1

Part of 2023 Caucasus Mathematical Olympiad

Problems(2)

the product of any $n$ odd consecutive positive integers is divisible by $45$.

Source: Caucasus MO 2023

Determine the least positive integer

n{}

for which the following statement is true: the product of any

n{}

odd consecutive positive integers is divisible by

45

Prove that if $q+q^{'} =r +r^{'}$, then $2n$ is a perfect square.

Source: Caucasus MO 2023

n{}

m

be positive integers,

n>m>1

n{}

m

have partial quotient

q

r

n = qm + r

r\in\{0,1,...,m-1\}

n-1

m

have partial quotient

q^{'}

r^{'}

. a) It appears that

q+q^{'} =r +r^{'} = 99

. Find all possible values of

n{}

. b) Prove that if

q+q^{'} =r +r^{'}

2n

is a perfect square.