21st Tournament of Towns

Spring 2000, O-level

Your total score is based on the three problems for which you earn the most points; the scores for the individual parts of a single problem are summed. Points for each problem are shown in brackets [ ].

Juniors

1. [3] Can the product of two consecutive positive integers it be equal to the product of two consecutive even positive integers?

V.Proizvolov

2. [4] In a trapezoid  ABCD  of area  1, the ratio of base  BC to base AD is 1:2. Let K be the midpoint of the diagonal AC. Let the line  DK  intersect the side  AB at the point L. Find the area of the quadrilateral BCKL.

M.Sonkin

3a. [2] Prove  that  one  can  color  the vertices of a  3n-gonal prism  using only three colors in such a way that every vertex isconnected by an edge to vertices of all three colors.
3b. [3] Prove  that if the  vertices of an  n-gonal  prism can be colored in three colors so that  every vertex is  connected by an edge to  vertices of all  three  colors, then  the  number  n  is divisible by 3.
(Reminder: the bases of an n-gonal prism are two equal n-gons.)

A.Shapovalov

4. [5] Can one place natural numbers at all vertices of a cube in such a way that for every pair  of numbers  connected by an edge, one will be divisible by the other,  and there are no other pairs of numbers with this property.

A.Shapovalov

Seniors

1. [3] A convex quadrangle was divided into four triangles by its diagonals. It turned out  that the sum of the  areas of  opposite triangles (that is triangles  having only one point in common) is equal to the sum of areas of the other two  triangles. Prove that one of  the diagonals  is  divided by the other diagonal into two equal parts.

folklore

2. [4] Two  opposite  faces of a cubic dice  are each marked by a point, two other  opposite faces, by two points, the last two, by three points. Eight  such cubes  were used to  construct a  2×2×2 cube and the total number of points on each of its six faces were counted. Could six consecutive numbers have been obtained?

A.Shapovalov

3. [4] Prove the inequality:

1^k+2^k+...+n^k < (n^2k-(n-1)^k)/(n^k-(n-1)^k)

for all positive integers n and k.

L.Emeljanov

4. Does there exist an infinite sequence of
   a) [3] real numbers,
   b) [3] integers
such that the sum of  every ten consecutive  numbers is positive, while for every n the sum of the first  10n+1 consecutive numbers is negative?

A.Tolpygo