Difference between revisions of "2016 USAMO Problems/Problem 5"

Latest revision as of 23:09, 11 October 2023

Problem

An equilateral pentagon $AMNPQ$ is inscribed in triangle $ABC$ such that $M\in\overline{AB},$ $Q\in\overline{AC},$ and $N, P\in\overline{BC}.$ Let $S$ be the intersection of lines $MN$ and $PQ.$ Denote by $\ell$ the angle bisector of $\angle MSQ.$

Prove that $\overline{OI}$ is parallel to $\ell,$ where $O$ is the circumcenter of triangle $ABC,$ and $I$ is the incenter of triangle $ABC.$

Solution 1

Let $D$ be the intersection of line $AI$ and the circumcircle of $\Delta ABC$ (other than $A$ ), then $OD\perp BC$ . Let $R$ be the point such that $NPQR$ is a rhombus. It follows that $OD\perp QR$ .

Since $AM=AQ$ , $AI\perp MQ$ , or $DI\perp MQ$ . It follows that $\angle ODI=\angle RQM$ .

Since $BO=OD$ , $MA=AQ$ , $\angle BOD=\angle MAQ$ , it follows that $\Delta BOD\sim\Delta MAQ$ , so $AQ/MQ=OD/BD$ .

It is given that $AQ=NP=RQ$ , and by basic properties of the incenter, $ID=BD$ . Therefore, $RQ/MQ=OD/ID$ , so $\Delta RQM\sim\Delta ODI$ . Since the rotation between the two triangles in 90 degrees, $OI\perp MR$ . However, $l$ is parallel to the bisector of $MNR$ , which is perpendicular to $MR$ , so we are done.

Solution 2

Write $\angle{JKL} = K$ for all $J,K,L$ chosen as distinct vertices of triangle $ABC$ . Define $a, b, c$ as sides opposite to angles $A, B$ , and $C$ , respectively. Place the triangle in the Euclidean plane with $A$ at the origin and $C$ on the positive x-axis. Assume without loss of generality that C is acute.

Consider the sides of the pentagon as vectors and note that $\overrightarrow{AM} + \overrightarrow{MN} + \overrightarrow{NP} = \overrightarrow{AQ} + \overrightarrow{QP} \qquad (1)$

Define $\delta$ and $\gamma$ as the angles made between the positive x-axis and $\overrightarrow{MN}$ and $\overrightarrow{QP}$ , respectively. Considering the x and y coordinates of the vectors in $(1)$ , it follows that $\cos \delta - \cos \gamma = 1 - \cos A - \cos C \qquad (2)$ $\sin \delta - \sin \gamma = \sin C - \sin A \qquad (3)$

Suppose $\sin C - \sin A = 0$ . Then $A = C$ , and the triangle is isosceles. In this case, it is clear by symmetry that $\overline{OI}$ is vertical. Further, since point $S$ exists, $\delta \neq \gamma$ , so $\delta + \gamma = 180$ and $\overrightarrow{MN} + \overrightarrow{QP}$ must be vertical as well.

For the remainder of the proof, assume $\sin C \neq \sin A$ . Note that $\frac{\cos y – \cos x}{\sin x - \sin y} = \frac{ (\cos y - \cos x)(\sin x + \sin y)}{ (\sin x - \sin y)(\sin x + \sin y)} = \frac {\sin x + \sin y}{\cos x + \cos y}$ whenever $x, y \in \mathbb{R}$ and $\sin x \neq \sin y$ . Note further that the slope of the line defined by the vector formed by summing vectors $(\cos x, \sin x)$ and $(\cos y, \sin y)$ is this expression. Since $\ell$ is parallel to $\overrightarrow{MN} + \overrightarrow{QP}$ , the slope of $\ell$ can be formed by dividing expressions in $(2)$ and $(3)$ and inverting the sign: $\frac{\cos A + \cos C – 1}{\sin C - \sin A} \qquad (4)$

Determine the coordinates of $I$ by drawing perpendiculars from $I$ to the sides and vertices of the triangle. By exploiting congruence between pairs of right triangles that share a vertex, one can partition $a,b,c$ into $p+q, p+r, q+r$ where $p,q,r$ are bases of these triangles that lie on the sides of triangle $ABC$ . From here it is clear that $I = \left(\frac{b + c – a}{2} , \frac{b + c – a}{2}\tan(A / 2)\right)$ .

To find the coordinates of $O$ , note that $\angle{OJK} + \angle{OJL} = \angle{J}$ and that $\angle {OJK} = \angle{OKJ}$ in any acute triangle $JKL$ . It easily follows that $\angle{OJK} = 90 – L$ . Note also that the perpendicular from $O$ to $\overline{JK}$ bisects $\overline{JK}$ . Hence, $O = \left(\frac{b}{2},\frac{b}{2} \cot B\right) \qquad (5)$ if triangle $ABC$ is acute.

