inclusionAI/Ring-lite-rl-data huggingface dataset:
Triangle \( ABC \) has \( AB=24 \), \( AC=26 \), and \( BC=22 \). Points \( D \) and \( E \) are located on \( \overline{AB} \) and \( \overline{AC} \), respectively, so that \( \overline{DE} \) is parallel to \( \overline{BC} \) and contains the center of the inscribed circle of triangle \( ABC \). Calculate \( DE \) and express it in the simplest form.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{275}{18}}
A volunteer organization is recruiting new members throughout the year. In the fall, they manage to increase their number by 5%. In the winter, membership grows by 14%. By the spring, however, membership falls by 19%. Finally, in the summer, membership decreases further by 7%. What is the overall percentage change in membership from the beginning of fall to the end of summer?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-9.83\%}
In a cycling competition with  $14$  stages, one each day, and  $100$  participants, a competitor was characterized by finishing  $93^{\text{rd}}$  each day.What is the best place he could have finished in the overall standings? (Overall standings take into account the total cycling time over all stages.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Given the function $f(x) = |2^{x} - 2|, (x \in (-1, 2))$, find the range of the function $y = f(x - 1)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[0, 2)}
A cyclist rides a bicycle 7 km at an average speed of 10 km/hr, then travels 10 km at an average speed of 7 km/hr. After that, the cyclist covers another 5 km at an average speed of 12 km/hr, and finally rides 8 km at an average speed of 6 km/hr. What is the average speed for the entire trip?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7.73}
25. Arrange the numbers $0,0,1,1,2,2,4,4$ in a row (0 can be placed first), so that no four consecutive numbers form 2014. There are $\qquad$ ways.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2401}
What is the sum of $x + y$ if the sequence $3, ~8, ~13, \ldots, ~x, ~y, ~33$ forms an arithmetic sequence?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{51}
In the context of the Joint Entrance Examination (JEE, a national level entrance test), determine the largest non-negative integer \( a \) for which the following limit holds:

\[
\lim_{x \to 1} \left( \dfrac{-ax + \sin(x-1) + a} { x + \sin(x-1) -1 } \right)^{\dfrac{1-x}{1-\sqrt x} } = \dfrac 1 4
\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Consider two fields with rectangular shapes rather than squares. The area of the first rectangular field is 1 hectare, and the second field has 1% greater width than the first field with a length that is 5% less than the length of the first field. Determine the difference in area between these two fields, given that the width of the first field is proportional to the square root of its length.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{405}
Determine the number of pairs \((a, b)\) of natural numbers such that \(b\) is a 3-digit number, \(a+1\) divides \(b-1\), and \(b\) divides \(a^2 + a + 2\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
The Crackhams were supposed to make their first stop in Baglmintster and spend the night at a family friend's house. This family friend in turn was to leave home at the same time as them and stop in London at the Crackhams' house. Both the Crackhams and the friend were traveling on the same road, looking out for each other, and they met 40 km from Baglmintster. That same evening, George came up with the following small puzzle:

- I realized that if, upon arrival at our destinations, each of our cars immediately turned back and set off in the opposite direction, we would meet 48 km from London.

