inclusionAI/Ring-lite-rl-data huggingface dataset:
Find the minimal integer \( k \) such that the number of \( q \)-torsion points of a non-supersingular elliptic curve \( E \) over \( GF(q) \) is equal to \( q \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Given the set $S=\{(x,y)|1\leqslant x\leqslant 10,1\leqslant y\leqslant 10,x\in \mathbb{N},y\in \mathbb{N}\}$, find the maximum number of elements in a subset $A\subseteq S$ such that for any $(a,b)\in A$ and $(s,t)\in A$, the condition $(a-s)(b-t)\leqslant 0$ holds. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{19}
Determine the number of tuples \((k_1, k_2, \ldots, k_{26})\) such that each \(k_i\) is either 0, 1, or 3, and the sum \(k_1 + k_2 + \ldots + k_{26} = 15\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{853423740}
Let  $F$  be a finite field having an odd number  $m$  of elements. Let  $p(x)$  be an irreducible (i.e. nonfactorable) polynomial over  $F$  of the form   $$ x^2+bx+c, ~~~~~~ b,c \in F. $$   For how many elements  $k$  in  $F$  is  $p(x)+k$  irreducible over  $F$ ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n = \frac{m-1}{2}}
Calculate the expectation of the Ito integral \( \int_0^3 B_t^3 \, dB_t \) using Ito's formula and the function \( f(t, B_t) = \frac{B_t^4}{4} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Consider the linear system of differential equations $\vec{y}'(t) = A\vec{y}(t)$, where $A$ is a real $2 \times 2$ constant matrix with a repeated eigenvalue. Assume that solution trajectories have horizontal tangents on the line $y_2 = 2y_1$ and vertical tangents on the line $y_1 = 0$. Given that $a_{21} = -4$ and the repeated eigenvalue is nonzero, determine the matrix $A$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\begin{bmatrix} 2 & 0 \\ -4 & 2 \end{bmatrix}}
A hall 36 m long and 15 m broad is to be paved with stones, each of a certain length and 5 dm broad. The number of stones required is 3600. What is the length of each stone?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Evaluate the sum \( \displaystyle \sum_{n=2}^{\infty}\log\left(1-\frac{1}{n^2}\right) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\ln 2}
Evaluate the integral \(\int_0^{2\pi}\int_{\pi/3}^{\pi/2}\int_{2\sec(\varphi)}^4\cos(\varphi)\sin(\varphi)\,\mathrm d\rho\,\mathrm d\varphi\,\mathrm d\theta\) using spherical coordinates.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\pi}
Points \( M, N, P, Q \) are taken on the diagonals \( D_1A, A_1B, B_1C, C_1D \) of the faces of cube \( ABCD A_1B_1C_1D_1 \) respectively, such that:

\[ D_1M: D_1A = BA_1: BN = B_1P: B_1C = DQ: DC_1 = \mu, \]

