Two positive integers $p,q \in \mathbf{Z}^{+}$ are given. There is a blackboard with $n$ positive integers written on it. A operation is to choose two same number $a,a$ written on the blackboard, and replace them with $a+p,a+q$. Determine the smallest $n$ so that such operation can go on infinitely.

There are eight rooms on the first floor of a hotel, with four rooms on each side of the corridor, symmetrically situated (that is each room is exactly opposite to one other room). Four guests have to be accommodated in four of the eight rooms (that is, one in each) such that no two guests are in adjacent rooms or in opposite rooms. In how many ways can the guests be accommodated?

Let $a>1$ be an odd positive integer. Find the least positive integer $n$ such that $2^{2000}$ is a divisor of $a^n-1$. [i]Mircea Becheanu [/i]

Given 5 nonzero vectors in three-dimensional Euclidean space, prove that the sum of their pairwise angles is at most $6\pi$.

Let $ABC$ be a triangle with $\angle BAC \neq 90^{\circ}.$ Let $O$ be the circumcenter of the triangle $ABC$ and $\Gamma$ be the circumcircle of the triangle $BOC.$ Suppose that $\Gamma$ intersects the line segment $AB$ at $P$ different from $B$, and the line segment $AC$ at $Q$ different from $C.$ Let $ON$ be the diameter of the circle $\Gamma.$ Prove that the quadrilateral $APNQ$ is a parallelogram.

Let $f(x)=\frac{\sin x}{x}$, for $x>0$, and let $n$ be a positive integer. Prove that $|f^{(n)}(x)|<\frac{1}{n+1}$, where $f^{(n)}$ denotes the $n^{\mathrm{th}}$ derivative of $f$. (Proposed by Alexander Bolbot, State University, Novosibirsk)

Prove that the function $f (x)$ satisfying the relation $|f (x) - f (y) | \le | x - y|^a$ for any real numbers $x, y$ and some number $a> 1$ is constant.

Let $S$ be the set of all positive integer divisors of $100,000.$ How many numbers are the product of two distinct elements of $S?$ $\textbf{(A) }98\qquad\textbf{(B) }100\qquad\textbf{(C) }117\qquad\textbf{(D) }119\qquad\textbf{(E) }121$

Compute $$\int_0^4\frac{x^4-4x+4}{1+2017^{x-2}}dx.$$

Given $1990$ piles of stones, containing $1, 2, 3, ... , 1990$ stones. A move is to take an equal number of stones from one or more piles. How many moves are needed to take all the stones?

In a tournament there are six teams that play each other twice. A team earns 3 points for a win, 1 point for a draw, and 0 points for a loss. After all the games have been played it turns out that the top three teams earned the same number of total points. What is the greatest possible number of total points for each of the top three teams? $\textbf{(A) }22\qquad\textbf{(B) }23\qquad\textbf{(C) }24\qquad\textbf{(D) }26\qquad\textbf{(E) }30$

Determine whether it is possible to partition $\mathbb{Z}$ into triples $(a,b,c)$ such that, for every triple, $|a^3b + b^3c + c^3a|$ is perfect square.

Let $ ABC$ be an acute triangle with orthocenter $ H$ and let $ X$ be an arbitrary point in its plane. The circle with diameter $ HX$ intersects the lines $ AH$ and $ AX$ at $ A_{1}$ and $ A_{2}$, respectively. Similarly, define $ B_{1}$, $ B_{2}$, $ C_{1}$, $ C_{2}$. Prove that the lines $ A_{1}A_{2}$, $ B_{1}B_{2}$, $ C_{1}C_{2}$ are concurrent. [hide][i]Remark[/i]. The triangle obviously doesn't need to be acute.[/hide]

