inclusionAI/Ring-lite-rl-data huggingface dataset:
In Smithtown, the ratio of right-handed people to left-handed people is 3 to 1 and the ratio of men to women is some ratio. If the number of right-handed men is maximized, then 25.00000000000001 percent of all the people in Smithtown are left-handed women. What is the ratio of men to women?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Determine the number of integer solutions to the equation \(\frac{xy}{z}+\frac{xz}{y}+\frac{yz}{x} = 6\), where \(x, y, z\) are non-zero integers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
Find the smallest integer value of \(a\) such that the function \(f(x) = \int_{0}^{x} |\log_{2}(\log_{3}(\log_{4}(\cos t + a)))| \, dt\) is increasing for all \(x \in \mathbb{R}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
It is known that the optimal amount of a certain material to be added is between 100g and 1100g. If the 0.618 method is used to arrange the experiment and the first and second trials are at points $x_1$ and $x_2$ ($x_1 > x_2$), then when $x_2$ is considered the better point, the third trial point $x_3$ should be __g (answer with a number).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{336}
For the NEMO, Kevin needs to compute the product:
\[
9 \times 99 \times 999 \times \cdots \times 999999999.
\]
Kevin takes exactly $ab$ seconds to multiply an $a$-digit integer by a $b$-digit integer. Compute the minimum number of seconds necessary for Kevin to evaluate the expression together by performing eight such multiplications.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{870}
If the proposition "For all $x$ in the interval $[-1, 1]$, the inequality $1 + 2^x + a \cdot 4^x < 0$ is false," then the minimum value of the real number $a$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-6}
Evaluate the integral \( \int_C \frac{\cos(z)}{z^2 - 4z + 5} \, dz \), where \( C \) is the contour consisting of the real line segment from \(-r\) to \(r\) and the upper semi-circle of radius \(r\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi \cos(2 + i)}
Two trains of different lengths are running in opposite directions, one at the rate of 40 km and the other at the rate of 50 kmph. The first train is 111 meters long. They will be completely clear of each other from the moment they meet in 11.039116870650348 seconds. What is the length of the second train?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{164.98}
What is the maximum number of distinct numbers from 1 to 1000 that can be selected so that the difference between any two selected numbers is not equal to 4, 5, or 6?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{400}
Given a positive integer $\mathrm{n}$, write the $\mathrm{n}$ positive integers $1, 2, \ldots, \mathrm{n}$ on paper, where some numbers are written in red and some in blue, with both colors used, and each number is only written in one color. For $1 \leq i \leq j \leq n$, if there are an odd number of blue numbers among $i, i+1, \ldots, j$, then the pair of numbers $(i, j)$ is called "good." Find the maximum possible number of good pairs.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left\lfloor \frac{(n+1)^2}{4} \right\rfloor}
Given the recurrence relation \(a_{n+1}a_{n} = a_{n}^{2} + 1\) with \(a_{0} = 5\), determine \(\left \lfloor{a_{1000}}\right \rfloor\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{45}
[ Area and orthogonal projection ]

Find the area of the section of a regular triangular prism ABCA1B1C1 by a plane passing through vertex C and the midpoint of side B1C1 of the base A1B1C1, and parallel to the diagonal AC1 of the lateral face AA1C1C, if the distance between the line AC1 and the cutting plane is 1, and the side of the base of the prism is $\sqrt{14}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{21}{4}}
Find the minimum value of \( \overline{AP} + \overline{AQ} \), where \( A(-3,2) \) is a point on the coordinate plane. A line \( L \) is perpendicular to the line \( x = y \). Line \( L \) intersects the curve \( y = 2^x \) at point \( P \) and the curve \( y = \log_2 x \) at point \( Q \). Determine the minimum value of the sum of the distances from \( A \) to \( P \) and \( Q \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5\sqrt{2}}
Isosceles trapezoid \(ABCD\) has parallel sides \(\overline{AD}\) and \(\overline{BC},\) with \(BC < AD\) and \(AB = CD.\) There is a point \(P\) in the plane such that \(PA=1, PB=2, PC=3,\) and \(PD=4.\) Find the value of \(\frac{BC}{AD}.\) The original answer is in \(\frac{k}{m}\) format, where \(\frac{k}{m}\) is in simplest form. Please give the value of \(k + m\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
From how many of the vertices of a convex $n$-gon can you choose four such that the quadrilateral they form has sides that are diagonals of the $n$-gon?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{n}{4}\binom{n-5}{3}}
$A,B,C$ are playing backgammon tournament. At first, $A$ plays with $B$. Then the winner plays with $C$. As the tournament goes on, the last winner plays with the player who did not play in the previous game. When a player wins two successive games, he will win the tournament. If each player has equal chance to win a game, what is the probability that $C$ wins the tournament?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$\frac{2}{7}$

