dungnv/qwen36-27b-length-traces huggingface dataset:
Let $T=$ -8, and let $i=\sqrt{-1}$. Compute the positive integer $k$ for which $(-1+i)^{k}=\frac{1}{2^{T}}$.
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 following in terms of $n$. $E(n, 2)$ for positive even integers $n$
Compute the number of ordered pairs $(x, y)$ of positive integers satisfying $x^{2}-8 x+y^{2}+4 y=5$.
Rectangle $A R M L$ has length 125 and width 8. The rectangle is divided into 1000 squares of area 1 by drawing in gridlines parallel to the sides of $A R M L$. Diagonal $\overline{A M}$ passes through the interior of exactly $n$ of the 1000 unit squares. Compute $n$.
Let $A R M L$ be a trapezoid with bases $\overline{A R}$ and $\overline{M L}$, such that $M R=R A=A L$ and $L R=$ $A M=M L$. Point $P$ lies inside the trapezoid such that $\angle R M P=12^{\circ}$ and $\angle R A P=6^{\circ}$. Diagonals $A M$ and $R L$ intersect at $D$. Compute the measure, in degrees, of angle $A P D$.
If $f(x)=x^{2}-x+2, g(x)=a x+b$, and $f(g(x))=9 x^{2}-3 x+2$, determine all possible ordered pairs $(a, b)$ which satisfy this relationship.
Compute the $2011^{\text {th }}$ smallest positive integer $N$ that gains an extra digit when doubled.
Let $m$ be a positive integer. Consider a $4 m \times 4 m$ array of square unit cells. Two different cells are related to each other if they are in either the same row or in the same column. No cell is related to itself. Some cells are coloured blue, such that every cell is related to at least two blue cells. Determine the minimum number of blue cells.
Circle $\omega$ is tangent to parallel lines $\ell_{1}$ and $\ell_{2}$ at $A$ and $B$ respectively. Circle $\omega_{1}$ is tangent to $\ell_{1}$ at $C$ and to $\omega$ externally at $P$. Circle $\omega_{2}$ is tangent to $\ell_{2}$ at $D$ and to $\omega$ externally at $Q$. Circles $\omega_{1}$ and $\omega_{2}$ are also externally tangent to each other. Given that $A Q=12$ and $D Q=8$, compute $C D$.
Let $S$ be the set of prime factors of the numbers you receive from positions 7 and 9 , and let $p$ and $q$ be the two least distinct elements of $S$, with $p<q$. Hexagon HEXAGO is inscribed in circle $\omega$, and every angle of $H E X A G O$ is $120^{\circ}$. If $H E=X A=G O=p$ and $E X=A G=O H=q$, compute the area of circle $\omega$.
Without using a calculator, determine positive integers $m$ and $n$ for which

$$
\sin ^{6} 1^{\circ}+\sin ^{6} 2^{\circ}+\sin ^{6} 3^{\circ}+\cdots+\sin ^{6} 87^{\circ}+\sin ^{6} 88^{\circ}+\sin ^{6} 89^{\circ}=\frac{m}{n}
$$

(The sum on the left side of the equation consists of 89 terms of the form $\sin ^{6} x^{\circ}$, where $x$ takes each positive integer value from 1 to 89.)
Let $T=7$. Given the sequence $u_{n}$ such that $u_{3}=5, u_{6}=89$, and $u_{n+2}=3 u_{n+1}-u_{n}$ for integers $n \geq 1$, compute $u_{T}$.
Let $T=(0,0), N=(2,0), Y=(6,6), W=(2,6)$, and $R=(0,2)$. Compute the area of pentagon $T N Y W R$.
All matters in the universe have fundamental properties called spin, besides their mass and charge. Spin is an intrinsic form of angular momentum carried by particles. Despite the fact that quantum mechanics is needed for a full treatment of spin, we can still study the physics of spin using the usual classical formalism. In this problem, we are investigating the influence of magnetic field on spin using its classical analogue.

The classical torque equation of spin is given by

$$
\boldsymbol{\tau}=\frac{d \boldsymbol{L}}{d t}=\boldsymbol{\mu} \times \boldsymbol{B}
$$

In this case, the angular momentum $\boldsymbol{L}$ represents the "intrinsic" spin of the particles, $\boldsymbol{\mu}$ is the magnetic moment of the particles, and $\boldsymbol{B}$ is magnetic field. The spin of a particle is associated with a magnetic moment via the equation

$$
\boldsymbol{\mu}=-\gamma \boldsymbol{L}
$$

where $\gamma$ is the gyromagnetic ratio.

