2022 AIME I Problems/Problem 1
Contents
Problem 1
Quadratic polynomials and
have leading coefficients
and
respectively. The graphs of both polynomials pass through the two points
and
Find
Solution 1 (Linear Polynomials)
Let Since the
-terms of
and
cancel, we conclude that
is a linear polynomial.
Note that
so the slope of
is
It follows that the equation of $$ (Error compiling LaTeX. Unknown error_msg)R(x)R(x)=-\frac12x+c$$ (Error compiling LaTeX. Unknown error_msg) for some constant
We substitute
into this equation to get
Therefore, the answer is
~MRENTHUSIASM
Solution 2 (Quadratic Polynomials)
Let
for some constants
and
We are given that
and we wish to find
We need to cancel
and
Since
we subtract
from
to get
~MRENTHUSIASM
See Also
2022 AIME I (Problems • Answer Key • Resources) | ||
Preceded by First Problem |
Followed by Problem 2 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
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