MathDB

11

Part of 2013 AIME Problems

Problems(2)

Ms. Math's Playing Blocks

Source: 2013 AIME I Problem 11

3/15/2013
Ms. Math's kindergarten class has 1616 registered students. The classroom has a very large number, NN, of play blocks which satisfies the conditions:
(a) If 1616, 1515, or 1414 students are present, then in each case all the blocks can be distributed in equal numbers to each student, and (b) There are three integers 0<x<y<z<140 < x < y < z < 14 such that when xx, yy, or zz students are present and the blocks are distributed in equal numbers to each student, there are exactly three blocks left over.
Find the sum of the distinct prime divisors of the least possible value of NN satisfying the above conditions.
modular arithmeticAIME
Number of $f$ for which $f(f(x))$ is constant

Source: AIME II 2013, Problem 11

4/4/2013
Let A={1,2,3,4,5,6,7}A = \left\{ 1,2,3,4,5,6,7 \right\} and let NN be the number of functions ff from set AA to set AA such that f(f(x))f(f(x)) is a constant function. Find the remainder when NN is divided by 10001000.
functionAMCAIMEAIME IIcombinatorics