Datasets:

shufanshen
/

SFT-Trajectories

prompt stringlengths 102 4.77k	response stringlengths 39 65.5k	ground_truth stringlengths 1 44.2M	teacher_verified bool 1 class	source_index int64 0 57k
Consider a holomorphic homeomorphism $f: U \rightarrow U$ of the unit disc $U$. Must $f$ extend to a continuous map $f: \overline{U} \rightarrow \overline{U}$? Provide a justification for your answer. Please reason step by step, and put your final answer within \boxed{}.	We are given a holomorphic homeomorphism $ f: U \to U $, where $ U = \{ z \in \mathbb{C} : \|z\| < 1 \} $ is the open unit disc. The question is whether $ f $ must extend to a continuous map $ \overline{f}: \overline{U} \to \overline{U} $, where $ \overline{U} $ is the closed unit disc. Let’s analyze this step b...	Yes	true	16,456
In triangle $ABC$, let $\omega$ be the incircle tangent to $BC, CA,$ and $AB$ at points $D, E,$ and $F$ respectively. A point $P$ on $EF$ is such that $PD$ is perpendicular to $BC$. The line $AP$ intersects $BC$ at $Q$. Circles $\omega_1$ and $\omega_2$ pass through $B$ and $C$ respect...	We are given triangle $ABC$ with side lengths: - $AB = 15$ - $BC = 14$ - $CA = 13$ Let’s solve this step-by-step using geometry, coordinate geometry, and properties of incircles, tangency, and circles. --- ### Step 1: Coordinates Setup We place triangle $ABC$ in the coordinate plane. Let: - $B = (0, 0)$ - $C ...	101	true	55,134
Determine whether the Galois group of the polynomial $ f(x) = (x^2-ax+b)(x^2-cx+d) $, where $ f(x) $ is a separable polynomial with rational coefficients, is always a subgroup of the Klein four group $ C_2 \times C_2 $. Please reason step by step, and put your final answer within \boxed{}.	We are given a polynomial $$ f(x) = (x^2 - ax + b)(x^2 - cx + d), $$ where $ f(x) $ is separable and has rational coefficients. We are to determine whether the Galois group of $ f(x) $ over $ \mathbb{Q} $ is always a subgroup of the Klein four-group $ C_2 \times C_2 $. --- ### Step 1: Understand...	Yes	true	7,223
Find all natural numbers $ N > 1 $ such that when all natural divisors of $ N $ are listed in ascending order, the sum of the greatest common divisors of each pair of adjacent divisors equals $ N-2 $. Please reason step by step, and put your final answer within \boxed{}.	We are given a natural number $ N > 1 $, and we are to find all such $ N $ such that: > When all natural divisors of $ N $ are listed in ascending order, the sum of the GCDs of each pair of adjacent divisors equals $ N - 2 $. Let’s denote the divisors of $ N $ in ascending order as: $$ d_1 = 1 < d_2 < d_3 < \...	3	true	55,997
Find the norm of the linear operator $T: C[0,1] \to \mathbb{R}$ defined by $T(f) = \int_0^1 2x f(x) \, dx$, where $C[0,1]$ is the space of continuous functions on the interval $[0,1]$ equipped with the supremum norm. Please reason step by step, and put your final answer within \boxed{}.	We are given a linear operator $ T: C[0,1] \to \mathbb{R} $ defined by $$ T(f) = \int_0^1 2x f(x) \, dx, $$ where $ C[0,1] $ is the space of continuous real-valued functions on $[0,1]$, equipped with the supremum norm: $$ \\|f\\|_\infty = \sup_{x \in [0,1]} \|f(x)\|. $$ We are to compute the operator norm of $ ...	1	true	35,875
Let $A$ and $B$ be two positive semi-definite $n \times n$ matrices such that $ \mathrm{Range}(B^{1/2}AB^{1/2}) = \mathrm{Range}(B) $ and $ \mathrm{Rank}(A) = \mathrm{Rank}(B) = n-1 $. Determine whether there exists a real symmetric matrix $X$ such that $A = XBX$. Provide a justification for your answer. Please rea...	We are given: - $ A $ and $ B $ are $ n \times n $ positive semi-definite matrices. - $ \mathrm{Rank}(A) = \mathrm{Rank}(B) = n - 1 $. - $ \mathrm{Range}(B^{1/2}AB^{1/2}) = \mathrm{Range}(B) $. - We are to determine whether there exists a real symmetric matrix $ X $ such that: $$ A = X B X. $$ We will a...	Yes	true	10,344
Given the group presentation $ \langle x, y \mid x^n = y^2 = 1, xy = yx^2 \rangle $, where $ n = 3k $ for some integer $ k $, determine the order of $ x $. Can the order of $ x $ be concluded to be exactly 3 under this condition? Please reason step by step, and put your final answer within \boxed{}.	We are given a group presentation: $$ \langle x, y \mid x^n = y^2 = 1, \, xy = yx^2 \rangle $$ with the condition that $ n = 3k $ for some integer $ k $, and we are to determine the order of $ x $ and decide whether the order of $ x $ must be exactly 3 under this condition. --- ### Step 1: Understand the re...	3	true	34,152
