# Honors Algebra 2 – Problems of the Week

## Honors Algebra 2 – Problems of the Week

```Due Septeber 6

The cost of three apples and two oranges is 2 cents more than the cost of two apples
and three oranges. By how much does the cost of one apple exceed the cost of one orange?

Due September 12

Find the largest integer k such that the polynomial x^2 + 8x + k can be factored
into the product of two linear polynomials, each of which has integral coefficients.

Due September 19

A certain product is sold as either a liquid or a powder. A survey revealed that:
One-third of the consumers interviewed do not use the powder form of the product;
Two-sevenths of the consumers interviewed do not use the liquid form of the product;
Four hundred twenty-seven of the consumers interviewed use both the liquid and powder
form of the product;
One-fifth of the consumers interviewed do not use the product at all.
How many consumers were interviewed for the survey?

Due September 26

Suppose that n is the product of four consecutive positive integers and that n is divisible by 7.
What is the largest integer that you can guarantee will be a divisor of n?

Due October 3

Pat and Chris have the same birthday.
Pat is twice as old as Chris was when Pat was as old as Chris is now.
If Pat is now 24 years old, how old is Chris?

Due October 10

Find a positive integer n such that the sum of n and its digits is 379.

Due October 17

In the isosceles trapezoid ABCD, with AB||CD,
diagonals AC and BD are perpendicular.
If AB = 4 and AD = 7, compute CD.```

```
Due October 24
A certain number leaves a remainder of 4 when divided by 5, a remainder of 5
when divided by 6, and a remainder of 6 when divided by 7.
Find the smallest number that satisfies these conditions.

Due October 31
One student in a class of girls and boys is to be chosen as class representative.
Any student is equally likely to be chosen, and the probability that a boy is chosen
is two-thirds of the probability that a girl is chosen. Determine the fraction of boys
in the class.
Due November 7
How many distinguishable rearrangements of the letters in the word CONTEST

Due November 14
How must one place the integers from 1 to 15 into each of the spaces
below in such a way that no number is repeated and the sum of the numbers
in any two consecutive spaces is a perfect square?```

```
Due November 21
For all real numbers, x,  f (2x) = x^2 - x + 3.
Express f(x) in terms of x.

Due November 28
A rectangle is divided into four sub-rectangles with areas 4, 7, 15, and x. Find x.

Due December 5
What is the smallest positive integer by which 252 can be multiplied so
that the result is a perfect cube?
Due December 12
In your basement are three light switches, each controlling a different
lamp upstairs. They are all in the off position, but you do not know which
switch controls each lamp. You may turn on any of the switches as often as
you like but may make only one trip upstairs. How can you determine which s
witch controls each lamp?

Due December 19
You are working in a store that has been very careless with the stock.
Three boxes of socks are each incorrectly labeled.
The labels say red socks, green socks, and red and green socks.
How can you re-label the boxes correctly by taking only one sock out of one box?

Due January 3
How many minutes is it before 6:00 if fifty minutes
ago it was four times as many minutes past 3:00?

Due January 9
Compute the product of [(2001 + 2)(2001 - 2) + (2001 - 2) (2001 + 2)] x
[(2001 - 2)(2001 + 2) - (2001 + 2)(2001 - 2)].

Due January 17
A certain water lily grew extremely quickly and doubled its surface area each day.
At the end of the thirtieth day, it had entirely covered the pond in which it lived.
If a second water lily, identical to the first, had been in the pond,
how long would the two lilies have taken to cover the entire pond?

Due January 23

What is the area of the triangle with
vertices A (3, 3), B (7, 5), and C (5, 6)?

Due January 30

Find all positive values of k for which the equations
3x + k = 2 and kx + 3 = 2 have a common solution for x.

Due February 6

The six-digit number 3730n5, with tens digit n, is divisible by 21.
What is n?

Due February 13

What are the possible dimensions for a rectangle with integer-valued side
lengths for which the numerical values of the area and the perimeter are equal?

Due February 27

How many five-digit positive integers are divisible by 4
and use each of the digits 1, 2, 3, 4, and 5 exactly once?

Due March 6

What proportion of all three-digit positive integers
have 7 as a digit?

Due March 13
The bases of an isosceles trapezoid are 17 cm and 25 cm,
and its base angles are 45°.
What is the trapezoid's altitude?

Due March 20

Two sides of a triangle measure 10 and 12 centimeters, and the
altitude to the shorter of the two sides measures 9 centimeters.
Find the length of the altitude to the longer of the two sides.
Hint - You can calculate the area of this triangle

Due March 27

Chuck is going on a four-day vacation. The probability of rain is 40 percent on Friday,
30 percent on Saturday, 60 percent on Sunday, and 50 percent on Monday.
What is the probability that Chuck has a rain-free vacation?

Due April 3

If the total area enclosed by these five congruent squares is 180 square centimeters,
what is the perimeter of the figure?

Due April 10

Each point in the hexagonal lattice shown is one unit from its nearest neighbor.
How many equilateral triangles have all three vertices in the lattice?

Due April 24

A letter of the alphabet is randomly selected. What is the probability that the letter
chosen does not occur in the name of a month?

Due May 1

The strips on the target shown are equal in width. If a randomly thrown dart lands on the target,
what is the probability that it will land within a black strip?

Due May 8
Three dice with faces numbered 1 through 6 are stacked as shown.
Seven of the eighteen faces are visible, leaving eleven faces hidden
on the back, on the bottom, and between faces.
What is the total number of dots not visible in this view?```

```

Due May 15
The first ring of squares around the center square contains
eight unit squares. The second ring contains sixteen unit squares.
Continuing this process, what is the number of unit squares in the
hundredth ring?

Due May 22

A camp stove uses up its gas in 195 minutes when it is sit on low.
It uses 7 1/2 times as much gas when it is set on high.
The stove has been set on low for 30 minutes.  How much longer
can the stove operate if it must be set on high to boil water?

Due May 30

This famous problem has been around for centuries and a variant of
it was used in the movie Die Hard with a Vengeance, starring
Bruce Willis and Samuel Jackson.  In this movie, the two characters
had to defuse a bomb by measuring exactly 4 gallons of water from a
5-gallon jug and a 3-gallon jug.  Can you figure out how to do this?

Due June 5

An artist has been commissioned to create 540 scale replicas, each one-foot tall,
of her most famous sculpture, which stands six feet tall.
If the original statue weighs 750 pounds, what is the total weight
of the 540 scale replicas?
Assume that the replicas are made from the same material and have the same density
as the original.