The non-negative numbers $ x_1, x_2, \ldots, x_5$ satisfy $ \sum_{i \equal{} 1}^5 \frac {1}{1 \plus{} x_i} \equal{} 1$. Prove that $ \sum_{i \equal{} 1}^5 \frac {x_i}{4 \plus{} x_i^2} \leq 1$.

Let $a>0,b>0,c>0$ and $\sqrt{1987+a}+\sqrt{1987+b}=2\sqrt{1987+c}$. Prove that $\frac{1}{2} (a+b )\ge c $.

Consider on the coordinate plane all rectangles whose (i) vertices have integer coordinates; (ii) edges are parallel to coordinate axes; (iii) area is $2^k$, where $k = 0,1,2....$ Is it possible to color all points with integer coordinates in two colors so that no such rectangle has all its vertices of the same color?

$\triangle PQR$ is isosceles and right angled at $R$. Point $A$ is inside $\triangle PQR$, such that $PA=11, QA=7$, and $RA=6$. Legs $\overline{PR}$ and $\overline{QR}$ have length $s=\sqrt{a+b\sqrt{2}}$, where $a$ and $b$ are positive integers. What is $a+b$?

Let $A$ be a set of positive integers such that for any $x,y\in A$, $$x>y\implies x-y\ge\frac{xy}{25}.$$Find the maximal possible number of elements of the set $A$.

Let $ABC$ be an acute scalene triangle with circumcircle $\omega$. Let $P$ and $Q$ be interior points of the sides $AB$ and $AC$, respectively, such that $PQ$ is parallel to $BC$. Let $L$ be a point on $\omega$ such that $AL$ is parallel to $BC$. The segments $BQ$ and $CP$ intersect at $S$. The line $LS$ intersects $\omega$ at $K$. Prove that $\angle BKP = \angle CKQ$. Proposed by [i]Ervin Macić, Bosnia and Herzegovina[/i]

A square is partitioned in $n^2\geq 4$ rectanles using $2(n-1)$ lines,$n-1$ of which,are parallel to the one side of the square,$n-1$ are parallel to the other side.Prove that we can choose $2n$ rectangles of the partition,such that,for each two of them,we can place the one inside the other (possibly with rotation).

Let $X_1X_2X_3$ be a triangle with $X_1X_2 = 4, X_2X_3 = 5, X_3X_1 = 7,$ and centroid $G$. For all integers $n \ge 3$, define the set $S_n$ to be the set of $n^2$ ordered pairs $(i,j)$ such that $1\le i\le n$ and $1\le j\le n$. Then, for each integer $n\ge 3$, when given the points $X_1, X_2, \ldots , X_{n}$, randomly choose an element $(i,j)\in S_n$ and define $X_{n+1}$ to be the midpoint of $X_i$ and $X_j$. The value of \[ \sum_{i=0}^\infty \left(\mathbb{E}\left[X_{i+4}G^2\right]\left(\dfrac{3}{4}\right)^i\right) \] can be expressed in the form $p + q \ln 2 + r \ln 3$ for rational numbers $p, q, r$. Let $|p| + |q| + |r| = \dfrac mn$ for relatively prime positive integers $m$ and $n$. Compute $100m+n$. Note: $\mathbb{E}(x)$ denotes the expected value of $x$. [i]Proposed by Yang Liu[/i]

Find all functions $f: (0, \infty) \to (0, \infty)$ such that \begin{align*} f(y(f(x))^3 + x) = x^3f(y) + f(x) \end{align*} for all $x, y>0$. [i]Proposed by Jason Prodromidis, Greece[/i]

Let $ABCD$ be a convex quadrilateral with $\angle DAB =\angle ABC =\angle BCD > 90^o$. The circle circumscribed around the triangle $ABC$ intersects the sides $AD$ and $CD$ at points $K$ and $L$, respectively, different from any vertex of the quadrilateral $ABCD$ . Segments $AL$ and $CK$ intersect at point $P$. Prove that $\angle ADB =\angle PDC$.

For which positive integers $n$ is there a permutation $(x_1,x_2,\ldots,x_n)$ of $1,2,\ldots,n$ such that all the differences $|x_k-k|$, $k = 1,2,\ldots,n$, are distinct?

Evaluate $\sum_{k=0}^{37}(-1)^k\binom{75}{2k}$.

