inclusionAI/Ring-lite-rl-data huggingface dataset:
A river has a right-angle bend. Except at the bend, its banks are parallel lines of distance  $a$  apart. At the bend the river forms a square with the river flowing in across one side and out across an adjacent side. What is the longest boat of length  $c$  and negligible width which can pass through the bend?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2a\sqrt{2}}
Evaluate the integral \( \int_{-1}^{1} \exp(x+e^{x})\,dx \), where \( \exp(x)=e^x \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e^e - e^{1/e}}
Four balls of radius $1$ are mutually tangent, three resting on the floor and the fourth resting on the others.
A tetrahedron, each of whose edges have length $s$, is circumscribed around the balls. Then $s$ equals
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2+2\sqrt{6}}
Calculate the limit: $$\lim_{n\to\infty}n\sin(5^n \pi \sqrt{2}\mathrm n!)$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Determine the values of \( \alpha \in \mathbb{R} \) for which the function \( g(x) = \frac{\ln(x)}{x^{\alpha}(x-1)} \) is integrable over the interval \((0,1)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-\infty, 1)}
Two trains 121 meters and 165 meters in length respectively are running in opposite directions, one at the rate of 80 km and the other at the rate of 55 kmph. In what time will they be completely clear of each other from the moment they meet?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7.63}
Let $\alpha = \cos^{-1} \left( \frac{3}{5} \right)$ and $\beta = \sin^{-1} \left( \frac{3}{5} \right)$.  
\[ \sum_{n=0}^{\infty}\sum_{m=0}^{\infty} \frac{\cos(\alpha n +\beta m)}{2^n3^m} \]
can be written as $\frac{A}{B}$ for relatively prime positive integers $A$ and $B$. Find $1000A + B$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15013}
Derek the Dolphin and Kevin the Frog are playing a game where they take turns taking coins from a stack of $N$ coins, except with one rule: The number of coins someone takes each turn must be a power of $6$. The person who cannot take any more coins loses. If Derek goes first, how many integers $N$ from $1$ to $6^{2021}$ inclusive will guarantee him a win? (Example: If $N = 37$, then a possible sequence of turns is: Derek takes one coin, Kevin takes $36$ coins, and Kevin wins.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$\frac{4}{7}(6^{2021} + 1)$
}
A strip is the region between two parallel lines. Let $A$ and $B$ be two strips in a plane. The intersection of strips $A$ and $B$ forms a parallelogram $P$. If strip $A$ is rotated by $60^\circ$ to form strip $A'$, the intersection of strips $A'$ and $B$ also forms a parallelogram with the same area as $P$. Determine the greatest possible value of one interior angle of parallelogram $P$ in degrees.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{120}
$ABC$  is triangle.  $l_1$ - line passes through  $A$  and parallel to  $BC$ ,  $l_2$  -  line passes through  $C$  and parallel to  $AB$ . Bisector of  $\angle B$  intersect  $l_1$  and  $l_2$  at  $X,Y$ .  $XY=AC$ . What  value can take  $\angle A- \angle C$  ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60^\circ}
Find the sum of the infinite series: $$\sum_{n=1}^{\infty}\left\lfloor\frac{3^{20}+2^{n-1}}{2^n}\right\rfloor$$ Given that you've identified a finite support for the sequence, how can you determine the sum?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3486784401}
Determine the smallest real value \( C \) such that for all functions \( f: [0,1] \to \mathbb{R} \) in the set \( M \), the inequality \( f(x) \le Cx \) holds for all \( x \in [0,1] \). The set \( M \) consists of functions satisfying the following properties:

- \( f(x) \ge 0 \) for all \( x \in [0,1] \),
- \( f(1) = 1 \),
- \( f(x+y) \ge f(x) + f(y) \) for all \( x, y \) such that \( x+y \in [0,1] \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Compute the limit \( \lim_{r\to 0}\frac{f(z+re^{i\alpha})-f(z)}{re^{i\alpha}} \) where \( f(z)=\bar{z} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e^{-2i\alpha}}
Find the derivative of the function \( g: M(2, \mathbb{R}) \to M(2, \mathbb{R}) \) defined by \( g(A) = A^2 \), where \( M(2, \mathbb{R}) \) is the vector space of \( 2 \times 2 \) real matrices.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{Dg(A)(H) = AH + HA}
Determine the number of rotational symmetries of the 120-cell.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7200}
