Difference between revisions of "1993 AHSME Problems/Problem 25"
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</asy> | </asy> | ||
− | Let <math>S</math> be the set of points on the rays forming the sides of a <math>120^\circ</math> angle, and let <math>P</math> be a fixed point inside the angle on the angle bisector. Consider all distinct equilateral triangles <math>PQR</math> with <math>Q</math> and <math>R</math> in <math>S</math>. (Points <math>Q</math> and <math>R</math> may be on the same ray, and switching the names of <math>Q</math> and <math>R</math> does not create a distinct triangle.) There are | + | Let <math>S</math> be the set of points on the rays forming the sides of a <math>120^{\circ}</math> angle, and let <math>P</math> be a fixed point inside the angle |
+ | on the angle bisector. Consider all distinct equilateral triangles <math>PQR</math> with <math>Q</math> and <math>R</math> in <math>S</math>. | ||
+ | (Points <math>Q</math> and <math>R</math> may be on the same ray, and switching the names of <math>Q</math> and <math>R</math> does not create a distinct triangle.) | ||
+ | There are | ||
− | <math>\text{(A) exactly 2 such triangles} \quad | + | <math>\text{(A) exactly 2 such triangles} \quad\ |
− | \text{(B) exactly 3 such triangles} \quad | + | \text{(B) exactly 3 such triangles} \quad\ |
− | \text{(C) exactly 7 such triangles} \quad | + | \text{(C) exactly 7 such triangles} \quad\ |
− | \text{(D) exactly 15 such triangles} \quad | + | \text{(D) exactly 15 such triangles} \quad\ |
\text{(E) more than 15 such triangles} </math> | \text{(E) more than 15 such triangles} </math> | ||
Revision as of 02:07, 27 September 2014
Problem
Let be the set of points on the rays forming the sides of a angle, and let be a fixed point inside the angle on the angle bisector. Consider all distinct equilateral triangles with and in . (Points and may be on the same ray, and switching the names of and does not create a distinct triangle.) There are
Solution
See also
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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