Difference between revisions of "2001 USAMO Problems/Problem 4"
5849206328x (talk | contribs) |
5849206328x (talk | contribs) m (→Solution) |
||
Line 11: | Line 11: | ||
Then, we wish to show | Then, we wish to show | ||
− | < | + | <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*}</ | + | \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 be a point in the plane of triangle such that the segments , , and are the sides of an obtuse triangle. Assume that in this triangle the obtuse angle opposes the side congruent to . Prove that is acute.
Solution
We know that and we wish to prove that . It would be sufficient to prove that Set , , , . Then, we wish to show
(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 |