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Old 10-29-2007   #71
Temperal
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Yeah, a Platonic solid.
 
Old 10-30-2007   #72
hunter34
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My attempt.

You know that the surface area of the tetrahedron is , making the side length . This gives the volume:

You all may laugh at this failing attempt.
 
Old 11-01-2007   #73
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Next problem?
 
Old 11-02-2007   #74
hunter34
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Wait is that right?

If , what is ?
 
Old 11-03-2007   #75
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, it's 18.
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Old 11-04-2007   #76
Temperal
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Okay, so here's the next one:

The 9-digit number abb,aba,ba3 is a multiple of 99 for some pair of digits a and b. What is b - a ?
 
Old 11-04-2007   #77
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Using properties for divisibilities by 9 and 11, we determine
a+b is 6 mod 9
b-a is 4 mod 11.
Since they are digits, b-a is 4.
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Old 11-05-2007   #78
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Good.

What is the smallest multiple of 24 that is a perfect cube?
 
Old 05-13-2008   #79
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(x+y)^6
Using Pascal's triangle, we could figure out the problem

x^6 + 6*x^5*y + 15*x^4*y*2 + 20*x^3*y*3 + 15*x^2*y^4 + 6*x*y^5 + y^6

Do you notice a pattern?
 
Old 05-13-2008   #80
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first, you factor 24 = 2^3 * 3
So the next perfect cube would be 2^a multiple of three * 3^3
The answer is 24 * 9, which is equal to 216

next question:
WITH FULL SOLUTIONS PLEASE

Fractions a/b and c/d are called neighbor fractions if their difference (ad - bc)/(bd) has a numerator of positive or negative 1, that is, ad - bc = positive or negative one. Prove that

If a/b and c/d are neighbor fractions, then (a + b)/(c + d) is between them and is a neighbor fraction for both a/b and c/d; moreover, no fraction e/f with positive integer "e" and "f" such that f is less than b + d is between a/b and c/d. [/img]
 
 

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