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All-Russian Olympiad Regional Round
2001 All-Russian Olympiad Regional Round
10.5
10.5
Part of
2001 All-Russian Olympiad Regional Round
Problems
(1)
ax^2+bx+c, (c-b)x^2 + (c- a)x +(a + b) - All-Russian MO 2001 Regional (R4) 10.5
Source:
9/26/2024
Given integers
a
a
a
,
b
b
b
and
c
c
c
,
c
≠
b
c\ne b
c
=
b
. It is known that the square trinomials
a
x
2
+
b
x
+
c
ax^2 + bx + c
a
x
2
+
b
x
+
c
and
(
c
−
b
)
x
2
+
(
c
−
a
)
x
+
(
a
+
b
)
(c-b)x^2 + (c- a)x + (a + b)
(
c
−
b
)
x
2
+
(
c
−
a
)
x
+
(
a
+
b
)
have a common root (not necessarily integer). Prove that
a
+
b
+
2
c
a+b+2c
a
+
b
+
2
c
is divisible by
3
3
3
.
algebra
polynomial
number theory
trinomial