inclusionAI/Ring-lite-rl-data huggingface dataset:
On a track for remote-controlled racing cars, racing car A completes the track in some seconds, while racing car B completes it in 24 seconds. If they both start at the same time, they will be side by side again after 168 seconds. How many seconds does it take for racing car A to complete the track?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Let  $N=10^6$ . For which integer  $a$  with  $0 \leq a \leq N-1$  is the value of \[\binom{N}{a+1}-\binom{N}{a}\] maximized?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{499499}
What is the maximum number of rooks that can be placed in an $8 \times 8 \times 8$ cube so that no two rooks attack each other?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{64}
During a town winter carnival, a group of friends - Steven, Stephanie, Sonya, Sam, Sue, and Sophie - decided to go ice skating. Steven, who is 20 years old and has been skating since he was 6, only fell down 3 times, but Stephanie, who is 24 and has only been skating for a year, had 13 more falls than Steven. Sonya, on the other hand, fell down 2 times less than half the number of times Stephanie fell. Sam, who is the same age as Stephanie but has been skating for 10 years, fell only once, whereas Sophie, who is two years younger than Sonya and has been skating for half as long as Sonya, fell four times more than Sam. Sue, the oldest in the group at 26 years old, has 15 years of skating experience. She fell one time more per year of her age difference with the youngest member of the group. Taking into account the complex relationships between the friends' ages, experience levels, and fall patterns, how many times did Sue fall?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
If $\sin(x-y)\cos x - \cos(x-y)\sin x = \frac{3}{5}$, then the value of $\tan 2y$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pm \frac{24}{7}}
Let $p(z) = az^2+bz+c$ be a quadratic polynomial with real coefficients. If $a > 0$, $|c| > 1$, and $b^2-2ac > 0$, find the number of roots of the function $g(z) = p(z) - e^{z}$ in the first quadrant of the complex plane.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
A rectangular piece of paper is divided into polygonal areas in the following way: at each step, one of the existing surfaces is cut by a straight line, resulting in two new areas. What is the minimum number of cuts needed such that among the obtained polygons, there are $251$ polygons with $11$ sides?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2007}
Find \( y'' \) if \( x = \ln t \) and \( y = \sin 2t \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-4 t^2 \sin 2 t + 2 t \cos 2 t}
Find the value of $a_0 + a_1 + a_2 + \cdots + a_6$ given that $(2-x)^7 = a_0 + a_1(1+x)^2 + \cdots + a_7(1+x)^7$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{129}
Right $\triangle ABC$ has $AB=3, BC=4,$ and $AC=5.$ Square $XYZW$ is inscribed in $\triangle ABC$ with $X$ and $Y$ on $\overline{AC}, W$ on $\overline{AB},$ and $Z$ on $\overline{BC}.$ What is the side length of the square?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{60}{37}$}
小明同学任选一个各项均为非零整数的等差数列 $\left\{a_{n}\right\}$ ，随后他列出 2020 个关于 $x$ 的一元二次方程:
$$
E_{i}: a_{i} x^{2}+a_{i+1} x+a_{i+2}=0(i=1,2, \cdots, 2020),
$$
再将上述每个方程的根都写在黑板上.求此时黑板上两两不同的实根的个数的最大可能值.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Compute the limit: $$\lim_{n\to\infty}\left(n-(\arccos(1/n)+\cdots+\arccos(n/n))\right).$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{4}}
Suppose $A B C$ is a triangle with circumcenter $O$ and orthocenter $H$ such that $A, B, C, O$, and $H$ are all on distinct points with integer coordinates. What is the second smallest possible value of the circumradius of $A B C$ ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{10}}
