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 [ ].
1a. [2] A right triangle made of paper is
folded along a straight line so that the vertex at the right
angle coincides with one of the other vertices. What is the ratio
into which the diagonals of the obtained quadrilateral divide
each other?
1b. [2] A right triangle of unit area made of
paper is folded along a straight line so that the vertex at the
right angle coincides with one of the other vertices. The
obtained quadrilateral is cut along the diagonal emanating from
the third vertex of the triangle. Find the area of the smallest
piece of paper thus obtained.
2. Triples of integers a,b,c
for which a+b+c=0 are considered. For
each such triple we calculate the integer
d = a1999 + b1999
+ c1999 .
a. [2] May it happen that d = 2 ?
b. [2] May it happen that d is prime? (An
integer greater than 1 is said to be prime if it has no divisors
other than itself and 1; here are the first prime numbers: 2, 3,
5, 7, 11, ... .)
3. [4] n straight lines are drawn in the plane so that each intersects exactly 1999 others. Find n. (Give all possible answers.)
4. [4] In Italy manufacturers produce watches in which the hour hand performs one revolution every 24 hours, while the minute hand performs 24 revolutions; the minute hand is longer than the hour hand. (In an ordinary watch the hour hand performs two revolutions every 24 hours, while the minute hand performs 24.) How many positions of the two hands and the zero notch can occur on the Italian watches in a 24 hour interval that are possible on the ordinary one? (The zero notch is the position marking 24 hours on an Italian watch and 12 hours on an ordinary one.)
5. [4] We are given 2 by 1 cardboard cards with a diagonal drawn on each. Thus there are two types of cards (since the diagonal can be drawn in two ways) and an unlimited supply of each type is available. Is it possible to choose 18 cards and arrange them in a 6 by 6 square so that the endpoints of the diagonals never coincide?
1. [4] The intersection point of the angle bisectors of a given triangle is joined to its vertices. As the result, the triangle is cut into three smaller triangles. One of these three triangles turned out to be similar to the given triangle. Find its angles.
2. [4] Prove that there exist infinitely many odd positive integers n for which the number 2n + n is composite. (A positive integer is called composite if it has divisors other than itself and 1.)
3. [4] n planes are drawn in space so that each intersects exactly 1999 others. Find n. (Give all possible answers)
4. [4] Is it possible to choose 50 intervals (possibly overlapping) on the number line so as to satisfy the following two conditions: (a) the lengths of the intervals are 1, 2, 3, ... , 50; (b) the endpoints of the intervals are all the integers from 1 to 100 inclusive?
5. [4] We are given 2 by 1 cardboard cards with a diagonal drawn on each. Thus there are two types of cards (since the dia- gonal can be drawn in two ways) and an unlimited supply of each type is available. Is it possible to choose 32 cards and arrange them in a 8 by 8 square so that the endpoints of the diagonals never coincide?