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1. **Rewrite the function using trigonometric identities:**
\[
f(x) = 1 - a \cos(x) - b \sin(x) - A \cos(2x) - B \sin(2x)
\]
We can use the angle addition formulas for sine and cosine:
\[
\cos(x + \theta) = \cos(x)\cos(\theta) - \sin(x)\sin(\theta)
\]
\[
\sin(x + \theta) = \sin(x)\cos(\theta)... | Let $a,b,A,B$ be given reals. We consider the function defined by \[ f(x) = 1 - a \cdot \cos(x) - b \cdot \sin(x) - A \cdot \cos(2x) - B \cdot \sin(2x). \] Prove that if for any real number $x$ we have $f(x) \geq 0$ then $a^2 + b^2 \leq 2$ and $A^2 + B^2 \leq 1.$ | math | AI-MO/NuminaMath-CoT/aops_forum | {'source': 'aops_forum', 'messages': [{'content': 'Let $a,b,A,B$ be given reals. We consider the function defined by \\[ f(x) = 1 - a \\cdot \\cos(x) - b \\cdot \\sin(x) - A \\cdot \\cos(2x) - B \\cdot \\sin(2x). \\] Prove that if for any real number $x$ we have $f(x) \\geq 0$ then $a^2 + b^2 \\leq 2$ and $A^2... | null | [
"Let $f(x) = 1 - a \\cos x - b \\sin x - A \\cos 2x - B \\sin 2x \\geq 0$ for all $x \\in \\mathbb{R}$.\nWe want to show that $a^2 + b^2 \\leq 2$ and $A^2 + B^2 \\leq 1$.\n\nConsider specific values of $x$.\nFor $x=0$, $f(0) = 1 - a - A \\geq 0$, so $a+A \\leq 1$.\nFor $x=\\pi$, $f(\\pi) = 1 - a \\cos \\pi - b \\si... | Let $P(x) = 1 - a \cos x - b \sin x - A \cos 2x - B \sin 2x$.
Since $P(x) \geq 0$ for all real $x$, by Fejer-Riesz theorem, there exists a polynomial $Q(z) = c_0 + c_1 z + c_2 z^2$ such that $P(x) = |Q(e^{ix})|^2$.
We have $|Q(e^{ix})|^2 = (c_0 + c_1 e^{ix} + c_2 e^{2ix})(\bar{c_0} + \bar{c_1} e^{-ix} + \bar{c_2} e^{-2... | Okay, so I need to prove that if the function f(x) = 1 - a cos(x) - b sin(x) - A cos(2x) - B sin(2x) is non-negative for all real numbers x, then the sums of the squares of the coefficients a and b must be less than or equal to 2, and similarly the sums of the squares of A and B must be less than or equal to 1. Hmm, al... |
128 | Given a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerator and denominator. For how many rational numbers between 0 and 1 will $20_{}^{}!$ be the resulting product? | math | qq8933/AIME_1983_2024 | {'ID': '1991-5', 'Year': 1991, 'Problem Number': 5, 'Part': None} | null | [
"\nThe problem asks for the number of rational numbers between 0 and 1 such that when the rational number is written as a fraction in lowest terms, the product of the numerator and the denominator is $20!$.\n\nLet the rational number be $r$, where $0 < r < 1$.\nWe can write $r$ as a fraction $\\frac{a}{b}$, where $... | Solution:
Let the rational number between 0 and 1 be $r$. When written as a fraction in lowest terms, let it be $\frac{a}{b}$, where $a$ and $b$ are positive integers, $\gcd(a, b) = 1$, and $0 < a < b$.
We are given that the product of the numerator and denominator is $20!$, so $a \times b = 20!$.
We need to find the ... | Okay, so I have this problem here: Given a rational number between 0 and 1, write it as a fraction in lowest terms and then calculate the product of the numerator and denominator. The question is asking how many such rational numbers exist where the product of the numerator and denominator is exactly 20!.
