MathDB

1

Part of 2015 Korea National Olympiad

Problems(2)

Pell's Equation Looking Equation

Source: 2015 Korean Mathematical Olmpiad P1, 2015 Korean Junior MO P2

11/1/2015
For a positive integer mm, prove that the number of pairs of positive integers (x,y)(x,y) which satisfies the following two conditions is even or 00.
(i): x2−3y2+2=16mx^2-3y^2+2=16m
(ii): 2y≤x−12y \le x-1
number theory
Familiar Functional Equation

Source: 2015 Korean Mathematical Olympiad P5

11/1/2015
Find all functions f:R→Rf: \mathbb{R} \rightarrow \mathbb{R} such that for all reals x,y,zx,y,z, we have (f(x)+1)(f(y)+f(z))=f(xy+z)+f(xz−y)(f(x)+1)(f(y)+f(z))=f(xy+z)+f(xz-y)
functionalgebrafunctional equation