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Difference between revisions of "2016 AIME II Problems/Problem 14"

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==Solution==
 
==Solution==
 
The inradius of <math>\triangle ABC</math> is <math>100\sqrt 3</math> and the circumradius is <math>200 \sqrt 3</math>. Now, consider the line perpendicular to plane <math>ABC</math> through the circumcenter of <math>\triangle ABC</math>. Note that <math>P,Q,O</math> must lie on that line to be equidistant from each of the triangle's vertices. Also, note that since <math>P, Q, O</math> are collinear, and <math>OP=OQ</math>, we must have <math>O</math> is the midpoint of <math>PQ</math>.  Now, Let <math>K</math> be the circumcenter of <math>\triangle ABC</math>, and <math>L</math> be the foot of the altitude from <math>A</math> to <math>BC</math>. We must have <math>\tan(\angle KLP+ \angle QLK)= \tan(120^{\circ})</math>. Setting <math>KP=x</math> and <math>KQ=y</math>, assuming WLOG <math>x>y</math>, we must have <math>(\tan(120^{\circ})=-\sqrt{3}=\dfrac{\dfrac{x+y}{100 \sqrt{3}}}{\dfrac{30000-xy}{30000}}</math>. Therefore, we must have <math>100(x+y)=xy-30000</math>. Also, we must have <math>(\dfrac{x+y}{2})^{2}=(\dfrac{x-y}{2})^{2}+120000</math> by the Pythagorean theorem, so we have <math>xy=120000</math>, so substituting into the other equation we have <math>90000=100(x+y)</math>, or <math>x+y=900</math>. Since we want <math>\dfrac{x+y}{2}</math>, the desired answer is <math>\boxed{450}</math>.
 
The inradius of <math>\triangle ABC</math> is <math>100\sqrt 3</math> and the circumradius is <math>200 \sqrt 3</math>. Now, consider the line perpendicular to plane <math>ABC</math> through the circumcenter of <math>\triangle ABC</math>. Note that <math>P,Q,O</math> must lie on that line to be equidistant from each of the triangle's vertices. Also, note that since <math>P, Q, O</math> are collinear, and <math>OP=OQ</math>, we must have <math>O</math> is the midpoint of <math>PQ</math>.  Now, Let <math>K</math> be the circumcenter of <math>\triangle ABC</math>, and <math>L</math> be the foot of the altitude from <math>A</math> to <math>BC</math>. We must have <math>\tan(\angle KLP+ \angle QLK)= \tan(120^{\circ})</math>. Setting <math>KP=x</math> and <math>KQ=y</math>, assuming WLOG <math>x>y</math>, we must have <math>(\tan(120^{\circ})=-\sqrt{3}=\dfrac{\dfrac{x+y}{100 \sqrt{3}}}{\dfrac{30000-xy}{30000}}</math>. Therefore, we must have <math>100(x+y)=xy-30000</math>. Also, we must have <math>(\dfrac{x+y}{2})^{2}=(\dfrac{x-y}{2})^{2}+120000</math> by the Pythagorean theorem, so we have <math>xy=120000</math>, so substituting into the other equation we have <math>90000=100(x+y)</math>, or <math>x+y=900</math>. Since we want <math>\dfrac{x+y}{2}</math>, the desired answer is <math>\boxed{450}</math>.
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Solution by Shaddoll

Revision as of 19:14, 17 March 2016

Equilateral $\triangle ABC$ has side length $600$. Points $P$ and $Q$ lie outside the plane of $\triangle ABC$ and are on opposite sides of the plane. Furthermore, $PA=PB=PC$, and $QA=QB=QC$, and the planes of $\triangle PAB$ and $\triangle QAB$ form a $120^{\circ}$ dihedral angle (the angle between the two planes). There is a point $O$ whose distance from each of $A,B,C,P,$ and $Q$ is $d$. Find $d$.

Solution

The inradius of $\triangle ABC$ is $100\sqrt 3$ and the circumradius is $200 \sqrt 3$. Now, consider the line perpendicular to plane $ABC$ through the circumcenter of $\triangle ABC$. Note that $P,Q,O$ must lie on that line to be equidistant from each of the triangle's vertices. Also, note that since $P, Q, O$ are collinear, and $OP=OQ$, we must have $O$ is the midpoint of $PQ$. Now, Let $K$ be the circumcenter of $\triangle ABC$, and $L$ be the foot of the altitude from $A$ to $BC$. We must have $\tan(\angle KLP+ \angle QLK)= \tan(120^{\circ})$. Setting $KP=x$ and $KQ=y$, assuming WLOG $x>y$, we must have $(\tan(120^{\circ})=-\sqrt{3}=\dfrac{\dfrac{x+y}{100 \sqrt{3}}}{\dfrac{30000-xy}{30000}}$. Therefore, we must have $100(x+y)=xy-30000$. Also, we must have $(\dfrac{x+y}{2})^{2}=(\dfrac{x-y}{2})^{2}+120000$ by the Pythagorean theorem, so we have $xy=120000$, so substituting into the other equation we have $90000=100(x+y)$, or $x+y=900$. Since we want $\dfrac{x+y}{2}$, the desired answer is $\boxed{450}$.

Solution by Shaddoll

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