Suppose that $x, y,$ and $z$ are positive integers with $xy=z^2 +1$. Prove that there exist integers $a, b, c,$ and $d$ such that $x=a^2 +b^2$, $y=c^2 +d^2$, and $z=ac+bd$.

If $a,b,c$ are positive real numbers, prove that $\frac{a^2b^2}{a+b}+\frac{b^2c^2}{b+c}+\frac{c^2a^2}{c+a}\le \frac{a^3+b^3+c^3}{2}$

A board of size $2015 \times 2015$ is covered with sub-boards of size $2 \times 2$, each of which is painted like chessboard. Each sub-board covers exactly $4$ squares of the board and each square of the board is covered with at least one square of a sub-board (the painted of the sub-boards can be of any shape). Prove that there is a way to cover the board in such a way that there are exactly $2015$ black squares visible. What is the maximum number of visible black squares?

Show that the power series for the function $$e^{ax} \cos bx,$$ where $a,b >0$, has either no zero coefficients or infinitely many zero coefficients.

Find the least possible cardinality of a set $A$ of natural numbers, the smallest and greatest of which are $1$ and $100$, and having the property that every element of $A$ except for $1$ equals the sum of two elements of $A$.

The incircle $\mathcal{C}_1$ of triangle $ABC$ touches the sides $AB$ and $AC$ at the points $D$ and $E$, respectively. The incircle $\mathcal{C}_2$ of the triangle $ADE$ touches the sides $AB$ and $AC$ at the points $P$ and $Q$, and intersects the circle $\mathcal{C}_1$ at the points $M$ and $n$. Prove that (a) the center of the circle $\mathcal{C}_2$ lies on the circle $\mathcal{C}_1$. (b) the four points $M,N,P,Q$ in appropriate order form a rectangle if and only if twice the radius of $\mathcal{C}_1$ is three times the radius of $\mathcal{C}_2$.

The $8$ eyelets for the lace of a sneaker all lie on a rectangle, four equally spaced on each of the longer sides. The rectangle has a width of $50$ mm and a length of $80$ mm. There is one eyelet at each vertex of the rectangle. The lace itself must pass between the vertex eyelets along a width side of the rectangle and then crisscross between successive eyelets until it reaches the two eyelets at the other width side of the rectrangle as shown. After passing through these final eyelets, each of the ends of the lace must extend at least $200$ mm farther to allow a knot to be tied. Find the minimum length of the lace in millimeters. [asy] size(200); defaultpen(linewidth(0.7)); path laceL=(-20,-30)..tension 0.75 ..(-90,-135)..(-102,-147)..(-152,-150)..tension 2 ..(-155,-140)..(-135,-40)..(-50,-4)..tension 0.8 ..origin; path laceR=reflect((75,0),(75,-240))*laceL; draw(origin--(0,-240)--(150,-240)--(150,0)--cycle,gray); for(int i=0;i<=3;i=i+1) { path circ1=circle((0,-80*i),5),circ2=circle((150,-80*i),5); unfill(circ1); draw(circ1); unfill(circ2); draw(circ2); } draw(laceL--(150,-80)--(0,-160)--(150,-240)--(0,-240)--(150,-160)--(0,-80)--(150,0)^^laceR,linewidth(1));[/asy]

Compute $ \int_0^1 x^{2n\plus{}1}e^{\minus{}x^2}dx\ (n\equal{}1,\ 2,\ \cdots)$ , then use this result, prove that $ \sum_{n\equal{}0}^{\infty} \frac{1}{n!}\equal{}e$.

Given an integer $n > 1$, prove that there exist distinct positive integers $a, b, c$ and $d$ such that $a + b = c + d$ and $\frac{a}{b}=\frac{nc}{d}$.

Prove that if opposite sides of a hexagon are parallel and the diagonals connecting opposite vertices have equal lengths, a circle can be circumscribed around the hexagon.

On the plane painted $101$ points in brown and another $101$ points in green so that there are no three lying on one line. It turns out that the sum of the lengths of all $5050$ segments with brown ends equals the length of all $5050$ segments with green ends equal to $1$, and the sum of the lengths of all $10201$ segments with multicolored equals $400$. Prove that it is possible to draw a straight line so that all brown points are on one side relative to it and all green points are on the other.

Let $ABCD$ be a rhombus with $\angle DAB = 60^o$. Let $K, L$ be points on its sides $AD$ and $DC$ and $M$ a point on the diagonal $AC$ such that $KDLM$ is a parallelogram. Prove that triangle $BKL$ is equilateral.

