MathDB

1

Part of 1990 Romania Team Selection Test

Problems(2)

set $\{k | f(k) < k\}$ is finite, the set $\{k | g(f(k)) \le k\}$ is infinite

Source: Romania BMO TST 1990 p1

2/19/2020
Let f:N→Nf : N \to N be a function such that the set {k∣f(k)<k}\{k | f(k) < k\} is finite. Prove that the set {k∣g(f(k))≤k}\{k | g(f(k)) \le k\} is infinite for all functions g:N→Ng : N \to N.
finite setInfinite setfunctionalgebra
[x/a]+[y/b]=[a^{n-1}/b]+[b^{n-1}/a]

Source: Romania IMO TST 1990 p1

2/19/2020
Let a,b,n be positive integers such that (a,b)=1(a,b) = 1. Prove that if (x,y)(x,y) is a solution of the equation ax+by=an+bnax+by = a^n + b^n then [xb]+[ya]=[an−1b]+[bn−1a]\left[\frac{x}{b}\right]+\left[\frac{y}{a}\right]=\left[\frac{a^{n-1}}{b}\right]+\left[\frac{b^{n-1}}{a}\right]
floor functionnumber theory