This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 6

1971 IMO Shortlist, 10
Prove that we can find an infinite set of positive integers of the from $2^n-3$ (where $n$ is a positive integer) every pair of which are relatively prime.
relatively prime sequence euler s theorem number theory construction modular arithmetic sequences

1985 IMO Longlists, 9
A polyhedron has $12$ faces and is such that: [b][i](i)[/i][/b] all faces are isosceles triangles, [b][i](ii)[/i][/b] all edges have length either $x$ or $y$, [b][i](iii)[/i][/b] at each vertex either $3$ or $6$ edges meet, and [b][i](iv)[/i][/b] all dihedral angles are equal. Find the ratio $x/y.$
polyhedron geometry 3d geometry euler s theorem angle solid geometry

1985 IMO Shortlist, 2
A polyhedron has $12$ faces and is such that: [b][i](i)[/i][/b] all faces are isosceles triangles, [b][i](ii)[/i][/b] all edges have length either $x$ or $y$, [b][i](iii)[/i][/b] at each vertex either $3$ or $6$ edges meet, and [b][i](iv)[/i][/b] all dihedral angles are equal. Find the ratio $x/y.$
polyhedron geometry 3d geometry euler s theorem angle solid geometry

1971 IMO Longlists, 35
Prove that we can find an infinite set of positive integers of the from $2^n-3$ (where $n$ is a positive integer) every pair of which are relatively prime.
relatively prime sequence euler s theorem number theory construction modular arithmetic sequences

1971 IMO, 3
Prove that we can find an infinite set of positive integers of the from $2^n-3$ (where $n$ is a positive integer) every pair of which are relatively prime.
relatively prime sequence euler s theorem number theory construction modular arithmetic sequences

2017 India PRMO, 28
Let $p,q$ be prime numbers such that $n^{3pq}-n$ is a multiple of $3pq$ for [b]all[/b] positive integers $n$. Find the least possible value of $p+q$.
fermat pseudo-primes carmichael s number euler s theorem number theory prime number Fermat's little theorem modular arithmetic parity prime numbers