MathDB

4

Part of 2015 Middle European Mathematical Olympiad

Problems(2)

For which m,n is (a^m+b^m) / (a^n+b^n) an integer

Source: MEMO 2015, problem I-4.

8/27/2015
Find all pairs of positive integers (m,n)(m,n) for which there exist relatively prime integers aa and bb greater than 11 such that am+bman+bn\frac{a^m+b^m}{a^n+b^n} is an integer.
number theoryrelatively primeExponentsfactorizationEuclidean algorithm
Number of regions in NxN board with diagonals

Source: MEMO 2015, problem T-4

8/28/2015
Let NN be a positive integer. In each of the N2N^2 unit squares of an N×NN\times N board, one of the two diagonals is drawn. The drawn diagonals divide the N×NN\times N board into KK regions. For each NN, determine the smallest and the largest possible values of KK.
[asy] draw((0,0)--(3,0)--(3,3)--(0,3)--cycle); draw((1,0)--(1,3), dotted); draw((2,0)--(2,3), dotted); draw((0,1)--(3,1), dotted); draw((0,2)--(3,2), dotted); draw((1,0)--(0,1)--(2,3)--(3,2)--(2,1)--(0,3)); draw((1,1)--(2,0)--(3,1)); label("11",(0.35,2)); label("22",(1,2.65)); label("33",(2,2)); label("44",(2.65,2.65)); label("55",(0.35,0.35)); label("66",(1.3,1.3)); label("77",(2.65,0.35)); label("Example with N=3N=3, K=7K=7",(0,-0.3)--(3,-0.3),S); [/asy]
combinatorics