Difference between revisions of "2001 USAMO Problems/Problem 4"

Latest revision as of 22:24, 18 June 2022

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

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

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.

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.

Solution 5

Let $M, N$ be midpoints of $AP$ and $BC$ , respectively. For the points $A, B, P, C$ ; let's apply Euler's quadrilateral formula, $AB^2 + BP^2 + PC^2 + CA^2 = AP^2 + BC^2 + 4MN^2 \geq AP^2 + BC^2 .$ Given that $AP^2 > BP^2 + PC^2$ . Thus, $AB^2 + AC^2 > BC^2 .$ and we get $\angle BAC$ is acute.

(Lokman GÖKÇE)

Solution 6

Without loss of generality, assume that in a Cartesian coordinate system, $A$ is at the point $(0, 0)$ and $C$ is at the point $(1,0)$ . Let $B$ be at the point $(b_x,b_y)$ and $P$ be at the point $(p_x,p_y)$ . Without loss of generality, also assume that $b_y>0$ .

Now, assume for contradiction that $\angle BAC$ is not acute. Since $PA$ , $PB$ , and $PC$ are the sides of an obtuse triangle, with $PA$ the longest side, it follows that $PA^2>PB^2+PC^2$ , implying that $p_x^2+p_y^2>(p_x-b_x)^2+(p_y-b_y)^2+(p_x-1)^2+p_y^2$ . This inequality simplifies to $b_x^2-2p_x b_x+b_y^2-2p_y b_y+p_x^2-2p_x+1+p_y^2<0$ . Note that since $p_x^2-2p_x+1$ and $b_y^2-2p_y b_y+p_y^2$ are both perfect squares, all terms of this inequality except for $-2p_x b_x$ are already guaranteed to be nonnegative.

If $p_x<0$ , then $P$ would be closer to $A$ than to $C$ , but since $PA^2=PB^2+PC^2$ , this is not possible. Therefore, $p_x \geq 0$ . Since $\angle BAC$ not being acute implies that $b_x \leq 0$ , it follows that $-2p_x b_x \geq 0$ . But now since all terms of $b_x^2-2p_x b_x+b_y^2-2p_y b_y+p_x^2-2p_x+1+p_y^2<0$ are guaranteed to be nonnegative, this entire expression cannot be negative, leading to a contradiction. Therefore, $\angle BAC$ is acute.

