MathDB

5

Part of 2018 Tuymaada Olympiad

Problems(2)

Product divisble by p^3

Source: Tuymaada 2018 Senior League/Problem 5

7/20/2018
A prime pp and a positive integer nn are given. The product (13+1)(23+1)...((n−1)3+1)(n3+1)(1^3+1)(2^3+1)...((n-1)^3+1)(n^3+1) is divisible by p3p^3. Prove that p≤n+1p \leq n+1.
Proposed by Z. Luria
number theorydivisibleprime numbers
Spectrometer test tells if a ball is zinc or copper

Source: Tuymaada 2018 Junior League/Problem 5

7/20/2018
9999 identical balls lie on a table. 5050 balls are made of copper, and 4949 balls are made of zinc. The assistant numbered the balls. Once spectrometer test is applied to 22 balls and allows to determine whether they are made of the same metal or not. However, the results of the test can be obtained only the next day. What minimum number of tests is required to determine the material of each ball if all the tests should be performed today?
Proposed by N. Vlasova, S. Berlov
combinatoricsgraph theory