Difference between revisions of "2020 AIME II Problems/Problem 11"

(Semi-automated contest formatting)
Line 21: Line 21:
 
https://youtu.be/BQlab3vjjxw ~ CNCM
 
https://youtu.be/BQlab3vjjxw ~ CNCM
 
==See Also==
 
==See Also==
{{AIME box|year=2020|n=II|num-b=10|num-a=2}}
+
{{AIME box|year=2020|n=II|num-b=10|num-a=12}}
 
[[Category:Intermediate Algebra Problems]]
 
[[Category:Intermediate Algebra Problems]]
 
{{MAA Notice}}
 
{{MAA Notice}}

Revision as of 18:31, 7 June 2020

Problem

Let $P(X) = x^2 - 3x - 7$, and let $Q(x)$ and $R(x)$ be two quadratic polynomials also with the coefficient of $x^2$ equal to $1$. David computes each of the three sums $P + Q$, $P + R$, and $Q + R$ and is surprised to find that each pair of these sums has a common root, and these three common roots are distinct. If $Q(0) = 2$, then $R(0) = \frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

Solution

Let $Q(x) = x^2 + ax + 2$ and $R(x) = x^2 + bx + c$. We can write the following: \[P + Q = 2x^2 + (a - 3)x - 5\] \[P + R = 2x^2 + (b - 3)x + (c - 7)\] \[Q + R = 2x^2 + (a + b)x + (c + 2)\] Let the common root of $P+Q,P+R$ be $r$; $P+R,Q+R$ be $s$; and $P+Q,Q+R$ be $t$. We then have that the roots of $P+Q$ are $r,t$, the roots of $P + R$ are $r, s$, and the roots of $Q + R$ are $s,t$.

By Vieta's, we have: \[(1) r + t = \dfrac{3 - a}{2}\] \[(2) r + s = \dfrac{3 - b}{2}\] \[(3) s + t = \dfrac{-a - b}{2}\] \[(4) rt = \dfrac{-5}{2}\] \[(5) rs = \dfrac{c - 7}{2}\] \[(6) st = \dfrac{c + 2}{2}\]

Video Solution

https://youtu.be/BQlab3vjjxw ~ CNCM

See Also

2020 AIME II (ProblemsAnswer KeyResources)
Preceded by
Problem 10
Followed by
Problem 12
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
All AIME Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png