If George is correct, what is the distance from London to Baglmintster?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{72 \text{ km}}
Determine the value of the discretization step $h$ for which the Euler explicit method is stable when applied to the differential equation $y'(x) + \frac{1}{4}y(x) = x$ with the initial condition $y(1) = 1$. Provide your answer as a specific value of $h$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Given a tetrahedron with vertices at coordinates \((3,1,5), (-7,0,13), (27,5,-13), (-17,4,21)\), find the minimum distance required to visit each vertex exactly once without crossing through the interior of the tetrahedron. Round your answer to the nearest hundredth.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{56.53}
Six boys stood equally spaced on a circle of radius 40 feet. Each boy walked to all of the other non-adjacent persons on the circle, shook their hands and then returned to his original spot on the circle before the next boy started his trip to shake hands with all of the other non-adjacent boys on the circle. After all six boys had done this, what is the least distance in feet that could have been traveled? Express your answer in simplest radical form.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{480 + 480\sqrt{3}\text{ feet}}
5. Natural numbers divisible by 3 were painted in two colors: red and blue, such that the sum of a blue and a red number is red, and the product of a blue and a red number is blue. In how many ways can the numbers be colored so that the number 546 is blue?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Calculate the minimal polynomial of the matrix $E(i,j) = I_n + \alpha I_{(i,j)}$, where $I_n$ is the identity matrix of order $n$, $I_{(i,j)}$ is the matrix with 1 in the $(i,j)$ position and 0 elsewhere ($i \neq j$), and $\alpha \in \mathbb{K}$ with $\mathbb{K}$ a field, $\alpha \neq 0$. Provide your answer in terms of $n$, $i$, $j$, and $\alpha$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(x - 1)^2}
In the rectangular coordinate plane $xOy$, the angle $\angle MON$ is formed by the ray $y = x$ $(x \geq 0)$ and the positive x-axis. Points $A(6,5)$ and $B(10,2)$ are fixed points within $\angle MON$. Points $P$ and $Q$ are movable points on the two sides of $\angle MON$ respectively. Determine the minimum value of the perimeter $l$ of the quadrilateral $ABQP$. \( \boxed{\text{Minimum Perimeter}} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5 + \sqrt{241}}
On $2020 \times 2021$ squares, cover the whole area with several L-Trominos and S-Tetrominos (Z-Tetrominos) along the square so that they do not overlap. The tiles can be rotated or flipped over. Find the minimum possible number of L-Trominos to be used.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2020}
Find the coordinates of the foci of the hyperbola $\frac{x^2}{8} - \frac{y^2}{17} = 1$. Express your answer as coordinates (x, y), including the ± symbol if necessary.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(±5,0)}
In a certain company, there are 100 shareholders, and any 66 of them own no less than 50% of the company's shares. What is the maximum percentage of all shares that one shareholder can own?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25}
What is the largest integer $n$ with no repeated digits that is relatively prime to $6$? Note that two numbers are considered relatively prime if they share no common factors besides $1$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{987654301}
Determine the number of distinct 7-permutations of the multiset \( \mathcal{M} = \{0^1,\ldots, 0^7,1^1,\ldots, 1^7, \cdots,9^1,\ldots, 9^7  \} \), where each digit from 0 to 9 appears 7 times. Two permutations are considered indistinguishable if they differ only by the exchange of identical digits. Calculate this number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10000000}
Determine the second homotopy group of the space obtained by removing a line and a point (where the line does not intersect the point) from \( \mathbb{R}^3 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{Z}}
A binary message is emitted by a source, where the probability of a '1' appearing is 0.6 and the probability of a '0' appearing is 0.4. The interpreter of the message incorrectly interprets 10% of the symbols. If the interpreter receives the message "101", what is the probability that this is the same message that was sent by the source? Provide your answer to three decimal places.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.743}
Evaluate $\int^4_1\frac{x-2}{(x^2+4)\sqrt x}dx$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
0}