and the lines \( MN \) and \( PQ \) are mutually perpendicular. Find \( \mu \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{\sqrt{2}}}
The decimal expansion of $37^9$ is $129A617B979C077$ for digits $A, B,$ and $C$. Find the three digit number $ABC$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
935}
Calculate the sum of the series \(\sum_{n=0}^{\infty} \frac{c_{n}d_{n}}{9^{n}}\), where \(c_{n} + id_{n} = (3 + i)^{n}\) and \(c_{n} - id_{n} = (3 - i)^{n}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{27}{37}}
Henry walks $\tfrac{3}{4}$ of the way from his home to his gym, which is $2$ kilometers away from Henry's home, and then walks $\tfrac{3}{4}$ of the way from where he is back toward home. Determine the difference in distance between the points toward which Henry oscillates from home and the gym.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{6}{5}}
Consider the function \( f(x,y,z) = 2^y3^za^xb^yc^z + 2^z3^xa^yb^zc^x + 2^x3^ya^zb^xc^y \) where \( a, b, c \in \mathbb{R} - \{0\} \) and \( x, y, z \in \mathbb{Z} \). Given that \( f(1,0,0) = 4 \) and \( f(2,0,0) = 6 \), find the maximum value of \( \lfloor a \rfloor^2 + \lfloor b \rfloor^2 + \lfloor c \rfloor^2 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
For which values of $c \in \mathbb{R}$ can the level set $M_c = \{(x, y) \mid x^2 + 2xy + 3y^2 = c\}$ be described locally as a $C^1$ graph at every point?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{c > 0}
Calculate the integral $\int_{-\infty}^\infty x^2e^{-\frac {(x-\mu)^2}{2\sigma^2}}dx$ using analytical methods.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(\mu^2 + \sigma^2)\sqrt{2\pi\sigma^2}}
Evaluate \( \lim\limits_{n\to +\infty}\int\limits_{[0,n]}\left(1-\frac{x}{n}\right)^ne^{\frac{x}{2}}dx \) using the Dominated Convergence Theorem.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Calculate the probability that a random permutation of the letters in the word POSSESSIONS contains exactly two occurrences of the substring "SS", with at least one letter separating the two occurrences.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{5}{22}}
Determine the minimum number of points in a set \( S \) required to ensure at least \( m \) intersection points between two polynomials \( P(x) \) and \( Q(x) \), given that both polynomials pass through at least \( p \) points in \( S \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2p - m}
If x gets 25% more than y and y gets some percentage more than z, the share of z out of Rs. 925 is Rs. 250. What is the percentage by which y gets more than z?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20\%}
A bag contains four balls, each labeled with one of the characters "美", "丽", "惠", "州". Balls are drawn with replacement until both "惠" and "州" are drawn, at which point the drawing stops. Use a random simulation method to estimate the probability that the drawing stops exactly on the third draw. Use a computer to randomly generate integer values between 0 and 3, with 0, 1, 2, and 3 representing "惠", "州", "美", and "丽" respectively. Each group of three random numbers represents the result of three draws. The following 16 groups of random numbers were generated:
232 321 230 023 123 021 132 220
231 130 133 231 331 320 122 233
Estimate the probability that the drawing stops exactly on the third draw.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{8}}
Given nine cards with the numbers $5,5,6,6,6,7,8,8,9$ written on them, form three three-digit numbers $A, B, C$, each of whose digits are different. What is the minimum value of the expression $A + B - C$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{149}
Find $$\sum_{i=1}^{100} i \cdot \gcd(i, 100).$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{31000}
Solve the inequality \( \sin x > \ln x \). Determine the solution set where the sine function is greater than the natural logarithm function.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(0, 2.219)}
Let $f: \mathbb{R} \to \mathbb{R}$ be a continuous function such that $f(tx) \geq f(x)$ for all $t > 0$ and $x \in \mathbb{R}$. Given $a, b \in \mathbb{R}$ with $a < b$, determine the value of $\int_a^b f(x) \, dx$. \( \text{Provide your answer as a single value.} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Evaluate the limit of the integral \( \int_{\pi}^0 \frac{e^{ire^{i\theta}}}{r^2e^{i2\theta}+1} d\theta \) as \( r \to 0 \), where the integral is taken along a small semicircular contour of radius \( r \) around the singularity at \( z=0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\pi}
