Mathcounts Marathon

Yeah, a Platonic solid.

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.

Next problem?

Wait is that right?

If , what is ?

[Sillysidley]

, it's 18.

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 ?

[Sillysidley]

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.

Good.

What is the smallest multiple of 24 that is a perfect cube?

(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?

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]