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Difference between revisions of "2020 AIME II Problems/Problem 11"
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==Problem==
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==Problem==  
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Let <math>P(X) = x^2 - 3x - 7</math>, and let <math>Q(x)</math> and <math>R(x)</math> be two quadratic polynomials also with the coefficient of <math>x^2</math> equal to <math>1</math>. David computes each of the three sums <math>P + Q</math>, <math>P + R</math>, and <math>Q + R</math> and is surprised to find that each pair of these sums has a common root, and these three common roots are distinct. If <math>Q(0) = 2</math>, then <math>R(0) = \fracmn</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m + n</math>.  
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Let <math>P(X) = x^2 - 3x - 7</math>, and let <math>Q(x)</math> and <math>R(x)</math> be two quadratic polynomials also with the coefficient of <math>x^2</math> equal to <math>1</math>. David computes each of the three sums <math>P + Q</math>, <math>P + R</math>, and <math>Q + R</math> and is surprised to find that each pair of these sums has a common root, and these three common roots are distinct. If <math>Q(0) = 2</math>, then <math>R(0) = \fracmn</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m + n</math>.
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==Solution==  
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==Video Solution==  
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==Solution==
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https://youtu.be/BQlab3vjjxw ~ CNCM  
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==Video Solution==
 
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https://youtu.be/BQlab3vjjxw ~ CNCM
 
 
==See Also==
 
==See Also==

Revision as of 18:12, 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 $R(0) = \fracmn$ (Error compiling LaTeX. Unknown error_msg), where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

Solution

Video Solution

https://youtu.be/BQlab3vjjxw ~ CNCM

See Also

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