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] Numbers are written in all 16 squares of a 4 by 4 table so that the sum of neighbors of each number is equal to 1 (neighboring squares are those with a common side). Find the sum of all the numbers in the table.
2. [3] ABCD is a parallelogram, M is the midpoint of side CD, H is the foot of the perpendicular from B to line AM. Prove that triangle BCH is isosceles.
3a. [2] 100 different numbers are written on the blackboard. Prove that you can choose 8 of them so that their arithmetical mean cannot be represented as the arithmetical mean of 9 of the written numbers.
3b. [4] 100 integers are written on the blackboard. It is known that for any 8 numbers one can find 9 numbers so that the arithmetical mean of the 8 numbers is equal to that of the 9. Prove that all the numbers are equal.
4. [5] It is known that among a set of 32 coins of similar appearance two are counterfeit. The counterfeit coins differ from the true ones by weight. (The true coins all have the same weight, the weights of the two counterfeit coins are equal to each other; it is not given whether the counterfeit coins are heavier or lighter than the real ones.) How can one divide the coins into two groups of equal total weight by performing no more than 4 weighting operations with a balance (the balance only shows which of two groups of coins is heavier or shows that they have the same total weight).
1. [3] Triangle ABC is inscribed in a circle. Chords are drawn from the point A, intersecting side BC at points K and L and the arc BC at points M and N. Prove that if a circle can be circumscribed about quadrangle KLNM, then triangle ABC was isosceles.
2. [4] Positive integers a,b,c,d satisfy the inequality ad-bc >1. Prove that at least one of the numbers a,b,c,d is not divisible by ad-bc.
3. [4] In each lateral face of a pentagonal prism at least one of the four angles is equal to F. Find all possible values of F. (The top face and bottom face have to be parallel and congruent, and the lateral faces are parallelograms.)
4. It is known that among a set of
(a [3]) 32
(b [2]) 22
coins of similar appearance two are counterfeit. The counterfeit coins differ from the true ones by weight. (The true coins all have the same weight, the weights of the two counterfeit coins are equal to each other; it is not given whether the counterfeit coins are heavier or lighter than the real ones.) How can one divide the coins into two groups of equal total weight by performing no more than 4 weighting operations with a balance (the balance only shows which of two groups of coins is heavier or shows that they have the same total weight).