MathDB

2

Part of 2016 Dutch IMO TST

Problems(3)

Nice problem

Source: Netherlands Team Selection Test 2016 Day 1-Problem 2

9/22/2016
In a 2n×2n2^n \times 2^n square with nn positive integer is covered with at least two non-overlapping rectangle pieces with integer dimensions and a power of two as surface. Prove that two rectangles of the covering have the same dimensions (Two rectangles have the same dimensions as they have the same width and the same height, wherein they, not allowed to be rotated.)
combinatoricsgeometryrectangle
(a^n + b^n + 1) is divisible by d for all positive integers n

Source: Dutch IMO TST 2016 p2

8/30/2019
Determine all pairs (a,b)(a, b) of integers having the following property: there is an integer d≥2d \ge 2 such that an+bn+1a^n + b^n + 1 is divisible by dd for all positive integers nn.
number theorydivisibledivisorexponential
sorting sums (a_i +a_j) in ascending order, arithmetic progression when?

Source: Dutch IMO TST 2016 day 3 p2

8/30/2019
For distinct real numbers a1,a2,...,ana_1,a_2,...,a_n, we calculate the n(n−1)2\frac{n(n-1)}{2} sums ai+aja_i +a_j with 1≤i<j≤n1 \le i < j \le n, and sort them in ascending order. Find all integers n≥3n \ge 3 for which there exist a1,a2,...,ana_1,a_2,...,a_n, for which this sequence of n(n−1)2\frac{n(n-1)}{2} sums form an arithmetic progression (i.e. the di erence between consecutive terms is constant).
arithmetic sequencealgebrasums