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MO-ARML – 11/23/14 Practice Question
1. If x, y, and z are positive numbers and x 
1
1
1 7
 4, y   1, z   , find the value of xyz.
y
z
x 3
2. A circle centered at A with a radius of 1 and a circle centered at B with
radius 4 are externally tangent. A third circle is tangent to the first two
circles and to one of their common external tangents as shown. Find the radius
of the third circle. (If you would like, use Descartes’ Circle Formula for four mutually externally
tangent circles with curvatures a, b, c, and d: (a + b + c + d)2 = 2(a2 + b2 + c2 + d2), where a
1
circle’s curvature is equal to
. A straight line can be considered as a circle
radius
with radius  and thus curvature 0.)
3. Circles A, B, and C are externally tangent to each other, and internally tangent
to circle D. Circles B and C are congruent. Circle A has radius 1 and passes
through the center of D. What is the radius of circle B? (To use Descartes’ Circle
Formula, we have to attach a ‘’ sign to the curvature of D because the three
other circles are internally tangent to D (same as keep D’s and negate all three others’.)
4. Find the number of order pairs of real numbers (a, b) such that (a  bi )2014 = a – bi.
5. Two distinct numbers a and b are chosen randomly from the set 2, 22 , 23 ,..., 224 , 225  . What is the
probability that log a b is an integer?
6. A point P is randomly selected from the rectangular region with vertices (0, 0), (0, 1), (2, 1), and
(2, 0). What is the probability that P is closer to the origin (0, 0) than the point (3, 1)?
7. How may perfect squares are divisors of the product 1!2!3!9!?
8. The graph of 2 x 2  xy  3 y 2  11x  20 y  40  0 is an
ellipse in the first quadrant of the xy-plane. Let a and b
y
be the maximum and minimum values of over all points
x
(x, y) on the ellipse. What is the value of a + b?
9. For each integer n  4, let an denote the base-n number 0.133n . The product a4a5a99 can be
m
expressed as
, where m and n are positive integers and n is as small as possible. Find m.
n!
10. Given a sequence S = a1 , a2 ,..., an  of n real numbers, where n > 1, let A(S) be the sequence
a a 
 a1  a2 a2  a3
,
,..., n 1 n  of n – 1 real numbers. Define A1(S) = A(S), and recursively

2
2
 2

 1 
Am(S) = Am-1(S) for 2  m  n  1. Suppose a > 0 and S = 1, a,..., a100  . If A100(S)=  50  , find
2 
the value of a.