Mathclub Problems 11-12

1. How many positive integers less than 50 have an odd number of positive integer divisors? (AHSME 1990 #11)

2. A positive integer N is a palindrome if the integer obtained by reversing the sequence of digits of N is equal to N. The year 1991 is the only year in the current century with the following two properties: (a) It is a palindrome. (b) It factors as a product of a 2-digit prime palindrome and a 3-digit prime palindrome. How many years in the millennium between 1000 and 2000 (including the year 1991) have properties (a) and (b)? (AHSME 1991 #17)

3. For a finite sequence A = ( a1 , a 2 ,..., a n ) of numbers, the Cesáro sum of A is defined to S + S 2 + ... + S n be 1 , where S k = a1 + a 2 + a3 + ... + a k and 1 ≤ k ≤ n . n If the Cesáro sum of the 99-term sequence (a1 , a 2 ,..., a 99 ) is 1000, what is the Cesáro sum of the 100-term sequence (1, a1 , a 2 ,..., a99 ) ? (AHSME 1992 #21)

4. An n-digit positive integer is cute if its n digits are an arrangement of the set {1,2,…,n} and its first k digits form an integer that is divisible by k, for k = 1,2,…,n. For example, 321 is a cute 3-digit integer because 1 divides 3, 2 divides 32, and 3 divides 321. How many cute 6-digit integers are there? (AHSME 1991 #26)

1 n (a + b n ) where a = 3 + 2 2 , b = 3 − 2 2 , and n = 0,1,2,... , then R12345 is 2 an integer. What is its units digit? (AHSME 1990 #30) 5. If Rn =

6. Find the smallest prime that is the fifth term of an increasing arithmetic sequence, all four preceding terms also being prime. (AIME 1999 #1)

7. Find the sum of all positive integers n for which n 2 − 19n + 99 is a perfect square. (AIME 1999 #3)

8. In an increasing sequence of four positive integers, the first three terms form an arithmetic progression, the last three terms form a geometric progression, and the first and fourth terms differ by 30. Find the sum of the four terms. (AIME 2003 #8)

9. Let N be the number of positive integers that are less than or equal to 2003 and whose base-2 representation has more 1’s than 0’s. Find the remainder when N is divided by 1000. (AIME 2003 #13)