Thread: Averages, Counting, and other questions (help?)

I'm studying for a math contest and not understanding some problems and the answers for them. Here are a few questions and what I don't understand about them:

Which of the following sets of whole numbers has the largest average?
(A)Multiples of 2 between 1 and 101. (B) Multiples of 3 between 1 and 101. (C) Multiples of 4 between 1 and 101. (D) Multiples of 5 between 1 and 101. (E) Multiples of 6 between 1 and 101
I'd think the answer would be E, but the answer key says that it is D. Any input?


The mean, median, unique mode, and range of a collection of eight integers are all equal to 8. the largest integer that can be an element in this collection is:
(A) 11. (B) 12. (C) 13. (D) 14. (E) 15.
I'm not entirely sure what a 'unique mode' is.
I'm fairly certain the highest number could be anything, because the lowest numbers could bring down the average. Their answer is D.


Thanks in advance!

It is reasonable to assume (E) for the first question, but the maximum number in that set is 96 versus 100 for the multiples of 5.

There are 16 multiples of 6 in that range with a sum of 816.
There are 20 multiples of 5 in that range with a sum of 1050.

I see... Thanks
Here's another:

A set of tiles numbered 1 through 100 is modified repeatedly by the following operation: remove all tiles numbered with a perfect square, and renumber the remaining tiles consecutively starting with 1. How many times must the operation be performed to reduce the number of tiles in the set to one?
I don't understand what they mean by this, I'd think that the answer would be an infinite amount of times, wouldn't you remove all of the tiles each time?
Their answer is 18.

Originally Posted by Quinch

Here's another:
A set of tiles numbered 1 through 100 is modified repeatedly by the following operation: remove all tiles numbered with a perfect square, and renumber the remaining tiles consecutively starting with 1. How many times must the operation be performed to reduce the number of tiles in the set to one?

The first time we remove 10: 1,4,9,…,81,100. Renumber and have 1,2,…,89,90.
This time we remove 9:1,4,9…,64,81. Renumber and have 1,2,…80,81.
Third time we remove 9 again.
So keep it up.

I see. How did you figure that each time you jump an odd number, from 1 to 4, 4, to 9, 9 to 16, 16 to 25, etc?

Originally Posted by Quinch

I see. How did you figure that each time you jump an odd number, from 1 to 4, 4, to 9, 9 to 16, 16 to 25, etc?

We remove the perfect squares.

Okay. Here are some more:


Square ABCD has sides of length 3. Segments of CM and CN divide the square's area into three equal parts. How long is segment CM?
(A) square root of 10. (B) square root of 12. (C) square root of 13. (D) square root of 14. (E) square root of 15.

How do you figure this type of problem? Their answer is A.

In rectangle ABCD, AD=1, P is on the line AB, and DB and DP trisect angle ADC. What is the perimeter of triangle BDP?
(A) 3 + square root of 3 divided by 3. (B) 2 + the square root of [4 times the square root of 33]. (C) 2 + 2 times the square root of 2. (D) (3 + 3 times the square root of 5) divided by 2. (E) 2 + (5 times the square root of 3 divided by 3)

How do you figure this problem? Their answer is B.

EDIT:
Here's another:

A checkerboard consists of one-inch squares. A square card, 1.5 inches on a side, is placed on the board so that it covers part or all of the area of each of N squares. The maximum possible value of N is
(A) 4 or 5. (B) 6 or 7. (C) 8 or 9. (D) 10 or 11. (E) 12 or more.
I would think that the answer would be C, because the square card would cover one whole square and part of the 8 squares surrounding it. Their answer is E.

Originally Posted by Quinch

Okay. Here are some more:

If you have a new problem start a new thread. Otherwise you risk that nobody will notice that you need some additional help!


Square ABCD has sides of length 3. Segments of CM and CN divide the square's area into three equal parts. How long is segment CM?
(A) square root of 10. (B) square root of 12. (C) square root of 13. (D) square root of 14. (E) square root of 15.

How do you figure this type of problem? Their answer is A.

...

1. |CM| = |CN|

2. the area of the square is 9 squ.units. Therefore the area of the triangle NDC must be 3 squ.units.

3. with DC = 3

4. Solve for ND. I've got ND = 2

5. Use Pythagorean theorem:

Originally Posted by Quinch

...A checkerboard consists of one-inch squares. A square card, 1.5 inches on a side, is placed on the board so that it covers part or all of the area of each of N squares. The maximum possible value of N is
(A) 4 or 5. (B) 6 or 7. (C) 8 or 9. (D) 10 or 11. (E) 12 or more.
I would think that the answer would be C, because the square card would cover one whole square and part of the 8 squares surrounding it. Their answer is E.

You have to turn the square card by 45°.

Since the diagonal of this square is larger than 2 inches there are 8 additional squares partly covered. See attachment.