MathDB

4

Part of 1961 All-Soviet Union Olympiad

Problems(3)

Stars in a 4x4 table

Source: 1961 All-Soviet Union Olympiad

8/4/2015
We are given a 4×44\times 4 table. a) Place 77 stars in the cells in such a way that the erasing of any two rows and two columns will leave at least one of the stars. b) Prove that if there are less than 77 stars, you can always find two columns and two rows such that erasing them, no star remains in the table.
combinatorics
Linear combination divisible by p

Source: 1961 All-Soviet Union Olympiad

8/4/2015
Given are arbitrary integers a,b,pa,b,p. Prove that there always exist relatively prime integers kk and ℓ\ell such that ak+bℓak+b\ell is divisible by pp.
number theoryrelatively prime
Maximum in equilateral triangle

Source: 1961 All-Soviet Union Olympiad

8/4/2015
Point PP and equilateral triangle ABCABC satisfy ∣AP∣=2|AP|=2, ∣BP∣=3|BP|=3. Maximize ∣CP∣|CP|.
geometryEquilateral Trianglemaximum value