Let $p$ be a prime greater than $2$. Prove that there is a prime $q < p$ such that $q^{p-1} - 1$ is not divisible by $p^2$

Let $x$ and $y$ be distinct positive integers below $15$. For any two distinct numbers $a, b$ from the set $\{2, x,y\}$, $ab + 1$ is always a positive square. Find all possible values of the square $xy + 1$.

In triangle $ABC$, point $M$ is the midpoint of $BC$, $P$ the point of intersection of the tangents at points $B$ and $C$ of the circumscribed circle of $ABC$, $N$ is the midpoint of the segment $MP$. The segment $AN$ meets the circumcircle $ABC$ at the point $Q$. Prove that $\angle PMQ = \angle MAQ$.

A polyomino is a connected figure constructed by joining one or more unit squares edge-to-edge. Determine, with proof, the number of non-congruent polyominoes with no holes, perimeter $180,$ and area $2024.$

Ruby has a non-negative integer $n$. In each second, Ruby replaces the number she has with the product of all its digits. Prove that Ruby will eventually have a single-digit number or $0$. (e.g. $86\rightarrow 8\times 6=48 \rightarrow 4 \times 8 =32 \rightarrow 3 \times 2=6$) [i]Proposed by Wong Jer Ren[/i]

In the triangle $ABC$, $\angle A$ is biggest. On the circumcircle of $\triangle ABC$, let $D$ be the midpoint of $\widehat{ABC}$ and $E$ be the midpoint of $\widehat{ACB}$. The circle $c_1$ passes through $A,B$ and is tangent to $AC$ at $A$, the circle $c_2$ passes through $A,E$ and is tangent $AD$ at $A$. $c_1$ and $c_2$ intersect at $A$ and $P$. Prove that $AP$ bisects $\angle BAC$. [hide="Diagram"][asy] /* File unicodetex not found. */ /* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki, go to User:Azjps/geogebra */ import graph; size(14.4cm); real labelscalefactor = 0.5; /* changes label-to-point distance */ pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ pen dotstyle = black; /* point style */ real xmin = -5.23, xmax = 9.18, ymin = -2.97, ymax = 4.82; /* image dimensions */ /* draw figures */ draw(circle((-1.32,1.36), 2.98)); draw(circle((3.56,1.53), 3.18)); draw((0.92,3.31)--(-2.72,-1.27)); draw(circle((0.08,0.25), 3.18)); draw((-2.72,-1.27)--(3.13,-0.65)); draw((3.13,-0.65)--(0.92,3.31)); draw((0.92,3.31)--(2.71,-1.54)); draw((-2.41,-1.74)--(0.92,3.31)); draw((0.92,3.31)--(1.05,-0.43)); /* dots and labels */ dot((-1.32,1.36),dotstyle); dot((0.92,3.31),dotstyle); label("$A$", (0.81,3.72), NE * labelscalefactor); label("$c_1$", (-2.81,3.53), NE * labelscalefactor); dot((3.56,1.53),dotstyle); label("$c_2$", (3.43,3.98), NE * labelscalefactor); dot((1.05,-0.43),dotstyle); label("$P$", (0.5,-0.43), NE * labelscalefactor); dot((-2.72,-1.27),dotstyle); label("$B$", (-3.02,-1.57), NE * labelscalefactor); dot((2.71,-1.54),dotstyle); label("$E$", (2.71,-1.86), NE * labelscalefactor); dot((3.13,-0.65),dotstyle); label("$C$", (3.39,-0.9), NE * labelscalefactor); dot((-2.41,-1.74),dotstyle); label("$D$", (-2.78,-2.07), NE * labelscalefactor); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* end of picture */[/asy][/hide]

