2011 AMC 12A Problems/Problem 25

Revision as of 00:21, 15 February 2011 by Moplam (talk | contribs) (Solution)

Problem

Triangle $ABC$ has $\angle BAC = 60^{\circ}$, $\angle CBA \leq 90^{\circ}$, $BC=1$, and $AC \geq AB$. Let $H$, $I$, and $O$ be the orthocenter, incenter, and circumcenter of $\triangle ABC$, repsectively. Assume that the area of pentagon $BCOIH$ is the maximum possible. What is $\angle CBA$?

$\textbf{(A)}\ 60^{\circ} \qquad \textbf{(B)}\ 72^{\circ} \qquad \textbf{(C)}\ 75^{\circ} \qquad \textbf{(D)}\ 80^{\circ} \qquad \textbf{(E)}\ 90^{\circ}$

Solution

25) Answer: (D) 80 degree

Given: $BC = 1$, $\angle BAC = 60^{\circ}$, $\angle CBA \le 90^{\circ}$, $AC \ge BC$

$H$, $I$, $O$ are orthocenter, incenter, and circumcenter. and $BOIHC$ has maximum area.

Find $\angle CBA$.


Solution:

1) Let's draw a circle with center $O$ (which will be the circumcircle of $\triangle ABC$. Since $\angle BAC = 60^{\circ}$, $\overline{BC}$ is a chord that intercept an arc of $120 ^{\circ}$

2) Draw any chord that can be $BC$, and lets define that as unit length.

3) Draw the diameter $\perp$ to $BC$. Let's call the interception of the diameter with $BC$ $M$ (because it is the midpoint) and interception with the circle $X$.

4) Note that OMB and XMC is fixed, hence the area is a constant. Thus, $XOIHC$ also achieved maximum area.


\textbf{Lemma)} $m\angle BOC = m \angle BIC = m \angle BHC = 120^{\circ}$

For $m\angle BOC$, we fixed it to $120^{\circ}$ when we drew the diagram.

Let $m\angle ABC = \beta$, $m\angle ACB = \gamma$


Now, lets isolate the points $A$,$B$,$C$, and $I$.

$m\angle IBC = \frac{\beta}{2}$, $m\angle ICB = \frac{\gamma}{2}$

$m\angle BIC = 180^{\circ} - \frac{\beta}{2} - \frac{\gamma}{2} = 180^{\circ} - \frac{180^{\circ} - 120^{\circ}}{2} = 120 ^{\circ}$


Now, lets isolate the points $A$,$B$,$C$, and $H$.

$m\angle HBC = \beta - 30^{\circ}$, $m\angle HCB = \gamma - 30^{\circ}$

$m\angle BHC = 180^{\circ} - \beta - \gamma + 60^{\circ} = 240^{\circ} - 120^{\circ} = 120^{\circ}$


Lemma proven. The lemma yields that BOIHC is a cyclic pentagon.

Since we got that XOIHC also achieved maximum area,

Let $m\angle XOI = x_1$, $m\angle OIH = x_2$, $m\angle IHC = x_3$, and the radius is $R$ (which will drop out.)

then area = $\frac{r^2}{2}(\sin x_1 + \sin x_2 + \sin x_3)$, where $x_1 + x_2 + x_3 = 60^\circ$

So we want to maximize $f(x_1, x_2) = \sin x_1 + \sin x_2 + \sin x_3$, Note that $x_3 = 60 ^\circ - x_1 - x_2$.

Let's do some multi-variable calculus.

$f_{x_1} = \cos x_1 - \cos (x_3)$, $f_{x_2} = \cos x_2 - \cos (x_3)$

If both partial is zero, then $x_1 = x_2 = x_3 = 20^\circ$, and it is very easy to show that $f(x_1, x_2)$ is maximum here with second derivative test (left for the reader).


Now, we need to verify that such situation exist and find the angle for this situation.

Let's extend $AI$ to the direction of $X$, since $AI$ is the angle bisector, $AI$ should intersection the midpoint of the arc, which is $X$. Hence, if such case exist, $m\angle AXB = m \angle ACB = 40 ^\circ$, which yield that $m\angle CBA = 80 ^\circ$.

If the angle is $80 ^\circ$, it is clear that since $I$ and $H$ are on the second circle (follow from lemma). $I$ will be at the right place. $H$ can be easily verify too.


Hence, the answer is $(D)$.

See also

2011 AMC 12A (ProblemsAnswer KeyResources)
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