A function $f$ is defined on the positive integers by \[\left\{\begin{array}{rcl}f(1) &=& 1, \\ f(3) &=& 3, \\ f(2n) &=& f(n), \\ f(4n+1) &=& 2f(2n+1)-f(n), \\ f(4n+3) &=& 3f(2n+1)-2f(n), \end{array}\right.\] for all positive integers $n$. Determine the number of positive integers $n$, less than or equal to 1988, for which $f(n) = n$.

In a board school, there are 9 subjects, 512 students, and 256 rooms (two people in each room.) For every student there is a set (a subset of the 9 subjects) of subjects the student is interested in. Each student has a different set of subjects, (s)he is interested in, from all other students. (Exactly one student has no subjects (s)he is interested in.) Prove that the whole school can line up in a circle in such a way that every pair of the roommates has the two people standing next to each other, and those pairs of students standing next to each other that are not roommates, have the following properties. One of the two students is interested in all the subjects that the other student is interested in, and also exactly one more subject.

Find all functions $f:\mathbb{N}\rightarrow \mathbb{N}$ such that for all $m,n\in\mathbb{N}$, \[(2^m+1)f(n)f(2^mn)=2^mf(n)^2+f(2^mn)^2+(2^m-1)^2n. \]

A function $f_n(x)\ (n=0,\ 1,\ 2,\ 3,\ \cdots)$ satisfies the following conditions: (i) $f_0(x)=e^{2x}+1$. (ii) $f_n(x)=\int_0^x (n+2t)f_{n-1}(t)dt-\frac{2x^{n+1}}{n+1}\ (n=1,\ 2,\ 3,\ \cdots).$ Find $\sum_{n=1}^{\infty} f_n'\left(\frac 12\right).$

A brick has the shape of a cube of size $2$ with one corner unit cube removed. Given a cube of side $2^{n}$ divided into unit cubes from which an arbitrary unit cube is removed, show that the remaining figure can be built using the described bricks.

Let $n > 1$ be an integer. Find, with proof, all sequences $x_1 , x_2 , \ldots , x_{n-1}$ of positive integers with the following three properties: (a). $x_1 < x_2 < \cdots < x_{n-1}$ ; (b). $x_i + x_{n-i} = 2n$ for all $i = 1, 2, \ldots , n - 1$; (c). given any two indices $i$ and $j$ (not necessarily distinct) for which $x_i + x_j < 2n$, there is an index $k$ such that $x_i + x_j = x_k$.

Prove that if $c$ is a positive number distinct from $1$ and $n$ a positive integer, then \[n^{2}\leq \frac{c^{n}+c^{-n}-2}{c+c^{-1}-2}. \]

Prove the inequality for all positive integer $n$ : \[ \left(\frac{2n-1}{e}\right)^{\frac{2n-1}{2}}<1\cdot 3\cdot 5\cdots (2n-1)<\left(\frac{2n+1}{e}\right)^{\frac{2n+1}{2}} \]

Let $p$ be a prime, $n$ be a positive integer, and let $\mathbb{Z}_{p^n}$ denote the set of congruence classes modulo $p^n.$ Determine the number of functions $f: \mathbb{Z}_{p^n} \to \mathbb{Z}_{p^n}$ satisfying the condition \[ f(a)+f(b) \equiv f(a+b+pab) \pmod{p^n} \] for all $a,b \in \mathbb{Z}_{p^n}.$

Given that $g(n) = \frac{1}{{2 + \frac{1}{{3 + \frac{1}{{... + \frac{1}{{n - 1}}}}}}}}$ and $k(n) = \frac{1}{{2 + \frac{1}{{3 + \frac{1}{{... + \frac{1}{{n - 1 + \frac{1}{n}}}}}}}}}$, for natural $n$. Prove that $\left| {g(n) - k(n)} \right| \le \frac{1}{{(n - 1)!n!}}$.

Prove that $1\cdot2\cdot3\cdots 2002<\left(\frac{2003}{2}\right)^{2002}.$

In a wagon, every $m \geq 3$ people have exactly one common friend. (When $A$ is $B$'s friend, $B$ is also $A$'s friend. No one was considered as his own friend.) Find the number of friends of the person who has the most friends.

$n$ people (with names $1,2,\dots,n$) are around a table. Some of them are friends. At each step 2 friend can change their place. Find a necessary and sufficient condition for friendship relation between them that with these steps we can always reach to all of posiible permutations.

Let $m$ circles intersect in points $A$ and $B$. We write numbers using the following algorithm: we write $1$ in points $A$ and $B$, in every midpoint of the open arc $AB$ we write $2$, then between every two numbers written in the midpoint we write their sum and so on repeating $n$ times. Let $r(n,m)$ be the number of appearances of the number $n$ writing all of them on our $m$ circles. a) Determine $r(n,m)$; b) For $n=2006$, find the smallest $m$ for which $r(n,m)$ is a perfect square. Example for half arc: $1-1$; $1-2-1$; $1-3-2-3-1$; $1-4-3-5-2-5-3-4-1$; $1-5-4-7-3-8-5-7-2-7-5-8-3-7-4-5-1$...