[u]Round 9[/u] [b]p25.[/b] For how many nonempty subsets of $\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16\}$ is the sum of the elements divisble by $32$? [b]p26.[/b] America declared independence in $1776$. Take the sum of the cubes of the digits of $1776$ and let that equal $S_1$. Sum the cubes of the digits of $S_1$ to get $S_2$. Repeat this process $1776$ times. What is $S_{1776}$? [b]p27.[/b] Every Golden Grahams box contains a randomly colored toy car, which is one of four colors. What is the expected number of boxes you have to buy in order to obtain one car of each color? [u]Round 10[/u] [b]p28.[/b] Let $B$ be the answer to Question $29$ and $C$ be the answer to Question $30$. What is the sum of the square roots of $B$ and $C$? [b]p29.[/b] Let $A$ be the answer to Question $28$ and $C$ be the answer to Question $30$. What is the sum of the sums of the digits of $A$ and $C$? [b]p30.[/b] Let $A$ be the answer to Question $28$ and $B$ be the answer to Question $29$. What is $A + B$? [u]Round 11[/u] [b]p31.[/b] If $x + \frac{1}{x} = 4$, find $x^6 + \frac{1}{x^6}$. [b]p32.[/b] Given a positive integer $n$ and a prime $p$, there is are unique nonnegative integers $a$ and $b$ such that $n = p^b \cdot a$ and $gcd (a, p) = 1$. Let $v_p(n)$ denote this uniquely determined $a$. Let $S$ denote the set of the first 20 primes. Find $\sum_{ p \in S} v_p \left(1 + \sum^{100}_{i=0} p^i \right)$. [b]p33. [/b] Find the maximum value of n such that $n+ \sqrt{(n - 1) +\sqrt{(n - 2) + ... +\sqrt{1}}} < 49$ (Note: there would be $n - 1$ square roots and $n$ total terms). [u]Round 12[/u] [b]p34.[/b] Give two numbers $a$ and $b$ such that $2015^a < 2015! < 2015^b$. If you are incorrect you get $-5$ points; if you do not answer you get $0$ points; otherwise you get $\max \{20-0.02(|b - a| - 1), 0\}$ points, rounded down to the nearest integer. [b]p35.[/b] Twin primes are prime numbers whose difference is $2$. Let $(a, b)$ be the $91717$-th pair of twin primes, with $a < b$. Let $k = a^b$, and suppose that $j$ is the number of digits in the base $10$ representation of $k$. What is $j^5$? If the correct answer is $n$ and you say $m$, you will receive $\max \left(20 - | \log \left(| \frac{m}{n} |\right), 0 \right)$ points, rounded down to the nearest integer. [b]p36.[/b] Write down any positive integer. Let the sum of the valid submissions (i.e. positive integer submissions) for all teams be $S$. One team will be chosen randomly, according to the following distribution: if your team's submission is $n$, you will be chosen with probability $\frac{n}{S}$ . The amount of points that the chosen team will win is the greatest integer not exceeding $\min \{K, \frac{ 10000}{S} \}$. $K$ is a predetermined secret value. PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h3157009p28696627]here [/url] and 5-8 [url=https://artofproblemsolving.com/community/c3h3157013p28696685]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

If the real numbers $x,y,z$ satisfies the equations $\frac{xyz}{x+y}=-1$, $\frac{xyz}{y+z}=1$, and $\frac{xyz}{z+x}=2$, what can $xyz$ be? $ \textbf{(A)}\ -\frac{8}{\sqrt {15}} \qquad\textbf{(B)}\ \frac{8}{\sqrt 5} \qquad\textbf{(C)}\ -8\sqrt{\frac{3}{5}} \qquad\textbf{(D)}\ \frac{7}{\sqrt{15}} \qquad\textbf{(E)}\ \text{None} $

Cut the $19 \times 20$ grid rectangle along the grid lines into several squares so that there are exactly four of them with odd sidelengths.

Problem: A person has seven friends and invites a different subset of three friends to dinner every night for one week (seven days). In how many ways can this be done so that all friends are invited at least once?

