Suppose that $a, b$ are natural numbers so that all the roots of $x^2 + ax - b$ and $x^2 - ax + b$ are integers. Show that exists a right triangle with integer sides, with $a$ the length of the hypotenuse and $b$ the area .

Suppose that the function $f:\mathbb{R}\to\mathbb{R}$ satisfies \[f(x^3 + y^3)=(x+y)(f(x)^2-f(x)f(y)+f(y)^2)\] for all $x,y\in\mathbb{R}$. Prove that $f(1996x)=1996f(x)$ for all $x\in\mathbb{R}$.

Let $ \omega$ be the circumcircle of $ ABC$ and let $ D$ be a fixed point on $ BC$, $ D\neq B$, $ D\neq C$. Let $ X$ be a variable point on $ (BC)$, $ X\neq D$. Let $ Y$ be the second intersection point of $ AX$ and $ \omega$. Prove that the circumcircle of $ XYD$ passes through a fixed point.

Note that $12^2=144$ ends in two $4$s and $38^2=1444$ end in three $4$s. Determine the length of the longest string of equal nonzero digits in which the square of an integer can end.

Let $x$ and $y$ be two real numbers such that $2 \sin x \sin y + 3 \cos y + 6 \cos x \sin y = 7$. Find $\tan^2 x + 2 \tan^2 y$.

For any integer $n \ge 2$, let $b_n$ be the least positive integer such that, for any integer $N$, $m$ divides $N$ whenever $m$ divides the digit sum of $N$ written in base $b_n$, for $2 \le m \le n$. Find the integer nearest to $b_{36}/b_{25}$.

Find all real polynomials $ f$ with $ x,y \in \mathbb{R}$ such that \[ 2 y f(x \plus{} y) \plus{} (x \minus{} y)(f(x) \plus{} f(y)) \geq 0. \]

In $\triangle ABC$, $AB = 40$, $BC = 60$, and $CA = 50$. The angle bisector of $\angle A$ intersects the circumcircle of $\triangle ABC$ at $A$ and $P$. Find $BP$. [i]Proposed by Eugene Chen[/i]

Let $U$ be a set of $m$ triangles. Prove that there exists a subset $W$ of $U$ which satisfies the following. (i). The number of triangles in $W$ is at least $0.45m^{\frac{4}{5}}$ (ii) There are no points $A, B, C, D, E, F$ such that triangles $ABC$, $BCD$, $CDE$, $DEF$, $EFA$, $FAB$ are all in $W$.

Let $$P(X)=a_{0}X^{n}+a_{1}X^{n-1}+\cdots+a_{n}$$ be a polynomial with real coefficients such that $a_{0}>0$ and $$a_{n}\geq a_{i}\geq 0,$$ for all $i=0,1,2,\ldots,n-1.$ Prove that if $$P^{2}(X)=b_{0}X^{2n}+b_{1}X^{2n-1}+\cdots+b_{n-1}X^{n+1}+\cdots+b_{2n},$$ then $P^2(1)\geq 2b_{n-1}.$

The number $21!=51,090,942,171,709,440,000$ has over $60,000$ positive integer divisors. One of them is chosen at random. What is the probability that it is odd? $\textbf{(A)} \frac{1}{21} \qquad \textbf{(B)} \frac{1}{19} \qquad \textbf{(C)} \frac{1}{18} \qquad \textbf{(D)} \frac{1}{2} \qquad \textbf{(E)} \frac{11}{21}$

Consider a $ 7\times 7$ numbers table $ a_{ij} \equal{} (i^2 \plus{} j)(i \plus{} j^2), 1\le i,j\le 7.$ When we add arbitrarily each term of an arithmetical progression consisting of $ 7$ integers to corresponding to term of certain row (or column) in turn, call it an operation. Determine whether such that each row of numbers table is an arithmetical progression, after a finite number of operations.

Let $\mathbb{Z}$ denote the set of all integers. Find all polynomials $P(x)$ with integer coefficients that satisfy the following property: For any infinite sequence $a_1$, $a_2$, $\dotsc$ of integers in which each integer in $\mathbb{Z}$ appears exactly once, there exist indices $i < j$ and an integer $k$ such that $a_i +a_{i+1} +\dotsb +a_j = P(k)$.

The number of revolutions of a wheel, with fixed center and with an outside diameter of $ 6$ feet, required to cause a point on the rim to go one mile is: $ \textbf{(A)}\ 880 \qquad\textbf{(B)}\ \frac{440}{\pi} \qquad\textbf{(C)}\ \frac{880}{\pi} \qquad\textbf{(D)}\ 440\pi \qquad\textbf{(E)}\ \text{none of these}$

[asy]defaultpen(linewidth(.8pt)); pair A = (0,11); pair B = (2,0); pair D = (4,0); pair E = (7,0); pair C = (13,0); label("$A$",A,N); label("$B$",B,SW); label("$C$",C,SE); label("$D$",D,S); label("$E$",E,S); label("$2$",midpoint(B--D),N); label("$3$",midpoint(D--E),NW); label("$6$",midpoint(E--C),NW); draw(A--B--C--cycle); draw(A--D); draw(A--E);[/asy] In triangle $ ABC$ in the adjoining figure, $ AD$ and $ AE$ trisect $ \angle BAC$. The lengths of $ BD$, $ DE$ and $ EC$ are $ 2$, $ 3$, and $ 6$, respectively. The length of the shortest side of $ \triangle ABC$ is $ \textbf{(A)}\ 2\sqrt{10}\qquad \textbf{(B)}\ 11\qquad \textbf{(C)}\ 6\sqrt{6}\qquad \textbf{(D)}\ 6\qquad \textbf{(E)}\ \text{not uniquely determined by the given information}$

Let $ABC$ be a triangle inscribed in a semicircle with center $O$ and diameter $BC.$ Two tangent lines to the semicircle at $A$ and $B$ intersect at $D.$ Prove that $DC$ goes through the midpoint of the altitude $AH$ of triangle $ABC.$