}
Determine the number of L-structures of size \( n \) for a language \( L \) that contains a single binary function symbol \( + \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n^{n^2}}
Given a Markov jump process for a gas station with a single gas pump, cars arrive at a rate of 20 vehicles per hour. Each car takes 6 minutes (1/10 hour) to fill up. The system states are 0, 1, or 2 cars at the pump. The transition rate matrix is:
$$
Q = \begin{bmatrix}
-20 & 20 & 0 \\
10 & -30 & 20 \\
0 & 10 & -10
\end{bmatrix}
$$
The stationary distribution is $(u_1, u_2, u_3) = \left(\frac{1}{7}, \frac{2}{7}, \frac{4}{7}\right)$. Calculate the average number of cars served by the gas station per hour.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{60}{7}}
Three numbers form a geometric progression; the sum of the numbers is 65, and their product is 3375. What are these numbers?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5, 15, 45}
Find the value of $c$ such that $6x^2 + cx + 16$ equals the square of a binomial.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8\sqrt{6}}
Any right triangle has the property that the sum of the areas of the squares on its two legs is equal to the area of the square on its hypotenuse. This is the famous Pythagorean theorem, also known in the West as the Pythagorean theorem. The Pythagorean theorem has a history of 4000 years, with hundreds of different proofs. The ancient Chinese mathematician Liu Hui from the Wei and Jin periods provided a concise and beautiful proof method shown in the left diagram below. The right diagram is the Liu Hui block puzzle based on this method:

The five blocks of the Liu Hui block puzzle can be assembled into the parallelogram shown on the right. If the right triangle's legs are 3 cm and 4 cm, what is the perimeter of the parallelogram in $\qquad$ centimeters?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
Two circles with equal radii intersect as shown. The area of the shaded region equals the sum of the areas of the two unshaded regions. If the area of the shaded region is $216\pi$, what is the circumference of each circle?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36\pi}
The worm consists of a white head and several segments, as shown in the figure.

When the worm is born, it has a head and one white segment. Each day, a new segment is added to the worm in one of the following ways:

- Either a white segment splits into a white and a gray segment.
- Or a gray segment splits into a gray and a white segment.

(On the fourth day, the worm reaches maturity and stops growing - its body consists of a head and four segments.)

How many different color variants of mature worms of this species can exist?

