dungnv/qwen36-27b-length-traces huggingface dataset:
user
$$

\begin{array}{ll}

g=9.8 \mathrm{~N} / \mathrm{kg} & G=6.67 \times 10^{-11} \mathrm{~N} \cdot \mathrm{m}^{2} / \mathrm{kg}^{2} \\

k=1 / 4 \pi \epsilon_{0}=8.99 \times 10^{9} \mathrm{~N} \cdot \mathrm{m}^{2} / \mathrm{C}^{2} & k_{\mathrm{m}}=\mu_{0} / 4 \pi=10^{-7} \mathrm{~T} \cdot \mathrm{m} / \mathrm{A} \\

c=3.00 \times 10^{8} \mathrm{~m} / \mathrm{s} & k_{\mathrm{B}}=1.38 \times 10^{-23} \mathrm{~J} / \mathrm{K} \\

N_{\mathrm{A}}=6.02 \times 10^{23}(\mathrm{~mol})^{-1} & R=N_{\mathrm{A}} k_{\mathrm{B}}=8.31 \mathrm{~J} /(\mathrm{mol} \cdot \mathrm{K}) \\

\sigma=5.67 \times 10^{-8} \mathrm{~J} /\left(\mathrm{s} \cdot \mathrm{m}^{2} \cdot \mathrm{K}^{4}\right) & e=1.602 \times 10^{-19} \mathrm{C} \\

1 \mathrm{eV}=1.602 \times 10^{-19} \mathrm{~J} & h=6.63 \times 10^{-34} \mathrm{~J} \cdot \mathrm{s}=4.14 \times \\

m_{e}=9.109 \times 10^{-31} \mathrm{~kg}=0.511 \mathrm{MeV} / \mathrm{c}^{2} & (1+x)^{n} \approx 1+n x \text { for }|x| \ll 1 \\

\sin \theta \approx \theta-\frac{1}{6} \theta^{3} \text { for }|\theta| \ll 1 & \cos \theta \approx 1-\frac{1}{2} \theta^{2} \text { for }|\theta| \ll 1

\end{array}

$$


$$

\begin{array}{lrr}

M_{\odot} & = & 1.989 \times 10^{30} \mathrm{~kg} \\

L_{\odot} & = & 3.828 \times 10^{26} \mathrm{~W} \\

R_{\text {earth }} & = & 1.5 \times 10^{11} \mathrm{~m} \\

\lambda_{\max } & = & 500 \mathrm{~nm}

\end{array}

$$

Extra Supplementary Reading Materials:

In this problem, use a particle-like model of photons: they propagate in straight lines and obey the law of reflection, but are subject to the quantum uncertainty principle. You may use small-angle approximations throughout the problem.



A photon with wavelength $\lambda$ has traveled from a distant star to a telescope mirror, which has a circular cross-section with radius $R$ and a focal length $f \gg R$. The path of the photon is nearly aligned to the axis of the mirror, but has some slight uncertainty $\Delta \theta$. The photon reflects off the mirror and travels to a detector, where it is absorbed by a particular pixel on a charge-coupled device (CCD).



Suppose the telescope mirror is manufactured so that photons coming in parallel to each other are focused to the same pixel on the CCD, regardless of where they hit the mirror. Then all small cross-sectional areas of the mirror are equally likely to include the point of reflection for a photon.

a. Find the standard deviation $\Delta r$ of the distribution for $r$, the distance from the center of the telescope mirror to the point of reflection of the photon.
Please reason step by step, and put your final answer within \boxed{}.


assistant
<think>
user
If $f(t)=\sin \left(\pi t-\frac{\pi}{2}\right)$, what is the smallest positive value of $t$ at which $f(t)$ attains its minimum value?
Please reason step by step, and put your final answer within \boxed{}.


assistant
<think>
user
A die, with the numbers $1,2,3,4,6$, and 8 on its six faces, is rolled. After this roll, if an odd number appears on the top face, all odd numbers on the die are doubled. If an even number appears on the top face, all the even numbers are halved. If the given die changes in this way, what is the probability that a 2 will appear on the second roll of the die?
Please reason step by step, and put your final answer within \boxed{}.


assistant
<think>
user
Computing electric fields

Electrostatics relies on multiple methods for computing electric fields and potentials. In this problem, we will explore two of them, Gauss's Law and Legendre polynomials.