Hmm, let me ... |
1. **Consider a countable subset \( S_0 \subseteq S \)**:
Let \( S_0 = \{x_1, x_2, \ldots\} \) be a countable subset of \( S \). Since \( S \) is a set of points in \( \mathcal{H} \) such that the distance between any two distinct points is \( d \), the same property holds for \( S_0 \).
2. **Find a point \( y \) f... | Let $ \mathcal{H}$ be an infinite-dimensional Hilbert space, let $ d>0$ , and suppose that $ S$ is a set of points (not necessarily countable) in $ \mathcal{H}$ such that the distance between any two distinct points in $ S$ is equal to $ d$ . Show that there is a point $ y\in\mathcal{H}$ such that
\[ \lef... | math | AI-MO/NuminaMath-CoT/aops_forum | {'source': 'aops_forum', 'messages': [{'content': 'Let $ \\mathcal{H}$ be an infinite-dimensional Hilbert space, let $ d>0$ , and suppose that $ S$ is a set of points (not necessarily countable) in $ \\mathcal{H}$ such that the distance between any two distinct points in $ S$ is equal to $ d$ . Show that ther... | null | [
"Let $S \\subset \\mathcal{H}$ such that for any $x_1, x_2 \\in S$ with $x_1 \\neq x_2$, we have $\\|x_1 - x_2\\| = d$. We need to find $y \\in \\mathcal{H}$ such that the set of vectors $v_x = \\frac{\\sqrt{2}}{d}(x-y)$ for $x \\in S$ is an orthonormal system.\n\nFor the set of vectors $\\{v_x\\}_{x \\in S}$ to be... | Let $x_0$ be an arbitrary point in $S$. Consider the set $S' = \{x - x_0 : x \in S, x \neq x_0\}$. For any $x, z \in S$ with $x \neq z$, we have $\|x-z\| = d$.
Consider any two distinct points $x_1, x_2 \in S$. We are given $\|x_1 - x_2\| = d$.
We want to find $y \in \mathcal{H}$ such that for all $x \in S$, $\|\frac{\... | Okay, so I need to show that in an infinite-dimensional Hilbert space H, given a set S of points where each pair of distinct points is exactly distance d apart, there exists a point y such that when we take each x in S, subtract y, scale by sqrt(2)/d, the resulting vectors form an orthonormal system. Hmm, okay.
First,... |
109 | "Find the remainder when $9 \\times 99 \\times 999 \\times \\cdots \\times \\underbrace{99\\cdots9}_(...TRUNCATED) | math | qq8933/AIME_1983_2024 | {'ID': '2010-I-2', 'Year': 2010, 'Problem Number': 2, 'Part': 'I'} | null | ["\nThe problem asks for the remainder when the product $P = 9 \\times 99 \\times 999 \\times \\cdot(...TRUNCATED) | "Solution:\nLet the product be $P = 9 \\times 99 \\times 999 \\times \\cdots \\times \\underbrace{99(...TRUNCATED) | "Okay, so I need to find the remainder when the product of a bunch of 9s is divided by 1000. The pro(...TRUNCATED) |
"Given a function $f$, define $N(f)$ to be the number of numbers that are in the same cycle as 1 (in(...TRUNCATED) | "We randomly choose a function $f:[n] \\rightarrow[n]$, out of the $n^{n}$ possible functions. We al(...TRUNCATED) | math | KbsdJames/Omni-MATH | "{'domain': ['Mathematics -> Discrete Mathematics -> Combinatorics'], 'difficulty': 5.0, 'answer': '(...TRUNCATED) | null | ["\nThe problem asks for the probability that a randomly chosen function $f:[n] \\rightarrow[n]$ and(...TRUNCATED) | "Solution:\nLet $\\mathcal{F}$ be the set of all functions $f:[n] \\rightarrow [n]$, so $|\\mathcal{(...TRUNCATED) | "Okay, so I have this probability problem here. Let me read it again and try to understand what it's(...TRUNCATED) |
