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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 $P(x) = 1 - a \cos x - b \sin x - A \cos 2x - B \sin 2x$.
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... | 6,382 | <|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... | <|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... | <|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... |
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 | 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... | 4,394 | <|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... | <|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... | <|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... |
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 $x_0$ 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... | 2,464 | <|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... | <|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... | <|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... |
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 | 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... | 4,262 | <|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 $1000$ . Let's think step by step and output the final answer within \boxed{}. As... | <|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 $1000$ . Let's think step by step and output the final answer within \boxed{}. T... | <|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 $1000$ . Think for 4262 tokens. <|im_end|> |
Given a function $f$, define $N(f)$ to be the number of numbers that are in the same cycle as 1 (including 1 itself), if there is one, and zero if there is no such cycle. The problem is equivalent to finding $\mathbb{E}(N(f)) / n$. Note that $P(N(f)=k)=\frac{n-1}{n} \cdot \frac{n-2}{n} \cdots \cdots \cdot \frac{n-k+1}{... | We randomly choose a function $f:[n] \rightarrow[n]$, out of the $n^{n}$ possible functions. We also choose an integer $a$ uniformly at random from $[n]$. Find the probability that there exist positive integers $b, c \geq 1$ such that $f^{b}(1)=a$ and $f^{c}(a)=1$. $\left(f^{k}(x)\right.$ denotes the result of applying... | math | KbsdJames/Omni-MATH | {'domain': ['Mathematics -> Discrete Mathematics -> Combinatorics'], 'difficulty': 5.0, 'answer': '\\frac{1}{n}', 'source': 'HMMT_2'} | null | Solution:
Let $\mathcal{F}$ be the set of all functions $f:[n] \rightarrow [n]$, so $|\mathcal{F}| = n^n$. We choose a function $f \in \mathcal{F}$ uniformly at random, and an integer $a \in [n]$ uniformly at random. The total number of pairs $(f, a)$ is $n^n \times n$.
The condition that there exist positive integers... | <|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
We randomly choose a function $f:[n] \rightarrow[n]$, out of the $n^{n}$ possible functions. We also choose an integer $a$ uniformly at random from $[n]$. Find the probability that there exist positive int... | 5,093 | <|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
We randomly choose a function $f:[n] \rightarrow[n]$, out of the $n^{n}$ possible functions. We also choose an integer $a$ uniformly at random from $[n]$. Find the probability that there exist positive int... | <|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
We randomly choose a function $f:[n] \rightarrow[n]$, out of the $n^{n}$ possible functions. We also choose an integer $a$ uniformly at random from $[n]$. Find the probability that there exist positive int... | <|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
We randomly choose a function $f:[n] \rightarrow[n]$, out of the $n^{n}$ possible functions. We also choose an integer $a$ uniformly at random from $[n]$. Find the probability that there exist positive int... |
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 | To compute the mean molecular speed (v) of radon (Rn), we can use the following formula derived from the kinetic theory of gases:
**v = √(8RT / (πM))**
Where:
* **v** is the mean molecular speed (in m/s)
* **R** is the ideal gas constant (8.314 J/mol·K)
* **T** is the absolute temperature (in Kelvin)
* **M** is the ... | <|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Compute the mean molecular speed v in the heavy gas radon (Rn) in m/s<|im_end|>
<|im_start|>assistant
<|im_start|>think
Here's a thinking process to arrive at the solution:
1. **Understand the Goal:** The... | 812 | <|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Compute the mean molecular speed v in the heavy gas radon (Rn) in m/s Let's think step by step and output the final answer within \boxed{}. As you work through the solution, explicitly verbalize your think... | <|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Compute the mean molecular speed v in the heavy gas radon (Rn) in m/s Let's think step by step and output the final answer within \boxed{}. Think for 812 tokens. <|im_end|> | <|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Compute the mean molecular speed v in the heavy gas radon (Rn) in m/s Think for 812 tokens. <|im_end|> |
1.3 | Two capacitors with capacitance values $C_{1}=2000 \pm 10 \mathrm{pF}$ and $C_{2}=3000 \pm 15 \mathrm{pF}$ are connected in series. The voltage applied across this combination is $V=5.00 \pm 0.02 \mathrm{~V}$. What is the percentage error in the calculation of the energy stored in this combination of capacitors? | math | daman1209arora/jeebench/phy | {'subject': 'phy', 'description': 'JEE Adv 2020 Paper 2', 'index': 15, 'type': 'Numeric'} | null | Given the capacitance values $C_{1}=2000 \pm 10 \mathrm{pF}$ and $C_{2}=3000 \pm 15 \mathrm{pF}$, and the voltage $V=5.00 \pm 0.02 \mathrm{~V}$.
