Let $ABC$ be a triangle such that $\angle A=90^{\circ }$ and $\angle B<\angle C$. The tangent at $A$ to the circumcircle $\omega$ of triangle $ABC$ meets the line $BC$ at $D$. Let $E$ be the reflection of $A$ in the line $BC$, let $X$ be the foot of the perpendicular from $A$ to $BE$, and let $Y$ be the midpoint of the segment $AX$. Let the line $BY$ intersect the circle $\omega$ again at $Z$. Prove that the line $BD$ is tangent to the circumcircle of triangle $ADZ$. [hide="comment"] [i]Edited by Orl.[/i] [/hide]

Because J’s cheetahs are everywhere, they are now running away. A particularly unintelligent one starts to run in a $720$ mile loop at $80$ miles per hour. J immediately starts to chase after it, starting at the same point, at $10$ miles per hour at $12:00$ PM, but realizes one hour later that it would be more wise to turn around and run in the opposite direction in the loop, so he does this. Assuming both maintain a constant speed, at what time do J and the cheetah collide? Round to the nearest minute, and be sure to include AM or PM.

[b]p1.[/b] Find the number of zeroes at the end of $20^{17}$. [b]p2.[/b] Express $\frac{1}{\sqrt{20} +\sqrt{17}}$ in simplest radical form. [b]p3.[/b] John draws a square $ABCD$. On side $AB$ he draws point $P$ so that $\frac{BP}{PA}=\frac{1}{20}$ and on side $BC$ he draws point $Q$ such that $\frac{BQ}{QC}=\frac{1}{17}$ . What is the ratio of the area of $\vartriangle PBQ$ to the area of $ABCD$? [b]p4.[/b] Alfred, Bill, Clara, David, and Emily are sitting in a row of five seats at a movie theater. Alfred and Bill don’t want to sit next to each other, and David and Emily have to sit next to each other. How many arrangements can they sit in that satisfy these constraints? [b]p5.[/b] Alex is playing a game with an unfair coin which has a $\frac15$ chance of flipping heads and a $\frac45$ chance of flipping tails. He flips the coin three times and wins if he flipped at least one head and one tail. What is the probability that Alex wins? [b]p6.[/b] Positive two-digit number $\overline{ab}$ has $8$ divisors. Find the number of divisors of the four-digit number $\overline{abab}$. [b]p7.[/b] Call a positive integer $n$ diagonal if the number of diagonals of a convex $n$-gon is a multiple of the number of sides. Find the number of diagonal positive integers less than or equal to $2017$. [b]p8.[/b] There are $4$ houses on a street, with $2$ on each side, and each house can be colored one of 5 different colors. Find the number of ways that the houses can be painted such that no two houses on the same side of the street are the same color and not all the houses are different colors. [b]p9.[/b] Compute $$|2017 -|2016| -|2015-| ... |3-|2-1|| ...||||.$$ [b]p10.[/b] Given points $A,B$ in the coordinate plane, let $A \oplus B$ be the unique point $C$ such that $\overline{AC}$ is parallel to the $x$-axis and $\overline{BC}$ is parallel to the $y$-axis. Find the point $(x, y)$ such that $((x, y) \oplus (0, 1)) \oplus (1,0) = (2016,2017) \oplus (x, y)$. [b]p11.[/b] In the following subtraction problem, different letters represent different nonzero digits. $\begin{tabular}{ccccc} & M & A & T & H \\ - & & H & A & M \\ \hline & & L & M & T \\ \end{tabular}$ How many ways can the letters be assigned values to satisfy the subtraction problem? [b]p12.[/b] If $m$ and $n$ are integers such that $17n +20m = 2017$, then what is the minimum possible value of $|m-n|$? [b]p13. [/b]Let $f(x)=x^4-3x^3+2x^2+7x-9$. For some complex numbers $a,b,c,d$, it is true that $f (x) = (x^2+ax+b)(x^2+cx +d)$ for all complex numbers $x$. Find $\frac{a}{b}+ \frac{c}{d}$. [b]p14.[/b] A positive integer is called an imposter if it can be expressed in the form $2^a +2^b$ where $a,b$ are non-negative integers and $a \ne b$. How many almost positive integers less than $2017$ are imposters? [b]p15.