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] Several successive positive integers were written out in one row in such order that the sum of any three integers standing one after the other in the row is divisible by the left-most number in the triple. What is the largest number of integers that can be written out if the last number in the row is odd?
2. Let ABC be an acute-angled
triangle, C' and A' be arbitrary points on
sides AB and BC, respectively, and C'
be the midpoint of side AC.
(a) [2] Prove that the area of triangle A'B'C'
is no greater than half of the area of triangle ABC.
(b) [2] Prove that the area of triangle A'B'C'
is equal to one fourth of the area of triangle ABC if
and only if at least one of the points A', C'
is the midpoint of the corresponding side.
3. [5] 100 weights of 1, 2, ... , 100 grams are placed on the two pans of a balance so that equilibrium is achieved. Prove that two weights can be removed from each pan so that equilibrium will be preserved.
4a. [3] On each of the little squares of the
upper and lower horizontal rows of an 8×8 chessboard there is a
pawn: white pawns on the lower row and black pawns on the upper
row. In each move one can shift any pawn vertically or
horizontally to any adjacent empty little square. In what minimal
number of moves can all the white pawns be moved to the upper row
and the black ones to the lower row?
4b. [4] Same question for a 7×7 board.
5. [8] Tireless Thomas and Jeremy construct a sequence. At first there is one positive integer in the sequence. Then they successively write new numbers in the sequence in the following way: Thomas obtains his number by adding, to the previous number, one of its (decimal) digits, while Jeremy obtains it by subtracting, from the previous number, one of its digits. Prove that some number in this sequence will be repeated no less than 100 times.
6. [9] Inside a rectangular piece of paper n rectangular holes with sides parallel to the sides of the paper have been cut out. Into what minimal number of rectangular pieces is it always possible to cut this piece of paper with holes? (Show that in any case it is possible to cut up the paper into the number of pieces that you assert, but there are examples when it is impossible to cut up the paper into a lesser number.)
1. [3] For what n can one place the integers 1 to n on a circle so that the sum of any two successive integers in the circle is divisible by the next one in clockwise order?
2. On a rectangular piece of paper there are
(a) [2] several marked points all on one
straight line;
(b) [3] three marked points.
We are allowed to fold the paper several times along a straight
line not including the marked points and then puncture the folded
paper with a needle. Show that this can be done so that all the
marked points are punctured and there are no extra holes.
3. [6] Tireless Thomas and Jeremy construct a sequence. At first there is one positive integer in the sequence. Then they successively write new numbers in the sequence in the following way: Thomas obtains his number by adding, to the previous number, one of its (decimal) digits, while Jeremy obtains it by subtracting, from the previous number, one of its digits. Prove that some number in this sequence will be repeated no less than 100 times.
4. The points K, L on sides
AC, CB of triangle ABC are the touch
points of the exinscribed circles. Prove that the line joining
the midpoints of KL and AB
(a) [3] divides the perimeter of triangle ABC
in half;
(b) [3] is parallel to the bisector of angle ACB.
5a. [4] 100 weights of 1, 2, ... , 100 grams
are placed on the two pans of a balance so that equilibrium is
achieved. Prove that two weights can be removed from each pan so
that equilibrium will be preserved.
5b. [4] Consider positive integers n
such that the weights 1, 2,... , n grams can be divided
into two parts of equal total weight. Is it true that for any
such n greater than 3 one can remove two weights from
each part so that the total weights remain equal?
6. [8] On a large chessboard, 2n little squares have been marked so that the rook (which moves only horizontally or vertically) can visit all the marked squares without jumping over any unmarked ones. Prove that the figure constituted by all the marked squares can be cut up into n rectangles.
7. [8] Prove that any convex polyhedron of 10n faces has at least n faces with the same number of sides.