can you solve this: Consider a function $f(x) = \sin(x) + \cos(x)$, which is a periodic function with period $2\pi$. Let $X$ be the smallest positive integer such that the function $g(x) = f(x) \cdot f(x+X)$ has a period of $\pi$. If the multiplicity of the solution to the equation $f(x) = 0$ in the interval $[0, 2\pi)$ is $m$, and we know that $\forall x \in [0, 2\pi), \exists y \in [0, 2\pi)$ such that $f(x) \cdot f(y) = 0$, then what is the value of $X$?

We are given the function $ f(x) = \sin(x) + \cos(x) $ and tasked with finding the smallest positive integer $ X $ such that the function $ g(x) = f(x) \cdot f(x+X) $ has a period of $ \pi $. Additionally, we are provided that the multiplicity of the solution to the equation $ f(x) = 0 $ in the interval $ [0, 2\pi) $ is $ m $, and that $ \forall x \in [0, 2\pi), \exists y \in [0, 2\pi) $ such that $ f(x) \cdot f(y) = 0 $.

### Step 1: Simplify $ f(x) $
We start by simplifying the function $ f(x) = \sin(x) + \cos(x) $.

Using the trigonometric identity $ \sin(x) + \cos(x) = \sqrt{2} \sin\left(x + \frac{\pi}{4}\right) $, we can express $ f(x) $ in terms of a single sine function:
\[
f(x) = \sqrt{2} \sin\left(x + \frac{\pi}{4}\right)
\]
Thus, the function $ f(x) $ is simply a sine wave, shifted by $ \frac{\pi}{4} $, and its zeros correspond to the zeros of $ \sin(x) $.

### Step 2: Find the zeros of $ f(x) $
The function $ f(x) = \sqrt{2} \sin\left(x + \frac{\pi}{4}\right) $ has zeros where the sine function is zero. That is, we need to solve
\[
\sin\left(x + \frac{\pi}{4}\right) = 0
\]
The general solution to this equation is:
\[
x + \frac{\pi}{4} = n\pi \quad \text{for some integer } n
\]
Thus,
\[
x = n\pi - \frac{\pi}{4}
\]
For $ x \in [0, 2\pi) $, the solutions are:
\[
x = \frac{3\pi}{4}, \frac{7\pi}{4}
\]
Therefore, the zeros of $ f(x) $ in $ [0, 2\pi) $ are $ \frac{3\pi}{4} $ and $ \frac{7\pi}{4} $.

### Step 3: Analyze $ g(x) = f(x) \cdot f(x+X) $
We are tasked with finding the smallest positive integer $ X $ such that $ g(x) = f(x) \cdot f(x+X) $ has a period of $ \pi $.

First, observe that $ f(x) $ itself has a period of $ 2\pi $, since $ \sin(x + \frac{\pi}{4}) $ is periodic with period $ 2\pi $.

Now, let's analyze the function $ f(x+X) $. The zeros of $ f(x+X) $ will occur when $ x+X $ satisfies the same zero conditions as $ f(x) $, namely:
\[
\sin\left((x+X) + \frac{\pi}{4}\right) = 0
\]
The key idea is that for $ g(x) = f(x) \cdot f(x+X) $ to have a period of $ \pi $, the zeros of $ f(x) $ and $ f(x+X) $ must be aligned in such a way that $ g(x) $ repeats every $ \pi $.

### Step 4: Determine the smallest $ X $
Since $ f(x) $ has zeros at $ \frac{3\pi}{4} $ and $ \frac{7\pi}{4} $, the function $ f(x+X) $ must have zeros that interlace with the zeros of $ f(x) $, leading to $ g(x) $ repeating every $ \pi $.

The smallest such $ X $ occurs when the shift aligns $ f(x+X) $ with $ f(x) $ such that the period of $ g(x) $ becomes $ \pi $. This happens when $ X = \pi $, because shifting by $ \pi $ causes the sine function to "flip," resulting in the desired periodic behavior.

### Step 5: Verify the multiplicity and conditions
The multiplicity $ m $ of the solution $ f(x) = 0 $ in $ [0, 2\pi) $ is $ 2 $ because there are two distinct zeros, $ \frac{3\pi}{4} $ and $ \frac{7\pi}{4} $.

The condition that $ \forall x \in [0, 2\pi), \exists y \in [0, 2\pi) $ such that $ f(x) \cdot f(y) = 0 $ is satisfied because for any $ x $, there exists a zero of $ f(y) $ within $ [0, 2\pi) $.

### Final Answer:
The smallest positive integer $ X $ is $ \boxed{1} $.

Consider a function f(x) = sin(x) + cos(x), which is a periodic function with period 2π. Let X be the smallest positive integer such that the function g(x) = f(x) ⋅ f(x+X) has a period of π. If the multiplicity of the solution to the equation f(x) = 0 in the interval [0, 2π) is m, and we know that ∀x ∈ [0, 2π), ∃y ∈ [0, 2π) such that f(x) ⋅ f(y) = 0, then what is the value of X?

Explore how to find the smallest positive integer X such that the function g(x) = f(x) ⋅ f(x+X), where f(x) = sin(x) + cos(x), has a period of π. This problem covers concepts in trigonometry, periodic functions, and the alignment of zeros, suitable for advanced high school students in Grades 11-12.

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