@@ Line 4: / Line 4: @@
 == Solution ==
+=== Solution 1 ===
 We know that <math>PB^2+PC^2 < PA^2</math> and we wish to prove that <math>AB^2 + AC^2 > BC^2</math>.
 It would be sufficient to prove that
@@ Line 11: / Line 11: @@
 Then, we wish to show
-<center><math>\begin{align*}
+<cmath>(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 </cmath>
-(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 \\
+<cmath>2p^2 + 2q^2 + 2x^2 + 2y^2 - 2p - 2px - 2qy + 2 \geq p^2 + q^2 + x^2 + y^2 - 2x + 1 </cmath>
-p^2 + 2q^2 + 2x^2 + 2y^2 - 2p - 2px - 2qy + 2 &\geq p^2 + q^2 + x^2 + y^2 - 2x + 1 \\
+<cmath>p^2 + q^2 + x^2 + y^2 + 2x - 2p - 2px - 2qy + 1 \geq 0 </cmath>
-p^2 + q^2 + x^2 + y^2 + 2x - 2p - 2px - 2qy + 1 &\geq 0 \\
+<cmath>(x-p)^2 + (q-y)^2 + 2(x-p) + 1 \geq 0 </cmath>
-(x-p)^2 + (q-y)^2 + 2(x-p) + 1 &\geq 0 \\
+<cmath>(x-p+1)^2 + (q-y)^2 \geq 0,</cmath>
-(x-p+1)^2 + (q-y)^2 &\geq 0,
-\end{align*}</math></center>
 which is true by the trivial inequality.
+=== Solution 2 ===
+Let <math>A</math> be the origin. For a point <math>Q</math>, denote by <math>q</math> the vector <math>\overrightarrow{AQ}</math>, and denote by <math>|q|</math> the length of <math>q</math>. The given conditions may be written as
+<cmath>|p - b|^2 + |p - c|^2 < |p|^2,</cmath>
+or
+<cmath>p\cdot p + b\cdot b + c\cdot c - 2p\cdot b - 2p\cdot c < 0.</cmath>
+Adding <math>2b\cdot c</math> on both sides of the last inequality gives
+<cmath>|p - b - c|^2 < 2b\cdot c.</cmath>
+Since the left-hand side of the last inequality is nonnegative, the right-hand side is positive. Hence
+<cmath>\cos\angle BAC = \frac{b\cdot c}{|b||c|} > 0,</cmath>
+that is, <math>\angle BAC</math> is acute.
+=== Solution 3 ===
+For the sake of contradiction, let's assume to the contrary that <math>\angle BAC</math>. Let <math>AB = c</math>, <math>BC = a</math>, and <math>CA = b</math>. Then <math>a^2\geq b^2 + c^2</math>. We claim that the quadrilateral <math>ABPC</math> is convex. Now applying the generalized Ptolemy's Theorem to the convex quadrilateral <math>ABPC</math> yields
+<cmath>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},</cmath>
+where the second inequality is by Cauchy-Schwarz. This implies <math>PA^2\leq PB^2 + PC^2</math>, in contradiction with the facts that <math>PA</math>, <math>PB</math>, and <math>PC</math> are the sides of an obtuse triangle and <math>PA > \max\{PB, PC\}</math>.
+We present two arguments to prove our claim.
+''First argument'': Without loss of generality, we may assume that <math>A</math>, <math>B</math>, and <math>C</math> are in counterclockwise order. Let lines <math>l_1</math> and <math>l_2</math> be the perpendicular bisectors of segments <math>AB</math> and <math>AC</math>, respectively. Then <math>l_1</math> and <math>l_2</math> meet at <math>O</math>, the circumcenter of triangle <math>ABC</math>. Lines <math>l_1</math> and <math>l_2</math> cut the plane into four regions and <math>A</math> is in the interior of one of these regions. Since <math>PA > PB</math> and <math>PA > PC</math>, <math>P</math> must be in the interior of the region that opposes <math>A</math>. Since <math>\angle BAC</math> is not acute, ray <math>AC</math> does not meet <math>l_1</math> and ray <math>AB</math> does not meet <math>l_2</math>. Hence <math>B</math> and <math>C</math> must lie in the interiors of the regions adjacent to <math>A</math>. Let <math>\mathcal{R}_X</math> denote the region containing <math>X</math>. Then <math>\mathcal{R}_A</math>, <math>\mathcal{R}_B</math>, <math>\mathcal{R}_P</math>, and <math>\mathcal{R}_C</math> are the four regions in counterclockwise order. Since <math>\angle BAC\geq 90^\circ</math>, either <math>O</math> is on side <math>BC</math> or <math>O</math> and <math>A</math> are on opposite sides of line <math>BC</math>. In either case <math>P</math> and <math>A</math> are on opposite sides of line <math>BC</math>. Also, since ray <math>AB</math> does not meet <math>l_2</math> and ray <math>AC</math> does not meet <math>l_1</math>, it follows that <math>\mathcal{R}_P</math> is entirely in the interior of <math>\angle BAC</math>. Hence <math>B</math> and <math>C</math> are on opposite sides of <math>AP</math>. Therefore <math>ABPC</math> is convex.
