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] On an unmoving escalator, a person descends faster than he ascends. Which is faster: to descend and ascend once on an upward-moving escalator, or to descend and ascend once on a downward-moving escalator? (It is assumed that all speeds mentioned here are constant, moreover that the speed of the escalator is the same whether moving up or down, and the speed of the person relative to the escalator is always greater than the speed of the escalator.)
2. [3] Prove that the equation x2+y2-z2=1997 has an infinite number of solutions in whole numbers x, y, z.
3. [4] In the square ABCD, points K and M lie on sides BC and CD respectively, such that AM is the bisector of angle KAD. Prove that the length of segment AK is equal to the sum of the lengths of segments DM and BK.
4. a) [2] What is the smallest number of
straight lines which can intersect all the squares of a 3×3
chessboard? Draw such a configuration of straight lines and prove
that no smaller number could suffice. (For a square to be
intersected, a line must pass through a point in the interior of
the square.)
b) [4] The same problem for a 4×4 board.
1. a) [2] What is the smallest number of
straight lines which can intersect all the squares of a 3×3
chessboard? Draw such a configuration of straight lines and prove
that no smaller number could suffice. (For a square to be
intersected, a line must pass through a point in the interior of
the square.)
b) [3] The same problem for a 4×4 board.
2. [3] Let a and b be two sides of a triangle. How can the third side, c, be chosen such that the points of tangency of the incircle and the excircle with side c divide that side into three equal segments? (The excircle is the circle tangent to side c and to the extensions of sides a and b.)
3. [4] Prove that the equation xy(x-y)+yz(y-z)+zx(z-x)=6 has an infinite number of solutions in whole numbers x, y, z.
4. [4] The maximum possible number of knights are positioned on a 5×5 chessboard such that no two attack one another. Prove that such a configuration is unique.