Algebra
Geometry
Number Theory
Counting & Probability
Wild Card
100
The sum of two natural numbers is 17,402. One of the two numbers is divisible by 10. If the units digit of that number is erased, the other number is obtained.
What is the difference of these two numbers?
14,238
100
Carl has 5 cubes each having side length 1, and Kate has 5 cubes each having side length 2. What is the total volume of these 10 cubes?
45 cubic units
100
The Duchess had a child on May 1st every two years until she had five children. This year the youngest is 1 and the ages of the children are 1, 3, 5, 7, and 9. Alice notices that the sum of the ages is a perfect square: 1+3+5+7+9 = 25. How old will the youngest be the next time the sum of the ages of the five children is a perfect square, and what is that perfect square?
16
100
A box contains 28 red balls, 20 green balls, 19 yellow balls, 13 blue balls, 11 white balls, and 9 black balls. What is the minimum number of balls that must be drawn from the box without replacement to guarantee that at least 15 balls of a single color will be drawn?
76
100
What is the value of (20 - 1 + 52 - 0)-1 x 5?
1/5
200
Assuming a=/=3, b=/=4, c=/=5, what is the value in simplest form of the following expression?
(a-3)/(5-c) * (b-4)/(3-a) * (c-5)/(4-b)
(sorry LaTeX is a paid feature and I'm not dishing out $20 :/)
-1
200
A rectangular floor that is 10 feet wide and 17 feet long is tiled with 170 one-foot square tiles. A bug walks from one corner to the opposite corner in a straight line. Including the first and the last tile, how many tiles does the bug visit?
26 tiles
200
What is the units digit of 32003?
7
200
How many ways can a student schedule 3 mathematics courses -- algebra, geometry, and number theory -- in a 6-period day if no two mathematics courses can be taken in consecutive periods? (What courses the student takes during the other 3 periods is of no concern here.)
24
200
A box contains a collection of triangular and square tiles. There are 25 tiles in the box, containing 84 edges total. How many square tiles are there in the box?
9
300
A flower bouquet contains pink roses, red roses, pink carnations, and red carnations. One third of the pink flowers are roses, three fourths of the red flowers are carnations, and six tenths of the flowers are pink. What percent of the flowers are carnations?
70%
300
The line 12x + 5y = 60 forms a triangle with the coordinate axes. What is the sum of the lengths of the altitudes of this triangle?
281/13
300
For how many (not necessarily positive) integer values of n is the value of 4000 * (2/5)^n an integer?
9
300
A radio program has a quiz consisting of 3 multiple-choice questions, each with 3 choices. A contestant wins if he or she gets 2 or more of the questions right. The contestant answers randomly to each question. What is the probability of winning?
7/27
300
How many rearrangements of abcd are there in which no two adjacent letters are also adjacent letters in the alphabet? For example, no such rearrangements could include either ab or ba.
2
400
The real number x satisfies the equation x + 1/x = √5. What is the value of x11 -7x7 + x3?
0
400
Rectangle ABCD has AB=3 and BC=4. Point E is the foot of the perpendicular from B to diagonal AC. What is the area of △AED?
54/25
400
Boris has an incredible coin-changing machine. When he puts in a quarter, it returns five nickels; when he puts in a nickel, it returns five pennies; and when he puts in a penny, it returns five quarters. Boris starts with just one penny. Which of the following amounts could Boris have after using the machine repeatedly?
$7.45
400
Three young brother-sister pairs from different families need to take a trip in a van. These six children will occupy the second and third rows in the van, each of which has three seats. To avoid disruptions, siblings may not sit right next to each other in the same row, and no child may sit directly in front of his or her sibling. How many seating arrangements are possible for this trip?
96
400
Alice has 24 apples. In how many ways can she share them with Becky and Chris so that each of the three people has at least two apples?
190
500
Which of the following describes the set of values of a for which the curves x2 + y2 = a2 and y = x2 - a in the real xy-plane intersect at exactly 3 points?
(a) a = 1/4, (b) 1/4 < a < 1/2, (c) a > 1/4, (d) a = 1/2, (e) a > 1/2
(e) a > 1/2
500
The diameter AB of a circle of radius 2 is extended to a point D outside the circle so that BD=3. Point E is chosen so that ED=5 and line ED is perpendicular to line AD. Segment AE intersects the circle at a point C between A and E. What is the area of △ABC?
140/37
500
There is a prime number p such that 16p + 1 is the cube of a positive integer. Find p.
307
500
Jason rolls three fair standard six-sided dice. Then he looks at the rolls and chooses a subset of the dice (possibly empty, possibly all three dice) to reroll. After rerolling, he wins if and only if the sum of the numbers face up on the three dice is exactly 7. Jason always plays to optimize his chances of winning. What is the probability that he chooses to reroll exactly two of the dice?
7/36
500
A line that passes through the origin intersects both the line x = 1 and the line y = 1 + (√3)x/3. The three lines create an equilateral triangle. What is the perimeter of the triangle?
3 + 2√3