Sign up to access problem solutions.

Already have an account? Log in here.

We have discovered a truly marvelous demonstration of this proposition that this margin is too narrow to contain.

An \(x \times y \times z\) rectangular prism is made out of \(xyz\) identical unit cubes. Divide these cubes into 2 mutually exclusive sets.

The first set contains all cubes which have at least one face that is a constituent of the exterior faces of the rectangular prism. These cubes will be called **external**.

The second set contains all cubes which are completely enclosed inside the rectangular prism. These cubes will be called **internal**.

An example is that in a \(3 \times 4 \times 7\) rectangular prism, there are 74 external cubes and 10 internal cubes.

How many different rectangular prisms are there, whose number of internal cubes is equal to the number of external cubes?

If you think the answer is infinite, type -1.

**Details and Assumptions**:

Rotations/Flips among any axis are considered the same.

Rectangular prisms can have square faces

Sign up to access problem solutions.

Already have an account? Log in here.

\[y^2(x^2+y^2-2xy-x-y) = (x+y)^2 (x-y)\]

Find the number of integer pairs \((x, y)\) that satisfy the above expression.

Sign up to access problem solutions.

Already have an account? Log in here.

\[ \large (p-4)(p+1)(p+3)=(n-4)(n+4) \]

Find the sum of all primes \(p\) satisfying the equation above, with \(n\) is a positive integer.

Sign up to access problem solutions.

Already have an account? Log in here.

*funny* if for each of its positive divisors \(d\), the number \(d+2\) is prime. Find the sum of all funny numbers that have the most quantity of divisors.

Sign up to access problem solutions.

Already have an account? Log in here.

Sign up to access problem solutions.

Already have an account? Log in here.

×

Problem Loading...

Note Loading...

Set Loading...