Subcontests
(8)IMC 2020 Problem 7
Let G be a group and n≥2 be an integer. Let H1,H2 be 2 subgroups of G that satisfy [G:H1]=[G:H2]=n and [G:(H1∩H2)]=n(n−1). Prove that H1,H2 are conjugate in G.Official definitions: [G:H] denotes the index of the subgroup of H, i.e. the number of distinct left cosets xH of H in G. The subgroups H1,H2 are conjugate if there exists g∈G such that g−1H1g=H2. IMC 2020 Problem 3
Let d≥2 be an integer. Prove that there exists a constant C(d) such that the following holds: For any convex polytope K⊂Rd, which is symmetric about the origin, and any ε∈(0,1), there exists a convex polytope L⊂Rd with at most C(d)ε1−d vertices such that
(1−ε)K⊆L⊆K.Official definitions: For a real α, a set T∈Rd is a convex polytope with at most α vertices, if T is a convex hull of a set X∈Rd of at most α points, i.e. T={x∈X∑txx∣tx≥0,x∈X∑tx=1}. Define αK={αx∣x∈K}. A set T∈Rd is symmetric about the origin if (−1)T=T. IMC 2020 Problem 1
Let n be a positive integer. Compute the number of words w that satisfy the following three properties.1. w consists of n letters from the alphabet {a,b,c,d}.2. w contains an even number of a's3. w contains an even number of b's.For example, for n=2 there are 6 such words: aa,bb,cc,dd,cd,dc.