Let $c: Q-\{0\} \to Q-\{0\}$ a function with the following properties (for all $x,y, a, b \in Q-\{0\}$ and $x \ne 1$): a) $c (x, 1- x) = 1$ b) $c (ab,y) = c (a,y)c(b, y)$ c) $c (y,ab) = c (y, a)c(y,b)$ Show that then $c (a,b) c(b,a) = 1 = c(a,-a)$ also holds.

Given real numbers $a, b, c$ such that $a + b - c = ab- bc - ca = abc = 8$. Find all possible values of $a$.

The numbers $1, 2, 3, ..., 99, 100$ are randomly arranged in the cells of a square table measuring $10\times 10$ (each number is used only once). Prove that there are three cells in the table whose sum of numbers does not exceed 1$82$. The centers of these cells form an isosceles right triangle, the legs of which are parallel to the edges of the table.

A convex $2019$-gon $A_1A_2...A_{2019}$ is cut into smaller pieces along its $2019$ diagonals of the form $A_iA_{i+3}$ for $1 \le i \le2019$, where $A_{2020} = A_1$, $A_{2021} = A_2$, and $A_{2022} = A_3$. What is the least possible number of resulting pieces?

Evaluate $ \int _ { 0 } ^ { a } d x / ( 1 + \cos x + \sin x ) $ for $ - \pi / 2 < a < \pi $. Use your answer to show that $ \int _ { 0 } ^ { \pi / 2 } d x / ( 1 + \cos x + \sin x ) = \ln 2 $.

Let $f:[0,1]\rightarrow [0,1]$ be a continuous and bijective function. Describe the set: \[A=\{f(x)-f(y)\mid x,y\in[0,1]\backslash\mathbb{Q}\}\] [hide="Note"] You are given the result that [i]there is no one-to-one function between the irrational numbers and $\mathbb{Q}$.[/i][/hide]

If the length of a rectangle is increased by $20\% $ and its width is increased by $50\% $, then the area is increased by $\text{(A)}\ 10\% \qquad \text{(B)}\ 30\% \qquad \text{(C)}\ 70\% \qquad \text{(D)}\ 80\% \qquad \text{(E)}\ 100\% $

a) Prove that for each positive integer $n$ there is a unique positive integer $a_n$ such that $$(1 + \sqrt5)^n =\sqrt{a_n} + \sqrt{a_n+4^n} . $$ b) Prove that $a_{2010}$ is divisible by $5\cdot 4^{2009}$ and find the quotient

Let $ABC$ be a right triangle with $AB = BC = 2$. Construct point $D$ such that $\angle DAC = 30^o$ and $\angle DCA = 60^o$, and $\angle BCD > 90^o$. Compute the area of triangle $BCD$.

Find the number of positive integer solutions to the equation $(x_1+x_2+x_3)^2(y_1+y_2) = 2548$.

Given a triangle with $a,b,c$ sides and with the area $1$ ($a \ge b \ge c$). Prove that $b^2 \ge 2$.

Let $S$ be a finite subset of a straight line. Say that $S$ has the [i]repeated distance property [/i] if every value of the distance between two points of $S$ (except the longest) occurs at least twice. Show that if $S$ has the [i]repeated distance property [/i] then the ratio of any two distances between two points of $S$ is rational.

Lucky starts doodling on a $5\times 5$ Bingo board. He puts his pencil at the center of the upper-left square (marked by ‘·’) and draws a continuous doodle ending on the Free Space, never going off the board or through a corner of a square. (See Figure 1.) (a) Is it possible for Lucky’s doodle to visit all squares exactly once? (The starting and ending squares are considered visited.) (b) Is it possible for Lucky’s doodle to visit all squares exactly twice?

assume that we have a n*n table we fill it with 1,...,n such that each number exists exactly n times prove that there exist a row or column such that at least $\sqrt{n}$ diffrent number are contained.

