dungnv/qwen36-27b-length-traces huggingface dataset:
Problem T3. Protostar formation

Let us model the formation of a star as follows. A spherical cloud of sparse interstellar gas, initially at rest, starts to collapse due to its own gravity. The initial radius of the ball is $r_{0}$ and the mass is $m$. The temperature of the surroundings (much sparser than the gas) and the initial temperature of the gas is uniformly $T_{0}$. The gas may be assumed to be ideal. The average molar mass of the gas is $\mu$ and its adiabatic index is $\gamma>\frac{4}{3}$. Assume that $G \frac{m \mu}{r_{0}} \gg R T_{0}$, where $R$ is the gas constant and $G$ is the gravitational constant.
Context question:
i. During much of the collapse, the gas is so transparent that any heat generated is immediately radiated away, i.e. the ball stays in thermodynamic equilibrium with its surroundings. What is the number of times, $n$, by which the pressure increases when the radius is halved to $r_{1}=0.5 r_{0}$ ? Assume that the gas density remains uniform.
Context answer:
\boxed{$n=8$}


Context question:
ii. Estimate the time $t_{2}$ needed for the radius to shrink from $r_{0}$ to $r_{2}=0.95 r_{0}$. Neglect the change of the gravity field at the position of a falling gas particle.
Context answer:
\boxed{$\sqrt{\frac{0.1 r_{0}^{3}}{G m}}$}


Context question:
iii. Assuming that the pressure remains negligible, find the time $t_{r \rightarrow 0}$ needed for the ball to collapse from $r_{0}$ down to a much smaller radius, using Kepler's Laws.
Context answer:
\boxed{$t_{r \rightarrow 0}=\pi \sqrt{\frac{r_{0}^{3}}{8 G m}}$}


Context question:
iv. At some radius $r_{3} \ll r_{0}$, the gas becomes dense enough to be opaque to the heat radiation. Calculate the amount of heat $Q$ radiated away during the collapse from the radius $r_{0}$ down to $r_{3}$.
Context answer:
\boxed{$Q=\frac{3 m R T_{0}}{\mu} \ln \frac{r_{0}}{r_{3}}$}

v. For radii smaller than $r_{3}$ you may neglect heat loss due to radiation. Determine how the temperature $T$ of the ball depends on its radius for $r<r_{3}$.
David builds a circular table; he then carves one or more positive integers into the table at points equally spaced around its circumference. He considers two tables to be the same if one can be rotated so that it has the same numbers in the same positions as the other. For example, a table with the numbers $8,4,5$ (in clockwise order) is considered the same as a table with the numbers 4, 5,8 (in clockwise order), but both tables are different from a table with the numbers 8, 5, 4 (in clockwise order). Given that the numbers he carves sum to 17 , compute the number of different tables he can make.
## Real Expansion



Consider a "real" monatomic gas consisting of $N$ atoms of negligible volume and mass $m$ in equilibrium inside a closed cubical container of volume $V$. In this "real" gas, the attractive forces between atoms is small but not negligible. Because these atoms have negligible volume, you can assume that the atoms do not collide with each other for the entirety of the problem.
Context question:
a. Consider an atom in the interior of this container of volume $V$. Suppose the potential energy of the interaction is given by


$$

u(r)= \begin{cases}0 & r<d \\ -\epsilon\left(\frac{d}{r}\right)^{6} & r \geq d\end{cases}

$$



where $d \ll V^{1 / 3}$ is the minimum allowed distance between two atoms. Assume the gas is uniformly distributed within the container, what is the average potential energy of this atom? Write your answer in terms of $a^{\prime}=\frac{2 \pi d^{3} \epsilon}{3}, N$, and $V$.
Context answer:
\boxed{$-2 a^{\prime} N / V$}


Context question:
b. What is the average potential energy of an atom near the boundary of the box? Assume that there is no interaction between atoms near the boundary and the box itself.
Context answer:
\boxed{$-a^{\prime} N / V$}


Context question:
c. Using Bernoulli's law $P+U+\rho v^{2} / 2=$ constant, with pressure $P$, potential energy density $U$, mass density $\rho$ and fluid velocity $v$, what is the pressure at the boundary of the box? Assume the interior pressure is given by the ideal gas law.
Context answer:
\boxed{$\frac{N k T}{V}-a^{\prime} \frac{N^{2}}{V^{2}}$}

d. Assuming most atoms are in the interior of the box, what is the total energy of the atoms in the box?
## Electroneering



An electron is a particle with charge $-q$, mass $m$, and magnetic moment $\mu$. In this problem we will explore whether a classical model consistent with these properties can also explain the rest energy $E_{0}=m c^{2}$ of the electron.