[/b] Evaluate the infinite sum $$\sum^{\infty}_{n=1} \frac{n(n +1)}{2^{n+1}}=\frac12 +\frac34+\frac68+\frac{10}{16}+\frac{15}{32}+...$$ [b]p16.[/b] Each face of a regular tetrahedron is colored either red, green, or blue, each with probability $\frac13$ . What is the probability that the tetrahedron can be placed with one face down on a table such that each of the three visible faces are either all the same color or all different colors? [b]p17.[/b] Let $(k,\sqrt{k})$ be the point on the graph of $y=\sqrt{x}$ that is closest to the point $(2017,0)$. Find $k$. [b]p18.[/b] Alice is going to place $2016$ rooks on a $2016 \times 2016$ chessboard where both the rows and columns are labelled $1$ to $2016$; the rooks are placed so that no two rooks are in the same row or the same column. The value of a square is the sum of its row number and column number. The score of an arrangement of rooks is the sumof the values of all the occupied squares. Find the average score over all valid configurations. [b]p19.[/b] Let $f (n)$ be a function defined recursively across the natural numbers such that $f (1) = 1$ and $f (n) = n^{f (n-1)}$. Find the sum of all positive divisors less than or equal to $15$ of the number $f (7)-1$. [b]p20.[/b] Find the number of ordered pairs of positive integers $(m,n)$ that satisfy $$gcd \,(m,n)+ lcm \,(m,n) = 2017.$$ [b]p21.[/b] Let $\vartriangle ABC$ be a triangle. Let $M$ be the midpoint of $AB$ and let $P$ be the projection of $A$ onto $BC$. If $AB = 20$, and $BC = MC = 17$, compute $BP$. [b]p22.[/b] For positive integers $n$, define the odd parent function, denoted $op(n)$, to be the greatest positive odd divisor of $n$. For example, $op(4) = 1$, $op(5) = 5$, and $op(6) =3$. Find $\sum^{256}_{i=1}op(i).$ [b]p23.[/b] Suppose $\vartriangle ABC$ has sidelengths $AB = 20$ and $AC = 17$. Let $X$ be a point inside $\vartriangle ABC$ such that $BX \perp CX$ and $AX \perp BC$. If $|BX^4 -CX^4|= 2017$, the compute the length of side $BC$. [b]p24.[/b] How many ways can some squares be colored black in a $6 \times 6$ grid of squares such that each row and each column contain exactly two colored squares? Rotations and reflections of the same coloring are considered distinct. [b]p25.[/b] Let $ABCD$ be a convex quadrilateral with $AB = BC = 2$, $AD = 4$, and $\angle ABC = 120^o$. Let $M$ be the midpoint of $BD$. If $\angle AMC = 90^o$, find the length of segment $CD$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $\triangle{ABC}$ be a triangle with incenter $I.$ The incircle of triangle $\triangle{ABC}$ touches $\overline{BC}$ at $D.$ Let $M$ be the midpoint of $\overline{BC},$ and let line $AI$ meet the circumcircle of triangle $\triangle{ABC}$ again at $L \neq A.$ Let $\omega$ be the circle centered at $L$ tangent to $AB$ and $AC.$ If $\omega$ intersects $\overline{AD}$ at point $P,$ prove that $\angle{IPM}=90^\circ.$

In certain country, the coins have the following values: $2^0, 2^1, 2^2,\dots 2^{10}$. A cash machine has $1000$ coins of each value and give the money using each coin(of each value) at most once. The customers order all the positive integers: $1,2,3,4,5,\dots$ (in this order) in coins. a) Determine the first integer, such that the cash machine cannot provide. b) In the moment that the first customer can not be attended, by the lack of coins, what are the coins which are not available in the cash machine?

We are given the triangle $ABC$ with an area of $2000$. Let $P,Q,R$ be the midpoints of the sides $BC$, $AC$, $AB$. Let $U,V,W$ be the midpoints of the sides $QR$, $PR$, $PQ$. The lengths of the line segments $AU$, $BV$, $CW$ are $x$, $y$, $z$. Show that there exists a triangle with side lengths $x$, $y$ and $z$ and caluclate it's area.

