Twenty-seven identical white cubes are assembled into a single cube, the outside of which is painted black. The cube is then disassembled and the smaller cubes thoroughly shuffled in a bag. A blindfolded man (who cannot feel the paint) reassembles the pieces into a cube. What is the probability that the outside of this cube is completely black?
Hint - Answer - Solution
Euler's totient function 
(n) is defined as the number of positive integers not exceeding n that are relatively prime to n, where 1 is counted as being relatively prime to all numbers. So, for example, (20) = 8 because the eight integers 1, 3, 7, 9, 11, 13, 17, and 19 are relatively prime to 20. The table below shows values of (n) for n 
20.
| n | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
 (n) | 1 | 1 | 2 | 2 | 4 | 2 | 6 | 4 | 6 | 4 | 10 | 4 | 12 | 6 | 8 | 8 | 16 | 6 | 18 | 8 |
Euler's totient valence function v(n) is defined as the number of positive integers k such that (k) = n. For instance, v(8) = 5 because only the five integers k = 15, 16, 20, 24, and 30 are such that (k) = 8. The table below shows values of v(n) for n 16. (For n not in the table, v(n) = 0.)
| n | v(n) | k such that (k) = n |
|---|---|---|
| 1 | 2 | 1, 2 |
| 2 | 3 | 3, 4, 6 |
| 4 | 4 | 5, 8, 10, 12 |
| 6 | 4 | 7, 9, 14, 18 |
| 8 | 5 | 15, 16, 20, 24, 30 |
| 10 | 2 | 11, 22 |
| 12 | 6 | 13, 21, 26, 28, 36, 42 |
| 16 | 6 | 17, 32, 34, 40, 48, 60 |
Evaluate v(21000).
Hint 1 - Hint 2 - Answer - Solution
Find the area of the largest semicircle that can be inscribed in the unit square.
Hint - Answer - Solution
In trapezoid¹ ABCD, with sides AB and CD parallel, 
DAB = 6° and ABC = 42°. Point X on side AB is such that AXD = 78° and CXB = 66°. If AB and CD are 1 inch apart, prove that AD + DX − (BC + CX) = 8 inches.
(1) A trapezoid is a quadrilateral with at least one pair of parallel sides. In some countries, such a quadrilateral is known as a trapezium.
Hint - Solution
Is 2n + 3n (where n is an integer) ever the square of a rational number?
Hint - Answer - Solution
Find all positive real solutions of the simultaneous equations:
Hint - Answer - Solution
Compute the infinite product
[sin(x) cos(x/2)]1/2 · [sin(x/2) cos(x/4)]1/4 · [sin(x/4) cos(x/8)]1/8 · ... ,
where 0 x 2
.
Hint - Answer - Solution
By Fermat's Little Theorem, the number x = (2p−1 − 1)/p is always an integer if p is an odd prime. For what values of p is x a perfect square?
Hint - Answer - Solution
Lagrange's Four-Square Theorem states that every positive integer can be written as the sum of at most four squares. For example, 6 = 22 + 12 + 12 is the sum of three squares. Given this theorem, prove that any positive multiple of 8 can be written as the sum of eight odd squares.
Hint - Solution
The absolute value of a real number is defined as its numerical value without regard for sign. So, for example, abs(2) = abs(−2) = 2.
The maximum of two real numbers is defined as the numerically bigger of the two. For example, max(2, −3) = max(2, 2) = 2.
Express:
Hint - Answer - Solution
