To solve this problem, we need to find the smallest value of $n$ such that any three numbers $a$, $b$, $c$ in the set $X$ can form the sides of an acute-angled triangle. The set $X$ consists of twenty positive integers: $n, n+2, n+4, \ldots, n+38$.

Key points to consider:

1. For three sides $a$, $b$, and $c$ to form a triangle, the following must hold:

   • $a + b > c$
   • $b + c > a$
   • $c + a > b$

2. For the triangle to be acute-angled, the square of the longest side should be less than the sum of the squares of the other two sides:

   • $c^2 < a^2 + b^2$ (assuming $c$ is the longest side).

Finding the smallest $n$:

The set $X = \{ n, n+2, n+4, \dots, n+38 \}$ contains 20 consecutive even numbers.

Let's consider the largest three numbers in the set, $a = n+34$, $b = n+36$, and $c = n+38$. For $n$ to be the smallest possible, we must ensure the triangle formed by these three numbers is acute-angled.

We must check the condition for an acute triangle: $(n+38)^2 < (n+34)^2 + (n+36)^2$ Expanding and simplifying: $n^2 + 76n + 1444 < n^2 + 68n + 1156 + n^2 + 72n + 1296$ Simplifying further: $n^2 + 76n + 1444 < 2n^2 + 140n + 2452$ Subtract $n^2 + 76n + 1444$ from both sides: $0 < n^2 + 64n + 1008$ This inequality is true for all positive $n$, so we need to check if this holds when $n$ is small enough to still satisfy the acute triangle condition.

Testing specific values of $n$:

To ensure that $X$ contains all positive numbers:

• $n + 38$ must be positive, so $n \geq 1$.

Next, let's consider the smallest $n$ such that the smallest elements in $X$ are also large enough to form an acute-angled triangle.

Try $n = 14$:

Set $X = \{ 14, 16, \dots, 52 \}$. The largest three numbers are 48, 50, and 52.

Check: $52^2 < 48^2 + 50^2$ $2704 < 2304 + 2500 = 4804$ This inequality is true, so the triangle formed is acute-angled.

Thus, the smallest $n$ that satisfies the conditions is $\boxed{14}$.

Would you like more details, or do you have any questions?

Here are five related questions:

1. What conditions must be met for three sides to form an obtuse-angled triangle?
2. How would the problem change if the numbers were odd instead of even?
3. What is the general method to determine if a set of numbers can form an acute-angled triangle?
4. How would the solution change if the set consisted of only 10 numbers?
5. Can you explain how the triangle inequality theorem applies in different scenarios?

Tip: When solving geometric problems involving triangles, always check the type of triangle (acute, right, or obtuse) based on the given conditions.