inclusionAI/Ring-lite-rl-data huggingface dataset:
Given that \( P(x) \) is a polynomial of degree 9 and satisfies \( P(k) = 2^k \) for each integer \( k \) from 1 to 10, determine the value of \( P(12) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4072}
In the diagram, \(PR, PS, QS, QT,\) and \(RT\) are straight line segments. \(QT\) intersects \(PR\) and \(PS\) at \(U\) and \(V\), respectively. If \(PU = PV\), \(\angle UPV = 24^\circ\), \(\angle PSQ = x^\circ\), and \(\angle TQS = y^\circ\), what is the value of \(x + y\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{78}
Evaluate the integral \( \int_0^1 f(x) \, dx \), where \( f(x) = \sum_{n=0}^{\infty} \frac{[2^n x] - 2[2^{n-1} x]}{3^n} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{4}}
In a right triangle \(ABC\) with angle \(A\) equal to \(30^{\circ}\), a circle with radius \(R\) is inscribed. A second circle, lying outside the triangle, touches the side \(BC\) and the extensions of the other two sides. Find the distance between the centers of these circles.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 R \sqrt{2}}
Determine the radius of convergence of the series: \[ f(z) = \frac{1}{2z^3} + \frac{1}{12z} - \frac{z}{240}. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Suppose that $f(x) = \frac{x}{x^2 - 2x + 2}$ and $g(x_1, x_2, \ldots , x_7) = f(x_1) + f(x_2) + \ldots + f(x_7)$. If $x_1, x_2, \ldots, x_7$ are non-negative real numbers with sum $5$, determine for how many tuples $(x_1, x_2, \ldots , x_7)$ does $g(x_1, x_2, \ldots , x_7)$ obtain its maximal value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
how long does a train 100 m long running at the speed of 65 km / hr takes to cross a bridge 145 m length ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13.57}
In a finite abelian group \( G \), the indices of two subgroups \( H_1 \) and \( H_2 \) are both \( m \), and the index of \( H_1 \) in \( H_2 \) is \( n \). Determine the index of \( H_1 \cap H_2 \) in \( G \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{mn}
Find the Radon-Nikodym derivative \( \frac{d\mu}{dm} \) where \( m \) is the Lebesgue measure on \([0,1]\), \( f(x)=x^2 \), and \( \mu(E)=m(f(E)) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2x}
I am trying to draw a diagonal from a vertex of an icosikaipentagon. How many diagonal lines can you draw?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{22}
How many different seven-digit natural numbers exist such that all digits are distinct, the number does not start with the digit 2, and the number 2018 can be formed by removing some of its digits?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1800}
Evaluate the expression for any positive integer $k$:
\[ T = 1\frac{\left( k \right)! \left( k \right)! }{(2k)!}[ (C_0)^2 -2(C_1)^2 + 3(C_2)^2 +\cdots + (-1)^{2k}(2k+1)(C_{2k})^2 ] \]
where $C_r$ denotes ${}^{n}C_r$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-1)^k (k + 1)}
Find the smallest integer \( n \) such that the system of equations \( \sin x_{1} + \sin x_{2} + \cdots + \sin x_{n} = 0 \) and \( \sin x_{1} + 2 \sin x_{2} + \cdots + n \sin x_{n} = 100 \) has a solution.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
Identify the group $G$ generated by the permutations $(1234)$ and $(3456)$ in the symmetric group $S_6$. The order of $G$ is 120. What is the group $G$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{S_5}
Evaluate the limit of the sequence \( \{y_n\} \) defined recursively by:

\[ y_n = y_{n-1} \cdot \left(3-\frac{\cos^{-1}(n) \cdot n^2 - 2}{4 + n^2}\right) \]

with the initial condition \( y_0 = 1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Determine the largest number among those $A$ values for which the following statement holds. No matter in what order we write down the first 100 positive integers, there will always be ten consecutive numbers in the sequence whose sum is at least $A$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{505}
Find the lower bound for the sum: \[\sum_{n=N^\delta}^N\sqrt{\ln(n)}\] where \(\delta\) is a positive constant less than 1.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\Omega(N \sqrt{\ln N})}