Determine the dimension of the quotient space $c_0/M$, where $c_0 = \{ (x_n) : x_n \in \mathbb{R}, x_n \to 0 \}$ and $M = \{(x_n) \in c_0 : x_1 + x_2 + \cdots + x_{10} = 0\}$. What is $\dim(c_0/M)$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Calculate the flux of the vector field \( \mathbf{F} = (3x^3y^2 + 3)\mathbf{i} + \left(\frac{y^2 + 2x}{3}\right)\mathbf{j} + (3yz^3 + 3)\mathbf{k} \) through the box with opposite corners at \((0, 1, 0)\) and \((2, 2, 1)\), where the faces are parallel to the coordinate planes. Use the divergence theorem to find the correct flux value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{67}
In a hockey tournament with $n$ teams, each team plays every other team exactly once. A win awards 2 points, a draw gives 1 point to each team, and a loss gives 0 points. Determine the maximum possible point difference between two consecutively ranked teams.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n}
Find the value of \( a \) such that \( l = m \), where \( l = \lim_{x \rightarrow a}\frac{x^x-a^x}{x-a} \) and \( m = \lim_{x \rightarrow a}\frac{a^x-x^a}{x-a} \), given that \( a > 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e^2}
Find the outward flux of the vector field \( V = x(x^2+y^2+z^2)^{-3/2}\mathbf{i} + y(x^2+y^2+z^2)^{-3/2}\mathbf{j} + z(x^2+y^2+z^2)^{-3/2}\mathbf{k} \) through the boundary of the rectangular box \( R \), where \( R \) consists of all points \((-1 \leq x \leq 1, -1 \leq y \leq 1, -1 \leq z \leq 1)\). Note that the vector field is not defined at the origin, so the Divergence Theorem cannot be directly applied to the box and its interior.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4\pi}
Given the differential equation $y'(t) = \frac{t}{4} - \frac{1}{y^2 + 1}$ with the initial condition $y(0) = 1$, and the function $x(t) = 1 - \frac{t}{2}$, approximate $|x(t) - y(t)|$ for $|t| < \frac{1}{2}$ using a corollary of Gronwall's theorem.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{t^3}{48}}
Evaluate the integral \( \int_0^a\!\!\!\int_0^a\frac{dx\,dy}{(x^2+y^2+a^2)^{\frac32}} \) using polar coordinates.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{6a}}
Determine the number of roots of the equation: \[ 10^{\lg \cos x} = \frac{1}{7} x - 1 \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Find the nine-digit number of the form \( \overline{a_{1} a_{2} a_{3} b_{1} b_{2} b_{3} a_{1} a_{2} a_{3}} \) that is equal to the square of the product of five different prime numbers, where \( \overline{b_{1} b_{2} b_{3}} = 2 \cdot \overline{a_{1} a_{2} a_{3}} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{225450225}
Find the coefficient \( a_{2^n} \) in the power series expansion of \( \left(\frac{x}{1-x^2}+\frac{3x^3}{1-x^6}+\frac{5x^5}{1-x^{10}}+\frac{7x^7}{1-x^{14}}+\cdots\right)^2 = \sum_{i=0}^{\infty}a_{i}x^i \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8^{n-1}}
Compute the integral \( \int_R (x^2-y^2)\,dx\,dy \) over the region \( R \) in the first quadrant defined by the inequalities \( 3 \geq y-x \geq 0 \) and \( 5 \geq xy \geq 2 \). Use the change of variables \( u = y-x \) and \( v = xy \) to evaluate the integral.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{27}{2}}
In the $xy$-coordinate plane, the $x$-axis and the line $y=x$ act as mirrors. A laser beam is shot from the point $(126, 21)$ towards a point on the positive $x$-axis. There are three points on the $x$-axis where the beam can be aimed such that it will bounce off the mirrors and return to $(126, 21)$. These points are $(126, 0)$, $(105, 0)$, and a third point $(d, 0)$. Find the value of $d$. (Note: When light bounces off a mirror, the angle of incidence equals the angle of reflection.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{111}
Determine the values of \( \alpha \) for which the series \( \sum_{n=0}^{\infty} \left( \frac{\alpha n}{n+1} \right)^n \) converges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-1, 1)}
Consider the double integral \( \int_0^{\infty} \int_0^{\infty} f(xy) g(x,y) \, dx \, dy \). Perform the substitution \( y=\frac{t}{x} \) with \( dy=\frac{dt}{x} \). Determine the correct bounds for the variable \( t \) in the transformed integral.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[0, \infty)}
In triangle $ABC$, a midline $MN$ that connects the sides $AB$ and $BC$ is drawn. A circle passing through points $M$, $N$, and $C$ touches the side $AB$, and its radius is equal to $\sqrt{2}$. The length of side $AC$ is 2. Find the sine of angle $ACB$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{2}}
Given that \(a_{1}, a_{2}, a_{3}, a_{4}\) are positive real numbers, satisfying:
$$
\sum_{i=1}^{4} i a_{i} \leqslant 10, \quad a_{i} \geqslant \frac{1}{2^{4-i}} \quad (i=1, 2, 3, 4).
