Mathcounts marathon!!!!

Ihatepie

1992 Nats Sprint

The product of three consecutive numbers divided by their sum is 33. What is the greatest of these three numbers?

Try to do this algebraically and without factoring anything with a power of three.

**Posted:** Wed Mar 19, 2008 11:43 am

shentang

I stink at latex so I am sorry if the solution is unclear.
Call the numbers x-1, x, and x+1

The product of these numbers is x^3-x
The sum of the numbers is 3x

So we set up an equation:
(x^3-x)/3x=33x
x^3-x=99x
Divide both sides by x.
x^2-1=99
Add 1
x^2=100
x=10
Since x+1 is the largest, 10+1=11 is the answer.
I will post another problem soon.

**Posted:** Wed Mar 19, 2008 12:36 pm

ZhangPeijin

Hao walked at a rate of 2mile\year for 16 years.
He then ran at a rate of 2ft\second for 16 minutes.
What was his average speed for those 16+ years?
Assume that theres 365.25 days in an year, and that theres exactly 5280ft in a mile.

**Posted:** Wed Mar 19, 2008 12:46 pm

_________________
Сфзшефдшяфешщт шы еру вшааукутсу иуецуут рудзштп нщгк Гтсду Офсл щаа ф рщкыу фтв рудзштп нщгк гтсду офсл щаа ф рщкыую

shentang

A triangle with vertices at (1234,4), (1238,4), and (1234,8) is rotated around the x-axis. What is the volume of the figure. Express your answer in simplest terms in pi.

Inspired by my math teacher. I am sure that this is solvable using mathcounts level skills.

**Posted:** Wed Mar 19, 2008 12:50 pm

ZhangPeijin

**Posted:** Wed Mar 19, 2008 12:52 pm

_________________
Сфзшефдшяфешщт шы еру вшааукутсу иуецуут рудзштп нщгк Гтсду Офсл щаа ф рщкыу фтв рудзштп нщгк гтсду офсл щаа ф рщкыую

Ihatepie

MC wouldn't have calculation probs like that.

Click to reveal hidden content

Edit: about shentang's problem

**Posted:** Wed Mar 19, 2008 12:54 pm

alanchou

**Posted:** Wed Mar 19, 2008 1:17 pm

_________________
Get a life and do some math.
ugh, 0.5 points off from USAMO Sad

Ihatepie

1999 Nats Sprint 11

If a phonographic record makes 33 1/3 revolutions per minute, how many revolutions will it make playing a record that is 3 minutes and 36 seconds long?

**Posted:** Wed Mar 19, 2008 5:59 pm

Yongyi781

**Posted:** Wed Mar 19, 2008 6:15 pm

_________________
"quoted and followed by itself is a quine" quoted and followed by itself is a quine.
"yields falsehood when preceded by its quotation" yields falsehood when preceded by its quotation.
(Almost) Red MOP 2009

SeventhSecret

*will revive with a new problem*

The mean of 6 integers is 5, and the median 6. What is the largest the mode could be, given that the mode is unique? (2002 State Sprint #30)

*edit* unless there's an open problem, which I don't think there is, or someone would've posted an answer by now.

**Posted:** Tue Mar 25, 2008 3:55 pm

wiseidiot

To get the largest mode we must have the other 3 numbers as small as possible.

We must include 6 because its the median. As for the other two, we just have 0 and 2(2 becuase the sum of the first three integers must be even ). The remaining two numbers must sum to 30-2-6=22. 22/2=11. So 11 is our greatest possible mode

edit: your problem says integers whcich means that the numbers can be negative. Do you mean whole numbers?

Next question:

A 3in x 5in piece of paper can be rolled to form a cylinder by taping either pair of paralell edges together. Whats the ratio of the volumes of the larger cylinder to the smaller cylinder obtained this way?Express as common Frac.

math154

Do you mean volume, surface area, or like height (which I really doubt)?

**Posted:** Tue Mar 25, 2008 4:08 pm

wiseidiot

srry, question was unclear. Ratio of the volumes.

This is fun. Two marathons going at the same time

**Posted:** Tue Mar 25, 2008 4:12 pm

SeventhSecret

**Posted:** Tue Mar 25, 2008 4:41 pm

wiseidiot

Okay, I'm reviving this marathon cuz no one is answering my problem:
Answer to the cylinder one was:
Click to reveal hidden content

New Problems:

1)If the prime factoarization of $N$ is: $2^63^{11}5^8$ , then what is the probability that a randomly selected factor of $N$ is a perfect cube? Express as common frac.

2)What is the sum of the months with the least and most relatively prime dates(A relatively prime date is when the month and date have to common factor other than one. EX: 12/31)

**Posted:** Wed Mar 26, 2008 5:13 am

vamathletes · Riemann Hypothesis Offline Joined: 28 Feb 2007 Posts: 318

1.

**Posted:** Wed Mar 26, 2008 7:20 am

wiseidiot

That was correct vamathletes, can you plz post new problems because this is a marathon and the solver has to post new ones.

**Posted:** Wed Mar 26, 2008 7:24 am

math154

2

And two problems,

1)How many perfect numbers are there from 1 to 100?

2)Joe flips 20 coins. What is the probability that he gets at least 5 heads? Express your answer as a common fraction in lowest terms.

**Posted:** Wed Mar 26, 2008 8:15 am

vamathletes · Riemann Hypothesis Offline Joined: 28 Feb 2007 Posts: 318

Sorry

#30 1995 Nationals Sprint:
Express $k$ as a common fraction in terms of $n$ , where $n$ is a integer greater than 1.
$\frac{k(n-2)!}{(n+1)!}=\frac{(n-1)!}{(n+2)!}$

**Posted:** Wed Mar 26, 2008 8:18 am

math154

Click to reveal hidden content
And I already posted a problem, so, yeah.

**Posted:** Wed Mar 26, 2008 8:25 am