6

Part of STEMS 2023 Math Cat A

Problems(2)

Computational Graph Theory Tree

Source: STEMS 2023 Maths CAT A Part A P6

5

vertices labelled

1,2,3,4,5

. For any two pairs of vertices

u, v

uv

is drawn with probability

1/2

. If the probability that the resulting graph is a tree is given by

\dfrac{p}{q}

p, q

are coprime, then find the value of

q^{1/10} + p

graph theorySTEMSComputationalprobabilityalgebra

Old ISL resurrected

Source: STEMS Mathematics 2023 Category B P4

Define a positive integer

n

to be a fake square if either

n = 1

n

can be written as a product of an even number of not necessarily distinct primes. Prove that for any even integer

k \geqslant 2

, there exist distinct positive integers

a_1

a_2, \cdots, a_k

such that the polynomial

(x+a_1)(x+a_2) \cdots (x+a_k)

takes ‘fake square’ values for all

x = 1,2,\cdots,2023

.
Proposed by Prof. Aditya Karnataki