Provide an example of a nonabelian group in which all proper subgroups are normal.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{Q_8}
Calculate the gradient of the function \( f(X) = \text{tr}((AX)^t (AX)) \) with respect to the matrix \( X \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 A^\top A X}
Martians love dancing dances that require holding hands. In the dance "Pyramid," no more than 7 Martians can participate, each with no more than three hands. What is the maximum number of hands that can be involved in the dance if each hand of one Martian holds exactly one hand of another Martian?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
Let $n$ be the number of days over a $13$ year span. Find the difference between the greatest and least possible values of $n$. Note: All years divisible by $4$ are leap years unless they are divisible by 100 but not $400$. For example, $2000$ and $2004$ are leap years, but $1900$ is not.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
A needle 2.5 cm long is dropped onto a piece of paper with parallel lines drawn 2.25 cm apart. What is the probability that the needle will lie entirely between two lines?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.314}
Let $cov(null)$ be the minimum cardinality of a set of nullsets of $\mathbb{R}$ so that their union equals $\mathbb{R}$. In a model $M[G]$ obtained from a ctm $M$ with $M\models CH$ by adding $\aleph_2$ many Cohen reals with forcing, we have $M[G]\models 2^{\omega}=\aleph_2$. What is the value of $cov(null)^{M[G]}$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\aleph_1$}
A can do a piece of work in 20 days, B can do it in some days, and C can do it in 10 days. They started the work together, and A leaves after 2 days and C leaves after 4 days from the beginning. The work lasts 15.000000000000002 days. How many days does it take for B to do the work alone?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
Given a positive integer S, determine the maximum value of N such that the sum of all elements in a 3D array A of dimensions N x N x N, where each element A[i][j][k] is defined as ik(i+j+k) for 1 <= i, j, k <= N, does not exceed S. Constraints: 3 <= S <= 10^{16}. For example, if S = 720, the maximum value of N is 3.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Let $S$ be the set of ordered pairs $(a, b)$ of positive integers such that $\gcd(a, b) = 1$. Compute \[ \sum_{(a, b) \in S} \left\lfloor \frac{300}{2a+3b} \right\rfloor. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7400}
In triangle $\triangle ABC$, $a$, $b$, $c$ are the opposite sides of angles $A$, $B$, $C$ respectively. Let $R$ be the radius of the circumcircle of $\triangle ABC$, and given that $2R(\sin^2 A - \sin^2 B) = (\sqrt{2}a - c)\sin C$, find the measure of angle $B$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\pi}{4}}
Evaluate the Cauchy principal value of the integral: $$\mrm{P.V.}\int_{0}^{\infty} \frac{\dd x}{x^3 - 1}.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\frac{\pi}{3\sqrt{3}}}
Given the joint probability density function of the lifetimes of two circuits $C1$ and $C2$ in an electronic device:
\[
f(x,y) = \begin{cases}
\frac{24}{(x+y)^4},  & \text{x,y > 1} \\
0, & \text{otherwise}
\end{cases}
\]
where $X$ and $Y$ are continuous random variables representing the lifetimes of $C1$ and $C2$, respectively. Calculate the expected value of the time until the device is completely corrupted.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
In the Cartesian coordinate system $xOy$, the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)$ has its right focus at $F(c, 0)$. If there is a line $l$ passing through the point $F$ that intersects the ellipse at points $A$ and $B$, such that $OA \perp OB$, find the range of values for the eccentricity $e=\frac{c}{a}$ of the ellipse.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left[ \frac{\sqrt{5} - 1}{2}, 1 \right)}
例3 设S={1,,2,⋯,15}.从S中取出n个子集$A _ { 1 }$,$A _ { 2 } , \cdots , A _ { n }$，满足下列条件：