Given a sequence $\{a_n\}$ where $a_n = n$, for each positive integer $k$, in between $a_k$ and $a_{k+1}$, insert $3^{k-1}$ twos (for example, between $a_1$ and $a_2$, insert three twos, between $a_2$ and $a_3$, insert $3^1$ twos, between $a_3$ and $a_4$, insert $3^2$ twos, etc.), to form a new sequence $\{d_n\}$. Let $S_n$ denote the sum of the first $n$ terms of the sequence $\{d_n\}$. Find the value of $S_{120}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{245}
Given the equation \(f(x-1) + f(x+3) = f(x+1) + f(x+5)\), determine the period of the function \(f(x)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
For the quadratic function $y=2mx^{2}+\left(8m+1\right)x+8m$ with respect to $x$, the graph intersects the $x$-axis. Find the range of $m$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{m \geq -\frac{1}{16}, m \neq 0}
Evaluate the integral \( \large \int_{0}^{\pi/2}\cos(2x)\ln^2(\tan(x/2))\mathrm{d}x \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi}
Given the function $f(x)=\frac{1}{2}ax^{2}-(x-1)e^{x}$, if for any real numbers $(x_{1}, x_{2}, x_{3})$ in the interval $[0,1]$, we have $f(x_{1})+f(x_{2})\geqslant f(x_{3})$, then the range of values for the real number $a$ is ________
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[1,4]}
Determine the cardinality of the set of all injective functions from an uncountable set to the set of bounded functions.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{\mathfrak{c}}}
The organizers of a fair projected a 25 percent increase in attendance this year over that of last year. Last year, the fair saw A attendees. However, due to an increase in ticket prices by $5 and unfavorable weather conditions, attendance this year actually decreased by 20 percent. The weather affected V percentage of the days of the fair, causing a reduction by W visitors per day due to the harsh conditions. Taking into account these factors, calculate the percent of the projected attendance that was the actual attendance this year.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{64\%}
In an equilateral triangle \(ABC\) with side length 1, points \(D\) and \(E\) are located on sides \(AB\) and \(AC\) respectively. When the triangle is folded along the line segment \(DE\), vertex \(A\) lands exactly on side \(BC\). Determine the minimum length of \(AD\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\sqrt{3} - 3}
The numbers $3, 6, 10, 15, \cdots$ are called triangular numbers because they can be arranged in the form of an equilateral triangle. Find the 19th triangular number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{210}
Given that $\cos 78^{\circ}$ is approximately $\frac{1}{5}$, what is the approximate value of $\sin 66^{\circ}$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.92}
Find the value of $Q(1)$ if $Q(x)$ is a polynomial satisfying $Q(x)-Q'(x) = (x-1)^n$, where $n$ is a positive integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n!}
In a regular tetrahedron $S-ABCD$, $O$ is the projection of the apex $S$ onto the base, $P$ is the midpoint of the lateral edge $SD$, and $SO=OD$. The angle formed by line $BC$ and plane $PAC$ is equal to _____.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{45^\circ}
There is a jar containing $m$ white balls and $n$ black balls $(m > n)$. Balls are drawn one by one without replacement. If at some point during the drawing process, the number of white balls drawn equals the number of black balls drawn, this is called a coincidence. Find the probability of having at least one coincidence.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2n}{m+n}}
Suppose Sachin borrows an amount X from a lender at an annual interest rate of R1% for T1 years. He immediately lends money to Rahul and Ravi at the annual interest rates of R2% and R3% for T2 and T3 years, respectively. Calculate the gain of Sachin in one year if:

