MathDB

1

Part of 1997 Estonia National Olympiad

Problems(4)

a^k = a^l + a^m+a^n diophantine in N+

Source: 1997 Estonia National Olympiad Final Round grade 10 p1

3/13/2020
Find: a) Any quadruple of positive integers (a,k,l,m)(a, k, l, m) such that ak=al+am,a^k = a^l + a^m, b) Any quintuple of positive integers (a,k,l,m,n)(a, k, l, m, n) for which ak=al+am+ana^k = a^l + a^m+a^n
diophantineDiophantine equationnumber theory
2^n = 7x^2 + y^2 , diophantine

Source: 1997 Estonia National Olympiad Final Round grade 9 p1

3/13/2020
Prove that for every integer n≥3n\ge 3 there are such positives integers xx and yy such that 2n=7x2+y22^n = 7x^2 + y^2
number theorydiophantineDiophantine equation
composite number criterion, exist a,b,x,y \in N+ : a+b = n , x/a + y/b= 1

Source: 1997 Estonia National Olympiad Final Round grade 11 p1

3/11/2020
Prove that a positive integer nn is composite if and only if there exist positive integers a,b,x,ya,b,x,y such that a+b=na+b = n and xa+yb=1\frac{x}{a}+\frac{y}{b}= 1.
number theoryCompositeprime
T(m,n) = gcd (m, n / gcd(m,n))

Source: 1997 Estonia National Olympiad Final Round grade 12 p1

3/11/2020
For positive integers mm and nn we define T(m,n)=gcd(m,ngcd(m,n))T(m,n) = gcd \left(m, \frac{n}{gcd(m,n)} \right) (a) Prove that there are infinitely many pairs (m,n)(m,n) of positive integers for which T(m,n)>1T(m,n) > 1 and T(n,m)>1T(n,m) > 1. (b) Do there exist positive integers m,nm,n such that T(m,n)=T(n,m)>1T(m,n) = T(n,m) > 1?
number theorygreatest common divisorinequalities