In this problem, the term "frequency" means angular frequency (rad/s), which is a scalar quantity. All bold letters represent vectors; otherwise they represent scalars.

Part A. Larmor precession
Context question:
1.  Prove that the magnitude of magnetic moment $\mu$ is always constant under the influence of a magnetic field $\boldsymbol{B}$. For a special case of stationary (constant) magnetic field, also show that the angle between $\boldsymbol{\mu}$ and $\boldsymbol{B}$ is constant.

(Hint: You can use properties of vector products.)
Context answer:
\boxed{证明题}

2.  A uniform magnetic field $\boldsymbol{B}$ exists and it makes an angle $\phi$ with a particle's magnetic moment $\boldsymbol{\mu}$. Due to the torque by the magnetic field, the magnetic moment $\boldsymbol{\mu}$ rotates around the field $\boldsymbol{B}$, which is also known as Larmor precession. Determine the Larmor precession frequency $\omega_{0}$ of the magnetic moment with respect to $\boldsymbol{B}=B_{0} \boldsymbol{k}$.
$$

\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}

$$

Context question:
a. Suppose you drop a block of mass $m$ vertically onto a fixed ramp with angle $\theta$ with coefficient of static and kinetic friction $\mu$. The block is dropped in such a way that it does not rotate after colliding with the ramp. Throughout this problem, assume the time of the collision is negligible.


i. Suppose the block's speed just before it hits the ramp is $v$ and the block slides down the ramp immediately after impact. What is the speed of the block right after the collision?
Context answer:
\boxed{$u=v(\sin \theta-\mu \cos \theta)$}


Context question:
ii. What is the minimum $\mu$ such that the speed of the block right after the collision is 0 ?
Context answer:
\boxed{$\mu=\tan \theta$}


Context question:
b. Now suppose you drop a sphere with mass $m$, radius $R$ and moment of inertia $\beta m R^{2}$ vertically onto the same fixed ramp such that it reaches the ramp with speed $v$.


i. Suppose the sphere immediately begins to roll without slipping. What is the new speed of the sphere in this case?
Context answer:
\boxed{$u=\frac{v \sin \theta}{1+\beta}$}

ii. What is the minimum coefficient of friction such that the sphere rolls without slipping immediately after the collision?
A set $S$ contains thirteen distinct positive integers whose sum is 120 . Compute the largest possible value for the median of $S$.
For a positive integer $n$, let $C(n)$ equal the number of pairs of consecutive 1's in the binary representation of $n$. For example, $C(183)=C\left(10110111_{2}\right)=3$. Compute $C(1)+C(2)+$ $C(3)+\cdots+C(256)$.
A bubble in the shape of a hemisphere of radius 1 is on a tabletop. Inside the bubble are five congruent spherical marbles, four of which are sitting on the table and one which rests atop the others. All marbles are tangent to the bubble, and their centers can be connected to form a pyramid with volume $V$ and with a square base. Compute $V$.
Let $T=$ 7. Compute the units digit of $T^{2023}+(T-2)^{20}-(T+10)^{23}$.
Determine all values of $x$ for which $2^{\log _{10}\left(x^{2}\right)}=3\left(2^{1+\log _{10} x}\right)+16$.
If $1, x, y$ is a geometric sequence and $x, y, 3$ is an arithmetic sequence, compute the maximum value of $x+y$.
Let $\mathbb{Z}_{>0}$ be the set of positive integers. Find all functions $f: \mathbb{Z}_{>0} \rightarrow \mathbb{Z}_{>0}$ such that

$$
m^{2}+f(n) \mid m f(m)+n
$$

for all positive integers $m$ and $n$.
The problem concerns investigation of transforming the electron microscope with magnetic guiding of the electron beam (which is accelerated with the potential difference $U=$ $511 \mathrm{kV}$ ) into a proton microscope (in which the proton beam is accelerated with the potential difference $-U$ ). For this purpose, solve the following two problems:

A. An electron after leaving a device, which accelerated it with the potential difference $U$, falls into a region with an inhomogeneous field $\boldsymbol{B}$ generated with a system of stationary coils $L_{1}, L_{2}, \ldots, L_{n}$. The known currents in the coils are $i_{1}, i_{2}, \ldots, i_{n}$, respectively.

