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2001 USAMO Problems/Problem 4

Revision as of 22:51, 6 September 2015 by Mathawesomeness777 (talk | contribs) (→‎Solution 1)

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

Solution 1

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 

\[(p-1)^2 + q^2 + (p-x)^2 + (q-y)^2 + 1 + x^2 + y^2 &\geq p^2 + q^2 + (x-1)^2 + y^2 \\
2p^2 + 2q^2 + 2x^2 + 2y^2 - 2p - 2px - 2qy + 2 &\geq p^2 + q^2 + x^2 + y^2 - 2x + 1 \\
p^2 + q^2 + x^2 + y^2 + 2x - 2p - 2px - 2qy + 1 &\geq 0 \\
(x-p)^2 + (q-y)^2 + 2(x-p) + 1 &\geq 0 \\
(x-p+1)^2 + (q-y)^2 &\geq 0,\] (Error compiling LaTeX. Unknown error_msg)

which is true by the trivial inequality.

Solution 2

Let $A$ be the origin. For a point $Q$, denote by $q$ the vector $\overrightarrow{AQ}$, and denote by $|q|$ the length of $q$. The given conditions may be written as \[|p - b|^2 + |p - c|^2 < |p|^2,\] or \[p\cdot p + b\cdot b + c\cdot c - 2p\cdot b - 2p\cdot c < 0.\] Adding $2b\cdot c$ on both sides of the last inequality gives \[|p - b - c|^2 < 2b\cdot c.\] Since the left-hand side of the last inequality is nonnegative, the right-hand side is positive. Hence \[\cos\angle BAC = \frac{b\cdot c}{|b||c|} > 0,\] that is, $\angle BAC$ is acute.

Solution 3

For the sake of contradiction, let's assume to the contrary that $\angle BAC$. Let $AB = c$, $BC = a$, and $CA = b$. Then $a^2\geq b^2 + c^2$. We claim that the quadrilateral $ABPC$ is convex. Now applying the generalized Ptolemy's Theorem to the convex quadrilateral $ABPC$ yields \[a\cdot PA\leq b\cdot PB + c\cdot PC\leq\sqrt{b^2 + c^2}\sqrt{PB^2 + PC^2}\leq a\sqrt{PB^2 + PC^2},\] where the second inequality is by Cauchy-Schwarz. This implies $PA^2\leq PB^2 + PC^2$, in contradiction with the facts that $PA$, $PB$, and $PC$ are the sides of an obtuse triangle and $PA > \max\{PB, PC\}$.

We present two arguments to prove our claim.

First argument: Without loss of generality, we may assume that $A$, $B$, and $C$ are in counterclockwise order. Let lines $l_1$ and $l_2$ be the perpendicular bisectors of segments $AB$ and $AC$, respectively. Then $l_1$ and $l_2$ meet at $O$, the circumcenter of triangle $ABC$. Lines $l_1$ and $l_2$ cut the plane into four regions and $A$ is in the interior of one of these regions. Since $PA > PB$ and $PA > PC$, $P$ must be in the interior of the region that opposes $A$. Since $\angle BAC$ is not acute, ray $AC$ does not meet $l_1$ and ray $AB$ does not meet $l_2$. Hence $B$ and $C$ must lie in the interiors of the regions adjacent to $A$. Let $\mathcal{R}_X$ denote the region containing $X$. Then $\mathcal{R}_A$, $\mathcal{R}_B$, $\mathcal{R}_P$, and $\mathcal{R}_C$ are the four regions in counterclockwise order. Since $\angle BAC\geq 90^\circ$, either $O$ is on side $BC$ or $O$ and $A$ are on opposite sides of line $BC$. In either case $P$ and $A$ are on opposite sides of line $BC$. Also, since ray $AB$ does not meet $l_2$ and ray $AC$ does not meet $l_1$, it follows that $\mathcal{R}_P$ is entirely in the interior of $\angle BAC$. Hence $B$ and $C$ are on opposite sides of $AP$. Therefore $ABPC$ is convex.

2001usamo4-1.png

Second argument: Since $PA > PB$ and $PA > PC$, $A$ cannot be inside or on the sides of triangle $PBC$. Since $PA > PB$, we have $\angle ABP > \angle BAP$ and hence $\angle BAC\geq 90^\circ > \angle BAP$. Hence $C$ cannot be inside or on the sides of triangle $BAP$. Symmetrically, $B$ cannot be inside or on the sides of triangle $CAP$. Finally, since $\angle ABP > \angle BAP$ and $\angle ACP > \angle CAP$, we have \[\angle ABP + \angle ACP > \angle BAC\geq 90^\circ\geq\angle ABC + \angle ACB.\] Therefore $P$ cannot be inside or on the sides of triangle $ABC$. Since this covers all four cases, $ABPC$ is convex.

Solution 4

Let $P$ be the origin in vector space, and let $a, b, c$ denote the position vectors of $A, B, C$ respectively. Then the obtuse triangle condition, $PA^2 > PB^2 + PC^2$, becomes $a^2 > b^2 + c^2$ using the fact that the square of a vector (the dot product of itself and itself) is the square of its magnitude. Now, notice that to prove $\angle{BAC}$ is acute, it suffices to show that $(a - b)(a - c) > 0$, or $a^2 - ab - ac + bc > 0$. But this follows from the observation that \[(-a + b + c)^2 \ge 0,\] which leads to \[2a^2 - 2ab - 2ac + 2bc > a^2 + b^2 + c^2 - 2ab - 2ac + 2bc \ge 0\] and therefore our desired conclusion.

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

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