Let us describe the electron as a thin spherical shell with uniformly distributed charge and radius $R$. Recall that the magnetic moment of a closed, planar loop of current is always equal to the product of the current and the area of the loop. For the electron, a magnetic moment can be created by making the sphere rotate around an axis passing through its center.
Context question:
a. If no point on the sphere's surface can travel faster than the speed of light (in the frame of the sphere's center of mass), what is the maximum magnetic moment that the sphere can have? You may use the integral:



$$

\int_{0}^{\pi} \sin ^{3} \theta d \theta=\frac{4}{3}

$$
Context answer:
\boxed{$\mu=q c R / 3$}

b. The electron's magnetic moment is known to be $\mu=q \hbar / 2 m$, where $\hbar$ is the reduced Planck constant. In this model, what is the minimum possible radius of the electron? Express your answer in terms of $m$ and fundamental constants.
$g=9.8 \mathrm{~N} / \mathrm{kg}$



$k=1 / 4 \pi \epsilon_{0}=8.99 \times 10^{9} \mathrm{~N} \cdot \mathrm{m}^{2} / \mathrm{C}^{2}$



$c=3.00 \times 10^{8} \mathrm{~m} / \mathrm{s}$



$N_{\mathrm{A}}=6.02 \times 10^{23}(\mathrm{~mol})^{-1}$



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



$1 \mathrm{eV}=1.602 \times 10^{-19} \mathrm{~J}$



$m_{e}=9.109 \times 10^{-31} \mathrm{~kg}=0.511 \mathrm{MeV} / \mathrm{c}^{2}$



$m_{p}=1.673 \times 10^{-27} \mathrm{~kg}=938 \mathrm{MeV} / \mathrm{c}^{2}$



$\sin \theta \approx \theta-\frac{1}{6} \theta^{3}$ for $|\theta| \ll 1$



$$

\begin{aligned}

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

& k_{\mathrm{m}}=\mu_{0} / 4 \pi=10^{-7} \mathrm{~T} \cdot \mathrm{m} / \mathrm{A} \\

& k_{\mathrm{B}}=1.38 \times 10^{-23} \mathrm{~J} / \mathrm{K} \\

& R=N_{\mathrm{A}} k_{\mathrm{B}}=8.31 \mathrm{~J} /(\mathrm{mol} \cdot \mathrm{K}) \\

& e=1.602 \times 10^{-19} \mathrm{C} \\

& h=6.63 \times 10^{-34} \mathrm{~J} \cdot \mathrm{s}=4.14 \times 10^{-15} \mathrm{eV} \cdot \mathrm{s} \\

& (1+x)^{n} \approx 1+n x \text { for }|x| \ll 1 \\

& \ln (1+x) \approx x \text { for }|x| \ll 1 \\

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

\end{aligned}

$$

$$

\begin{array}{lrl}

\text { velocity parameter } & \beta & =\frac{v}{c} \\

\text { Lorentz factor } & \gamma & =\frac{1}{\sqrt{1-\beta^{2}}} \\

\text { relativistic momentum } & p & =\gamma \beta m c \\

\text { relativistic energy } & E & =\gamma m c^{2} \\

\text { relativistic doppler shift } & \frac{f}{f_{0}} & =\sqrt{\frac{1-\beta}{1+\beta}}

\end{array}

$$

Extra Supplementary Reading Materials:

Relativistic particles obey the mass energy relation



$$

E^{2}=(p c)^{2}+\left(m c^{2}\right)^{2}

$$



where $E$ is the relativistic energy of the particle, $p$ is the relativistic momentum, $m$ is the mass, and $c$ is the speed of light.



A proton with mass $m_{p}$ and energy $E_{p}$ collides head on with a photon which is massless and has energy $E_{b}$. The two combine and form a new particle with mass $m_{\Delta}$ called $\Delta$, or "delta". It is a one dimensional collision that conserves both relativistic energy and relativistic momentum.

a. Determine $E_{p}$ in terms of $m_{p}, m_{\Delta}$, and $E_{b}$. You may assume that $E_{b}$ is small.
For each positive integer $N$, an Eden sequence from $\{1,2,3, \ldots, N\}$ is defined to be a sequence that satisfies the following conditions:

(i) each of its terms is an element of the set of consecutive integers $\{1,2,3, \ldots, N\}$,

(ii) the sequence is increasing, and

(iii) the terms in odd numbered positions are odd and the terms in even numbered positions are even.

