6

Part of 2013 AIME Problems

Problems(2)

Melinda's Math Textbooks and Boxes

Source: 2013 AIME I Problem 6

Melinda has three empty boxes and

12

textbooks, three of which are mathematics textbooks. One box will hold any three of her textbooks, one will hold any four of her textbooks, and one will hold any five of her textbooks. If Melinda packs her textbooks into these boxes in random order, the probability that all three mathematics textbooks end up in the same box can be written as

\frac{m}{n}

m

n

Are relatively prime positive integers. Find

m+n

probabilityMATHCOUNTSAMCAIMEnumber theoryrelatively prime#6

1000N, 1000N+1, ..., 1000N+999 has no squares

Source: AIME II 2013, Problem 6

Find the least positive integer

N

such that the set of

1000

consecutive integers beginning with

1000 \cdot N

contains no square of an integer.

modular arithmeticAMCnumber theoryPerfect SquaresAIME