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[Solutions] Baltic Way Mathematical Competition 2001

  1. 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?
  2. Let $n\ge 2$ 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.
  3. 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 $A = B$?
  4. Let $p$ and $q$ be two different primes. Prove that \[\left\lfloor\frac{p}{q}\right\rfloor+\left\lfloor\frac{2p}{q}\right\rfloor+\left\lfloor\frac{3p}{q}\right\rfloor+\ldots +\left\lfloor\frac{(q-1)p}{q}\right\rfloor=\frac{1}{2}(p-1)(q-1) \]
  5. Let $2001$ given points on a circle be coloured either red or green. In one step all points are recoloured simultaneously in the following way: If both direct neighbours of a point $P$ have the same colour as $P$, then the colour of $P$ remains unchanged, otherwise $P$ obtains the other colour. Starting with the first colouring $F_1$, we obtain the colourings $F_2,F_3 ,\ldots .$ after several recolouring steps. Prove that there is a number $n_0\le 1000$ such that $F_{n_0}=F_{n_0 +2}$. Is the assertion also true if $1000$ is replaced by $999$?
  6. The points $A, B, C, D, E$ lie on the circle $c$ in this order and satisfy $AB\parallel EC$ and $AC\parallel ED$. The line tangent to the circle $c$ at $E$ meets the line $AB$ at $P$. The lines $BD$ and $EC$ meet at $Q$. Prove that $|AC|=|PQ|$.
  7. Given a parallelogram $ABCD$. A circle passing through $A$ meets the line segments $AB, AC$ and $AD$ at inner points $M,K,N$, respectively. Prove that \[|AB|\cdot |AM | + |AD|\cdot |AN|=|AK|\cdot |AC|\]
  8. Let $ABCD$ be a convex quadrilateral, and let $N$ be the midpoint of $BC$. Suppose further that $\angle AND=135^{\circ}$. Prove that $$|AB|+|CD|+\frac{1}{\sqrt{2}}\cdot |BC|\ge |AD|.$$
  9. Given a rhombus $ABCD$, find the locus of the points $P$ lying inside the rhombus and satisfying $\angle APD+\angle BPC=180^{\circ}$.
  10. In a triangle $ABC$, the bisector of $\angle BAC$ meets the side $BC$ at the point $D$. Knowing that $|BD|\cdot |CD|=|AD|^2$ and $\angle ADB=45^{\circ}$, determine the angles of triangle $ABC$.
  11. The real-valued function $f$ is defined for all positive integers. For any integers $a>1, b>1$ with $d=\gcd (a, b)$, we have \[f(ab)=f(d)\left(f\left(\frac{a}{d}\right)+f\left(\frac{b}{d}\right)\right).\] Determine all possible values of $f(2001)$.
  12. Let $a_1, a_2,\ldots , a_n$ be positive real numbers such that $$\sum_{i=1}^na_i^3=3, \quad \sum_{i=1}^na_i^5=5.$$ Prove that $$\sum_{i=1}^na_i>\frac{3}{2}.$$
  13. Let $a_0, a_1, a_2,\ldots $ be a sequence of real numbers satisfying $a_0=1$ and $a_n=a_{\lfloor 7n/9\rfloor}+a_{\lfloor n/9\rfloor}$ for $n=1, 2,\ldots .$ Prove that there exists a positive integer $k$ with $a_k<\frac{k}{2001!}$.
  14. There are $2n$ cards. On each card some real number $x$, $(1\le x\le 2n)$, 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 $\frac{n}{n+1}\le\frac{s_1}{s_2}\le 1$.
  15. Let $a_0,a_1,a_2,\ldots $ be a sequence of positive real numbers satisfying $i\cdot a_2\ge (i + 1)\cdot a_{i_1}a_{i+1}$ for $i=1, 2, \ldots $ Furthermore, let $x$ and $y$ be positive reals, and let $b_i=xa_i+ya_{i-1}$ for $i=1, 2, \ldots .$ Prove that the inequality $i\cdot b_2\ge (i + 1)\cdot b_{i-1}b_{i+1}$ holds for all integers $i\ge 2$.
  16. 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 $f(2002)$?
  17. Let $n$ be a positive integer. Prove that at least $2^{n-1}+n$ numbers can be chosen from the set $\{1, 2, 3,\ldots ,2^n\}$ such that for any two different chosen numbers $x$ and $y$, $x+y$ is not a divisor of $x\cdot y$.
  18. Let $a$ be an odd integer. Prove that $a^{2^m}+2^{2^m}$ and $a^{2^n}+2^{2^n}$ are relatively prime for all positive integers $n$ and $m$ with $n\not= m$.
  19. What is the smallest positive odd integer having the same number of positive divisors as $360$?
  20. From a sequence of integers $(a, b, c, d)$ each of the sequences \[(c, d, a, b),\quad (b, a, d, c),\quad (a + nc, b + nd, c, d),\quad (a + nb, b, c + nd, d)\] 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?

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MOlympiad: [Solutions] Baltic Way Mathematical Competition 2001
[Solutions] Baltic Way Mathematical Competition 2001
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