Problem 1940.1.

In a set of objects, each has one of two colors and one of two shapes. There is at least one object of each color and at least one object of each shape. Prove that there exist two objects in the set that are different both in color and in shape.

Problem 1943.1.

Prove that in any group of people, the number of those who know an odd number of the others in the group is even. Assume that "knowing" is a symmetric relation.

Problem 1933.2.

Sixteen squares of an 8×8 chessboard are chosen so that there are exactly two in each row and two in each column. Prove that eight white pawns and eight black pawns can be placed on these sixteen squares so that there is one white pawn and one black pawn in each row and in each column.

Problem 1930.2.

A straight line is drawn across an 8 × 8 chessboard. It is said to pierce a square if it passes through an interior point of the square. At most how many of the 64 squares can this line pierce?

Problem 1930.1.

How many five-digit multiples of 3 end with the digit 6?

Problem 1929.1.

In how many ways can the sum of 100 fillér be made up with coins of denominations 1, 2, 10, 20 and 50 fillér?