Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "2022 SSMO Speed Round Problems/Problem 9"

(Created page with "==Problem== Find the sum of the maximum and minimum values of <math>8x^2+7xy+5y^2</math> under the constraint that <math>3x^2+5xy+3y^2 = 88.</math> ==Solution== We want to m...")
 
 
(One intermediate revision by the same user not shown)
Line 1: Line 1:
 
==Problem==
 
==Problem==
−
Find the sum of the maximum and minimum values of <math>8x^2+7xy+5y^2</math> under the constraint that <math>3x^2+5xy+3y^2 = 88.</math>  
+
 
 +
Consider a triangle <math>ABC</math> such that <math>AB=13</math>, <math>BC=14</math>, <math>CA=15</math> and a square <math>WXYZ</math> such that <math>Y</math> and <math>Z</math> lie on <math>\overleftrightarrow{BC}</math>, <math>W</math> lies on <math>\overleftrightarrow{AB}</math>, and <math>X</math> lies on <math>\overleftrightarrow{CA}</math>. Suppose further that <math>W</math>, <math>X</math>, <math>Y</math>, and <math>Z</math> are distinct from <math>A</math>, <math>B</math>, and <math>C</math>. Let <math>O</math> be the center of <math>WXYZ</math>. If <math>AO</math> intersects <math>BC</math> at <math>P</math>, then the sum of all values of <math>\frac{BP}{CP}</math> can be expressed as <math>\frac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>.
  
 
==Solution==
 
==Solution==
−
We want to maximize <math>k</math> such
 
−
<cmath>
 
−
    8x^2+7xy+5y^2 = \frac{k}{88}(3x^2+5xy+3y^2)
 
−
</cmath>
 
−
or, if <math>a = \frac{x}{y}</math>
 
−
<cmath>
 
−
    8a^2+7a+5 = \frac{k}{88}(3a^2+5a+3)
 
−
</cmath>
 
−
which has discriminant in <math>a</math> of
 
−
<cmath>
 
−
    121(-k^2 + 688k - 78144)
 
−
</cmath>
 
−
so the sum of the extremes of <math>k</math> are <math>\boxed{688}</math>.
 

Latest revision as of 19:14, 2 May 2025

Problem

Consider a triangle $ABC$ such that $AB=13$, $BC=14$, $CA=15$ and a square $WXYZ$ such that $Y$ and $Z$ lie on $\overleftrightarrow{BC}$, $W$ lies on $\overleftrightarrow{AB}$, and $X$ lies on $\overleftrightarrow{CA}$. Suppose further that $W$, $X$, $Y$, and $Z$ are distinct from $A$, $B$, and $C$. Let $O$ be the center of $WXYZ$. If $AO$ intersects $BC$ at $P$, then the sum of all values of $\frac{BP}{CP}$ can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Solution

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2022_SSMO_Speed_Round_Problems/Problem_9&oldid=247995"