Counting
How many permutations of {1, 2, 3, 4} are there such that 1 comes before 2?
12
The area of an equilateral triangle, in square inches, is the same as its perimeter, in inches. What is its perimeter?
12 √3 inches
What is the product of the roots of the equation 2x2 + 3x + 14 = 0?
7
What is the last digit of 3 to the power 2026?
9
How many three-digit numbers satisfy the property that the middle digit is the average of the first and the last digits?
45
What is the area of a triangle whose sides measure 13 ft, 14 ft, and 15 ft?
84 ft2
What should you add to both sides of the equation 4x2 + 7x = 10 in order to complete the square?
49 / 16
What reduced fraction has the decimal expansion 0.1232323..., where after the initial 1, the pattern 23 repeats forever?
61 / 495
How many paths from (0, 0) to (5, 5), taking steps of one unit in the positive x or y direction at a time, do not pass through the points (2, 3) or (3, 2)?
52
A rectangle is inscribed in a circle. There is a central angle of 120° between the two vertices of one of the rectangle's sides, which has length 8 ft. What is the area of the circle?
64 π / 3 ft2
For how many distinct integer values of k does the equation 5x2+k2x+18 = 0 have no real solutions?
9
When Bella divides her skittles into groups of 3, there is 1 left over. When she divides her skittles into groups of 5, there are 3 left over. When she divides her skittles into groups of 7, there are 5 left over. If Bella has at least 1000 skittles, what is the least number of skittles she could have?
1048
The sides of a square are to each be painted with one of 5 colors. In how many ways can this be done, if two colorings which differ only by rotating the square count as the same?
165
Triangle ABC has lengths AB = 7, BC = 8, and an area of 21 square units. If x is the measure of angle B, then find tan(x)2.
9 / 7
For x > 0, simplify -ln(2/x) + ln(4/x) - ln(8/x)..... - ln(512/x) + ln(1024/x).
ln(32), or 5 ln(2)
In base ten, the integer 40909 is a multiple of 11. In which other base b is 40909b a multiple of 11b?
21