[u]Round 5[/u] [b]p13.[/b] A $2016 \times 2016$ chess board is cut into $k \ge 1$ rectangle(s) with positive integer sidelengths. Let $p$ be the sum of the perimeters of all $k$ rectangles. Additionally, let $m$ and $M$ be the minimum and maximum possible value of $\frac{p}{k}$, respectively. Determine the ordered pair $(m,M)$. [b]p14.[/b] For nonnegative integers $n$, let $f (n)$ be the product of the digits of $n$. Compute $\sum^{1000}_{i=1}f (i )$. [b]p15.[/b] How many ordered pairs of positive integers $(m,n)$ have the property that $mn$ divides $2016$? [u]Round 6[/u] [b]p16.[/b] Let $a,b,c$ be distinct integers such that $a +b +c = 0$. Find the minimum possible positive value of $|a^3 +b^3 +c^3|$. [b]p17.[/b] Find the greatest positive integer $k$ such that $11^k -2^k$ is a perfect square. [b]p18.[/b] Find all ordered triples $(a,b,c)$ with $a \le b \le c$ of nonnegative integers such that $2a +2b +2c = ab +bc +ca$. [u]Round 7[/u] [b]p19.[/b] Let $f :N \to N$ be a function such that $f ( f (n))+ f (n +1) = n +2$ for all positive integers $n$. Find $f (20)+ f (16)$. [b]p20.[/b] Let $\vartriangle ABC$ be a triangle with area $10$ and $BC = 10$. Find the minimum possible value of $AB \cdot AC$. [b]p21.[/b] Let $\vartriangle ABC$ be a triangle with sidelengths $AB = 19$, $BC = 24$, $C A = 23$. Let $D$ be a point on minor arc $BC$ of the circumcircle of $\vartriangle ABC$ such that $DB =DC$. A circle with center $D$ that passes through $B$ and $C$ interests $AC$ again at a point $E \ne C$. Find the length of $AE$. [u]Round 8[/u] [b]p22.[/b] Let $m =\frac12 \sqrt{2+\sqrt{2+... \sqrt2}}$, where there are $2014$ square roots. Let $f_1(x) =2x^2 -1$ and let $f_n(x) = f_1( f_{n-1}(x))$. Find $f_{2015}(m)$. [b]p23.[/b] How many ordered triples of integers $(a,b,c)$ are there such that $0 < c \le b \le a \le 2016$, and $a +b-c = 2016$? [b]p24.[/b] In cyclic quadrilateral $ABCD$, $\angle B AD = 120^o$,$\angle ABC = 150^o$,$CD = 8$ and the area of $ABCD$ is $6\sqrt3$. Find the perimeter of $ABCD$. PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h3158461p28714996]here [/url] and 9-12 [url=https://artofproblemsolving.com/community/c3h3162282p28763571]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Yan is somewhere between his home and the stadium. To get to the stadium he can walk directly to the stadium, or else he can walk home and then ride his bicycle to the stadium. He rides $ 7$ times as fast as he walks, and both choices require the same amount of time. What is the ratio of Yan's distance from his home to his distance from the stadium? $ \textbf{(A)}\ \frac {2}{3}\qquad \textbf{(B)}\ \frac {3}{4}\qquad \textbf{(C)}\ \frac {4}{5}\qquad \textbf{(D)}\ \frac {5}{6}\qquad \textbf{(E)}\ \frac {6}{7}$

The catheti $AC$ and $BC$ in a right-angled triangle $ABC$ have lengths $b$ and $a$, respectively. A circle centered at $C$ is tangent to hypotenuse $AB$ at point $D$. The tangents to the circle through points $A$ and $B$ intersect the circle at points $E$ and $F$, respectively (where $E$ and $F$ are both different from $D$). Express the length of the segment $EF$ in terms of $a$ and $b$.

Each person in Cambridge drinks a (possibly different) $12$ ounce mixture of water and apple juice, where each drink has a positive amount of both liquids. Marc McGovern, the mayor of Cambridge, drinks $\frac{1}{6}$ of the total amount of water drunk and $\frac{1}{8}$ of the total amount of apple juice drunk. How many people are in Cambridge?