If triangle $JKL$ is obtuse at $\angle J$ , then it can be similarly shown that $\angle{OKL} = \angle{OLK} = J – 90$ but that the remaining angles of this form are still $90-L$ and $90-K$ . It easily follows that $(5)$ holds if $\angle A$ is obtuse. If $\angle B$ is obtuse then $\angle OAC = B – 90$ and the $y$ coordinate of $O$ is $-\frac{b}{2} \tan{B-90}$ . From this, $(5)$ follows in this case as well.

We can conclude the slope of $\overline {OI}$ is $\left(\frac{b \cot {B} – (b + c – a) \tan(A/2)}{a – c}\right) = \left( \frac {\cos B – \tan{(A/2)}\sin B}{\sin A - \sin C}\right) + \tan(A/2) \quad (6)$ by the Law of Sines and rearrangement.

Setting $(6) = (4)$ is equivalent to $1 - \cos A - \cos C = \cos B + \tan(A/2)(\sin A - \sin B - \sin C)$

Since $\tan(A/2) = \frac{\sin A}{1 + \cos A}$ , this equation is equivalent to $(1 + \cos A)(\cos A + \cos B + \cos C – 1) = (\sin A)(\sin B + \sin C - \sin A)$

This equation is equivalent to $\cos(A + B) + \cos(A + C) + \cos B + \cos C = 0$ which is evident.

Solution 3

Let $A', B',$ and $C'$ be the arc midpoints of $BC, CA, AB,$ respectively. Let $E$ be crosspoint of $AI$ and $B'C'.$

Therefore $O$ is the circumcenter of triangle $A'B'C'.$

Points $A, I, E,$ and $A'$ are collinear. $\angle A'EB' = \frac {\overset{\Large\frown} {AC'}+\overset{\Large\frown} {B'C} +\overset{\Large\frown} {CA'}}{2} = 90^\circ \implies AA' \perp B'C'$ $\implies I$ is orthocenter of $\triangle A'B'C' \implies$

$IO$ is the Euler line of $\triangle A'B'C'.$

Let $G$ be the centroid of $\triangle A'B'C' \implies G$ lies on line $IO$ $\implies \overline {OG} = \frac {\overline{OA'} + \overline{OB'} + \overline{OC'}}{3}$ is paraller to $\overline{OI}.$ $\overline{OB'} \perp \overline{AC} \implies \overline{OB'} \perp \overline{QA}.$

Similarly $\overline{OC'} \perp \overline{AM}, \overline{OA'} \perp \overline{NP},$ rotation from $\overline{OA'}$ to $\overline{NP}$ , from $\overline{OB'}$ to $\overline{QA},$ and from $\overline{OC'}$ to $\overline{AM}$ is in clockwise direction, $|\overline{QA}|=| \overline{AM}| =|\overline{NP}|, |\overline{OA'}| = |\overline{OB'}| = |\overline{OC'}| \implies$

$\overline{QA} + \overline{AM} + \overline{NP} = \overline{QM} + \overline{NP}$ is perpendicular to $|\overline{OI}.$

$|\overline{MN}| = |\overline{QP}|$ therefore in accordance with Claim $\overline{MN} + \overline{QP}$ is parallel to $\overline{OI}.$

This sum is parallel to $\ell,$ , so we are done.

Claim

Let $|\overline{BA}|= |\overline{CD}|, \overline{CB} + |\overline{BA} + |\overline{AD} = |\overline{CD}.$ Then $(\overline{BA} + \overline{CD}) \perp (\overline{CB} + \overline{AD}).$

Proof

$(\overline{BA} + \overline{CD}) \cdot (\overline{CB} + \overline{AD}) = (\overline{BA} + \overline{CD}) \cdot (\overline{CB} + \overline{CD} – \overline{CB} – \overline{BA}) = |\overline{CD}|^2 – |\overline{BA}|^2 = 0.$

vladimir.shelomovskii@gmail.com, vvsss

Video Solution by MOP 2024

https://youtu.be/OnGdqIvOBd4

~r00tsOfUnity

@@ Line 76: / Line 76: @@
 '''vladimir.shelomovskii@gmail.com, vvsss'''
+==Video Solution by MOP 2024==
+https://youtu.be/OnGdqIvOBd4
+~r00tsOfUnity
 ==See also==
 {{USAMO newbox|year=2016|num-b=4|num-a=6}}
+{{MAA Notice}}

2016 USAMO (Problems • Resources)
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