inclusionAI/Ring-lite-rl-data huggingface dataset:
Laura's House is a 20-mile round trip from her school. The supermarket is an additional 10 miles farther away from the school. Laura also visits her local gym 3 times a week, which is 5 miles from her house, and visits a friend's house every Saturday, located 12 miles away from her house. If Laura drives to school every morning, to the supermarket two afternoons a week, and includes her trips to the gym and her friend's house, how many miles does Laura drive per week?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{234}
Find the largest \( k \) for which there exists a permutation
 \( \left( {{a}_{1},{a}_{2},\ldots ,{a}_{2022}}\right) \) of integers from 1 to 2022 such that
 for at least \( k \) distinct \( i \) with \( 1 \leq  i \leq  {2022} \) the number
 \( \frac{{a}_{1} + {a}_{2} + \ldots  + {a}_{i}}{1 + 2 + \ldots  + i} \) is an integer larger than 1 .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
k = 1011
}
For $k \in \{0, 1, \ldots, 9\}$, let $\epsilon_k \in \{-1, 1\}$. If the minimum possible value of $\sum_{i=1}^9 \sum_{j=0}^{i-1} \epsilon_i \epsilon_j 2^{i+j}$ is $m$, find $|m|$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{174762}
A number, when divided by the sum of a certain number and 445, gives 2 times their difference as quotient and 50 as remainder. The number is 220050. What is the first number?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{555}
Use Plancharel's theorem to evaluate the integral \( \int_{-\infty}^{\infty}\frac{\cos^2 x}{x^2}\,dx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi}
A positive integer $n$ is called \textit{good} if $2 \mid \tau(n)$ and if its divisors are $1 = d_1 < d_2 < \ldots < d_{2k-1} < d_{2k} = n$, then $d_{k+1} - d_k = 2$ and $d_{k+2} - d_{k-1} = 65$. Find the smallest \textit{good} number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2024}
Determine the number of subgroups isomorphic to \(\mathbb{Z}_{2} \times \mathbb{Z}_{2}\) in the dihedral group \(D_8 = \langle \sigma, \rho \; | \; \rho^8 = \sigma^2 = e \text{ and } \sigma\rho\sigma = \rho^{-1} \rangle\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Moor has $3$ different shirts, labeled $T$, $E$, and $A$. Across $5$ days, the only days Moor can wear shirt $T$ are days $2$ and $5$. How many different sequences of shirts can Moor wear across these $5$ days?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{72}
Determine the splitting field of the polynomial \((x^3+x^2+1)(x^3+x+1)(x^2+x+1)\) over the finite field \(\mathbb{F}_2\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{F}_{2^6}}
Suppose that $a$ and $b$ are natural numbers such that \[ p = \frac{b}{4} \sqrt{\frac{2a-b}{2a+b}} \] is a prime number. What is the maximum possible value of $p$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Find the value of $a$ in the inequality $ax^2 + 2x + 20$ with the solution set $(\-\frac{1}{2}, \frac{1}{3})$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-12}
Evaluate the double integral \( \iint_D (x + y)\, dx \, dy \) where \(D\) is the trapezoidal region with vertices \((0,0), (5,0), (5/2, 5/2), (5/2, -5/2)\), using the transformations \( x = 2u + 3v \) and \( y = 2u - 3v\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{125}{4}}
Find the number of distinct elements in the group defined by the presentation \( \langle x,y \mid x^3=y^2=(xy)^3=e \rangle \) by determining a normal form for its elements.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Let $R$  be a set of nine distinct integers. Six of the elements are $2$, $3$, $4$, $6$, $9$, and $14$. What is the number of possible values of the median of $R$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Suppose you have $n$ red balls, $n$ green balls, and an arbitrary number of white balls. Each ball is selected into a set with a probability of $p = \frac{1}{n}$ and excluded with a probability of $1 - p = 1 - \frac{1}{n}$, independently of one another. Determine a constant lower bound (independent of $n$) on the probability that the set will contain at least one green ball but no red balls.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{3}{16}}
Find the sum of the infinite series: \[ \sum_{n=0}^\infty \frac{2n^7 + n^6 + n^5 + 2n^2}{n!} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2013e}
Determine the values of $p$ for which the integral \( \int_0^{\infty} \frac{1-\cos x}{x^p} \, dx \) converges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(1, 3)}
Evaluate the limit of the infinite product: \[ \prod_{p \in \mathbb{P}} \frac{p^4+1}{p^4-1} \] where \( \mathbb{P} \) is the set of prime numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{7}{6}}
The base of the tetrahedron $S-ABCD$ is a square $ABCD$ with side length 1, and $SO$ is the height of the pyramid with $SO = \frac{1}{2}$. If $SA = SD$ and $SB = SC$, what is the dihedral angle between the planes $ASB$ and $CSD$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{90^\circ}
Find the value of \( \beta \) such that the function \( u(x_1, x_2, \ldots, x_n) = \left(\sum_{i=1}^{n}x_{i}^2\right)^{\beta/2} \) satisfies \( \nabla^2 u = 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 - n}
In triangle \(ABC\), it is known that \(\angle BAC = \alpha\) and \(AC = b\). The incircle touches the sides \(AB\) and \(BC\) at points \(M\) and \(N\), respectively. The angle bisector of \(\angle BAC\) intersects the line \(MN\) at point \(K\). Find the distance from point \(K\) to the line \(AC\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{b}{2} \sin(\alpha)}