167.0 | Compute the mean molecular speed v in the heavy gas radon (Rn) in m/s | math | TIGER-Lab/TheoremQA/float | {'Answer_type': 'float', 'Picture': None} | null | ["Here's a thinking process to arrive at the solution:\n\n1. **Understand the Goal:** The request is(...TRUNCATED) | "To compute the mean molecular speed (v) of radon (Rn), we can use the following formula derived fro(...TRUNCATED) | "Okay, so I need to compute the mean molecular speed of radon gas, right? Hmm, let me remember how t(...TRUNCATED) |
1.3 | "Two capacitors with capacitance values $C_{1}=2000 \\pm 10 \\mathrm{pF}$ and $C_{2}=3000 \\pm 15 \\(...TRUNCATED) | math | daman1209arora/jeebench/phy | {'subject': 'phy', 'description': 'JEE Adv 2020 Paper 2', 'index': 15, 'type': 'Numeric'} | null | ["Let $C_1$ and $C_2$ be the capacitance values of the two capacitors, and $\\Delta C_1$ and $\\Delt(...TRUNCATED) | "Given the capacitance values $C_{1}=2000 \\pm 10 \\mathrm{pF}$ and $C_{2}=3000 \\pm 15 \\mathrm{pF}(...TRUNCATED) | "Okay, so I have this problem here about two capacitors connected in series, and I need to find the (...TRUNCATED) |
181 | "One base of a trapezoid is $100$ units longer than the other base. The segment that joins the midpo(...TRUNCATED) | math | qq8933/AIME_1983_2024 | {'ID': '2000-II-6', 'Year': 2000, 'Problem Number': 6, 'Part': 'II'} | null | ["\nThe problem asks for the value of $\\lfloor x^2/100 \\rfloor$, where $x$ is the length of a segm(...TRUNCATED) | "Solution:\nLet the lengths of the two bases of the trapezoid be $b_1$ and $b_2$, with $b_1 > b_2$. (...TRUNCATED) | "Okay, let me try to tackle this problem step by step. First, I need to understand what's being aske(...TRUNCATED) |
"1. **Defining the Sparse Set \\( S \\) and the Turing Machine \\( M \\)**\n\n We need to construc(...TRUNCATED) | "Let's say a language $L \\subseteq \\{0,1\\}^*$ is in $\\textbf{P}_{angel}$ if there exists a p(...TRUNCATED) | math | AI-MO/NuminaMath-CoT/aops_forum | "{'source': 'aops_forum', 'messages': [{'content': \"Let's say a language $L \\\\subseteq \\\\{0,1\(...TRUNCATED) | null | ["Let's break down each part of the problem and build the solutions.\n\n**Part 1: Combining Sparse S(...TRUNCATED) | "Let's solve each part step by step.\n\n**Part 1: Combining Sparse Sets**\n\nWe are given $k$ sparse(...TRUNCATED) | "Okay, let's tackle these problems one by one. I need to make sure I understand each part before jum(...TRUNCATED) |
"(a) Let the number of promoted participants be \\( P \\) and the number of repeaters be \\( R \\). (...TRUNCATED) | "In a mathematics test number of participants is $N < 40$ . The passmark is fixed at $65$ . The te(...TRUNCATED) | math | AI-MO/NuminaMath-CoT/aops_forum | "{'source': 'aops_forum', 'messages': [{'content': 'In a mathematics test number of participants is (...TRUNCATED) | null | ["Let $N$ be the total number of participants, $N < 40$.\nLet $P$ be the number of promoted particip(...TRUNCATED) | "Solution:\nLet $N$ be the total number of participants. The passmark is $65$.\nInitial state:\nAver(...TRUNCATED) | "Okay, let me try to tackle this problem step by step. It's a bit complex, but I'll break it down. \(...TRUNCATED) |
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