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Home » Mathematics » Middle School Contests (MATHCOUNTS, AMC 8, MOEMS, Ages 10-13)
 
Middle School Marathon
Moderators: AIME15, frost13, isabella2296, LadyKn1ght, nebula42, rcv, solafidefarms
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AIME15
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#561
It seems we need a little reminder: take the time to read MAYBE THE FIVE POSTS ABOVE YOURS.

Thank you.

Current problem: The cheerleading team bought five bags of apples to sell at a track meet. Each bag cost $3.20 and had an average of eight apples. The team sold the apples for 75 cents each. All but six apples were sold. What was the team's profit?
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PostPosted: Sun Sep 20, 2009 4:50 pm  Back to top 
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pytheagle
Riemann Hypothesis
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#562
It's been a while since I came back to math. I spent the summer and a month after school starting playing games on the computer! And by the way, congratulations AIME15 on becoming a moderator.

Solution
We have 5\times8=40 apples in total meaning we sold 40-6=34 of these. We made a total of 34\times.75=25.50 dollars. However, we spent 3.20\times5=16 dollars on purchasing the apples, so our profit is 25.50-16=\boxed{9.50}.


New Problem
How many integers satisfy the equation 3n^3-12n=0? (Source: MATHCOUNTS}
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PostPosted: Sun Sep 20, 2009 4:56 pm  Back to top 
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AIME15
Birch & Swinnerton Dyer
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#563
Solution

We have 3n(n^2-4)=3n(n+2)(n-2)=0, for an answer of \boxed{3}.

Alternatively, we see that it is a cubic equation and immediately guess 3 since it's a Countdown problem.


New Problem

Given that \frac{a}{b}=12^9 and \frac{b}{c}=12^8, what is \frac{a}{c}? Express your answer in the form p^{k_1}q^{k_2}, where p,q are primes.

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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 |

PostPosted: Sun Sep 20, 2009 7:25 pm  Back to top 
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AceOfDiamonds
Riemann Hypothesis
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#564
Solution
This is just \frac{a}{b}*\frac{b}{c} as the b's will cancel out.
so
12^9*12^8=12^{17}
now it's
2^{34}*3^17

New Problem

If a train going at 48 mph slows down to 40 mph, what is the percent change?
Express your answer as a mixed number.


PostPosted: Sun Sep 20, 2009 7:32 pm  Back to top 
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mathfreak77
Poincare Conjecture
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#565
Solution
From 48 mph to 40 mph is a 8 mph decrease. Since 8 is 8/48=0.1666...% of 48, the percent change is \boxed{0.1\overline6\%\text{ decrease}}

NP
The three sides of a triangle are all integers, not necessarily different. If two are 7 and 10, how many different lengths can the third one be?
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PostPosted: Sun Sep 20, 2009 9:15 pm  Back to top 
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AwesomeToad
Yang-Mills Theory
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#566
Solution
By the Triangle inequality the third side x must satisfy x+7>10 and x<10+7 so 3<x<17 and x=\{4,5,6,\cdots,15,16\}
So there are 13 possible x

New Problem
Find the value of \sqrt{50\cdot51\cdot52\cdot53+1}
No calculator

PostPosted: Mon Sep 21, 2009 3:55 am  Back to top 
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AIME15
Birch & Swinnerton Dyer
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#567
Please keep problems to school and chapter level for now.

New Problem
Amy has 12 apples and Burt has 16 apples. If Amy gives \frac{1}{3} of her apples to Burt, and Burt gives \frac{1}{4} of his apples (after the first exchange) to Carlos, how many apples does Carlos have?
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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 |

PostPosted: Mon Sep 21, 2009 6:44 am  Back to top 
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AwesomeToad
Yang-Mills Theory
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#568
Solution
After the exchange between Amy and Burt, Burt has 20 apples.
Then Carlos gets 5

New Problem
I am x years old. In y years I will be 15 years old. y years ago, I was 5 years old. How old am I now?

PostPosted: Mon Sep 21, 2009 8:10 am  Back to top 
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basketball9
Hodge Conjecture
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#569
result
x+y=15 and x-y=5 so 2x=20 and x=10

np: you and fve friends divide 1500 hundred dollars equally how many dollars does each person get
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PostPosted: Mon Sep 21, 2009 11:33 am  Back to top 
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Riemann Hypothesis
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#570
Solution

Dividing 1500 by 6 gives \boxed{250}.

New Problem

What is 7 \frac{5}{7} \% of the epxression 17.7 - 2 \frac{1}{9} +2.3? Express your answer as a decimal to the nearest hundredth.

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PostPosted: Mon Sep 21, 2009 4:21 pm  Back to top 
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Dojo
Navier-Stokes Equations
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#571
LaTeX wrote:

What is 7 \frac {5}{7} \% of the epxression 17.7 - 2 \frac {1}{9} + 2.3? Express your answer as a decimal to the nearest hundredth.

Solution

17.7-2 \dfrac{1}{9}+2.3=18\dfrac{1}{9}

\dfrac{163}{9}\times \dfrac{\frac{54}{7}}{100}=\dfrac{27}{350}=\dfrac{4401}{3150}\approx\boxed{1.40}

NP

Audrey got her learner’s permit for driving an automobile on Saturday, June 21. She can get her
driver’s license in as little as six months after she gets her learner’s permit. In other words, she
could get it on the 21st of the sixth month after June. If the Department of Licensing is the only
place she can get her driver’s license and they are closed on Sundays and Mondays, what is the
earliest date that she can get her driver’s license? Give the month and day.

