Difference between revisions of "1993 AHSME Problems/Problem 25"

Revision as of 02:36, 27 September 2014

Problem

$[asy] draw((0,0)--(1,sqrt(3)),black+linewidth(.75),EndArrow); draw((0,0)--(1,-sqrt(3)),black+linewidth(.75),EndArrow); draw((0,0)--(1,0),dashed+black+linewidth(.75)); dot((1,0)); MP("P",(1,0),E); [/asy]$

Let $S$ be the set of points on the rays forming the sides of a $120^{\circ}$ angle, and let $P$ be a fixed point inside the angle on the angle bisector. Consider all distinct equilateral triangles $PQR$ with $Q$ and $R$ in $S$ . (Points $Q$ and $R$ may be on the same ray, and switching the names of $Q$ and $R$ does not create a distinct triangle.) There are

$\text{(A) exactly 2 such triangles} \quad\\ \text{(B) exactly 3 such triangles} \quad\\ \text{(C) exactly 7 such triangles} \quad\\ \text{(D) exactly 15 such triangles} \quad\\ \text{(E) more than 15 such triangles}$

Solution

$\fbox{E}$

1993 AHSME (Problems • Answer Key • Resources)
Preceded by Problem 24	Followed by Problem 26
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30
All AHSME Problems and Solutions

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Difference between revisions of "1993 AHSME Problems/Problem 25"

Revision as of 02:36, 27 September 2014

Problem

Solution

See also

@@ Line 1: / Line 1: @@
 == Problem ==
 <asy>
-draw((0,0)--(1,sqrt(3)),black+linewidth(.75));
+draw((0,0)--(1,sqrt(3)),black+linewidth(.75),EndArrow);
-draw((0,0)--(1,-sqrt(3)),black+linewidth(.75));
+draw((0,0)--(1,-sqrt(3)),black+linewidth(.75),EndArrow);
 draw((0,0)--(1,0),dashed+black+linewidth(.75));
 dot((1,0));