For example, the four Eden sequences from $\{1,2,3\}$ are

$$
\begin{array}{llll}
1 & 3 & 1,2 & 1,2,3
\end{array}
$$
Determine the number of Eden sequences from $\{1,2,3,4,5\}$.
## Electroneering



An electron is a particle with charge $-q$, mass $m$, and magnetic moment $\mu$. In this problem we will explore whether a classical model consistent with these properties can also explain the rest energy $E_{0}=m c^{2}$ of the electron.



Let us describe the electron as a thin spherical shell with uniformly distributed charge and radius $R$. Recall that the magnetic moment of a closed, planar loop of current is always equal to the product of the current and the area of the loop. For the electron, a magnetic moment can be created by making the sphere rotate around an axis passing through its center.
Context question:
a. If no point on the sphere's surface can travel faster than the speed of light (in the frame of the sphere's center of mass), what is the maximum magnetic moment that the sphere can have? You may use the integral:



$$

\int_{0}^{\pi} \sin ^{3} \theta d \theta=\frac{4}{3}

$$
Context answer:
\boxed{$\mu=q c R / 3$}


Context question:
b. The electron's magnetic moment is known to be $\mu=q \hbar / 2 m$, where $\hbar$ is the reduced Planck constant. In this model, what is the minimum possible radius of the electron? Express your answer in terms of $m$ and fundamental constants.
Context answer:
\boxed{$R=\frac{3}{2} \frac{\hbar}{m c}$}

c. Assuming the radius is the value you found in part (b), how much energy is stored in the electric field of the electron? Express your answer in terms of $E_{0}=m c^{2}$ and the fine structure constant,



$$

\alpha=\frac{q^{2}}{4 \pi \epsilon_{0} \hbar c} \approx \frac{1}{137}

$$
Let $T=5$. Compute the smallest positive integer $n$ such that there are at least $T$ positive integers in the domain of $f(x)=\sqrt{-x^{2}-2 x+n}$.
Suppose that $u$ and $v$ are distinct numbers chosen at random from the set $\{1,2,3, \ldots, 30\}$. Compute the probability that the roots of the polynomial $(x+u)(x+v)+4$ are integers.
An $\boldsymbol{n}$-label is a permutation of the numbers 1 through $n$. For example, $J=35214$ is a 5 -label and $K=132$ is a 3 -label. For a fixed positive integer $p$, where $p \leq n$, consider consecutive blocks of $p$ numbers in an $n$-label. For example, when $p=3$ and $L=263415$, the blocks are 263,634,341, and 415. We can associate to each of these blocks a $p$-label that corresponds to the relative order of the numbers in that block. For $L=263415$, we get the following:

$$
\underline{263} 415 \rightarrow 132 ; \quad 2 \underline{63415} \rightarrow 312 ; \quad 26 \underline{341} 5 \rightarrow 231 ; \quad 263 \underline{415} \rightarrow 213
$$

Moving from left to right in the $n$-label, there are $n-p+1$ such blocks, which means we obtain an $(n-p+1)$-tuple of $p$-labels. For $L=263415$, we get the 4 -tuple $(132,312,231,213)$. We will call this $(n-p+1)$-tuple the $\boldsymbol{p}$-signature of $L$ (or signature, if $p$ is clear from the context) and denote it by $S_{p}[L]$; the $p$-labels in the signature are called windows. For $L=263415$, the windows are $132,312,231$, and 213 , and we write

$$
S_{3}[263415]=(132,312,231,213)
$$

More generally, we will call any $(n-p+1)$-tuple of $p$-labels a $p$-signature, even if we do not know of an $n$-label to which it corresponds (and even if no such label exists). A signature that occurs for exactly one $n$-label is called unique, and a signature that doesn't occur for any $n$-labels is called impossible. A possible signature is one that occurs for at least one $n$-label.

