MathDB

2

Part of 2014 JBMO TST - Turkey

Problems(2)

3m balls distributed into 8 boxes

Source: Turkey JBMO TST 2014 P2

6/21/2014
3m3m balls numbered 1,1,1,2,2,2,3,3,3,…,m,m,m1, 1, 1, 2, 2, 2, 3, 3, 3, \ldots, m, m, m are distributed into 88 boxes so that any two boxes contain identical balls. Find the minimal possible value of mm.
ceiling functioncombinatorics proposedcombinatorics
(a^3+b)(b^3+a)=2^c

Source: Turkey JBMO TST 2014 P6

6/21/2014
Find all triples of positive integers (a,b,c)(a, b, c) satisfying (a3+b)(b3+a)=2c(a^3+b)(b^3+a)=2^c.
modular arithmeticnumber theory proposednumber theory