inclusionAI/Ring-lite-rl-data huggingface dataset:
A company takes out a loan of a hundred thousand yuan at the beginning of this year, with an annual interest rate of r. They plan to repay a fixed amount at the end of each year and to pay off the loan within five years. Calculate the annual repayment amount in terms of a and r, using a geometric series approach. Express your answer as a formula.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{ar(1+r)^5}{(1+r)^5-1}}
How many solutions does the equation $\tan{(2x)} = \cos{\left(\frac{x}{2}\right)}$ have on the interval $[0, 2\pi]$? Write your answer as an integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Circle $S_1$ has a radius of $5$. Circle $S_2$ has a radius of $7$ and its center lies on $S_1$. Circle $S_3$ has an integer radius and its center lies on $S_2$. If the center of $S_1$ lies on $S_3$, how many possible values are there for the radius of $S_3$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
In the land of Chaina, people pay each other in the form of links from chains. Fiona, originating from Chaina, has an open chain with  $2018$  links. In order to pay for things, she decides to break up the chain by choosing a number of links and cutting them out one by one, each time creating  $2$  or  $3$  new chains. For example, if she cuts the  $1111$ th link out of her chain first, then she will have  $3$  chains, of lengths  $1110$ ,  $1$ , and  $907$ . What is the least number of links she needs to remove in order to be able to pay for anything costing from  $1$  to  $2018$  links using some combination of her chains?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Evaluate the limit \( \lim_{n \to \infty }\frac{x_{n+1}}{x_{n}} \), where \( x_{n}=\int_{0}^{1}x^{n}(1-x)^{n}dx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{4}}
Use Horner's method to calculate the value of $v_1$ for the polynomial function $f(x) = 3x^4 + 2x^2 + x + 4$ when $x=10$. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
Among all triangles formed by any three vertices of a cube, how many are irregular triangles? Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
Evaluate the first three digits to the right of the decimal point in the decimal representation of $\left(10^{987} + 1\right)^{8/3}$ using the Binomial Expansion.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{666}
Let $x$ and $y$ satisfy the constraint conditions. If the maximum value of the objective function $z = abx + y$ (with $a>0$, $b>0$) is 8, then the minimum value of $a+b$ is _________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4\sqrt{2}}
It is known that the optimal addition amount of a certain material is between 10g and 110g. If the 0.618 method is used to arrange the experiment, the addition amount for the second trial can be ___ g.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{48.2}
The hypotenuse of a right triangle whose legs are consecutive even numbers is 50 units. What is the sum of the lengths of the two legs?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{70}
A circle touches the extensions of two sides \( AB \) and \( AD \) of a square \( ABCD \) with a side length of 4 cm. From point \( C \), two tangents are drawn to this circle. Find the radius of the circle if the angle between the tangents is \( 60^{\circ} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4 (\sqrt{2} + 1)}
In a regular triangular prism, the area of the cross-section passing through the lateral edge and perpendicular to the opposite lateral face is $Q$. The side length of the prism's base is $a$. Find the total surface area of the prism.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{3}\left(0.5 a^{2}+2 Q\right)}
In a certain variety show, there are 3 dance and song performances, 2 comedy skits, and 1 cross-talk performance. The order of the performances needs to be arranged such that performances of the same type are not adjacent. Calculate the number of possible arrangements.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{120}
