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 [ ].
1. [3] Can one find 10 successive positive integers such that the sum of their squares is equal to the sum of squares of the next 9 integers?
2. [3] For what positive integer values of n can an equilateral triangle of side n be cut up into pieces each for which is a trapezoid with sides 1, 1, 1, 2 ?
3a. [2] Can it happen that in a group of 10
girls and 9 boys all the girls are acquainted with a different
number of boys while all the boys are acquainted with the same
number of girls?
3b. [2] What if there are 11 girls and 10 boys?
4. [4] A circle intersects each side of a rhombus in two points, dividing the side into three segments. Let us go around the rhombus clockwise beginning with a vertex and paint these segments successively in red, white, and blue. Prove that the sum of lengths of the blue segments equals that of the red ones.
1. [3] Consider three nonintersecting and nonparallel edges a, b, c of a cube. Find the locus of points inside the cube equidistant from a, b, c.
2. [3] Can a paper circle be cut up into pieces with scissors and rearranged into a square of the same area? (Only a finite number of cuts, along segments and circular, arcs are allowed.)
3. [4] The parabola y = x2 is drawn in the coordinate plane xOy, then the axes are erased (the parabola remains, but the origin O is not shown on it). Reconstruct the axes with compass and straight edge.
4. [4] For what n>1 it can happen that in a group of n+1 girls and n boys all the girls are acquainted with a different number of boys while all the boys are acquainted with the same number of girls?