$1)$. Prove a graph with $2n$ vertices and $n+2$ edges has an independent set of size $n$ (there are $n$ vertices such that no two of them are adjacent ). $2)$.Find the number of graphs with $2n$ vertices and $n+3$ edges , such that among any $n$ vertices there is an edge connecting two of them

Joao Salta-Pocinhas jumps $1$ meter in the first jump, $2$ meters in the second, $4$ meters in the third, . . ., $2^{n-1}$ meters in jump number $n$. Is there any possibility for Joao to choose the directions of his jumps in order to get back to the starting point?

Let $G$ be a graph on $n\geq 6$ vertices and every vertex is of degree at least 3. If $C_{1}, C_{2}, \dots, C_{k}$ are all the cycles in $G$, determine all possible values of $\gcd(|C_{1}|, |C_{2}|, \dots, |C_{k}|)$ where $|C|$ denotes the number of vertices in the cycle $C$.

Given a cube. On each edge of the cube, we write a number, either $1$ or $-1$. For each face of the cube, we multiply the four numbers on the edges of this face, and write the product on this face. Finally, we add all the eighteen numbers that we wrote down on the edges and face of the cube. What is the smallest possible sum that we can get?

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 $

Determine all sequences of non-negative integers $a_1, \ldots, a_{2016}$ all less than or equal to $2016$ satisfying $i+j\mid ia_i+ja_j$ for all $i, j\in \{ 1,2,\ldots, 2016\}$.

Find all real $x > 0$ and integer $n > 0$ so that $$ \lfloor x \rfloor+\left\{ \frac{1}{x}\right\}= 1.005 \cdot n.$$

Let $ABCDE$ be a convex pentagon such that angles $CAB$, $BCA$, $ECD$, $DEC$ and $AEC$ are equal. Prove that $CE$ bisects $BD$.

Alex launches his boat into a river and heads upstream at a constant speed. At the same time at a point $8$ miles upstream from Alex, Alice launches her boat and heads downstream at a constant speed. Both boats move at $6$ miles per hour in still water, but the river is owing downstream at $2\frac{3}{10}$ miles per hour. Alex and Alice will meet at a point that is $\frac{m}{n}$ miles from Alex's starting point, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

Find all positive integer $ n$ satisfying the following condition: There exist positive integers $ m$, $ a_1$, $ a_2$, $ \cdots$, $ a_{m\minus{}1}$, such that $ \displaystyle n \equal{} \sum_{i\equal{}1}^{m\minus{}1} a_i(m\minus{}a_i)$, where $ a_1$, $ a_2$, $ \cdots$, $ a_{m\minus{}1}$ may not distinct and $ 1 \leq a_i \leq m\minus{}1$.

The faces of a box with integer edge lengths are painted green. The box is partitioned into unit cubes. Find the dimensions of the box if the number of unit cubes with no green face is one third of the total number of cubes.

Let $k$ and $d$ be positive integers. Prove that there exists a positive integer $N$ such that for every odd integer $n>N$, the digits in the base-$2n$ representation of $n^k$ are all greater than $d$.

For a point $ M$ inside an equilateral triangle $ ABC$, let $ D,E,F$ be the feet of the perpendiculars from $ M$ onto $ BC,CA,AB$, respectively. Find the locus of all such points $ M$ for which $ \angle FDE$ is a right angle.

(A.Zaslavsky) A convex polygon can be divided into 2008 congruent quadrilaterals. Is it true that this polygon has a center or an axis of symmetry?

Find all positive integers $x$ for which there exists a positive integer $y$ such that $\dbinom{x}{y}=1999000$

The numbers $p, 4p^2 + 1,$ and $6p^2 + 1$ are primes. Determine $p.$

Let $O$ be the circumcenter and $H$ the orthocenter of an acute triangle $ABC$. Show that there exist points $D$, $E$, and $F$ on sides $BC$, $CA$, and $AB$ respectively such that \[ OD + DH = OE + EH = OF + FH\] and the lines $AD$, $BE$, and $CF$ are concurrent.

Find $n$ so that $(4^{n+7})^3=(2^{n+23})^4.$

Which of the following values is largest? $\textbf{(A) }2+0+1+7\qquad\textbf{(B) }2 \times 0 +1+7\qquad\textbf{(C) }2+0 \times 1 + 7\qquad\textbf{(D) }2+0+1 \times 7\qquad\textbf{(E) }2 \times 0 \times 1 \times 7$