由 $1,2, \cdots, 9$ 这九个正整数构成的所有圆排列中, 任意相邻两数之积均不超过 $60$ 的圆排列的个数为 $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21600}
Identify the field with 4 elements.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{F}_4}
4. Given the complex sequence $\left\{a_{n}\right\}$ satisfies: $a_{n+1}^{2}-a_{n} a_{n+1}+$ $a_{n}^{2}=0(n \geqslant 0)$, and $a_{n+1} \neq a_{n-1}(n \geqslant 1), a_{1}=1$, then $\sum_{n=0}^{2006} a_{n}$ is $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Person A and Person B ride bicycles simultaneously from point $A$ to point $B$. The speed of Person A's bicycle is 1.2 times the speed of Person B's bicycle. After riding 5 kilometers, Person B's bicycle has a malfunction and the time spent fixing it is equivalent to riding $\frac{1}{6}$ of the total distance. After fixing the malfunction, Person B's speed increases by $60\%$, and both Person A and Person B arrive at point $B$ at the same time. What is the distance between points $A$ and $B$ in kilometers?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{45 \text{ kilometers}}
Evaluate the integral \( \int_{-\infty}^{\infty}\frac{\cos x}{4x^2-\pi^2}dx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{1}{2}}
In a class of 25 students, it is known that the number of boy friends for any two girls in the class does not match. What is the maximum number of girls that can be in this class?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
Determine the range of Kendall's tau that can be used in the Frank copula.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-1, 1)}
A solid right prism $PQRSTU$ has a height of 20, as shown. Its bases are equilateral triangles with side length 15. Points $M$, $N$, and $O$ are the midpoints of edges $PQ$, $QR$, and $RS$, respectively. Determine the perimeter of triangle $MNO$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32.5}
In the cave of Ali Baba, there is a lot of gold and diamonds. A full sack of gold weighs 200 kg, and a full sack of diamonds weighs 40 kg. Ali Baba can carry 100 kg in one trip. A kilogram of gold costs 20 dinars, and a kilogram of diamonds costs 60 dinars. How much money can he get for the gold and diamonds carried in one sack (in one trip)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3000 \ \text{dinars}}
How many sequences of $5$ positive integers $(a, b, c, d, e)$ satisfy the inequality $abcde \leq a + b + c + d + e \leq 10$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{116}
Of the mathematicians with a finite Erdős number (including those who are no longer alive), what is their average Erdős number according to the Erdős Number Project?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4.65}
Determine the center of the $k$-algebra $A(k)$ with generators $x$ and $y$ and the defining relation $yx - xy = 1$, given that the characteristic of $k$ is greater than 0.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{k[x^p, y^p]}
Find the directional derivative of the function \( f(x,y) = \frac{x^3}{x^2 + y^2} \) for \((x,y) \neq (0,0)\) and \( f(0,0) = 0 \) in the direction of \((1,1)\) at the point \((0,0)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\sqrt{2}}{4}}
Evaluate the limit of the sequence as \( n \to \infty \): \[ \lim_{n \to \infty} \frac{\sqrt{n}}{\left|c+\frac{d}{\sqrt{n}}\right|+|c|}\left(\left| c+\frac{d}{\sqrt{n}}\right|-|c|\right) \] where \( c, d \in \mathbb{R} \) and \( c \neq 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{d}{2c}}
Let $z$ be a complex number that satisfies the equation\[\frac{z-4}{z^2-5z+1} + \frac{2z-4}{2z^2-5z+1} + \frac{z-2}{z^2-3z+1} = \frac{3}{z}.\]Over all possible values of $z$, find the sum of the values of\[\left| \frac{1}{z^2-5z+1} + \frac{1}{2z^2-5z+1} + \frac{1}{z^2-3z+1} \right|.\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{11}{6}}
Suppose $f(x)$ and $g(x)$ are nonconstant smooth functions satisfying the equations

$$\frac{f'(x)}{f(x)} + \frac{g'(x)}{g(x)} = 1$$

$$ \frac{f''(x)}{f(x)} + \frac{g''(x)}{g(x)} = \frac{f'''(x)}{f(x)} + \frac{g'''(x)}{g(x)}$$

