MathDB

2

Part of 2013 Danube Mathematical Competition

Problems(2)

a+b-c-d be a multiple of 2013, 6 numbers from a set of 63 < = 2012

Source: Danube 2013 junior p2

7/22/2019
Consider 6464 distinct natural numbers, at most equal to 20122012. Show that it is possible to choose four of them, denoted as a,b,c,da,b,c,d such that a+b−c−d a+b-c-d to be a multiple of 20132013
number theorycombinatoricsmultipleSum
p divides a^2+ab+b^2 and a^n+b^n+c^n, but not a+b+c, prove n, p-1 not coprime

Source: Danube 2013 p2

7/22/2019
Let a,b,c,na, b, c, n be four integers, where n≥2\ge 2, and let pp be a prime dividing both a2+ab+b2a^2+ab+b^2 and an+bn+cna^n+b^n+c^n, but not a+b+ca+b+c. for instance, a≡b≡−1(mod  3),c≡1(mod  3),na \equiv b \equiv -1 (mod \,\, 3), c \equiv 1 (mod \,\, 3), n a positive even integer, and p=3p = 3 or a=4,b=7,c=−13,n=5a = 4, b = 7, c = -13, n = 5, and p=31p = 31 satisfy these conditions. Show that nn and p−1p - 1 are not coprime.
number theorycoprimeprimedivides