Uniform charge distributions

Let us consider a hollow conducting sphere of radius $R$ charged with the electric charge $Q$, uniformly distributed on its surface. In order to calculate its potential, we can use Gauss's Law, which states that the flux of the electric field $d F=\mathbf{E} \cdot \mathbf{d} \mathbf{A}$ across a closed surface is proportional to the charge enclosed by that surface: $F=Q / \varepsilon_{0}$. We have denoted $\mathbf{d A}=d A \mathbf{n}$ the elementary oriented (towards the exterior) surface element.

(a) Compute the electric potential inside and outside the sphere.
Please reason step by step, and put your final answer within \boxed{}.


assistant
<think>
user
The positive integers $a$ and $b$ have no common divisor larger than 1 . If the difference between $b$ and $a$ is 15 and $\frac{5}{9}<\frac{a}{b}<\frac{4}{7}$, what is the value of $\frac{a}{b}$ ?
Please reason step by step, and put your final answer within \boxed{}.


assistant
<think>
user
For every real number $x$, define $\lfloor x\rfloor$ to be equal to the greatest integer less than or equal to $x$. (We call this the "floor" of $x$.) For example, $\lfloor 4.2\rfloor=4,\lfloor 5.7\rfloor=5$, $\lfloor-3.4\rfloor=-4,\lfloor 0.4\rfloor=0$, and $\lfloor 2\rfloor=2$.
Determine a polynomial $p(x)$ so that for every positive integer $m>4$,

$$
\lfloor p(m)\rfloor=\left\lfloor\frac{1}{3}\right\rfloor+\left\lfloor\frac{2}{3}\right\rfloor+\left\lfloor\frac{3}{3}\right\rfloor+\ldots+\left\lfloor\frac{m-2}{3}\right\rfloor+\left\lfloor\frac{m-1}{3}\right\rfloor
$$

(The sum has $m-1$ terms.)

A polynomial $f(x)$ is an algebraic expression of the form $f(x)=a_{n} x^{n}+a_{n-1} x^{n-1}+\cdots+a_{1} x+a_{0}$ for some integer $n \geq 0$ and for some real numbers $a_{n}, a_{n-1}, \ldots, a_{1}, a_{0}$.
Please reason step by step, and put your final answer within \boxed{}.


assistant
<think>
user
Let $p$ be a prime number. If $p$ years ago, the ages of three children formed a geometric sequence with a sum of $p$ and a common ratio of 2 , compute the sum of the children's current ages.
Please reason step by step, and put your final answer within \boxed{}.


assistant
<think>
user
Computing electric fields

Electrostatics relies on multiple methods for computing electric fields and potentials. In this problem, we will explore two of them, Gauss's Law and Legendre polynomials.

Uniform charge distributions

Let us consider a hollow conducting sphere of radius $R$ charged with the electric charge $Q$, uniformly distributed on its surface. In order to calculate its potential, we can use Gauss's Law, which states that the flux of the electric field $d F=\mathbf{E} \cdot \mathbf{d} \mathbf{A}$ across a closed surface is proportional to the charge enclosed by that surface: $F=Q / \varepsilon_{0}$. We have denoted $\mathbf{d A}=d A \mathbf{n}$ the elementary oriented (towards the exterior) surface element.
Context question:
(a) Compute the electric potential inside and outside the sphere.
Context answer:
\boxed{$\Phi_{-}=\frac{Q}{4 \pi \varepsilon_{0} R}$,$\Phi_{+}=\frac{Q}{4 \pi \varepsilon_{0} r}$}


Extra Supplementary Reading Materials:

Legendre polynomials and non-uniform charge distributions

Legendre polynomials are a type of orthogonal polynomials essential in mathematical physics. One of their applications is in computing electric potentials for more complicated charge configurations. We will denote the $n$-th Legendre polynomial (having degree $n$ ) as $P_{n}$. Legendre polynomials are defined on $[-1,1]$ and we can express their scalar product as

$$
\left\langle P_{m}(x), P_{n}(x)\right\rangle=\int_{-1}^{1} P_{m}(x) P_{n}(x) d x
\tag{1}
$$

The first two Legendre polynomials are $P_{0}(x)=1$ and $P_{1}(x)=x$.
Context question:
(a) Knowing that Legendre polynomials are orthogonal $\left(\left\langle P_{m}(x), P_{n}(x)\right\rangle=0\right.$ if $m \neq n)$ and $\operatorname{deg} P_{n}(x)=n$, obtain $P_{2}(x)$ and $P_{3}(x)$. For reaching the usual and most convenient form of these polynomials, divide your results by the norm: $\left\|P_{n}(x)\right\|=\frac{2}{2 n+1}$.