Hint: What could a two-day-old worm look like?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
What is the weight of 3.75 moles of Na2Ca(CO3)2?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{772.8}
You have a $3 \times 2021$ chessboard from which one corner square has been removed. You also have a set of $3031$ identical dominoes, each of which can cover two adjacent chessboard squares. Let $m$ be the number of ways in which the chessboard can be covered with the dominoes, without gaps or overlaps. What is the remainder when $m$ is divided by $19$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
A hollow cylinder with a very thin wall and a block are placed at rest at the top of a plane with inclination  $\theta$  above the horizontal. The cylinder rolls down the plane without slipping, and the block slides down the plane; it is found that both objects reach the bottom of the plane simultaneously. Find the coefficient of kinetic friction between the block and the plane in terms of  $\theta$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{2} \tan \theta}
Calculate \( \lim _{ n\to \infty } \int _{ |x|<n }{ \int _{ |y|<n }{ \sin(x^2+y^2)\,dx\,dy } } \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi}
Determine the number of automorphisms of the group \( \Bbb Z \oplus \Bbb Z_3 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Let $f$ and $g$ be meromorphic functions. Suppose $f$ has a pole of order $k$ at $z_0$ and $g$ has a zero of order $m$ at $z_0$. Determine the condition on $k$ and $m$ for the function $f(z)g(z)$ to have a removable singularity at $z_0$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{m \geq k}
Determine the range of values for \( \beta > 0 \) such that the random walk defined by \( S_0 = 0 \) and \( S_n = \varepsilon_1 + \cdots + \varepsilon_n, n \geq 1 \), with i.i.d increments \( \lbrace \varepsilon_n \rbrace \) having the distribution \( P(\varepsilon_1 = -1) = 1 - C_{\beta} \) and \( P (\varepsilon_1 > t) = C_{\beta}e^{-t^{\beta}}, t \geq 0 \), where \( C_{\beta} \in (0,1) \) and \( \mathbb{E}\varepsilon_1 = -1/2 \), is heavy-tailed.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(0, 1)}
For composite number $n$, let $f(n)$ denote the sum of the least three divisors of $n$, and $g(n)$ the sum of the greatest two divisors of $n$. Find all composite numbers $n$, such that $g(n)=(f(n))^m$ ($m\in N^*$).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$n= 2^{m+1}3^{m-1}$  ($m \in \mathbb{N}^*$)

}
Let $M$ be the intersection of the diagonals $AC$ and $BD$ of cyclic quadrilateral $ABCD$. If $|AB|=5$, $|CD|=3$, and $m(\widehat{AMB}) = 60^\circ$, what is the circumradius of the quadrilateral?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$ \dfrac {7\sqrt 3}{3}$}
Find all positive integers \( n \) such that \( f(n^2+2) + f(n^2+5) = 2n-4 \), where \( f(m) \) is the \( \left\lfloor \frac{p(m)+1}{2} \right\rfloor \)th smallest prime divisor of \( m \) and \( p(m) \) is the number of distinct prime divisors of \( m \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Let \( a \) be an element of a group with infinite order. Determine the number of generators of the cyclic group \( \langle a \rangle \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Evaluate the integral \( \int\limits_{-\pi}^\pi\int\limits_{-\pi}^\pi\int\limits_{-\pi}^\pi{\det}^2\begin{Vmatrix}\sin \alpha x&\sin \alpha y&\sin \alpha z\\\sin \beta x&\sin \beta y&\sin \beta z\\\sin \gamma x&\sin \gamma y&\sin \gamma z\end{Vmatrix} \text{d}x\,\text{d}y\,\text{d}z \) where \(\alpha,\beta,\gamma\) are integers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6\pi^3}
