Category: math competition

The positive integers A,B,A-B,and A+B are all prime numbers. The sum of these four primes is

(A) even (B) divisible by 3 (C) divisible by 5 (D) divisible by 7 (E) prime

ANSWER: E

The product of three consecutive positive integers is 8 times their sum. What is the sum of the squares?

(A) 50 (B) 77 (C) 110 (D) 149 (E) 194

ANSWER: B

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

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

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

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

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

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

The mean, median, unique mode, and range of a collection of eight integers are all equal to 8. The largest integer that can be an element of this collection is

(A) 11 (B) 12 (C) 13 (D) 14 (E) 15

ANSWER: D

A 3*3*3 cube is made of 27 normal dice. Each die’s opposite sides sum to 7. What is the smallest possible sum of all of the values visible on the 6 faces of the large cube ?

(A) 60 (B) 72 (C) 84 (D) 90 (E) 96

ANSWER: D