A line $r$ contains the points $A$, $B$, $C$, $D$ in that order. Let $P$ be a point not in $r$ such that $\angle{APB} = \angle{CPD}$. Prove that the angle bisector of $\angle{APD}$ intersects the line $r$ at a point $G$ such that: $\frac{1}{GA} + \frac{1}{GC} = \frac{1}{GB} + \frac{1}{GD}$

Is it possible to choose $1983$ distinct positive integers, all less than or equal to $10^5$, no three of which are consecutive terms of an arithmetic progression?

[u]Set 1[/u] [b]p1.[/b] Find $1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11$. [b]p2.[/b] Find $1 \cdot 11 + 2 \cdot 10 + 3 \cdot 9 + 4 \cdot 8 + 5 \cdot 7 + 6 \cdot 6$. [b]p3.[/b] Let $\frac{1}{1\cdot 2} +\frac{1}{2\cdot 3} +\frac{1}{3\cdot 4} +\frac{1}{4\cdot 5} +\frac{1}{5\cdot 6} +\frac{1}{6\cdot 7} +\frac{1}{7\cdot 8} +\frac{1}{8\cdot 9} +\frac{1}{9\cdot 10} +\frac{1}{10\cdot 11} =\frac{m}{n}$ , where $m$ and $n$ are positive integers that share no prime divisors. Find $m + n$. [u]Set 2[/u] [b]p4.[/b] Define $0! = 1$ and let $n! = n \cdot (n - 1)!$ for all positive integers $n$. Find the value of $(2! + 0!)(1! + 8!)$. [b]p5.[/b] Rachel’s favorite number is a positive integer $n$. She gives Justin three clues about it: $\bullet$ $n$ is prime. $\bullet$ $n^2 - 5n + 6 \ne 0$. $\bullet$ $n$ is a divisor of $252$. What is Rachel’s favorite number? [b]p6.[/b] Shen eats eleven blueberries on Monday. Each day after that, he eats five more blueberries than the day before. For example, Shen eats sixteen blueberries on Tuesday. How many blueberries has Shen eaten in total before he eats on the subsequent Monday? [u]Set 3[/u] [b]p7.[/b] Triangle $ABC$ satisfies $AB = 7$, $BC = 12$, and $CA = 13$. If the area of $ABC$ can be expressed in the form $m\sqrt{n}$, where $n$ is not divisible by the square of a prime, then determine $m + n$. [b]p8.[/b] Sebastian is playing the game Split! on a coordinate plane. He begins the game with one token at $(0, 0)$. For each move, he is allowed to select a token on any point $(x, y)$ and take it off the plane, replacing it with two tokens, one at $(x + 1, y)$, and one at $(x, y + 1)$. At the end of the game, for a token on $(a, b)$, it is assigned a score $\frac{1}{2^{a+b}}$ . These scores are summed for his total score. Determine the highest total score Sebastian can get in $100$ moves. [b]p9.[/b] Find the number of positive integers $n$ satisfying the following two properties: $\bullet$ $n$ has either four or five digits, where leading zeros are not permitted, $\bullet$ The sum of the digits of $n$ is a multiple of $3$. [u]Set 4[/u] [b]p10.[/b] [i]A unit square rotated $45^o$ about a vertex, Sweeps the area for Farmer Khiem’s pen. If $n$ is the space the pigs can roam, Determine the floor of $100n$.[/i] If $n$ is the area a unit square sweeps out when rotated 4$5$ degrees about a vertex, determine $\lfloor 100n \rfloor$. Here $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$. [img]https://cdn.artofproblemsolving.com/attachments/b/1/129efd0dbd56dc0b4fb742ac80eaf2447e106d.png[/img] [b]p11.[/b][i] Michael is planting four trees, In a grid, three rows of three, If two trees are close, Then both are bulldozed, So how many ways can it be?[/i] In a three by three grid of squares, determine the number of ways to select four squares such that no two share a side. [b]p12.[/b] [i]Three sixty-seven Are the last three digits of $n$ cubed. What is $n$?[/i] If the last three digits of $n^3$ are $367$ for a positive integer $n$ less than $1000$, determine $n$. [u]Set 5[/u] [b]p13.[/b] Determine $\sqrt[4]{97 + 56\sqrt{3}} + \sqrt[4]{97 - 56\sqrt{3}}$. [b]p14. [/b]Triangle $\vartriangle ABC$ is inscribed in a circle $\omega$ of radius $12$ so that $\angle B = 68^o$ and $\angle C = 64^o$ . The perpendicular from $A$ to $BC$ intersects $\omega$ at $D$, and the angle bisector of $\angle B$ intersects $\omega$ at $E$. What is the value of $DE^2$? [b]p15.[/b] Determine the sum of all positive integers $n$ such that $4n^4 + 1$ is prime. [u]Set 6[/u] [b]p16.[/b] Suppose that $p, q, r$ are primes such that $pqr = 11(p + q + r)$ such that $p\ge q \ge r$. Determine the sum of all possible values of $p$. [b]p17.[/b] Let the operation $\oplus$ satisfy $a \oplus b =\frac{1}{1/a+1/b}$ . Suppose $$N = (...((2 \oplus 2) \oplus 2) \oplus ... 2),$$ where there are $2018$ instances of $\oplus$ . If $N$ can be expressed in the form $m/n$, where $m$ and $n$ are relatively prime positive integers, then determine $m + n$. [b]p18.[/b] What is the remainder when $\frac{2018^{1001} - 1}{2017}$ is divided by $2017$? PS. You had better use hide for answers. Last sets have been posted [url=https://artofproblemsolving.com/community/c4h2777307p24369763]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

