MathDB

2

Part of ICMC 3

Problems(2)

ICMC 2019/20 Round 1, Problem 2

Source: Imperial College Mathematics Competition 2019/20 - Round 1

8/7/2020
Find integers aa and bb such that
ab=30(20200)−31(20202)+32(20204)−⋯+31010(20202020).a^b=3^0\binom{2020}{0}-3^1\binom{2020}{2}+3^2\binom{2020}{4}-\cdots+3^{1010}\binom{2020}{2020}.
proposed by the ICMC Problem Committee
college contestscomplex numbersSumroots of unity
ICMC 2019/20 Round 2, Problem 2

Source: Imperial College Mathematics Competition 2019/20 - Round 2

8/7/2020
Let R2\mathbb{R}^2 denote the set of points in the Euclidean plane. For points A,P∈R2A,P\in\mathbb{R}^2 and a real number kk, define the dilation of AA about PP by a factor of kk as the point P+k(A−P)P+k(A-P). Call a sequence of point A0,A1,A2,…∈R2A_0, A_1, A_2,\ldots\in\mathbb{R}^2 unbounded if the sequence of lengths ∣A0−A0∣,∣A1−A0∣,∣A2−A0∣,…\left|A_0-A_0\right|,\left|A_1-A_0\right|,\left|A_2-A_0\right|,\ldots has no upper bound. Now consider nn distinct points P0,P1,…,Pn−1∈R2P_0,P_1,\ldots,P_{n-1}\in\mathbb{R}^2, and fix a real number rr. Given a starting point A0∈R2A_0\in\mathbb{R}^2, iteratively define Ai+1A_{i+1} by dilating AiA_i about PjP_j by a factor of rr, where jj is the remainder of ii when divided by nn.
Prove that if ∣r∣≥1\left|r\right|\geq 1, then for any starting point A0∈R2A_0\in\mathbb{R}^2, the sequence A0,A1,A2,…A_0,A_1,A_2,\ldots is either periodic or unbounded.
Proposed by the ICMC Problem Committee
college contestsgeometry