MathDB

4

Part of ICMC 3

Problems(2)

ICMC 2019/20 Round 1, Problem 4

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

8/7/2020
Let n be a non-negative integer. Define the decimal digit product D(n)D(n) inductively as follows:
- If nn has a single decimal digit, then let D(n)=nD(n) = n.
- Otherwise let D(n)=D(m)D(n) = D(m), where mm is the product of the decimal digits of nn.
Let Pk(1)P_k(1) be the probability that D(i)=1D(i) = 1 where ii is chosen uniformly randomly from the set of integers between 1 and kk (inclusive) whose decimal digit products are not 0.
Compute lim⁡k→∞Pk(1)\displaystyle\lim_{k\to\infty} P_k(1).
proposed by the ICMC Problem Committee
college contests
ICMC 2019/20 Round 2, Problem 4

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

8/7/2020
Let S={S1,S2,…,Sn}\mathcal{S}=\left\{S_1,S_2,\ldots,S_n\right\} be a set of n≥2020n\geq 2020 distinct points on the Euclidean plane, no three of which are collinear. Andy the ant starts at some point Si1S_{i_1} in S\mathcal{S} and wishes to visit a series of 2020 points {Si1,Si2,…,Si2020}⊆S\left\{S_{i_1},S_{i_2},\ldots,S_{i_{2020}}\right\}\subseteq\mathcal{S} in order, such that ij>iki_j>i_k whenever j>kj>k. It is known that ants can only travel between points in S\mathcal{S} in straight lines, and that an ant's path can never self-intersect.
Find a positive integer nn such that Andy can always fulfill his wish.
(Lower n will be awarded more marks. Bounds for this problem may be used as a tie-breaker, should the need to do so arise.)
Proposed by the ICMC Problem Committee
college contests