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

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

3/6/15

Two circles of radius 70 and 73 are placed on the x-axis, and both are tangent to each other.  Then another circle is constructed so that it is tangent to the two larger circles and the x-axis.  The original circles are labeled 0 and 1, and the new circle is labeled 2.  Two new circles are constructed, one with tangency to 0, 2 and 2, 1.  We continue this process ad infinum.  The first set of circles is called the first layer, the second circle is Layer 1, etc.  What is the sum of the radii to the -.5 from layer 1 to layer 7?

Hint: This is a lot of words, so draw out a diagram.  Use right triangles to get a relationship between radii of different layers.

Level: 7

3/5/15

A unit square P has two random points selected on its sides.  Find the probability that the two points are at least 1/2 apart.

Hint: Do casework.  What percentage of the time do the points end up on the same side?  What is the probability then?

Hint 2: You can't use normal Probability formulas here because of infinity.  But there is one useful tool that can handle infinity very easily: geometry.

Level: 4

3/4/15

In quadrilateral ABCD, <DAC is 98 degrees, <DBC is 82 degrees, <BCD is 70 degrees, and BC = AD. Find <ACD.

Hint: Draw a diagram, and build similar triangles.

Level: 3

3/3/15

There are a 100 points picked on a plane.  No three points are collinear.  Prove that at most 70% of all possible triangles between any three points are acute.

Hint: Try to find the percentage of acute triangles for 4 points, and continue from there.

Level: 5

3/2/15

A circle of radius 16 has 650 random points selected within it.  Prove that one can put a washer (a washer is a ring with an outer radius and inner radius) with outer radius 3 and inner radius 2 on the circle so that al least 10  of the selected points are covered.

Hint: Expected Value

Level: 6

3/1/15

Suppose there are 1978 delegates from 6 different countries meeting at a conference.  Each of the delegates is numbered a number from 1 to 1978.  Prove that there is a delegate whose number is the sum of two delegates number from his own country.

Hint: Use the Pigeonhole principle in order to determine the most expansive delegate crew. (I am being vague on purpose)

Level: 6

2/28/15

Problem:

Find the number of functions that satisfy the following conditions:

1.  f: Z -> {'green','blue'}

2. f (x + 22) = f (x)

3. if f (x) = 'green' then f (x + 2) = 'blue'

(HMMT February 2015)

Hint: This problem can be simplified by looking at a smaller domain by condition 2

Hint 2: Split the new domain in 2

Hint 3: Translate the problem into binary (Fibonacci?)

Level: 5