5

Part of 2021 May Olympiad

Problems(2)

100 numbers and 1 perfect cube

Source: May Olympiad L2 #5

Prove that there are

100

distinct positive integers

n_1,n_2,\dots,n_{99},n_{100}

\frac{n_1^3+n_2 ^3+\dots +n_{100}^3}{100}

is a perfect cube.

Sum of 16 < Sum of 10

Source: May Olympiad 2021 #5 L1

36

consecutive positive integers in a white paper(in ascending order), next he computes the sum of digits of each one of

36

numbers(in the order) and writes the first

16

results in a red paper and the last

10

results in a blue paper. Determine if Bob can choose the

36

integers, such that the sum of the numbers in the red paper is less than or equal to sum of the numbers in the blue paper.