dungnv/qwen36-27b-length-traces huggingface dataset:
Consider the function $f(x)=x^{2}-2 x$. Determine all real numbers $x$ that satisfy the equation $f(f(f(x)))=3$.
$$

\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:

Radiation pressure from the sun is responsible for cleaning out the inner solar system of small particles.


a. The force of radiation on a spherical particle of radius $r$ is given by



$$

F=P Q \pi r^{2}

$$



where $P$ is the radiation pressure and $Q$ is a dimensionless quality factor that depends on the relative size of the particle $r$ and the wavelength of light $\lambda$. Throughout this problem assume that the sun emits a single wavelength $\lambda_{\max }$; unless told otherwise, leave your answers in terms of symbolic variables.
Context question:
i. Given that the total power radiated from the sun is given by $L_{\odot}$, find an expression for the radiation pressure a distance $R$ from the sun.
Context answer:
\boxed{$P=\frac{L \odot}{4 \pi R^{2} c}$}

ii. Assuming that the particle has a density $\rho$, derive an expression for the ratio $\frac{F_{\text {radiation }}}{F_{\text {gravity }}}$ in terms of $L_{\odot}$, mass of sun $M_{\odot}, \rho$, particle radius $r$, and quality factor $Q$.
What are all values of $x$ such that

$$
\log _{5}(x+3)+\log _{5}(x-1)=1 ?
$$
Compute $\cos ^{3} \frac{2 \pi}{7}+\cos ^{3} \frac{4 \pi}{7}+\cos ^{3} \frac{8 \pi}{7}$.
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?
If $f(x)=\sin ^{2} x-2 \sin x+2$, what are the minimum and maximum values of $f(x)$ ?
Determine the smallest number $M$ such that the inequality

$$
\left|a b\left(a^{2}-b^{2}\right)+b c\left(b^{2}-c^{2}\right)+c a\left(c^{2}-a^{2}\right)\right| \leq M\left(a^{2}+b^{2}+c^{2}\right)^{2}
$$

holds for all real numbers $a, b, c$.
Elizabeth is in an "escape room" puzzle. She is in a room with one door which is locked at the start of the puzzle. The room contains $n$ light switches, each of which is initially off. Each minute, she must flip exactly $k$ different light switches (to "flip" a switch means to turn it on if it is currently off, and off if it is currently on). At the end of each minute, if all of the switches are on, then the door unlocks and Elizabeth escapes from the room.

Let $E(n, k)$ be the minimum number of minutes required for Elizabeth to escape, for positive integers $n, k$ with $k \leq n$. For example, $E(2,1)=2$ because Elizabeth cannot escape in one minute (there are two switches and one must be flipped every minute) but she can escape in two minutes (by flipping Switch 1 in the first minute and Switch 2 in the second minute). Define $E(n, k)=\infty$ if the puzzle is impossible to solve (that is, if it is impossible to have all switches on at the end of any minute).

For convenience, assume the $n$ light switches are numbered 1 through $n$.
Find the $E(2020,1993)$
$\quad$ Suppose that $p$ and $q$ are two-digit prime numbers such that $p^{2}-q^{2}=2 p+6 q+8$. Compute the largest possible value of $p+q$.
If $3\left(8^{x}\right)+5\left(8^{x}\right)=2^{61}$, what is the value of the real number $x$ ?
Problem3. Simplest model of gas discharge

An electric current flowing through a gas is called a gas discharge. There are many types of gas discharges including glow discharge in lighting lamps, arc discharge inwelding and the well known spark discharge that occurs between the clouds and the earth in the form of lightning.

Part A. Non-self-sustained gas discharge 

In this part of the problem the so-called non-self-sustained gas discharge is studied. To maintain it a permanent operationan external ionizer is needed, which creates $Z_{\text {ext }}$ pairs of singly ionized ions and free electrons per unit volume and per unit time uniformly in the volume.