X = Rs. 5000
R1 = 4% p.a.
T1 = 2 years
R2 = 6.25% p.a. (25/4%)
T2 = 2 years
R3 = 7.5% p.a.
T3 = 3 years

(Note that only Rahul repays the loan within the same period as the original borrowed amount, which is 2 years. Ravi repays the loan after 3 years.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{487.50}
Given four numbers, the sum of every three numbers among them is 17, 21, 25, and 30, respectively. Find these four numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14, 10, 6, 1}
For some integer values of n, the value of the expression 4n + 7 is an integer greater than a certain number and less than 40. There are 10 such integer values of n. What is the lower bound if the expression is greater than 1?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Compute $$2 \sqrt{2 \sqrt[3]{2 \sqrt[4]{2 \sqrt[5]{2 \cdots}}}}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{e-1}}
Let $T$ be the set of ordered triples $(x,y,z)$, where $x,y,z$ are integers with $0\leq x,y,z\leq9$. Players $A$ and $B$ play the following guessing game. Player $A$ chooses a triple $(x,y,z)$ in $T$, and Player $B$ has to discover $A$[i]'s[/i] triple in as few moves as possible.  A [i]move[/i] consists of the following: $B$ gives $A$ a triple $(a,b,c)$ in $T$, and $A$ replies by giving $B$ the number $\left|x+y-a-b\right |+\left|y+z-b-c\right|+\left|z+x-c-a\right|$. Find the minimum number of moves that $B$ needs to be sure of determining $A$[i]'s[/i]  triple.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
In a large bag of decorative ribbons, $\frac{1}{4}$ are yellow, $\frac{1}{3}$ are purple, $\frac{1}{8}$ are orange, and the remaining 45 ribbons are silver. How many of the ribbons are orange?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{19}
A chess king is placed in the bottom-left corner of a chessboard. Two players take turns moving the king. On each turn, the king can be moved one square to the right, one square up, or one square diagonally "up-right." The player who places the king in the top-right corner of the board wins the game. Assuming both players play optimally, which player will win?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{First player}
The numbers from 1 to 1996 are written sequentially as follows: 12345678910111213... up to 1996. Find how many zeros appear in this sequence.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{489}
Given a positive integer \(a\), there exists a quadratic trinomial with integer coefficients, where \(a\) is the coefficient of the quadratic term, and it has two distinct positive roots both less than 1. Find the smallest value of \(a\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Jane is considering buying a sweater priced at $50. A store is offering a 10% discount on the sweater. After applying the discount, Jane needs to pay a state sales tax of 7.5%, and a local sales tax of 7%. Calculate the difference between the state and local sales taxes that Jane has to pay.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.225}
A vertical wooden pillar has a rope tied to its top. The part of the rope hanging down from the top of the pillar to the ground measures 4 feet. When pulling the rope, it runs out when 8 feet away from the base of the pillar. Calculate the total length of the rope. Express your answer in feet.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
A sum was put at simple interest at a certain rate for 9 years. Had it been put at a higher rate, it would have fetched $1,350 more. The sum is $3,000. What is the difference in percentage between the original rate and the higher rate?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5\%}
20. Suppose that $a, b, c$ are real numbers such that
$$
\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=4\left(\frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a}\right)=\frac{c}{a+b}+\frac{a}{b+c}+\frac{b}{c+a}=4 .
$$

Determine the value of $a b c$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{49}{23}}
For any natural values \( m \) and \( n \) (with \( n > 1 \)), a function is defined as \( f\left(\frac{m}{n}\right)=\frac{\sqrt[n]{3^{m}}}{\sqrt[n]{3^{m}}+3} \). Compute the sum

\[ f\left(\frac{1}{2020}\right)+f\left(\frac{2}{2020}\right)+f\left(\frac{3}{2020}\right)+\cdots+f\left(\frac{4039}{2020}\right)+f\left(\frac{4040}{2020}\right) \].
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2020}
We can label the squares of an 8 x 8 chessboard from 1 to 64 in 64! different ways. For each way, we find \( D \), the largest difference between the labels of two squares that are adjacent (orthogonally or diagonally). What is the smallest possible \( D \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
In triangle $ABC$, point $F$ is on segment $AB$ such that $CF$ bisects $\angle ACB$. Points $D$ and $E$ are on line $CF$ such that lines $AD$ and $BE$ are perpendicular to $CF$. Let $M$ be the midpoint of $AB$. If $ME = 13$, $AD = 15$, and $BE = 25$, find $AC + CB$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{104}
1. Given that $A D$ and $B E$ are the angle bisectors of $\triangle A B C$, and they intersect at point $I$, with $I D=I E$. If $\angle C A B=80^{\circ}$, then $\angle A B C=$ $\qquad$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40}
If $A$ is a square matrix satisfying $$A^{T}A = I,$$ write the value of $|A|$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Gloria wants to buy the $129,000 mountain cabin that her friend Alfonso is selling. She only has $150 in cash. She intends to raise the remaining amount by selling her mature trees for lumber. She has 20 cypress trees, 600 pine trees, and 24 maple trees. She will get $100 for each cypress tree, a certain amount for a maple tree, and $200 per pine tree. After paying Alfonso for the cabin, Gloria has $350 left. How much does Gloria get for each maple tree?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$300}
How many 9-digit numbers, divisible by 2, can be formed by permuting the digits of the number 231157152?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3360}
Each of the twelve letters in "STATISTICS" is written on its own square tile and placed in a bag. What is the probability that a tile randomly selected from the bag will have a letter on it that is in the word "MIST"? Express your answer as a common fraction.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2}{3}}
What is the covariance of the process $X(t) = \int_0^t B(u)\,du$ where $B$ is a standard Brownian motion? i.e., I wish to find $E[X(t)X(s)]$, for $0<s<t<\infty$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{ts^2}{2} - \frac{s^3}{6}}
Richard starts with the string $HHMMMMTT$. A move consists of replacing an instance of $HM$ with $MH$, replacing an instance of $MT$ with $TM$, or replacing an instance of $TH$ with $HT$. Compute the number of possible strings he can end up with after performing zero or more moves.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{70}
Let \( a, b, c, d, e, f \) be integers selected from the set \( \{1,2, \ldots, 100\} \), uniformly and at random with replacement. Set
\[ M = a + 2b + 4c + 8d + 16e + 32f. \]
What is the expected value of the remainder when \( M \) is divided by 64?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{31.5}
Evaluate the limit:

