BALTIC-WAY 2001

A set of 8 problems was prepared for an examination. Each student was given 3 of them. No two students received more than one common problem. What is the largest possible number of students?

Let $n \geqslant 2$ be a positive integer. Find whether there exist $n$ pairwise nonintersecting nonempty subsets of $\{1,2,3, \ldots\}$ such that each positive integer can be expressed in a unique way as a sum of at most $n$ integers, all from different subsets.

The numbers $1,2, \ldots, 49$ are placed in a $7 \times 7$ array, and the sum of the numbers in each row and in each column is computed. Some of these 14 sums are odd while others are even. Let $A$ denote the sum of all the odd sums and $B$ the sum of all even sums. Is it possible that the numbers were placed in the array in such a way that $A=B$ ?

Let $p$ and $q$ be two different primes. Prove that

$$\left\lfloor\frac{p}{q}\right\rfloor+\left\lfloor\frac{2 p}{q}\right\rfloor+\left\lfloor\frac{3 p}{q}\right\rfloor+\ldots+\left\lfloor\frac{(q-1) p}{q}\right\rfloor=\frac{1}{2}(p-1)(q-1) .$$

(Here $\lfloor x\rfloor$ denotes the largest integer not greater than $x$.)

Let 2001 given points on a circle be colored either red or green. In one step all points are recolored simultaneously in the following way: If both direct neighbors of a point $P$ have the same color as $P$, then the color of $P$ remains unchanged, otherwise $P$ obtains the other color. Starting with the first coloring $F_{1}$, we obtain the colorings $F_{2}, F_{3}, \ldots$ after several recoloring steps. Prove that there is a number $n_{0} \leqslant 1000$ such that $F_{n_{0}}=F_{n_{0}+2}$. Is the assertion also true if 1000 is replaced by 999 ?

The points $A, B, C, D, E$ lie on the circle $c$ in this order and satisfy $A B \| E C$ and $A C \| E D$. The line tangent to the circle $c$ at $E$ meets the line $A B$ at $P$. The lines $B D$ and $E C$ meet at $Q$. Prove that $|A C|=|P Q|$.

Given a parallelogram $A B C D$. A circle passing through $A$ meets the line segments $A B, A C$ and $A D$ at inner points $M, K, N$, respectively. Prove that

$$|A B| \cdot|A M|+|A D| \cdot|A N|=|A K| \cdot|A C| .$$

Let $A B C D$ be a convex quadrilateral, and let $N$ be the midpoint of $B C$. Suppose further that $\angle A N D=135^{\circ}$. Prove that

$$|A B|+|C D|+\frac{1}{\sqrt{2}} \cdot|B C| \geqslant|A D|$$

Given a rhombus $A B C D$, find the locus of the points $P$ lying inside the rhombus and satisfying $\angle A P D+\angle B P C=180^{\circ}$.

In a triangle $A B C$, the bisector of $\angle B A C$ meets the side $B C$ at the point $D$. Knowing that $|B D| \cdot|C D|=|A D|^{2}$ and $\angle A D B=45^{\circ}$, determine the angles of triangle $A B C$.

The real-valued function $f$ is defined for all positive integers. For any integers $a>1, b>1$ with $d=\operatorname{gcd}(a, b)$, we have

$$f(a b)=f(d) \cdot\left(f\left(\frac{a}{d}\right)+f\left(\frac{b}{d}\right)\right),$$

Determine all possible values of $f(2001)$.

Let $a_{1}, a_{2}, \ldots, a_{n}$ be positive real numbers such that $\sum_{i=1}^{n} a_{i}^{3}=3$ and $\sum_{i=1}^{n} a_{i}^{5}=5$. Prove that $\sum_{i=1}^{n} a_{i}>\frac{3}{2}$.

Let $a_{0}, a_{1}, a_{2}, \ldots$ be a sequence of real numbers satisfying $a_{0}=1$ and $a_{n}=a_{\lfloor 7 n / 9\rfloor}+a_{\lfloor n / 9\rfloor}$ for $n=1,2, \ldots$ Prove that there exists a positive integer $k$ with $a_{k}<\frac{k}{2001 !}$.

(Here $\lfloor x\rfloor$ denotes the largest integer not greater than $x$.)

There are $2 n$ cards. On each card some real number $x, 1 \leqslant x \leqslant 2$, is written (there can be different numbers on different cards). Prove that the cards can be divided into two heaps with sums $s_{1}$ and $s_{2}$ so that $\frac{n}{n+1} \leqslant \frac{s_{1}}{s_{2}} \leqslant 1$.

Let $a_{0}, a_{1}, a_{2}, \ldots$ be a sequence of positive real numbers satisfying $i \cdot a_{i}^{2} \geqslant(i+1) \cdot a_{i-1} a_{i+1}$ for $i=1,2, \ldots$ Furthermore, let $x$ and $y$ be positive reals, and let $b_{i}=x a_{i}+y a_{i-1}$ for $i=1,2, \ldots$ Prove that the inequality $i \cdot b_{i}^{2}>(i+1) \cdot b_{i-1} b_{i+1}$ holds for all integers $i \geqslant 2$.

Let $f$ be a real-valued function defined on the positive integers satisfying the following condition: For all $n>1$ there exists a prime divisor $p$ of $n$ such that

$$f(n)=f\left(\frac{n}{p}\right)-f(p)$$

Given that $f(2001)=1$, what is the value of $f(2002)$ ?

Let $n$ be a positive integer. Prove that at least $2^{n-1}+n$ numbers can be chosen from the set $\left\{1,2,3, \ldots, 2^{n}\right\}$ such that for any two different chosen numbers $x$ and $y, x+y$ is not a divisor of $x \cdot y$.

Let $a$ be an odd integer. Prove that $a^{2^{n}}+2^{2^{n}}$ and $a^{2^{m}}+2^{2^{m}}$ are relatively prime for all positive integers $n$ and $m$ with $n \neq m$.

What is the smallest positive odd integer having the same number of positive divisors as 360 ?

From a sequence of integers $(a, b, c, d)$ each of the sequences

$$(c, d, a, b),(b, a, d, c),(a+n c, b+n d, c, d),(a+n b, b, c+n d, d),$$

for arbitrary integer $n$ can be obtained by one step. Is it possible to obtain $(3,4,5,7)$ from $(1,2,3,4)$ through a sequence of such steps?