Determine the limit of the sequence defined by the recursive logarithms:
\[ t_1 = \ln(n) \]
\[ t_2 = \ln(n - t_1) = \ln(n - \ln(n)) \]
\[ t_3 = \ln(n - t_2) = \ln(n - \ln(n - \ln(n))) \]
\[ \ldots \]
\[ t_k = \ln(n - t_{k-1}) \]
Find the value to which this sequence converges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n - W(e^n)}
Six soccer teams are competing in a tournament in Waterloo. Every team is to play three games, each against a different team. How many different schedules are possible?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{70}
Given two matrices, $\mathrm{M}$, a $3 \times 2$ matrix, and $\mathrm{N}$, a $2 \times 3$ matrix, such that their product $\mathrm{MN}$ equals the matrix $\begin{pmatrix}8&2&-2\\2&5&4\\-2&4&5\end{pmatrix}$ and $\mathrm{NM}$ is invertible, find the determinant of $\mathrm{NM}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{81}
The distances from point A to point B by the river and by the channel are both equal to 1 km. The speed of the current in the channel is $V$ km/h, while in the river it is $(2V+1)$ km/h. The current in both the river and the channel flows from A to B. If the difference between the travel times of the boat through the channel from B to A and back is added to the travel time of a raft along the river from A to B, the sum is exactly 1 hour. By how many kilometers per hour is the speed of the boat greater than the speed of the current in the channel? The value of $V$ is not given. The answer should be a number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Let $p,q$ be two-digit number neither of which are divisible by $10$. Let $r$ be the four-digit number by putting the digits of $p$ followed by the digits of $q$ (in order). As $p,q$ very, a computer prints $r$ on the screen if $\gcd(p,q) = 1$ and $p+q$ divides $r$. Suppose that the largest number that is printed by the computer is $N$. Determine the number formed by the last two digits of $N$ (in the same order).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
13
}
Evaluate the triple integral \( \iiint_D (4-x^2) \, dV \) over the solid region \( D \) bounded by the cylinder \( x^2 + y^2 = 4 \) and the planes \( z = 0 \) and \( z = 4-x^2 \). Use cylindrical coordinates to find the volume of the region.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40\pi}
In a diagnostic math test for high school seniors at a certain school, the scores (in points) follow a normal distribution $N(100, 10^{2})$. Let $X$ be the score of a randomly selected student. Let event $A$ be defined as the student's score being $80 < X \leqslant 100$, and event $B$ be defined as the student's score being $70 < X \leqslant 90$. Find the conditional probability of event $B$ given that event $A$ has occurred, denoted as $P(B|A)=\_\_\_\_\_\_$ (express the result as a fraction). 

Given data: $P(\mu -\sigma < X \leqslant \mu + \sigma) = 0.68$, $P(\mu -2\sigma < X \leqslant \mu + 2\sigma) = 0.95$, $P(\mu -3\sigma < X \leqslant \mu + 3\sigma) = 0.99$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{27}{95}}
There are five students, A, B, C, D, and E, arranged to participate in the volunteer services for the Shanghai World Expo. Each student is assigned one of four jobs: translator, guide, etiquette, or driver. Each job must be filled by at least one person. Students A and B cannot drive but can do the other three jobs, while students C, D, and E are capable of doing all four jobs. The number of different arrangements for these tasks is _________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{108}
Evaluate the contour integral \( \int_C \frac{dw}{e^w-1} \) over a loop \( C \) contained in the annulus \( 0<|z|<2\pi \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\pi i}
A mouse named Jerry decided to gift a cat named Tom a birthday cake in the shape of an $8 \times 8$ square. He placed fish in three pieces marked with the letter 'P', sausage in two pieces marked with the letter 'K', and both fish and sausage in one piece, which was not marked. All other pieces are without filling. Jerry also informed Tom that in any $6 \times 6$ square, there are at least 2 pieces with fish, and in any $3 \times 3$ square, there is at most one piece with sausage.

