International mathematical olympiad “Formula of Unity” / “The Third ...
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International mathematical olympiad “Formula of Unity” / “The Third ...
International mathematical olympiad. “Formula of Unity” / “The ... Also there are
twenty book covers with prices from 10 cents to 1 dollar in this shop. There no two
...
International mathematical olympiad “Formula of Unity” / “The Third Millenium” 2013/2014 year 1st round, grade R7* 1. There is a pile of identical cards, each card contains numbers from 1 to 9. Bill took one card and secretly marked 4 numbers on it. Mark can do the same operation with several of remaining cards. After that, boys show their cards to each other. Mark wins if he has a card where at least two marked numbers coincide with Bill’s numbers. Find the smallest number of cards Mark should use to win the game and find the way to fill them. 2. There are ten candles on a round cake. After four cuts, the cake was divided into 10 parts with exactly one candle on each of them. How many candles could be on each of two parts obtained after the first cut? You should find all possibilities and prove that there are no other variants. 3. A magician has 7 pink cards and 7 blue cards. Numbers from 0 to 6 are written on pink cards. There is number 1 on the first blue card, and on each next blue card the number is 7 times bigger than on the previous one. The magician puts his cards by pairs (each blue card with a pink one). Then spectators multiply numbers in each pair and sum up all the products. The sense of trick is to obtain the prime number as the result. Find the way for the magician to arrange the cards for this trick, or prove that there is no way to do it. 4. John has six dice. Their faces are painted into six colours (each face has one colour). All dice are painted in the same way. John made a column from all the dice and looked at it from four sides. Could he make such a column that, while looking from each side, all the faces have different colours? 5. Numbers from 1 to 77 are written in a circle in some order. Find the smallest possible value of the sum of absolute values of differences between each two adjacent numbers. 6. In a bookstore, there are twenty books with prices ranging from 7 to 10 dollars. Also there are twenty book covers with prices from 10 cents to 1 dollar in this shop. There no two items which cost the same. Is it always possible to buy two books with book covers paying for each book with the cover the same amount?