Find the cardinality of the set \( B = \{ g \in C[0, 1] : g(0) = 0, g(1) = 1, |g'(t)| \geq 1 \text{ for all } t \in [0, 1] \} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Let \( f: (0,\infty )\rightarrow \mathbb{R} \) and \( F(x) = \int_{0}^{x}tf(t)dt \). If \( F(x^2)= x^{4} + x^{5} \), find the value of \( \sum_{r=1}^{12}f(r^{2}) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{219}
2019 numbers are arranged in a circle. For any two neighboring numbers \( x \) and \( y \), the inequalities \( |x - y| \geq 2 \) and \( x + y \geq 6 \) hold. Determine the smallest possible sum of the numbers arranged in the circle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6060}
Evaluate the integral of the function \( f(x,y) = x^2 + y^2 \) over the region \( R \) bounded by the curves \( y = x^2 \), \( y = x^2 + 4 \), \( x = y \), and \( x = y + 2 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{3}{35}}
Ana, Bob, and Cao bike at constant rates of $8.6$ meters per second, $6.2$ meters per second, and $5$ meters per second, respectively. They all begin biking at the same time from the northeast corner of a rectangular field whose longer side runs due west. Ana starts biking along the edge of the field, initially heading west, Bob starts biking along the edge of the field, initially heading south, and Cao bikes in a straight line across the field to a point $D$ on the south edge of the field. Cao arrives at point $D$ at the same time that Ana and Bob arrive at $D$ for the first time. The ratio of the field's length to the field's width to the distance from point $D$ to the southeast corner of the field can be represented as $p : q : r$, where $p$, $q$, and $r$ are positive integers with $p$ and $q$ relatively prime. Find $p+q+r$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{61}
Evaluate the expression \(\frac{I}{J}\), where:
\[I=\int_{0}^{1}\frac{x^{\frac{5}{2}}(1-x)^{\frac{7}{2}}}{12}\,dx\]
and
\[J=\int_{0}^{1}\frac{x^{\frac{5}{2}}(1-x)^{\frac{7}{2}}}{(x+3)^8}\,dx\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{864\sqrt{3}}
Find the smallest positive integer \( n \) such that the symmetric group \( S_n \) has a subgroup of order \( p^2 \), where \( p \) is a prime number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Each face of a cube is given a single narrow stripe painted from the center of one edge to the center of the opposite edge.  The choice of the edge pairing is made at random and independently for each face.  What is the probability that there is a continuous stripe encircling the cube?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3}{16}}
Determine the number of independent parameters required to specify all possible conditional probabilities for a set of $n$ events $X_1, X_2, ..., X_n$. Consider all probabilities such as $P(X_i)$, $P(X_i, X_j)$ for $i \neq j$, $P(X_i | X_j)$ for $i \neq j$, $P(X_i, X_j, X_k)$ for $i \neq j \neq k$, $P(X_i | X_j, X_k)$ for $i \neq j \neq k$, and $P(X_i, X_j | X_k)$ for $i \neq j \neq k$. How many of these probabilities must be specified to determine all others using the standard laws of probability?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^n - 1}
If a matrix $X$ has $m$ rows and $n$ columns and is row orthogonal with rank $m$, what is the minimum number of columns that a new matrix $Y$ must have to ensure that it also has rank $m$, assuming that the columns of $Y$ are sampled from $X$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{m}
设 $\alpha, \beta$ 为实数, 若对任意实数 $x, y, z$. 有 $a(x y+y z+z x) \leq M \leq \beta\left(x^{2}+y^{2}+z^{2}\right)$ 恒成立, 其中 $M=\sqrt{x^{2}+x y+y^{2}} \cdot \sqrt{y^{2}+y z+z^{2}}+\sqrt{y^{2}+y z+z^{2}} \cdot \sqrt{z^{2}+z x+x^{2}}+\sqrt{z^{2}+z x+x^{2}} \cdot \sqrt{x^{2}+x y+y^{2}}$.求 $\alpha$ 的最大值和 $\beta$ 的最小值.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$3,3$}
Josh takes a walk on a rectangular grid of \( n \) rows and 3 columns, starting from the bottom left corner. At each step, he can either move one square to the right or simultaneously move one square to the left and one square up. In how many ways can he reach the center square of the topmost row?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{n-1}}
The polynomial  $f(x)=x^{2007}+17x^{2006}+1$  has distinct zeroes  $r_1,\ldots,r_{2007}$ .  A polynomial  $P$  of degree  $2007$  has the property that  $P\left(r_j+\dfrac{1}{r_j}\right)=0$  for  $j=1,\ldots,2007$ .  Determine the value of  $P(1)/P(-1)$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{289/259}
Evaluate the limit: $$\lim_{n\to\infty} \frac{\int_0^{\pi/4}(\tan x)^{2n}dx}{\int_0^{\pi/4}(\tan x)^{2n+1}dx}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