What is the minimum number of cake pieces Tom needs to eat to guarantee that he gets the piece with both fish and sausage?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Compute the intersection number \( \iota^*\mathcal{O}(2,3) \cdot \iota^*\mathcal{O}(2,3) \) for the smooth subvariety \( \iota:X=V_+(f)\subset\mathbb{P}^2\times\mathbb{P}^1 \) where \( f\in H^0(\mathcal{O}(1,2)) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
Two people play a math game. Player A selects a group of single-digit integers \( x_{1}, x_{2}, \cdots, x_{n} \) as the secret numbers, which can be positive or negative. Player B can ask questions about the sum \( a_{1} x_{1} + a_{2} x_{2} + \cdots + a_{n} x_{n} \), where \( a_{1}, a_{2}, \cdots, a_{n} \) can be any set of numbers. Determine the minimum number of questions Player B needs to ask in order to guess the secret numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Triangles $ABC$ and $AFG$ have areas $3012$ and $10004$, respectively, with $B=(0,0),$ $C=(335,0),$ $F=(1020, 570),$ and $G=(1030, 580).$ Find the sum of all possible $x$-coordinates of $A$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1800}
Calculate the expected number of times the number 2 is rolled when repeatedly rolling a fair die until the sequence "2, 4, 6" appears.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36}
Given that $O$ is the origin of the coordinate system, the focus of the parabola $y^{2}=4x$ is $F$, and $P$ is a point on the parabola. If $|PF|=3$, find the area of $\triangle OPF$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt {2}}
Determine all four-digit numbers $\overline{abcd}$ such that

$$a(a+b+c+d)(a^{2}+b^{2}+c^{2}+d^{2})(a^{6}+2b^{6}+3c^{6}+4d^{6})=\overline{abcd}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2010}
a can run 2 km distance in 2 min . , while b can run this distance in 2 min 40 sec . by how much distance can a beat b ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.5}
Find the remainder when the expression \( S = \sum_{i=0}^{99} (n+i)^6 + 2^{2^{2558}} + 1 \) is divided by 100, given that \( 2^{2^{2558}} \equiv 16 \pmod{100} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{67}
Ana, Bob, and Cao bike at constant rates of $8.6$ meters per second, $6.2$ meters per second, and $5$ meters per second, respectively. They all begin biking at the same time from the northeast corner of a rectangular field whose longer side runs due west. Ana starts biking along the edge of the field, initially heading west, Bob starts biking along the edge of the field, initially heading south, and Cao bikes in a straight line across the field to a point D on the south edge of the field. Cao arrives at point D at the same time that Ana and Bob arrive at D for the first time. The ratio of the field's length to the field's width to the distance from point D to the southeast corner of the field can be represented as $p : q : r$, where $p$, $q$, and $r$ are positive integers with p and q relatively prime. Find $p + q + r$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{061}
A student sits for 6 examination papers, each worth 100 marks. In how many ways can the student score a total of 240 marks across all papers?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4188528351}
Exercise 11. A group of 20 dwarves coming out of the mine sit around a round table to count the gold nuggets each has mined. They make the following observations:

$\triangleright$ The difference in the number of nuggets between two neighboring dwarves is always 2 or 3.

$\triangleright$ All the dwarves have mined a different number of nuggets.