+<center>[[File:2001usamo4-1.png]]</center>
+''Second argument'': Since <math>PA > PB</math> and <math>PA > PC</math>, <math>A</math> cannot be inside or on the sides of triangle <math>PBC</math>. Since <math>PA > PB</math>, we have <math>\angle ABP > \angle BAP</math> and hence <math>\angle BAC\geq 90^\circ > \angle BAP</math>. Hence <math>C</math> cannot be inside or on the sides of triangle <math>BAP</math>. Symmetrically, <math>B</math> cannot be inside or on the sides of triangle <math>CAP</math>. Finally, since <math>\angle ABP > \angle BAP</math> and <math>\angle ACP > \angle CAP</math>, we have
+<cmath>\angle ABP + \angle ACP > \angle BAC\geq 90^\circ\geq\angle ABC + \angle ACB.</cmath>
+Therefore <math>P</math> cannot be inside or on the sides of triangle <math>ABC</math>. Since this covers all four cases, <math>ABPC</math> is convex.
+===Solution 4===
+Let <math>P</math> be the origin in vector space, and let <math>a, b, c</math> denote the position vectors of <math>A, B, C</math> respectively. Then the obtuse triangle condition, <math>PA^2 > PB^2 + PC^2</math>, becomes <math>a^2 > b^2 + c^2</math> 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 <math>\angle{BAC}</math> is acute, it suffices to show that <math>(a - b)(a - c) > 0</math>, or <math>a^2 - ab - ac + bc > 0</math>. But this follows from the observation that
+<cmath>(-a + b + c)^2 \ge 0,</cmath>
+which leads to
+<cmath>2a^2 - 2ab - 2ac + 2bc > a^2 + b^2 + c^2 - 2ab - 2ac + 2bc \ge 0</cmath>
+and therefore our desired conclusion.
+===Solution 5===
+Let <math>M, N</math> be midpoints of <math>AP</math> and <math>BC</math>, respectively. For the points <math>A, B, P, C</math>; let's apply Euler's quadrilateral formula,
+<cmath> AB^2 + BP^2 + PC^2 + CA^2 = AP^2 + BC^2 + 4MN^2 \geq AP^2 + BC^2 .</cmath>
+Given that <math>AP^2 > BP^2 + PC^2</math>. Thus,
+<cmath> AB^2 + AC^2 > BC^2 .</cmath>
+and we get <math>\angle BAC</math> is acute.
+(Lokman GÖKÇE)
+===Solution 6===
+Without loss of generality, assume that in a Cartesian coordinate system, <math>A</math> is at the point <math>(0, 0)</math> and <math>C</math> is at the point <math>(1,0)</math>. Let <math>B</math> be at the point <math>(b_x,b_y)</math> and <math>P</math> be at the point <math>(p_x,p_y)</math>. Without loss of generality, also assume that <math>b_y>0</math>.
+Now, assume for contradiction that <math>\angle BAC</math> is not acute. Since <math>PA</math>, <math>PB</math>, and <math>PC</math> are the sides of an obtuse triangle, with <math>PA</math> the longest side, it follows that <math>PA^2>PB^2+PC^2</math>, implying that <math>p_x^2+p_y^2>(p_x-b_x)^2+(p_y-b_y)^2+(p_x-1)^2+p_y^2</math>. This inequality simplifies to <math>b_x^2-2p_x b_x+b_y^2-2p_y b_y+p_x^2-2p_x+1+p_y^2<0</math>. Note that since <math>p_x^2-2p_x+1</math> and <math>b_y^2-2p_y b_y+p_y^2</math> are both perfect squares, all terms of this inequality except for <math>-2p_x b_x</math> are already guaranteed to be nonnegative.
+If <math>p_x<0</math>, then <math>P</math> would be closer to <math>A</math> than to <math>C</math>, but since <math>PA^2=PB^2+PC^2</math>, this is not possible. Therefore, <math>p_x \geq 0</math>. Since <math>\angle BAC</math> not being acute implies that <math>b_x \leq 0</math>, it follows that <math>-2p_x b_x \geq 0</math>. But now since all terms of <math>b_x^2-2p_x b_x+b_y^2-2p_y b_y+p_x^2-2p_x+1+p_y^2<0</math> are guaranteed to be nonnegative, this entire expression cannot be negative, leading to a contradiction. Therefore, <math>\angle BAC</math> is acute.
 == 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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