There are$n$ balls numbered $1,2,\cdots,n$, respectively. They are painted with $4$ colours, red, yellow, blue, and green, according to the following rules: First, randomly line them on a circle. Then let any three clockwise consecutive balls numbered $i, j, k$, in order. 1) If $i>j>k$, then the ball $j$ is painted in red; 2) If $i<j<k$, then the ball $j$ is painted in yellow; 3) If $i<j, k<j$, then the ball $j$ is painted in blue; 4) If $i>j, k>j$, then the ball $j$ is painted in green. And now each permutation of the balls determine a painting method. We call two painting methods distinct, if there exists a ball, which is painted with two different colours in that two methods. Find out the number of all distinct painting methods.

We have $2p-1$ integer numbers, where $p$ is a prime number. Prove that we can choose exactly $p$ numbers (from these $2p-1$ numbers) so that their sum is divisible by $p$.

Let $M$ be the set of all integers from $1$ to $2013$. Each subset of $M$ is given one of $k$ available colors, with the only condition that if the union of two different subsets $A$ and $B$ is $M$, then $A$ and $B$ are given different colors. What is the least possible value of $k$?

The jury of an Olympiad has $30$ members in the beginning. Each member of the jury thinks that some of his colleagues are competent, while all the others are not, and these opinions do not change. At the beginning of every session a voting takes place, and those members who are not competent in the opinion of more than one half of the voters are excluded from the jury for the rest of the olympiad. Prove that after at most $15$ sessions there will be no more exclusions. (Note that nobody votes about his own competence.)

We are given a finite set of segments of the same line. Prove that we can color each segment red or blue such that, for each point $p$ on the line, the number of red segments containing $p$ differs from the number of blue segments containing $p$ by at most $1$.

Let $u_1$, $u_2$, $u_3$, $\dots$ be a sequence of integers satisfying the recurrence relation $u_{n + 2} = u_{n + 1}^2 - u_n$. Suppose $u_1 = 39$ and $u_2 = 45$. Prove that 1986 divides infinitely many terms of the sequence.

Consider two odd natural numbers $a$ and $b$ where $a$ is a divisor of $b^2+2$ and $b$ is a divisor of $a^2+2.$ Prove that $a$ and $b$ are the terms of the series of natural numbers $\langle v_n\rangle$ defined by \[v_1 = v_2 = 1; v_n = 4v_ {n-1}-v_{n-2} \ \ \text{for} \ n\geq 3.\]

Given a polynomial $f(x)$ with rational coefficients, of degree $d \ge 2$, we define the sequence of sets $f^0(\mathbb{Q}), f^1(\mathbb{Q}), \ldots$ as $f^0(\mathbb{Q})=\mathbb{Q}$, $f^{n+1}(\mathbb{Q})=f(f^{n}(\mathbb{Q}))$ for $n\ge 0$. (Given a set $S$, we write $f(S)$ for the set $\{f(x)\mid x\in S\})$. Let $f^{\omega}(\mathbb{Q})=\bigcap_{n=0}^{\infty} f^n(\mathbb{Q})$ be the set of numbers that are in all of the sets $f^n(\mathbb{Q})$, $n\geq 0$. Prove that $f^{\omega}(\mathbb{Q})$ is a finite set. [i]Dan Schwarz, Romania[/i]

Let $n\ge 2$ be a given integer. Show that the number of strings of length $n$ consisting of $0'$s and $1'$s such that there are equal number of $00$ and $11$ blocks in each string is equal to \[2\binom{n-2}{\left \lfloor \frac{n-2}{2}\right \rfloor}\]

Let $A$ be a ring with $2^n+1$ elements, where $n$ is a positive integer and let \[ M = \{ k \in\mathbb{Z} \mid k \geq 2, \ x^k =x , \ \forall \ x\in A \} . \] Prove that the following statements are equivalent: a) $A$ is a field; b) $M$ is not empty and the smallest element in $M$ is $2^n+1$. [i]Marian Andronache[/i]

Let $a_1,a_2,\dots,a_n$ be positive real numbers whose product is $1$. Show that the sum \[\textstyle\frac{a_1}{1+a_1}+\frac{a_2}{(1+a_1)(1+a_2)}+\frac{a_3}{(1+a_1)(1+a_2)(1+a_3)}+\cdots+\frac{a_n}{(1+a_1)(1+a_2)\cdots(1+a_n)}\] is greater than or equal to $\frac{2^n-1}{2^n}$.