11

Part of 2009 AIME Problems

Problems(2)

Triangles with integer area

Source: 2009 AIME I #11

Consider the set of all triangles

OPQ

O

is the origin and

P

Q

are distinct points in the plane with nonnegative integer coordinates

(x,y)

such that 41x\plus{}y \equal{} 2009. Find the number of such distinct triangles whose area is a positive integer.

geometryanalytic geometryvectorcalculusintegrationabsolute valueAMC

Log Difference Range

Source: AIME 2009II Problem 11

For certain pairs

(m,n)

of positive integers with

m\ge n

there are exactly

50

distinct positive integers

k

such that |\log m \minus{} \log k| < \log n. Find the sum of all possible values of the product

mn