When an external ionizer is switched on, the number of electrons and ions starts to grow. Unlimited increase in the number densities of electrons and ions in the gas is prevented by the recombination process in which a free electron recombineswith an ion to form a neutral atom. The number of recombining events $Z_{\text {rec }}$ that occurs in the gas per unit volume and per unit time is given by

$$
Z_{\mathrm{rec}}=r n_{e} n_{i},
$$

where $r$ is a constant called the recombination coefficient, and $n_{e}, n_{i}$ denotethe electron and ion number densities, respectively.

Suppose that at time $t=0$ the external ionizer is switched on and the initial number densities of electrons and ions in the gas are both equal to zero. Then, the electron number density $n_{e}(t)$ depends on timetas follows:

$$
n_{e}(t)=n_{0}+a \tanh b t
$$

where $n_{0}, a$ and $b$ are some constants, and $\tanh x$ stands for the hyperbolic tangent.

A1 Find $_{0}, a, b$ and express them in terms of $Z_{\text {ext }}$ and $r$.
Let $\mathbb{Z}_{>0}$ denote the set of positive integers. For any positive integer $k$, a function $f: \mathbb{Z}_{>0} \rightarrow \mathbb{Z}_{>0}$ is called $k$-good if $\operatorname{gcd}(f(m)+n, f(n)+m) \leqslant k$ for all $m \neq n$. Find all $k$ such that there exists a $k$-good function.
A bag contains 20 lavender marbles, 12 emerald marbles, and some number of orange marbles. If the probability of drawing an orange marble in one try is $\frac{1}{y}$, compute the sum of all possible integer values of $y$.
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?
Let $T=184$. A group of $n$ friends goes camping; two of them are selected to set up the campsite when they arrive and two others are selected to take down the campsite the next day. Compute the smallest possible value of $n$ such that there are at least $T$ ways of selecting the four helpers.
Let $T=2$. Compute the product of $-T-i$ and $i-T$, where $i=\sqrt{-1}$.
Find the largest possible integer $k$, such that the following statement is true:

Let 2009 arbitrary non-degenerated triangles be given. In every triangle the three sides are colored, such that one is blue, one is red and one is white. Now, for every color separately, let us sort the lengths of the sides. We obtain

$$
\begin{aligned}
b_{1} \leq b_{2} \leq \ldots \leq b_{2009} & \text { the lengths of the blue sides } \\
r_{1} \leq r_{2} \leq \ldots \leq r_{2009} & \text { the lengths of the red sides, } \\
\text { and } \quad & w_{1} \leq w_{2} \leq \ldots \leq w_{2009} \quad \text { the lengths of the white sides. }
\end{aligned}
$$

Then there exist $k$ indices $j$ such that we can form a non-degenerated triangle with side lengths $b_{j}, r_{j}, w_{j}$.
A bag contains some blue and some green hats. On each turn, Julia removes one hat without looking, with each hat in the bag being equally likely to be chosen. If it is green, she adds a blue hat into the bag from her supply of extra hats, and if it is blue, she adds a green hat to the bag. The bag initially contains 4 blue hats and 2 green hats. What is the probability that the bag again contains 4 blue hats and 2 green hats after two turns?
$$

\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 10^{-15} \mathrm{eV} \cdot \mathrm{s} \\

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

\end{array}

$$

Extra Supplementary Reading Materials:

Electric Slide



Two large parallel plates of area $A$ are placed at $x=0$ and $x=d \ll \sqrt{A}$ in a semiconductor medium. The plate at $x=0$ is grounded, and the plate at $x=d$ is at a fixed potential $-V_{0}$, where $V_{0}>0$. Particles of positive charge $q$ flow between the two plates. You may neglect any dielectric effects of the medium.
Context question:
a. For large $V_{0}$, the velocity of the positive charges is determined by a strong drag force, so that



$$

v=\mu E

$$



where $E$ is the local electric field and $\mu$ is the charge mobility.

