P6

Part of 2024 Bangladesh Mathematical Olympiad

Problems(2)

Minimum possible value

Source: BdMO 2024 Secondary National P6 Higher Secondary National P3

a_1, a_2, \ldots, a_{2024}

be a permutation of

1, 2, \ldots, 2024

. Find the minimum possible value of

\sum_{i=1} ^{2023} \Big[(a_i+a_{i+1})\Big(\frac{1}{a_i}+\frac{1}{a_{i+1}}\Big)+\frac{1}{a_ia_{i+1}}\Big]

Proposed by Md. Ashraful Islam Fahim

inequalitiesalgebrapermutation

Polynomial with a constructor sequence

Source: BdMO 2024 Higher Secondary National P6

Find all polynomials

P(x)

for which there exists a sequence

a_1, a_2, a_3, \ldots

of real numbers such that

a_m + a_n = P(mn)

for any positive integer

m

n

algebrapolynomialSubstitution