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SW-ARML Practice 10-14-12
1.
The rational number r is the largest number less than 1 whose base-7 expansion consists of two
p
distinct repeating digits, r = (0.ABABAB…)7. Written as a reduced fraction, r = . Compute
q
p + q. [Hint: If it were base-10, r would be 0.989898….]
2.
Compute the least positive integer n such that the set of angles {123, 246, …., n123} contains
at least one angle in each of the four quadrants.
3.
Let set A be a 90-element subset of {1, 2, 3, 4, …., 98, 99, 100} and let S be the sum of the
elements of A. Find the number of possible values of S. [Hint: Find the largest and smallest sums
first.]
4.
Find the least positive integer such that when its leftmost digit is deleted, the resulting integer is
1
of the original integer. [Hint: An integer can be expressed as 10na + b if a is the leftmost
29
digit.]
5.
Let (a1, a2, a3, …, a12) be a permutation of {1, 2, 3, …, 12} for which a1> a2 > a3 > a3 > a5 > a6
and a6 < a7 < a8 < a9 < a10 < a11 < a12. An example of such a permutation is
(10, 7, 5, 3, 2, 1, 4, 6, 8, 9, 11, 12). Find the number of such permutations. [Hint: What must a6
be?]
6.
 200 
 200 
200!

What is the largest 2–digit prime factor of the integer 
? [Hint: 
]


 100 
 100  100!100!
7.
Let S be the set of real numbers that can be represented as repeating decimals of the form 0.abc ,
where a, b, c are distinct digits. Find the sum of the elements of S. [Hint: All 0.abc have the same
denominator as fractions.]
8.
The lengths of the sides of a triangle with positive area are log 12, log 75, and log n, where n is a
positive integer. Find the number of possible values of n. [Hint: The Triangle Inequality.]
9.
Let P be the product of the first 100 positive odd integers. Find the largest integer k such that P is
1 2  3 10
10!
 5
divisible by 3k. [Hint: For example, 13579 =
.]
2  4  6 10 2  5!
10. A sequence is defined as follows:
a1 = a2 = a3 = 1, and for all positive integers n, an+3 = an+2 + an+1 + an. Given that a28 = 6090307,
a29 = 11201821, and a30 = 20603361, find the remainder when

28
a  a1  a2 
k 1 k
 a28 is
divided by 1000.
1