(1)$\vert A _ { i } \vert = 7$(i=1,2,⋯,n),

(2)$\vert A _ { i } \cap A _ { j } \vert \leq 3$(1≤i&lt;j≤n)(,

（3）对S中任何3元子集M，存在某个$A _ { k }$，使$M \subset A _ { k } .$

求这样的子集个数n的最小值．（第40届IMO中国国家队选拔考试题）
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
Given a cube of unit side. Let  $A$  and  $B$  be two opposite vertex. Determine the radius of the sphere, with center inside the cube, tangent to the three faces of the cube with common point  $A$  and tangent to the three sides with common point  $B$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{2}}
Find the last three digits of the sum of all the real values of $m$ such that the ellipse $x^2+xy+y^2=m$ intersects the hyperbola $xy=n$ only at its two vertices, as $n$ ranges over all non-zero integers $-81\le n \le 81$. 
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{284
}
Three vertices of parallelogram $ABCD$ are $A(-1, 3), B(2, -1), D(7, 6)$ with $A$ and $D$ diagonally opposite. Calculate the product of the coordinates of vertex $C$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40}
Let \( n \) be a positive integer. A sequence of \( n \) positive integers \( a_{1}, a_{2}, \cdots, a_{n} \) is called a "full sequence" if it satisfies the following conditions: For any positive integer \( k \) (\( k \geq 2 \)), if \( k \) is an element of the sequence, then \( k-1 \) is also an element of the sequence, and the first appearance of \( k-1 \) in the sequence is before the last appearance of \( k \). For each \( n \), how many "full sequences" are there?
(43rd IMO Preliminary Problem)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n!}
p runs at a certain percentage faster than q, so p gives q a 50-meter head start. If the race ends in a tie, p ran 300 meters in the race. What is the percentage by which p runs faster than q?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20\%}
Find a complex number $z=\_\_\_\_\_\_$ that satisfies the following two conditions:① $|z|=\sqrt{7}$; ② The point corresponding to $z$ in the complex plane is located in the second quadrant.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2+\sqrt{3}i}
A frog starts climbing from the bottom of a 12-meter deep well at 8:00 AM. For every 3 meters it climbs up, it slides down 1 meter due to the slippery walls. The time to slide down 1 meter is one-third the time taken to climb up 3 meters. At 8:17 AM, the frog reaches 3 meters from the well's top for the second time. Determine the total time the frog spends climbing from the bottom to the top of the well in minutes.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{22}
For an integer $n$, let $f_9(n)$ denote the number of positive integers $d\leq 9$ dividing $n$.  Suppose that $m$ is a positive integer and $b_1,b_2,\ldots,b_m$ are real numbers such that $f_9(n)=\textstyle\sum_{j=1}^mb_jf_9(n-j)$ for all $n>m$.  Find the smallest possible value of $m$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{28}
2. Let $n$ be a positive integer. If $n$ is divisible by 2010 and exactly one of the digits of $n$ is even, find the smallest possible value of $n$.
(1 mark)
Let $n$ be a positive integer. If $n$ is divisible by 2010 and exactly one of the digits of $n$ is even, find the smallest possible value of $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{311550}
A freight train leaves the town of Jenkinsville at 1:00 PM traveling due east at constant speed. Jim, a hobo, sneaks onto the train and falls asleep. At the same time, Julie leaves Jenkinsville on her bicycle, traveling along a straight road in a northeasterly direction (but not due northeast) at 10 miles per hour. At 1:12 PM, Jim rolls over in his sleep and falls from the train onto the side of the tracks. He wakes up and immediately begins walking at 3.5 miles per hour directly towards the road on which Julie is riding. Jim reaches the road at 2:12 PM, just as Julie is riding by. What is the speed of the train in miles per hour?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{62.5 \, \text{mph}}
Suppose that $(a_{1}, \ldots, a_{20})$ and $(b_{1}, \ldots, b_{20})$ are two sequences of integers such that the sequence $(a_{1}, \ldots, a_{20}, b_{1}, \ldots, b_{20})$ contains each of the numbers $1, \ldots, 40$ exactly once. What is the maximum possible value of the sum $\sum_{i=1}^{20} \sum_{j=1}^{20} \min (a_{i}, b_{j})$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5530}
Given an integer \( n \geq 2 \). There are \( n \) sets written on the blackboard. The following operation is performed: select two sets \( A \) and \( B \) on the blackboard that do not contain each other, erase them, and then write \( A \cap B \) and \( A \cup B \). This is referred to as one operation. Continue performing this operation until, in any two sets, one set contains the other. Find the maximum possible number of operations for all initial states and operation methods.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{n(n-1)}{2}}
Jackie can do 5 push-ups in 10 seconds. As she gets fatigued, her push-up rate decreases by 20% for each subsequent 10-second interval. How many push-ups can she do in one minute if she takes a 7-second break at the 22-second mark and a 12-second break at the 38-second mark, regaining 10% of her push-up rate during each break?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
