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] There is $500 in a bank. Two bank operations are allowed: one can either take $300 from the bank, or put $198 into the bank. These operations can be repeated; however, there is no money other than that initially in the bank. What is the maximal amount of money that can be borrowed from the bank? How can this be done?
2. [4] Let O be the intersection point of the diagonals of a parallelogram ABCD. Prove that if the line BC is tangent to the circle passing through the points A, B, and O, then the line CD is tangent to the circle passing through the points B, C and O.
3. [4] Two players play the following game. The first player starts by writing either 0 or 1, and then proceeds to add either 0 or 1 to the right of the existing digits until there are 1999 digits. After each digit is written down by the first player (except for the very first one), the second player chooses arbitrary two digits among those already written and exchanges them. Can the second player guarantee that after his last move the line of digits will be symmetric?
4. [6] n diameters divide a disk into 2n equal sectors. Half of the sectors is colored blue, and the other half is colored red (in arbitrary order). Blue sectors are numbered from 1 to n in the counterclockwise direction, starting from an arbitrary blue sector, and red sectors are numbered from 1 to n in the clockwise direction, starting from an arbitrary red sector. Prove that there is a semi-disk containing sectors with all numbers from 1 to n.
5. [6] The segments AB and AC are tangent at points, respectively, P and Q, to the circle inscribed in a triangle ABC. R and S are the middle points of the edges, respectively, AC and BC, and T is the intersection point of the lines PQ and RS. Prove that T belongs to the bisector of the angle B of the triangle.
6. [9] A move of the rook consists of passing to a neighboring cell in either the horizontal, or the vertical direction. After 64 moves the rook visited all cells of the 8×8 chessboard and returned back to the initial cell. Prove that the number of moves in the vertical direction and the number of moves in the horizontal direction are distinct.
1. [4] A convex polyhedron body drifts in a sea. Can it happen that 90% of its volume is below the water level, while more than half of its surface area is above the water level?
2. [4] Let ABCD be a convex quadrangle inscribed in a circle centered at a point O. Let F be the second intersection point of the circles circumscribed around the triangles ABO and CDO. Prove that the circle passing through the points A, F and D passes also through the intersection point of the segments AC and BD.
3. [5] Find all pairs (x,y) of integers satisfying the following condition: both numbers x3+y and x+y3 are divisible by x2+y2.
4. [5] n diameters divide a disk into 2n equal sectors. Half of the sectors is colored blue, and the other half is colored red (in arbitrary order). Blue sectors are numbered from 1 to n in the counterclockwise direction, starting from an arbitrary blue sector, and red sectors are numbered from 1 to n in the clockwise direction, starting from an arbitrary red sector. Prove that there is a semi-disk containing sectors with all numbers from 1 to n.
5. For every non-negative integer i,
define the number M(i) as follows: write i down
as a binary number, so that we have a string of zeroes and ones;
if the number of ones in this string is even, then set M(i)
= 0, otherwise set M(i) = 1. (The first terms
of the sequence M(i), i = 0,1,2,...
are 0,1,1,0,1,0,0,1,...)
a) [2] Consider the finite sequence M(0),
M(1), ..., M(1000). Prove that there are at
least 320 terms in this sequence which are equal to their
neighbor on the right: M(i) = M(i+1).
b) [5] Consider the finite sequence M(0),
M(1), ..., M(1000000). Prove that the number of
terms M(i) such that M(i) = M(i+7)
is at least 450000.
6. [8] A move of the rook consists of passing to a neighboring cell in either the horizontal, or the vertical direction. After 64 moves the rook visited all cells of the 8×8 chessboard and returned back to the initial cell. Prove that the number of moves in the vertical direction and the number of moves in the horizontal direction are distinct.