a rectangle and which are colored in at most 2 colors.

Does there exist $2017$ consecutive positive integers, none of which could be written as $a^2 + b^2$ for some integers $a, b$? Justify your answer.