Combinatorics
Lance, Sally, Joy, Fred, Justin, and Patrick are chosen for the team. In how many ways can the three starters be chosen?
The sum of the measures of the first three interior angles of a pentagon is 345. The measure of the fourth angle is the average of the measures of the first three. Compute the number of degrees in the measure of the fifth angle.
80
Real numbers x and y satisfy the equation x2 + y2 = 10x - 6y - 34. What is x+y?
2
What is the units digit of 13^{2012}?
1
What is the expected value of the product of two six-sided dice?
49/4 or 12.25
A singles tournament had six players. Each player played every other player only once, with no ties. If Helen won 4 games, Ines won 3 games, Janet won 2 games, Kendra won 2 games and Lara won 2 games, how many games did Monica win?
2
An equilateral triangle of side length 10 is completely filled in by non-overlapping equilateral triangles of side length 1. How many small triangles are required?
100
What non-zero real value for x satisfies (7x)^14=(14x)^7
2/7
What is the value of 4 x (-1+2-3+4-5+6-7+...+1000)?
2000
Harold tosses a nickel four times. The probability that he gets at least as many heads as tails is?
11/16
If x, y, and z are positive integers, how many values of x, y, and z satisfy: x + y + z = 8?
21
An equilateral triangle and a regular hexagon have equal perimeters. If the area of the triangle is 4, what is the area of the hexagon?
6
How many integers between 1 and 2010 inclusive are divisible by neither 3 nor 5?
1072
How many 4-digit numbers greater than 1000 are there that use the four digits of 2012?
9
Two fair coins are to be tossed once. For each head that results, one fair die is to be rolled. What is the probability that the sum of the die rolls is odd? (Note that if no die is rolled, the sum is 0.)
3/8
A set of tiles numbered 1 through 100 is modified repeatedly by the following operation: remove all tiles numbered with a perfect square, and renumber 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?
18
The midpoints of a regular hexagon are joined to form another regular hexagon inside. What is the ratio of the area of the inner hexagon to the area of the outer hexagon?
3/4
The number (811_)9 (where the 9 represents base-9 number) is a perfect square. What must the last digit be?
7
For each positive integer n, the mean of the first n terms of a sequence is n. What is the 2008th term of the sequence?
4015
Six distinct positive integers are randomly chosen between 1 and 2006, inclusive. What is the probability that some pair of these integers has a difference that is a multiple of 5?
1
George has 7 pennies, 1 nickel, 1 dime, 1 quarter, and 1 half-dollar. How many unique sums can he make with the coins he has, given he must use at least 2 coins?
91
If log(xy3) = 1 and log(x2y) = 1, what is log(xy)?
3/5
How many perfect squares are divisors of the product 1! * 2! * 3! * 4! * 5!
10
There are 6 positive and 8 negative numbers. Four numbers are chosen at random and multiplied. What is the probability that the product is a positive number?
505/1001