Evaluate the limit $ f(x) = \lim_{n \to \infty} \left(n \int_0^{\frac{\pi}{4}} (\tan x)^n \, dx\right) $ using appropriate mathematical techniques. Please reason step by step, and put your final answer within \boxed{}.	We are asked to evaluate the limit: $$ f(x) = \lim_{n \to \infty} \left(n \int_0^{\frac{\pi}{4}} (\tan x)^n \, dx\right) $$ Note: The notation $ f(x) $ is misleading here since the expression does not depend on $ x $ — it's a limit of an integral involving $ n $. So we'll interpret this as evaluating: $$ \lim_{n \to...	\dfrac{1}{2}	true	30,806
Does there exist a degree 6 irreducible polynomial in $ \mathbb{Q}[x] $ whose Galois group is isomorphic to $ S_3 $, the symmetric group of degree 3? Please reason step by step, and put your final answer within \boxed{}.	We are asked whether there exists a degree 6 irreducible polynomial in $ \mathbb{Q}[x] $ whose Galois group is isomorphic to $ S_3 $, the symmetric group on 3 letters. Let’s analyze this step by step. --- ### Step 1: Understand the structure of $ S_3 $ - $ S_3 $ is the symmetric group on 3 elements. - It ha...	Yes	true	5,115
Let $C \subset \mathbb{R}^2$ be an algebraic curve defined by an irreducible polynomial $P$. Suppose $Q \in \mathbb{R}[x,y]$ is another polynomial such that $Q(x,y) = 0$ on a set $A \subset C$ where $A$ has a limit point. Does it follow that $P$ divides $Q$? Please reason step by step, and put your final answer within ...	We are given the following: - $ C \subset \mathbb{R}^2 $ is an algebraic curve defined by an irreducible polynomial $ P \in \mathbb{R}[x, y] $. So, $$ C = \{(x, y) \in \mathbb{R}^2 \mid P(x, y) = 0\}. $$ - $ Q \in \mathbb{R}[x, y] $ is another polynomial such that $ Q(x, y) = 0 $ on a set $ A \subset C $, and $ ...	Yes	true	30,964
Find the smallest real number $ C $ such that the inequality \[ C(x_1^{2005} +x_2^{2005} + \cdots + x_5^{2005}) \geq x_1x_2x_3x_4x_5(x_1^{125} + x_2^{125}+ \cdots + x_5^{125})^{16} \] holds for all positive real numbers $ x_1, x_2, x_3, x_4, x_5 $. Please reason step by step, and put your final answer within \boxed...	We are given the inequality: $$ C(x_1^{2005} + x_2^{2005} + \cdots + x_5^{2005}) \geq x_1x_2x_3x_4x_5(x_1^{125} + x_2^{125} + \cdots + x_5^{125})^{16} $$ for all positive real numbers $ x_1, x_2, x_3, x_4, x_5 $, and we are to find the smallest real number $ C $ such that this holds. --- ### Step 1: Understand ...	5^{15}	true	55,725
Evaluate the limit: \[ \lim_{n \to \infty} \frac{n \log_2( \log_2 n)}{3^{\log_2 n^2}}. \] Please reason step by step, and put your final answer within \boxed{}.	We are asked to evaluate the limit: $$ \lim_{n \to \infty} \frac{n \log_2( \log_2 n)}{3^{\log_2 n^2}}. $$ Let’s simplify this expression step by step. --- ### Step 1: Simplify the exponent in the denominator We have: $$ 3^{\log_2 n^2} $$ First, simplify $\log_2 n^2$: $$ \log_2 n^2 = 2 \log_2 n $$ So the denomina...	0	true	17,251
Evaluate the improper integral $ \int_{0}^{\infty}\frac{\sin^2t}{t^2}dt $ using Laplace transforms. Please reason step by step, and put your final answer within \boxed{}.	We are asked to evaluate the improper integral $$ \int_{0}^{\infty} \frac{\sin^2 t}{t^2} \, dt $$ using Laplace transforms. Let's proceed step by step. --- ### Step 1: Use a known identity for $\sin^2 t$ Recall the trigonometric identity: $$ \sin^2 t = \frac{1 - \cos(2t)}{2} $$ So the integral becomes: $$ \...	\dfrac{\pi}{2}	true	38,857
Evaluate the contour integral $ \oint_{\|z\|=1} \|z-1\|^2 \, dz $, where $ z $ is a complex number and the integration is performed along the circle of radius 1 centered at the origin. Please reason step by step, and put your final answer within \boxed{}.	We are asked to evaluate the contour integral: $$ \oint_{\|z\|=1} \|z - 1\|^2 \, dz $$ where the contour is the unit circle $ \|z\| = 1 $, traversed once counterclockwise. --- ### Step 1: Understand the integrand The integrand is $ \|z - 1\|^2 $. Since $ z $ is a complex variable on the unit circle, we can write $ z = e^{...	-2\pi i	true	29,084