How many of the letters in MATHCOUNTS have a horizontal line of symmetry?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Let  $k$  be a fixed positive integer. The  $n$ th derivative of  $\tfrac{1}{x^k-1}$  has the form  $\tfrac{P_n(x)}{(x^k-1)^{n+1}}$ , where  $P_n(x)$  is a polynomial. Find  $P_n(1)$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ (-1)^n n! k^n }
For a positive integer n, let  $w(n)$  denote the number of distinct prime
divisors of n. Determine the least positive integer k such that $2^{w(n)} \leq k \sqrt[4]{n}$ 
for all positive integers n.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Consider the set  $S$  of permutations of  $1, 2, \dots, 2022$  such that for all numbers  $k$  in the
permutation, the number of numbers less than  $k$  that follow  $k$  is even.

For example, for  $n=4; S = \{[3,4,1,2]; [3,1,2,4]; [1,2,3,4]; [1,4,2,3]\}$ If  $|S| = (a!)^b$  where  $a, b \in \mathbb{N}$ , then find the product  $ab$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2022}
Determine the Krull dimension of the injective hull $E = E_R(R/\mathfrak{m})$, where $(R, \mathfrak{m})$ is a Noetherian local ring.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
How many subsets of {1,2,3,4,5,6,7,8,9,10,11,12} have the property that no two of its elements diﬀer by more than 5? For example, count the sets {3}, {2,5,7}, and {5,6,7,8,9} but not the set {1,3,5,7}.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{256}
Evaluate the limit: $$\lim_{x\to0}\frac{x\sin{(\sin{x})}-\sin^2{x}}{x^6}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{18}}
Find the area of the part of the cylinder \(x^2+y^2=2ay\) that lies inside the sphere \(x^2+y^2+z^2=4a^2\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16a^2}
In the triangle \(ABC\), it is known that \(AB=BC\) and \(\angle BAC=45^\circ\). The line \(MN\) intersects side \(AC\) at point \(M\), and side \(BC\) at point \(N\). Given that \(AM=2 \cdot MC\) and \(\angle NMC=60^\circ\), find the ratio of the area of triangle \(MNC\) to the area of quadrilateral \(ABNM\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{7 - 3\sqrt{3}}{11}}
Determine the Galois group of the polynomial \( f(X) = X^4 + 2X^2 + 4 \) over \( \mathbb{Q} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{V_4}
Two rectangles, each measuring 7 cm in length and 3 cm in width, overlap to form the shape shown on the right. What is the perimeter of this shape in centimeters?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{28}
Find the coefficient of \(x^8\) in the expansion of \((1-2x+3x^2-4x^3+5x^4-6x^5+7x^6)^6\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{75048}
Find the mass percentage of Cl in Chlorous acid
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{51.78\%}
Given a positive integer $n$, it can be shown that every complex number of the form $r+si$, where $r$ and $s$ are integers, can be uniquely expressed in the base $-n+i$ using the integers $0,1,2,\ldots,n^2$ as digits. That is, the equation
$r+si=a_m(-n+i)^m+a_{m-1}(-n+i)^{m-1}+\cdots +a_1(-n+i)+a_0$
 is true for a unique choice of non-negative integer $m$ and digits $a_0,a_1,\ldots,a_m$ chosen from the set $\{0,1,2,\ldots,n^2\}$, with $a_m\ne 0$. We write
$r+si=(a_ma_{m-1}\ldots a_1a_0)_{-n+i}$
 to denote the base $-n+i$ expansion of $r+si$. There are only finitely many integers $k+0i$ that have four-digit expansions
$k=(a_3a_2a_1a_0)_{-3+i}~~$
$~~a_3\ne 0.$
 Find the sum of all such $k$,
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{490}
Denis painted some of the faces of his cubes gray. Vova selected 10 cubes so that all of them had different patterns of painting and then stacked them as shown in the figure. How many white faces does the top cube have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Find the value of $n$ that satisfies the equation: $$\frac{_{365}P_n}{365^n} \approx \frac{1}{5}.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{34}
A table  $110\times 110$  is given, we define the distance between two cells  $A$  and  $B$  as the least quantity of moves to move a chess king from the cell  $A$  to cell  $B$ . We marked  $n$  cells on the table  $110\times 110$  such that the distance between any two cells is not equal to  $15$ . Determine the greatest value of  $n$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6050}
