Math Problem : HSTEST'09

1. Simplify ½ + ⅓ + ¼

2. A quart of liquid contains 10% alcohol, and another 3-quart bottle full of liquid contains 30% alcohol. They are mixed together. What is the percentage of alcohol in the mixture?

3. You run the first mile at 10 miles per hour, and the second mile at 8 miles per hour. How many miles per hour is your average speed for the two-mile run?

4. What is the circumference of a circle inside of which is inscribed a triangle with side lengths 3, 4, and 5? (Note that the triangle is inside the circle.)

5. If all people eat the same amount of pizza, and a pizza 12 inches in diameter serves two people, how many inches in diameter should each of two pizzas be in order to serve three people? (Pizzas are circular and are eaten entirely.)

6. How many integers between 90 and 100 are prime?

7. If f(x) = (3x+1) : (2x – 1) , then write a formula in simplified form for f(a + 2). There should be no parentheses in your formula.

8. Circle B passes through the center of circle A and is tangent to it. Circle C passes through the center of circle B and is tangent to it. What fraction of the area of circle A lies inside circle B but outside circle C?

9. How many integers between 1 and 99, inclusive, are not divisible by either 3 or 7? (This means that they are not divisible by 3 and also they are not divisible by 7.)

10. The number 1234567891011....585960, which consists of the first 60 positive integers written in order to form a single number with 111 digits, is modi¯ed by removing 100 of its digits. What is the smallest number which can be obtained in this way? (If the number begins with some 0-digits, you may write it either with or without the 0's. Credit will be given for both answers.)

11. Point P is inside regular octagon ABCDEFGH so that triangle ABP is equilateral.

How many degrees are in angle APC?

12. What 5-digit number 32a1b is divisible by 156? (Here a and b represent digits.)

13. Mary paid $480 to purchase a certain number of items, but the nice vendor gave her two extra. This decreased the price per item by $1. How many items did she receive (including the two extra)?

14. There is only one positive integer n for which the number obtained by removing the last three digits of n exactly equals the cube root of n. What is this integer n?

15. In how many ways can a group of 16 people be divided into eight pairs? (You may write your answer as a product of integers without multiplying out. You may also use symbols such as factorials or exponentials.)