SW-ARML 3-11-12
1.
How many four-digit integers have the sum of their two leftmost digits equals the sum of
their two rightmost digits?
2.
For each positive integer k, let Sk denote the increasing arithmetic sequence of integers
whose first term is 1 and whose common difference is k. For example, S3 is the sequence 1,
4, 7, 10, … For how many values of k does Sk contain the term 1001?
3.
Find the least positive integer such that when its leftmost digit is deleted, the resulting
1
integer is
of the original integer.
29
4.
How many positive integers have exactly three proper divisors (positive integral divisors
excluding itself), each of which is less than 50?
5.
The digits of a positive integer n are four consecutive integers in decreasing order when read
from left to right. What is the sum of the possible remainders when n is divided by 37?
6.
The equation 2333x–2 + 2111x+2 = 2222x+1 + 1 has three real roots. Given that their sum is
m
,
n
where m and n are relatively prime positive integers, find m + n.
7.
Corresponding terms of two arithmetic progressions are multiplied to give the sequence
1440, 1716, 1848, ... . Find the eighth term.
8.
How many positive integer divisors of 20122012 are divisible by exactly 2012 positive
integers? (Hint: The number of divisors of n = paqb is (a + 1)((b + 1), where p, q, … are
distinct primes.)
9.
The roots of x4 – x3 – x2 – 1 = 0 are a, b, c, d. Find p(a) + p(b) + p(c) + p(d), where
p(x) = x6 – x5 – x3 – x2 – x. (Hint: x4 – x3 – x2 – 1 = (x+1)(x3 – 2x2 + x – 1); General Facts:
If r1, r2, r3 are roots of p(x) = x3 + x2 + x + , then r1 + r2 + r3 = –, r1r2 + r1r3 + r2r3 = ,
r1r2r3 = –.)
10. Let C be the coefficient of x2 in the expansion of the product
(1 – x)(1 + 2x)(1 – 3x)(1 + 14x)(1 – 15x). Find C.
1