inclusionAI/Ring-lite-rl-data huggingface dataset:
Jeannie hikes the 12 miles to Mount Overlook at a pace of 4 miles per hour on level ground. The trail up the mountain has an elevation gain of 3,000 feet. Her pace decreases by 1 mile per hour for every 1,000 feet of elevation gain. After reaching Mount Overlook, she continues to hike another 9 miles to Lake Astonishing at a pace of 4 miles per hour but with 2,000 feet of elevation loss. Her pace increases by 1 mile per hour for every 1,000 feet of elevation loss. She then returns to her starting point on a different trail that is 15 miles long with 5,000 feet of elevation change (3,000 feet up and 2,000 feet down). Her uphill pace decreases by 1 mile per hour for every 1,000 feet of elevation gain, and her downhill pace increases by 1 mile per hour for every 1,000 feet of elevation loss.

How long did her entire hike take, in hours?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{23.5}
On the sides \(AB\) and \(AD\) of square \(ABCD\), points \(E\) and \(F\) are marked such that \(BE : EA = AF : FD = 2022 : 2023\). Segments \(EC\) and \(FC\) intersect the diagonal \(BD\) of the square at points \(G\) and \(H\), respectively. Find the ratio \(GH : BD\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{12271519}{36814556}}
What is the maximum number of rooks that can be placed in an \(8 \times 8 \times 8\) cube so that they do not attack each other?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{64}
The distance from a point $A$ on the parabola $y^{2}=4x$ to its focus $F$ is $4$. What is the slope of line $AF$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{±\sqrt{3}}
If n is an integer, f(n) = f(n-1) - n. The value of f(6) is 1. What is the value of f(n-1) when f(n) = 4?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
The axis cross-section $SAB$ of a cone with an equal base triangle side length of 2, $O$ as the center of the base, and $M$ as the midpoint of $SO$. A moving point $P$ is on the base of the cone (including the circumference). If $AM \perp MP$, then the length of the trajectory formed by point $P$ is ( ).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{\sqrt{7}}{2}$}
In $\vartriangle ABC, \angle A = 52^o$ and $\angle B = 57^o$. One circle passes through the points $B, C$, and the incenter of $\vartriangle ABC$, and a second circle passes through the points $A, C$, and the circumcenter of $\vartriangle ABC$. Find the degree measure of the acute angle at which the two circles intersect.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{59}
Dana creates a "rectangular spiral" by drawing line segments of lengths, in cm, of \(1,1,2,2,3,3,4,4, \ldots\). Dana's pen runs out of ink after the total of all the lengths he has drawn is \(3000 \, \text{cm}\). Find the length of the longest line segment that Dana draws. Express your answer as a single integer (length in cm).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{54}
In $\triangle ABC$, $\angle ABC = 90^{\circ}$, and $BC = 8$. Points $D$ and $E$ lie on $AB$ and $AC$ respectively, such that $CD = DE$ and $\angle DCB = \angle EDA$. If the area of $\triangle EDC$ is 50, what is the length of $AB$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{56}
Find the largest value for the constant $\beta$ in the inequality: $$xy^2 \ge 3x^3+\beta y^3$$ assuming $x,y$ are positive numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2/9}
Find the value of \(m + n\) where \(m\) and \(n\) are integers defined as follows: The positive difference between the roots of the quadratic equation \(5x^2 - 9x - 14 = 0\) can be expressed as \(\frac{\sqrt{m}}{n}\), with \(m\) not divisible by the square of any prime number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{366}
Let  $ABC$  be a triangle such that  $AB = 7$ , and let the angle bisector of  $\angle BAC$  intersect line  $BC$  at  $D$ . If there exist points  $E$  and  $F$  on sides  $AC$  and  $BC$ , respectively, such that lines  $AD$  and  $EF$  are parallel and divide triangle  $ABC$  into three parts of equal area, determine the number of possible integral values for  $BC$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
Anička received a rectangular cake for her birthday. She cut the cake with two straight cuts. The first cut was made such that it intersected both longer sides of the rectangle at one-third of their length. The second cut was made such that it intersected both shorter sides of the rectangle at one-fifth of their length. Neither cut was parallel to the sides of the rectangle, and at each corner of the rectangle, there were either two shorter segments or two longer segments of the divided sides joined.