What is the greatest possible difference in nuggets between the dwarf who found the most and the one who found the least?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{29}
In the parallelogram \(ABCD\), side \(AB\) is equal to 6, and the height dropped to the base \(AD\) is 3. The angle bisector of \(\angle BAD\) intersects side \(BC\) at point \(M\), with \(MC = 4\). Point \(N\) is the intersection of the bisector \(AM\) and diagonal \(BD\). Find the area of triangle \(BNM\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{27}{8}}
Mary uses plastic grocery bags that can hold a maximum of twenty pounds each. She needs to distribute the weight of her groceries evenly between two bags. She buys 4 pounds of green beans, 6 pounds of milk, twice the amount of carrots as green beans, 3 pounds of apples, and 1 pound of bread. Additionally, she purchases a 5-pound bag of rice, but the rice must be placed in one of the bags without any of the other items. How many more pounds of groceries can Mary fit in the remaining bag?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Evaluate the limit: $$ \lim_{n\rightarrow\infty}\frac{1+n+\frac{n^{2}}{2!}+\cdots +\frac{n^{n}}{n!}}{e^{n}} $$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
The numbers $1, 2, \ldots, 33$ are written on the board. A student performs the following procedure: choose two numbers from those written on the board such that one of them is a multiple of the other. After choosing, the student deletes the two numbers and writes on the board their greatest common divisor (GCD). The student repeats this procedure until only numbers without multiples remain on the board. Determine how many numbers remain on the board when the student can no longer repeat the procedure.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{17}
An ant moves on the following lattice, beginning at the dot labeled $A$. Each minute he moves to one of the dots neighboring the dot he was at, choosing from among its neighbors at random. What is the probability that after 5 minutes he is at the dot labeled $B$? [asy]
draw((-2,0)--(2,0));
draw((0,-2)--(0,2));
draw((1,1)--(1,-1)--(-1,-1)--(-1,1)--cycle);
dot((0,0)); dot((1,0)); dot((2,0)); dot((-1,0)); dot((-2,0)); dot((0,1)); dot((0,2)); dot((0,-1)); dot((0,-2)); dot((1,1)); dot((1,-1)); dot((-1,-1)); dot((-1,1));
label("$A$",(0,0),SW);
label("$B$",(0,1),NE);
[/asy]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{4}}
Find a constant \( d \) such that the inequality \( \prod_{p < x} \left(1 - \frac{n}{p} \right) \geq d(\log(x))^{-n} \) holds for all \( x \) and \( n \), using Merten's Theorem and the provided inequality.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e^{-\gamma}}
Let $\Phi$ be a homogeneous Poisson point process (PPP) of intensity $\lambda$ on $\mathbb{R}$. Consider the ordered points $\{x_0, x_1, \dotsc\} = \Phi \cap \mathbb{R}^+$, such that $0 < x_0 < x_1 < \dotsc$. Find the complementary cumulative distribution function (CCDF) of the ratio $x_0/x_k$, where $x_k$ is Erlang distributed with probability density function $f_{x_k}(x) = \frac{\lambda^{k+1}x^k}{k!}e^{-\lambda x}$. Provide your answer in terms of $\lambda$ and $k$. 
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(1 - r)^k}
Let \( g(t) = t \) for all \( t \in [-1,1] \) and \( f(t) = t^{1/n} \), where \( n \) is a large odd integer. Consider the operator \( T: C[-1,1] \rightarrow \mathbb{R} \), defined by \( T_g(f) = \int_{-1}^1 g(t)f(t)dt \). Given the norm \( \|f\| = \max_{t \in [-1,1]} |f(t)| \), find \( \|T\| \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
In a local election, 6 initial candidates received the following number of votes: 15236, 20689, 12359, 30682, 25213, and 18492. Before the final tally, the candidates who received 12359 and 18492 votes had to withdraw due to personal reasons. The votes for these two candidates were redistributed to the remaining candidates based on the following rules: 60% of their votes were given to the candidate who initially had 15236 votes, 25% were given to the candidate with 30682 votes, and the remaining 15% were distributed to the candidate with 20689 votes. What % of the total votes did the winning candidate get in the final tally of the election?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{31.3\%}
A and B play a game with the following rules: In the odd-numbered rounds, A has a winning probability of \(\frac{3}{4}\), and in the even-numbered rounds, B has a winning probability of \(\frac{3}{4}\). There are no ties in any round, and the game ends when one person has won 2 more rounds than the other. What is the expected number of rounds played until the game ends?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{16}{3}}
