15 of the cells of a chessboard 8x8 are chosen. We draw the segments which unite the centers of every two of the chosen squares. Prove that among these segments there are four segments which have the same length.

Let $n$ be a natural number. Prove that, \[ \left\lfloor \frac{n}{1} \right\rfloor+ \left\lfloor \frac{n}{2} \right\rfloor + \cdots + \left\lfloor \frac{n}{n} \right\rfloor + \left\lfloor \sqrt{n} \right\rfloor \] is even.

The players on a basketball team made some three-point shots, some two-point shots, and some one-point free throws. They scored as many points with two-point shots as with three-point shots. Their number of successful free throws was one more than their number of successful two-point shots. The team's total score was 61 points. How many free throws did they make? $ \textbf{(A)}\ 13 \qquad \textbf{(B)}\ 14 \qquad \textbf{(C)}\ 15 \qquad \textbf{(D)}\ 16 \qquad \textbf{(E)}\ 17 $

Four identical squares and one rectangle are placed together to form one large square as shown. The length of the rectangle is how many times as large as its width? [asy]unitsize(8mm); defaultpen(linewidth(.8pt)); draw(scale(4)*unitsquare); draw((0,3)--(4,3)); draw((1,3)--(1,4)); draw((2,3)--(2,4)); draw((3,3)--(3,4));[/asy]$ \textbf{(A)}\ \frac {5}{4} \qquad \textbf{(B)}\ \frac {4}{3} \qquad \textbf{(C)}\ \frac {3}{2} \qquad \textbf{(D)}\ 2 \qquad \textbf{(E)}\ 3$

Find all ordered triplets $(p,q,r)$ of positive integers such that $p$ and $q$ are two (not necessarily distinct) primes, $r$ is even, and \[p^3+q^2=4r^2+45r+103.\]

Given a simple, connected graph with $n$ vertices and $m$ edges. Prove that one can find at least $m$ ways separating the set of vertices into two parts, such that the induced subgraphs on both parts are connected.

Given that the base-$17$ integer $\overline{8323a02421_{17}}$ (where a is a base-$17$ digit) is divisible by $\overline{16_{10}}$, find $a$. Express your answer in base $10$. [i]Proposed by Jonathan Liu[/i]

Given a table $4\times 4$. a) Find, how $7$ stars can be put in its fields in such a way, that erasing of two arbitrary lines and two columns will always leave at list one of the stars. b) Prove that if there are less than $7$ stars, You can always find two columns and two rows, such, that if you erase them, no star will remain in the table.

A spring has an equilibrium length of $2.0$ meters and a spring constant of $10$ newtons/meter. Alice is pulling on one end of the spring with a force of $3.0$ newtons. Bob is pulling on the opposite end of the spring with a force of $3.0$ newtons, in the opposite direction. What is the resulting length of the spring? (A) $\text{1.7 m}$ (B) $\text{2.0 m}$ (C) $\text{2.3 m}$ (D) $\text{2.6 m}$ (E) $\text{8.0 m}$

We are given a bag of sugar, a two-pan balance, and a weight of $1$ gram. How do we obtain $1$ kilogram of sugar in the smallest possible number of weighings?

For a positive integer $n$ write down all its positive divisors in increasing order: $1=d_1<d_2<\ldots<d_k=n$. Find all positive integers $n$ divisible by $2019$ such that $n=d_{19}\cdot d_{20}$. [i](I. Gorodnin)[/i]

Let $ABC$ be an isosceles triangle with $AC = BC$. The point $D$ lies on the side $AB$ such that the semicircle with diameter $BD$ and center $O$ is tangent to the side $AC$ in the point $P$ and intersects the side $BC$ at the point $Q$. The radius $OP$ intersects the chord $DQ$ at the point $E$ such that $5 \cdot PE = 3 \cdot DE$. Find the ratio $\frac{AB}{BC}$ .

The points $A$, $B$, $C$, $D$, and $E$ lie in one plane and have the following properties: $AB = 12, BC = 50, CD = 38, AD = 100, BE = 30, CE = 40$. Find the length of the segment $ED$.

The real numbers $a,b$ and $c$ satisfy the inequalities $|a|\ge |b+c|,|b|\ge |c+a|$ and $|c|\ge |a+b|$. Prove that $a+b+c=0$.

