Problem 1

The arithmetic mean of the nine numbers in the set??is a?-digit number?, all of whose digits are distinct. The number??does not contain the digit

Problem 2

What is the value of

when??

Problem 3

For how many positive integers??is??a prime number?

Problem 4

Let??be a positive integer such that??is an integer. Which of the following statements is?not?true:

Problem 5

Let??and??be the degree measures of the five angles of a pentagon. Suppose that??and??and??form an arithmetic sequence. Find the value of?.

Problem 6

Suppose that??and??are nonzero real numbers, and that the equation??has solutions??and?. Then the pair??is

Problem 7

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

Problem 8

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

Problem 9

If??are positive real numbers such that??form an increasing arithmetic sequence and??form a geometric sequence, then?is

Problem 10

How many different integers can be expressed?as?the sum of three distinct members of the set??

Problem 11

The positive integers??and??are all prime numbers. The sum of these four primes is

Problem 12

For how many integers??is??the square of an integer?

Problem 13

The sum of??consecutive positive integers is a perfect square. The smallest possible value of this sum is

Problem 14

Four distinct circles are drawn in a plane. What is the maximum number of points where at least two of the circles intersect?

Problem 15

How many four-digit numbers??have the property that the three-digit number obtained by removing the leftmost digit is one ninth of??

Problem 16

Juan rolls a fair regular octahedral die marked with the numbers??through?. Then Amal rolls a fair six-sided die. What is the probability that the product of the two rolls is a multiple of 3?

Problem 17

Andy’s lawn has twice as much area as Beth’s lawn and three times as much area as Carlos’ lawn. Carlos’ lawn mower cuts half as fast as Beth’s mower and one third as fast as Andy’s mower. If they all start to mow their lawns at the same time, who will finish first?

Problem 18

A point??is randomly selected from the rectangular region with vertices?. What is the probability that??is closer to the origin than it is to the point??

Problem 19

If??and??are positive real numbers such that??and?, then??is

Problem 20

Let??be a right-angled triangle with?. Let??and??be the midpoints of legs??and?, respectively. Given that??and?, find?.

Problem 21

For all positive integers??less than?, let

Calculate?.

Problem 22

For all integers??greater than?, define?. Let??and?. Then?equals

Problem 23

In?, we have??and?. Side??and the median from??to??have the same length. What is??

Problem 24

A convex quadrilateral??with area??contains a point??in its interior such that?. Find the perimeter of?.

?

Problem 25

Let?, and let??denote the set of points??in the coordinate plane such thatThe area of??is closest to?