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PostPosted: Mon Sep 21, 2009 10:27 pm  Back to top 
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goldenboy1.618
Poincare Conjecture
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#572
Solution
We see that December 21 is 30 + 31 + 31 + 30 + 31 + 30 = 183 days from June 21. As 183\equiv1\pmod7, December 20 is a Saturday, 21 is Sunday, 22 is Monday, and 23 is Tuesday. She can go December 23, Tuesday.


NP
Kelly draws square ABCD with sides 6. She then constructs equilateral triangles ABX and BCY. Find the distance XY in simplest radical form.
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PostPosted: Tue Sep 22, 2009 11:34 am  Back to top 
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$LaTeX$
Riemann Hypothesis
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#573
@Dojo, your solution is incorrect. The correct solution is7 \frac{5}{7}\%(17.7.+2.3-2\frac{1}{9})=\frac{54}{7}\%(20-\frac{19}{9})=\frac{54}{7}\cdot\frac{161}{9}\%=6\cdot23\%=138\%=\bo....

@goldenboy: I don't quite get your problem. Is x and y inside the square or outside?
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PostPosted: Tue Sep 22, 2009 1:15 pm  Back to top 
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Math Champion
Yang-Mills Theory
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#574
Solution

The height of the equalateral triangles would be \frac {\sqrt {3}}{2}\cdot6 = 3\sqrt {3}. If the points are outside the square, then the answer is 6\sqrt {3} + 6. If one of them is inside and one of them is outside then the answer is 6. If they are both inside the square, then the answer is 3\sqrt {3} - (6 - 3\sqrt {3}) = 6\sqrt {3} - 6.


NP

If the value of x + \frac {1}{x} is 4, what is x^8 + \frac {1}{x^8}?

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PostPosted: Tue Sep 22, 2009 2:33 pm  Back to top 
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Erdong171
P versus NP
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#575
Solution
Square the equation to get x^2+2+\frac{1}{x^2}=16, which simplifies to x^2+\frac{1}{x^2}=14. Repeating the same process gets x^4+\frac{1}{x^4}=194. Repeating one last time gets the equation x^8+\frac{1}{x^8}=\boxed{37634}


NP
If x+\frac{1}{x}=2, find x^{12}+\frac{1}{x^{12}}.

PostPosted: Tue Sep 22, 2009 5:51 pm  Back to top 
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AwesomeToad
Yang-Mills Theory
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#576
Since x+\frac{1}{x}=2 we have x^2+2+\frac{1}{x^2}=4 and x^2+\frac{1}{x^2}=2 Therefore x^4+\frac{1}{x^4}=4-2=2 Therefore (x^4+\frac{1}{x^4})^3=x^{12}+\frac{1}{x^{12}}+3\cdot4=64sothe expression is 64-12=52

PostPosted: Wed Sep 23, 2009 8:23 am  Back to top 
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$LaTeX$
Riemann Hypothesis
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#577
Correct Solution: x=1, so x^{12}+\frac{1}{x^{12}}=2

New Problem:
Alexa has a mixture consisting of 8 ounces of red jelly beans and
13 ounces of green jelly beans. What is the minimum number of
ounces of white jelly beans she must buy so that, when added to the
old mixture, her new mixture is an integer number of pounds?

BTW: Don't be a hypocrite Awesome Toad....
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PostPosted: Wed Sep 23, 2009 3:16 pm  Back to top 
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AwesomeToad
Yang-Mills Theory
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#578
Solution
21 ounces total right now

Since a pound is 16 ounces we need 32 pounds. Therefore we need \boxed{11} ounces of jelly beans.


NP
The sum \binom{12}{3} + \binom{12}{4} can be written as \binom{x + 9}{y + 1}
Find \binom{x}{y}

@latex:
Wow didn't realize x=1, I'm used to problems with nonreal roots Blush

EDIT:
Thanks for pointing out that AIME15, I will post a new problem Blush
New Problem (Hopefully this is right)
Suppose there is a square table. There are 8 seats around the table, 2 on each side.Assuming the seats are distinguishable, how many distinct seatings are possible?
Last edited by AwesomeToad on Thu Sep 24, 2009 8:12 am; edited 1 time in total 
PostPosted: Thu Sep 24, 2009 3:55 am  Back to top 
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AIME15
Birch & Swinnerton Dyer
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#579
AwesomeToad: Your problem does not have distinct solutions.

For example, \binom{x+9}{y+1}=\binom{13}{4}=\binom{13}{9}=\binom{\binom{12}{3}+\binom{12}{4}}{1}=\binom{\binom{12}{3}+\binom{12}{4}}{\bino....
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PostPosted: Thu Sep 24, 2009 6:40 am  Back to top 
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$AceofDiamonds$
P versus NP
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#580
Okay so if there's 8 chairs and each are disquinishable then the number of chair combinations should be

40320.

Ps. some one teach me how to change my username because it just says the password does not match the name required

np:

find the sum of the digits in base 10 for 24base7 and 30 base7
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PostPosted: Thu Sep 24, 2009 1:32 pm  Back to top 
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