Find all real solutions of the equation \[ \left(1+x^2\right)\left(1+x^3\right)\left(1+x^5\right)=8x^5 \]

For each prime $p$, construct a graph $G_p$ on $\{1,2,\ldots p\}$, where $m\neq n$ are adjacent if and only if $p$ divides $(m^{2} + 1-n)(n^{2} + 1-m)$. Prove that $G_p$ is disconnected for infinitely many $p$

$X_1, X_2, . . . , X_{2015}$ are $2015$ points in the plane such that for all $1 \le i, j \le 2015$, the line segment $X_iX_{i+1} = X_jX_{j+1}$ and angle $\angle X_iX_{i+1}X_{i+2} = \angle X_jX_{j+1}X_{j+2}$ (with cyclic indices such that $X_{2016} = X_1$ and $X_{2017} = X_2$). Given fixed $X_1$ and $X_2$, determine the number of possible locations for $X_3$.

Let $x$ be a real number such that $\sin^{10}x+\cos^{10} x = \tfrac{11}{36}$. Then $\sin^{12}x+\cos^{12} x = \tfrac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

The vertices of an equilateral triangle lie on sides $ AB$, $CD$ and $EF$ of a regular hexagon $ABCDEF$. Prove that the triangle and the hexagon have a common centre. (N Sedrakyan, Yerevan )

An equilateral triangle with side length $1$ has the same area as a square with side length $s$. Find $s$. $\textbf{(A) }\dfrac{\sqrt[4]3}2\hspace{14em}\textbf{(B) }\dfrac{\sqrt[4]3}{\sqrt2}\hspace{14em}\textbf{(C) }1$ $\textbf{(D) }\dfrac34\hspace{14.9em}\textbf{(E) }\dfrac43\hspace{14.9em}\textbf{(F) }\sqrt3$ $\textbf{(G) }\dfrac{\sqrt6}2$

Let $T=\text{TNFTPP}$. For natural numbers $k,n\geq 2$, we define $S(k,n)$ such that \[S(k,n)=\left\lfloor\dfrac{2^{n+1}+1}{2^{n-1}+1}\right\rfloor+\left\lfloor\dfrac{3^{n+1}+1}{3^{n-1}+1}\right\rfloor+\cdots+\left\lfloor\dfrac{k^{n+1}+1}{k^{n-1}+1}\right\rfloor.\] Compute the value of $S(10,T+55)-S(10,55)+S(10,T-55)$.

Let $ABC$ be an acute triangle , with $AB \neq AC$ and denote its orthocenter by $H$ . The point $D$ is located on the side $BC$ and the circumcircles of the triangles $ABD$ and $ACD$ intersects for the second time the lines $AC$ , respectively $AB$ in the points $E$ respectively $F$. If we denote by $P$ the intersection point of $BE$ and $CF$ then show that $HP \parallel BC$ if and only if $AD$ passes through the circumcenter of the triangle $ABC$.

One writes 268 numbers around a circle, such that the sum of 20 consectutive numbers is always equal to 75. The number 3, 4 and 9 are written in positions 17, 83 and 144 respectively. Find the number in position 210.

Let $ABC$ be an arbitrary triangle. Let $E,E$ be two points on $AB,AC$ respectively such that their distance to the midpoint of $BC$ is equal. Let $P$ be the second intersection of the triangles $ABC,AEF$ circumcircles . The tangents from $E,F$ to the circumcircle of $AEF$ intersect each other at $K$. Prove that : $\angle KPA = 90$

