1. [3] Does there exist a set of 10 natural numbers, none of which is divisible by any of the others, but such that the square of each number is divisible by all of the others?
2. [3] ABCD is a parallelogram. Point M is found on side AB or its extension such that angle MAD = angle AMO (O is the point of intersection of the diagonals of the parallelogram). Prove that MD = MC.
3. [4] Six dice are strung on a rigid wire (the wire passes through two opposite faces of each die). Each die can be rotated independently of the others. Prove that it is always possible to rotate the dice and then place the wire horizontally on a table in such a way that the six-digit number formed by their top faces is divisible by 7. (The faces of a die are numbered from 1 to 6; the sum of the numbers on opposite faces is always equal to 7.)
4. [4] A traveller visited a village whose inhabitants either always tell the truth, or always lie. The villagers stood in a circle, facing inward, and each villager announced whether the person standing to his right is a truth-teller. On the basis of this information, the traveller was able to determine what fraction of the villagers were liars. What was this fraction?
5. [7] A square is divided into 25 small squares. We draw diagonals of some of the small squares such that no two diagonals share a common point (not even a common endpoint). What is the largest possible number of diagonals that we can draw?
6. [8] 10 people are sitting at a round table. There are some nuts in front of each of them, 100 nuts in all. After a certain signal each person passes some of his nuts to the person sitting to his right. If he has an even number of nuts, he passes half of them; otherwise, he passes one nut plus half of the remaining nuts. This procedure is repeated over and over again. Prove that eventually everyone has exactly 10 nuts.
1. [4] Prove the inequality:
a3/(a2+ab+b2)
+ b3/(b2+bc+c2)
+ c3/(c2+ca+a2)
> (a+b+c)/3
(a, b, c are positive).
2. [4] A square of side 1 is divided into rectangles. We choose one of the two smaller sides of each rectangle (if the rectangle is a square, then we can choose any of the four sides). Prove that the sum of the lengths of all the chosen sides is at least 1.
3. a) [2] The numbers 1, 2, 4, 8, 16, 32, 64,
128 are written on a blackboard. We are allowed to erase any two
numbers and write their difference instead (this is always a
nonnegative number). After 7 repetitions of this procedure only a
single number will remain. Could this number be 97?
b) [3] The numbers 1, 21, 22,
23, ..., 210 are written on a blackboard.
We are allowed to erase any two numbers and write their
difference instead (this is always a nonnegative number). After
several repetitions of this procedure only a single number will
remain. What values could this number have?
4. [5] ABCD is a quadrilateral. Point M is found such that triangles AMB and CMD are isoceles (AM = MB, CM = MD) and angle AMB = angle CMD = 120o. Prove that there exists a point N such that triangles BNC and DNA are equilateral.
5. [6] A "labyrinth" is an 8x8 chessboard with barriers between some neighboring squares. If a rook can traverse the entire board without jumping over any barriers, the labyrinth is "good"; otherwise, it is "bad". Are there more good labyrinths or more bad labyrinths?
6. a) [6] Two people perform a card trick.
The first performer takes 5 cards from a 52-card deck (previously
shuffled by a member of the audience), looks at them, and
arranges them in a row from left to right: one face down (not
necessarily the first one), the others face up. The second
performer guesses the card which is face down. Prove that the
performers can agree on a system which always makes this possible.
b) [6] For their second trick, the first
performer arranges four cards in a row, face up; the fifth card
is kept hidden. Can they still agree on a system which enables
the second performer to guess the hidden card?