Algebra
How many real solutions does the equation have?
(x^4)-5(x^2)+4=0
4
The two acute angles of a right triangle are in the ratio 2:3. What is the measure of the smaller angle?
36
How many positive divisors does 72 have?
12
A bag has 4 red and 6 blue marbles. Two are drawn without replacement. What is the probability both are red?
2/15
A merchant marks an item up 50% from cost, then sells at a 10% discount off the marked price, making a $70 profit. What was the cost price?
$200
If x+(1/x)=4, what is the value of (x^2)+(1/(x^2))?
14
A triangle has side lengths 13, 14, and 15. What is its area?
84
What is the smallest positive integer that is divisible by each of the integers from 1 through 10?
2520
A group contains 8 students. In how many ways can a team of 3 students be selected?
56
A two-digit number has digit sum 11. Reversing its digits gives a number 45 less than the original. What is the original number?
83
Find the sum of all real solutions to the equation:
|x-2|+|x-6|=8
8
A square is inscribed in a right triangle with legs 6 and 8, with two sides of the square lying along the two legs. What is the side length of the square?
24/7
How many ordered pairs of positive integers (x,y) satisfy 1/x+1/y=1/12?
15
A committee of 4 is chosen from 6 men and 4 women. What is the probability it has exactly 2 men and 2 women?
3/7
Machines A, B, C alone finish a job in 6, 8, and 12 hours respectively. All three work together for 2 hours, then A stops. How many more hours do B and C need to finish?
6/5 hrs = 1.2 hrs
The equation x^2-5x+1=0 has two real roots a and b.
What is the value of a^2b+ab^2?
5
A regular hexagon has area 24sqrt(3). What is the length of its longest diagonal?
8
What is the largest power of 2 that divides 100!?
97
A bag contains 4 red marbles, 3 blue marbles, and 2 green marbles. Three marbles are selected at random without replacement. What is the probability that exactly two different colors are represented?
55/84
Two cars start together and travel in opposite directions; Car B is 10 mph faster than Car A. After 3 hours they are 270 miles apart. How far apart are they after 5 total hours?
450
Positive real numbers x and y satisfy
x+y=6 and xy=6.
What is the value of x^4+y^4?
504
Three circles of radius 4 are mutually externally tangent. What is the area of the region bounded by all three circles (the small curved triangle between them)?
16sqrt(3)-8pi
How many positive integers less than 1000 are divisible by neither 2, 3, nor 5?
266
How many ways can 6 people be seated around a round table if two of them refuse to sit next to each other?
72
A group of people is planning a round-robin tournament in which every participant plays exactly one game against every other participant. If 45 games are played in total, how many participants are in the tournament?
10