3/16/15

Find the number of positive integers n less than 1000 for which there exists a positive real number x such that n = x*[x].

Note: [x] is the greatest integer less than or equal to x.

Hint: What kind of number can x be?  What kind of numbers can be represented by x*[x]?

Level: 5

3/15/15

In a group of nine people each person shakes hands with exactly two of the other people from the group. Let N be the number of ways this handshaking can occur. Consider two handshaking arrangements different if and only if at least two people who shake hands under one arrangement do not shake hands under the other arrangement. Find the remainder when N is divided by 1000.

Hint: Use combinatorics and mods

Level: 4

3/14/15 (PI DAY!)

Because I am not a big fan of pi (LETS GO TAU), here is the easiest question I have ever posted on this blog.

What is the minimum value of 1/(sinx+cosx)?

Hint: What are we celebrating today?!

Level: 1

3/13/15

Find the solutions to the equation: cos^2 x + cos^2 2x + cos^2 3x = 1

Hint: u substitution

Level: 3

3/12/15

Let S be a set with 2002 elements, and let N be an integer with 0<N<2^2012. Prove that it is possible to color every subset of S either blue or red so that the following conditions hold:

(a) the union of any two red subsets is red;

(b) the union of any two blue subsets is blue;

(c) there are exactly N red subsets.

Hint: Use Induction

Level: 5

3/11/15

(a,b) = gcd(a,b) and [a,b] = lcd(a,b).

Prove (a,b,c)^2/(a,b)*(b,c)*(c,a) = [a,b,c]^2/[a,b]*[b,c]*[c,a]

Hint: Use combinatorial identities

Level: 6

3/10/15

Consider the cube whose vertices are the eight points (x, y, z) for which each of x, y, and z is either 0 or 1. How many ways are there to color its vertices black or white such that, for any vertex, if all of its neighbors are the same color then it is also that color? Two vertices are neighbors if they are the two endpoints of some edge of the cube.

Hint: Split the cube.  Casework much easier?

Level: 7

3/9/15

Today's problem is an easy one (a little easter egg for y'all)

There are 16 boys in a line.  The first boy picks a number in (0,1].  The next picked a number in (0,2].  This continued till the last boy picked a number in (0,16].  What is the probability that the numbers picked by the boys are strictly increasing?

Note: number implies real number

Hint: Do each case (first boy vs second boy) at a time. What did we use in a previous problem to find probability with infinite ranges? Check out 3/5/15.

Level: 4

3/8/15

An equilateral triangle of side length 2 is drawn on a plane.  4 circles are drawn on the plane as well.  The first is the circumcircle of the triangle, and the rest have the same radius as the first, and are centered at each of the vertices of the triangle.  Find the area of the region only occupied by exactly two of the 4 circles.

Hint: Draw the diagram.  Which circles can fit into the description?

Hint 2: Now that you know what the region is, there are a few parts of the diagram whose area is hard to calculate.  Is there any way to cancel them out?

Level: 5

3/7/15

Find the sum from n = 1 to infinity of phi(n)/n^3.  You can use the following facts.  pi^2 / 6 = sum from n = 1 to infinity of 1/n^2, and the sum from n = 1 to infinity of 1/n^3 = a. (Note: phi(n) is the number of integers below n and above one that satisfy gcd(n,k) = 1)

Hint: Start with a given, and work it out till you get the desired expression.

Level: 6