i. In the steady state, there is a nonzero but time-independent density of charges between the two plates. Let the charge density at position $x$ be $\rho(x)$. Use charge conservation to find a relationship between $\rho(x), v(x)$, and their derivatives.
Context answer:
$v \frac{\mathrm{d} \rho}{\mathrm{d} x}+\rho \frac{\mathrm{d} v}{\mathrm{~d} x}=0$


Extra Supplementary Reading Materials:

a. For large $V_{0}$, the velocity of the positive charges is determined by a strong drag force, so that



$$

v=\mu E

$$



where $E$ is the local electric field and $\mu$ is the charge mobility.
Context question:
ii. Let $V(x)$ be the electric potential at $x$. Derive an expression relating $\rho(x), V(x)$, and their derivatives. (Hint: start by using Gauss's law to relate the charge density $\rho(x)$ to the derivative of the electric field $E(x)$.)
Context answer:
$$

\frac{d^{2} V}{d x^{2}}=-\frac{\rho}{\epsilon_{0}}

$$

iii. Suppose that in the steady state, conditions have been established so that $V(x)$ is proportional to $x^{b}$, where $b$ is an exponent you must find, and the current is nonzero. Derive an expression for the current in terms of $V_{0}$ and the other given parameters.
Suppose that $a, b, c, d$ are positive real numbers satisfying $(a+c)(b+d)=a c+b d$. Find the smallest possible value of

$$
S=\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}
$$
Emma goes to the store to buy apples and peaches. She buys five of each, hands the shopkeeper one $\$ 5$ bill, but then has to give the shopkeeper another; she gets back some change. Jonah goes to the same store, buys 2 apples and 12 peaches, and tries to pay with a single $\$ 10$ bill. But that's not enough, so Jonah has to give the shopkeeper another $\$ 10$ bill, and also gets some change. Finally, Helen goes to the same store to buy 25 peaches. Assuming that the price in cents of each fruit is an integer, compute the least amount of money, in cents, that Helen can expect to pay.
For each positive integer $n$, let $T(n)$ be the number of triangles with integer side lengths, positive area, and perimeter $n$. For example, $T(6)=1$ since the only such triangle with a perimeter of 6 has side lengths 2,2 and 2 .
Determine the values of $T(10), T(11)$ and $T(12)$.
A system consisted of two conductor bodies is immersed in a uniform dielectric and weakly conducting liquid. When a constant voltage difference is applied between both conductors, the system has both electric and magnetic fields. In this problem we will investigate this system.

1.  First consider an infinitely long line with charge per unit length $\lambda$ in vacuum. Calculate the electric field $\mathbf{E}(\mathbf{r})$ due to the line.
Consider the system of equations:

