MathDB

7

Part of 2011 AIME Problems

Problems(2)

Nonnegative integer solutions

Source:

3/18/2011
Find the number of positive integers mm for which there exist nonnegative integers x0,x1,…,x2011x_0,x_1,\ldots,x_{2011} such that mx0=∑k=12011mxk. m^{x_0}=\sum_{k=1}^{2011}m^{x_k}.
modular arithmeticalgebrapolynomialfunctionAIMEnumber theory
Green and red marbles [2011.II.7]

Source:

3/31/2011
Ed has five identical green marbles and a large supply of identical red marbles. He arranges the green marbles and some of the red marbles in a row and finds that the number of marbles whose right hand neighbor is the same color as themselves equals the number of marbles whose right hand neighbor is the other color. An example of such an arrangement is GGRRRGGRG. Let mm be the maximum number of red marbles for which Ed can make such an arrangement, and let NN be the number of ways in which Ed can arrange the m+5m+5 marbles to satisfy the requirement. Find the remainder when NN is divided by 1000.
AMC