MathDB

1

Part of ICMC 3

Problems(2)

ICMC 2019/20 Round 1, Problem 1

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

8/7/2020
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
college contests
ICMC 2019/20 Round 2, Problem 1

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

8/7/2020
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
college contestsgroup theory