SW-ARML 3-11-12
- How many four-digit integers have the sum of their two leftmost digits equals the sum of their two rightmost digits?
- 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?
- Find the least positive integer such that when its leftmost digit is deleted, the resulting integer is of the original integer.
- How many positive integers have exactly three proper divisors (positive integral divisors excluding itself), each of which is less than 50?
- 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 nis divided by 37?
- The equation 2333x–2 + 2111x+2 = 2222x+1 + 1 has three real roots. Given that their sum is ,where m and n are relatively prime positive integers, find m + n.
- Corresponding terms of two arithmetic progressions are multiplied to give the sequence 1440, 1716, 1848, ... . Find the eighth term.
- 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.)
- 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 = –.)
- 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.
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