Suppose that $P(x)$ is a monic cubic polynomial with integer roots, and suppose that $\frac{P(a)}{a}$ is an integer for exactly $6$ integer values of $a$. Suppose furthermore that exactly one of the distinct numbers $\frac{P(1) + P(-1)}{2}$ and $\frac{P(1) - P(-1)}{2}$ is a perfect square. Given that $P(0) > 0$, find the second-smallest possible value of $P(0).$ [i]Proposed by Andrew Wu[/i]

The incircle of a scalene triangle $ABC$ touches the sides $BC, CA$, and $AB$ at points $D, E$, and $F$, respectively. Triangles $APE$ and $AQF$ are constructed outside the triangle so that \[AP =PE, AQ=QF, \angle APE=\angle ACB,\text{ and }\angle AQF =\angle ABC.\]Let $M$ be the midpoint of $BC$. Find $\angle QMP$ in terms of the angles of the triangle $ABC$. [i]Iran, Shayan Talaei[/i]

A circle can be inscribed in the convex quadrilateral $ ABCD$. The incircle touches the sides $ AB, BC, CD, DA$ at $ A_1, B_1, C_1, D_1$ respectively. The points $ E, F, G, H$ are the midpoints of $ A_1B_1, B_1C_1, C_1D_1, D_1A_1$ respectively. Prove that the quadrilateral $ EFGH$ is a rectangle if and only if $ A, B, C, D$ are concyclic.

The wheel shown is spun twice, and the randomly determined numbers opposite the pointer are recorded. The first number is divided by $ 4$, and the second number is divided by $ 5$. The first remainder designates a column, and the second remainder designates a row on the checkerboard shown. What is the probability that the pair of numbers designates a shaded square? [asy]unitsize(1.5cm); defaultpen(linewidth(.8pt)+fontsize(15pt)); draw(Circle(origin,1)); for(int i = 0;i < 6; ++i) { draw(origin--dir(60i+30)); } label("$7$",midpoint(origin--(dir(0))),E); label("$1$",midpoint(origin--(dir(60))),NE); label("$6$",midpoint(origin--(dir(120))),NW); label("$3$",midpoint(origin--(dir(180))),W); label("$9$",midpoint(origin--(dir(240))),SW); label("$2$",midpoint(origin--(dir(300))),SE); draw((2,0)--(3.5,0)--(3.5,1)--(2,1)--cycle); draw((2,0)--(3.5,0)--(3.5,-1)--(2,-1)--cycle); pair[] V = {(2.5,0.5),(2,0),(3,0),(2.5,-0.5),(2,-1),(3,-1)}; for(int i = 0; i <= 5; ++i) { pair A = V[i]; path p = A--(A.x,A.y + 0.5)--(A.x + 0.5,A.y + 0.5)--(A.x + 0.5, A.y)--cycle; fill(p,mediumgray); draw(p); } path pointer = (-2.5,-0.125)--(-2.5,0.125)--(-1.2,0.125)--(-1.05,0)--(-1.2,-0.125)--cycle; fill(pointer,mediumgray); draw(pointer); label("Pointer",(-1.85,0),fontsize(10pt)); label("$4$",(2,0.5),2N + 2W); label("$3$",(2,0),2N + 2W); label("$2$",(2,-0.5),2N + 2W); label("$1$",(2,-1),2N + 2W); label("$1$",(2,-1),2S + 2E); label("$2$",(2.5,-1),2S + 2E); label("$3$",(3,-1),2S + 2E);[/asy]$ \textbf{(A)}\ \frac {1}{3}\qquad \textbf{(B)}\ \frac {4}{9}\qquad \textbf{(C)}\ \frac {1}{2}\qquad \textbf{(D)}\ \frac {5}{9}\qquad \textbf{(E)}\ \frac {2}{3}$

Let a, b, c be positive integers. Prove that the inequality \[ (x\minus{}y)^a(x\minus{}z)^b(y\minus{}z)^c \ge 0 \] holds for all reals x, y, z if and only if a, b, c are even.

Let $X=\{1,2,3,...,10\}$. Find the number of pairs of $\{A,B\}$ such that $A\subseteq X, B\subseteq X, A\ne B$ and $A\cap B=\{5,7,8\}$.