Anička ate the piece of cake marked in grey. Determine what portion of the cake this was.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2}{15}}
Solve the equation:
$$(1 + x + x^ 2 )(1 + x + . . . + x^{10}) = (1 + x + . . . + x^6 ) ^2.$$
Please write the sum of all solutions for $x$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
在正方体的8个顶点中任取3个顶点构成一个三角形，求该三角形为等腰三角形的概率。答案的格式为\frac{k}{m}，请给出k+m的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
If a = 105 and a ^ 5 = 3^3 × 5^2 × 7^2 × 11^2 × 13 × c, what is the value of c?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{385875}
The area of the figure formed by the parabola $y=x^2$ and its tangent line at $x=1$ and the x-axis is \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{12}}
A polyhedron with vertices at the midpoints of the edges of a cube is called a cuboctahedron. The cross-section of a cuboctahedron with a plane results in a regular polygon. What is the maximum number of sides this polygon can have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Nathaniel and Obediah play a game in which they take turns rolling a fair six-sided die and keep a
running tally of the sum of the results of all rolls made. A player wins if, after he rolls, the number on the running tally is a multiple of 7. Play continues until either player wins, or else indenitely. If Nathaniel goes first, determine the probability that he ends up winning.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{5}{11}}
Use the division algorithm to find the gcd of $f(x)= 3x^3 + 2x^2 -x +1$ and $g(x)= x^2 + 2x -3$ in $\mathbb Z[x]$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
For nonnegative integer $n$, the following are true:
- $f(0) = 0$
- $f(1) = 1$
- $f(n) = f\left(n - \frac{m(m-1)}{2}\right) - f\left(\frac{m(m+1)}{2} - n\right)$ for integer $m$ satisfying $m \geq 2$ and $\frac{m(m-1)}{2} < n \leq \frac{m(m+1)}{2}$.