$$
Define \(f=\sum_{i=1}^{4} \frac{1}{1+a_{i}^{i}}\). Find the minimum value of \(f\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
In triangle $ABC$, the lateral sides $AB$ and $BC$ are equal. A line parallel to the base $AC$ intersects side $AB$ at point $D$ and side $BC$ at point $E$, and each of the segments $AD$, $EC$, and $DE$ is equal to 2. Point $F$ is the midpoint of segment $AC$, and point $G$ is the midpoint of segment $EC$, connected by a straight line segment. It is known that the measure of angle $GFC$ is $\beta$. Find the area of triangle $ABC$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(1+2\cos2\beta)^{2}\tan2\beta}
3. If real numbers $a, b, x$ and $y$ satisfy the system of equations
$$
\left\{\begin{array}{l}
a x+b y=3, \\
a x^{2}+b y^{2}=7, \\
a x^{3}+b y^{3}=6, \\
a x^{4}+b y^{4}=42 .
\end{array}\right.
$$

Find the value of $a x^{5}+b y^{5}$.
(8th American Invitational Mathematics Examination)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
For the power function $y = (m^2 - m - 1)x^{2m+1}$, if it is a decreasing function for $x \in (0, +\infty)$, then the value of the real number $m$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
Today, Ivan the Confessor prefers continuous functions $f:[0,1] \rightarrow \mathbb{R}$ satisfying $f(x)+f(y) \geq|x-y|$ for all pairs $x, y \in[0,1]$. Find the minimum of $\int_{0}^{1} f$ over all preferred functions. The original answer is in \frac{m}{n} format, please give the value of m + n.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Professors Alpha, Beta, Gamma, and Delta choose their chairs so that each professor will be between two students. Given that there are 13 chairs in total, determine the number of ways these four professors can occupy their chairs.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1680}
25. $[\mathbf{1 4}]$ Let $n$ be an integer greater than 3 . Let $R$ be the set of lattice points $(x, y)$ such that $0 \leq x, y \leq n$ and $|x-y| \leq 3$. Let $A_{n}$ be the number of paths from $(0,0)$ to $(n, n)$ that consist only of steps of the form $(x, y) \rightarrow(x, y+1)$ and $(x, y) \rightarrow(x+1, y)$ and are contained entirely within $R$. Find the smallest positive real number that is greater than $\frac{A_{n+1}}{A_{n}}$ for all $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2+\sqrt{2}}
Problem 3. Points $A$ and $B$, located on a circular avenue, are connected by a straight highway segment 4 km long, which is the diameter of the circular avenue. A pedestrian left his house and started walking along the avenue from point $A$. After 1 hour, he realized he had forgotten his keys and asked a neighbor-cyclist to bring them as quickly as possible. What is the minimum time it will take for the pedestrian to receive the keys if the cyclist's speed on the highway is 15 km/h, on the avenue - 20 km/h, and the pedestrian's speed is 7 km/h? The pedestrian can walk towards the cyclist.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{4\pi-7}{27}}
Let $M$ and $N$ be any two points on the parabola $C$: $y^{2}=2px(p > 0)$. The coordinates of point $E$ are $(-\lambda,0)$ where $\lambda > 0$. If the minimum value of $\overrightarrow{EM}\cdot \overrightarrow{EN}$ is $0$, then $\lambda=$ ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{p}{2}}
How many  $5$ -digit numbers  $N$  (in base  $10$ ) contain no digits greater than  $3$  and satisfy the equality  $\gcd(N,15)=\gcd(N,20)=1$ ? (The leading digit of  $N$  cannot be zero.)