(assume $f$ and $g$ are always nonzero). Compute the value of

$$ \frac{f''(2015)}{f'(2015)} + \frac{g''(2015)}{g'(2015)}.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2}{3}}
For each integer  $i=0,1,2, \dots$ , there are eight balls each weighing  $2^i$  grams. We may place balls as much as we desire into given  $n$  boxes. If the total weight of balls in each box is same, what is the largest possible value of  $n$ ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
What is the smallest number of points that can be chosen on a circle of length 1956 such that for each of these points there is exactly one chosen point at a distance of 1 and exactly one chosen point at a distance of 2 (distances are measured along the circle)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1304}
Determine the value of the series $-1 + 2 + 3 + 4 - 5 - 6 - 7 - 8 - 9 + \ldots + 10000$, where the signs change after each perfect square.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1000000}
Let $X$ be a smooth projective curve in $\mathbb{P}^2(\mathbb{C})$ of degree $4$ and $p, q, r \in X$. Compute $\dim L(p+q+r)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine the cardinality of the set of all triangular subsets in the Sierpinski triangle, denoted as T, where T is a subset of the power set P(S) of the Sierpinski triangle S, and each element T is a triangle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\aleph_0}
Two 10-digit integers are called neighbours if they diﬀer in exactly one digit (for example, integers $1234567890$ and $1234507890$ are neighbours). Find the maximal number of elements in the set of 10-digit integers with no two integers being neighbours.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9 \cdot 10^8}
A regular 12-gon is inscribed in a circle of radius 12.  The sum of the lengths of all sides and diagonals of the 12-gon can be written in the form $a + b \sqrt{2} + c \sqrt{3} + d \sqrt{6},$ where $a^{}_{}$, $b^{}_{}$, $c^{}_{}$, and $d^{}_{}$ are positive integers.  Find $a + b + c + d^{}_{}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{720}
Calculate the curvature at each point of the Lobachevsky plane \(\mathbb{R}^2_+ = \{(x,y) \in \mathbb{R}^2; y > 0\}\) with the metric given by \(g_{11} = g_{22} = \frac{1}{y^2}, g_{12} = 0\). Use the formula \(K = \frac{R_{1212}}{g_{11}g_{22}-g^2_{12}}\), where \(R_{klij} = \frac{1}{2}(\frac{d^2}{dx^jdx^k}g_{il} + \frac{d^2}{dx^idx^l}g_{jk} - \frac{d^2}{dx^jdx^l}g_{ik} - \frac{d^2}{dx^idx^l}g_{jl} + g_{rs}\Gamma^r_{jk}\Gamma^s_{il} - g_{rs}\Gamma^r_{jl}\Gamma^s_{ik} )\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
An empty fuel tank is filled with brand Z gasoline. When the tank is 3/4 empty, it is filled with brand X gasoline. When the tank is half empty again, it is filled with brand Z gasoline. When the tank is half empty again, it is filled with brand X gasoline. At this time, what percent of the gasoline in the tank is brand Z?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{31.25\%}
A lame king is a chess piece that can move from a cell to any cell that shares at least one vertex with it, except for the cells in the same column as the current cell. A lame king is placed in the top-left cell of a \(7 \times 7\) grid. Compute the maximum number of cells it can visit without visiting the same cell twice (including its starting cell).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{43}
A person has a rectangular piece of paper that is 40 cm by 30 cm. They cut out 10 pieces of paper, each 5 cm by 5 cm, from this rectangle. The sides of these 10 small pieces are parallel to the corresponding sides of the rectangle, and they do not overlap. What is the perimeter of the remaining figure in centimeters?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{240 \, \text{cm}}
At a school, 6 subjects are offered, and students can enroll in any combination of these subjects. It is observed that for any pair of subjects, fewer than 5 students are enrolled in both, and fewer than 5 students are enrolled in neither. What is the maximum possible number of students at the school?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
