Difference between revisions of "2010 AMC 12B Problems/Problem 23"
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x^2 - (c+d)x + cd &= a, b | x^2 - (c+d)x + cd &= a, b | ||
\end{align*}</cmath> | \end{align*}</cmath> | ||
− | The two possible equations are then <math>x^2 - (c+d)x + cd-a=0</math> and <math>x^2 - (c+d)x + cd-b=0</math>. The solutions are <math>-23, -21, -17, -15</math>. From Vieta's we know that the total sum <math>2(c+d) = -76 \implies c+d = -38</math> so the roots are paired <math>-23, -15</math> and <math>-21, -17</math>. | + | The two possible equations are then <math>x^2 - (c+d)x + cd-a=0</math> and <math>x^2 - (c+d)x + cd-b=0</math>. The solutions are <math>-23, -21, -17, -15</math>. From Vieta's we know that the total sum <math>2(c+d) = -76 \implies c+d = -38</math> so the roots are paired <math>-23, -15</math> and <math>-21, -17</math>. Then, <math>cd - a = 23*15</math> and <math>cd - b = 21*17</math>. |
We can similarly get that <math>ab - c = 59*49</math> and <math>ab - d = 57*51</math>, and <math>a+b = -108</math>. Add the first two equations to get <cmath>2cd - (a+b) = 23*15 + 21*17 \implies cd = \frac{23*15+21*17 - 108}{2} = 297.</cmath> This means <math>Q(x) = x^2 + 38x + 297</math>. | We can similarly get that <math>ab - c = 59*49</math> and <math>ab - d = 57*51</math>, and <math>a+b = -108</math>. Add the first two equations to get <cmath>2cd - (a+b) = 23*15 + 21*17 \implies cd = \frac{23*15+21*17 - 108}{2} = 297.</cmath> This means <math>Q(x) = x^2 + 38x + 297</math>. | ||
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~[https://artofproblemsolving.com/wiki/index.php/User:Isabelchen isabelchen] | ~[https://artofproblemsolving.com/wiki/index.php/User:Isabelchen isabelchen] | ||
+ | |||
+ | ==Video Solution by MOP 2024== | ||
+ | https://youtu.be/FPhaUQoRtPs | ||
+ | |||
+ | ~r00tsOfUnity | ||
==See Also== | ==See Also== |
Latest revision as of 17:29, 30 June 2024
Contents
Problem
Monic quadratic polynomial and
have the property that
has zeros at
and
, and
has zeros at
and
. What is the sum of the minimum values of
and
?
Solution 1
. Notice that
has roots
, so that the roots of
are the roots of
. For each individual equation, the sum of the roots will be
(symmetry or Vieta's). Thus, we have
, or
. Doing something similar for
gives us
.
We now have
. Since
is monic, the roots of
are "farther" from the axis of symmetry than the roots of
. Thus, we have
, or
. Adding these gives us
, or
. Plugging this into
, we get
.
The minimum value of
is
, and the minimum value of
is
. Thus, our answer is
, or answer
.
Solution 2 (Bash)
Let and
.
Then is
, which simplifies to:
We can find by simply doing
and
to get:
The sum of the zeros of is
. From Vieta, the sum is
. Therefore,
.
The sum of the zeros of is
. From Vieta, the sum is
. Therefore,
.
Plugging in, we get:
Let's tackle the coefficients, which is the sum of the six double-products possible. Since
gives the sum of these six double products of the roots of
, we have:
Similarly with , we get:
Thus, our polynomials are and
.
The minimum value of happens at
, and is
.
The minimum value of happens at
, and is
.
The sum of these minimums is . -srisainandan6
Solution 3 (Mild Bash)
Let and
. Notice that the roots of
are
and the roots of
are
Then we get:
The two possible equations are then
and
. The solutions are
. From Vieta's we know that the total sum
so the roots are paired
and
. Then,
and
.
We can similarly get that and
, and
. Add the first two equations to get
This means
.
Once more, we can similarly obtain Therefore
.
Now we can find the minimums to be and
Summing, the answer is
~Leonard_my_dude~
Solution 4
Let ,
.
Notice how the coefficient for has to be the same for the two quadratics that are multiplied to create
, and
.
,
,
,
,
,
,
,
,
,
.
Video Solution by MOP 2024
~r00tsOfUnity
See Also
2010 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 22 |
Followed by Problem 24 |
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 12 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.