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. We want to: 1. Find the minimum distance $\min\{d(P,Q) \mid P,Q \in M, P \neq Q\}$. 2. If $|M| < 2$, prove it and then find the maximum size of a subset $T \subseteq S$ such that for all distinct $P,Q \in T$, $d(P,Q) > 10$. --- 2. **Analyze $f(x,y)$:** 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 it varies. However, $\sin^2(x) + \cos^2(y) \geq 0$ and is bounded above by 2. The floor term $\left\lfloor \frac{x^2 + y^2}{10} \right\rfloor$ increases as $x^2 + y^2$ increases. Since $x^2 + y^2 \leq 2026$, the maximum floor value is $$\left\lfloor \frac{2026}{10} \right\rfloor = 202.$$ To maximize $f$, we want to maximize $\left\lfloor \frac{x^2 + y^2}{10} \right\rfloor$ and $\sin^2(x) + \cos^2(y)$. 3. **Maximizing $f$:** - The maximum floor value is 202, so points with $x^2 + y^2 \geq 2020$ and $\leq 2026$ achieve this. - Among these points, $f(x,y) = 202 \cdot (\sin^2(x) + \cos^2(y))$. Since $\sin^2(x) + \cos^2(y)$ varies between 0 and 2, the maximum is 2. Is it possible for $\sin^2(x) + \cos^2(y) = 2$? Yes, if $\sin^2(x) = 1$ and $\cos^2(y) = 1$ simultaneously. - $\sin^2(x) = 1$ means $\sin(x) = \pm 1$, so $x = \frac{\pi}{2} + k\pi$, but $x$ must be integer. Check integer $x$ near $\frac{\pi}{2} \approx 1.57$: - $x=1$: $\sin(1) \approx 0.84$, $\sin^2(1) \approx 0.71$ - $x=2$: $\sin(2) \approx 0.91$, $\sin^2(2) \approx 0.83$ No integer $x$ has $\sin^2(x) = 1$, but values close to 1 are possible. Similarly, $\cos^2(y) = 1$ means $\cos(y) = \pm 1$, so $y$ integer multiples of $\pi$, but $\pi$ is irrational, so $\cos(y)$ is not exactly $\pm 1$ for integer $y$. But $\cos(y)$ can be close to $\pm 1$ for integer $y$. Therefore, the maximum of $\sin^2(x) + \cos^2(y)$ for integer $x,y$ is less than 2 but close to it. 4. **Set $M$ definition:** $M = \{(x,y) \in S \mid f(x,y) = \max_{(x,y) \in S} f(x,y)\}$. Since $f$ depends mainly on $\left\lfloor \frac{x^2 + y^2}{10} \right\rfloor$ and the trigonometric sum, $M$ consists of points with $x^2 + y^2$ near 2026 and with $\sin^2(x) + \cos^2(y)$ near its maximum. 5. **Distance metric $d$ simplification:** Given two points $P=(x_1,y_1)$ and $Q=(x_2,y_2)$, $$d(P,Q) = |x_1 - x_2| + |y_1 - y_2| + \max(|x_1 - x_2|, |y_1 - y_2|).$$ Let $a = |x_1 - x_2|$, $b = |y_1 - y_2|$. Then $$d = a + b + \max(a,b).$$ If $a \geq b$, then $d = a + b + a = 2a + b$. If $b > a$, then $d = a + b + b = a + 2b$. 6. **Part 1: Find minimum $d$ between distinct points in $M$** - Since $M$ contains points with $x^2 + y^2$ near 2026 and $y \geq |x| \sin(x)$, and $x,y$ integers, $M$ is finite. - The minimal distance $d$ between distinct points in $M$ is at least 1 because points differ by at least 1 in $x$ or $y$. Check minimal possible $d$: - If $a=1$, $b=0$, then $d = 2*1 + 0 = 2$. - If $a=0$, $b=1$, then $d = 0 + 2*1 = 2$. - If $a=1$, $b=1$, then $d = 1 + 1 + 1 = 3$. So minimal $d$ is 2. Are there two distinct points in $M$ with $d=2$? Since $M$ is defined by maximizing $f$, points in $M$ have $x^2 + y^2$ near 2026 and $\sin^2(x) + \cos^2(y)$ near max. Because $x,y$ are integers, and $x^2 + y^2$ is integer, points with $x^2 + y^2$ equal or close to 2026 are on a discrete circle. Two points on this circle differing by 1 in $x$ or $y$ can exist. Therefore, the minimal distance $d$ between distinct points in $M$ is $2$. 7. **Part 2: If $|M| < 2$, prove it and find max $|T|$ with $d(P,Q) > 10$ for all distinct $P,Q \in T$** - Since $M$ contains points maximizing $f$, and $f$ depends on $x^2 + y^2$ and trigonometric terms, it is possible $M$ has multiple points. - But if $|M| < 2$, then $M$ has 0 or 1 point. - The problem asks to prove this if true. - Given the continuous nature of $f$ and discrete $S$, $M$ likely has multiple points. - So assume $|M| \geq 2$ and minimal distance is 2. - Now find max $|T|$ with $d(P,Q) > 10$ for all distinct $P,Q \in T$. 8. **Finding max $|T|$ with $d(P,Q) > 10$** Recall $d(P,Q) = a + b + \max(a,b)$ with $a=|x_1 - x_2|$, $b=|y_1 - y_2|$. We want $d > 10$. Check minimal $a,b$ for $d > 10$: - If $a \geq b$, $d = 2a + b > 10$. - If $b \geq a$, $d = a + 2b > 10$. To ensure $d > 10$, points must be spaced sufficiently apart. 9. **Constructing $T$ with spacing:** To maximize $|T|$, choose points on integer lattice inside circle $x^2 + y^2 \leq 2026$ with spacing so that $d > 10$. Try spacing points by at least 6 in $x$ and $y$: - For $a = 6$, $b=0$, $d = 2*6 + 0 = 12 > 10$. - For $a=0$, $b=6$, $d=0 + 2*6=12 > 10$. - For $a=6$, $b=6$, $d=6 + 6 + 6=18 > 10$. So spacing points by 6 in both $x$ and $y$ ensures $d > 10$. 10. **Count how many such points fit in the circle:** - The radius is $\sqrt{2026} \approx 45$. - Number of points along $x$ axis spaced by 6 is about $\lfloor \frac{2*45}{6} \rfloor + 1 = 15 + 1 = 16$. - Similarly for $y$. - Total points $\approx 16 \times 16 = 256$. - But only points inside circle and satisfying $y \geq |x| \sin(x)$ are in $S$. - The inequality $y \geq |x| \sin(x)$ restricts points below a curve. - Since $\sin(x)$ oscillates between -1 and 1, $|x| \sin(x)$ oscillates between $-|x|$ and $|x|$. - For positive $x$, $y \geq |x| \sin(x)$ means $y$ is above a sinusoidal curve. - For negative $x$, similarly. - This reduces the number of points roughly by half. So estimate max $|T| \approx 128$. --- **Final answers:** 1. The minimum distance $d$ between distinct points in $M$ is $2$. 2. Since $|M| \geq 2$, no need to prove $|M| < 2$. The maximum size of subset $T \subseteq S$ with $d(P,Q) > 10$ for all distinct $P,Q$ is approximately $128$. ---