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SW-ARML Practice 10-12-13
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5.
Find the value of a2 + a4 + a6 +    + a100 if a1, a2, a3, …, a100 is an arithmetic sequence with
common difference 1, and a1 + a2 + a3 +    + a100 = 137.
 200 
What is the largest 2-digit prime factor of the integer 
?
 100 
9 x 2 sin 2 x  4
Find the minimum value of
for 0 < x < .
x sin x
A point P is chosen at random in the interior of a unit square S. Let d(P) denote the distance
1
1
m
from P to the closest side of S. The probability that  d ( P )  is equal to , where m
5
3
n
and n are relatively prime positive integers. Find m + n.
Positive numbers x, y, and z satisfy xyz = 1081 and  log x  (log yz)   log y  (log z)  168 .
Find
6.
 log x    log y 
2
2
 (log z)2 .
Let N be the number of ordered pairs of nonempty sets A and B that have the following
properties:
 A  B = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12},
 A  B = ,

The number of elements of A is not an element of A,

The number of elements of B is not an element of B.
Find N.
7.
Find the number of second-degree polynomials f(x) with integer coefficients and integer
zeros for which f(0) = 2013.
8.
Jar A contains four liters of a solution that is 45% acid. Jar B contains five liters of a
solution that is 48% acid. Jar C contains one liter of a solution that is k% acid. From jar C,
m
liters of the solution is added to jar A, and the remainder of the solution in jar C is added
n
to jar B. At the end both jar A and jar B contain solutions that are 50% acid. Given that m
and n are relatively prime positive integers, find k + m + n.
9.
What is the product of the real roots of the equation x2 + 18x + 30 = 2 x2  18x  45 ?
10. Let an = 6n + 8n for all positive integers n. Determine the remainder upon dividing a83 by 49.
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