In a group of some students, each student is registered for at least one of 3 classes – history, math, and English. 20 - one students are registered for history, 20 - 6 students are registered for math, and thirty - 4 students are registered for English. If only 3 students are registered for all 3 classes, and 7 students are registered for exactly two classes, how many students are there in the group?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{46}
The ship decided to determine the depth of the ocean at its location. The signal sent by the echo sounder was received on the ship 8 seconds later. The speed of sound in water is 1.5 km/s. Determine the depth of the ocean.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6000 \, \text{m}}
Given two positive integers \( m \) and \( n \) satisfying \( 1 \leq m \leq n \leq 40 \), find the number of pairs \( (m, n) \) such that their product \( mn \) is divisible by 33.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{64}
Find the last two digits of the decimal representation of $9^{8^{7^{\cdot^{\cdot^{\cdot^{2}}}}}}$. Express your answer as a two-digit number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21}
a wire in the form of a circle of radius 3.5 m is bent in the form of a rectangule , whose length and breadth are in the ratio of 6 : 5 . what is the area of the rectangle ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(735 * π^2) / 242}
Find the mass of the solid $E$ enclosed by the planes $z = 0$, $z = x + y + 5$, and the cylinders $x^2 + y^2 = 4$, $x^2 + y^2 = 9$. The density at any point is equal to its distance from the $yz$-plane.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{380}{3}}
The speed of a boat in still water is 65 km/hr and the rate of the current is 15 km/hr. The boat travelled downstream for a certain amount of time and covered a distance of 33.33 km. How long did the boat travel downstream?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25}
Let $z$ be a complex number satisfying $(z+\frac{1}{z})(z+\frac{1}{z}+1)=1$. Evaluate $(3z^{100}+\frac{2}{z^{100}}+1)(z^{100}+\frac{2}{z^{100}}+3)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36}
Rob and Mark plan to meet at the national park. It takes 1 hour for Rob to get to the park and it takes three times as much time for Mark to get there. Rob lives in a time zone 2 hours behind Mark's time zone. If Rob leaves his home at 11 a.m. local time, at what local time should Mark leave his home so that they both arrive at the national park at the same time?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
4. Among the 21 numbers $8,9, \cdots, 28$, the probability that the sum of three randomly selected different numbers is exactly divisible by 9 is $\qquad$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{15}{133}}
Jerry starts at $0$ on the real number line. He tosses a fair coin $8$ times. When he gets heads, he moves $1$ unit in the positive direction; when he gets tails, he moves $1$ unit in the negative direction. The probability that he reaches $4$ at some time during this process is $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. What is the value of $a + b$? (For example, he succeeds if his sequence of tosses is $HTHHHHHH.$)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{151}
Among the 99 positive integers from 1, 2, ..., 99, determine the minimum value of k such that any selection of k numbers must include at least two numbers a and b (a != b) satisfying 1/2 <= b/a <= 2.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
What is the volume of a cuboctahedron with a side length of 1?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{5 \sqrt{2}}{3}}
For the smallest value of $n$, the following condition is met: if $n$ crosses are placed in some cells of a $6 \times 6$ table in any order (no more than one cross per cell), three cells will definitely be found forming a strip of length 3 (vertical or horizontal) in each of which a cross is placed.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25}
7. Point $P$ is located inside a square $A B C D$ of side length 10 . Let $O_{1}, O_{2}, O_{3}, O_{4}$ be the circumcenters of $P A B, P B C, P C D$, and $P D A$, respectively. Given that $P A+P B+P C+P D=23 \sqrt{2}$ and the area of $O_{1} O_{2} O_{3} O_{4}$ is 50 , the second largest of the lengths $O_{1} O_{2}, O_{2} O_{3}, O_{3} O_{4}, O_{4} O_{1}$ can be written as $\sqrt{\frac{a}{b}}$, where $a$ and $b$ are relatively prime positive integers. Compute $100 a+b$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16902}
Evaluate the integral \( \int_{\partial B_1(0)} P(x, y) \, dA_y \) for the Poisson kernel \( P(x, y) \) defined as \[ P(x,y)=\frac{1}{\omega_n R} \frac{R^2-|x|^2}{|x-y|^n}, \] where \( x \) is in the open ball \( B_R(0) \) in \( \mathbb{R}^n \) with \( |x|<R \), \( y \) lies on the boundary of \( B_R(0) \) with \( |y|=R \), and \( \omega_n \) is the surface area of an \( n \)-dimensional unit sphere.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
On a $5 \times 5$ grid, cover the area with several S-Tetrominos (also known as Z-Tetrominos) such that in every square, there are two or fewer tiles covering it (tiles can overlap). Determine the maximum possible number of squares that can be covered by at least one tile.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
In trapezoid $KLMN$, base $KL$ is $40 \text{ cm}$ and base $MN$ is $16 \text{ cm}$. Point $P$ lies on segment $KL$ such that segment $NP$ divides the trapezoid into two parts with equal areas.

