5

Part of 2024 Caucasus Mathematical Olympiad

Problems(2)

Ratio of two products

Source: Caucasus MO 2024, Juniors P5

Alex calculated the value of function

f(n) = n^2 + n + 1

for each integer from

1

100

. Marina calculated the value of function

g(n) = n^2-n+1

for the same numbers. Who of them has greater product of values and what is their ratio?

Source: Caucasus MO 2024, Seniors P5

a, b, c

be reals and consider three lines

y=ax+b, y=bx+c, y=cx+a

. Two of these lines meet at a point with

x

1

. Show that the third one passes through a point with two integer coordinates.