ALGEBRA
The average of five numbers is 12. One of the numbers is removed, and the average of the remaining four is 10. What number was removed?
20
A rectangle has perimeter 36, and its length is twice its width. What is its area?
72
What is the sum of all positive divisors of 28?
56
A fair coin is flipped 4 times. What is the probability that exactly two of the flips are heads?
3/8
If 20% of a number is 14, what is 50% of that number?
35
A real number x satisfies x + 1/x = 5. What is x^2 + 1/x^2?
23
Right triangle ABC has a right angle at C, with AC = 6 and BC = 8. What is the length of the altitude from C to the hypotenuse AB?
24/5 (that is, 4.8)
What is the remainder when 3^2026 is divided by 7?
4
How many distinguishable arrangements are there of the letters in the word BANANA?
60
What is the value of (log base 2 of 8) + (log base 3 of 81)?
7
Real numbers a and b satisfy a + b = 7 and a^3 + b^3 = 91. What is ab?
12
A circle is inscribed in a square of side length 10. A second square is inscribed in that circle. What is the area of the second square?
50
How many zeros are at the end of 2026! when it is written out in full?
505
A committee of 3 people is chosen at random from a group of 5 boys and 4 girls. What is the probability that the committee includes at least one girl?
37/42
What is the sum of the first 20 terms of the arithmetic sequence 2, 5, 8, 11, ... ?
610
In triangle ABC, point D lies on BC with BD:DC = 2:3, and point E lies on AC with AE:EC = 1:2. Segments AD and BE meet at P. What is the ratio AP:PD?
5 : 4
How many ordered pairs of positive integers (x, y) satisfy 1/x + 1/y = 1/12?
15
How many four-digit positive integers have digits that strictly increase from left to right?
126
What is the units digit of 7^(7^7)?
3
Nonzero real numbers a, b, c satisfy a + b + c = 0 and a^2 + b^2 + c^2 = 26. What is a^4 + b^4 + c^4?
338
In triangle ABC, the median from A and the median from B are perpendicular. If CA = 6 and CB = 8, what is AB?
2*sqrt(5)
What is the sum of all positive integers n for which n + 1 divides n^2 + 7?
11 (from n = 1, 3, 7)
A fair coin is flipped 8 times. What is the probability that no two heads occur on consecutive flips?
55/256
The polynomial x^3 - 6x^2 + 11x - 6 has three roots. What is the sum of the reciprocals of those roots?
11/6