Given circle $O$: $x^{2}+y^{2}=4$, if a line $l$ that does not pass through the origin $O$ intersects the circle $O$ at points $P$ and $Q$, and the slopes of lines $OP$, $PQ$, and $OQ$ form a geometric sequence, then the slope of line $l$ is _______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{±1}
Find a polynomial \( p(x) \) of degree \( \le 2 \) that approximates \( f(x) = x^{1/3} \) and minimizes the error \[ E = (f(-1)-p(-1))^2 + \int_{-1}^1 (f(x)-p(x))^2 \, dx + (f(1)-p(1))^2. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{15}{14}x}
Let the positive integer \( n \) be expressed as the sum of the reciprocals of four different positive integers. How many such \( n \) exist? Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Find the largest possible value of the product $pqr$, where $p$, $q$, and $r$ are prime numbers satisfying the conditions $p+q<111$ and $\frac{p+q}{r}=p-q+r$. What is the largest possible value of $pqr$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2014}
There are relatively prime positive integers $p$ and $q$ such that $\frac{p}{q} = \sum_{n=3}^{\infty} \frac{1}{n^5 - 5n^3 + 4n}$. Find $p+q$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{97}
In an acute triangle \( {ABC} \) ,let \( D \) be a point on \( \left\lbrack  {AC}\right\rbrack \) and \( E \) be a point on \( \left\lbrack  {AB}\right\rbrack \) such that
 \( \angle {ADB} = \angle {AEC} = {90}^{ \circ  } \) . If perimeter of triangle \( {AED} \) is 9, circumradius of \( {AED} \) is
 \( \frac{9}{5} \) and perimeter of triangle \( {ABC} \) is 15 , then \( \left| {BC}\right| \) is
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{24}{5}}
Let  $n \geq 2$  be an integer. Lucia chooses  $n$  real numbers  $x_1,x_2,\ldots,x_n$  such that  $\left| x_i-x_j \right|\geq 1$  for all  $i\neq j$ . Then, in each cell of an  $n \times n$  grid, she writes one of these numbers, in such a way that no number is repeated in the same row or column. Finally, for each cell, she calculates the absolute value of the difference between the number in the cell and the number in the first cell of its same row. Determine the smallest value that the sum of the  $n^2$  numbers that Lucia calculated can take.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{n^3 - n}{3}}
Given a blackboard with 128 ones, in each step you can erase any two numbers \(a\) and \(b\) on the blackboard and replace them with \(a \cdot b + 1\). After performing this operation 127 times, leaving only one number, find the last digit of the maximum possible value of this number, denoted as \(A\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Find the sum of all values of $n$ such that the function \( g(x) = 100|f(x)| - \sum_{k=1}^n |f(x^k)| \) is differentiable over \( \mathbb{R} \), where \( f(x) = e^{x+1} - 1 \) and \( n \in \mathbb{N} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{39}
Calculate the area enclosed by the curve $g(x)$, the lines $x=-3$ and $x=5$, and the $x$-axis, where $g(x)$ is the inverse function of $f(x)=x^3+3x+1$. Use techniques of definite integrals to find the area.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{9}{2}}
What is the simple closed form of the integral given by $$\int_{0}^{\infty} \frac{\cos\left(\frac{2x}{\pi}\right) - \cos^2\left(\frac{2x}{\pi}\right)}{x^2} \cdot \ln(x) \, \mathrm dx \tag1$$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\ln(2)}
Let $\sigma_1 : \mathbb{N} \to \mathbb{N}$ be a function that takes a natural number $n$, and returns the sum of the positive integer divisors of $n$. For example, $\sigma_1(6) = 1 + 2 + 3 + 6 = 12$. What is the largest number $n$ such that $\sigma_1(n) = 1854$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1234}
Find an analytical function $f(z)$ such that $f(z + \sin i) = f(z)$ and $f(z + \cos i) = f(z)$. Provide the function that satisfies these conditions.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{f(z) = C}
An urn initially contains 5 white and 7 black balls. Each time a ball is drawn, its color is noted and it is placed back in the urn along with two balls of the same color. Compute the probability that the first two balls selected are black given that the next two balls drawn are white.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{7}{32}}
Create a six-digit number using the digits $1$, $2$, $3$, $4$, $5$, $6$ without repetition, such that the digit $1$ is not at either end, and among the three even digits $2$, $4$, $6$, exactly two are adjacent. Find the number of such six-digit numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{288}
We color all the sides and diagonals of a regular polygon $P$ with 43 vertices either red or blue. Each vertex is an endpoint of 20 red segments and 22 blue segments. A triangle formed by vertices of $P$ is called monochromatic if all of its sides have the same color. Suppose there are 2022 blue monochromatic triangles. How many red monochromatic triangles are there?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{859}