3

Part of 1973 Bundeswettbewerb Mathematik

Problems(2)

Fractional parts of square roots can be anything

Source: Bundeswettbewerb Mathematik 1973, round 1, problem 3

n

a_{1}, a_{2},..., a_{n}

in that order. Does there exist a positive integer such that the first

n

decimal digits after the dot of that number's square root are exactly those given digits¿ Give reason for your answer.

Tiling rectangular rooms with 2×2 and 4×1.

Source: Bundeswettbewerb Mathematik 1973, round 2, problem 3

For covering the floor of a rectangular room rectangular tiles of sizes

2 \times 2

4 \times 1

were used. Show that it's not possible to cover the floor if there is one plate less of one type and one more of the other type.

modular arithmeticcombinatorics proposedcombinatorics