question string | solution string | thinking_trajectories string | attempt string | grades bool |
|---|---|---|---|---|
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? | 128 | We need to parse the problem: "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?" So we consider rational numbers r in (0,1), reduced fraction a/b where ... | The reduced form of a rational number \( \dfrac{a}{b}\;(0<a<b) \) has \(\gcd(a,b)=1\).
If
\[
a\cdot b = 20! ,
\]
then every prime factor of \(20!\) must belong entirely to either \(a\) or to \(b\); otherwise the two numbers would share a common prime factor.
Hence each distinct prime divisor of \(20!\) can be assi... | true |
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? | 128 | We need to parse problem: "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?" So we consider rational numbers between 0 and 1 inclusive? Probably open in... | \[
\boxed{128}
\] | true |
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? | 128 | Alright, so I need to figure out how many rational numbers between 0 and 1 have a product of the numerator and denominator (when written in lowest terms) equal to 20 factorial. Let me start by understanding the problem.
First, a rational number between 0 and 1 can be written as a fraction \(\frac{a}{b}\) where \(0 < a... | Given a rational number between 0 and 1, we need to write it as a fraction in lowest terms \(\frac{a}{b}\) and calculate the product of the resulting numerator and denominator. We need to determine how many such rational numbers exist where the product \(a \times b = 20!\).
1. **Prime Factorization of 20!**:
- The ... | true |
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? | 128 | We need to find all rational numbers $\frac{p}{q}$ where $0 < \frac{p}{q} < 1$, $\gcd(p, q) = 1$, and $p \cdot q = 20!$.
Since $\gcd(p, q) = 1$ and $p \cdot q = 20!$, we need to find all ways to factor $20!$ into two coprime factors $p$ and $q$ where $p < q$ (since $\frac{p}{q} < 1$).
First, let's find the prime fact... | Looking at this problem, I need to find all fractions $\frac{p}{q}$ where:
- $0 < \frac{p}{q} < 1$ (so $p < q$)
- $\gcd(p,q) = 1$ (fraction is in lowest terms)
- $p \cdot q = 20!$
First, let me find the prime factorization of $20!$.
For each prime $p \leq 20$, the exponent of $p$ in $20!$ is:
$$\sum_{k=1}^{\infty} \l... | true |
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... | 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... | We need to analyze the problem: infinite-dimensional Hilbert space H, d>0, S is a set of points (not necessarily countable) such that distance between any two distinct points in S is equal to d. Show that there exists a point y∈H such that the set { (√2/d)(x - y) : x∈S } is an orthonormal system of vectors in H.
Thus ... | \[
\boxed{
\begin{aligned}
&\text{Let }H\text{ be an infinite–dimensional Hilbert space, }d>0,\\
&\text{and }S\subset H\text{ satisfy } \|x-z\|=d \;(x\neq z,\;x,z\in S).\\[4pt]
&\text{Pick a point }x_{0}\in S\text{ and put }v_{x}=x-x_{0}\;(x\in S).\\[2pt]
&\text{Then } \|v_{x}\|=d \text{ and for }x\neq z\\
&\qquad\|v_{... | true |
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... | 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... | We need to parse the problem: In an infinite-dimensional Hilbert space \(\mathcal{H}\), we have a set S (maybe uncountable) of points such that the distance between any two distinct points in S is equal to d > 0. So S is an equidistant set (pairwise distances all equal to d). Need to show there exists a point y in \(\m... | \[
\boxed{\text{There exists }y\in\mathcal H\text{ such that }
\Bigl\{\frac{\sqrt2}{d}\,(x-y):x\in S\Bigr\}
\text{ is an orthonormal system.}}
\]
--------------------------------------------------------------------
### 1. Geometry of a set whose points are pairwise at distance \(d\)
Fix an element \(a\in S\) and pu... | true |
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... | 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's think about this problem. We have a set $S$ of points in a Hilbert space where any two distinct points are at distance $d$ from each other. We need to find a point $y$ such that when we translate all points by $-y$ and scale by $\frac{\sqrt{2}}{d}$, we get an orthonormal system.
For the vectors $\frac{\sqrt{2}}{... | Looking at this problem, I need to find a point $y$ such that the translated and scaled vectors form an orthonormal system.
For the vectors $\frac{\sqrt{2}}{d}(x-y)$ to form an orthonormal system, I need:
1. $\left\|\frac{\sqrt{2}}{d}(x-y)\right\| = 1$ for all $x \in S$
2. $\left\langle \frac{\sqrt{2}}{d}(x_1-y), \fra... | true |
"Let $ \\mathcal{H}$ be an infinite-dimensional Hilbert space, let $ d>0$ , and suppose that $ S(...TRUNCATED) | "1. **Consider a countable subset \\( S_0 \\subseteq S \\)**:\n Let \\( S_0 = \\{x_1, x_2, \\ldots(...TRUNCATED) | "We need to analyze the problem: Let H be an infinite-dimensional Hilbert space, d>0, S is a set (ma(...TRUNCATED) | "Let \n\n\\[\n\\mathcal H\\ \\text{ be an infinite‑dimensional (real) Hilbert space},\\qquad d>0,(...TRUNCATED) | true |
"Find the remainder when $9 \\times 99 \\times 999 \\times \\cdots \\times \\underbrace{99\\cdots9}_(...TRUNCATED) | 109 | "We need to compute product of numbers: 9, 99, 999, ..., up to number consisting of 999 nines (i.e.,(...TRUNCATED) | "\\[\nP=\\prod_{k=1}^{999}\\bigl(10^{k}-1\\bigr)=9\\cdot 99\\cdot 999\\cdots\\underbrace{99\\ldots 9(...TRUNCATED) | true |
"Find the remainder when $9 \\times 99 \\times 999 \\times \\cdots \\times \\underbrace{99\\cdots9}_(...TRUNCATED) | 109 | "We need to compute product of numbers: 9, 99, 999, ..., up to a number consisting of 999 copies of (...TRUNCATED) | "\\[\n\\begin{aligned}\nP &= 9\\cdot 99\\cdot 999\\cdots\\underbrace{99\\ldots 9}_{\\text{999 nines}(...TRUNCATED) | true |
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