Download SWMO-ARML Practice Materials February 7, 2016

SWMO-ARML Practice Materials
February 7, 2016
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
Find the ratio of the area to the perimeter of the circumscribed
pentagon when the inscribed circle has a radius of 2.
11. In ABC , AB = 86 and AC = 97. A circle with center A and radius AB intersects BC at
points B and X. Moreover, BX and CX have integer lengths. What is BC?
(A) 11
(B) 28
(C) 33
(D) 61
(E) 72
12. Real numbers x and y satisfy the equation x2  y 2  10x  6 y  34 . What is x + y?
(A) 1
(B) 2
(C) 3
(D) 6
(E) 8
13. Define a b  a 2b  ab 2 . Which of the following describes the set of points  x, y  for
which x y  y x ?
(A)
(B)
(C)
(D)
(E)
A finite set of points
One line
Two parallel lines
Two intersecting lines
Three lines
14. Positive integers a, b, c, and d satisfy a  b  c  d , a  b  c  d  2010 , and
a 2  b2  c 2  d 2  2010 . Find the number of possible values of a.
15. Maya lists all the positive divisors of 20102 . She then randomly selects two distinct
divisors from this list. Let p be the probability that exactly one of the selected divisors is
a perfect square. The probability p can be expressed in the form m , where m and n are
n
relatively prime positive integers. Find m + n.