1. How many positive integers from 1 to 500 yield a remainder of 1 when divided by 7?

2. If the perimeter of an isosceles right triangle is
, what is the length of the hypotenuse of the triangle?

3. Jack and Henrietta were each paid a dollars in advance to do a job. Jack spent 10 hours on the job, and Henrietta spent 8 hours on the job. If Henrietta gave Jack b dollars of her advance so that their hourly wages were equal, what is a in terms of b?

4. Jar A contains 20% alcohol and 80% water. Jar B contains 40% alcohol and 60% water. If jars A and B are combined to form a solution that is 25% alcohol and 75% water, then the volume of jar A is what fraction of the resulting mixture?

5. If a fair-sided coin is flipped four times, what is the probability that the coin will land on heads exactly three times?

6. What is the greatest integer p such that 15^p is a factor of 25!?

7. Forty-five people were polled about their preferences for pizza, tacos, and salad. Sixteen people stated that they liked pizza, 19 people stated that they liked tacos, and 22 people stated that they liked salad. Eight people stated that they liked both pizza and tacos, 7 people stated that they liked both tacos and salad, and 9 people stated that they liked both pizza and salad. If 39 people preferred at least one of the foods, and the number of people who stated that they liked all three equals the number of people who stated that they don’t like any of the foods, how many of the 45 people stated that they liked only one of the foods?

8. Working together at their respective constant rates, machines A and B filled an empty pool in 5 hours. Working together at their respective constant rates, machines A and C filled the same pool in 8 hours. Working together at their respective constant rates, machines B and C filled the same pool in 10 hours. How many hours would it take machines A, B, and C, working together but independently, to fill the pool?

9. The infinite sequence x₁, x₂, . . . , x_n is such that x₁ = 3, x₂ = 5, x₃ = –1, x₄ = 7, and x_n = x_n–4, for all n > 4. What is the sum of the first 102 terms of the sequence?

10. A certain set of 100 numbers has an average (arithmetic mean) of 10 and a standard deviation of x, where x is positive. Which of the following pairs of numbers, if added to the set, must reduce the standard deviation?

11. From a group of four married couples, three individuals will be selected to form a committee. If no spouses can be on the committee, how many groups can be selected?

12. A certain regular hexagon has side lengths of 6. What is the area of the hexagon?

13. Which of the following is not a factor of 5⁸ – 2⁸?

14. If 10²⁷ – 3 is expressed in base-10 notation, what is the sum of the digits?

15. If x and y are positive integers and
, then what is the value of x + y ?