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Problem 57

Using the letters A,M,O,S,and U, we can form five-letter “words”. If these “words” are arranged in alphabetical order, then the “word” USAMO occupies position

(A) 112 (B) 113 (C) 114 (D) 115 (E) 116

ANSWER: D

Problem 56

Suppose July of year N has five Mondays. Which of the following must occur five times in the August of year N? (Note: Both months have 31 days)

(A) Monday (B) Tuesday (C) Wednesday (D) Thursday (E) Friday

ANSWER: D

Problem 55

The arithmetic mean of the nine numbers in the set {9,99,999,9999,…,999999999} is a 9-digit number M, all of whose digits are distinct. The number M does not contain the digit

(A) 0 (B) 2 (C) 4 (D) 6 (E) 8

ANSWER: A

Problem 54

Tina randomly selects two distinct numbers from the set {1,2,3,4,5}, and Sergio randomly selects a number from the set {1,2,…,10}. What is the probability that Sergio’s number is larger than the sum of the two numbers chosen by Tina ?

(A) 2/5 (B) 9/20 (C) 1/2 (D) 11/20 (E) 24/25

ANSWER: A

Problem 53

Points A,B,C, and D lie on a line, in that order, with AB=CD and BC=12. Point E is not on the line, and BE=CE=10. The perimeter of triangle AED is twice the perimeter of triangle BEC. Find AB.

(A) 15/2 (B) 8 (C) 17/2 (D) 9 (E) 19/2

ANSWER: D

Problem 52

A set of tiles numbered 1 through 100 is modified repeatedly by the following operation: remove all tiles numbered with a perfect square, and renumbered the remaining tiles consecutively starting with 1. How many times must the operation be performed to reduce the number of tiles in the set to one ?

(A) 10 (B) 11 (C) 18 (D) 19 (E) 20

ANSWER:C