inclusionAI/Ring-lite-rl-data huggingface dataset:
A light ray is emitted from point $A(-3,3)$, reflected off a point $P$ on the $x$-axis, and then intersects with the circle $(x-1)^{2}+(y-1)^{2}=2$ at point $B$. Determine the minimum value of $|AP|+|PB|$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3\sqrt{2}}
Given point $M(x_{0},2-x_{0})$, suppose there exists a point $N$ on circle $O$ : $x^{2}+y^{2}=1$ such that $\angle OMN=30^{\circ}$, then the range of real number $x_{0}$ is _____.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[0,2]}
The polynomial $x^4 + ax^3 + bx^2 - 32x$, where$ a$ and $b$ are real numbers, has roots that form a square in the complex plane. Compute the area of this square.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8
}
Petya and Vasya came up with ten fifth-degree polynomials. Then, Vasya sequentially called out consecutive natural numbers (starting from some number), and Petya substituted each called number into one of the polynomials of his choice, writing the resulting values on the board from left to right. It turned out that the numbers written on the board formed an arithmetic progression (in this order). What is the maximum number of numbers Vasya could call out?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50}
If x = - 6 and y = - 3, the value of a certain number ( x - y ) ^ 2 - xy is 18. What is the value of that certain number?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-9}
When \(0 < x < 1\), given \( f(x) = \frac{x}{\log x} \), what is the correct relationship between \( f(x) \), \( f(x^2) \), and \( f^2(x) \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\( f(x) < f(x^2) < f^2(x) \)}
已知不全相等的三个实数 $a, b, c$, 满足 $a=a b+c, b=b c+a, c=c a+b$ ，则 $a+b+$ $c=$ $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Consider a sequence of random variables $(X_n)$, where each $X_n$ has the distribution: $P(X_n = 1 - \frac{1}{n}) = P(X_n = 1 + \frac{1}{n}) = \frac{1}{2}$ for $n \in \mathbb{N}^*$. Determine the limiting distribution of this sequence as $n$ approaches infinity.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\delta_1}
Evaluate the limit: $$\lim_{n\to \infty} \sum_{i=1} ^ {n} \left(1- \frac{2i}{n} \right) ^ {\frac{n}{i}}.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e^{-2}}
Let \( g : \mathbb{R} \to \mathbb{R} \) be a function such that
\[ g(g(x) - y) = 2g(x) + g(g(y) - g(-x)) + y \] for all real numbers \( x \) and \( y \).

Let \( n \) be the number of possible values of \( g(2) \), and let \( s \) be the sum of all possible values of \( g(2) \). Find \( n \times s \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2}
In a math competition, the possible scores for each problem for each participating team are 0 points, 3 points, or 5 points. By the end of the competition, the sum of the total scores of three teams is 32 points. If the total score of any single team can reach 32 points, how many different possible combinations are there for the total scores of these three teams?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{255}
Find the maximum value of \( \boldsymbol{x}^T \boldsymbol{A} \boldsymbol{x} \) subject to the constraints \( \boldsymbol{q}^T \boldsymbol{x} = 1 \) and \( x_i \geq 0 \) for all \( i \), where \( \boldsymbol{x} \in \mathbb{R}^n \), \( \boldsymbol{A} \) is a positive definite matrix, and \( \boldsymbol{q} \) is a given vector. For example, if \( \boldsymbol{A} = \begin{bmatrix} 2 & 0 \\ 0 & 1 \end{bmatrix} \) and \( \boldsymbol{q} = \begin{bmatrix} 1 \\ 1 \end{bmatrix} \), determine the solution that maximizes the objective function.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
John and Ingrid have different income sources with varying tax rates. John makes $57,000 from his job, where he pays a tax rate of 30%, and he also earns $11,000 from rental income, which is taxed at a rate of 25%. On the other hand, Ingrid makes $72,000 from her employment, which attracts a 40% tax, and she earns $4,500 from her investments, which are subject to a 15% tax rate. Calculate their combined weighted average tax rate.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{34.14\%}
Evaluate the integral \( \int _C \frac{\exp(z^2)}{z^2\left(z-1-i\right)}dz \), where \( C \) consists of the contour \(|z|=2\) oriented counterclockwise and \(|z|=1\) oriented clockwise.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi e^{2i}}
Triangle $\triangle ABC$ has circumcenter $O$ and incircle $\gamma$. Given that $\angle BAC = 60^\circ$ and $O$ lies on $\gamma$, if \[ \tan B \tan C = a + \sqrt{b} \] for positive integers $a$ and $b$, find $100a + b$. 