Evaluate the contour integral $ \frac{1}{2\pi i} \oint_y \frac{f(C)}{(C-z)^2}dC $, where $ f: \mathbb{C} \to \mathbb{C} $ is holomorphic and $ y $ is a closed circle centered at $ z \in \mathbb{C} $, traced anticlockwise. Determine if the value of this integral is $ f'(z) $. Please reason step by step, and pu...	We are asked to evaluate the contour integral: $$ \frac{1}{2\pi i} \oint_\gamma \frac{f(C)}{(C - z)^2} \, dC $$ where: - $ f: \mathbb{C} \to \mathbb{C} $ is holomorphic (entire, in fact, since domain is all of $ \mathbb{C} $), - $ \gamma $ is a closed circle centered at $ z \in \mathbb{C} $, traced **anticloc...	f'(z)	true	15,262
Given a finite group $G$ and a subgroup $H$ of $G$, determine if there exist elements $x, y \in G \setminus H$ such that the number of left cosets of $H$ containing $x$ is different from the number of right cosets of $H$ containing $y$. Answer with "Yes" or "No". Please reason step by step, and put your final answer wi...	We are given a finite group $ G $ and a subgroup $ H \leq G $. We are to determine whether there exist elements $ x, y \in G \setminus H $ such that: > The number of left cosets of $ H $ containing $ x $ is different from the number of right cosets of $ H $ containing $ y $. Let’s analyze this step by step. ...	No	true	49,062
Given the function $ f(x) = e^{\frac{x^2}{2}} + \int_0^x tf(t)dt $, determine which of the following intervals contains $ f(\sqrt{2}) $: A) $ 5 < f(\sqrt{2}) < 6 $ B) $ 2 < f(\sqrt{2}) < 3 $ C) $ 3 < f(\sqrt{2}) < 4 $ D) $ 4 < f(\sqrt{2}) < 5 $ Please reason step by step, and put your final answer wi...	We are given the function: $$ f(x) = e^{\frac{x^2}{2}} + \int_0^x t f(t)\,dt $$ We are to estimate $ f(\sqrt{2}) $ and determine which interval it lies in. --- ### Step 1: Understand the structure of $ f(x) $ The function is defined in terms of itself via an integral. This suggests we may need to differentiate $ f...	A	true	48,613
A parabola has its focus at point $F$ and its vertex at point $V$, with $VF = 10$ units. A chord $AB$ of length $100$ units passes through $F$. Given the parabola equation $y^2 = 4ax$, determine the area of triangle $VAB$. Please reason step by step, and put your final answer within \boxed{}.	We are given: - A parabola with equation $ y^2 = 4ax $ - Focus at point $ F $ - Vertex at point $ V $ - Distance $ VF = 10 $ units - A chord $ AB $ of length 100 units passes through the focus $ F $ - We are to find the area of triangle $ VAB $ --- ### Step 1: Understand the standard parabola $ y^2 = 4ax $ For ...	100\sqrt{10}	true	22,395
Calculate the remainder when $73^{1567}$ is divided by 990 using the Chinese Remainder Theorem. Please reason step by step, and put your final answer within \boxed{}.	We are asked to compute the remainder when $ 73^{1567} $ is divided by 990, i.e., $$ 73^{1567} \mod 990 $$ We'll use the Chinese Remainder Theorem (CRT). First, factor 990: $$ 990 = 2 \times 3^2 \times 5 \times 11 $$ So, we will compute $ 73^{1567} \mod 2 $, $ \mod 9 $, $ \mod 5 $, and $ \mod 11 $, then com...	127	true	13,587
Given real numbers $a_{1}, a_{2}, \ldots, a_{7}$ such that $a_{1} = a_{7} = 0$, determine if there is always an index $k \leq 5$ such that the inequality $a_{k} + a_{k+2} \leq a_{k+1} \sqrt{3}$ holds. Please reason step by step, and put your final answer within \boxed{}.	We are given real numbers $ a_1, a_2, \ldots, a_7 $ such that: - $ a_1 = a_7 = 0 $ - We are to determine whether there is always an index $ k \leq 5 $ such that: $$ a_k + a_{k+2} \leq a_{k+1} \sqrt{3} $$ We are to decide whether this inequality must hold for at least one $ k = 1, 2, 3, 4, 5 $, **no matt...	Yes	true	53,216
Determine the spectrum of the operator $T: C[0,1] \to C[0,1]$ defined by $Tf(x) = \int_0^x f(t) \, dt$. Please reason step by step, and put your final answer within \boxed{}.	We are given a linear operator $ T: C[0,1] \to C[0,1] $ defined by $$ Tf(x) = \int_0^x f(t)\,dt. $$ Our goal is to determine the spectrum of $ T $, denoted $ \sigma(T) $, where $ T $ acts on the space $ C[0,1] $ of continuous real- (or complex-) valued functions on $[0,1]$, equipped with the supremum norm: $...	\{0\}	true	46,745

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OPSFT SFT Trajectories

Two SFT trajectory datasets used in the OPSFT release.

deepmath/ — DeepMath trajectories generated with Qwen3-30B (train.parquet, validation.parquet)
eurus/ — Eurus trajectories generated with Qwen3-30B-A3B (non-thinking) (train.parquet, validation.parquet)