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Baltic Way Mathematical Competition 1990 Solutions

  1. Numbers $1, 2, \dots , n$ are written around a circle in some order. What is the smallest possible sum of the absolute differences of adjacent numbers?
  2. The squares of a squared paper are enumerated as shown on the picture. \[\begin{array}{|c|c|c|c|c|c} \ddots &&&&&\\ \hline 10&\ddots&&&&\\ \hline 6&9&\ddots&&&\\ \hline 3&5&8&12&\ddots&\\ \hline 1&2&4&7&11&\ddots\\ \hline \end{array}\] Devise a polynomial $p(m, n)$ in two variables such that for any $m, n \in \mathbb{N}$ the number written in the square with coordinates $(m, n)$ is equal to $p(m, n)$.
  3. Given $a_0 > 0$ and $c > 0$, the sequence $(a_n)$ is defined by \[a_{n+1}=\frac{a_n+c}{1-ca_n}\quad\text{for }n=1,2,\dots\] Is it possible that $a_0, a_1, \dots , a_{1989}$ are all positive but $a_{1990}$ is negative?
  4. Prove that, for any real numbers $a_1, a_2, \dots , a_n$, \[ \sum_{i,j=1}^n \frac{a_ia_j}{i+j-1}\ge 0.\]
  5. Let $*$ be an operation, assigning a real number $a * b$ to each pair of real numbers $(a, b)$. Find an equation which is true (for all possible values of variables) provided the operation $*$ is commutative or associative and which can be false otherwise.
  6. Let $ABCD$ be a quadrilateral with $AD = BC$ and $$\angle DAB + \angle ABC = 120^\circ.$$ An equilateral triangle $DPC$ is erected in the exterior of the quadrilateral. Prove that the triangle $APB$ is also equilateral.
  7. The midpoint of each side of a convex pentagon is connected by a segment with the centroid of the triangle formed by the remaining three vertices of the pentagon. Prove that these five segments have a common point.
  8. It is known that for any point $P$ on the circumcircle of a triangle $ABC$, the orthogonal projections of $P$ onto $AB,BC,CA$ lie on a line, called a Simson line of $P$. Show that the Simson lines of two diametrically opposite points $P_1$ and $P_2$ are perpendicular.
  9. Two congruent triangles are inscribed in an ellipse. Are they necessarily symmetric with respect to an axis or the center of the ellipse?
  10. A segment $AB$ is marked on a line $t$. The segment is moved on the plane so that it remains parallel to $t$ and that the traces of points $A$ and $B$ do not intersect. The segment finally returns onto $t$. How far can point $A$ now be from its initial position?
  11. Prove that the modulus of an integer root of a polynomial with integer coefficients cannot exceed the maximum of the moduli of the coefficients.
  12. Let $m$ and $n$ be positive integers. Show that $25m+ 3n$ is divisible by $83$ if and only if so is $3m+ 7n$.
  13. Show that the equation $x^2-7y^2 = 1$ has infinitely many solutions in natural numbers.
  14. Do there exist $1990$ pairwise coprime positive integers such that all sums of two or more of these numbers are composite numbers?
  15. Prove that none of the numbers $2^{2^n}+ 1$, $n = 0, 1, 2, \dots$ is a perfect cube.
  16. A closed polygonal line is drawn on a unit squared paper so that its vertices lie at lattice points and its sides have odd lengths. Prove that its number of sides is divisible by $4$.
  17. There are two piles with $72$ and $30$ candies. Two students alternate taking candies from one of the piles. Each time the number of candies taken from a pile must be a multiple of the number of candies in the other pile. Which student can always assure taking the last candy from one of the piles?
  18. Numbers $1, 2,\dots , 101$ are written in the cells of a $101\times 101$ square board so that each number is repeated $101$ times. Prove that there exists either a column or a row containing at least $11$ different numbers.
  19. What is the largest possible number of subsets of the set $\{1, 2, \dots , 2n+1\}$ such that the intersection of any two subsets consists of one or several consecutive integers?
  20. A creative task: propose an original competition problem together with its solution.

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MOlympiad: Baltic Way Mathematical Competition 1990 Solutions
Baltic Way Mathematical Competition 1990 Solutions
MOlympiad
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