# Category Archives: Problem solving

## Children Sharing Money

Al, Bee, Cecil, and Di have , and , respectively. Their father proposed that Al and Bee share their wealth equally, and then Bee and Cecil do likewise, and then Cecil and Di. Their mother’s plan is the same except … Continue reading

## Cottages on a Straight Road

There are four cottages on a straight road. The distance between Ted’s and Alice’s cottages is km. Both Bob’s and Carol’s cottages are twice as far from Alice’s as they are from Ted’s. Find the distance between Bob’s and Carol’s … Continue reading

## Partners in Integers

Two integers are said to be partners if both are divisible by the same set of prime numbers. Find the number of positive integer less than that have no partners less than . Source: NCTM Mathematics Teacher, August 2006 Solution … Continue reading

## Water in Glass Box

A glass box cm, closed on all six sides, is partially filled with colored water. When the box is placed on one of its sides, the water level is cm above the table. If the box is placed on one … Continue reading

## Making Triangles

You have six sticks of lengths , and cm. Find the number of noncongruent triangles that can be formed using three of these sticks as sides. Source: NCTM Mathematics Teacher, August 2006 Solution Three sticks make a triangle. There are … Continue reading

## U.S. Senate Committee

There are members in the U.S. Senate ( from each state). In how many ways can a committee of senators be formed if no state may be represented more than once? Source: NCTM Mathematics Teacher, August 2006 Solution We want … Continue reading

## Interest Earned

At the end of every month, Elle deposits into a savings account with an annual interest rate of percent, compounded monthly. How much interest will be earned at the end of four years? Source: NCTM Mathematics Teacher, August 2006 Solution … Continue reading

## Sum of Three Integers

For certain positive integers , and , . If and are prime, find . Source: SCVMA Math Olympiad 2010 Solution Since and are odd, and are odd which implies that is even. Hence is even because the sum is even. … Continue reading

## Cot(C) and Sine(2C)

If (a positive integer), express as a fraction involving . Source: SCVMA Math Olympiad 2010 Solution thus Answer:

## Sum of Infinite Geometric Series

The sum of the infinite geometric series is , and the sum of the series whose terms are the squares of the terms of this series is . Find the sum of the infinite geometric series Source: SCVMA Math Olympiad … Continue reading