The maximum score for each problem is shown in brackets [ ]. Your total score is based on the three problems for which you earn the highest scores. The scores for the individual parts of a single problem are summed.
1. [3] Anya, Borya, and Vasya listed words that could be formed from a given set of letters. They each listed a different number of words: Anya listed the most, Vasya the least. Then they were awarded points for their words. Words listed by two of them scored 1 point, while words listed by only one of them scored 2 points. Words common to all three of them were crossed out. Is it possible that Vasya got the highest score, and Anya the lowest?
2. [3] A chess king tours an entire 8×8 chess board, visiting each square exactly once and returning at last to his starting position. Prove that he made an even number of diagonal moves.
3. [3] AB and CD are segments lying on the two sides of an angle: O is the vertex of the angle, A is between O and B, and C is between O and D. The line connecting the midpoints of segments AD and BC intersects the sides of the angle at points M and N (M, A, and B are on one side of the angle; N, C, and D are on the other). Prove that OM/ON = AB/CD.
4. [4] For every three-digit number, we take the product of its digits; then we add all of these products together. What is the result? (Clarification: we take the product of all the digits of a three-digit number, so if at least one of the digits is a 0, the product is 0.)
5. [5] Pinocchio claims that he can divide an isoceles triangle into three triangles, any two of which can be put together to form a new isoceles triangle. Is Pinocchio lying?
1. [3] Pinocchio claims that he can take some non-right triangles which are similar to one another and put them together to form a rectangle. Do you believe him? (Some of the triangles can be congruent.)
2. [3] For every four-digit number, we take the product of its digits; then we add all of these products together. What is the result? (Clarification: we take the product of all the digits of a four-digit number, so if at least one of the digits is a 0, the product is 0.)
3. [3] What is the maximum number of colors that can be used to paint an 8×8 chessboard such that every square is adjacent to at least two other squares of its own color? (Each square is painted with only one color.)
4. [4] For some positive numbers A, B,
C, and D, the system of equations
x2 + y2 = A
|x| + |y| = B
has m solutions, and the system of equations
x2 + y2 + z2
= A
|x| + |y| + |z| = B
has n solutions. It is known that m>n>1.
Find m and n.
5. [5] A circle with center O is inscribed in an angle. Let A be the reflection of O through one of the sides of the angle. Tangents to the circle through A intersect the other side of the angle at points B and C. Prove that the circumcircle of triangle ABC lies on the bisector of the original angle.