How many $6$-digit positive integers whose digits are different from $0$ are there such that each number generated by rearranging the digits of the original number is always divisible by $7$? $ \textbf{(A)}\ 11 \qquad\textbf{(B)}\ 77 \qquad\textbf{(C)}\ 133 \qquad\textbf{(D)}\ 166 \qquad\textbf{(E)}\ 255 $

$(1 + \sqrt 3)^2$ may be written as $a + b \sqrt 3$ for certain integers $a$ and $b$. What is $a + b$? $\textbf{(A) }1\qquad\textbf{(B) }2\qquad\textbf{(C) }4\qquad\textbf{(D) }6\qquad\textbf{(E) }7$

The coordinates of the vertices of isosceles trapezoid $ABCD$ are all integers, with $A=(20,100)$ and $D=(21,107).$ The trapezoid has no horizontal or vertical sides, and $\overline{AB}$ and $\overline{CD}$ are the only parallel sides. The sum o f the absolute values of all possible slopes for $\overline{AB}$ is $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$

For each problem, you will be asked to submit two integers. The first value that you submit represents what you think the correct answer to the problem is. The second value that you submit represents [b]how many teams[/b] you think will submit the correct answer. For example, consider 0. What is $32\div 2 \times 4 +3?$ The correct answer would be $67$. If you think every team will get it right, you should submit the number of teams competing at PUMaC. Therefore, a viable submission for the first entry could be $67$ and $n$ for the second, where $n$ is the number of teams taking the team round. There are $\mathbf{72}$ [b]teams[/b] signed up to take this round at PUMaC: 45 in A division and 27 in B division. You will receive $\left(\min\left(\tfrac{a}{b},\tfrac{b}{a}\right)\right)^2$ points for your guess, where $a$ is the number of teams that correctly answered the question and $b$ is the number of teams you guessed would get it correct (Note that in the case that no teams answer correctly or you guess $0$, you will receive $0$ points).

Consider a circle $C_1$ and a point $O$ on it. Circle $C_2$ with center $O$, intersects $C_1$ in two points $P$ and $Q$. $C_3$ is a circle which is externally tangent to $C_2$ at $R$ and internally tangent to $C_1$ at $S$ and suppose that $RS$ passes through $Q$. Suppose $X$ and $Y$ are second intersection points of $PR$ and $OR$ with $C_1$. Prove that $QX$ is parallel with $SY$.