MathDB

Problem 1

Part of 1997 Slovenia National Olympiad

Problems(3)

k=m+2mn+n iff 2k+1 composite (Slovenia 1997 1st Grade P1)

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5/2/2021
Let kk be a positive integer. Prove that: (a) If k=m+2mn+nk=m+2mn+n for some positive integers m,nm,n, then 2k+12k+1 is composite. (b) If 2k+12k+1 is composite, then there exist positive integers m,nm,n such that k=m+2mn+nk=m+2mn+n.
number theory
m+n-1|m^2+n^2-1 implies m+n-1 composite (Slovenia 1997 3rd Grade P1)

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5/3/2021
Suppose that m,nm,n are integers greater than 11 such that m+n−1m+n-1 divides m2+n2−1m^2+n^2-1. Prove that m+n−1m+n-1 cannot be a prime number.
number theoryDivisibility
p+q-pq=154, p,q prime (Slovenia 1997 4th Grade P1)

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5/3/2021
Marko chose two prime numbers aa and bb with the same number of digits and wrote them down one after another, thus obtaining a number cc. When he decreased cc by the product of aa and bb, he got the result 154154. Determine the number cc.
Diophantine equationnumber theory