Number Theory
Let g be the greatest common divisor of 84 and 126. Find the greatest prime factor of g.
7
How many different arrangements are there of the letters in the word CAT?
6
You toss two fair dice. What is the probability of getting a “double,” that is, two 1’s or two 2’s or two 3’s or two 4’s or two 5’s or two 6’s? Express your answer as a common fraction.
1/6
Find the diagonal length of a rectangle with width 5 and length 12?
13
72
How many positive divisors does 72 have?
12
How many 3-digit numbers can be formed using the digits 1, 2, 3, 4, and 5 if repetition is not allowed?
60
Aleph, Beth, and Gimel compete every day to see who can get to school first. Aleph wins with probability 1/2, Beth wins with probability 1/3, and Gimel wins with probability 1/6. Today Gimel did not win. What is the probability that Beth won?
2/5
Find the surface area of a cube with volume 125.
150
The sum of 4 consecutive integers is 182. What is the largest of the 4 integers?
What is the greatest prime factor of 13^12 + 13^13 + 13^14?
61
How many of the integers from 1 to 100, inclusive, have at least one digit equal to 9?
18
Two different numbers are chosen at random from 1, 2, 3, ...,10. What is the probability that one of the numbers is twice the other? Express the answer as a fraction in lowest terms.
1/9
A regular n-sided polygon has 27 diagonals. What is the value of n?
9
If x + y = 10 and xy = 21, what is the value of x^2 + y^2?
Call a prime p lonely if neither p − 2 nor p + 2 is prime. What is the smallest lonely odd prime?
23
How many 5-digit numbers have digits that are strictly increasing from left to right?
126
Two fair dice are rolled. What is the average value of the non-negative 25 difference between the numbers showing on the two dice? Express the answer as a common fraction.
35/18
What is the area of the triangle formed by the line 6y + 7x = 126, the x-axis, and the y-axis?
189
James delivers newspapers. Today, he started his job at 9 am. At 10 am he realized that 225 newspapers were remaining and at 1 pm there were 90 newspapers left to deliver. Assume James delivered newspapers at a constant rate today. How many newspapers is James supposed to deliver today?
270
What is the remainder when 14 + 34 + 54 + 74 + 94 + 114 + 134 + 154 + 174 + 194 is divided by 10?
8
How many positive integers less than 2026 are divisible by 2 and 5 but not by 10?
Alphonse and Beth play the following game. Gamal, a neutral third party, flips one fair coin repeatedly. He continues until sometime or other during the tossing either (i) the sequence “HH” (two heads in a row) appears, in which case Alphonse wins or (ii) the sequence “TH” (tail followed by a head) appears, in which case Beth wins. What is the probability that Beth wins? Express your answer as a common fraction.
3/4
An archery target face is made of three concentric circles with radii 30 cm, 60 cm and 90 cm as seen below. An archer shoots two arrows hitting the target. What is the probability both of them land in the unshaded region? Express your answer as a fraction in lowest terms.
1/9
Suppose that there are two operations • and # between two numbers, which are given by x • y = xy + 1 and x#y = xy + x. For what value of y is x • y = y#x?
1