MathDB

4

Part of 2017 All-Russian Olympiad

Problems(3)

Relatively prime sums

Source: All Russian Olympiad 2017,Day1,grade 9,P4

5/3/2017
Are there infinite increasing sequence of natural numbers, such that sum of every 2 different numbers are relatively prime with sum of every 3 different numbers?
number theoryrelatively prime
Table with cells

Source: All Russian Olympiad 2017,Day2,grade 9,P8

5/3/2017
Every cell of 100×100100\times 100 table is colored black or white. Every cell on table border is black. It is known, that in every 2×22\times 2 square there are cells of two colors. Prove, that exist 2×22\times 2 square that is colored in chess order.
combinatorics
Magic trick

Source: All Russian Olympiad 2017,Day1,grade 11,P4

5/1/2017
Magicman and his helper want to do some magic trick. They have special card desk. Back of all cards is common color and face is one of 20172017 colors. Magic trick: magicman go away from scene. Then viewers should put on the table n>1n>1 cards in the row face up. Helper looks at these cards, then he turn all cards face down, except one, without changing order in row. Then magicman returns on the scene, looks at cards, then show on the one card, that lays face down and names it face color. What is minimal nn such that magicman and his helper can has strategy to make magic trick successfully?
combinatorics