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 for any two natural numbers a and b the equality LCM(a,a+5)=LCM(b,b+5) implies that a=b.
2. [4] John and Mary each have an 8 by 8 square divided into 1 by 1 cells. They have painted an equal number of cells on their respective squares in blue. Prove that one can cut up the two squares into 2 by 1 dominos in such a way that it then possible to reassemble John's dominos, as well as Mary's, into two new squares with the same pattern of blue cells.
3. [5] Segment AB intersects two equal circles, is parallel to the line joining their centers, and all the intersection points of the segment and the circles lie between A and B. From the point A tangents to the circle nearest to A were drawn, from the point B, tangents to the circle nearest to B. It turned out that the quadrilateral formed by the four tangents contains both circles. Prove that it is circumscribed.
4. [6] All the diagonals of a regular 25-gon are drawn. Prove that there are no 9 diagonals among them passing through the same interior point of the 25-gon.
5. [7] There were 20 beads of 10 colors, two of each color. They were placed in 10 boxes. It is known that one bead may be chosen from each of the boxes so that each color is represented. Prove that the number of such choices is a nonzero power of 2.
6. [7] A gang of robbers took away a bag of coins from a rich merchant. Each coin is worth an integer number of pennies. It is known that if any single coin is removed from the bag, then the remaining coins can be divided fairly among the robbers (i.e., so that they all get the same worth of coins in pennies). Prove that after one coin is removed, the number of coins remaining is divisible by the number of robbers.
1a. [2] Prove that if LCM(a,a+5)=LCM(b,b+5),
where a and b are natural numbers, then a=b.
1b. [3] Can we have LCM(a,b)=LCM(a+c,b+c),
where a,b, and c are natural numbers?
2. [4] Segment AB intersects two equal circles, is parallel to the line joining their centers, and all the intersection points of the segment and the circles lie between A and B. From the point A tangents to the circle nearest to A were drawn, from the point B, tangents to the circle nearest to B. It turned out that the quadrilateral formed by the four tangents contains both circles. Prove that it is circumscribed.
3. [5] Nine numbers are arranged in a square
table: a1, a2, a3,
b1, b2, b3,
c1, c2, c3.
It is known that the six numbers obtained by summing the rows and
columns of the table are equal: a1+a2+a3=b1+b2+b3=c1+c2+c3=a1+b1+c1=a2+b2+c2=a3+b3+c3.
Prove that the sum of products of numbers in the rows is equal to
the sum of products of numbers in the columns:
a1·b1·c1+a2·b2·c2+a3·b3·c3=a1·a2·a3+b1·b2·b3+c1·c2·c3.
4. [6] Twelve places were prepared at a round table for members of the Jury, with name tags at each place. Professor K., being absent-minded, instead of occupying his place, sat down at the next place (clockwise). All the other Jury members in succession either occupied their prescribed place or, if it was already occupied, sat down at the first free place in clockwise order. The resulting seating arrangement depends on the order in which the Jury members come to the table. How many different seating arrangements are possible?
5. [7] The sum of the length, width, and height of a rectangular parallelipiped will be called its "size". Can it happen that one rectangular parallelipiped contains another one of greater size?
6. [8] The function f(x)=(x2+ax+b)/(x2+cx+d), where the trinomials
x2+ax+b and x2+cx+d
have no common roots, is given. Prove that the next two
statements are equivalent:
(i) there is a numerical interval without any values of f(x);
(ii) f(x) can be represented in the form f(x)=f1(f2(...fn-1(fn(x)...))
where each of the functions fj(x)
has one of the three forms kjx+bj,
1/x, x2.