$$
\begin{aligned}
c+d & =2000 \\
\frac{c}{d} & =k
\end{aligned}
$$

Determine the number of integers $k$ with $k \geq 0$ for which there is at least one pair of integers $(c, d)$ that is a solution to the system.
Determine all prime numbers $p$ and all positive integers $x$ and $y$ satisfying $x^{3}+y^{3}=$ $p(x y+p)$.
Let $N$ be the least integer greater than 20 that is a palindrome in both base 20 and base 14 . For example, the three-digit base-14 numeral (13)5(13) ${ }_{14}$ (representing $13 \cdot 14^{2}+5 \cdot 14^{1}+13 \cdot 14^{0}$ ) is a palindrome in base 14 , but not in base 20 , and the three-digit base-14 numeral (13)31 14 is not a palindrome in base 14 . Compute the base-10 representation of $N$.
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.
Serge likes to paddle his raft down the Speed River from point $A$ to point $B$. The speed of the current in the river is always the same. When Serge paddles, he always paddles at the same constant speed. On days when he paddles with the current, it takes him 18 minutes to get from $A$ to $B$. When he does not paddle, the current carries him from $A$ to $B$ in 30 minutes. If there were no current, how long would it take him to paddle from $A$ to $B$ ?
For a finite set $A$ of positive integers, we call a partition of $A$ into two disjoint nonempty subsets $A_{1}$ and $A_{2}$ good if the least common multiple of the elements in $A_{1}$ is equal to the greatest common divisor of the elements in $A_{2}$. Determine the minimum value of $n$ such that there exists a set of $n$ positive integers with exactly 2015 good partitions.
Xenia and Sergey play the following game. Xenia thinks of a positive integer $N$ not exceeding 5000. Then she fixes 20 distinct positive integers $a_{1}, a_{2}, \ldots, a_{20}$ such that, for each $k=1,2, \ldots, 20$, the numbers $N$ and $a_{k}$ are congruent modulo $k$. By a move, Sergey tells Xenia a set $S$ of positive integers not exceeding 20 , and she tells him back the set $\left\{a_{k}: k \in S\right\}$ without spelling out which number corresponds to which index. How many moves does Sergey need to determine for sure the number Xenia thought of?
Let $A$ be the number you will receive from position 7 , and let $B$ be the number you will receive from position 9 . There are exactly two ordered pairs of real numbers $\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right)$ that satisfy both $|x+y|=6(\sqrt{A}-5)$ and $x^{2}+y^{2}=B^{2}$. Compute $\left|x_{1}\right|+\left|y_{1}\right|+\left|x_{2}\right|+\left|y_{2}\right|$.
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 $\frac{a}{b}$ ?
Lucy starts by writing $s$ integer-valued 2022-tuples on a blackboard. After doing that, she can take any two (not necessarily distinct) tuples $\mathbf{v}=\left(v_{1}, \ldots, v_{2022}\right)$ and $\mathbf{w}=\left(w_{1}, \ldots, w_{2022}\right)$ that she has already written, and apply one of the following operations to obtain a new tuple:

$$
\begin{aligned}
& \mathbf{v}+\mathbf{w}=\left(v_{1}+w_{1}, \ldots, v_{2022}+w_{2022}\right) \\
& \mathbf{v} \vee \mathbf{w}=\left(\max \left(v_{1}, w_{1}\right), \ldots, \max \left(v_{2022}, w_{2022}\right)\right)
\end{aligned}
$$

and then write this tuple on the blackboard.

It turns out that, in this way, Lucy can write any integer-valued 2022-tuple on the blackboard after finitely many steps. What is the smallest possible number $s$ of tuples that she initially wrote?
Suppose that $x$ and $y$ are real numbers with $3 x+4 y=10$. Determine the minimum possible value of $x^{2}+16 y^{2}$.
For how many integers $k$ with $0<k<18$ is $\frac{5 \sin \left(10 k^{\circ}\right)-2}{\sin ^{2}\left(10 k^{\circ}\right)} \geq 2$ ?
In each square of a garden shaped like a $2022 \times 2022$ board, there is initially a tree of height 0 . A gardener and a lumberjack alternate turns playing the following game, with the gardener taking the first turn:

- The gardener chooses a square in the garden. Each tree on that square and all the surrounding squares (of which there are at most eight) then becomes one unit taller.
- The lumberjack then chooses four different squares on the board. Each tree of positive height on those squares then becomes one unit shorter.

