Subcontests
(8)Computational Graph Theory Tree
There are 5 vertices labelled 1,2,3,4,5. For any two pairs of vertices u,v, the edge uv
is drawn with probability 1/2. If the probability that the resulting graph is a tree is given by qp where p,q are coprime, then find the value of q1/10+p. Old ISL resurrected
Define a positive integer n to be a fake square if either n=1 or n can be written as a product of an even number of not necessarily distinct primes. Prove that for any even integer k⩾2, there exist distinct positive integers a1, a2,⋯,ak such that the polynomial (x+a1)(x+a2)⋯(x+ak) takes ‘fake square’ values for all x=1,2,⋯,2023.Proposed by Prof. Aditya Karnataki Computational Geometric Ratio
Given a triangle ABC with angles ∠A=60∘,∠B=75∘,∠C=45∘, let H be its orthocentre, and O be its circumcenter. Let F be the midpoint of side AB, and Q be the foot of the perpendicular from B onto AC. Denote by X the intersection point of the lines FH and QO. Suppose the ratio of the length of FX and the circumradius of the triangle is given by da+bc, then find the value of 1000a+100b+10c+d. Computational Set Construction
Consider the set S of permutations of 1,2,…,2022 such that for all numbers k in the
permutation, the number of numbers less than k that follow k is even.For example, for n=4;S={[3,4,1,2];[3,1,2,4];[1,2,3,4];[1,4,2,3]}If ∣S∣=(a!)b where a,b∈N, then find the product ab. Computational board game
The following 100 numbers are written on the board: 21−1,22−1,23−1,…,2100−1.Alice chooses two numbers a,b, erases them and writes the number a+b+2ab−1 on the board. She keeps doing this until a single number remains on the board.If the sum of all possible numbers she can end up with is qp where p,q are coprime, then what
is the value of log2(p+q)?