In this power question, you will be asked to analyze some of the properties of labels and signatures.
Compute the 3 -signature for 52341.
Oi-Lam tosses three fair coins and removes all of the coins that come up heads. George then tosses the coins that remain, if any. Determine the probability that George tosses exactly one head.
Find the smallest real constant $C$ such that for any positive real numbers $a_{1}, a_{2}, a_{3}, a_{4}$ and $a_{5}$ (not necessarily distinct), one can always choose distinct subscripts $i, j, k$ and $l$ such that

$$
\left|\frac{a_{i}}{a_{j}}-\frac{a_{k}}{a_{l}}\right| \leqslant C .\tag{1}
$$
Suppose that the functions $f(x)$ and $g(x)$ satisfy the system of equations

$$
\begin{aligned}
f(x)+3 g(x) & =x^{2}+x+6 \\
2 f(x)+4 g(x) & =2 x^{2}+4
\end{aligned}
$$

for all $x$. Determine the values of $x$ for which $f(x)=g(x)$.
Where is the neutrino?

When two protons collide with a very high energy at the Large Hadron Collider (LHC), several particles may be produced as a result of that collision, such as electrons, muons, neutrinos, quarks, and their respective anti-particles. Most of these particles can be detected by the particle detector surrounding the collision point. For example, quarks undergo a process called hadronisation in which they become a shower of subatomic particles, called "jet". In addition, the high magnetic field present in the detectors allows even very energetic charged particles to curve enough for their momentum to be determined. The ATLAS detector uses a superconducting solenoid system that produces a constant and uniform 2.00 Tesla magnetic field in the inner part of the detector, surrounding the collision point. Charged particles with momenta below a certain value will be curved so strongly that they will loop repeatedly in the field and most likely not be measured. Due to its nature, the neutrino is not detected at all, as it escapes through the detector without interacting.

Data: Electron rest mass, $m=9.11 \times 10^{-31} \mathrm{~kg}$; Elementary charge, $e=1.60 \times 10^{-19} \mathrm{C}$;

Speed of light, $c=3.00 \times 10^{8} \mathrm{~m} \mathrm{~s}^{-1}$; Vacuum permittivity, $\epsilon_{0}=8.85 \times 10^{-12} \mathrm{~F} \mathrm{~m}^{-1}$

Part A.ATLAS Detector physics

A.1 Derive an expression for the cyclotron radius, $r$, of the circular trajectory of an electron acted upon by a magnetic force perpendicular to its velocity, and express that radius as a function of its kinetic energy, $K$; charge modulus, $e$; mass, $m$; and magnetic field, $B$. Assume that the electron is a non-relativistic classical particle.
A circle $\omega$ of radius 1 is given. A collection $T$ of triangles is called good, if the following conditions hold:

(i) each triangle from $T$ is inscribed in $\omega$;

(ii) no two triangles from $T$ have a common interior point.

Determine all positive real numbers $t$ such that, for each positive integer $n$, there exists a good collection of $n$ triangles, each of perimeter greater than $t$.
A computer is programmed to choose an integer between 1 and 99, inclusive, so that the probability that it selects the integer $x$ is equal to $\log _{100}\left(1+\frac{1}{x}\right)$. Suppose that the probability that $81 \leq x \leq 99$ is equal to 2 times the probability that $x=n$ for some integer $n$. What is the value of $n$ ?
Let $n$ be a positive integer. Determine the smallest positive integer $k$ with the following property: it is possible to mark $k$ cells on a $2 n \times 2 n$ board so that there exists a unique partition of the board into $1 \times 2$ and $2 \times 1$ dominoes, none of which contains two marked cells.
Let $x$ be a positive real number such that $\log _{\sqrt{2}} x=20$. Compute $\log _{2} \sqrt{x}$.
2. Trajectory of a point mass 

A point mass on the ground is thrown with initial velocity $\vec{v}_{0}$ that makes an angle $\theta$ with the horizontal. Assuming that air friction is negligible,
Context question:
(a) What value of $\theta$ maximizes the range?
Context answer:
\boxed{$\frac{\pi}{4}$}


Context question:
(b) What value of $\theta$ maximizes the surface area under the trajectory curve?
Context answer:
\boxed{$\pi / 3$}


Context question:
(c) What is the answer for (a), if the point mass is thrown from an apartment of height $h$ ?


