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] The sequence {xn}
is defined by these conditions:
x1 = 19; x2 = 97; xn+2
= xn - 1/xn+1.
Prove that some term of this sequence is equal to 0. Find the
index of this term.
2. [3] M is the midpoint of side BC of triangle ABC. Construct line L, intersecting the triangle and parallel to BC, such the segment of L between sides AC and BC is the hypotenuse of a right triangle with third vertex M.
3. [5] Initially there is a checker on every square of a 1×n board. The first move consists of moving any checker to an adjacent square (either of two adjacent squares, if the checker isn't at an end of the board), thereby creating a stack of two checkers. Each subsequent move consists of moving any stack in either direction, as many squares as the stack has checkers (within the boundaries of the board); if a stack lands on a non-empty square, it is placed on top of the stack already there, forming a single stack. Prove that it is possible to stack all the checkers on one square in n-1 moves.
4. [5] Two circles intersect at points A and B. A common tangent touches the first circle at point C and the second at point D. Let B be the nearer of the intersection points to line CD. Line CB intersects the second circle again at point E. Prove that AD bisects angle CAE.
5. [8] Each face of a cube is exactly the same size as each square of a chessboard. The cube is colored black and white, placed on one of the squares of the chessboard, and rolled such that each square of the chessboard is visited exactly once. Can this procedure be performed in such a way that the color of the visited square and the color of the bottom face of the cube always coincide?
6. [9] Points on each side of a equilateral triangle divide the side into 10 equal segments. Lines parallel to the sides of the triangle are drawn through all of these points, dividing the original triangle into 100 small triangles, or "cells." The cells between any two adjacent parallel lines form a "stripe." What is the maximum number of cells that can be chosen such that no two chosen cells belong to a single stripe in any of the three possible orientations?
1. [4] CM and BN are medians of triangle ABC. Points P and Q are chosen on sides AB and AC respectively, such that the bisector of angle C of the triangle also bisects angle MCP, and the bisector of angle B also bisects angle NBQ. It turns out that AP = AQ. Does it follow that ABC is isosceles?
2. Which of the following are true?
a) [1] If a polygon can be divided into two
congruent polygons by a broken line segment, it can be divided
into two congruent polygons by a straight line segment.
b) [2] If a convex polygon can be divided into
two congruent polygons by a broken line segment, it can be
divided into two congruent polygons by a straight line segment.
c) [4] If a convex polygon can be divided into
two polygons by a broken line segment, one of which can be mapped
to the other by means of an orientation-preserving isometry (that
is, a rotation followed by a parallel translation), then it can
be divided into two polygons by a straight line segment, one of
which can be mapped to the other by means of an orientation-
preserving isometry.
3. All expressions of the form +11/2+21/2+...+991/2+1001/2
(with every possible combination of signs + or -) are
multiplied together. Prove that the result is:
a) [3] a whole number,
b) [3] the square of a whole number.
4. a) [4] Some identical napkins are arranged
on a table (with overlappings), each in the shape of a regular
hexagon. Each napkin has one side which is parallel to a fixed
line. Is it always possible to hammer some nails into the table
such that every napkin is nailed and moreover, each napkin is
nailed by only one nail?
b) [4] The same question for regular pentagons.
5. [8] Dima invented a secret code in which every letter is replaced by a word no longer than 10 letters. A code is called "good" if every encoded word can be decoded in only one way. Serjozha (with the help of a computer) checked that for Dima's code, every possible word of no more than 10000 letters can be decoded in only one way. Does it follow that Dima's code is good? (Please note that Dima and Serjozha are Russian, so they use the Cyrillic alphabet, which has 33 letters! A word is any sequence of letters, whether or not it means anything.)
6. Points on each side of a equilateral
triangle divide the side into n equal segments. Lines parallel to
the sides of the triangle are drawn through all of these points,
dividing the original triangle into n2 small
triangles, or "cells." The cells between any two
adjacent parallel lines form a "stripe."
a) [7] What is the maximum number of cells that
can be chosen such that no two chosen cells belong to a single
stripe in any of the three possible orientations, if n=10?
b) [7] The same question for n=9.