Thirty-one books are arranged from left to right in order of increasing prices. The price of each book differs by $\$2$ from that of each adjacent book. For the price of the book at the extreme right a customer can buy the middle book and the adjacent one. Then: $ \textbf{(A)}\ \text{The adjacent book referred to is at the left of the middle book}$ $\qquad\textbf{(B)}\ \text{The middle book sells for \$36 }$ $\qquad\textbf{(C)}\ \text{The cheapest book sells for \$4 }$ $\qquad\textbf{(D)}\ \text{The most expensive book sells for \$64 }$ $\qquad\textbf{(E)}\ \text{None of these is correct } $

A set of $n$ positive integers is said to be [i]balanced[/i] if for each integer $k$ with $1 \leq k \leq n$, the average of any $k$ numbers in the set is an integer. Find the maximum possible sum of the elements of a balanced set, all of whose elements are less than or equal to $2017$.

On a table, there is an empty bag and a chessboard containing exactly one token on each square. Next to the table is a large pile that contains an unlimited supply of tokens. Using only the following types of moves what is the maximum possible number of tokens that can be in the bag? $\bullet$ Type 1: Choose a non-empty square on the chessboard that is not in the rightmost column. Take a token from this square and place it, along with one token from the pile, on the square immediately to its right. $\bullet$ Type 2: Choose a non-empty square on the chessboard that is not in the bottommost row. Take a token from this square and place it, along with one token from the pile, on the square immediately below it. $\bullet$ Type 3: Choose two adjacent non-empty squares. Remove a token from each and put them both into the bag.

A cylindrical hole of radius $r$ is bored through a cylinder of radiues $R$ ($r\leq R$) so that the axes intersect at right angles. i) Show that the area of the larger cylinder which is inside the smaller one can be expressed in the form $$S=8r^2\int_{0}^{1} \frac{1-v^{2}}{\sqrt{(1-v^2)(1-m^2 v^2)}}\;dv,\;\; \text{where} \;\; m=\frac{r}{R}.$$ ii) If $K=\int_{0}^{1} \frac{1}{\sqrt{(1-v^2)(1-m^2 v^2)}}\;dv$ and $E=\int_{0}^{1} \sqrt{\frac{1-m^2 v^2}{1-v^2 }}dv$. show that $$S=8[R^2 E - (R^2 - r^2 )K].$$

This problem is about pairs of consecutive whole numbers satisfying the property that one of the numbers is a perfect square and the other one is the double of a perfect square. (a) The smallest such pairs are $(0,1)$ and $(8,9)$, Indeed $0=2 \cdot 0^2$ and $1=1^2$; $8=2 \cdot 2^2$ and $9=3^2$. Show that there are infinitely many pairs of the form $(2a^2,b^2)$ where the smaller number is the double of a perfect square satisfying the given property. (b) Find a pair of integers satisfying the property that is not in the form given in the first part, that is, find a pair of integers such that the smaller one is a perfect square and the larger one is the double of a perfect square.

Solve the system of equations:$$\left\{ \begin{array}{l} ({x^2} + y)\sqrt {y - 2x} - 4 = 2{x^2} + 2x + y\\ {x^3} - {x^2} - y + 6 = 4\sqrt {x + 1} + 2\sqrt {y - 1} \end{array} \right.(x,y \in \mathbb{R}).$$

Determine the sum of absolute values for the complex roots of $ 20 x^8 \plus{} 7i x^7 \minus{}7ix \plus{} 20.$

Find the smallest positive number $\lambda $ , such that for any complex numbers ${z_1},{z_2},{z_3}\in\{z\in C\big| |z|<1\}$ ,if $z_1+z_2+z_3=0$, then $$\left|z_1z_2 +z_2z_3+z_3z_1\right|^2+\left|z_1z_2z_3\right|^2 <\lambda .$$

In triangle $ABC$, let $\omega$ be the excircle opposite to $A$. Let $D, E$ and $F$ be the points where $\omega$ is tangent to $BC, CA$, and $AB$, respectively. The circle $AEF$ intersects line $BC$ at $P$ and $Q$. Let $M$ be the midpoint of $AD$. Prove that the circle $MPQ$ is tangent to $\omega$.

Determine all ordered pairs $(a,p)$ of positive integers, with $p$ prime, such that $p^a+a^4$ is a perfect square. [i]Proposed by Tahjib Hossain Khan, Bangladesh[/i]