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{408}
Determine the dimension of the space of planes in \(\Bbb R^3\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Given two random variables $X$ and $Y$, where $X$ follows a uniform distribution $U(1, 2)$, and the conditional distribution of $Y$ given $X$ is an exponential distribution $\operatorname{Exp}(X)$, find the joint distribution of $X$ and $Y$ conditioned on the event $X \leq 1.5$. Provide your answer in terms of the joint probability density function.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2x e^{-xy}}
Find a function $h$ such that $h$ is orthogonal to both $x$ and $x^2$ with respect to the inner product $\langle f,g \rangle = \int_{-1}^{1} g(x)f(x) \, dx$ defined on $\mathcal{C}[-1,1]$. Normalizing $h$ is not required.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5x^3 - 3x}
In the isosceles triangle \(ABC\), points \(K\) and \(L\) are marked on the equal sides \(AB\) and \(BC\) respectively so that \(AK = KL = LB\) and \(KB = AC\). Find the size of angle \(ABC\). Express your answer in degrees as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36^\circ}
Let $G{}$ be an arbitrary finite group, and let $t_n(G)$ be the number of functions of the form \[f:G^n\to G,\quad f(x_1,x_2,\ldots,x_n)=a_0x_1a_1\cdots x_na_n\quad(a_0,\ldots,a_n\in G).\]Determine the limit of $t_n(G)^{1/n}$ as $n{}$ tends to infinity.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{|G|}{|Z(G)|}}
$24! = 620,448,401,733,239,439,360,000$ ends in four zeros, and $25!=15,511,210,043,330,985,984,000,000$ ends in six zeros. Thus, there is no integer $n$ such that  $n!$ ends in exactly five zeros. Let $S$ be the set of all $k$ such that for no integer n does $n!$ end in exactly $k$ zeros. If the numbers in $S$ are listed in increasing order, 5 will be the first number. Find the 100th number in that list.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{504}
In an acute triangle \(ABC\), determine the values of \(\alpha\) such that point \(A\) is equidistant from the circumcenter \(O\) and the orthocenter \(H\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60^\circ}
Find the monic real polynomial $f(x)$ such that $f(1/x) = f''(x) = c$ as $x$ tends to infinity, and $f(x) \ge f(1)$ for all real $x$. Determine $f(x)$. 
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{x^2 - 2x + 2}
Among six different quadratic trinomials, differing by permutation of coefficients, what is the maximum number that can have two distinct roots?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Compute the unique positive integer \( n \) such that
\[
3 \cdot 2^3 + 4 \cdot 2^4 + 5 \cdot 2^5 + \dots + n \cdot 2^n = 2^{n + 11}.
