Let $s_n$ be the number of solutions to $a_1 + a_2 + a_3 +a _4 + b_1 + b_2 = n$, where $a_1,a_2,a_3$ and $a_4$ are elements of the set $\{2, 3, 5, 7\}$ and $b_1$ and $b_2$ are elements of the set $\{ 1, 2, 3, 4\}$. Find the number of $n$ for which $s_n$ is odd. [i]Author: Alex Zhu[/i] [hide="Clarification"]$s_n$ is the number of [i]ordered[/i] solutions $(a_1, a_2, a_3, a_4, b_1, b_2)$ to the equation, where each $a_i$ lies in $\{2, 3, 5, 7\}$ and each $b_i$ lies in $\{1, 2, 3, 4\}$. [/hide]

For any positive integer $n$, we define the integer $P(n)$ by : $P(n)=n(n+1)(2n+1)(3n+1)...(16n+1)$. Find the greatest common divisor of the integers $P(1)$, $P(2)$, $P(3),...,P(2016)$.

Find all positive integers $x,y$ such that $202x + 4x^2 = y^2$.

Define $f(x) = |x-1|$. Determine the number of real numbers $x$ such that $f(f(\cdots f(f(x))\cdots )) = 0$, where there are $2018$ $f$'s in the equation. [i]Proposed by Yannick Yao

Let $a,b,c,d$ be positive integers such that $ad \neq bc$ and $gcd(a,b,c,d)=1$. Let $S$ be the set of values attained by $\gcd(an+b,cn+d)$ as $n$ runs through the positive integers. Show that $S$ is the set of all positive divisors of some positive integer.

A $45 \times 45$ grid has had the central unit square removed. For which positive integers $n$ is it possible to cut the remaining area into $1 \times n$ and $n\times 1$ rectangles?

Let $ABC$ be an acute triangle with circumcircle $\omega$ and circumcenter $O$. The perpendicular from $A$ to $BC$ intersects $BC$ and $\omega$ at $D$ and $E$, respectively. Let $F$ be a point on the segment $AE$, such that $2 \cdot FD = AE$. Let $l$ be the perpendicular to $OF$ through $F$. Prove that $l$, the tangent to $\omega$ at $E$, and the line $BC$ are concurrent. Proposed by [i] Stefan Lozanovski, Macedonia[/i]

Let be two nontrivial rings linked by an application ($ K\stackrel{\vartheta }{\mapsto } L $) having the following properties: $ \text{(i)}\quad x,y\in K\implies \vartheta (x+y) = \vartheta (x) +\vartheta (y) $ $ \text{(ii)}\quad \vartheta (1)=1 $ $ \text{(iii)}\quad \vartheta \left( x^3\right) =\vartheta^3 (x) $ [b]a)[/b] Show that if $ \text{char} (L)\ge 4, $ and $ K,L $ are fields, then $ \vartheta $ is an homomorphism. [b]b)[/b] Prove that if $ K $ is a noncommutative division ring, then it’s possible that $ \vartheta $ is not an homomorphism.

In the figure, triangle $ADE$ is produced from triangle $ABC$ by a rotation by $90^o$ about the point $A$. If angle $D$ is $60^o$ and angle $E$ is $40^o$, how large is then angle $u$? [img]https://1.bp.blogspot.com/-6Fq2WUcP-IA/Xzb9G7-H8jI/AAAAAAAAMWY/hfMEAQIsfTYVTdpd1Hfx15QPxHmfDLEkgCLcBGAsYHQ/s0/2009%2BMohr%2Bp1.png[/img]

Eleven pirates find a treasure chest. When they split up the coins in it, they find that there are 5 coins left. They throw one pirate overboard and split the coins again, only to find that there are 3 coins left over. So, they throw another pirate over and try again. This time, the coins split evenly. What is the least number of coins there could have been?

Let $x, y$, and $z$ be real numbers satisfying $x + y + z = xyz.$ Prove that \[x(1 - y^2)(1 - z^2) + y(1 -z^2)(1 - x^2) + z(1 - x^2)(1 - y^2) = 4xyz.\]

Each of the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 is used only once to make two five-digit numbers so that they have the largest possible sum. Which of the following could be one of the numbers? $\textbf{(A)}\hspace{.05in}76531 \qquad \textbf{(B)}\hspace{.05in}86724 \qquad \textbf{(C)}\hspace{.05in}87431 \qquad \textbf{(D)}\hspace{.05in}96240 \qquad \textbf{(E)}\hspace{.05in}97403 $

If $m,~n,~p,$ and $q$ are real numbers and $f(x)=mx+n$ and $g(x)=px+q$, then the equation $f(g(x))=g(f(x))$ has a solution $\textbf{(A) }\text{for all choices of }m,~n,~p, \text{ and } q\qquad$ $\textbf{(B) }\text{if and only if }m=p\text{ and }n=q\qquad$ $\textbf{(C) }\text{if and only if }mq-np=0\qquad$ $\textbf{(D) }\text{if and only if }n(1-p)-q(1-m)=0\qquad$ $\textbf{(E) }\text{if and only if }(1-n)(1-p)-(1-q)(1-m)=0$