Now assume that we have a frictional force that is proportional to the velocity vector, such that the equation of motion is as follows

$$
\frac{d \vec{v}}{d t}=\vec{g}-\beta \vec{v}
$$
Context answer:
\boxed{$\arcsin \left(\frac{1}{\sqrt{2}} \frac{v_{0}}{\sqrt{v_{0}^{2}+g h}}\right)$}

(d) Supposing that $\beta<<\frac{g}{v_{0}}$, find the duration of the motion $T$.
In a sequence of $n$ consecutive positive integers, where $n>1$, an element of the sequence is said to be cromulent if it is relatively prime to all other numbers in the sequence. Every element of a sequence with $n=2$ is cromulent because any two consecutive integers are relatively prime to each other.
Find the maximum and minimum possible number of cromulent elements in a sequence of $n$ consecutive positive integers with $n=6$;
Given a positive integer $n \geq 2$, determine the largest positive integer $N$ for which there exist $N+1$ real numbers $a_{0}, a_{1}, \ldots, a_{N}$ such that

(1) $a_{0}+a_{1}=-\frac{1}{n}$, and

(2) $\left(a_{k}+a_{k-1}\right)\left(a_{k}+a_{k+1}\right)=a_{k-1}-a_{k+1}$ for $1 \leq k \leq N-1$.
Suppose that the function $g$ satisfies $g(x)=2 x-4$ for all real numbers $x$ and that $g^{-1}$ is the inverse function of $g$. Suppose that the function $f$ satisfies $g\left(f\left(g^{-1}(x)\right)\right)=2 x^{2}+16 x+26$ for all real numbers $x$. What is the value of $f(\pi)$ ?
Square $O P Q R$ has vertices $O(0,0), P(0,8), Q(8,8)$, and $R(8,0)$. The parabola with equation $y=a(x-2)(x-6)$ intersects the sides of the square $O P Q R$ at points $K, L, M$, and $N$. Determine all the values of $a$ for which the area of the trapezoid $K L M N$ is 36 .
Let $T=60$ . Lydia is a professional swimmer and can swim one-fifth of a lap of a pool in an impressive 20.19 seconds, and she swims at a constant rate. Rounded to the nearest integer, compute the number of minutes required for Lydia to swim $T$ laps.
In square $A B C D$ with diagonal $1, E$ is on $\overline{A B}$ and $F$ is on $\overline{B C}$ with $\mathrm{m} \angle B C E=\mathrm{m} \angle B A F=$ $30^{\circ}$. If $\overline{C E}$ and $\overline{A F}$ intersect at $G$, compute the distance between the incenters of triangles $A G E$ and $C G F$.
Let $n$ be a positive integer, and set $N=2^{n}$. Determine the smallest real number $a_{n}$ such that, for all real $x$,

$$
\sqrt[N]{\frac{x^{2 N}+1}{2}} \leqslant a_{n}(x-1)^{2}+x
$$
Let $T=3$, and let $K=T+2$. Compute the largest $K$-digit number which has distinct digits and is a multiple of 63.
Let $\mathbb{Q}_{>0}$ denote the set of all positive rational numbers. Determine all functions $f: \mathbb{Q}_{>0} \rightarrow \mathbb{Q}_{>0}$ satisfying

$$
f\left(x^{2} f(y)^{2}\right)=f(x)^{2} f(y)
\tag{*}
$$

for all $x, y \in \mathbb{Q}_{>0}$.
Let $T=256$. Let $\mathcal{R}$ be the region in the plane defined by the inequalities $x^{2}+y^{2} \geq T$ and $|x|+|y| \leq \sqrt{2 T}$. Compute the area of region $\mathcal{R}$.
Suppose that noncongruent triangles $A B C$ and $X Y Z$ are given such that $A B=X Y=10, B C=$ $Y Z=9$, and $\mathrm{m} \angle C A B=\mathrm{m} \angle Z X Y=30^{\circ}$. Compute $[A B C]+[X Y Z]$.
Let $T=27/2$. The sequence $a_{1}, a_{2}, a_{3}, \ldots$, is arithmetic with $a_{16}=13$ and $a_{30}=20$. Compute the value of $k$ for which $a_{k}=T$.
Let $T=95$. Triangle $A B C$ has $A B=A C$. Points $M$ and $N$ lie on $\overline{B C}$ such that $\overline{A M}$ and $\overline{A N}$ trisect $\angle B A C$, with $M$ closer to $C$. If $\mathrm{m} \angle A M C=T^{\circ}$, then $\mathrm{m} \angle A C B=U^{\circ}$. Compute $U$.
In a $999 \times 999$ square table some cells are white and the remaining ones are red. Let $T$ be the number of triples $\left(C_{1}, C_{2}, C_{3}\right)$ of cells, the first two in the same row and the last two in the same column, with $C_{1}$ and $C_{3}$ white and $C_{2}$ red. Find the maximum value $T$ can attain.
Each positive integer less than or equal to 2019 is written on a blank sheet of paper, and each of the digits 0 and 5 is erased. Compute the remainder when the product of the remaining digits on the sheet of paper is divided by 1000 .
Let $n$ and $k$ be fixed positive integers of the same parity, $k \geq n$. We are given $2 n$ lamps numbered 1 through $2 n$; each of them can be on or off. At the beginning all lamps are off. We consider sequences of $k$ steps. At each step one of the lamps is switched (from off to on or from on to off).