$$\lim_{n \to \infty} \int_{\epsilon}^1\frac {n+1}{2^{n+1}}\frac{(1+t)^{n+1}-(1-t)^{n+1}}t\,dt$$

where $\epsilon \in (0, 1)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
2. 8 girls and 25 boys stand in a circle, with at least two boys standing between any two girls. How many different arrangements are there (arrangements that coincide by rotating the circle are considered the same)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16!\times25!/9!}
Three circles with radii \(\frac{1}{2}\), \(\frac{1}{2}\), and \(\sqrt{\frac{13-6 \sqrt{3}}{13}}\) are positioned such that the triangle formed by the centers of these circles is equilateral with a side length of \(\frac{\sqrt{3}}{2}\). Find the radius of the circumcircle around the triangle, each vertex of which is the intersection point of two of these circles, and is farther from the center of the third circle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 \sqrt{3} - 3}
The sum of the absolute values of the terms of a finite arithmetic progression is 100. If all its terms are increased by 1 or all its terms are increased by 2, the sum of the absolute values of the terms of the resulting progression will also be 100. What values can the quantity $n^{2} d$ take under these conditions, where $d$ is the common difference of the progression and $n$ is the number of its terms?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{400}
Calculate the integral \( I \) over the region \( Q \) in \( \mathbb{R}^3 \) defined by:

\[ I = \int_{Q} e^{-x-y-z} \, dV \]