Let $ABC$ be the triangle with vertices located at the center of masses of Vincent Huang's house, Tristan Shin's house, and Edward Wan's house; here, assume the three are not collinear. Let $N = 2017$, and define the $A$-ntipodes to be the points $A_1,\dots, A_N$ to be the points on segment $BC$ such that $BA_1 = A_1A_2 = \cdots = A_{N-1}A_N = A_NC$, and similarly define the $B$, $C$-ntipodes. A line $\ell_A$ through $A$ is called a qevian if it passes through an $A$-ntipode, and similarly we define qevians through $B$ and $C$. Compute the number of ordered triples $(\ell_A, \ell_B, \ell_C)$ of concurrent qevians through $A$, $B$, $C$, respectively.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
6049
}
Determine the cardinality of the set of chains of \((\mathcal{P}(\mathbb{N}), \subseteq)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{2^{\aleph_0}}}
Determine the values of $x$ for which the series \( \sum\limits_{n=1}^\infty \frac{\sin(2nx+x)-\sin(2nx-x)}{n} \) converges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{R}}
There are three alloys made of nickel, copper, and manganese. The first alloy contains 30% nickel and 70% copper, the second one has 10% copper and 90% manganese, and the third one consists of 15% nickel, 25% copper, and 60% manganese. A new alloy of these three metals, containing 40% manganese, needs to be created. What values can the percentage content of copper take in the new alloy?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{from 40\% to 43 \frac{1}{3}\%}
Determine the number of distinct topologies on a set with 4 elements.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{355}
Two trains of different lengths run at the speed of 68 kmph and 40 kmph in opposite directions in parallel tracks. The time which they take to cross each other is 11.999040076793857 seconds. If the longer train is 200 m long, how long is the shorter train?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{159.97}
Determine the smallest number of same-suited cards, n, out of 9 cards (2 cards for each of 2 players and 5 community cards) such that there is a 50% probability that at least one player has a flush.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Given that $\tan x + 2\tan y + 3\tan z = 40$ for $x, y, z \in \left(\frac{\pi}{2}, \frac{3\pi}{2}\right)$, find the minimum value of $\tan^2 x + \frac{\tan^2 y}{4} + \frac{\tan^2 z}{9}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{800}{49}}
Given the block matrix \( \pmatrix{A&B\\ C&D} \), determine the expression for its determinant if both pairs \((A,C)\) and \((A,B)\) commute.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\det(AD - BC)}
In the 10 by 10 grid of squares shown, point \( P \) can be at any of the 41 points of intersection of pairs of gridlines inside (and not on) \(\triangle F G H\). For each possible location of \( P \), exactly three triangles are formed: \(\triangle F P G\), \(\triangle G P H\), \(\triangle H P F\). How many of these triangles have an area that is exactly half of the area of \(\triangle F G H\)? Provide your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
The odd positive integers $1,3,5,7,\cdots,$ are arranged into in five columns continuing with the pattern shown on the right. Determine the column number in which 1985 appears.