What should the currents $i_{1}$ ', $i_{2}$ ', $\ldots, i_{n}$ ' in the coils $L_{1}, L_{2}, \ldots, L_{n}$ be, in order to guide the proton (initially accelerated with the potential difference $-U$ ) along the same trajectory (and in the same direction) as that of the electron?

HINT: The problem can be solved by finding a condition under which the equation describing the trajectory is the same in both cases. It may be helpful to use the relation:

$$
\boldsymbol{p} \frac{d}{d t} \boldsymbol{p}=\frac{1}{2} \frac{d}{d t} \boldsymbol{p}^{2}=\frac{1}{2} \frac{d}{d t} p^{2}
$$
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{$Q=\frac{3 m R T_{0}}{\mu} \ln \frac{r_{0}}{r_{3}}$}


Context question:
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 $r<r_{3}$.
Context answer:
\boxed{$T(r)=T_{0}\left(\frac{r_{3}}{r}\right)^{3 \gamma-3}$}

vi. Eventually we cannot neglect the effect of the pressure on the dynamics of the gas and the collapse stops at $r=r_{4}$ (with $r_{4} \ll r_{3}$ ). However, the radiation loss can still be neglected and the temperature is not yet high enough to ignite nuclear fusion. The pressure of such a protostar is not uniform anymore, but rough estimates with inaccurate numerical prefactors can still be done. Estimate the final radius $r_{4}$ and the respective temperature $T_{4}$.
## Toffee Pudding



A box of mass $m$ is at rest on a horizontal floor. The coefficients of static and kinetic friction between the box and the floor are $\mu_{0}$ and $\mu$ (less than $\mu_{0}$ ), respectively. One end of a spring with spring constant $k$ is attached to the right side of the box, and the spring is initially held at its relaxed length. The other end of the spring is pulled horizontally to the right with constant velocity $v_{0}$. As a result, the box will move in fits and starts. Assume the box does not tip over.
Context question:
a. Calculate the distance $s$ that the spring is stretched beyond its rest length when the box is just about to start moving.
Context answer:
\boxed{$s=\mu_{0} m g / k$}


Context question:
b. Let the box start at $x=0$, and let $t=0$ be the time the box first starts moving. Find the acceleration of the box in terms of $x, t, v_{0}, s$, and the other parameters, while the box is moving.
Context answer:
\boxed{$a=\frac{k}{m}\left(s+v_{0} t-x\right)-\mu g$}


Extra Supplementary Reading Materials:

The position of the box as a function of time $t$ as defined in part (b) is



$$

x(t)=\frac{v_{0}}{\omega}(\omega t-\sin \omega t)+(1-r) s(1-\cos \omega t),

$$



where $\omega=\sqrt{k / m}$ and $r=\mu / \mu_{0}$. This expression applies as long as the box is still moving, and you can use it in the parts below. Express all your answers in terms of $v_{0}, \omega, s$, and $r$.

c. Find the time $t_{0}$ when the box stops for the first time.
Let $P(x)=x^{2}+T x+800$, and let $r_{1}$ and $r_{2}$ be the roots of $P(x)$. The polynomial $Q(x)$ is quadratic, it has leading coefficient 1, and it has roots $r_{1}+1$ and $r_{2}+1$. Find the sum of the coefficients of $Q(x)$.
The taxicab distance between points $\left(x_{1}, y_{1}, z_{1}\right)$ and $\left(x_{2}, y_{2}, z_{2}\right)$ is given by

$$
d\left(\left(x_{1}, y_{1}, z_{1}\right),\left(x_{2}, y_{2}, z_{2}\right)\right)=\left|x_{1}-x_{2}\right|+\left|y_{1}-y_{2}\right|+\left|z_{1}-z_{2}\right| .
$$

The region $\mathcal{R}$ is obtained by taking the cube $\{(x, y, z): 0 \leq x, y, z \leq 1\}$ and removing every point whose taxicab distance to any vertex of the cube is less than $\frac{3}{5}$. Compute the volume of $\mathcal{R}$.
A bakery sells small and large cookies. Before a price increase, the price of each small cookie is $\$ 1.50$ and the price of each large cookie is $\$ 2.00$. The price of each small cookie is increased by $10 \%$ and the price of each large cookie is increased by $5 \%$. What is the percentage increase in the total cost of a purchase of 2 small cookies and 1 large cookie?
Let $T=11$. In ARMLvania, license plates use only the digits 1-9, and each license plate contains exactly $T-3$ digits. On each plate, all digits are distinct, and for all $k \leq T-3$, the $k^{\text {th }}$ digit is at least $k$. Compute the number of valid ARMLvanian license plates.
## 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{$\frac{N k T}{V}-a^{\prime} \frac{N^{2}}{V^{2}}$}


