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] Prove that if a2 +b2-ab = c where a, b, c are positive numbers, then (a-c)(b-c) < 0.
2. [3] Two nonintersecting circles with centers O1 and O2 are given. Their common exterior tangent line touches them at the points A1 and A2 , respectively; the segment O1O2 intersects the circles at the points B1 and B2 , respectively; C is the intersection point of the lines A1B1 and A2B2 ; D is a point on the line A1A2 such that CD is perpendicular to B1B2 . Prove that A1D = DA2.
3. [3] Prove that from any sequence of 1996 real numbers a1 , a2, ... , a1996 one can choose one or several numbers standing successively one after another so that their sum differs from an integer by no more than 0.001.
4. [5] A rook stands at a corner of an m×n squared board. Two players move the rook in turn (vertically or horizontally through any numbers of squares). As the rook moves, it paints the squares that it visits (stopping or passing through). The rook is not allowed to pass through or stop at the painted squares. The player who cannot move, loses. Who has a guaranteed win: the first player (who starts the game) or the other, and how should he/she play?
5. Eight students were asked to solve 8
problems (the same ones).
a) [3] Each problem was solved by 5 students.
Prove that one can find two students so that each problem was
solved by at least one of them.
b) [3] If each problem was solved by 4 students,
then it is possible that no such a pair of students exists. Prove
this.
6. In the equilateral triangle ABC ,
let D be a point on the side AB such that AD
= AB/n . Prove that the
sum of n-1 angles DP1A , DP2A
, ... , DPn-1A where P1
, P2 , ... , Pn-1
are the points dividing the side BC into n
equal parts, is equal to 30o if
a) [3] n=3;
b) [5] n is arbitrary integer, n>2
.
1. [4] Does there exist a cube in space such that the distances of its vertices from the given plane are equal to 0, 1, 2, 3, 4, 5, 6, 7 ?
2. [5] The square 0<x<1, 0<y<1 is given on the plane Oxy. A grasshopper sitting at a point M outside the square with noninteger coordinates jumps to the new point symmetrical to M with respect to the leftmost vertex of the square (from the grasshopper's point of view). Prove that the grasshopper repeating these jumps will never reach the distance more than 10d from the center C of the square, where d is the distance between the initial position M and the center C.
3. The angle A of an isosceles
triangle ABC (AB=AC) is equal to a.
Let D be a point on the side AB such that AD
= AB/n . Find the sum of n-1
angles DP1A , DP2A
, ... , DPn-1A where P1
, P2 , ... , Pn-1
are the points dividing the side BC into n
equal parts if
a) [3] n=3;
b) [4] n is arbitrary integer, n>2
.
4. [6] There are two very strict laws in the country of Militaria. (A) Anyone who is shorter than 80% (or more) of his 'neighbors' (i.e., men living at the distance less then r from him) are free from the military service. (B) Anyone who is taller than 80% (or more) of his 'neighbors' (men living at the distance less then R from him) are allowed to serve in the police. A nice thing is that each man X may choose his own (possibly different) positive numbers r=r(X) and R=R(X). Can it turn that 90% (or more) of the men in Militaria are free from the army and, at the same time, 90% (or more) of the men in Militaria are allowed to serve in the police? (The places of living of the men are fixed points on the plane. Of course the sets of neighbors must be nonempty.)
5. Prove that there exist an infinite number
of triples n-1, n, n+1 such that
a) [3] n is representable as the sum of
two squares of (positive) natural numbers but n-1 and n+1
are not;
b) [5] each of these three numbers is
representable in the form of the sum of two squares.
6. At first all 2 rows of a 2 times n
table were filled with all different n-tuples of numbers
+1 and -1 . Then some of the numbers replaced by 0's.
Prove that one can choose a (nonempty) set of rows such that:
a) [4] the sum of all the numbers in all the
chosen rows is equal to 0;
b) [5] the sum of all the chosen rows is equal
to the zero row (the rows are added like vectors; in other words,
the sum of numbers in each column of the chosen rows is equal to
0).