3/26/15

Given a triangle ABC, we can "fold it in half" to get a new triangle: pick a vertex, e.g. A, and fold so that segments AB and AC line up. The same can be done from vertex B and vertex C, so there are three different ways to fold a triangle in half (Figure 1).

If the angles at vertices A, B and C are α, β, and γ respectively, with α ≤ β ≤ γ, what are the angles of the three possible triangles that can be obtained by folding ABC in half?

You can fold a 45-45-90 triangle at the right angle to obtain another 45-45-90 triangle. Describe all triangles that can be folded in half to get a similar triangle.

You can fold a 30-60-90 triangle to obtain a 30-30-120 triangle, then fold it again to get another 30-60-90 triangle. Describe all triangles that can be folded in half twice to get a similar triangle.

Prove that no matter how many times you fold a 40-60-80 triangle, you can never get another 40-60-80 triangle.

Hint: Solve the first bullet, and the rest will come out of the first result.

Level: 5

3/25/15

Let S be the set of all ordered triple of integers (A1,A2,A3) with 1<=A1,A2,A3<=10. Each ordered triple in S generates a sequence according to the rule An = An-1 * |An-2 - An-3| for all n=>4. Find the number of such sequences for which An = 0 for some n.

(Note: An is A sub n)

Hint:  Under what conditions does An = 0?  Lots of casework ahead.

Level: 8

3/24/15

With all angles measured in degrees, the product 

, where m and n are integers greater than 1. Find m+n.

Hint: Use trig identities, and the answer comes out nicely

Level: 7

3/23/15

A block of wood has the shape of a right circular cylinder with radius  and height , and its entire surface has been painted blue. Points A and B are chosen on the edge of one of the circular faces of the cylinder so that arc AB on that face measures 120 degrees. The block is then sliced in half along the plane that passes through point A, point B, and the center of the cylinder, revealing a flat, unpainted face on each half. The area of one of these unpainted faces is a*pi + b*sqrt(c), where a, b, and c are integers and c is not divisible by the square of any prime. Find a+b+c.

Hint: Use angle relations and right triangles.

Level: 7

3/22/15

Three concentric circles of radius 3, 4, and 5 exist within the plane.  Find the area of the largest equilateral triangle that has one vertex on each circle.

Hint: Use rotation and right triangles.  What are the angle measures of an equalateral triangle?

Level: 6

3/21/15

Consider all 1000-element subsets of the set {1, 2, 3, ... , 2015}. From each such subset choose the least element. The arithmetic mean of all of these least elements is p/q, where p and q are relatively prime positive integers. Find p+q.

Hint: Combinatorics.  Hockey Stick.

Level: 5

3/20/15

If r and s are the roots of x^2+x+7 = 0, compute the numerical value of 2r^2 + rs + s^2 + r + 7.

Hint: Vieta

Level: 3

3/19/15

Several points are plotted on a line.  Every possible line segment between any two points plotted is drawn.  One point on the line has 80 segments going through it, and one has 90.  How many points have been plotted?

Hint: How do you calculate the number of line segments that go through a point?

Level: 1

3/18/15

In O'Jax cereal, there is one gift included in every box.  A pencil comes 50% of the time, a doll comes 15% of the time, a latern comes 30% of the time, and a cardboard game comes 5% of the time.  What is the expected number of boxes one needs to buy to get all of the gifts?

Hint: Find a mathematical expression for the probability of getting any prize, multiply them, and predict the expected value.

Level: 5

3/17/15

 Suppose P(x) is a degree n monic polynomial with integer coefficients such that 2013 divides P(r) for exactly 1000 values of r between 1 and 2013 inclusive. Find the minimum value of n.

Hint: Prime factorize 2013.  What does this do?  Can you set up mod equations?  How do they relate to the question?

Level: 4