Powerful Problem (hint)

A few weeks ago I posted this “powerful” problem:

Solve $\left(x^2-5x+5\right)^{\left(x^2-9x+20\right)}=1$

Now, allow me to give you a major hint. Consider the simpler equation

$a^b=1$

What are the possible values of $a$ and $b$? Here are the possible combinations:

• $a=1$ and $b$ is anything
• $b=0$ and $a$ is any nonzero number

And here’s the tricky one that most people forget:

• $a=-1$ and $b$ is even

You now have enough information to solve the original equation. I think you’ll be delighted with the solution!

Powerful Problem

I love this problem. I love it because it seems so complicated at first, just because we don’t teach students how to attack problems like this in Algebra class. There aren’t any “traditional” methods of attacking it, just a little mathematical reasoning/logic. Here it is:

Solve $\left(x^2-5x+5\right)^{\left(x^2-9x+20\right)}=1$

And this is my new “super duper” problem which I post throughout the year on my board (I use a lot of the same problems each year). I first saw this problem at Messiah College where one of my professors shared it–either Dr. Phillippy or Dr. Brubaker, I can’t remember which.

So give it a try. It’s sure to delight you. My Precalculus class was sharp enough to solve it today in one period (albeit, while I was teaching about a completely different topic :-)).

Interesting Cube Problem

If the cube has a volume of 64, what is the area of the green parallelogram? (Assume points I and J are midpoints.)

Go ahead, work it out. Then, go here for a more in depth discussion, including a video explanation. Also, see my very simple solution in the comments on that page. (My Precalculus students should especially take note!)

And, welcome, SAT Math Blog, to the internet! Thanks for pointing us to this great problem and creating the nice diagram above.

Soda Mixing Problem (revisited)

I posted a problem back in December that I never got back to answering. Sorry about that. The problem statement was:

Two jars contain an equal volume of soda. One contains Sprite, the other Coca Cola. You take a small amount of Coca Cola from the Coca Cola jar and add it to the Sprite jar. After uniformly mixing this concoction, you take a small amount out and put it back in the Coca Cola jar, restoring both jars to their original volumes. After having done this, is there more Coca Cola in the Sprite jar or more Sprite in the Coca Cola jar? Or, are they equally contaminated?

I have had the worked out solution for a while, just haven’t posted it until now. I’m relatively new with $\LaTeX$, but I’ve typed up the solution here, if you want all the gory details :-).  And yes, Peekay, you got the right answer!

“Real” Cool Solution

This week I start complex numbers with my Precalculus class. This is a fun problem I found here (you can also go there for the solution when you’re ready).

Given $x^2+x+1=0$, find $x^3$.

Really Interesting Limit Problem (again)

I just posted this problem the other day.

I made this java applet to help visualize the problem. Give it a try. Use the slider to let r go to zero and see where point R goes. You’re sure to figure it out just by playing with the diagram. Now, can you prove it?

Really Fun Limit Problem

Here’s a great problem that a student brought to me today. For those who’ve been wanting a ‘problem of the month,’ here you go:

The figure shows a fixed circle $C_1$ with equation $\left(x-1\right)^2+y^2=1$ and a shrinking circle $C_2$ with radius $r$ and center the origin (in red). $P$ is the point $(0,r)$, $Q$ is the upper point of intersection of the two circles, and $R$ is the point of intersection of the line $PQ$ and the $x$-axis. What happens to $R$ as $C_2$ shrinks, that is, as $r\rightarrow 0^{+}$

Who am I? (hint)

I posted the following problem back on December 3. I thought I’d post the solution, but then I decided maybe to just give you a hint. I’ve emboldened each true statement. The other statement in each pair is false. I did a lot of trial and error, making lists of numbers and crossing things off, narrowing it down. I didn’t have a great strategy, so see if you can do better. Can you figure out the number now?

There are five true and five false statements about the secret number. Each pair of statements contains one true and one false statement. Find the trues, find the falses, and find the number.

1a. I have 2 digits
1b. I am even

2a. I contain a “7”
2b. I am prime

3a. I am the product of two consecutive odd integers
3b. I am one more than a perfect square

4a. I am divisible by 11
4b. I am one more than a perfect cube

5a. I am a perfect square
5b. I have 3 digits

Soda Mixing Problem

Here’s a good puzzle for you!

Two jars contain an equal volume of soda. One contains Sprite, the other Coca Cola. You take a small amount of Coca Cola from the Coca Cola jar and add it to the Sprite jar. After uniformly mixing this concoction, you take a small amount out and put it back in the Coca Cola jar, restoring both jars to their original volumes. After having done this, is there more Coca Cola in the Sprite jar or more Sprite in the Coca Cola jar? Or, are they equally contaminated?

This problem has been stated in many different ways, with various liquids. I’ve phrased it in my own way. If you search around the internet, you can find the solution. But I want you to think it through. Give an answer and see if you can justify it! (I’ll post my solution later.)

Who Am I?

I’m reposting this great puzzle, found originally at The Math Less Traveled blog. Enjoy!

There are five true and five false statements about the secret number. Each pair of statements contains one true and one false statement. Find the trues, find the falses, and find the number.

1a. I have 2 digits
1b. I am even

2a. I contain a “7”
2b. I am prime

3a. I am the product of two consecutive odd integers
3b. I am one more than a perfect square

4a. I am divisible by 11
4b. I am one more than a perfect cube

5a. I am a perfect square
5b. I have 3 digits