MathDB

1

Part of 1999 Estonia National Olympiad

Problems(4)

for odd prime p, p^2(p^2 -1999) is divisible by 6 but not by 12

Source: 1999 Estonia National Olympiad Final Round grade 9 p1

3/13/2020
Prove that if pp is an odd prime, then p2(p2−1999)p^2(p^2 -1999) is divisible by 66 but not by 1212.
divisibleprimesoddnumber theory
(m - n)^2 =4mn/(m + n - 1) diophantine

Source: 1999 Estonia National Olympiad Final Round grade 11 p1

3/11/2020
Find all pairs of integers (m,n)(m, n) such that (m−n)2=4mnm+n−1(m - n)^2 =\frac{4mn}{m + n - 1}
diophantineDiophantine equationnumber theory
a^2 + b = b^{1999} diophantine

Source: 1999 Estonia National Olympiad Final Round grade 10 p1

3/11/2020
Find all pairs of integers (a,ba, b) such that a2+b=b1999a^2 + b = b^{1999} .
number theorydiophantineDiophantine equation
2^a7^b=2^c7^d (mod 15) iff 3^a5^b =3^c5^d (mod 16)

Source: 1999 Estonia National Olympiad Final Round grade 12 p1

3/11/2020
Let a,b,ca, b, c and dd be non-negative integers. Prove that the numbers 2a7b2^a7^b and 2c7d2^c7^d give the same remainder when divided by 1515 iff the numbers 3a5b3^a5^b and 3c5d3^c5^d give the same remainder when divided by 1616.
number theoryremainderpower of 2power of 3