Let us now consider a sphere of radius $R$ centered at the origin. Suppose a point charge $q$ is put at the origin and that this is the only charge inside or outside the sphere. Furthermore, the potential is $\Phi=V_{0} \cos \theta$ on the surface of the sphere.

We know that we can write the potential induced by the charge on the sphere (without taking into account $q$ ) in the following way:

$$
\begin{array}{ll}
\Phi_{-}=\sum_{n=0}^{\infty} A_{n} r^{n} P_{n}(\cos \theta), & r<R \\
\Phi_{+}=\sum_{n=0}^{\infty} \frac{B_{n}}{r^{n+1}} P_{n}(\cos \theta), & r>R
\end{array}
$$
Context answer:
\boxed{$P_{2}(x)=C_{2}\left(x^{2}+\lambda_{1} x+\lambda_{0}\right)$ , $P_{3}=\frac{1}{2}\left(5 x^{3}-3 x\right)$}

(b) Compute the electric potential both inside and outside the sphere.
Please reason step by step, and put your final answer within \boxed{}.


assistant
<think>
user
The geometric sequence with $n$ terms $t_{1}, t_{2}, \ldots, t_{n-1}, t_{n}$ has $t_{1} t_{n}=3$. Also, the product of all $n$ terms equals 59049 (that is, $t_{1} t_{2} \cdots t_{n-1} t_{n}=59049$ ). Determine the value of $n$.

(A geometric sequence is a sequence in which each term after the first is obtained from the previous term by multiplying it by a constant. For example, $3,6,12$ is a geometric sequence with three terms.)
Please reason step by step, and put your final answer within \boxed{}.


assistant
<think>
user
Find (in closed form) the difference between the number of positive integers at most $2^{2017}$ with even weight and the number of positive integers at most $2^{2017}$ with odd weight.
Please reason step by step, and put your final answer within \boxed{}.


assistant
<think>
user
Let $T=8$. Let $f(x)=a x^{2}+b x+c$. The product of the roots of $f$ is $T$. If $(-2,20)$ and $(1,14)$ lie on the graph of $f$, compute $a$.
Please reason step by step, and put your final answer within \boxed{}.


assistant
<think>
user
In a soccer league with 5 teams, each team plays 20 games(that is, 5 games with each of the other 4 teams). For each team, every game ends in a win (W), a loss (L), or a tie (T). The numbers of wins, losses and ties for each team at the end of the season are shown in the table. Determine the values of $x, y$ and $z$.

| Team | W | L | T |
| :---: | ---: | ---: | ---: |
| A | 2 | 15 | 3 |
| B | 7 | 9 | 4 |
| C | 6 | 12 | 2 |
| D | 10 | 8 | 2 |
| E | $x$ | $y$ | $z$ |
Please reason step by step, and put your final answer within \boxed{}.


assistant
<think>
user
4. A complex dance 

In this problem, we will solve a number of differential equations corresponding to very different physical phenomena that are unified by the idea of oscillation. Oscillations are captured elegantly by extending our notion of numbers to include the imaginary unit number $i$, strangely defined to obey $i^{2}=-1$. In other words, rather than using real numbers, it is more convenient for us to work in terms of complex numbers.