In a classroom there are $20$ rows of $22$ desks each $(22$ desks have noone in front of them). The $440$ contestants of a certain regional math contest are going to sit at the desks. Before the exam, the organizers left some amount of sweets on each desk, which amount can be any positive integer. When the students go into the room, just before sitting down they look at the desk behind them, the one on the left and the one diagonally opposite to the right of theirs, thus seeing how many sweets each one has; if there is no desk in any of these directions, they simply ignore that position. Then they sit and watch their own sweets. A student gets angry if any of the desks he saw has more than one candy more than his. The organizers managed to distribute the sweets in such a way that no student gets angry. Prove that there are $8$ students with the same amount of sweets.

The map of a country is in the form of a convex polygon with $n (n\geq5)$ sides, such as any 3 diagonals do not concur inside the polygon. Some of the diagonals are roads, which allow movement in both directions and the other diagonals are not roads. There are cities on the intersection points of any two diagonals inside the polygon and at least one of the two diagonals is a road. Prove that you can move from any city to any other city using at most 3 roads.

In the subtraction PURPLE $-$ COMET $=$ MEET each distinct letter represents a distinct decimal digit, and no leading digit is $0$. Find the greatest possible number represented by PURPLE.

Let $ a, $ $ b $, and $ c $ be sides in a triangle, a $ h_a, $ $ h_b $, and $ h_c $ are the corresponding altitudes. Prove that $h ^ 2_a + h ^ 2_b + h ^ 2_c \leq \frac{3}{4} (a ^ 2 + b ^ 2 + c ^ 2). $ When is the equation valid?

Specify at least one right triangle $ABC$ with integer sides, inside which you can specify a point $M$ such that the lengths of the segments $MA, MB, MC$ are integers. Are there many such triangles, none of which are are similar?

Suppose that $ a$, $ b$, and $ c$ are positive real numbers such that $ a^{\log_3 7} \equal{} 27$, $ b^{\log_7 11} \equal{} 49$, and $ c^{\log_{11} 25} \equal{} \sqrt {11}$. Find \[ a^{(\log_3 7)^2} \plus{} b^{(\log_7 11)^2} \plus{} c^{(\log_{11} 25)^2}. \]

Let be a triangle $ ABC, $ and three points $ A',B',C' $ on the segments $ BC,CA, $ respectively, $ AB, $ such that the lines $ AA',BB',CC' $ are concurent at $ M. $ Name $ a,b,c,x,y,z $ the areas of the triangles $ AB'M,BC'M,CA'M,AC'M,BA'M, $ respectively, $ CB'M. $ Show that: [b]a)[/b] $ abc=xyz $ [b]b)[/b] $ ab+bc+ca=xy+yz+zx $ [i]Bogdan Enescu[/i] and [i]Marcel Chiriță[/i]

Let $I,H,O$ be the incenter, centroid, and circumcenter of the nonisosceles triangle $ABC$. Prove that $AI \parallel HO$ if and only if $\angle BAC =120^{\circ}$.

Let $N$ be the positive integer $7777\ldots777$, a $313$-digit number where each digit is a $7$. Let $f(r)$ be the leading digit of the $r{ }$th root of $N$. What is$$f(2) + f(3) + f(4) + f(5)+ f(6)?$$ $(\textbf{A})\: 8\qquad(\textbf{B}) \: 9\qquad(\textbf{C}) \: 11\qquad(\textbf{D}) \: 22\qquad(\textbf{E}) \: 29$

Let $p,q$ be positive reals with sum 1. Show that for any $n$-tuple of reals $(y_1,y_2,...,y_n)$, there exists an $n$-tuple of reals $(x_1,x_2,...,x_n)$ satisfying $$p\cdot \max\{x_i,x_{i+1}\} + q\cdot \min\{x_i,x_{i+1}\} = y_i$$ for all $i=1,2,...,2017$, where $x_{2018}=x_1$.

Square $ABCD$ has center $O$, $AB=900$, $E$ and $F$ are on $AB$ with $AE<BF$ and $E$ between $A$ and $F$, $m\angle EOF =45^\circ$, and $EF=400$. Given that $BF=p+q\sqrt{r}$, wherer $p,q,$ and $r$ are positive integers and $r$ is not divisible by the square of any prime, find $p+q+r$.

Define an operation $*$ for positve real numbers as $a*b=\dfrac{ab}{a+b}$. Then $4*(4*4)$ equals: $\textbf{(A)}\ \frac{3}{4} \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ \dfrac{4}{3} \qquad \textbf{(D)}\ 2 \qquad \textbf{(E)}\ \dfrac{16}{3} $