\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1025}
A robotic vacuum cleaner "Orthogonal" decided to clean the coordinate plane. It starts at point $O$, located at the origin, and travels a distance $X_{1}$ along a straight line. At the point where it stops, the vacuum turns and describes a circle with its center at the origin. After completing the circle, it continues straight without changing direction for a distance $X_{2}$. Then it turns to face the origin again to describe another circle centered at the origin, and after completing this circle, it travels another straight path for a distance of $X_{3}$. This pattern continues. It is known that the distances $X_{1}, X_{2}, X_{3}, \ldots$ follow a certain distribution with an expected value $a$ and variance $d$. Find the expected value of the area of the circle bounded by the $n$-th circle described by the vacuum cleaner.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi n \left( d + a^2 \right)}
A pyramid has a square base with side of length 1 and has lateral faces that are equilateral triangles. A cube is placed within the pyramid so that one face is on the base of the pyramid and its opposite face has all its edges on the lateral faces of the pyramid. What is the volume of this cube?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5\sqrt{2} - 7}
Alice picks a number from the set \( \{1, 2, \ldots, 1000000\} \). Bob can ask yes/no questions to determine Alice's number, but Alice is allowed to lie once. What is the minimum number of questions Bob must ask to guarantee he correctly identifies the number?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25}
Determine the number of idempotent matrices in the matrix space $M_2(\mathbb{F}_p)$. An idempotent matrix $A$ satisfies $A^2 = A$. Calculate the total number of such matrices in this space.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{p^2 + p + 2}
Evaluate the lambda calculus expression: 
\[(\lambda x.\lambda y.x + ( \lambda x.x+1) (x+y)) ( \lambda z.z-4\ 5)\ 10\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
Evaluate the limit: $$\lim_{n \to \infty} \frac{1}{n} \int_{0}^{n} \frac{x \ln(1 + \frac{x}{n})}{1 + x} \, dx.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 \ln 2 - 1}
Find the coefficient of \(x^{24}\) in the expansion of \((1 + x + x^2 + x^3 + x^4 + x^5)^8\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{98813}
A sequence $(a_n)$ is defined by means of the recursion\[a_1 = 1, a_{n+1} = \frac{1 + 4a_n +\sqrt{1+ 24a_n}}{16}.\]Find an explicit formula for $a_n.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
\[  \frac{(3 + 2^{2 - n})^2 - 1}{24} \]}
What digit must be placed in place of the "$?" in the number $888\ldots 88?999\ldots 99$ (where the $8$ and $9$ are each written $50$ times) so that the resulting number is divisible by $7$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
On an auto trip, the distance read from the instrument panel was $450$ miles. With snow tires on for the return trip over the same route, the reading was $440$ miles. Find, to the nearest hundredth of an inch, the increase in radius of the wheels if the original radius was $15$ inches.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
0.34
}
Given that the line $l_{1}$: $3x-(k+2)y+3=0$ is parallel to the line $l_{2}$: $kx+(2k-3)y+1=0$, find $k=$____.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-9}
设函数 f:\bN_+\rightarrow\bN_+，满足对任意正整数 s,t，均有 f(t^2f(s))=sf(t)^2. 求 f(1998) 的最小可能值.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{120}
Determine the interval of convergence for the power series:

$$
1-\frac{x}{2 \cdot 2}+\frac{x^{2}}{3 \cdot 2^{2}}-\frac{x^{3}}{4 \cdot 2^{3}}+\ldots+(-1)^{n} \frac{x^{n}}{(n+1) 2^{n}}+\ldots
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-2, 2]}
Evaluate the limit: $$\lim_{n\to\infty}\left(\int _{0}^{n} \frac{\mathrm dx}{x^{x^n}}\right)^n$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Let $f(n) = 1 \times 3 \times 5 \times \cdots \times (2n-1)$. Compute the remainder when $f(1) + f(2) + f(3) + \cdots + f(2016)$ is divided by $100$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
Among the four faces of a pyramid, the maximum number of right-angled triangles is ____.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Solve the equation for real and positive parameters $a$, $b$, and $c$:

$$\sqrt{a+bx}+\sqrt{b+cx}+\sqrt{c+ax}=\sqrt{b-ax}+\sqrt{c-bx}+\sqrt{a-cx}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Let the odd function $f(x)$ defined on $\mathbb{R}$ satisfy $f(2-x) = f(x)$, and it is monotonically decreasing on the interval $[0, 1)$. If the equation $f(x) = -1$ has real roots in the interval $[0, 1)$, then the sum of all real roots of the equation $f(x) = 1$ in the interval $[-1, 7]$ is $\_\_\_\_\_\_$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
