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] A 20×20×20 block is cut up into 8000 non-overlapping unit cubes and a number is assigned to each . It is known that in each column of 20 cubes parallel to any edge of the block, the sum of their numbers is equal to 1. The number of one of the unit cubes is 10. Three 1×20×20 slices parallel to the faces of the block contain this unit cube. Find the sum of all numbers of the cubes outside of these slices.
2. [3] The units-digit of the square of an integer is 9 and the tens-digit of this square is 0. Prove that the hundred-digit is even.
3. [4] In trianle ABC the points A', B' and C' lie on the sides BC, CA and AB, respectively. It is known that angle AC'B = angle B'A'C, angle CB'A' = angle A'C'B and angle BA'C' = angle C'B'A. Prove that A', B' and C' are the midpoints of the corresponding sides.
4. [4] Twelve candidates for mayor participate in a TV talk show. At some point a candidate said: "One lie has been told". Another said: "Now two lies have been told". "Now three lies", said a third. This continued until the twelfth said: "Now twelve lies have been told". At this point the moderator ended the discussion. It turned out that at least one of the candidates correctly stated the number of lies told before he made the claim. How many lies were actually told by the candidates?
5. [5] Let n and m be given positive integers. In one move, a chess piece called an (n,m)-crocodile goes n squares horizontally or vertically and then goes m squares in a perpendicular direction. Prove that the squares of an infinite chessboard can be painted in black and white so that this chess piece always moves from a black to a white one or vice-versa.
1. [3] Nineteen weights of mass 1 gm, 2 gm, 3 gm, ..., 19 gm are given. Nine are made of iron, nine are of bronze and one is pure gold. It is known that the total mass of all the iron weights is 90 gm more than the total mass of the bronze ines. Find the mass of the gold weight.
2. [3] On the plane are n paper disks of radius 1 whose boundaries all pass through a certain point, which lies inside the region covered by the disks. Find the perimeter of this region.
3. [4] On an 8×8 chessboard, 17 cells are marked. Prove that one can always choose two cells among the marked ones so that a Knight will need at least three moves to go from one of the chosen cells to the other.
4. [4] Among all sets of real numbers {x1, x2, ...,x20} from the open interval (0,1) such that x1x2...x20 = (1-x1)(1-x2)...(1-x20), find the one for which x1x2...x20 is maximal.
5. The intelligence quotient (IQ) of a
country is defined as the average IQ of its entire population. It
is assumed that the total population and individual IQ's remain
constant throughout.
(a) [1] A group of people from country A
has emigrated to country B. Show that it can happen that
as a result, the IQ's of both countries will increase.
(b) [3] After this, a group of people from B,
which may include immigrants from A, emigrates to A.
Can it happen that the IQ's of both countries will increase again?
(c) [2] A group of people from country A
has emigrated to country B, and a group of people from B
has emigrated to country C. It is known that as a result,
the IQ's of all three countries have increased. After this, a
group of people from C emigrates to B and a
group of people from B emigrates to A. Can it
happen that the IQ's of all three countries will increase again?