Find the sum of the series \( \sum_{n=1}^\infty \frac{3^n}{n^3} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
5. Islandland consists of ten islands, some of which are connected by two-way air routes. If you choose any 9 islands, you can fly around them one by one and return to the starting island at the end. Find the minimum number of air routes that can exist in this country.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
Mrs. Choi purchased a house for some amount. Five years later, she sold it for a 20% profit and got a 5% broker's commission from the original price. The house sold for $100,000. What was the original price of the house?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$86,956.52}
Square \(ABCD\) has a side length of 4. Point \(E\) is the midpoint of \(AB\), and \(F\) is a variable point on side \(BC\). Triangles \(\triangle ADE\) and \(\triangle DCF\) are folded along \(DE\) and \(DF\) respectively, so that points \(A\) and \(C\) coincide at point \(A'\). Find the maximum distance from point \(A'\) to the plane \(DEF\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{4\sqrt{5}}{5}}
Petya starts with an acute-angled triangle with angles \(3^\circ\), \(88^\circ\), and \(89^\circ\). He performs an operation called "breaking," where a point \(O\) is placed inside the triangle such that \(OA = OB = OC\), dividing it into three smaller triangles \(OAB\), \(OAC\), and \(OBC\). Petya continues to break one of the smaller acute-angled triangles until all resulting triangles are obtuse-angled. How many triangles does he obtain in total?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
Determine the number of ways to distribute $2n$ objects of 3 different kinds among two people such that each person receives exactly $3n$ objects.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3n^2 + 3n + 1}
Find the coefficient of \(x^{10}\) in the expansion of \(\left(\frac{1 - x^3 + x^4}{1-x}\right)^5\) using generating functions.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{531}
Determine the values of \( n \) for which the cyclotomic extension \( \mathbb{Q}(\zeta_n) \) over \( \mathbb{Q} \) contains the imaginary unit \( i \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4 \mid n}
Oleg drew an empty $50 \times 50$ table and wrote a number above each column and to the left of each row. It turned out that all 100 written numbers are different, with 50 being rational and the remaining 50 being irrational. Then, in each cell of the table, he recorded the sum of the numbers written next to its row and column ("addition table"). What is the maximum number of sums in this table that could be rational numbers?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1250}
Evaluate \( \lim_{n \to \infty} \int_{0}^1 [x^n + (1-x)^n ]^{1/n} \, \mathrm{d}x \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{3}{4}}
Determine the values of the positive parameter $a$ for which the integral \( \int_0^\infty \frac{\sin x}{x^{a}+x^{2a}}\, \mathrm{d}x \) is convergent.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(0, 2)}
Evaluate the integral \( \int_0^{2 \pi} \ln(1+e^{i \theta}) \, d\theta \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Determine the number of 3-tuples \((x, y, z) \in \mathbb{N}^3\) such that \(\mathrm{lcm}(x, y) = \mathrm{lcm}(y, z) = \mathrm{lcm}(z, x) = 2^4 3^3 5^2\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{910}
For a given positive integer $k$, denote the square of the sum of its digits by $f_1(k)$. Define the function recursively as $f_{n+1}(k) = f_1(f_n(k))$. Determine the value of $f_{1991}(2^{1990})$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{256}
A large rectangle consists of three identical squares and three identical small rectangles. The perimeter of a square is 24, and the perimeter of a small rectangle is 16. What is the perimeter of the large rectangle?

