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 "2001 USAMO Problems/Problem 4"
m (→‎Solution)
Line 11: Line 11:
 
Then, we wish to show
 
Then, we wish to show
  
−
<cmath>\begin{align*}
+
<center><math>\begin{align*}
 
(p-1)^2 + q^2 + (p-x)^2 + (q-y)^2 + 1 + x^2 + y^2 &\geq p^2 + q^2 + (x-1)^2 + y^2 \\
 
(p-1)^2 + q^2 + (p-x)^2 + (q-y)^2 + 1 + x^2 + y^2 &\geq p^2 + q^2 + (x-1)^2 + y^2 \\
 
2p^2 + 2q^2 + 2x^2 + 2y^2 - 2p - 2px - 2qy + 2 &\geq p^2 + q^2 + x^2 + y^2 - 2x + 1 \\
 
2p^2 + 2q^2 + 2x^2 + 2y^2 - 2p - 2px - 2qy + 2 &\geq p^2 + q^2 + x^2 + y^2 - 2x + 1 \\
Line 17: Line 17:
 
(x-p)^2 + (q-y)^2 + 2(x-p) + 1 &\geq 0 \\
 
(x-p)^2 + (q-y)^2 + 2(x-p) + 1 &\geq 0 \\
 
(x-p+1)^2 + (q-y)^2 &\geq 0,
 
(x-p+1)^2 + (q-y)^2 &\geq 0,
−
\end{align*}</cmath>
+
\end{align*}</math></center>
  
 
which is true by the trivial inequality.
 
which is true by the trivial inequality.

Revision as of 18:00, 19 April 2010

Problem

Let $P$ be a point in the plane of triangle $ABC$ such that the segments $PA$, $PB$, and $PC$ are the sides of an obtuse triangle. Assume that in this triangle the obtuse angle opposes the side congruent to $PA$. Prove that $\angle BAC$ is acute.

Solution

We know that $PB^2+PC^2 < PA^2$ and we wish to prove that $AB^2 + AC^2 > BC^2$. It would be sufficient to prove that \[PB^2+PC^2+AB^2+AC^2 \geq PA^2 + BC^2.\] Set $A(0,0)$, $B(1,0)$, $C(x,y)$, $P(p,q)$. Then, we wish to show

$\begin{align*}

(p-1)^2 + q^2 + (p-x)^2 + (q-y)^2 + 1 + x^2 + y^2 &\geq p^2 + q^2 + (x-1)^2 + y^2 \\ 2p^2 + 2q^2 + 2x^2 + 2y^2 - 2p - 2px - 2qy + 2 &\geq p^2 + q^2 + x^2 + y^2 - 2x + 1 \\ p^2 + q^2 + x^2 + y^2 + 2x - 2p - 2px - 2qy + 1 &\geq 0 \\ (x-p)^2 + (q-y)^2 + 2(x-p) + 1 &\geq 0 \\ (x-p+1)^2 + (q-y)^2 &\geq 0,

\end{align*}$ (Error compiling LaTeX. Unknown error_msg)

which is true by the trivial inequality.

See also

2001 USAMO (Problems • Resources)
Preceded by
Problem 3 		Followed by
Problem 5
1 • 2 • 3 • 4 • 5 • 6
All USAMO Problems and Solutions
Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2001_USAMO_Problems/Problem_4&oldid=34340"