Context question:
d. Assuming most atoms are in the interior of the box, what is the total energy of the atoms in the box?
Context answer:
\boxed{$\frac{3}{2} N k T-a^{\prime} N^{2} / V$}


Extra Supplementary Reading Materials:

Now consider an insulated partitioned container with two sections, each of volume $V$. We fill one side of the container with $N$ atoms of this "real" gas at temperature $T$, which the other side being a vacuum. We then quickly remove the partition and let the gas expand to fill the entirety of the partitioned container. During this expansion, the energy of the gas remains unchanged.

e. What is the final temperature of the gas after the expansion?
Let $a=19, b=20$, and $c=21$. Compute

$$
\frac{a^{2}+b^{2}+c^{2}+2 a b+2 b c+2 c a}{a+b+c}
$$
In rectangle $P A U L$, point $D$ is the midpoint of $\overline{U L}$ and points $E$ and $F$ lie on $\overline{P L}$ and $\overline{P A}$, respectively such that $\frac{P E}{E L}=\frac{3}{2}$ and $\frac{P F}{F A}=2$. Given that $P A=36$ and $P L=25$, compute the area of pentagon $A U D E F$.
Find all positive integers $n \geqslant 2$ for which there exist $n$ real numbers $a_{1}<\cdots<a_{n}$ and a real number $r>0$ such that the $\frac{1}{2} n(n-1)$ differences $a_{j}-a_{i}$ for $1 \leqslant i<j \leqslant n$ are equal, in some order, to the numbers $r^{1}, r^{2}, \ldots, r^{\frac{1}{2} n(n-1)}$.
Suppose that, for some angles $x$ and $y$,

$$
\begin{aligned}
& \sin ^{2} x+\cos ^{2} y=\frac{3}{2} a \\
& \cos ^{2} x+\sin ^{2} y=\frac{1}{2} a^{2}
\end{aligned}
$$

Determine the possible value(s) of $a$.
Let $r=1$ and $R=5$. A circle with radius $r$ is centered at $A$, and a circle with radius $R$ is centered at $B$. The two circles are internally tangent. Point $P$ lies on the smaller circle so that $\overline{B P}$ is tangent to the smaller circle. Compute $B P$.
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 two other 6-labels with the same 4-signature as 462135.
Two fair dice, each having six faces numbered 1 to 6 , are thrown. What is the probability that the product of the two numbers on the top faces is divisible by 5 ?
Let $T=5$. If $|T|-1+3 i=\frac{1}{z}$, compute the sum of the real and imaginary parts of $z$.
Let $T=12$. Each interior angle of a regular $T$-gon has measure $d^{\circ}$. Compute $d$.
The Local Area Inspirational Math Exam comprises 15 questions. All answers are integers ranging from 000 to 999, inclusive. If the 15 answers form an arithmetic progression with the largest possible difference, compute the largest possible sum of those 15 answers.
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 following in terms of $n$. $E(n, n-2)$ for $n \geq 5$
$\quad$ Let $a$ and $b$ be real numbers such that

$$
a^{3}-15 a^{2}+20 a-50=0 \quad \text { and } \quad 8 b^{3}-60 b^{2}-290 b+2575=0
$$

Compute $a+b$.
Six solid regular tetrahedra are placed on a flat surface so that their bases form a regular hexagon $\mathcal{H}$ with side length 1 , and so that the vertices not lying in the plane of $\mathcal{H}$ (the "top" vertices) are themselves coplanar. A spherical ball of radius $r$ is placed so that its center is directly above the center of the hexagon. The sphere rests on the tetrahedra so that it is tangent to one edge from each tetrahedron. If the ball's center is coplanar with the top vertices of the tetrahedra, compute $r$.
Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying the equation

$$
f(x+f(x+y))+f(x y)=x+f(x+y)+y f(x)\tag{1}
$$

for all real numbers $x$ and $y$.
Given a positive integer $n$, find the smallest value of $\left\lfloor\frac{a_{1}}{1}\right\rfloor+\left\lfloor\frac{a_{2}}{2}\right\rfloor+\cdots+\left\lfloor\frac{a_{n}}{n}\right\rfloor$ over all permutations $\left(a_{1}, a_{2}, \ldots, a_{n}\right)$ of $(1,2, \ldots, n)$.