MathDB

Problem 3

Part of 2002 Federal Math Competition of S&M

Problems(3)

n choose k=2002

Source: S&M 2002 1st Grade P3

5/14/2021
Find all pairs (n,k)(n,k) of positive integers such that (nk)=2002\binom nk=2002.
number theoryDiophantine equation
(2^m-1)^2|2n-1 iff m(2^m-1)|n

Source: S&M 2002 2nd Grade P3

5/14/2021
Let mm and nn be positive integers. Prove that the number 2n−12n-1 is divisible by (2m−1)2(2^m-1)^2 if and only if nn is divisible by m(2m−1)m(2^m-1).
number theoryDivisibility
Nice geometric inequality

Source: Yugoslavia 2002

8/20/2005
Let ABCD ABCD be a rhombus with \angle BAD \equal{} 60^{\circ}. Points S S and R R are chosen inside the triangles ABD ABD and DBC DBC, respectively, such that \angle SBR \equal{} \angle RDS \equal{} 60^{\circ}. Prove that SR2≥AS⋅CR SR^2\geq AS\cdot CR.
inequalitiesgeometryrhombustrigonometrycircumcirclegeometric transformationgeometry proposed