1975 USAMO Problems/Problem 4: Difference between revisions

Revision as of 08:42, 13 May 2011

Problem

Two given circles intersect in two points $P$ and $Q$ . Show how to construct a segment $AB$ passing through $P$ and terminating on the two circles such that $AP\cdot PB$ is a maximum.

$[asy] size(150); defaultpen(fontsize(7)); pair A=(0,0), B=(10,0), P=(4,0), Q=(3.7,-2.5); draw(A--B); draw(circumcircle(A,P,Q)); draw(circumcircle(B,P,Q)); label("A",A,(-1,1));label("P",P,(0,1.5));label("B",B,(1,1));label("Q",Q,(-0.5,-1.5)); [/asy]$

Solution

by Vo Duc Dien

Let $E$ and $F$ be the centers of the small and big circles, respectively, and $r$ and $R$ be their respective radii.

Let $M$ and $N$ be the feet of $E$ and $F$ to $AB$ , and $\alpha = \angle APE$ and $\epsilon = \angle BPF$

We have:

$AP \times PB = 2r \cos{\alpha} \times 2R \cos{\epsilon} = 4 rR \cos{\alpha} \cos{\epsilon}$

$AP\times PB$ is maximum when the product $\cos{\alpha} \cos{\epsilon}$ is a maximum.

We have $\cos{\alpha} \cos{\epsilon}= \frac{1}{2} [\cos(\alpha +\epsilon) + \cos(\alpha -\epsilon)]$

But $\alpha +\epsilon = 180^{\circ} - \angle EPF$ and is fixed, so is $\cos(\alpha +\epsilon)$ .

So its maximum depends on $cos(\alpha -\epsilon)$ which occurs when $\alpha=\epsilon$ . To draw the line $AB$ :

Draw a circle with center $P$ and radius $PE$ to cut the radius $PF$ at $H$ . Draw the line parallel to $EH$ passing through $P$ . This line meets the small and big circles at $A$ and $B$ , respectively.

@@ Line 12: / Line 12: @@
 </asy>
+==Solution==
 by Vo Duc Dien
-Let E and F be the centers of the small and big circles, respectively, and r and R be their respective radii.
+Let <math>E</math> and <math>F</math> be the centers of the small and big circles, respectively, and <math>r</math> and <math>R</math> be their respective radii.
-Let M and N be the feet of E and F to AB, and α = ∠APE and ε = ∠BPF
+Let <math>M</math> and <math>N</math> be the feet of <math>E</math> and <math>F</math> to <math>AB</math>, and <math>\alpha = \angle APE</math> and <math>\epsilon = \angle BPF</math>
 We have:
-AP × PB = 2r cosα × 2R cosε = 4 rR cosα cosε
+<cmath>AP \times PB = 2r \cos{\alpha} \times 2R \cos{\epsilon} = 4 rR \cos{\alpha} \cos{\epsilon}</cmath>
-AP × PB is maximum when the product cosα cosε is a maximum.
-We have cosα cosε = ½ [cos(α + ε) + cos(α - ε)]
+<math>AP\times PB</math> is maximum when the product <math>\cos{\alpha} \cos{\epsilon}</math> is a maximum.
-But α + ε = 180° - ∠EPF and is fixed, so is cos(α + ε)
+We have <math>\cos{\alpha} \cos{\epsilon}= \frac{1}{2} [\cos(\alpha +\epsilon) + \cos(\alpha -\epsilon)]</math>
-So its maximum depends on cos(α - ε) which occurs when α = ε. To draw the line AB:
+But <math>\alpha +\epsilon = 180^{\circ} - \angle EPF</math> and is fixed, so is <math>\cos(\alpha +\epsilon)</math>.
-Draw a circle with center P and radius PE to cut the radius PF at H. Draw the line parallel to EH passing through P. This line meets the small and big circles at A and B, respectively.
+So its maximum depends on <math>cos(\alpha -\epsilon)</math> which occurs when <math>\alpha=\epsilon</math>. To draw the line <math>AB</math>:
-Solution with graph posted at
+Draw a circle with center <math>P</math> and radius <math>PE</math> to cut the radius <math>PF</math> at <math>H</math>. Draw the line parallel to <math>EH</math> passing through <math>P</math>. This line meets the small and big circles at <math>A</math> and <math>B</math>, respectively.
-http://www.cut-the-knot.org/wiki-math/index.php?n=MathematicalOlympiads.USA1975Problem4
 ==See also==
+[http://www.cut-the-knot.org/wiki-math/index.php?n=MathematicalOlympiads.USA1975Problem4 Solution with graph at Cut the Knot]
 {{USAMO box|year=1975|num-b=3|num-a=5}}

1975 USAMO (Problems • Resources)
Preceded by Problem 3	Followed by Problem 5
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1975 USAMO Problems/Problem 4: Difference between revisions

Revision as of 08:42, 13 May 2011

Problem

Solution

See also