1977 USAMO Problems/Problem 3
Problem
If and
are two of the roots of
, prove that
is a root of
.
Solution
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a,b,c,d are roots of equation then by vietas relation
let us suppose
are roots of
.
then sum of roots
sum taken two at a time
similarly we prove for the roots taken three four five and six at a time
to prove
are roots of second equation
Given the roots of the equation
.
First, .
Then and
.
Remains or
.
Let and
, so
(1).
Second, is a root,
and
is a root,
.
Multiplying: or
.
Solving .
In (1): .
or
.
Conclusion: is a root of
.
See Also
1977 USAMO (Problems • Resources) | ||
Preceded by Problem 2 |
Followed by Problem 4 | |
1 • 2 • 3 • 4 • 5 | ||
All USAMO Problems and Solutions |
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