MathDB

4

Part of 2018 Korea Junior Math Olympiad

Problems(1)

Counting solutions of diophantine equation

Source: KJMO 2018 p4

7/26/2019
For a positive integer nn, denote p(n)p(n) to be the number of nonnegative integer tuples (x,y,z,w)(x,y,z,w) such that x+2y+2z+3w=nx+2y+2z+3w=n. Also, denote q(n)q(n) to be the number of nonnegative integer tuples (a,b,c,d)(a,b,c,d) such that
(i) a+b+c+d=na+b+c+d=n (ii) a≥b≥da \ge b \ge d (iii) a≥c≥da \ge c \ge d
Prove that for all nn, p(n)=q(n)p(n) = q(n).
combinatoricsnumber theoryDiophantine equation