Three cards, each with a positive integer written on it, are lying face-down on a table. Casey, Stacy, and Tracy are told that (a) the numbers are all different, (b) they sum to 13, and (c) they are in increasing order, left to right First, Casey looks at the number on the leftmost card and says, "I don't have enough information to determine the other two numbers." Then Tracy looks at the number on the rightmost card and says, "I don't have enough information to determine the other two numbers." Finally, Stacy looks at the number on the middle card and says, "I don't have enough information to determine the other two numbers." Assume that each perosn knows that the other two reason perfectly and hears their comments. What number is on the middle card? $ \textbf{(A)}\ 2\qquad \textbf{(B)}\ 3\qquad \textbf{(C)}\ 4\qquad \textbf{(D)}\ 5$ $ \textbf{(E)}\ \text{There is not enough information to determine the number.}$

Find the sum of all positive integers $a=2^{n}3^{m},$ where $n$ and $m$ are non-negative integers, for which $a^{6}$ is not a divisor of $6^{a}.$

Let $ABC$ be a triangle such that the coordinates of the points $A$ and $B$ are rational numbers. Prove that the coordinates of $C$ are rational if, and only if, $\tan A$, $\tan B$, and $\tan C$, when defined, are all rational numbers.

A function $ f:\mathbb{R}_{\ge 0}\longrightarrow\mathbb{R} $ has the property that $ \lim_{x\to\infty } \frac{1}{x^2}\int_0^x f(t)dt=1. $ [b]a)[/b] Give an example of what $ f $ could be if it's continuous and $ f/\text{id.} $ doesn't have a limit at $ \infty . $ [b]b)[/b] Prove that if $ f $ is nondecreasing then $ f/\text{id.} $ has a limit at $ \infty , $ and determine it.

What is the largest possible value of $|a_1 - 1| + |a_2-2|+...+ |a_n- n|$ where $a_1, a_2,..., a_n$ is a permutation of $1,2,..., n$?

Determine all $ f:R\rightarrow R $ such that $$ f(xf(y)+y^3)=yf(x)+f(y)^3 $$

Find all real numbers $ a,b,c,d$ such that \[ \left\{\begin{array}{cc}a \plus{} b \plus{} c \plus{} d \equal{} 20, \\ ab \plus{} ac \plus{} ad \plus{} bc \plus{} bd \plus{} cd \equal{} 150. \end{array} \right.\]

An $n\times n\times n$ cube is divided into unit cubes. We are given a closed non-self-intersecting polygon (in space), each of whose sides joins the centers of two unit cubes sharing a common face. The faces of unit cubes which intersect the polygon are said to be distinguished. Prove that the edges of the unit cubes may be colored in two colors so that each distinguished face has an odd number of edges of each color, while each nondistinguished face has an even number of edges of each color. [i]M. Smurov[/i]

Consider a function $f$ on nonnegative integers such that $f(0)=1, f(1)=0$ and $f(n)+f(n-1)=nf(n-1)+(n-1)f(n-2)$ for $n \ge 2$. Show that \[\frac{f(n)}{n!}=\sum_{k=0}^n \frac{(-1)^k}{k!}\]

Given is a chessboard 8x8. We have to place $n$ black queens and $n$ white queens, so that no two queens attack. Find the maximal possible $n$. (Two queens attack each other when they have different colors. The queens of the same color don't attack each other)

Let $a$ and $b$ be integers. Define $a$ to be a power of $b$ if there exists a positive integer $n$ such that $a = b^n$. Define $a$ to be a multiple of $b$ if there exists an integer $n$ such that $a = bn$. Let $x$, $y$ and $z$ be positive integer such that $z$ is a power of both $x$ and $y$. Decide for each of the following statements whether it is true or false. Prove your answers. (a) The number $x + y$ is even. (b) One of $x$ and $y$ is a multiple of the other one. (c) One of $x$ and $y$ is a power of the other one. (d) There exist an integer $v$ such that both $x$ and $y$ are powers of $v$ (e) For each power of $x$ and for each power of $y$, an integer $w$ can be found such that $w$ is a power of each of these powers. (f) There exists a positive integer $k$ such that $x^k > y$.