We say that a tree is majestic if its height is at least $10^{6}$. Determine the largest number $K$ such that the gardener can ensure there are eventually $K$ majestic trees on the board, no matter how the lumberjack plays.
In $\triangle A B C, D$ is on $\overline{A C}$ so that $\overline{B D}$ is the angle bisector of $\angle B$. Point $E$ is on $\overline{A B}$ and $\overline{C E}$ intersects $\overline{B D}$ at $P$. Quadrilateral $B C D E$ is cyclic, $B P=12$ and $P E=4$. Compute the ratio $\frac{A C}{A E}$.
## 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{$R=\frac{3}{2} \frac{\hbar}{m c}$}


Context question:
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}

$$
Context answer:
\boxed{$U_{E}=\frac{1}{3} \alpha E_{0}$}

d. Roughly estimate the total energy stored in the magnetic field of the electron, in terms of $E_{0}$ and $\alpha$. (Hint: one way to do this is to suppose the magnetic field has roughly constant magnitude inside the sphere and is negligible outside of it, then estimate the field inside the sphere.)
Determine the largest real number $a$ such that for all $n \geqslant 1$ and for all real numbers $x_{0}, x_{1}, \ldots, x_{n}$ satisfying $0=x_{0}<x_{1}<x_{2}<\cdots<x_{n}$, we have

$$
\frac{1}{x_{1}-x_{0}}+\frac{1}{x_{2}-x_{1}}+\cdots+\frac{1}{x_{n}-x_{n-1}} \geqslant a\left(\frac{2}{x_{1}}+\frac{3}{x_{2}}+\cdots+\frac{n+1}{x_{n}}\right) .\tag{1}
$$
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 $a_{n}, a_{n-1}, \ldots, a_{1}, a_{0}$.
Compute all ordered pairs of real numbers $(x, y)$ that satisfy both of the equations:

$$
x^{2}+y^{2}=6 y-4 x+12 \quad \text { and } \quad 4 y=x^{2}+4 x+12
$$
In triangle $A B C, A B=5, A C=6$, and $\tan \angle B A C=-\frac{4}{3}$. Compute the area of $\triangle A B C$.
In each town in ARMLandia, the residents have formed groups, which meet each week to share math problems and enjoy each others' company over a potluck-style dinner. Each town resident belongs to exactly one group. Every week, each resident is required to make one dish and to bring it to his/her group.

It so happens that each resident knows how to make precisely two dishes. Moreover, no two residents of a town know how to make the same pair of dishes. Shown below are two example towns. In the left column are the names of the town's residents. Adjacent to each name is the list of dishes that the corresponding resident knows how to make.

| ARMLton |  |
| :--- | :--- |
| Resident | Dishes |
| Paul | pie, turkey |
| Arnold | pie, salad |
| Kelly | salad, broth |


| ARMLville |  |
| :--- | :--- |
| Resident | Dishes |
| Sally | steak, calzones |
| Ross | calzones, pancakes |
| David | steak, pancakes |

The population of a town $T$, denoted $\operatorname{pop}(T)$, is the number of residents of $T$. Formally, the town itself is simply the set of its residents, denoted by $\left\{r_{1}, \ldots, r_{\mathrm{pop}(T)}\right\}$ unless otherwise specified. The set of dishes that the residents of $T$ collectively know how to make is denoted $\operatorname{dish}(T)$. For example, in the town of ARMLton described above, pop(ARMLton) $=3$, and dish(ARMLton) $=$ \{pie, turkey, salad, broth\}.

A town $T$ is called full if for every pair of dishes in $\operatorname{dish}(T)$, there is exactly one resident in $T$ who knows how to make those two dishes. In the examples above, ARMLville is a full town, but ARMLton is not, because (for example) nobody in ARMLton knows how to make both turkey and salad.

Denote by $\mathcal{F}_{d}$ a full town in which collectively the residents know how to make $d$ dishes. That is, $\left|\operatorname{dish}\left(\mathcal{F}_{d}\right)\right|=d$.
Compute $\operatorname{pop}\left(\mathcal{F}_{17}\right)$.
Let $n$ be a positive integer such that $\frac{3+4+\cdots+3 n}{5+6+\cdots+5 n}=\frac{4}{11}$. Compute $\frac{2+3+\cdots+2 n}{4+5+\cdots+4 n}$.
Quadrilateral $A R M L$ is a kite with $A R=R M=5, A M=8$, and $R L=11$. Compute $A L$.