Let $N$ be the number of $k$-step sequences ending in the state: lamps $1, \ldots, n$ on, lamps $n+1, \ldots, 2 n$ off.

Let $M$ be the number of $k$-step sequences leading to the same state and not touching lamps $n+1, \ldots, 2 n$ at all.

Find the ratio $N / M$.
Thurka bought some stuffed goats and some toy helicopters. She paid a total of $\$ 201$. She did not buy partial goats or partial helicopters. Each stuffed goat cost $\$ 19$ and each toy helicopter cost $\$ 17$. How many of each did she buy?
Let $k \geqslant 2$ be an integer. Find the smallest integer $n \geqslant k+1$ with the property that there exists a set of $n$ distinct real numbers such that each of its elements can be written as a sum of $k$ other distinct elements of the set.
A cubic sequence is a sequence of integers given by $a_{n}=n^{3}+b n^{2}+c n+d$, where $b, c$ and $d$ are integer constants and $n$ ranges over all integers, including negative integers.

Determine the possible values of $a_{2015} \cdot a_{2016}$ for a cubic sequence satisfying the condition in part (a).
The points $(1,2,3)$ and $(3,3,2)$ are vertices of a cube. Compute the product of all possible distinct volumes of the cube.
For each positive integer $N$, let $P(N)$ denote the product of the digits of $N$. For example, $P(8)=8$, $P(451)=20$, and $P(2023)=0$. Compute the least positive integer $n$ such that $P(n+23)=P(n)+23$.
Determine the greatest positive integer $k$ that satisfies the following property: The set of positive integers can be partitioned into $k$ subsets $A_{1}, A_{2}, \ldots, A_{k}$ such that for all integers $n \geq 15$ and all $i \in\{1,2, \ldots, k\}$ there exist two distinct elements of $A_{i}$ whose sum is $n$.
The product of the first five terms of a geometric progression is 32 . If the fourth term is 17 , compute the second term.
One face of a $2 \times 2 \times 2$ cube is painted (not the entire cube), and the cube is cut into eight $1 \times 1 \times 1$ cubes. The small cubes are reassembled randomly into a $2 \times 2 \times 2$ cube. Compute the probability that no paint is showing.
Let $S=15$ and let $M=10$ . Sam and Marty each ride a bicycle at a constant speed. Sam's speed is $S \mathrm{~km} / \mathrm{hr}$ and Marty's speed is $M \mathrm{~km} / \mathrm{hr}$. Given that Sam and Marty are initially $100 \mathrm{~km}$ apart and they begin riding towards one another at the same time, along a straight path, compute the number of kilometers that Sam will have traveled when Sam and Marty meet.
Alice fills the fields of an $n \times n$ board with numbers from 1 to $n^{2}$, each number being used exactly once. She then counts the total number of good paths on the board. A good path is a sequence of fields of arbitrary length (including 1) such that:

(i) The first field in the sequence is one that is only adjacent to fields with larger numbers,

(ii) Each subsequent field in the sequence is adjacent to the previous field,

(iii) The numbers written on the fields in the sequence are in increasing order.

Two fields are considered adjacent if they share a common side. Find the smallest possible number of good paths Alice can obtain, as a function of $n$.
A circle passes through the origin and the points of intersection of the parabolas $y=x^{2}-3$ and $y=-x^{2}-2 x+9$. Determine the coordinates of the centre of this circle.
An integer $N$ is worth 1 point for each pair of digits it contains that forms a prime in its original order. For example, 6733 is worth 3 points (for 67,73 , and 73 again), and 20304 is worth 2 points (for 23 and 03). Compute the smallest positive integer that is worth exactly 11 points. [Note: Leading zeros are not allowed in the original integer.]