Determine the length of segment $KP$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{28 \, \text{cm}}
John drives 1,000 miles a month, totaling 12,000 miles a year. He needs to get an oil change every 3,000 miles, and the price of an oil change fluctuates throughout the year. In the first quarter, the price is $55, in the second quarter, it is $45, in the third quarter, it is $50, and in the fourth quarter, it is $40. John receives one free oil change per year through his auto shop's loyalty program.

Additionally, he needs to get his tires rotated every 6,000 miles, which costs $40 per rotation. John is also eligible for a 10% discount on tire rotations, as he holds a membership with a local automobile association.

Furthermore, he needs to replace his brake pads every 24,000 miles, which costs $200. The local auto shop offers a biennial membership for $60, which grants a 20% discount on brake pad replacements.

Calculate John's total annual car maintenance expenses for oil changes, tire rotations, and brake pad replacements, taking into account fluctuating oil change prices, the discount on tire rotations, and the biennial membership for brake pad replacement discounts.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$317$}
Evaluate the limit: \[ \lim _{x\rightarrow0}\frac{1-\cos\left(\frac{1-\cos x \cos 2x}{x^2}-\frac {5}{2}\right)\cos2x}{x^2} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
How many four-digit integers $abcd$, with $a \not\equiv 0$, have the property that the three two-digit integers $ab<bc<cd$ form an increasing arithmetic sequence? One such number is $4692$, where $a=4$, $b=6$, $c=9$, and $d=2$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{17