The LCM of two numbers is 76176 and HCF is 116. If one of the numbers is 8128, then what is the other number?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1087}
If the line \( x = \frac{\pi}{4} \) intercepts the curve \( C: (x - \arcsin a)(x - \arccos a) + (y - \arcsin a)(y + \arccos a) = 0 \) at a chord of length \( d \), find the minimum value of \( d \) as \( a \) varies.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\pi}{2}}
What is the $\iint_G x^2 z $ where $G$ is the closed cylinder $x^2 + y^2 = 4$, and $0 \le z \le 3$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36π}
Arrange $\frac{n(n+1)}{2}$ different numbers randomly into a triangular array as shown in the diagram. Let $M_k$ be the maximum number in the $k$-th row (from top to bottom). Find the probability that $M_1 < M_2 < \cdots < M_n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2^n}{(n+1)!}}
Given the functions \(f(x)=\frac{x+2}{1-2x}\) and \(g(x)=\frac{2x+1}{2-x}\), find the minimum number of compositions of \(f\) and \(g\), alternating between the two functions, that result in the identity function.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Given a hyperbola \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1\) with left and right foci \(F_{1} 、 F_{2}\), a line passing through \(F_{2}\) with an inclination angle of \(\frac{\pi}{4}\) intersects the hyperbola at point \(A\). If the triangle \(	riangle F_{1} F_{2} A\) is an isosceles right triangle, find the eccentricity of the hyperbola. Express your answer as a single value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{2}+1}
For \(x \in (-1,1)\), determine the smallest positive integer \(n\) such that the following inequality is satisfied:

\[\left|e^{2x}-\sum_{k=0}^n \frac{(2x)^k}{k!}\right| \leq \frac{|e^{2x}|}{10^{10}}\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
Evaluate the expression to 8 decimal places: $$ 16\left(\frac{1}{5}-\frac{1}{3} \times \frac{1}{5^{3}}+\frac{1}{5} \times \frac{1}{5^{5}}-\frac{1}{7} \times \frac{1}{5^{7}}+\frac{1}{9} \times \frac{1}{5^{9}}-\frac{1}{11} \times \frac{1}{5^{11}}\right)-4\left(\frac{1}{239}-\frac{1}{3} \times \frac{1}{239^{3}}\right) $$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3.14159265}
Consider the function $f:[1,3]\to\mathbf{R}$ given by $f(x)=-x^4+8x^3+ax^2+bx+d$, where $a$, $b$, and $d$ are real constants. Determine the values of $d$ such that $f$ has exactly 3 stationary points in the interval $(1,3)$ and satisfies $f(1)=f(3)=0$. Provide your answer as a set of values for $d$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-15, -9)}
Find the derivative of the function \( f(z) = z \text{Re}(z) + \bar{z} \text{Im}(z) + \bar{z} \) at the point \( z_0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 - i}
For every $a \in \mathbb N$ denote by $M(a)$ the number of elements of the set
\[ \{ b \in \mathbb N | a + b \text{  is a divisor of } ab \}.\]
Find $\max_{a\leq 1983} M(a).$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{121}
Calculate the sum \( \sum_{k=1}^{240} f(k) \), where the function \( f(n) \) is defined as follows for each positive integer \( n \):
\[
f(n) = \begin{cases} 
0, & \text{if } n \text{ is a perfect square,} \\
\left\lfloor \frac{1}{\{\sqrt{n}\}} \right\rfloor, & \text{if } n \text{ is not a perfect square,}
\end{cases}
\]
with \( \lfloor x \rfloor \) denoting the greatest integer less than or equal to \( x \), and \( \{x\} = x - \lfloor x \rfloor \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{768}
Let $p$ be an odd prime, and let $(p^p)!=mp^k$ for some positive integers $m$ and $k$. Find in terms of $p$ the number of ordered pairs $(m,k)$ satisfying $m+k\equiv0\pmod p$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\dfrac{p^{p-1} + p - 2}{p - 1}$}
If the function $y=-x^{3}+ax$ is strictly decreasing on $\left[1,+\infty \right)$, then the range of values for $a$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left(-\infty, 3\right]}
Determine the number of differentiable functions on $(0, \infty)$ that satisfy the equation $f^{-1} = f'$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Find a subgroup $H$ of the multiplicative group $\Bbb{R}\setminus\{0\}$ such that for any $x, y \in \Bbb{R}\setminus\{0\}$, $xy^{-1} \in H$ if and only if $x$ and $y$ have the same sign.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{R}^+}
