Let $n$ be a positive integer. Using the integers from $1$ to $4n$ inclusive, pairs are to be formed such that the product of the numbers in each pair is a perfect square. Each number can be part of at most one pair, and the two numbers in each pair must be different. Determine, for each $n$, the maximum number of pairs that can be formed.

Given that the polynomial $P(x) = x^5 - x^2 + 1$ has $5$ roots $r_1, r_2, r_3, r_4, r_5$. Find the value of the product $Q(r_1)Q(r_2)Q(r_3)Q(r_4)Q(r_5)$, where $Q(x) = x^2 + 1$.

The number $d_1d_2\cdots d_9$ has nine (not necessarily distinct) decimal digits. The number $e_1e_2\cdots e_9$ is such that each of the nine $9$-digit numbers formed by replacing just one of the digits $d_i$ in $d_1d_2\cdots d_9$ by the corresponding digit $e_i \;\;(1 \le i \le 9)$ is divisible by $7$. The number $f_1f_2\cdots f_9$ is related to $e_1e_2\cdots e_9$ is the same way: that is, each of the nine numbers formed by replacing one of the $e_i$ by the corresponding $f_i$ is divisible by $7$. Show that, for each $i$, $d_i-f_i$ is divisible by $7$. [For example, if $d_1d_2\cdots d_9 = 199501996$, then $e_6$ may be $2$ or $9$, since $199502996$ and $199509996$ are multiples of $7$.]

Given an integer $n>1$ and a $n\times n$ grid $ABCD$ containing $n^2$ unit squares, each unit square is colored by one of three colors: Black, white and gray. A coloring is called [i]symmetry[/i] if each unit square has center on diagonal $AC$ is colored by gray and every couple of unit squares which are symmetry by $AC$ should be both colred by black or white. In each gray square, they label a number $0$, in a white square, they will label a positive integer and in a black square, a negative integer. A label will be called $k$-[i]balance[/i] (with $k\in\mathbb{Z}^+$) if it satisfies the following requirements: i) Each pair of unit squares which are symmetry by $AC$ are labelled with the same integer from the closed interval $[-k,k]$ ii) If a row and a column intersectes at a square that is colored by black, then the set of positive integers on that row and the set of positive integers on that column are distinct.If a row and a column intersectes at a square that is colored by white, then the set of negative integers on that row and the set of negative integers on that column are distinct. a) For $n=5$, find the minimum value of $k$ such that there is a $k$-balance label for the following grid [asy] size(4cm); pair o = (0,0); pair y = (0,5); pair z = (5,5); pair t = (5,0); dot("$A$", y, dir(180)); dot("$B$", z); dot("$C$", t); dot("$D$", o, dir(180)); fill((0,5)--(1,5)--(1,4)--(0,4)--cycle,gray); fill((1,4)--(2,4)--(2,3)--(1,3)--cycle,gray); fill((2,3)--(3,3)--(3,2)--(2,2)--cycle,gray); fill((3,2)--(4,2)--(4,1)--(3,1)--cycle,gray); fill((4,1)--(5,1)--(5,0)--(4,0)--cycle,gray); fill((0,3)--(1,3)--(1,1)--(0,1)--cycle,black); fill((2,5)--(4,5)--(4,4)--(2,4)--cycle,black); fill((2,1)--(3,1)--(3,0)--(2,0)--cycle,black); fill((2,1)--(3,1)--(3,0)--(2,0)--cycle,black); fill((4,3)--(5,3)--(5,2)--(4,2)--cycle,black); for (int i=0; i<=5; ++i) { draw((0,i)--(5,i)^^(i,0)--(i,5)); } [/asy] b) Let $n=2017$. Find the least value of $k$ such that there is always a $k$-balance label for a symmetry coloring.

$ ABC$ is a triangle with $ \angle BAC \equal{} 10{}^\circ$, $ \angle ABC \equal{} 150{}^\circ$. Let $ X$ be a point on $ \left[AC\right]$ such that $ \left|AX\right| \equal{} \left|BC\right|$. Find $ \angle BXC$. $\textbf{(A)}\ 15^\circ \qquad\textbf{(B)}\ 20^\circ \qquad\textbf{(C)}\ 25^\circ \qquad\textbf{(D)}\ 30^\circ \qquad\textbf{(E)}\ 35^\circ$

A bug starts at vertex $A$ of triangle $ABC$. Six times, the bug travels to a randomly chosen adjacent vertex. For example, the bug could go from $A$, to $B$, to $C$, back to $B$, and back to $C$. What is the probability that the bug ends up at $A$ after its six moves?

If three of the roots of the quartic polynomial $f(x) = x^4 + ax^3 + bx^2 + cx + d$ are $0$, $2$, and $4$, and the sum of $a$, $b$, and $c$ is at most $12$, then find the largest possible value of $f(1)$.