}
Calculate the probability of obtaining an infinite sequence of the same number when rolling a fair six-sided die infinitely many times.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Given an equilateral triangle with side length \( a \), find the segment connecting a vertex of the triangle to a point that divides the opposite side in the ratio \( 2:1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{a\sqrt{7}}{3}}
A frog starts jumping from the point (1,2) and makes a series of jumps, each of which is parallel to one of the coordinate axes and is 1 unit in length. The direction of each jump (up, down, right, or left) is chosen at random. The frog stops jumping when it reaches the boundary of the square with vertices (0,0), (0,4), (4,4), and (4,0). What is the probability that the frog ends its jumps on a vertical side of the square? Express your answer as a fraction.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{5}{8}}
双曲线 $x^{2}-y^{2}=1$ 的右半支与直线 $x=100$ 围成的区域内部（不含边界）整点 （纵横坐标均为整数的点）的个数是 $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9800}
Square $EFGH$ has a side length of $40$. Point $Q$ lies inside the square such that $EQ = 16$ and $FQ = 34$. The centroids of $\triangle{EFQ}$, $\triangle{FGQ}$, $\triangle{GHQ}$, and $\triangle{HEQ}$ are the vertices of a convex quadrilateral. Calculate the area of this quadrilateral.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3200}{9}}
Consider the set of points S with Cartesian coordinates (x, y) that satisfy all the following five conditions:
(i) a/2 ≤ x ≤ 2a.
(ii) a/2 ≤ y ≤ 2a.
(iii) x + y ≥ a.
(iv) x + a ≥ y.
(v) y + a ≥ x.

Determine the number of edges of the polygon that forms the boundary of the set S. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
The average age of 6 boys is 18 years, and their ages are in proportion x : x+2 : x+4 : 2x-3 : 2x-1 : 2x+1. What is the age in years of the eldest boy, given that the youngest boy is twice as old as his sister who is currently 5 years old?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21}
A man buys three items: a cycle for Rs. 2300, a scooter for Rs. 12,000, and a motorbike for Rs. 25,000. He sells the cycle at a loss of 30%, the scooter at a profit of 25%, and the motorbike at a profit of 15%. What is the combined selling price of these three items?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{45,360}
设函数 $f_{0}(x)=|x|, f_{1}(x)=\left|f_{0}(x)-1\right|, f_{2}(x)=\left|f_{1}(x)-2\right|$, 则函数 $y=f_{2}(x)$ 的图象与 $x$ 轴所围成图形中的封闭部分的面积是 $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Let the set \( A = \{1, 2, \cdots, 2016\} \). For any 1008-element subset \( X \) of \( A \), if there exist \( x \) and \( y \in X \) such that \( x < y \) and \( x \mid y \), then \( X \) is called a "good set". Find the largest positive integer \( a \) (where \( a \in A \)) such that any 1008-element subset containing \( a \) is a good set.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1008}
Three numbers,  $a_1$ ,  $a_2$ ,  $a_3$ , are drawn randomly and without replacement from the set  $\{1, 2, 3, \dots, 1000\}$ .  Three other numbers,  $b_1$ ,  $b_2$ ,  $b_3$ , are then drawn randomly and without replacement from the remaining set of 997 numbers.  Let  $p$  be the probability that, after a suitable rotation, a brick of dimensions  $a_1 \times a_2 \times a_3$  can be enclosed in a box of dimensions  $b_1 \times b_2 \times b_3$ , with the sides of the brick parallel to the sides of the box.  If  $p$  is written as a fraction in lowest terms, what is the sum of the numerator and denominator?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21}
Given a circle O: x^2 + y^2 = 1 and a point P(-1, sqrt(3)), where A and B are two moving points on circle O, find the maximum value of angle APB in radians.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\pi}{3}}
Eva draws an equilateral triangle and its altitudes. In the first step, she draws the center triangle of the equilateral triangle. In the second step, she draws the center triangle of this center triangle, and so on. After each step, Eva counts all triangles whose sides lie completely on the drawn lines. What is the minimum number of center triangles she must have drawn so that the figure contains more than 2022 such triangles?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Each face of a $6 \times 6 \times 6$ cube is divided into $1 \times 1$ cells. The cube is covered with $2 \times 2$ squares such that each square covers exactly four cells, no squares overlap, and each cell is covered by an equal number of squares. What is the maximum value of this equal number? (A square can be folded over an edge.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
设整数 $n \geq 2$. 某天 $n$ 个人组成了一个社团，每人向政府交 1 元注册费。接下来每天都发生如下事情：每一个前一天组建的社团都分裂成两个团体，其中含至少两人的团体组建成一个新社团，每个新社团的成员向政府交 1 元注册费，只有一名成员的团体不再组建新社团。上述过程直到不能继续进行下去为止。当$n=6$时，求政府总共收到的注册费的最大值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
A circle is circumscribed around a right triangle \(ABC\) with legs \(AC = 5\) and \(BC = 12\). Points \(E\) and \(G\) are the midpoints of the smaller arcs \(AC\) and \(BC\) of this circle, respectively. Point \(F\) is the midpoint of the arc \(AB\) that does not contain point \(C\).

