Source: NCTM Mathematics Teacher, August 2006

**Solution
**Father’s plan

Al, Bee, and Cecil end up with more money.

Mother’s plan

Only Al ends up with more money.

Two children end up with more money under father’s plan than under mother’s plan.

**Answer**:

]]>

Source: NCTM Mathematics Teacher, August 2006

**Solution
**

Suppose the four cottages of Bob, Ted, Carol, and Alice are located on a straight line with respective coordinates , and such that .

Recall that distance between real numbers and on the number line.

distance between Ted’s and Alice’s cottages

distance between Bob’s and Alice’s cottages

distance between Carol’s and Alice’s cottages

distance between Carol’s and Ted’s cottages

Given that , and , we want to find the distance between Bob’s and Carol’s cottages.

implies

Calculate the distance between Bob’s and Ted’s cottages

implies

Add to both sides of the equation

Substitute the value of from Eq.

which implies

Calculate the distance between Ted’s and Carol’s cottages

Calculate the distance between Bob’s and Carol’s cottages. From Eqs. and

km

**Answer**: km

]]>

Source: NCTM Mathematics Teacher, August 2006

**Solution
**

are partners; all are divisible by

are partners; all are divisible by

are partners; all are divisible by

are partners; all are divisible by

integers have no partners:

**Answer**: 12

]]>

Source: NCTM Mathematics Teacher, August 2006

**Solution**

When the box is placed on one of on its sides, the volume of water = . If represents the level of water above the table when the box is placed on one of its sides, the same volume . Hence the water level cm.

**Answer**: cm

**Alternative solution
**If we filled the box completely full with water, the box will look full no matter how we set it on the table. The height of the water level equals the height of the box , or cm depending on which side it is placed on. Similarly, if we filled the box half full, the box will look half full no matter how we set it on the table. The height of the water level equals half the height of the box , or depending on which side it is placed on. Since the water level equals when the box height equals , the ratio of water level to box height is . So when the box is placed on one of its sides, the water level equals cm.

]]>

Source: NCTM Mathematics Teacher, August 2006

**Solution
**Three sticks make a triangle. There are ways to choose three sticks out of six. Of these ways only satisfy the Triangle Inequality theorem to make triangles, namely

and none of them are congruent.

**Answer**:

]]>

Source: NCTM Mathematics Teacher, August 2006

**Solution**

We want to organize the senators into -member committees with each state sending only one senator to the committees.

Ways to choose states from states =

Ways to send one of two senators from each of the five states =

Ways to form committees =

**Answer**:

**Alternative solution**

We select senators from a group of to form -member committees. First, we pick one senator from ; second, we pick one from the remaining (not because we cannot have senators from the same state); third, we pick one from , etc. Hence the number of ways to pick senators = . Each way is an ordering (permutation) of the senators and the number of orderings of the objects in a set of objects equals . Since the order of selection is not important, we divide the number of ways by 5! to get

]]>

Source: NCTM Mathematics Teacher, August 2006

**Solution**

To compute the total interest earned at the end of four years we compute the interest earned by each individual deposit during that time. Since some of the later deposits are invested in less than a year and the interest is compounded monthly, we calculate the interest earned in units of months. The interest rate per month is . The first invested for months grows to , the second invested for months grows to , etc. The interests earned by each individual deposit are listed below

Month

Month

Month

Month

Month

Month

Total interest earned

is a geometric sequence with first term , common ratio , and number of terms .

**Answer**:

]]>

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.

implies that is an even perfect fifth power less that .

Case 1:

Trial and error method: looking for prime integers and such that

too low

still too low

better

too high

Maybe we should try and closer to each other like and

Case 2: Time to try

Try

Bingo! and

**Answer**:

]]>

Source: SCVMA Math Olympiad 2010

**Answer**:

]]>

Source: SCVMA Math Olympiad 2010

**Solution**

is the sum of an infinite geometric series with first term = , common ratio = , and sum = .

is the sum of an infinite geometric series with first term = , common ratio = , and sum = .

Substitute the value of from Eq. into Eq.

Solving for using the quadratic formula yields (not possible) or .

Substitute the value of into Eq.

Let represent the sum of the infinite geometric series

Collecting the positive and negative terms

is the sum of an infinite geometric series with first term = , common ratio = , and sum = .

is the sum of an infinite geometric series with first term= , common ratio = , and sum = .

Substitute the values of and

**Answer**:

]]>