Determine all integers $n\ge 2$ with the following property: the number $2^k\cdot n-1$ is prime for all $k\in\{2,3,\ldots,n\}$.

Is it true that if two linear subspaces $V$ and $W$ of a Hilbert space are closed, then their sum $V+W$ is also closed?

A mixture of 30 liters of paint is $25\%$ red tint, $30\%$ yellow tint, and $45\%$ water. Five liters of yellow tint are added to the original mixture. What is the percent of yellow tint that is the mixture? $\textbf{(A)}\ 25 \qquad \textbf{(B)}\ 35 \qquad \textbf{(C)}\ 40\qquad \textbf{(D)}\ 45 \qquad \textbf{(E)}\ 50$

When Cirno walks into her perfect math class today, she sees a polynomial $P(x)=1$ (of degree 0) on the blackboard. As her teacher explains, for her pop quiz today, she will have to perform one of the two actions every minute: (a) Add a monomial to $P(x)$ so that the degree of $P$ increases by 1 and $P$ remains monic; (b) Replace the current polynomial $P(x)$ by $P(x+1)$. For example, if the current polynomial is $x^2+2x+3$, then she will change it to $(x+1)^2+2(x+1)+3=x^2+4x+6$. Her score for the pop quiz is the sum of coefficients of the polynomial at the end of 9 minutes. Given that Cirno (miraculously) doesn't make any mistakes in performing the actions, what is the maximum score that she can get? [i]Proposed by Yannick Yao[/i]

The product of $\sqrt[3]{4}$ and $\sqrt[4]{8}$ equals $\textbf{(A) }\sqrt[7]{12}\qquad\textbf{(B) }2\sqrt[7]{12}\qquad\textbf{(C) }\sqrt[7]{32}\qquad\textbf{(D) }\sqrt[12]{32}\qquad\textbf{(E) }2\sqrt[12]{32}$

Prove that the number $1\underbrace{\hbox{0...0}}_{\hbox{49}}5\underbrace{\hbox{0...0}}_{\hbox{99}}1$ is not the cube of any integer.

Prove that there exists a unique point $M$ on the side $AD$ of a convex quadrilateral $ABCD$ such that \[\sqrt{S_{ABM}}+\sqrt{S_{CDM}} = \sqrt{S_{ABCD}}\] if and only if $AB\parallel CD$.

What is the smallest number over 9000 that is divisible by the first four primes?

A regular 60-gon is given. What is the maximum size of a subset of its vertices containing no isosceles triangles?

Side $\overline{AB}$ of $\triangle ABC$ has length $10$. The bisector of angle $A$ meets $\overline{BC}$ at $D$, and $CD = 3$. The set of all possible values of $AC$ is an open interval $(m,n)$. What is $m+n$? $ \textbf{(A) }16 \qquad \textbf{(B) }17 \qquad \textbf{(C) }18 \qquad \textbf{(D) }19 \qquad \textbf{(E) }20 \qquad $

Find the shortest distance between the point $(6,12)$ and the parabola given by the equation $x=\frac{y^2}{2}$

Give rational numbers $x, y$ such that $(x^2 + y^2 - 2) (x + y)^2 + (xy + 1)^2 = 0 $ Prove that $\sqrt{1 + xy}$ is a rational number.

Consider the sequence $ \left( x_n \right)_{n\ge 1} $ having $ x_1>1 $ and satisfying the equation $$ x_1+x_2+\cdots +x_{n+1} =x_1x_2\cdots x_{n+1} ,\quad\forall n\in\mathbb{N} . $$ Show that this sequence is convergent and find its limit.

Find all positive integers $(x, n)$ such that $x^{n}+2^{n}+1$ divides $x^{n+1}+2^{n+1}+1$.

A frog starts at $0$ on a number line and plays a game. On each turn the frog chooses at random to jump $1$ or $2$ integers to the right or left. It stops moving if it lands on a nonpositive number or a number on which it has already landed. If the expected number of times it will jump is $\tfrac{p}{q}$ for relatively prime positive integers $p$ and $q$, find $p+q$. [i]Proposed by Michael Kural[/i]

Decide whether for every arrangement of the numbers $1,2,3, . . . ,15$ in a sequence one can color these numbers with at most four different colors in such a way that the numbers of each color form a monotone subsequence.

The points of a circle are colored by three colors. Prove that there exist infinitely many isosceles triangles inscribed in the circle whose vertices are of the same color.

Let $AB$ be a diameter of the circle $\Omega$ with center $O{}$. The points $C, D, X$ and $Y{}$ are chosen on $\Omega$ so that the segments $CX$ and $DX$ intersect the segment $AB$ at points symmetric with respect to $O{}$, and $XY\parallel AB$. Let the lines $AB{}$ and $CD{}$ intersect at the point $E$. Prove that the tangent to $\Omega$ through $Y{}$ passes through $E{}$.