[asy]
int i,j;
for(i=0; i<4; i=i+1) {
label(string(16*i+1), (2*1,-2*i));
label(string(16*i+3), (2*2,-2*i));
label(string(16*i+5), (2*3,-2*i));
label(string(16*i+7), (2*4,-2*i));
}
for(i=0; i<3; i=i+1) {
for(j=0; j<4; j=j+1) {
label(string(16*i+15-2*j), (2*j,-2*i-1));
}}
dot((0,-7)^^(0,-9)^^(2*4,-8)^^(2*4,-10));
for(i=-10; i<-6; i=i+1) {
for(j=1; j<4; j=j+1) {
dot((2*j,i));
}}
[/asy]

Express your answer as a column number (e.g., first, second, third, etc.).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\text{first}}
Calculate the variance of the integral \( \int_0^1 X(t) \, dt \), where \( X(t) \) is a stationary random process with expected value \( E[X(t)] = m \) and covariance function \( r_X(\tau) = 2e^{-\lvert \tau \rvert} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{4}{e}}
Let \(\zeta = e^{2\pi i / 7}\). Determine the degree of the field extension \([ \mathbb{Q}(\zeta) : \mathbb{Q}(\zeta) \cap \mathbb{R}]\].
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Simplify $(\sqrt{32})(\sqrt[5]{64})$ to the simplest radical form. Express any number inside a radical by its prime factorization.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$8(\sqrt[10]{2^{7}})$}
Given Abby finished the softball season with a total of 45 hits, among which were 2 home runs, 3 triples, and 7 doubles, calculate the percentage of her hits that were singles.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{73.33\%}
Ah Hua and Ah Hua did the same number of questions. Each correct answer adds 10 points, and each wrong answer deducts 5 points. In the end, Ah Hua's score is 30 points higher than Ah Hua's score. Given that Ah Hua got 5 questions correct, how many questions did Ah Hua get correct?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Given that $X$ and $Y$ are jointly distributed continuous random variables with $Y$ being positive and $E[Y^3]=27$, and the conditional distribution of $X$ given $Y=y$ is uniform on $(1-y, 1+y)$, find the expected value $E[X^3]$. Express your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Given 8 planes in space, for each pair of planes, the line of their intersection is noted. For each pair of these noted lines, the point of their intersection is noted (if the lines intersect). What is the maximum number of noted points that could be obtained?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{56}
Find a generator of the ideal generated by $3$ and $2 - 2\sqrt{-2}$ in the ring $\mathbb{Z}[\sqrt{-2}]$. The generator should be an element of the ideal with minimal norm. The norms of $3$ and $2 - 2\sqrt{-2}$ are $9$ and $12$, respectively. Determine this generator.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1 - \sqrt{-2}}
A clock shows the time as some hour in the morning. If the minute hand gains 7 minutes every hour, the clock will gain 63 minutes by 6 p.m. What time does the clock show initially?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9:00}
Evaluate the integral: \[ u = \int_0^\infty \frac{dx}{x^4 + 7x^2 + 1} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{6}}
Michelle is at the bottom-left corner of a $6 \times 6$ lattice grid, at $(0,0)$. The grid also contains a pair of one-time-use teleportation devices at $(2,2)$ and $(3,3)$. The first time Michelle moves to one of these points, she is instantly teleported to the other point and the devices disappear. If she can only move up or to the right in unit increments, in how many ways can she reach the point $(5,5)$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{280}
Ten people form a circle.  Each picks a number and tells it to the two neighbors adjacent to them in the circle.  Then each person computes and announces the average of the numbers of their two neighbors. The figure shows the average announced by each person (not the original number the person picked.)

The number picked by the person who announced the average $6$ was
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Find the expected number of additional coin flips needed to achieve a run of \( n+1 \) successive heads, given that it took \( k \) flips to achieve a run of \( n \) successive heads. Express this expectation as \( E(X_{n+1} | X_n = k) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{n+1}}
$ABCD$ is a rectangle with an area of 24. Find the area of the shaded part if $DQ = QC$ and $AP:PD = 1:2$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Determine the convergence domain of the series \(\sum_{n=1}^\infty \frac{z^n - (\overline{z})^n}{n}.\)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\{ z \in \mathbb{C} \mid |z| \leq 1 \}}
Fill the numbers 1 through 6 into the three equations in the diagram, such that each number is used exactly once, $\mathrm{A}+\mathrm{B}$ is a multiple of 2, $\mathrm{C}+\mathrm{D}$ is a multiple of 3, $\mathrm{E}+\mathrm{F}$ is a multiple of 5. Find the smaller number between $\mathrm{C}$ and $\mathrm{D}$. (Enter specific value)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
In the diagram, use the given angle measurements and geometric properties to calculate the value of \(x\). Express your answer in degrees as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60}
A random walker on the real line starts at position 25 and takes independent normal steps with mean 0 and variance 4. What is the probability that the walker will be at a position greater than 32 at 12 seconds? Express this probability as \( P(X_{12} > 32) \), where each step \( \Delta X_i \) is an independent normally distributed random variable with mean 0 and variance 4.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.1562}
How many times does the digit 9 appear in the list of all integers from 1 to 1000?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{300}
A school has carried out a variety of colorful club activities. Some students in a class signed up for two clubs, calligraphy and chess. The class monitor went to the mall to buy brushes and chess sets (both items) for the students, spending a total of $360. Each brush costs $15, and each chess set costs $20. Calculate the number of different purchasing plans that satisfy the given conditions.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Let the sequence of real numbers \(a_{0}, a_{1}, \cdots\) satisfy \(a_{0} = 0\) and \(a_{1} = 1\). For every positive integer \(n \geq 2\), there exists a positive integer \(k\) (where \(1 \leq k \leq n\)) such that
\[ 
a_{n} = \frac{a_{n-1} + a_{n-2} + \cdots + a_{n-k}}{k}.
\]

Find the maximum value of \(a_{2018} - a_{2017}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2016}{2017^2}}