Monthly Archives: January 2012

Circles and Pythagoras

Let and be integers. If is a circle of radius and is a circle of radius , and if , and , are tangent to each other and both are tangent to the same line, find the radius of the … Continue reading

Posted in Problem solving | Tagged , , , , , , | Leave a comment

Carnival Chance

You are at a carnival and decide to play a game that can win you a beautiful stuffed teddy bear. For one dollar, you get to randomly pick two numbered balls out of a jar without replacement and without looking. … Continue reading

Posted in Problem solving | Tagged , , , , , | Leave a comment

Three Tangent Circles

Three mutually tangent circles of radius one are surrounded by a larger circle that is simultaneously tangent to all three. What is the radius of the larger circle? Source: http://www.mathcircles.org SOLUTION The centers of the three mutually tangent circles form … Continue reading

Posted in Problem solving | Tagged , , , , , , , , | Leave a comment

January Sequence

In the sequence of numbers 1, 3, 2, . . . , each term after the first two is defined to be equal to the term preceding it minus the term preceding that. Find the sum of the first 1001 … Continue reading

Posted in Problem solving | Tagged , , , , | Leave a comment

Twelve Pennies

In how many ways can 12 pennies be put in 5 purses? What if none of the purses can be empty? Source: http://www.mathcircles.org SOLUTION Because pennies are pennies, it is easy to accept that pennies are non-distinguishable objects. However, purses are … Continue reading

Posted in Problem solving | Tagged , , , , | 1 Comment

Circle, Square, and Triangle

A circle, square, and triangle are drawn overlapping in the same plane. What is the maximum possible number of points of intersection created by the three overlapping figures? Note: the image does not represent the maximum possible points. Source: http://www.mathcontest.olemiss.eduContinue reading

Posted in Problem solving | Tagged , , , , | 6 Comments

Necklaces

In how many ways can a necklace be made using 5 identical red beads and 2 identical blue beads? Source: http://www.mathcircles.org SOLUTION Imagine the 7 positions of the beads are represented by the 7 letters . Of these 7 positions … Continue reading

Posted in Problem solving | Tagged , , , , | Leave a comment

Regions in The Plane

With zero line the plane is in one piece (or region). One line cuts the plane into 2 regions. Two lines (not parallel) cut the plane into 4 regions. Find the number of regions formed by drawing lines in the … Continue reading

Posted in Problem solving | Tagged , , , , , , , , | Leave a comment