MathDB
Problems
Contests
National and Regional Contests
India Contests
India EGMO TST
India EGMO 2022 TST
6
6
Part of
India EGMO 2022 TST
Problems
(1)
Bob writes polynomials
Source: India EGMO 2022 TST P6
11/28/2021
Suppose
P
(
x
)
P(x)
P
(
x
)
is a non-constant polynomial with real coefficients, and even degree. Bob writes the polynomial
P
(
x
)
P(x)
P
(
x
)
on a board. At every step, if the polynomial on the board is
f
(
x
)
f(x)
f
(
x
)
, he can replace it with 1.
f
(
x
)
+
c
f(x)+c
f
(
x
)
+
c
for a real number
c
c
c
, or 2. the polynomial
P
(
f
(
x
)
)
P(f(x))
P
(
f
(
x
))
.Can he always find a finite sequence of steps so the final polynomial on the board has exactly
2020
2020
2020
real roots? What about
2021
2021
2021
?~Sutanay Bhattacharya
algebra
polynomial