1. **Problem statement:** We have a set $S$ of integer-coordinate points $(x,y)$ satisfying three conditions: - $x^2 + y^2 \leq 2026$ - $y \geq |x| \sin(x)$ - $x, y \in \mathbb{Z}$ Define a function $f: S \to \mathbb{R}$ by $$f(x,y) = \left\lfloor \frac{x^2 + y^2}{10} \right\rfloor \cdot (\sin^2(x) + \cos^2(y))$$ where $\lfloor \cdot \rfloor$ is the floor function. Define a weighted distance metric $d$ on $\mathbb{R}^2$ by $$d((x_1,y_1),(x_2,y_2)) = |x_1 - x_2| + |y_1 - y_2| + \max\{|x_1 - x_2|, |y_1 - y_2|\}.$$ Let $M$ be the subset of $S$ where $f$ attains its maximum value. **Question 1:** Find the minimum distance $\min\{d(P,Q) \mid P,Q \in M, P \neq Q\}$. 2. **Step 1: Understand $f(x,y)$** - Note that $\sin^2(x) + \cos^2(y)$ is always between 0 and 2, but since $\sin^2(\theta) + \cos^2(\theta) = 1$ for the same angle, here $x$ and $y$ differ, so the sum is between 0 and 2. - However, for any $x,y$, $\sin^2(x) + \cos^2(y) \leq 2$ and is always positive. 3. **Step 2: Maximize $f(x,y)$ over $S$** - Since $f(x,y) = \left\lfloor \frac{x^2 + y^2}{10} \right\rfloor \cdot (\sin^2(x) + \cos^2(y))$, the dominant factor is $\left\lfloor \frac{x^2 + y^2}{10} \right\rfloor$ because it grows with $x^2 + y^2$. - The maximum $x^2 + y^2$ in $S$ is at most 2026. - So the maximum floor value is $\left\lfloor \frac{2026}{10} \right\rfloor = 202$. 4. **Step 3: Find points in $S$ with $x^2 + y^2$ close to 2026 and satisfying $y \geq |x| \sin(x)$** - Since $x,y$ are integers, $x^2 + y^2$ must be an integer less or equal to 2026. - The maximum floor value is 202, so points with $x^2 + y^2$ in $[2020,2026]$ will have floor 202. 5. **Step 4: Evaluate $\sin^2(x) + \cos^2(y)$ for these points** - Since $\sin^2(x) + \cos^2(y)$ is always positive and less than or equal to 2, the maximum value of $f$ is approximately $202 \times$ (max of $\sin^2(x) + \cos^2(y)$). - The maximum of $\sin^2(x) + \cos^2(y)$ is 2, achieved when $\sin^2(x) = 1$ and $\cos^2(y) = 1$. - $\sin^2(x) = 1$ when $x = \frac{\pi}{2} + k\pi$, but $x$ is integer, so check integer $x$ near these values. - $\cos^2(y) = 1$ when $y = k\pi$, but $y$ integer, so $y=0$ is a candidate. 6. **Step 5: Identify $M$** - $M$ consists of points $(x,y)$ in $S$ with $x^2 + y^2$ near 2026 and $f(x,y)$ maximal. - Because $f$ depends mainly on $x^2 + y^2$, $M$ is the set of points on the circle $x^2 + y^2$ close to 2026 satisfying $y \geq |x| \sin(x)$. 7. **Step 6: Compute minimum distance $d$ between distinct points in $M$** - The metric is $$d(P,Q) = |x_1 - x_2| + |y_1 - y_2| + \max(|x_1 - x_2|, |y_1 - y_2|).$$ - For integer points, the minimal nonzero difference in $x$ or $y$ is 1. - The minimal $d$ between distinct points is at least when $|x_1 - x_2|$ and $|y_1 - y_2|$ are minimal positive integers. 8. **Step 7: Calculate minimal $d$ for minimal coordinate differences** - Case 1: $|x_1 - x_2|=1$, $|y_1 - y_2|=0$: $$d = 1 + 0 + \max(1,0) = 1 + 0 + 1 = 2.$$ - Case 2: $|x_1 - x_2|=0$, $|y_1 - y_2|=1$: $$d = 0 + 1 + \max(0,1) = 1 + 1 = 2.$$ - Case 3: $|x_1 - x_2|=1$, $|y_1 - y_2|=1$: $$d = 1 + 1 + \max(1,1) = 2 + 1 = 3.$$ So the minimal possible $d$ between distinct points is 2. 9. **Step 8: Check if $M$ has at least two points** - Since $S$ contains many integer points on the circle $x^2 + y^2 \leq 2026$ and the inequality $y \geq |x| \sin(x)$ is not very restrictive, there are multiple points in $M$. - Therefore, the minimal distance between distinct points in $M$ is 2. **Final answer for question 1:** $$\boxed{2}$$