MathDB

ICMC 3

Part of ICMC

Subcontests

(6)
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ICMC 2019/20 Round 1, Problem 4

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

ICMC 2019/20 Round 2, Problem 4

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
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ICMC 2019/20 Round 1, Problem 3

Consider a grid of points where each point is coloured either white or black, such that no two rows have the same sequence of colours and no two columns have the same sequence of colours. Let a table denote four points on the grid that form the vertices of a rectangle with sides parallel to those of the grid. A table is called balanced if one diagonal pair of points are coloured white and the other diagonal pair black.
Determine all possible values of k≥2k \geq 2 for which there exists a colouring of a k×2019k\times 2019 grid with no balanced tables.
proposed by the ICMC Problem Committee

ICMC 2019/20 Round 2, Problem 3

Let R\mathbb{R} denote the set of real numbers. A subset S⊆RS\subseteq\mathbb{R} is called dense if any non-empty open interval of R\mathbb{R} contains at least one element in SS. For a function f:R→Rf:\mathbb{R}\to\mathbb{R}, let Of(x)\mathcal{O}_f(x) denote the set {x,f(x),f(f(x)),…}\left\{x,f(x),f(f(x)),\ldots\right\}.
(a) Is there a function g:R→Rg:\mathbb{R}\to\mathbb{R}, continuous everywhere in R\mathbb{R} such that Og(x)\mathcal{O}_g(x) is dense for all x∈Rx\in\mathbb{R} for all x∈Rx\in\mathbb{R}?
(b) Is there a function h:R→Rh:\mathbb{R}\to\mathbb{R}, continuous at all but a single x0∈Rx_0\in\mathbb{R}, such that Oh(x)\mathcal{O}_h(x) is dense for all x∈Rx\in\mathbb{R}?
Proposed by the ICMC Problem Committee
2
2

ICMC 2019/20 Round 1, Problem 2

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

ICMC 2019/20 Round 2, Problem 2

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
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ICMC 2019/20 Round 1, Problem 1

Alice and Bob play a game on a sphere which is initially marked with a finite number of points. Alice and Bob then take turns making moves, with Alice going first:
- On Alice’s move, she counts the number of marked points on the sphere, nn. She then marks another n+1n + 1 points on the sphere.
- On Bob’s move, he chooses one hemisphere and removes all marked points on that hemisphere, including any marked points on the boundary of the hemisphere.
Can Bob always guarantee that after a finite number of moves, the sphere contains no marked points?
(A hemisphere is the region on a sphere that lies completely on one side of any plane passing through the centre of the sphere.)
proposed by the ICMC Problem Committee

ICMC 2019/20 Round 2, Problem 1

An [I]automorphism of a group (G,∗)\left(G,*\right) is a bijective function f:G→Gf:G\to G satisfying f(x∗y)=f(x)∗f(y)f(x*y)=f(x)*f(y) for all x,y∈Gx,y\in G. Find a group (G,∗)(G,*) with fewer than (201.6)2=40642.56(201.6)^2=40642.56 unique elements and exactly 201622016^2 unique automorphisms.
Proposed by the ICMC Problem Committee