In  $ \vartriangle ABC$ ,  $\angle B = 46^o$  and  $\angle C = 48^o$  . A circle is inscribed in   $ \vartriangle ABC$  and the points of tangency are connected to form  $PQR$ . What is the measure of the largest angle in  $\vartriangle P QR$ ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{67^\circ}
Using each of the eight digits $1, 3, 4, 5, 6, 7, 8,$ and $9$ exactly once, form a three-digit number $A$, two two-digit numbers $B$ and $C$ (where $B < C$), and a one-digit number $D$. The numbers are such that $A + D = B + C = 143$. In how many ways can this be done?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
Let $A$, $B$, and $C$ be independent random variables, each uniformly distributed between $-1$ and $1$. What is the probability that $A+B+C\geq1$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1/6}
10.248. To two externally tangent circles with radii $R$ and $r$, a secant is constructed such that the circles intercept three equal segments on it. Find the lengths of these segments.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{2}\sqrt{\frac{14Rr-R^2-r^2}{3}}}
Let \( a \) and \( b \) be real numbers such that \( a + b = 1 \). Then, the minimum value of 
\[ 
f(a, b) = 3 \sqrt{1 + 2a^2} + 2 \sqrt{40 + 9b^2} 
\] 
is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5 \sqrt{11}}
将 $1,2,3,4,5,6$ 随机排成一行, 记为 $a, b, c, d, e, f$, 则 $a b c+def$ 是偶数的概率为 $\qquad$.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{19}
Let \( a, b, c, d, e \) be nonnegative integers such that \( 625a + 250b + 100c + 40d + 16e = 15^3 \). What is the maximum possible value of \( a + b + c + d + e \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{153}
Given the parabola $x^{2}=y$ with focus $F(0, \frac{1}{4})$, and points $A$ and $B$ on the parabola such that $|AF|+|BF|=3$, calculate the distance from the midpoint of line segment $AB$ to the x-axis. Express your answer as the absolute value of the y-coordinate of the midpoint, in fractional form.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{5}{4}}
A queue has two people, Person 1 and Person 2. Person 1 enters the queue first, and their impatience is an exponential random variable with rate $θ$. Person 2 enters the queue $s$ minutes later, and their impatience is also an exponential random variable, independent of Person 1's impatience, with rate $μ$. What is the probability that Person 1 will abandon the queue before Person 2 does?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1-\frac{\mu  e^{-\theta  s}}{\theta +\mu }}
Given four non-collinear points on a plane, what is the maximum number of acute triangles that can be formed using these points? Provide your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Given a polynomial equation \(z^{4} + a z^{3} + b z^{2} + c z + d = 0\) with real coefficients and all its roots lying on the unit circle in the complex plane centered at the origin, calculate the sum of the reciprocals of these roots. Express your answer in terms of the coefficients of the polynomial.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-a}
${{\cos }^{2}}\theta +{{\cos 2}}\left( \theta +120{}^\circ  \right)+{{\cos }^{2}}\left( \theta +240{}^\circ  \right)$的值是______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{3}{2}$}
7. (4 points) With the number written on the board, one of the following operations is allowed:
1) If there is a digit in the original number that is not equal to 9 and has two neighboring digits greater than 0, you can increase this digit by 1, and decrease the neighboring digits by 1.
2) Subtract 1 from any non-zero digit except the last one, and add 3 to the next digit.
3) Decrease any sufficiently large digit by 7.

If, as a result of any of these operations, there are zeros in the number on one or more of the leading positions, they are automatically discarded.

Initially, the number consisting of one hundred nines was written on the board. In the end, a single-digit number remained. Which one exactly?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
An equilateral triangle is divided into smaller equilateral triangles with side length 1 by lines parallel to its sides, dividing each side into \( n \) parts (in the diagram \( n = 5 \)). 

What is the maximum number of length 1 segments, with endpoints at the vertices of these smaller triangles, that can be marked so that no triangle is formed where all sides are marked segments?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n(n+1)}
13. Let $n$ be the smallest positive integer of 4 digits greater than or equal to 2016 that has the following property: there exists a positive integer $S$ such that

$$
S=\sqrt{a+\sqrt{b+\sqrt{c+\sqrt{d+S}}}}
$$

where $a, b, c, d$ are, in order, the thousands, hundreds, tens, and units digits of $n$. What is the value of $n$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2167}
If @ is a binary operation defined as the difference between an integer n and the product of n and a certain number, then the largest positive integer n such that the outcome of the binary operation of n is less than 10 is 2. What is the number that n is multiplied by in the product?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-3}