where \( Q = \{ (x, y, z) : x + y > 0, \ y + z > 0, \ x + z > 0 \} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
The monotonic decreasing interval of the function $y=2^{x^2-2x+1}$ is ______________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-\infty,1]}
Evaluate the limit \(\displaystyle\lim_{n\to\infty}\sqrt[n]{|1-|z|^n|}\) for \(z\in\mathbb{C}\) with \(|z|\ne 1\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\max\{1, |z|\}}
Consider the L-shaped region formed by three unit squares joined at their sides, as shown below. Two points $A$ and $B$ are chosen independently and uniformly at random from inside the region. The probability that the midpoint of $\overline{AB}$ also lies inside this L-shaped region can be expressed as $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{35}
Let $L = \mathbb{Q}(\sqrt[4]{2}, \omega)$ where $\omega = e^{\frac{2\pi i}{8}}$. Determine the degree of the field extension $[L: \mathbb{Q}(\sqrt[4]{2})]$. \( \boxed{\text{Your answer here}} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Find a tight asymptotic upper bound for the recurrence relation \( p(n) = \sqrt{n}p(\sqrt{n}) + \sqrt{n} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{O(n)}
In triangle \(ABC\), points \(M\) and \(N\) are marked on the sides \(AB\) and \(BC\) respectively, such that \(BM = BN\). A line is drawn through point \(M\) perpendicular to \(BC\), and another line is drawn through point \(N\) perpendicular to \(AB\). These lines intersect at point \(O\). The extension of segment \(BO\) intersects side \(AC\) at point \(P\) and divides it into segments \(AP = 5\) and \(PC = 4\). Find the length of segment \(BP\), given that \(BC = 6\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
A man saves 10% of his monthly salary. If on account of dearness of things he is to increase his monthly expenses by 5%, he is only able to save Rs. 400 per month. What is his monthly salary?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7272.73}
In how many ways can we change the sign $ *$ with $ +$ or $ -$, such that the following equation is true?
\[ 1 *2*3*4*5*6*7*8*9*10=29\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
The sequence \( a_1, a_2, a_3, \ldots \) is defined as follows: \( a_1 = 1 \). For \( n \geq 2 \), \( a_n \) is the smallest integer greater than \( a_{n-1} \) such that there do not exist \( 1 \leq i, j, k \leq n \) (not necessarily distinct) such that \( a_i + a_j = 3a_k \). Find \( a_{1998} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4494}
Determine all values of $p$ for which the divergence of the vector field $\vec{F} = r^p\vec{r}$ is zero, where $\vec{r}=x\hat{i}+y\hat{j}+z\hat{k}$ and $r = \|\vec{r}\|$. Provide your answer in terms of $p$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-3}
Jamie, a history enthusiast with a passion for social and educational movements, is organizing a series of educational talks about influential figures in history. Jamie has invited 5 speakers, each specializing in a different movement: civil rights, feminism, environmentalism, educational reform, and labor rights. 

Each speaker will give two talks, one in the morning and one in the afternoon. Jamie plans to offer 3 history-themed snacks for each talk: one representing the era, one representing the movement, and one representing the speaker's favorite historical figure. 

If each snack costs $2 and Jamie expects 20 people to attend each talk, how much will Jamie spend on snacks for all the talks combined?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1200}
Solve the equation \(\frac{x^3+2x}{x^2-1}=\sqrt{x^2-\frac{1}{x}}\) for \(x\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{1}{2}}
Calculate the volume enclosed by the paraboloids \(z = x^2 + (y-1)^2\) and \(z = \frac{5}{2} - x^2 - y^2\) for \(x \geq 0\). Use a triple integral to find the volume.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{2}}
Evaluate the double integral \( \iint_{R} f \cdot g \, dx \, dy \) over the circular disk \( R \) with boundary \( x^2 + y^2 = 1 \), where the vector fields \( f \) and \( g \) are defined as follows:

\[ f(x,y) = v(x,y) \mathbf{i} + u(x,y) \mathbf{j} \]
\[ g(x,y) = \left(\frac{\partial u}{\partial x} - \frac{\partial u}{\partial y}\right) \mathbf{i} + \left(\frac{\partial v}{\partial x} - \frac{\partial v}{\partial y}\right) \mathbf{j} \]

Given that on the boundary of \( R \), \( u(x,y) = 1 \) and \( v(x,y) = y \), find the value of the integral.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\pi}
已知离心率为$\frac{2}{3}$的椭圆$C:\frac{{{x}^{2}}}{{{a2}}}+\frac{{{y}^{2}}}{{{b2}}}=1\left( a>b>0 \right)$的左、右焦点分别为${{F}_{1}}$、${{F2}}$，点$P$在椭圆$C$上，点$M$为$\Delta P{{F}_{1}}{{F2}}$的内心，且$\Delta MP{{F}_{1}}$、$\Delta MP{{F2}}$、$\Delta M{{F}_{1}}{{F2}}$的面积分别为${{S}_{\Delta MP{{F1}}}}$、${{S}_{\Delta MP{{F2}}}}$、${{S}_{\Delta M{{F1}}{{F2}}}}$，若${{S}_{\Delta MP{{F1}}}}+3{{S}_{\Delta MP{{F2}}}}=2{{S}_{\Delta M{{F1}}{{F2}}}}$，则$\frac{{{S}_{\Delta MP{{F1}}}}}{{{S}_{\Delta MP{{F2}}}}}$的值为__________．
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}