2002 AMC 10B Problems/Problem 20
Contents
Problem
Let ,
, and
be real numbers such that
and
. Then
is
Solution
Solution 1
Rearranging, we get and
Squaring both, and
are obtained.
Adding the two equations and dividing by gives
, so
.
Solution 2
The easiest way is to assume a value for and then solve the system of equations. For
, we get the equations
and
Multiplying the second equation by
, we have
Adding up the two equations yields
, so
We obtain
after plugging in the value for
.
Therefore,
which corresponds to
.
This time-saving trick works only because we know that for any value of
,
will always be constant (it's a contest), so any value of
will work. This is also called without loss of generality or WLOG.
Solution 3 (fakesolve)
Notice that the coefficients of and
are pretty similar (15s for reading and noticing), so let
gives
, and
(10s writing). Since the desired quantity simplifies to
, the
term of the quadratics after squaring gets canceled by adding up the squares of the two equations because they have the same coefficients but opposite sign (15s mind-binom). This simplifies to
, or
(15s writing and addition and fraction simplification and (B) circling and submission)
Video Solution
https://www.youtube.com/watch?v=3Oq21r5OezA ~David
See Also
2002 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 19 |
Followed by Problem 21 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AMC 10 Problems and Solutions |
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