Given a rectangle $ ABCD,$ let $ AB \equal{} b, AD \equal{} a ( a\geq b),$ three points $ X,Y,Z$ are put inside or on the boundary of the rectangle, arbitrarily. Find the maximum of the minimum of the distances between any two points among the three points. (Denote it by $ a,b$)

Let $ABC$ is a triangle with $\angle BAC=\frac{\pi}{6}$ and the circumradius equal to 1. If $X$ is a point inside or in its boundary let $m(X)=\min(AX,BX,CX).$ Find all the angles of this triangle if $\max(m(X))=\frac{\sqrt{3}}{3}.$

Let $S = \{ a_0, a_1, a_2, a_3, \dots \}$ be a set of positive integers with $1 = a_0 < a_1 < a_2 < a_3 < \dots$. For a subset $T$ of $S$, let $\sigma(T)$ be the sum of the elements of $T$. For instance, $\sigma(\{1, 2, 3\}) = 6$. By convention, $\sigma(\emptyset) = 0$, where $\emptyset$ denotes an empty set. Call a number $n$ representable if there exists a subset $T$ of $S$ such that $\sigma(T) = n$. We aim to prove for any set $S$ satisfying $a_{k+1} \le 2a_k$ for every $k \ge 0$, that all non-negative integers are representable. (a) Prove there is a unique value of $a_1$, and find this value. Use this to determine, with proof, all possible sets $\{a_0, a_1, a_2, a_3 \}$. (Hint: there are 7 possible sets.) [Not for credit] I recommend that you show that for all 7 sets in part (a), every integer between $0$ and $a_3 - 1$ is representable. (Note that this does not depend on the values of $a_4, a_5, a_6, \dots$.) (b) Show that if $a_k \le n \le a_{k+1} - 1$, then $0 \le n - a_k \le a_k - 1$. (c) Prove that any non-negative integer is representable.

Given natural $a,b,n$. It is known, that for every natural $k$ ($k\ne b$) the number $a-k^n$ is divisible by $b-k$. Prove that $$a=b^n$$

Let $\vartriangle ABC$ be a triangle with $BC = 7$, $CA = 6$, and, $AB = 5$. Let $I$ be the incenter of $\vartriangle ABC$. Let the incircle of $\vartriangle ABC$ touch sides $BC$, $CA$, and $AB$ at points $D,E$, and $F$. Let the circumcircle of $\vartriangle AEF$ meet the circumcircle of $\vartriangle ABC$ for a second time at point $X\ne A$. Let $P$ denote the intersection of $XI$ and $EF$. If the product $XP \cdot IP$ can be written as $\frac{m}{n}$ for relatively prime positive integers $m, n$, find $m + n$.

Denote $f(m) =\sum_{k=1}^m (-1)^k cos \frac{k\pi}{2 m + 1}$ For which positive integers $m$ is $f(m)$ rational?

For all positive integers $n$, show that: $$ \dfrac1n \sum^n _{k=1} \dfrac{k \cdot k! \cdot {n\choose k}}{n^k} = 1$$

Let positive real numbers $ a,b$ satisfy $ b \minus{} a > 2.$ Prove that for any two distinct integers $ m,n$ belonging to $ [a,b),$ there always exists non-empty set $ S$ consisting of certain integers belonging to $ [ab,(a \plus{} 1)(b \plus{} 1))$ such that $ \frac {\displaystyle\prod_{x\in S}}{mn}$ is square of a rational number.

Define $f: \mathbb{N} \rightarrow \mathbb{N}$ by $$f(n) = \sum \frac{(1+\sum_{i=1}^{n} t_i)!}{(1+t_1) \cdot \prod_{i=1}^{n} (t_i!) }$$ where the sum runs through all $n$-tuples such that $\sum_{j=1}^{n}j \cdot t_j=n$ and $t_j \ge 0$ for all $1 \le j \le n$. Given a prime $p$ greater than $3$, prove that $$\sum_{1 \le i < j <k \le p-1 } \frac{f(i)}{i \cdot j \cdot k} \equiv \sum_{1 \le i < j <k \le p-1 } \frac{2^i}{i \cdot j \cdot k} \pmod{p}.$$

In how many ways can the numbers $0,1,2,\dots , 9$ be arranged in such a way that the odd numbers form an increasing sequence, also the even numbers form an increasing sequence? $ \textbf{(A)}\ 126 \qquad\textbf{(B)}\ 189 \qquad\textbf{(C)}\ 252 \qquad\textbf{(D)}\ 315 \qquad\textbf{(E)}\ \text{None} $

Let $ABC$ be a triangle with circumcircle $\omega$, incenter $I$, and $A$-excenter $I_A$. Let the incircle and the $A$-excircle hit $BC$ at $D$ and $E$, respectively, and let $M$ be the midpoint of arc $BC$ without $A$. Consider the circle tangent to $BC$ at $D$ and arc $BAC$ at $T$. If $TI$ intersects $\omega$ again at $S$, prove that $SI_A$ and $ME$ meet on $\omega$. [i]Amol Aggarwal.[/i]