1960 AMC 12 Problem 35
Attempt Problem 35 of the 1960 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1960 AMC 12 solutions, or check the answer key.
All problems are used with official legal permission of the Mathematical Association of America (MAA).
35.
From point outside a circle, with a circumference of units, a tangent is drawn. Also from a secant is drawn dividing the circle into unequal arcs with lengths and It is found that the length of the tangent, is the mean proportional between and If and are integers, then may have the following number of values:
zero
one
two
three
infinitely many
Answer: C
Small Hint:
Use and
Big Hint:
Test the integer values excluding equal arcs
Solution:
The arc lengths satisfy while the mean-proportional condition gives For integral from through the possible products, up to symmetry, are and The last comes from equal arcs and is excluded. The remaining squares give and two values.
Thus, the correct answer is C.
Problem 35 in Other Years
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