Let \( f(z) = \sin\left(z + \mathrm{e}^{3z}\right) \). Find \( \frac{\partial f}{\partial \bar{z}}(z) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Given that \( \det(X) = 1 \) for the matrix \( X = \begin{bmatrix} a & 1 & d \\ b & 1 & e \\ c & 1 & f \end{bmatrix} \) and \( \det(Y) = 4 \) for the matrix \( Y = \begin{bmatrix} a & 1 & d \\ b & 2 & e \\ c & 3 & f \end{bmatrix} \), find the determinant of the matrix \( Z = \begin{bmatrix} a & 5 & d \\ b & 7 & e \\ c & 9 & f \end{bmatrix} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
Given the vector field \( \vec{F}(x,y) = (y + xg(x), y^2) \) and the condition \( \vec{F}(1,1) = (3,1) \), find the function \( g(x) \) such that the partial derivatives \( \vec{F}_x \) and \( \vec{F}_y \) are orthogonal. Determine \( g(x) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2}{x}}
An urn contains 4 red balls and 3 blue balls. Picking a red ball earns you $1, while picking a blue ball costs you $1. You may stop picking balls at any time. What is the expected value of this game if you play optimally?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{58}{35}}
Compute the fundamental group of the space obtained by taking two copies of the 2-sphere $S^2$, denoted as $X_1$ and $X_2$, and a circle $S^1$, denoted as $X_3$. Identify the north pole $N_i$ of each $X_i$ with a point $n$ on $X_3$ and the south pole $S_i$ with a point $s$ on $X_3$, for $i = 1, 2$. What is the fundamental group of the resulting space?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{Z}}
Determine the radius of convergence for the complex series \(\sum_{n=0}^\infty c_nz^n\), where the coefficients \(c_n\) are given by \(c_n=(1/n!)\int_1^\infty t^{-1}e^{-t}(\log t)^ndt\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Brian goes fishing a certain number of times as often as Chris, but catches 2/5 times fewer fish than Chris per trip. If Brian caught 400 fish every time he went fishing, they caught 13600 fish altogether when Chris went fishing 10 times. What is the ratio of the number of times Brian goes fishing to the number of times Chris goes fishing?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{26}{15}}
James buys 4 packs of sodas with 10 sodas each, 3 packs of juice with 8 juice boxes each, and 2 packs of water bottles with 15 bottles each. He also buys 7 additional energy drinks. He already has 12 sodas and 5 juice boxes at home. James drinks 3 sodas, 2 juice boxes, and 1 water bottle from Monday to Wednesday. From Thursday to Sunday, he drinks 2 sodas, 4 juice boxes, 1 water bottle, and 1 energy drink per day. Assuming that he drinks at the same rate throughout the entire week, how many total beverages does he drink by the end of the week?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50}
Find the limit of the sequence \(a_n = \frac{\sqrt[n]{\binom{2n+3}{n+2}}}{\sqrt[n]{(n+2)!}}.\)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Calculate the spectral radius of the Volterra operator $A$ acting on $L^2[0,1]$, defined by $$A(f)(x)=\int_0^x f(t) dt,$$ using the spectral radius formula for bounded linear operators: $$\rho(A)=\lim_{n\rightarrow \infty} \|A^n\|^{1/n}.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
At an international conference, “red” world countries, “blue” world countries, and "green" world countries are the participants. The ratio of “red” world participants to “blue” world participants is 3:2, and the ratio of “blue” world participants to "green" world participants is 5:4. If one-third of “red” world participants are left-handed, two-thirds of “blue” world participants are left-handed, and one-fourth of "green" world participants are ambidextrous (and not considered left-handed), then what is the fraction of the participants who are left-handed, considering participants from all three worlds?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{35/99}
Find the continuous least squares trigonometric polynomial \( S_{2}(x) \) for \( f(x) = 2x - 1 \) on the interval \([-\pi, \pi]\). Express \( S_{2}(x) \) in the form \( \frac{a_{0}}{2} + a_{1}\cos(x) + a_{2}\cos(2x) + b_{1}\sin(x) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1 + 4\sin(x)}
Determine the number of positive integers \( x \) less than 10000 such that the difference \( 2^x - x^2 \) is not divisible by 7.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7142}
Let \( S \) be the set of all nondegenerate triangles formed from the vertices of a regular octagon with side length 1. Find the ratio of the largest area of any triangle in \( S \) to the smallest area of any triangle in \( S \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3 + 2\sqrt{2}}
Evaluate the integral \( \int_C \frac{e^{1/z^2}}{z-1} \, dz \), where \( C \) is the positively oriented circle \( |z| = \frac{1}{2} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\pi i (1 - e)}
Determine the dimension of the vector space \( \mathbb{C}[[x,y]]/(2xy-y^2,x^2-2xy) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}