Let $\triangle A_0B_0C_0$ be a triangle whose angle measures are exactly $59.999^\circ$, $60^\circ$, and $60.001^\circ$. For each positive integer $n$, define $A_n$ to be the foot of the altitude from $A_{n-1}$ to line $B_{n-1}C_{n-1}$. Likewise, define $B_n$ to be the foot of the altitude from $B_{n-1}$ to line $A_{n-1}C_{n-1}$, and $C_n$ to be the foot of the altitude from $C_{n-1}$ to line $A_{n-1}B_{n-1}$. Find the least positive integer $n$ for which $\triangle A_nB_nC_n$ is obtuse.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
Let be given a triangle  $ABC$  and points  $D,M,N$  belong to  $BC,AB,AC$ , respectively. Suppose that  $MD$  is parallel to  $AC$  and  $ND$  is parallel to  $AB$ . If  $S_{\vartriangle BMD} = 9$  cm  $^2, S_{\vartriangle DNC} = 25$  cm $^2$ , compute  $S_{\vartriangle AMN}$ ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
In $\triangle ABC$, $BD$ is a median. $CF$ intersects $BD$ at $E$ so that $\overline{BE}=\overline{ED}$. Point $F$ is on $AB$. Then, if $\overline{BF}=5$, $\overline{BA}$ equals:
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$15$}
Let  $u, v$  be real numbers. The minimum value of  $\sqrt{u^2+v^2} +\sqrt{(u-1)^2+v^2}+\sqrt {u^2+ (v-1)^2}+ \sqrt{(u-1)^2+(v-1)^2}$  can be written as  $\sqrt{n}$ . Find the value of  $10n$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{80}
In the diagram, each scale shows the total mass (weight) of the shapes on that scale. Using the information provided by the scales, calculate the mass (weight) of a triangle. Express your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Determine the number of loops in the curve defined by \(y^4 = b^2(y^2 - x^2)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Find the smallest positive number $\lambda $ , such that for any complex numbers ${z_1},{z_2},{z_3}\in\{z\in C\big| |z|<1\}$ ,if  $z_1+z_2+z_3=0$, then $$\left|z_1z_2 +z_2z_3+z_3z_1\right|^2+\left|z_1z_2z_3\right|^2 <\lambda .$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Given a linearly independent set of functionals in the dual space of a vector space \( V \), determine the nullity of a linear transformation from \( V \) to \( \mathbb{R}^m \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dim(V) - m}
An isosceles right triangle has a leg length of 36 units. Starting from the right angle vertex, an infinite series of equilateral triangles is drawn consecutively on one of the legs. Each equilateral triangle is inscribed such that their third vertices always lie on the hypotenuse, and the opposite sides of these vertices fill the leg. Determine the sum of the areas of these equilateral triangles.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{324}
In the diagram, \( AE \) and \( BD \) are straight lines that intersect at \( C \). Given that \( BD = 16 \), \( AB = 9 \), \( CE = 5 \), and \( DE = 3 \), calculate the length of \( AC \). Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
Determine the number of non-trivial ring homomorphisms from the ring of integers modulo 20, $\mathbb{Z}_{20}$, to the ring of integers modulo 44, $\mathbb{Z}_{44}$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Solve the recurrence relation: \[ T(n) = 2T(n - 1) + (\log n)T([\log n]) + 1 \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{T(n) = \Theta(2^n)}
Evaluate whether the rising factorial $0^{\bar{0}}$ is equal to 1 using the relationship $a^{\bar{n}}=\frac{\Gamma(a+n)}{\Gamma(a)}$. Consider the case when $a=n=0$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Evaluate the limit by expressing it as a definite integral: $$\lim_{n\to\infty}\frac{\pi}{2n}\left[\cos\left(\frac{\pi}{2n}\right)+\cos\left(\frac{\pi}{n}\right)+\cos\left(\frac{3\pi}{2n}\right)+\cdots+\cos\left(\frac{(n-1)\pi}{2n}\right)\right]$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Evaluate the limit: $$\lim_{{x \to 0^+}} \frac{\log(x+\sqrt{1+x^2})-\sin x}{(2^{x^2}-1)(\sin(e^{x^2}-1))}.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
There are $n$ mathematicians seated around a circular table with $n$ seats numbered $1,$ $2,$ $3,$ $...,$ $n$ in clockwise order. After a break they again sit around the table. The mathematicians note that there is a positive integer $a$ such that
 ($1$) for each $k,$ the mathematician who was seated in seat $k$ before the break is seated in seat $ka$ after the break (where seat $i + n$ is seat $i$);
 ($2$) for every pair of mathematicians, the number of mathematicians sitting between them after the break, counting in both the clockwise and the counterclockwise directions, is different from either of the number of mathematicians sitting between them before the break.