Exponentials are usually associated with rapid growth or decay. However, with the inclusion of complex numbers, imaginary "growth" and "decay" can be translated into oscillations by the Euler identity:

$$
e^{i \theta}=\cos \theta+i \sin \theta
\tag{1}
$$
Context question:
(a) The usual form of Newton's second law $(\vec{F}=m \vec{a})$ breaks down when we go into a rotating frame, where both the centrifugal and Coriolis forces become important to account for. Newton's second law then takes the form

$$
\vec{F}=m(\vec{a}+2 \vec{v} \times \vec{\Omega}+\vec{\Omega} \times(\vec{\Omega} \times \vec{r}))
\tag{2}
$$

For a particle free of forces confined to the $x-y$ plane in a frame which rotates about the $z$ axis with angular frequency $\Omega$, this becomes the complicated-looking system of differential equations,

$$
\begin{aligned}
& 0=\ddot{x}+2 \Omega \dot{y}-\Omega^{2} x \\
& 0=\ddot{x}-2 \Omega \dot{x}-\Omega^{2} y
\end{aligned}
\tag{3}
$$

where dots represent time derivatives.

Defining $\eta=x+i y$, show that Equations 3 are equivalent to the following single (complex) equation:

$$
0=\ddot{\eta}-2 i \Omega \dot{\eta}-\Omega^{2} \eta
\tag{4}
$$
Context answer:
\boxed{证明题}


Context question:
(b) Equation 4 is a version of the damped harmonic oscillator, and can be solved by guessing a solution $\eta=\alpha e^{\lambda t}$.

Plugging in this guess, what must $\lambda$ be?
Context answer:
\boxed{$\lambda=i \Omega$}


Context question:
(c) Using your answer to part (b), and defining $\alpha=A e^{i \phi}$ where $A$ and $\phi$ are real, find $\mathbf{x}(\mathbf{t})$ and $\mathbf{y}(\mathbf{t})$.

This is the trajectory for a particle which is stationary with respect to the symmetry axis. While not required for this problem, an additional guess would reveal that $\eta=\beta t e^{\lambda t}$ is also a solution.
Context answer:
\boxed{$x(t)=A \cos (\Omega t+\phi)$ , $y(t)=A \sin (\Omega t+\phi)$}

(d) The one-dimensional diffusion equation (also called the "heat equation") is given (for a free particle) by

$$
\frac{\partial \psi}{\partial t}=a \frac{\partial^{2} \psi}{\partial x^{2}}
\tag{5}
$$

A spatial wave can be written as $\sim e^{i k x}$ (larger $k$ 's correspond to waves oscillating on smaller length scales). Guessing a solution $\psi(x, t)=A e^{i k x-i \omega t}$, find $\omega$ in terms of k. A relationship of this time is called a "dispersion relation."
Please reason step by step, and put your final answer within \boxed{}.


assistant
<think>
user
Let $T=5$. Compute the number of positive divisors of the number $20^{4} \cdot 11^{T}$ that are perfect cubes.
Please reason step by step, and put your final answer within \boxed{}.


assistant
<think>
user
Determine all values of $x$ for which $2+\sqrt{x-2}=x-2$.
Please reason step by step, and put your final answer within \boxed{}.


assistant
<think>
user
Let $\mathcal{S}$ be the set of integers from 0 to 9999 inclusive whose base- 2 and base- 5 representations end in the same four digits. (Leading zeros are allowed, so $1=0001_{2}=0001_{5}$ is one such number.) Compute the remainder when the sum of the elements of $\mathcal{S}$ is divided by 10,000.
Please reason step by step, and put your final answer within \boxed{}.


assistant
<think>
user
Compute the number of quadratic functions $f(x)=a x^{2}+b x+c$ with integer roots and integer coefficients whose graphs pass through the points $(0,0)$ and $(15,225)$.
Please reason step by step, and put your final answer within \boxed{}.


assistant
<think>
user
The arithmetic sequences $a_{1}, a_{2}, a_{3}, \ldots, a_{20}$ and $b_{1}, b_{2}, b_{3}, \ldots, b_{20}$ consist of 40 distinct positive integers, and $a_{20}+b_{14}=1000$. Compute the least possible value for $b_{20}+a_{14}$.
Please reason step by step, and put your final answer within \boxed{}.


assistant
<think>
user
Suppose that $\triangle A B C$ is right-angled at $B$ and has $A B=n(n+1)$ and $A C=(n+1)(n+4)$, where $n$ is a positive integer. Determine the number of positive integers $n<100000$ for which the length of side $B C$ is also an integer.
Please reason step by step, and put your final answer within \boxed{}.


assistant
<think>