Contest Corner Solutions: CC301–CC310

[摘要] All natural numbers are coloured using 100 di erent colours. Provethat you cannd several (no less than 2) di erent numbers, all of the same colour,that have a product with exactly 1000 di erent natural divisors.Originally 2017 Savin Open Contest, Problem 7 (by E. Bakaev).We received two solutions. We present the solution by Richard Hess.Consider the set of numbers p91; p92; : : : ; p9n, where each pi is a distinct prime andn 201. If we colour each number in this set with any of 100 colours, then bythe pigeonhole principle there will be at least three numbers with the same colour.The product of the three numbers has exactly 1000 natural divisors.

[学科分类] 数学（综合）