 Find the number of possible values of $n$ with $1 < n < 1000.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{332}
Find the annihilator operator of the function \( f(x) = x(\sin x + 3x - x^{-1}) - \cos x \), where the operator is \( \frac{d}{dt} \equiv D \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{D^3 (D^2 + 1)^2}
Determine the complexity of the function \( f(n) = f(7n/15) + f(8n/15) + n^2 \) for all \( n \geq 0 \) using the Akra-Bazzi theorem.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\Theta(n^2)}
The new PUMaC tournament hosts  $2020$  students, numbered by the following set of labels  $1, 2, . . . , 2020$ . The students are initially divided up into  $20$  groups of  $101$ , with each division into groups equally likely. In each of the groups, the contestant with the lowest label wins, and the winners advance to the second round. Out of these  $20$  students, we chose the champion uniformly at random. If the expected value of champion’s number can be written as  $\frac{a}{b}$ , where  $a, b$  are relatively prime integers, determine  $a + b$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2123}
Cynthia loves Pokemon and she wants to catch them all. In Victory Road, there are a total of  $80$  Pokemon. Cynthia wants to catch as many of them as possible. However, she cannot catch any two Pokemon that are enemies with each other. After exploring around for a while, she makes the following two observations:
1. Every Pokemon in Victory Road is enemies with exactly two other Pokemon.
2. Due to her inability to catch Pokemon that are enemies with one another, the maximum number of the Pokemon she can catch is equal to  $n$ .
What is the sum of all possible values of  $n$ ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{469}
Evaluate the infinite series \( \sum_{r=1}^{\infty} \frac{r^3 + (r^2 + 1)^2}{(r^4 + r^2 + 1)(r^2 + r)} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{3}{2}}
Given an element \( v = (v_1, \dots, v_d) \in \mathbb{Z}^d \) with \( \gcd(v_1, \dots, v_d) = 1 \), there exists a regular integer matrix \( A \in \mathop{GL}_d(\mathbb{Z}) \) such that \( A(v) = (1, 0, \dots, 0) \). Assume \( |v_1| + \dots + |v_d| = n \). Determine the smallest \( \ell \in \mathbb{N} \) such that the image \( A\left([-1, 1]^d\right) \) is contained in the cube \( [-\ell, \ell]^d \). Find the asymptotic behavior of \( \ell(n) \) up to a multiplicative constant.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\Theta(n)}
Let $A B C$ be a triangle where $A B=9, B C=10, C A=17$. Let $\Omega$ be its circumcircle, and let $A_{1}, B_{1}, C_{1}$ be the diametrically opposite points from $A, B, C$, respectively, on $\Omega$. Find the area of the convex hexagon with the vertices $A, B, C, A_{1}, B_{1}, C_{1}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1155}{4}}
Evaluate the triple integral $\iiint_S {z{\sqrt{x^2+y^2}}}\,dx\,dy\,dz$ over the region $S$ bounded by $x^2+y^2=1$, $z=0$, and $z=x$ with $z \ge 0$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{20}}
If \( a < b \), then which inequality sign ( \( > \) or \( < \) ) should be placed between \( \frac{1}{a} \) and \( \frac{1}{b} \) ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{>}
有 20 个队参加全国足球冠军决赛，为了使任何三个队中都有两个队相互比赛过，问最少要进行多少场比赛?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{90}
Which of the following numbers is an odd integer, contains the digit 5, is divisible by 3, and lies between \(12^{2}\) and \(13^{2}\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{165}