question stringlengths 40 2.36k | solution stringlengths 2.33k 80.3k | answer stringlengths 1 191 |
|---|---|---|
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? | 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... | 128 |
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... | Okay, so I need to show that there exists a point y in the Hilbert space H such that when I take each x in S, subtract y, scale by sqrt(2)/d, the resulting set is an orthonormal system. That means each vector has length 1 and any two distinct vectors are orthogonal.
First, let me recall that an orthonormal system requ... | \text{Such a point } y \text{ exists as described} |
Find the remainder when $9 \times 99 \times 999 \times \cdots \times \underbrace{99\cdots9}_{\text{999 9's}}$ is divided by $1000$ . | Alright, so I have this problem here: I need to find the remainder when the product of numbers like 9, 99, 999, and so on, up to a number with 999 nines, is divided by 1000. Hmm, okay, let's unpack this step by step.
First, the problem is asking for the remainder of a very large product divided by 1000. When dealing w... | 109 |
Compute the mean molecular speed v in the heavy gas radon (Rn) in m/s | Okay, I need to calculate the mean molecular speed of radon gas. Hmm, first, I remember there's a formula for the root mean square speed, which is related to temperature and molar mass. Wait, but the question specifically says "mean molecular speed"—is that the same as the root mean square speed, or is it a different k... | 169 |
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? | Okay, so I need to find the percentage error in the calculation of the energy stored in two capacitors connected in series. The capacitors have capacitance values C1 = 2000 ± 10 pF and C2 = 3000 ± 15 pF, and the voltage applied is V = 5.00 ± 0.02 V. Hmm, alright, let's start by recalling how to calculate the equivalent... | 0.88\% |
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... | Okay, let's see. I need to solve this problem about a trapezoid with bases differing by 100 units. The segment connecting the midpoints of the legs divides the trapezoid into two regions with areas in the ratio 2:3. Then, we need to find x, which is the length of the segment that divides the trapezoid into two equal ar... | 181 |
In a mathematics test number of participants is $N < 40$ . The passmark is fixed at $65$ . The test results are
the following:
The average of all participants is $66$ , that of the promoted $71$ and that of the repeaters $56$ .
However, due to an error in the wording of a question, all scores are increased by ... | Alright, so I'm trying to solve this math problem about participants in a test. Let me first try to understand what's being asked here.
We have a math test with N participants, where N is less than 40. The pass mark is 65. Originally, the average score of all participants is 66. The average of those who passed (promot... | 36 |
"Kathy has $5$ red cards and $5$ green cards. She shuffles the $10$ cards and lays out $5$ of the ca(...TRUNCATED) | "Okay, let's see. We need to find the probability that when Kathy lays out 5 cards from a shuffled d(...TRUNCATED) | 157 |
"Square $AIME$ has sides of length $10$ units. Isosceles triangle $GEM$ has base $EM$ , and the are(...TRUNCATED) | "Okay, so I need to solve this geometry problem where there's a square named AIME with sides of 10 u(...TRUNCATED) | 25 |
"Consider the following two person game. A number of pebbles are situated on the table. Two players (...TRUNCATED) | "Okay, so I need to prove that in this pebble game, there are infinitely many starting numbers of pe(...TRUNCATED) | There are infinitely many initial situations where the second player can force a win. |
End of preview. Expand in Data Studio
s1K-1.1 (DeepSeek-R1 traces) — format cho SegmentSelectiveSFT
Chuyen doi tu simplescaling/s1K-1.1
bang prepare_s1k.py (default flags) trong repo SegmentSelectiveSFT.
Moi dong jsonl co 3 truong:
| Truong | Nguon |
|---|---|
question |
question |
solution |
deepseek_thinking_trajectory (long-CoT trace cua R1) |
answer |
\\boxed{...} cuoi cung trong trace, fallback ve solution cua s1K |
Giu 934 / 1000 mau — bo cac mau khong co trace, khong co dap an, hoac dap an dai hon 200 ky tu.
from datasets import load_dataset
ds = load_dataset("baesad/s1K-1.1-deepseek-cot", split="train")
- Downloads last month
- 76