Interactive version of problem and solution.

## Friday, January 30, 2009

### January 30, 2009

Four distinct circles are drawn in a plane. What is the maximum number of points where at least two of the circles intersect?

Interactive version of problem and solution.

Interactive version of problem and solution.

## Thursday, January 29, 2009

### January 29, 2009

Using the letters A, M, O, S, and U, we can form 120 five-letter "words". If these "words" are arranged in alphabetical order, then the "word" USAMO occupies what position?

Interactive version of problem and solution.

## Wednesday, January 28, 2009

### January 28, 2009

How many four-digit positive integers have at least one digit that is a 2 or a 3?

Interactive version of problem and solution.

Interactive version of problem and solution.

## Tuesday, January 27, 2009

### January 27, 2009

Three friends have a total of 6 identical pencils, and each one has at least one pencil. In how many ways can this happen?

## Monday, January 26, 2009

### January 26, 2008

What is the value of (3

Interactive version of problem and solution.

*x*– 2)(4*x*+ 1) – (3*x*– 2)4*x*+ 1 when*x*= 4?Interactive version of problem and solution.

## Friday, January 23, 2009

### January 23, 2009

An 8-foot by 10-foot floor is tiled with square tiles of size 1 foot by 1 foot. Each tile has a pattern consisting of four white quarter circles of radius 1/2 foot centered at each corner of the tile. The remaining portion of the tile is shaded. How many square feet of the floor are shaded?

## Thursday, January 22, 2009

### January 22, 2009

How many non-congruent right triangles with positive integer leg lengths have areas that are numerically equal to 3 times their perimeters?

## Wednesday, January 21, 2009

### January 21, 2009

A line with slope 3 intersects a line with slope 5 at the point (10, 15). What is the distance between the

Interactive version of problem and solution.

*x*-intercepts of these two lines?Interactive version of problem and solution.

## Friday, January 16, 2009

### January 16, 2009

A circle of radius

Interactive version of problem and solution.

*r*is concentric with and outside a regular hexagon of side length 2. The probability that three entire sides of the hexagon are visible from a randomly chosen point on the circle is 1/2. What is*r*?Interactive version of problem and solution.

## Thursday, January 15, 2009

### January 15, 2009

There are two values of a for which the equation 4

Interactive version of problem and solution.

*x*^{2}+*ax*+ 8*x*+ 9 = 0 has only one solution for*x*. What is the sum of those values of*a*?Interactive version of problem and solution.

## Wednesday, January 14, 2009

### January 14, 2009

Brenda and Sally run in opposite directions on a circular track, starting at diametrically opposite points. They first meet after Brenda has run 100 meters. They next meet after Sally has run 150 meters past their first meeting point. Each girl runs at a constant speed. What is the length of the track in meters?

Interactive version of problem and solution.

Interactive version of problem and solution.

## Tuesday, January 13, 2009

### January 13, 2009

The sum of the digits of a two-digit number is subtracted from the number. The units digit of the result is 6. How many two-digit numbers have this property?

Interactive version of problem and solution.

Buy Hexaflexagons, Probability Paradoxes, and the Tower of Hanoi

## Monday, January 12, 2009

### January 12, 2009

## Friday, January 9, 2009

### January 9, 2009

The parallelogram bounded by the lines

Interactive version of problem and solution.

*y = ax + c*,*y**=**ax + d*,*y = bx + c*, and*y = bx + d*has area 18. The parallelogram bounded by the lines*y = ax + c*,*y = ax -*d,*y = bx + c*, and*y = bx - d*has area 72. Given that*a, b, c,*and*d*are positive integers, what is the smallest possible value of*a + b + c + d*?Interactive version of problem and solution.

## Thursday, January 8, 2009

### January 8, 2009

The first 2007 positive integers are each written in base 3. How many of these base-3 representations are palindromes? (A palindrome is a number that reads the same forward and backward.)

## Wednesday, January 7, 2009

### January 7, 2009

Centers of adjacent faces of a unit cube are joined to form a regular octahedron. What is the volume of this octahedron?

Interactive version of problem and solution.

Interactive version of problem and solution.

## Tuesday, January 6, 2009

### January 6, 2009

A bug starts at one vertex of a cube and moves along the edges of the cube according to the following rule. At each vertex the bug will choose to travel along one of the three edges emanating from that vertex. Each edge has equal probability of being chosen, and all choices are independent. What is the probability that after seven moves the bug will have visited every vertex exactly once?

Interactive version of problem and solution.

## Monday, January 5, 2009

### January 5, 2009

What is the measure of the acute angle formed by the hands of a clock at 4:20 a.m.?

Interactive version of problem and solution.

Interactive version of problem and solution.

## Friday, January 2, 2009

### January 2, 2009

A wooden cube

Interactive version of problem and solution.

*n*units on a side is painted red on all six faces and then cut into*n*^{3}unit cubes. Exactly one-fourth of the total number of faces of the unit cubes are red. What is*n*?Interactive version of problem and solution.

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