*Based on a proposal by Yannick Yao*
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{256}
There are $n$ parking spaces along a road, and $n$ drivers each driving a car. Each driver parks their car in their preferred parking space. If their preferred space is already occupied, they park in the nearest available space along the road. If both their preferred and subsequent spaces are occupied, they drive away and do not park. How many sequences $\left(a_{1}, a_{2}, \cdots, a_{n}\right)$ exist such that every parking space is occupied? Here, $a_{i}$ represents the preferred parking space of the $i$-th driver, and the values of $a_{1}, a_{2}, \cdots, a_{n}$ do not need to be distinct.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(n+1)^{n-1}}
A chemical factory needs to sequentially add two types of chemical raw materials during production. It is known that there are 5 types of raw materials available, but materials A and B cannot be used at the same time, and when adding materials in sequence, if material A is used, it must be added first. Find the total number of different adding schemes available. Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
Given a right quadrilateral pyramid \( V-ABCD \) with the height equal to half the length of \( AB \), \( M \) is the midpoint of the lateral edge \( VB \), and \( N \) is a point on the lateral edge \( VD \) such that \( DN=2VN \). Find the cosine of the angle between the skew lines \( AM \) and \( BN \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{11}}{11}}
The flag-raising time at Dongfanghong Elementary School in 2012 varies depending on the date as follows: From January 1 to January 10, it is fixed at 7:13 AM. From January 11 to June 6, it gradually advances from 7:13 AM to 4:46 AM, advancing by 1 minute every day. From June 7 to June 21, it is fixed at 4:46 AM. What is Dongfanghong Elementary School's flag-raising time today (March 11)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6:12}
5. We define a "table" of type $\left(n_{1}, n_{2}, \ldots, n_{m}\right), n_{1} \geq n_{2} \geq \ldots \geq n_{m}$ as follows: We place $n_{1}$ squares horizontally next to each other, then we place $n_{2}$ squares horizontally next to each other below the $n_{1}$ squares starting from the first square on the left. We continue this process until we get a "table" with $n_{1}$ squares in the first row, $n_{2}$ squares in the second row, $\ldots, n_{m}$ squares in the $m$-th row, with a total of $n=n_{1}+n_{2}+\ldots+n_{m}$ squares. The rows of placed squares always start from the square below the leftmost square of the first row (see example). We fill the obtained "table" with numbers from 1 to n such that in each row and each column, the numbers from left to right and from top to bottom, respectively, are in increasing order. Below is an example of a table of type $(5,4,2,1)$ and one of its fillings.

| 1 | 3 | 4 | 7 | 11 |
| :---: | :---: | :---: | :---: | :---: |
| 2 | 5 | 10 | 12 |  |
| 6 | 9 |  |  |  |
| 8 |  |  |  |  |
|  |  |  |  |  |

Determine the number of fillings of a "table" of type $(4,3,2)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{168}
Let $C_1$ and $C_2$ be circles defined by:
\[
(x - 10)^2 + y^2 = 36
\]
and
\[
(x + 15)^2 + y^2 = 81
\]
respectively. Find the length of the shortest line segment $\overline{PQ}$ that is tangent to $C_1$ at $P$ and to $C_2$ at $Q$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
It takes 40 minutes to freeze ice cubes. Once they're frozen, it takes 3 minutes per smoothie to turn them into smoothies. Additionally, you have to chop fruits for each smoothie: 2 minutes for apples, 3 minutes for bananas, 4 minutes for strawberries, 5 minutes for mangoes, and 6 minutes for pineapples. Assuming you need 1 of each fruit for 5 different smoothies, how long does it take to make the 5 smoothies?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{115}
Evaluate the limit:
$$\lim_{n \to \infty} n\int_0^1 \frac{x^n}{4x + 5} \, dx$$
Given that the sequence $(I_n)_{n \geq 1}$ is defined by $I_n = \int_0^1 \frac{x^n}{4x + 5} \, dx$ and is known to be decreasing and convergent to $0$, determine the limit's value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1/9}
Given \( A, B, C, D \in\{1, 2, \cdots, 6\} \), and each is distinct from the others. If the curves \( y = Ax^{2} + B \) and \( y = Cx^{2} + D \) intersect, then there are \(\quad\) different ways to choose \( A, B, C, D \) (the intersection of the curves is independent of the order of \( A, B, C, D \); for example, \( A=3, B=2, C=4, D=1 \) and \( A=4, B=1, C=3, D=2 \) are considered the same).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{90}
Let's say the function q is defined by the formula q = (5 w / (4 h (z ^ 2))) + kx, where k is a constant and x is an additional variable. By what factor will q be multiplied if w is quadrupled, h is doubled, z is tripled, and x is halved, while keeping k constant?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2/9}
Find the smallest positive integer \( n \) such that if \( n \) squares of a \( 1000 \times 1000 \) chessboard are colored, then there will exist three colored squares whose centers form a right triangle with sides parallel to the edges of the board.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1999}