Let $ABC$ be a triangle with $AB = 5$, $AC = 8$, and $BC = 7$. Let $D$ be on side $AC$ such that $AD = 5$ and $CD = 3$. Let $I$ be the incenter of triangle $ABC$ and $E$ be the intersection of the perpendicular bisectors of $\overline{ID}$ and $\overline{BC}$. Suppose $DE = \frac{a\sqrt{b}}{c}$ where $a$ and $c$ are relatively prime positive integers, and $b$ is a positive integer not divisible by the square of any prime. Find $a+b+c$. [i]Proposed by Ray Li[/i]

Three circles pass through a point $P$, and the second points of their intersection $A, B, C$ lie on a straight line. Let $A_1 B_1, C_1$ be the second meets of lines $AP, BP, CP$ with the corresponding circles. Let $C_2$ be the intersections of lines $AB_1$ and $BA_1$. Let $A_2, B_2$ be defined similarly. Prove that the triangles $A_1B_1C_1$ and $A_2B_2C_2$ are equal,

Find all triples $(p, x, y)$ consisting of a prime number $p$ and two positive integers $x$ and $y$ such that $x^{p -1} + y$ and $x + y^ {p -1}$ are both powers of $p$. [i]Proposed by Belgium[/i]

Suppose that $a$ is an irrational number. (a) If there is a real number $b$ such that both $(a+b)$ and $ab$ are rational numbers, show that $a$ is a quadratic surd. ($a$ is a quadratic surd if it is of the form $r+\sqrt{s}$ or $r-\sqrt{s}$ for some rationals $r$ and $s$, where $s$ is not the square of a rational number). (b) Show that there are two real numbers $b_1$ and $b_2$ such that i) $a+b_1$ is rational but $ab_1$ is irrational. ii) $a+b_2$ is irrational but $ab_2$ is rational. (Hint: Consider the two cases, where $a$ is a quadratic surd and $a$ is not a quadratic surd, separately).

Find all positive integers $a$ for which $a^{2000}-1$ is divisible by $10$.

Prove that for any positive integers $a$, $b$, $c$, at least one of the numbers $a^3b+1$, $b^3c+1$, $c^3a+1$ is not divisible by $a^2+b^2+c^2$.

Let $(a_m)$ be a sequence satisfying $a_n \geq 0$, $n=0,1,2,\ldots$ Suppose there exists $A >0$, $a_m - a_{m+1}$ $\geq A a_m ^2$ for all $m \geq 0$. Prove that there exists $B>0$ such that \begin{align*} a_n \le \frac{B}{n} \qquad \qquad \text{for }1 \le n \end{align*}

Let $\triangle ABC $ be an acute triangle.$O$ denote its circumcenter.Points $D$,$E$,$F$ are the midpoints of the sides $BC$,$CA$,and $AB$.Let $M$ be a point on the side $BC$ . $ AM \cap EF = \big\{ N \big\} $ . $ON \cap \big( ODM \big) = \big\{ P \big\} $ Prove that $M'$ lie on $\big(DEF\big)$ where $M'$ is the symmetrical point of $M$ thought the midpoint of $DP$.

Consider an acute triangle $ABC$, points $E,F$ are the feet of the perpendiculars from $B$ and $C$ in $\triangle ABC$. Points $I$ and $J$ are the projections of points $F,E$ on the line $BC$, points $K,L$ are on sides $AB,AC$ respectively such that $IK \parallel AC$ and $JL \parallel AB$. Prove that the lines $IE$,$JF$,$KL$ are concurrent.

Consider two non-parallel half-planes $\pi,\pi'$ with the common boundary line $p.$ Four different points $A,B,C,D$ are given in the half-plane $\pi.$ Similarly, four points $A',B',C',D'\in\pi'$ are given such that $AA'\parallel BB'\parallel CC'\parallel DD'$. Moreover, none of these points lie on $p$ and the points $A,B,C,D'$ form a tetrahedron. Show that the points $A',B',C',D$ also form a tetrahedron with the same volume as $ABCD'.$

Determine the number of integer solutions $(x, y, z)$, with $0 \le x, y, z \le 100$, for the equation $$(x - y)^2 + (y + z)^2 = (x + y)^2 + (y - z)^2.$$

We say a triangle in the coordinate plane is [i]integral[/i] if its three vertices have integer coordinates and if its three sides have integer lengths. (a) Find an integral triangle with perimeter of $42$. (b) Is there an integral triangle with perimeter of $43$?

Let $ABCD$ be a cyclic quadrilateral with circumradius $100\sqrt{3}$ and $AC=300$. If $\angle DBC = 15^{\circ}$, then find $AD^2$. [i]Proposed by Anand Iyer[/i]