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Preliminary Test
You have 2 hours to finish this test (at maximum). Please pace yourself wisely! You can use a calculator.

If you do not know what some characters mean, here is a sheet of symbols:

"^" is a caret, which is used in exponents. So, 2^3=8, where it is 2 to the power of 3.
A cube number is a number that can be expressed as a^3 for some integer a. It is the third power of something.
The "sqrt()" is a symbol for a square root. So, sqrt(9)=3.
Angle ABC is going to be the angle formed by line segments AB and BC.
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Pace yourself wisely!
You have 2 hours! For a list of symbols, look at the very top.
1) What is the sum of the first 61 positive integers?
1 point
2) Let a, b, and c be numbers. Simplify the expression (a-(b-c))-((a-b)-c)
1 point
3) What is the sum of the digits of 333x999?
1 point
4) Express 3^16 as a power of 1/9. So, write it as (1/9)^a for some number a.
1 point
5) Express 91 as the sum of 2 positive cubes. So, write it as x^3+y^3 for some numbers x and y.
1 point
6) Express (2^5)^6/(4^3) as a power of 2. So, write it as 2^x for some number x
1 point
7) If the 4-digit number 72d2 is divisible by 6, what is the largest possible value of the digit d?
1 point
8) A light flashes every 1 minute 15 seconds. Another flashes every 1 minute 40 seconds. Suppose they flash together at a certain time. What is the shortest amount of time that will elapse before both lights will flash together again? Answer in minutes.
1 point
9) When the six-digit number 3456n7 is divided by 8, the remainder is 5. List both possible values of the digit n, separated by a comma.
1 point
10) The reciprocal of 5 plus the reciprocal of 7 is the reciprocal of what mixed number? With mixed numbers, write your answer as a b/c. Notice that there is a space in between the fraction and the whole number.
1 point
11) Loki, Moe, Nick, and Ott are good friends. Ott had no money, but the others did. Moe gave Ott one-fifth of his money, Loki gave Ott one-fourth of his money, and Nick gave Ott one-third of his money. Each gave Ott the same amount of money. Which fractional part of the group's money does Ott now have?
1 point
12) Six more than double a number equals twelve less than half the number. What is the number?
1 point
13) A number x is twice its reciprocal. What is x^6?
1 point
14) Express the sum 1/2+0.1/2+1/0.2 as a decimal.
1 point
15) What is the smallest positive integer k such that k/660 can be expressed as a terminating decimal?
1 point
16) Boston is 295 miles from NYC along a certain route. A car starts from Boston at 1:00 pm and travels along the route toward NY at a steady rate of 50 mph. Another car starts from NY at 1:30 pm and travels along this route toward Boston at a steady rate of 40 mph. At what time do the cars pass each other?
1 point
17) Kim was elected class president. She received 3 votes for every 2 that Amy got. No one else ran. However, if 8 of the people who voted for Kim had voted for Amy instead, Kim would have received only 1 vote for every 2 that Amy would have gotten. How many people voted?
1 point
18) Jack had a bag of 128 apples. He sold 25% of them to Jill. Next he sold 25% of those remaining to June. Of those apples still in his bag, he gave the most shiny one to his teacher. How many apples did Jack have then?
1 point
19) Find x if 20% of x equals 12% of (x+20).
1 point
20) Find all values of h such that 3sqrt(27)/h=h/(27sqrt(3)). Separate them by a comma.
1 point
21) If angle A=20 and angle AFG=angle AGF in the diagram below, then how many degrees is B+D?
1 point
Captionless Image
22) In the quadrilateral ABCD angle BAD and angle CDA are divided into 3 equal angles as shown. What is the measure of angle AFD in degrees?
1 point
Captionless Image
23) Points A, B, C, and D lie on a line in alphabetical order. If AB:BD=5:7 and AC:CD=13:11, determine the ratio AB:BC:CD.
1 point
24) Teri has sticks of length 9, 12, and 14 inches. She cuts the same amount off of each stick. After cutting this amount off each stick, she is no longer able to make a triangle by attaching the sticks end to end. What is the smallest amount Teri could have cut off of each stick?
1 point
25) A garden is laid out in the fashion shown in the diagram below. If only the shaded isosceles right triangles are used for planting, what is the total area, in square feet, that is to be used for planting?
1 point
Captionless Image
26) A 25-foot ladder reaches 24 feet up the side of a building. Then the top of the ladder slides down 4 feet while the bottom slides horizontally away from the wall. How many additional feet does the bottom of the ladder slide out from the base of the building?
1 point
27) The mean, median, and mode of the five numbers 5, 7, 8, A, B are equal. The list only has 1 mode. Find all possible values of A+B, separated by a comma.
1 point
28) There are 7 married couples at a party. At the start of the party, every person shakes hands once with every other person except his or her spouse. How many handshakes are there?
1 point
29) Two fair 6-sided dice are rolled. What is the probability that at least one of the dice shows a 1?
1 point
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