MathDB

2

Part of 2013 239 Open Mathematical Olympiad

Problems(2)

kn = 1+2+...+n (mod 10^100)

Source: 239 2013 S2

8/7/2020
For some 9999-digit number kk, there exist two different 100100-digit numbers nn such that the sum of all natural numbers from 11 to nn ends in the same 100100 digits as the number knkn, but is not equal to it. Prove that k−3k-3 is divisible by 55.
number theory
Union of subsets

Source: 239 2013 J2

8/7/2020
In the set AA with nn elements, [2n]+2[\sqrt{2n}]+2 subsets are chosen such that the union of any three of them is equal to AA. Prove that the union of any two of them is equal to AA as well.