Find the smallest integer $n$ such that $f(n) = 4$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{646}
Let $g_{1}(x)=\frac{1}{3}\left(1+x+x^{2}+\cdots\right)$ for all values of $x$ for which the right hand side converges. Let $g_{n}(x)=g_{1}\left(g_{n-1}(x)\right)$ for all integers $n \geq 2$. What is the largest integer $r$ such that $g_{r}(x)$ is defined for some real number $x$ ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
The new paint can was weighed 24 kilograms (kg). After using half the paint, I reweighed the can and it weighed 14 kilograms (kg). What is the weight in kilograms (kg) of the paint in the new can, excluding the weight of the can?\
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
If a, b, c, d, e, f, g, h, i, and j are integers and (ab + cdef + ghij) < 0, then what is the maximum number w of integers that can be negative?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
There are 15 members in a family. It takes 180 days for a woman to complete the work, however, a man completes it in 120 days. They take 17 days to complete the work if men work on alternate days and women work every third day. If all of them started together on the 1st day, how many women are there in the family?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Given the following numbers: 20172017, 20172018, 20172019, 20172020, and 20172021. Is there a number among them that is relatively prime to all the others? If so, which one?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20172019}
已知 $a_{n}=\mathrm{C}_{200}^{n} \cdot(\sqrt[3]{6})^{200-n} \cdot\left(\frac{1}{\sqrt{2}}\right)^{n}(n=1,2, \cdots, 95)$, 则数列 $\left\{a_{n}\right\}$ 中整数项的个数为 $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
Given that ABCD is a unit square and P is the midpoint of side AB, the square is folded along PC and PD to form a tetrahedron PACD (where A coincides with B). Calculate the volume of tetrahedron PACD. Express your answer as a fraction with a square root.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{3}}{24}}
Given that the function f(x) has two zero points (x1, x2) where x1 < 0 < x2 < 1, and the function g(x) = x - ln(x^2), find the maximum number of real roots of the equation f[g(x)] = 0. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
In the geometric sequence $\{a_n\}$, if $a_2a_5 = -\frac{3}{4}$ and $a_2 + a_3 + a_4 + a_5 = \frac{5}{4}$, calculate the value of $\frac{1}{a_2} + \frac{1}{a_3} + \frac{1}{a_4} + \frac{1}{a_5}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\frac{5}{3}}
From the set $\{1, 2, 3, 4, \ldots, 20\}$, select four different numbers $a, b, c, d$ such that $a+c=b+d$. If the order of $a, b, c, d$ does not matter, calculate the total number of ways to select these numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{525}
At exactly noon, a truck left a village for the city, and at the same time, a car left the city for the village. If the truck had left 45 minutes earlier, they would have met 18 kilometers closer to the city. If the car had left 20 minutes earlier, they would have met $k$ kilometers closer to the village. Find $k$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Baker made some cakes and 148 pastries. He sold 15 cakes and 103 pastries. Baker still has 45 pastries. How many cakes did Baker make initially?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
Determine the limiting value of the infinite product $$P = \prod_{k=2}^{\infty} \frac{k^3 - 1}{k^3 + 1}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2/3}
Positive integers \(a\), \(b\), and \(c\) are randomly and independently selected with replacement from the set \(\{1, 2, 3, \cdots, 2010\}\). Find the probability that \(abc + ab + a\) is divisible by \(3\). The original answer is in \(\frac{k}{m}\) format, please give the value of \(k + m\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40}
There are two fair dice and their sides are positive integers \( a_{1}, \ldots, a_{6} \) and \( b_{1}, \ldots, b_{6} \), respectively. After throwing them, the probability of getting a sum of \( 2, 3, 4, \ldots, 12 \) respectively is the same as that of throwing two normal fair dice. Suppose that \( a_{1}+\cdots+a_{6} < b_{1}+\cdots+b_{6} \). What is \( a_{1}+\cdots+a_{6} \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
On the Cartesian plane $\mathbb{R}^{2}$, a circle is said to be nice if its center is at the origin $(0,0)$ and it passes through at least one lattice point (i.e. a point with integer coordinates). Define the points $A=(20,15)$ and $B=(20,16)$. How many nice circles intersect the open segment $A B$ ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Peter knows that the distance an airplane travels between Spain and Russia is 7019 km, while the distance between Spain and Germany is 1615 km. Peter took a flight from Spain, which made a stopover in Germany, while being there, he receives a call from his brother asking him how many kilometers he has to fly to get to Russia and then return to Spain.  What will Peter answer?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12423}
The number $1$ is special. The number $2$ is special because it is relatively prime to $1$. The number $3$ is not special because it is not relatively prime to the sum of the special numbers less than it, $1 + 2$. The number $4$ is special because it is relatively prime to the sum of the special numbers less than it. So, a number bigger than $1$ is special only if it is relatively prime to the sum of the special numbers less than it. Find the twentieth special number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{28}
We take $100$ consecutive natural numbers $a_1, a_2, \ldots, a_{100}$. Determine the last two digits of the number $a_1^8 + a_2^8 + \ldots + a_{100}^8$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
A school's sports department has 2 identical basketballs and 3 identical volleyballs to distribute among 4 different classes, with each class receiving 1 ball. Calculate the total number of different methods to distribute the balls.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Complex numbers \(a\), \(b\), \(c\) form an equilateral triangle with side length 24 in the complex plane. If \(|a + b + c| = 48\), find \(|ab + ac + bc|\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{768}
Armen paid \$190 to buy movie tickets for a group of \( t \) people, consisting of some adults and some children. Movie tickets cost \$5 for children and \$9 for adults. How many children's tickets did he buy?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
A group of adventurers is showing their loot. It is known that exactly 4 adventurers have rubies; exactly 10 have emeralds; exactly 6 have sapphires; and exactly 14 have diamonds. Additionally, it is known that:

- If an adventurer has rubies, they also have either emeralds or diamonds (but not both simultaneously).
- If an adventurer has emeralds, they also have either rubies or sapphires (but not both simultaneously).