The perimeter of a figure is the sum of the lengths of all its sides.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{52}
If an investment is made with a principal of $5000, at an interest rate of 3% per annum, compounded semi-annually, in how many years will the initial sum double?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{23.28}
Two circles with radii 5 and 4 touch externally. A line that is tangent to the smaller circle at point $A$ intersects the larger circle at points $B$ and $C$, such that $A B = B C$. Find $A C$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Using systematic sampling, a survey of 32 individuals is to be conducted from a pool of 960 people. They are randomly assigned numbers from 1 to 960. After grouping, the number 9 is selected through simple random sampling in the first group. Individuals with assigned numbers in the range [1, 460] will complete questionnaire A, those in the range [461, 761] will complete questionnaire B, and the rest will complete questionnaire C. Determine the number of individuals who will receive questionnaire B.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Calculate the sum $$\sum_{k=0}^{n}\binom{n}{k}\left(\binom{m+k+1}{m-n}+(-1)^{k+1}\binom{m+2(n-k)+1}{m}\right)$$ for any integers $m$ and $n$ satisfying $m\ge n\ge 0$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Petya came up with 1004 monic quadratic trinomials \( f_{1}, \ldots, f_{1004} \), among the roots of which all integers from 0 to 2007 appear. Vasya considers all possible equations \( f_{i} = f_{j} \) ( \( i \neq j \) ), and for each root found, Petya pays Vasya one ruble. What is the smallest possible income for Vasya?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Find the number of quadruples $(a, b, c, d)$ of integers with absolute value at most 5 such that $\left(a^{2}+b^{2}+c^{2}+d^{2}\right)^{2}=(a+b+c+d)(a-b+c-d)\left((a-c)^{2}+(b-d)^{2}\right)$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{49}
the average marks of a class of 28 students is 40 and that of another class of 50 students is 60 . find the average marks of all the students ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{52.82}
1. In the cells of a $4 \times 6$ table, natural numbers are arranged such that all ten sums of these numbers in the rows and columns of the table are distinct. Find the smallest possible value of the sum of all the numbers in the table.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{43}
Given that $F_1$ and $F_2$ are the left and right foci of the hyperbola C: $$\frac {x^{2}}{a^{2}}$$- $$\frac {y^{2}}{b^{2}}$$\=1 ($a>0$, $b>0$), and point P is on the right branch of C. If ray PT bisects ∠$F_1$PF$_2$, a line is drawn through the origin O parallel to PT intersecting P$F_1$ at point M, and $|MP|= $$\frac {1}{3}$$|F_1F_2|$, then find the eccentricity of C.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac {3}{2}}
Point $A$ lies on the line $y = \frac{8}{15} x - 10$, and point $B$ lies on the parabola $y = x^2$. What is the shortest length of segment $AB$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2234}{255}}
Given matrices \(A\) and \(B\) of size \(n \times n\) such that \(A^2 = B^2 = 0\) and \(AB = BA\), find the smallest natural number \(x\) such that \((A+B)^x = 0\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Express $y$ in terms of $x$ using the Lambert W function for the equation $y^y = x$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{y = e^{W(\ln x)}}
The Fibonacci numbers are defined recursively by $F_0=0, F_1=1,$ and $F_i=F_{i-1}+F_{i-2}$ for $i \ge 2.$ Given $15$ wooden blocks of weights $F_2, F_3, \ldots, F_{16},$ compute the number of ways to paint each block either red or blue such that the total weight of the red blocks equals the total weight of the blue blocks.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32
}
Given a symmetric graph $G(V,E)$ with $n$ vertices and $m$ edges, and a $2\text{-factor}$ with edge set $F$, represented as an unordered sequence $\big( (u_1,v_1),\,\dots,\,(u_n,v_n)\big)$, determine the complexity of finding a permutation of the $n$ vertices that has the same cycles as $F$. Specifically, store the vertices in an array $\boldsymbol{a}$ such that $(i,\boldsymbol{a}[i])\in F$. What is the complexity of this task?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{O(n)}
For how many ordered pairs of positive integers(x,y),with \( y < x \leq  {100} \) ,are both \( \frac{x}{y} \) and \( \frac{x + 1}{x + 1} \) integers?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
85
}
Determine the smallest integer $n \in \mathbb{N}$ such that for any two sets $\{ x_1, x_2, \ldots, x_n \}$ and $\{ y_1, y_2, \ldots, y_n \}$ of $n$ distinct points on the unit circle $\mathbb{S}^1$, there exists a unique automorphism $f: \mathbb{D} \rightarrow \mathbb{D}$ satisfying $f(x_i) = y_i$ for all $i$. What is the value of $n$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
In the complex vector space \(\mathbb{C}^d\), consider two orthonormal bases \(\mathbf{B_1}\) and \(\mathbf{B_2}\) that are mutually unbiased. For any non-zero vector \(|\alpha\rangle \in \mathbb{C}^d\), determine the maximum number of vectors from the combined set \(\mathbf{B_1} \cup \mathbf{B_2}\) that can be orthogonal to \(|\alpha\rangle\). What is the maximum cardinality of this set?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{d - 1}