Geometry
A rectangle has one side of length 8 and a diagonal of length 10. What is the area of the rectange?
48
How many distinct arrangements of the letters of the word LEVEL are there?
30
What is the units digit of 2026^{2026}+2027^{2027}?
9
The roots of the quadratic x^2-11x+k=0 differ by 5. What is k?
24
A sequence is defined by a_1 = 2, a_2 = 3, and a_{n+1} = a_n - a_{n-1} for n ≥ 2. What is a_1 + a_2 + ... + a_{2026}?
4
A circle is tangent to both the x-axis and the y-axis and passes through the point (2,9). What is the sum of all possible radii of such a circle?
22
How many 3-element subsets of {1,2,...,10} contain no two consecutive integers?
A positive integer n has the form 2^a + 3^b, where a and b are nonnegative integers. If the sum of the positive divisors of n is 403, what is n?
144
Find the real number x satisfying sqrt(x+3) + sqrt(x-2) = 5.
6
tan^2(10) + tan^2(50) + tan^2(70)
9
Square ABCD has a side length 12. Point E lies on side BC with BE=5. Segment AE meets diagonal BD at F . The area of triangle BEF can be written as m/n for relatively prime positive integers m and n. Find m + n.
167
Three distinct numbers are chosen at random from {1,2,...,12}. The probability that their sum is divisible by 3 is m/n, where m and n are relatively prime positive integers. Find m+n.
74
How many positive integers n ≤ 360 satisfy n^2 ≡ 1 (mod 360)?
16
A function f is defined for all real numbers x ≠ 0,1 ad satisfies
f(x) = f(1/(1-x)) = x
for every such x. What is f(2)?
7/4
Twelve people sit evenly spaced around a circular table. Each is either a knight, who always tells the truth, or a knave, who always lies. Each person says "Both of the people sitting next to me are knaves." What is the sum of all possible values for the number of knights at the table?
15
A right triangle has legs of lengths 20 and 21. Let O be its circumcenter and I its incenter. Then OI^2 = m/n for relatively prime positive integers m and n. Find m + n.
149
In how many ways can three identical rooks be placed on a 5x5 chessboard so that no two rooks share a row or a column, and no rook lies on the main diagonal (the squares running from the top-left corner to the bottom-right corner)?
320
Find the smallest positive integer n for which the equation x^2-y^2=n has exactly 8 solutions (x,y) in positive integers.
480
The sum
sum_{n=1}^19(n/(n^4+n^2+1))
can be written as p/q for relatively prime positive integers p and q. Find p+q.
571
Let w = cos(2π/7) + isin(2π/7). The sum
sum_{k=0}^6(1/(2-w^k))
can be written as p/q for relatively prime positive integers p and q. Find p + q.
575
In triangle ABC, AB=6, AC=9, and BC=10. The bisector of angle BAC meets BC at D and meets the circumcircle of triangle ABC again at M which is not equal to A. Then AM^2 = m/n for relatively prime positive integers m and n. Find m + n
491
Let N be the number of subsets of {1,2,...,20}, including the empty set, that do not contain two elements whose difference is exactly 7. Find the remainder when N is divided by 1000.
875
How many ordered pairs (a,b) of positive integers with a,b ≤ 100 satisfy (a+b)|ab?
170
Let f(x) = ax^2 + bx + c be a quadratic with real coefficients such that |f(x)| ≤ 1 for every x in the interval 0 ≤ x ≤ 1. Find the maximum possible value of |a|+|b|+|c|.
17
Define a sequence by a_1 = 1 and a_{n+1} = a_n + floor(sqrt(a_n)) for n
≥ 1. Find a_{100}.
2273