What is the minimum number of adventurers that can be in such a group?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
Let $f(x)=x^{4}+14 x^{3}+52 x^{2}+56 x+16$. Let $z_{1}, z_{2}, z_{3}, z_{4}$ be the four roots of $f$. Find the smallest possible value of $|z_{a} z_{b}+z_{c} z_{d}|$ where $\{a, b, c, d\}=\{1,2,3,4\}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Three people are sitting in a row of eight seats. If there must be an empty seat on each side of each person, the number of different seating arrangements is ____.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
Define the annoyingness of a permutation of the first \(n\) integers to be the minimum number of copies of the permutation that are needed to be placed next to each other so that the subsequence \(1,2, \ldots, n\) appears. For instance, the annoyingness of \(3,2,1\) is 3, and the annoyingness of \(1,3,4,2\) is 2. A random permutation of \(1,2, \ldots, 2022\) is selected. Compute the expected value of the annoyingness of this permutation.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2023}{2}}
Find the natural number "a" for which:

$$\sum_{k=0}^n f(a+k)=16(2^n-1)$$

where the function $f$ satisfies the relation

$$f(x+y)=f(x)*f(y)$$

for all natural numbers $x,y$ and $f(1)=2$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
In the Cartesian coordinate system, the point $P\left(a-3,a+3\right)$ cannot be in the ________ quadrant.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{fourth}
In a 6-story building, three people, A, B, and C, enter the elevator on the first floor. Find the total number of possible situations in which exactly one person wants to get off at the 6th floor, and A does not get off on the 2nd floor. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{65}
If \((1.0025)^{10}\) is calculated accurate to five decimal places, what is the digit in the fifth decimal place?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Anna works everyday and gets her salary on a weekly basis. If Anna didn't go to work on 2 days this week, her deducted salary was 985 dollars. What is her usual weekly salary?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$2462.50}
Example 10 Find the smallest positive integer $n$, such that in any two-coloring of the complete graph $K_{n}$ with $n$ vertices, there always exist $m$ monochromatic edges, no two of which share a common endpoint.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3m-1}
There are $22$ chairs around a round table. Find the minimum number $n$ such that for any group of $n$ people sitting at the table, there will always be two people with exactly $2$ or $8$ chairs between them.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Evaluate $\displaystyle\int_{-\infty}^\infty\frac{\sin(t)}{t(1+t^2)}\,\mathrm{d}t.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\pi(1-e^{-1})$}
Evaluate the limit: \[ \lim_{k\rightarrow \infty}\binom{2k+1}{3}\frac{1}{2^{2(2k-1)}}\left[\binom{2k-2}{k-2}-\binom{2k-2}{k-1}\right]^2. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{3\pi}}
For which positive integers \(n\) do there always exist positive integers \(a, b, c, d\) such that \(x < \frac{a}{b} < \frac{c}{d} < y\) and \(a^n + b^n = c^n + d^n\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine the order of the quotient module $M/K$ for the ring $D = \mathbb{Z}[\sqrt{-2}]$ and the free $D$-module $M = D^3$, where $K$ is the submodule generated by $(1, 1 + \sqrt{-2}, 2), (0, 1, 3), (2, 0, 1)$. Provide your answer as a number or expression.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{81}
Evaluate the limit: $$\lim_{n \to \infty} \left( \lim_{x \to 0} \left(1+\sin^2(x)+\sin^2(2x)+\cdots+\sin^2(nx)\right)^\frac{1}{n^3\cdot x^2}\right)$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e^{\frac{1}{3}}}
4. A rectangle of 19 cm $\times$ 65 cm is divided by lines parallel to its sides into squares with a side of 1 cm. Into how many parts will this rectangle be divided if we also draw its diagonal?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1318}