Find the area of quadrilateral \(AEGF\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{117}{2}}
Let $f:\mathbb{N}\to\mathbb{N}$ be a function. Derive an upper bound for $$\frac{a}{b}+\frac{b}{a},$$ given that $$a < b$$ and $$\frac{f(b)/b}{f(a)/a}<2.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
# Problem 3.

In triangle $A B C$, the bisector $B E$ and the median $A D$ are equal and perpendicular. Find the area of triangle $A B C$, if $A B=\sqrt{13}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Evaluate the integral \( I = \int_{-\frac{\pi }{2}}^{\frac{\pi }{2}} \frac{\cos (x)}{\gamma ^{H_x-H_{-x}}+1} \, dx \), where \( \gamma \) is Euler's gamma and \( H_x \) is the harmonic number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
An even function $f(x)$ defined on $\mathbb{R}$ satisfies $f(x+2) = -f(x)$ and is monotonic in the interval $[0,2]$. If there exists $x \in (0,2)$ such that $f(x) = 0$, then the sum of all roots of the equation $f(x) = 0$ in the interval $[2002, 2010]$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8024}
The New York Public Library requires patrons to choose a 4-digit Personal Identification Number (PIN) to access its online system. (Leading zeros are allowed.) The PIN is not allowed to contain either of the following two forbidden patterns:

1. A digit that is repeated 3 or more times in a row. For example, $0001$ and $5555$ are not PINs, but $0010$ is a PIN.
2. A pair of digits that is duplicated. For example, $1212$ and $6363$ are not PINs, but $1221$ and $6633$ are PINs.

How many distinct possible PINs are there?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9720}
Given \( 0 \leqslant a_{k} \leqslant 1 \) for \( k = 1, 2, \cdots, 2002 \), and let \( a_{2003} = a_{1} \) and \( a_{2004} = a_{2} \), find the maximum value of \( \sum_{k=1}^{20002} \left( a_{k} - a_{k+1} a_{k+2} \right) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1001}
In a regular quadrilateral pyramid $P-A B C D$, the dihedral angle between a lateral face and the base is $\alpha$, and the dihedral angle between two adjacent lateral faces is $\beta$. Find the value of $2 \cos \beta + \cos 2 \alpha$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
What is the largest integer $n$ for which

$$
\frac{\sqrt{7}+2 \sqrt{n}}{2 \sqrt{7}-\sqrt{n}}
$$

is an integer?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{343}
Mom made 32 chocolate chip cookies. It took some time to mix the ingredients and 16 minutes to bake. Julie and Matt ate 9 cookies after dinner. There were 23 cookies left. How many minutes did it take to mix the ingredients?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
The sum of positive numbers \(a, b, c,\) and \(d\) does not exceed 4. Find the maximum value of the expression

\[
\sqrt[4]{a^{2}(a+b)}+\sqrt[4]{b^{2}(b+c)}+\sqrt[4]{c^{2}(c+d)}+\sqrt[4]{d^{2}(d+a)}
\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4 \sqrt[4]{2}}
Determine the smallest integer \( n > 4 \) such that there exists a graph with \( n \) vertices, no triangles, and for every pair of non-adjacent vertices, there are exactly two vertices adjacent to both.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
Determine how the prime number 2 splits in the splitting field of the polynomial \(X^3 - 3X + 1 = 0\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(2)}