Middle School Marathon

Math Champion

Hello! I would like to start a new marathon, which is a combination of the MOEMS marathon and the MATHCOUNTS marathons that are currently going on. In this thread, you can post problems ranging from very easy MOEMS problems all the way up to hard national MATHCOUNTS problems. Ill first start off with a problem:

NP: An isosceles triangle has a height of 12 cm and an area of 24 square cm. How many cm's are in the perimeter of the triangle? Express your answer in simplest radical form.

P.S.: Whoever posts the solution to a problem, please try and post a new problem.

Thanks.

**Posted:** Fri Jul 10, 2009 12:07 pm

_________________
~Math Champion
If you don't believe that this post was helpful, please message me and tell me how I can improve my posts.

PowerOfPi

We have (b=base) $\frac 12 \cdot 12 \cdot b=24 \implies b=4$ . Now, using half the base and the height, we find the other two sides to be each $\sqrt{2^2+12^2}=\sqrt{148}=2\sqrt{37}$ . So the final answer is:

$\boxed{4+4\sqrt{37}}$

New Problem

When the positive integers up to 2007 are alternately subtracted and added (1 – 2 + 3 – 4 + … + 2001 – 2002 + 2003 – 2004 + 2005 – 2006 + 2007), what is the value of the expression?

Source: 2007 State Countdown

**Posted:** Fri Jul 10, 2009 12:21 pm

_________________
Sign up for MOCK AMC!

Math Champion

Solution

NP

**Posted:** Fri Jul 10, 2009 12:25 pm

_________________
~Math Champion
If you don't believe that this post was helpful, please message me and tell me how I can improve my posts.

PowerOfPi

It is just the sum of [probability that a cube with [variable] sides is chosen] $\times$ [the probability that it comes up with a painted side]:

$\frac{24}{64} \cdot \frac 36+ \frac {24}{64} \cdot \frac26+ \frac {8}{64} \cdot \frac16=\frac38 \cdot \frac12 + \frac38 \cdot...$

New Problem

1 out of 6 Yummy Cereal boxes has a bobblehead of Zac Efron inside. If Katie buys 3 Yummy Cereal boxes, what is the probability that at least one will have a Zac Efron bobblehead inside? Express your answer as a common fraction.

**Posted:** Fri Jul 10, 2009 1:35 pm

_________________
Sign up for MOCK AMC!

cgyao15

Ok, i will find the probablity that it doesn't occur.
$\ (\frac {5}{6})^{3}= \frac {125}{216}$
Then we subtract from total.
$\ 1-\frac {125}{216}=\frac {91}{216}$
NP
Define a blarghish number as a number when added and subtracted by $\ 1$ , results in a perfect square find the smallest blargish number.

**Posted:** Fri Jul 10, 2009 1:43 pm

_________________
Check out my new avatar made by Izzy. PM her to get one!

RoFlLoLcOpT

Clarify please...What do you mean by "added and subtracted by $1$ "?

**Posted:** Fri Jul 10, 2009 1:50 pm

Math Champion

**Posted:** Fri Jul 10, 2009 1:52 pm

_________________
~Math Champion
If you don't believe that this post was helpful, please message me and tell me how I can improve my posts.

PowerOfPi

Yeah, I mixed it up. It should be:

$\frac {24}{64} \cdot \frac 16 + \frac {24}{64} \cdot \frac26 + \frac {8}{64} \cdot \frac36 = \frac38 \cdot \frac16 + \frac38 ...$

**Posted:** Fri Jul 10, 2009 1:55 pm

_________________
Sign up for MOCK AMC!

Math Champion

Well, since the other guy's problem doens't make sense,

NP

**Posted:** Fri Jul 10, 2009 2:39 pm

_________________
~Math Champion
If you don't believe that this post was helpful, please message me and tell me how I can improve my posts.

tennis123

Solution?
I only got one solution though. Is it right? If it is, I'll post a new problem.

**Posted:** Fri Jul 10, 2009 2:54 pm

_________________
"A question that sometimes drives me hazy: am I or are the others crazy?" -Albert Einstein

Math Champion

Nope

. The other solution is smaller.

**Posted:** Fri Jul 10, 2009 2:57 pm

_________________
~Math Champion
If you don't believe that this post was helpful, please message me and tell me how I can improve my posts.

AIME15

Solution

New Problem:

How many positive integers less than $1000$ are not divisible by $5$ nor $7$ ?

**Posted:** Sat Jul 11, 2009 7:31 am

_________________
GOALS: MC School: 46 | Chapter: 46 | State: 46 | National: 46 | AMC 8: 25 | AMC 10: 150 | AMC 12: 150 | AIME: 15 | USAMO: 42 | TST: 63 | IMO: 42 |

jxl28

Solution

New Problem

**Posted:** Sat Jul 11, 2009 7:39 am

King! · Riemann Hypothesis Offline Joined: 27 Mar 2009 Posts: 351

**Posted:** Sat Jul 11, 2009 8:38 am

_________________
♣Hm♣ Add some zest to the humdrum of your everyday life!(Mafia games in my blog)

cgyao15

Well, suface area is $\ n^{2}*6$
Where 6 is the number of faces, n^2 is surface area of 1 face.
Area is $\ n^{3}$
so $\ n^{3}\div\ n^{2}=n=6$
$\ 6^{3}=216$

Find $\ P(n,n)$ for any number n? P means choose size r for a quantity n.

**Posted:** Sat Jul 11, 2009 8:48 am

_________________
Check out my new avatar made by Izzy. PM her to get one!

pytheagle

I don't understand your question, but maybe the answer is $\boxed{1}$ ?

New Problem
Consider a unit circle centered at the origin. Let us draw a square with side length $4$ and centered $(1,1)$ . Find the area of intersection between the circle and the square.

**Posted:** Sat Jul 11, 2009 9:02 am

_________________
The Secret of Being a Geek
Avatar made by isabella2296, PM her if you want one.

Math Champion

Actually, for cgyao15's problem, it should be $n!$ because he's asking for permutations, not combinations, so order DOES matter.

**Posted:** Sat Jul 11, 2009 9:48 am

_________________
~Math Champion
If you don't believe that this post was helpful, please message me and tell me how I can improve my posts.

SuperNerd123

So this is what it looks like (red is intersection)

import graph;

size(100);

fill((-1,-1)--(-1,3)--(3,3)--(3,-1)--cycle,blue);
fill(Circle((0,0),1),red);

for (int i=-4; i&l...

So we see the intersection is simply the circle.

The area of the cirlce is $\boxed{\pi}$

so the np is the one that cgyao15 wrote

**Posted:** Sat Jul 11, 2009 10:40 am

Math Champion

I already solved the one he wrote, so:

NP: What is the vertex of the graph of $y=x^2-19x+47$ ?

**Posted:** Sat Jul 11, 2009 2:41 pm

_________________
~Math Champion
If you don't believe that this post was helpful, please message me and tell me how I can improve my posts.

PowerOfPi

What do you mean by "vertex"? Do you mean x-intercepts?

EDIT: Sorry, my bad. I can't find a non-calculus way of doing it, so I won't post a solution.

_________________
Sign up for MOCK AMC!