MathDB
Problems
Contests
National and Regional Contests
Sweden Contests
Swedish Mathematical Competition
1994 Swedish Mathematical Competition
5
5
Part of
1994 Swedish Mathematical Competition
Problems
(1)
a_1^2 > 2ka_2/(k-1) if a polynomial had k distinct real roots
Source: 1994 Swedish Mathematical Competition p5
4/2/2021
The polynomial
x
k
+
a
1
x
k
−
1
+
a
2
x
k
−
2
+
.
.
.
+
a
k
x^k + a_1x^{k-1} + a_2x^{k-2} +... + a_k
x
k
+
a
1
x
k
−
1
+
a
2
x
k
−
2
+
...
+
a
k
has
k
k
k
distinct real roots. Show that
a
1
2
>
2
k
a
2
k
−
1
a_1^2 > \frac{2ka_2}{k-1}
a
1
2
>
k
−
1
2
k
a
2
.
algebra
polynomial