MathDB

1

Part of 2000 Tuymaada Olympiad

Problems(3)

Number of positive divisors and integer part.

Source: Tuymaada 2000, day 1, problem 1.

4/30/2007
Let d(n)d(n) denote the number of positive divisors of nn and let e(n)=[2000n]e(n)=\left[2000\over n\right] for positive integer nn. Prove that d(1)+d(2)+⋯+d(2000)=e(1)+e(2)+⋯+e(2000).d(1)+d(2)+\dots+d(2000)=e(1)+e(2)+\dots+e(2000).
number theory proposednumber theory
colouring the plane

Source: Tuymaada 2000

7/2/2004
Can the plane be coloured in 2000 colours so that any nondegenerate circle contains points of all 2000 colors?
analytic geometryLaTeXtopologyset theorycombinatorics solvedcombinatorics
erasing digits from 188188...188, largest multiple of 7

Source: Tuymaada 2000 Juniors 1

4/28/2019
Given the number 188188...188188188...188 (number 188188 is written 101101 times). Some digits of this number are crossed out. What is the largest multiple of 77, that could happen?
number theoryDigitsmultiplemaximum