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. [2] In an acute triangle, the smallest angle is 1/5 of the greatest angle and each angle contains an integral number of degrees. Find these angles.
2. Does there exist an integer n
such that all three numbers
a) [2] n-96, n, n+96;
b) [2] n-1996, n, n+1996
are (positive) prime?
3. [4] Let PQ be the projection of the inscribed circle of a right triangle ABC on its hypotenuse AB. Find the angle PCQ.
4. Consider closed broken lines of 6 links
with all 6 verticies lying on a circle.
a) [3] Draw such a line with the maximal
possible number of selfintersection points, and
b) [3] Prove that it cannot have more
selfintersection points.
5. Two persons play an (O-X)-game on
the squared 10×10 board according to the following rules. At
first they fill all 100 cells of the board with their signs
writing them in turn (the first who starts the game writes X's
and the other writes O's). Then they calculate two
numbers C and Z: C is the total number
of fives of consecutive X's standing on a line parallel
to the sides or diagonals of the board (so that 6 consecutive X's
add 2 points to C, 7 consecutive X's add 3
points etc.), and Z is the total number of fives of
zeroes. The first player wins if C>Z, loses
if C<Z and makes a draw if C=Z.
Does the first player have a strategy using which:
a) [3] he never loses;
b) [3] he always wins (no matter how his
opponent plays)?
1. [3] A hundred persons have answered the question: "Do you think that the new president will be better than the recent one?" Among them a persons said "better", b said "the same" and c said "worse". Sociologists calculate two estimations of 'social optimism': m = a + b/2 and n = a-c. Find n if m=40.
2. [3] Nine digits: 1, 2, 3, , 9 are written in arbitrary order (like a 9-digit number). Consider all triples of consecutive digits and find the sum of these seven 3-digit numbers. What is the maximal possible value of the sum?
3. [4] Consider the product of the following 100 multipliers: 1!, 2!, ... , 100!. Can one delete one of the multipliers so that the product of all the others will be a perfect square? (n! means the product 1·2·3· ... ·n; 1!=1.)
4. [4] Is it possible to cut the space into regular tetrahedrons and regular octahedrons? (All faces of these polyhedrons are regular triangles: 4 for a tetrahedron and 8 for an octahedron).
5. On the sides of a triangle ABC
outside it, the sqares ABMN, BCKL, ACPQ
are constructed. On the segments NQ and PK the
squares NQZT and PKXY are constructed. The
difference of the areas of the squares ABMN and BCKL
equals d. Find the difference of the areas of the
squares NQZT and PKXY
a) [3] if the angle ABC is right;
b) [2] for an arbitrary triangle ABC.