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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... | <|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
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... | <|begin_of_text|><|start_header_id|>user<|end_header_id|>
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... |
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 ... | <|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
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 r... | <|begin_of_text|><|start_header_id|>user<|end_header_id|>
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?<|eot_id|><|start_header_id|>assistant<... |
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{\... | <|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
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 tw... | <|begin_of_text|><|start_header_id|>user<|end_header_id|>
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 tha... |
109 | Find the remainder when $9 \times 99 \times 999 \times \cdots \times \underbrace{99\cdots9}_{\text{999 9's}}$ is divided by $1000$ . | 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 $P = 9 \\times 99 \\times 999 \\times \\cdots \\times \\underbrace{99\\cdots9}_{\\text{999 9's}}$ is divided by $1000$.\n\nThe terms in the product are of the form $10^k - 1$, where $k$ ranges from $1$ to $999$.\nThe terms are $10^1 - 1, 10^2 - 1, 10^3 - 1, \\l... | Solution:
Let the product be $P = 9 \times 99 \times 999 \times \cdots \times \underbrace{99\cdots9}_{\text{999 9's}}$.
The terms in the product can be written as $10^k - 1$, where $k$ ranges from $1$ to $999$.
The product is $P = (10^1 - 1)(10^2 - 1)(10^3 - 1) \cdots (10^{999} - 1)$.
We need to find the remainder when... | <|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Find the remainder when $9 \times 99 \times 999 \times \cdots \times \underbrace{99\cdots9}_{\text{999 9's}}$ is divided by $1000$ .<|im_end|>
<|im_start|>assistant
<|im_start|>think
The problem asks for t... | <|begin_of_text|><|start_header_id|>user<|end_header_id|>
Find the remainder when $9 \times 99 \times 999 \times \cdots \times \underbrace{99\cdots9}_{\text{999 9's}}$ is divided by $1000$ .<|eot_id|><|start_header_id|>assistant<|end_header_id|>
The problem asks for the remainder when the product $P = 9 \times 99 \ti... |
"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) | "<|im_start|>system\nYou are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>\(...TRUNCATED) | "<|begin_of_text|><|start_header_id|>user<|end_header_id|>\n\nWe randomly choose a function $f:[n] \(...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) | "<|im_start|>system\nYou are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>\(...TRUNCATED) | "<|begin_of_text|><|start_header_id|>user<|end_header_id|>\n\nCompute the mean molecular speed v in (...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) | "<|im_start|>system\nYou are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>\(...TRUNCATED) | "<|begin_of_text|><|start_header_id|>user<|end_header_id|>\n\nTwo capacitors with capacitance values(...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) | "<|im_start|>system\nYou are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>\(...TRUNCATED) | "<|begin_of_text|><|start_header_id|>user<|end_header_id|>\n\nOne base of a trapezoid is $100$ units(...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) | "<|im_start|>system\nYou are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>\(...TRUNCATED) | "<|begin_of_text|><|start_header_id|>user<|end_header_id|>\n\nLet's say a language $L \\subseteq \\(...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) | "<|im_start|>system\nYou are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>\(...TRUNCATED) | "<|begin_of_text|><|start_header_id|>user<|end_header_id|>\n\nIn a mathematics test number of partic(...TRUNCATED) |
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