Let $ x, y, z$ be real numbers such that $ x \plus{} y \plus{} z \geq xyz$. Find the smallest possible value of $ \frac {x^2 \plus{} y^2 \plus{} z^2}{xyz}$.

From twenty different books on mathematics and physics, sets are made containing $5$ books on mathematics and $5$ books on physics each. How many math books should there be for the largest number of possible sets?

Let $ (a,b,c)$ be the real solution of the system of equations $ x^3 \minus{} xyz \equal{} 2$, $ y^3 \minus{} xyz \equal{} 6$, $ z^3 \minus{} xyz \equal{} 20$. The greatest possible value of $ a^3 \plus{} b^3 \plus{} c^3$ can be written in the form $ \frac{m}{n}$, where $ m$ and $ n$ are relatively prime positive integers. Find $ m \plus{} n$.

Let $ABCD$ be a isosceles trapezoid such that $AB || CD$ and all of its sides are tangent to a circle. $[AD]$ touches this circle at $N$. $NC$ and $NB$ meet the circle again at $K$ and $L$, respectively. What is $\dfrac {|BN|}{|BL|} + \dfrac {|CN|}{|CK|}$? $ \textbf{(A)}\ 4 \qquad\textbf{(B)}\ 6 \qquad\textbf{(C)}\ 8 \qquad\textbf{(D)}\ 9 \qquad\textbf{(E)}\ 10 $

Prove that if $m=5^n+3^n+1$ is a prime, then $12$ divides $n$.

Find the largest integer that divides $p^4 - 1$ for all primes $p > 4$

Prove that there exists a unique polynomial function f with positive integer coefficients such that $f(1) = 6$ and $f(2) = 2016$.

A bag contains 4 red marbles, 5 yellow marbles, and 6 blue marbles. Three marbles are to be picked out randomly (without replacement). What is the probability that exactly two of them have the same color?

Let's take a $ABCD$ rectangle of paper; the side $AB$ measures $5$ cm and the side $BC$ measures $9$ cm. We do three folds: 1.We take the $AB$ side on the $BC$ side and call $P$ the point on the $BC$ side that coincides with $A$. A right trapezoid $BCDQ$ is then formed. 2. We fold so that $B$ and $Q$ coincide. A $5$-sided polygon $RPCDQ$ is formed. 3. We fold again by matching $D$ with $C$ and $Q$ with $P$. A new right trapezoid $RPCS$. After making these folds, we make a cut perpendicular to $SC$ by its midpoint $T$, obtaining the right trapezoid $RUTS$. Calculate the area of the figure that appears as we unfold the last trapezoid $RUTS$.

Let $21$ pieces, some white and some black, be placed on the squares of a $3\times 7$ rectangle. Prove that there always exist four pieces of the same color standing at the vertices of a rectangle.

King William is located at $(1, 1)$ on the coordinate plane. Every day, he chooses one of the eight lattice points closest to him and moves to one of them with equal probability. When he exits the region bounded by the $x, y$ axes and $x+y = 4$, he stops moving and remains there forever. Given that after an arbitrarily large amount of time he must exit the region, the probability he ends up on $x+y = 4$ can be expressed as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$. [i]Proposed by Andrew Wen[/i]

Let $n\geqslant 2$ be a positive integer. Paul has a $1\times n^2$ rectangular strip consisting of $n^2$ unit squares, where the $i^{\text{th}}$ square is labelled with $i$ for all $1\leqslant i\leqslant n^2$. He wishes to cut the strip into several pieces, where each piece consists of a number of consecutive unit squares, and then [i]translate[/i] (without rotating or flipping) the pieces to obtain an $n\times n$ square satisfying the following property: if the unit square in the $i^{\text{th}}$ row and $j^{\text{th}}$ column is labelled with $a_{ij}$, then $a_{ij}-(i+j-1)$ is divisible by $n$. Determine the smallest number of pieces Paul needs to make in order to accomplish this.

In a mathematics test number of participants is $N < 40$. The passmark is fixed at $65$. The test results are the following: The average of all participants is $66$, that of the promoted $71$ and that of the repeaters $56$. However, due to an error in the wording of a question, all scores are increased by $5$. At this point the average of the promoted participants becomes $75$ and that of the non-promoted $59$. (a) Find all possible values ​​of $N$. (b) Find all possible values ​​of $N$ in the case where, after the increase, the average of the promoted had become $79$ and that of non-promoted $47$.