Determine the minimum number of charts required to cover a torus, given that each chart must consist of simply connected open sets.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Determine the maximum number of integer solutions \( k \) to the equation \( p(k) = k^3 \), where \( p(x) \) is a polynomial with integer coefficients and \( p(100) = 100 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Determine the cardinality of the quotient ring of Gaussian integers when factored by a non-zero element with relatively prime components.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a^2 + b^2}
Determine the maximum depth of subnormality for subgroups of a nilpotent group with nilpotency class $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n}
Evaluate the limit: $$\lim_{x \to \frac{\pi}{2}} \biggl|\frac{1}{(1+\sin x)\log(\sin x)}+\frac{1}{\cos^2x} \biggl| $$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{4}}
Let $A$ be a square matrix consisting of $1$s and $0$s. If $A^2=0$, determine the minimum number of $1$s that $A$ can have.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Given the equation of the conic $5x^2 + 6xy + 5y^2 = 8$, find the equation of its auxiliary circle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{x^2 + y^2 = 4}
Calculate the flux of the curl of the vector field \( F(x, y, z) = (x - yz, xz, y) \) across the surface \( S \), which is obtained by rotating the graph of \( x = 2^z + 3^z \) for \( z \in [0, 1] \) around the \( z \)-axis, with normal vectors oriented outward.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50\pi}
For any positive integers $a$ and $b$ with $b>1$, let $s_b(a)$ be the sum of the digits of $a$ when it is written in base $b$. Suppose $n$ is a positive integer such that 
$$
\sum_{i=1}^{\lfloor\log_{23}n\rfloor}s_{20}\left(\left\lfloor\frac{n}{23^i}\right\rfloor\right)=103\mathrm{\quad and}\sum_{i=1}^{\lfloor\log_{20}n\rfloor}s_{23}\left(\left\lfloor\frac{n}{20^i}\right\rfloor\right)=115
$$
Compute $s_{20}(n) - s_{23}(n)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
81
}
Consider a random walk on a finite state space $0, 1, \\ldots, N$ with absorbing barriers at states $0$ and $N$. At each state $1, \\ldots, N-1$, the probability of moving to the adjacent states is $\\frac{1}{2}$ each. At states $0$ and $N$, the process is absorbed, meaning it stays in the same state with probability $1$. How many stationary measures does this Markov chain have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Determine the minimal dimension of an affine subspace in \(\mathbb{R}^n\) that can divide an open, contractible set \(U \subseteq \mathbb{R}^n\) into disconnected components, where \(n \ge 3\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n - 1}
How many distinguishable large equilateral triangles can be formed using four congruent equilateral triangles, each with a different color on each side, considering that the colors on both sides of the triangles can be used?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Given that \(a < b\) and \(\frac{f(b)/b}{f(a)/a}<2\), find a tighter upper bound for the expression \(\frac{a}{b}+\frac{b}{a}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{5}{2}}
Evaluate the limit \( \lim_{n\to\infty}\int_{1}^{1+\delta}\frac{n}{x^2(1+x^n)}dx \) using Lebesgue's Dominated Convergence Theorem or the Sandwich Theorem, where \( \delta > 0 \) is a fixed number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\ln 2}
Find the values of \( \alpha \) for which the matrix \( A_\alpha \) has a repeated eigenvalue.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Determine the number of distinct solutions to the equation \( xyz + yvv = xxzy \), where each letter represents a different digit from 0 to 9. Each digit must be unique to a letter.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}