(11/29/2023) Assuming that the shaded region is a square, find the area of the shaded region.

(11/7/2023) There are a set of red points ((0,0), (0,1), (0,2), ..., (0,7)) and set of blue points ((1,0), (1,1), (1,2), ..., (1,7)). Each red point is randomly connected to a blue point. What is the expected number of groups formed, if a group is a region where someone can visit the entire region without lifting his pencil up.

(???) In triangle ABC, AB=488. Point E is on AB so that BE=CE=BC. Given that AC=12, find the area of triangle AEC.

(???) In an increasing sequence of 15 integers, the average of the first 10 integers is 50 and the average of the last 10 integers is 500. If the first integer is 1 and the last integer is 1000, find the maximum value of the average of the entire sequence.

(???) Let f(x) be the number of trailing zeroes of x!. Let g(x)=5x+1. Find the sum of the digits of f(1)+f(g(1))+f(g^2(1))+...+f(g^10(1)) given that 5^11=48828125.

(10/30/2023) Prove that \sum_{a=-n}^{n} |x+a| = x^2+n(n+1) as n and x approach infinity

(9/27/2023) Let f(x)=\sum_{p=2}^{\infty} p*v_p(x) for all primes p. Find all values of 4 digit integer abcd such that f(0.abcd...)=0 and lcm(abcd,810000)=39690000.

(8/18/2023) Let a,b,c be the roots of x^3-11x^2+14x-2. Find (sqrt(a^2+b^2+ab)+sqrt(a^2+c^2+ac)+sqrt(b^2+c^2+bc))(-sqrt(a^2+b^2+ab)+sqrt(a^2+c^2+ac)+sqrt(b^2+c^2+bc))(sqrt(a^2+b^2+ab}-sqrt(a^2+c^2+ac}+sqrt(b^2+c^2+bc))(sqrt(a^2+b^2+ab)+sqrt(a^2+c^2+ac)-sqrt(b^2+c^2+bc))