[u]Round 1[/u] [b]p1.[/b] This year, the Mathathon consists of $7$ rounds, each with $3$ problems. Another math test, Aspartaime, consists of $3$ rounds, each with $5$ problems. How many more problems are on the Mathathon than on Aspartaime? [b]p2.[/b] Let the solutions to $x^3 + 7x^2 - 242x - 2016 = 0 $be $a, b$, and $c$. Find $a^2 + b^2 + c^2$. (You might find it helpful to know that the roots are all rational.) [b]p3.[/b] For triangle $ABC$, you are given $AB = 8$ and $\angle A = 30^o$ . You are told that $BC$ will be chosen from amongst the integers from $1$ to $10$, inclusive, each with equal probability. What is the probability that once the side length $BC$ is chosen there is exactly one possible triangle $ABC$? [u]Round 2 [/u] [b]p4.[/b] It’s raining! You want to keep your cat warm and dry, so you want to put socks, rain boots, and plastic bags on your cat’s four paws. Note that for each paw, you must put the sock on before the boot, and the boot before the plastic bag. Also, the items on one paw do not affect the items you can put on another paw. How many different orders are there for you to put all twelve items of rain footwear on your cat? [b]p5.[/b] Let $a$ be the square root of the least positive multiple of $2016$ that is a square. Let $b$ be the cube root of the least positive multiple of $2016$ that is a cube. What is $ a - b$? [b]p6.[/b] Hypersomnia Cookies sells cookies in boxes of $6, 9$ or $10$. You can only buy cookies in whole boxes. What is the largest number of cookies you cannot exactly buy? (For example, you couldn’t buy $8$ cookies.) [u]Round 3 [/u] [b]p7.[/b] There is a store that sells each of the $26$ letters. All letters of the same type cost the same amount (i.e. any ‘a’ costs the same as any other ‘a’), but different letters may or may not cost different amounts. For example, the cost of spelling “trade” is the same as the cost of spelling “tread,” even though the cost of using a ‘t’ may be different from the cost of an ‘r.’ If the letters to spell out $1$ cost $\$1001$, the letters to spell out $2$ cost $\$1010$, and the letters to spell out $11$ cost $\$2015$, how much do the letters to spell out $12$ cost? [b]p8.[/b] There is a square $ABCD$ with a point $P$ inside. Given that $PA = 6$, $PB = 9$, $PC = 8$. Calculate $PD$. [b]p9.[/b] How many ordered pairs of positive integers $(x, y)$ are solutions to $x^2 - y^2 = 2016$? [u]Round 4 [/u] [b]p10.[/b] Given a triangle with side lengths $5, 6$ and $7$, calculate the sum of the three heights of the triangle. [b]p11. [/b]There are $6$ people in a room. Each person simultaneously points at a random person in the room that is not him/herself. What is the probability that each person is pointing at someone who is pointing back to them? [b]p12.[/b] Find all $x$ such that $\sum_{i=0}^{\infty} ix^i =\frac34$. PS. You should use hide for answers. Rounds 5-7 have been posted [url=https://artofproblemsolving.com/community/c4h2782837p24446063]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $S$ be a nonempty set of positive integers. We say that a positive integer $n$ is [i]clean[/i] if it has a unique representation as a sum of an odd number of distinct elements from $S$. Prove that there exist infinitely many positive integers that are not clean.

Write the sum $\sum_{i=0}^{n}{\frac{(-1)^i\cdot\binom{n}{i}}{i^3 +9i^2 +26i +24}}$ as the ratio of two explicitly defined polynomials with integer coefficients.

Let \[ S_1 \equal{} \{ (x,y)\ | \ \log_{10} (1 \plus{} x^2 \plus{} y^2)\le 1 \plus{} \log_{10}(x \plus{} y)\} \]and \[ S_2 \equal{} \{ (x,y)\ | \ \log_{10} (2 \plus{} x^2 \plus{} y^2)\le 2 \plus{} \log_{10}(x \plus{} y)\}. \]What is the ratio of the area of $ S_2$ to the area of $ S_1$? $ \textbf{(A) } 98\qquad \textbf{(B) } 99\qquad \textbf{(C) } 100\qquad \textbf{(D) } 101\qquad \textbf{(E) } 102$

There is a set of 1000 switches, each of which has four positions, called $A, B, C,$ and $D.$ When the position of any switch changes, it is only from $A$ to $B,$ from $B$ to $C,$ from $C$ to $D,$ or from $D$ to $A.$ Initially each switch is in position $A.$ The switches are labeled with the 1000 different integers $2^x3^y5^z,$ where $x, y,$ and $z$ take on the values $0, 1, \ldots, 9.$ At step $i$ of a 1000-step process, the $i$th switch is advanced one step, and so are all the other switches whose labels divide the label on the $i$th switch. After step 1000 has been completed, how many switches will be in position $A$?