Find all positive integers a<b<c (a is less than or equal to b, b is less than or equal to c) such that:
arctan (1/a) + arctan (1/b) + arctan (1/c) = (pi/4)
To solve this problem, we need to understand the properties of the arctangent function and how it behaves.
First, let's rewrite the given equation:
arctan(1/a) + arctan(1/b) + arctan(1/c) = π/4
Now, the arctangent function has several important properties:
1. arctan(x) + arctan(1/x) = π/2 for any positive real number x.
2. The arctangent function is strictly increasing for x > 0.
3. The arctangent function is strictly decreasing for x < 0.
Using property 1, we can rewrite the equation as:
arctan(1/a) + arctan(1/b) + arctan(1/c) = arctan(a) + arctan(b) + arctan(c) = π/2
Now, let's analyze the equation further. Since the arctangent function is strictly increasing for positive arguments, we can conclude that a < b < c.
To find all possible sets of positive integers (a, b, c) that satisfy the equation, we need to consider different cases:
Case 1: a = 1
If a = 1, then arctan(a) = arctan(1) = π/4. This means:
arctan(b) + arctan(c) = π/2 - π/4 = π/4
We can find all pairs (b, c) such that arctan(b) + arctan(c) = π/4. For example:
- b = 1, c = 2: arctan(1) + arctan(2) = π/4
- b = 2, c = 3: arctan(2) + arctan(3) = π/4
- b = 3, c = 4: arctan(3) + arctan(4) = π/4
- and so on...
Case 2: a = 2
If a = 2, then arctan(a) = arctan(2) < π/4. This means b and c must be greater than 2. However, it is not possible to find values of b and c such that arctan(a) + arctan(b) + arctan(c) = π/4.
Hence, the only positive integers that satisfy the given equation are when a = 1, b = 1, and c = 2 or any permutation of these values.
Therefore, the solutions are:
(1, 1, 2)
(1, 2, 1)
(2, 1, 1)