The equivalent capacitance $C_{eq}$ for capacitors in series is given by:
$\frac{1}{C_{eq}} = \frac{1}{C_1} + \frac{1}{C_2} = \frac{C_1 + C_2}{C_1 C_2}$
$C_{eq} = \frac{C_1 C... | <|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Two capacitors with capacitance values $C_{1}=2000 \pm 10 \mathrm{pF}$ and $C_{2}=3000 \pm 15 \mathrm{pF}$ are connected in series. The voltage applied across this combination is $V=5.00 \pm 0.02 \mathrm{~... | 2,417 | <|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Two capacitors with capacitance values $C_{1}=2000 \pm 10 \mathrm{pF}$ and $C_{2}=3000 \pm 15 \mathrm{pF}$ are connected in series. The voltage applied across this combination is $V=5.00 \pm 0.02 \mathrm{~... | <|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Two capacitors with capacitance values $C_{1}=2000 \pm 10 \mathrm{pF}$ and $C_{2}=3000 \pm 15 \mathrm{pF}$ are connected in series. The voltage applied across this combination is $V=5.00 \pm 0.02 \mathrm{~... | <|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Two capacitors with capacitance values $C_{1}=2000 \pm 10 \mathrm{pF}$ and $C_{2}=3000 \pm 15 \mathrm{pF}$ are connected in series. The voltage applied across this combination is $V=5.00 \pm 0.02 \mathrm{~... |
181 | One base of a trapezoid is $100$ units longer than the other base. The segment that joins the midpoints of the legs divides the trapezoid into two regions whose areas are in the ratio $2: 3$ . Let $x$ be the length of the segment joining the legs of the trapezoid that is parallel to the bases and that divides the trape... | math | qq8933/AIME_1983_2024 | {'ID': '2000-II-6', 'Year': 2000, 'Problem Number': 6, 'Part': 'II'} | null | Solution:
Let the lengths of the two bases of the trapezoid be $b_1$ and $b_2$, with $b_1 > b_2$. We are given that $b_1 = b_2 + 100$.
The segment that joins the midpoints of the legs (the midline) divides the trapezoid into two smaller trapezoids with heights $h/2$. The lengths of the parallel sides of the upper trape... | <|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
One base of a trapezoid is $100$ units longer than the other base. The segment that joins the midpoints of the legs divides the trapezoid into two regions whose areas are in the ratio $2: 3$ . Let $x$ be t... | 4,732 | <|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
One base of a trapezoid is $100$ units longer than the other base. The segment that joins the midpoints of the legs divides the trapezoid into two regions whose areas are in the ratio $2: 3$ . Let $x$ be t... | <|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
One base of a trapezoid is $100$ units longer than the other base. The segment that joins the midpoints of the legs divides the trapezoid into two regions whose areas are in the ratio $2: 3$ . Let $x$ be t... | <|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
One base of a trapezoid is $100$ units longer than the other base. The segment that joins the midpoints of the legs divides the trapezoid into two regions whose areas are in the ratio $2: 3$ . Let $x$ be t... |
1. **Defining the Sparse Set \( S \) and the Turing Machine \( M \)**
We need to construct a sparse set \( S \) and a deterministic polynomial time Turing Machine \( M \) with oracle access to \( S \) such that given an input \( \langle x, i \rangle \), the TM \( M \) will accept it if and only if \( x \in S_i \).
... | Let's say a language $L \subseteq \{0,1\}^*$ is in $\textbf{P}_{angel}$ if there exists a polynomial $p : \mathbb{N} \mapsto \mathbb{N}$ , a sequence of strings $\{\alpha_n\}_{n \in \mathbb{N}}$ with $\alpha_n \in \{0,1\}^{p(n)}$ , and a deterministic polynomial time Turing Machine $M$ such that for every $x... | math | AI-MO/NuminaMath-CoT/aops_forum | {'source': 'aops_forum', 'messages': [{'content': "Let's say a language $L \\subseteq \\{0,1\\}^*$ is in $\\textbf{P}_{angel}$ if there exists a polynomial $p : \\mathbb{N} \\mapsto \\mathbb{N}$ , a sequence of strings $\\{\\alpha_n\\}_{n \\in \\mathbb{N}}$ with $\\alpha_n \\in \\{0,1\\}^{p(n)}$ , and a determi... | null | Let's solve each part step by step.
**Part 1: Combining Sparse Sets**
We are given $k$ sparse sets $S_1, S_2, \ldots, S_k$. For each $S_i$, there exists a polynomial $p_i: \mathbb{N} \mapsto \mathbb{N}$ such that $|S_i^{=n}| \leq p_i(n)$.
We need to construct a sparse set $S$ and a deterministic polynomial time TM $... | <|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Let's say a language $L \subseteq \{0,1\}^*$ is in $\textbf{P}_{angel}$ if there exists a polynomial $p : \mathbb{N} \mapsto \mathbb{N}$ , a sequence of strings $\{\alpha_n\}_{n \in \mathbb{N}}$ wit... | 1,795 | <|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Let's say a language $L \subseteq \{0,1\}^*$ is in $\textbf{P}_{angel}$ if there exists a polynomial $p : \mathbb{N} \mapsto \mathbb{N}$ , a sequence of strings $\{\alpha_n\}_{n \in \mathbb{N}}$ wit... | <|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Let's say a language $L \subseteq \{0,1\}^*$ is in $\textbf{P}_{angel}$ if there exists a polynomial $p : \mathbb{N} \mapsto \mathbb{N}$ , a sequence of strings $\{\alpha_n\}_{n \in \mathbb{N}}$ wit... | <|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Let's say a language $L \subseteq \{0,1\}^*$ is in $\textbf{P}_{angel}$ if there exists a polynomial $p : \mathbb{N} \mapsto \mathbb{N}$ , a sequence of strings $\{\alpha_n\}_{n \in \mathbb{N}}$ wit... |
"(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 | "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) | 5,090 | "<|im_start|>system\nYou are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>\(...TRUNCATED) | "<|im_start|>system\nYou are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>\(...TRUNCATED) | "<|im_start|>system\nYou are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>\(...TRUNCATED) |
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