inclusionAI/Ring-lite-rl-data huggingface dataset:
If x = y = 2z, and the value of x is 7.999999999999999, what is the product of x, y, and z?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{255.9999999999998}
Given the sets $S=\{x\mid (x-2)^{2} > 9\}$ and $T=\{x\mid a < x < a+8\}$, with $S\cup T=\mathbb{R}$, find the range of values for the real number $a$ in interval notation.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-3,-1)}
Last year, Springtown Hardware ordered 3 claw hammers and 2 ball-peen hammers in June, 4 claw hammers and 3 ball-peen hammers in July, 6 claw hammers and 7 ball-peen hammers in August, and 9 claw hammers and 11 ball-peen hammers in September. If the pattern continues for each type of hammer separately and there is a 5% increase in the combined orders due to seasonal demand in October, how many hammers will the store order in total in October (considering both types)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
Given the complex numbers $z\_1=a^2-2-3ai$ and $z\_2=a+(a^2+2)i$, if $z\_1+z\_2$ is a purely imaginary number, determine the value of the real number $a$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2}
$N$ is a $50$-digit number (in the decimal scale). All digits except the $26^{\text{th}}$ digit (from the left) are $1$. If $N$ is divisible by $13$, what is the value of the $26^{\text{th}}$ digit?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
The center of a balloon is observed by two ground observers at angles of elevation of $45^{\circ}$ and $22.5^{\circ}$, respectively. The first observer is to the south, and the second one is to the northwest of the point directly under the balloon. The distance between the two observers is 1600 meters. How high is the balloon floating above the horizontal ground?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{500}
If 7 < x < 9 < y < some number, then the greatest possible positive integer difference of x and y is 7. What is that number?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
The numbers \(1, 2, 3, \ldots, 29, 30\) are written in a row in a random order, and their partial sums are calculated: the first sum \(S_1\) is equal to the first number, the second sum \(S_2\) is equal to the sum of the first and second numbers, \(S_3\) is equal to the sum of the first, second, and third numbers, and so on. The last sum \(S_{30}\) is equal to the sum of all the numbers. What is the highest possible count of odd numbers among the sums \(S_{1}, S_{2}, \ldots, S_{30}\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{23}
A positive integer $n$ is defined as a \textit{stepstool number} if $n$ has one less positive divisor than $n + 1$. For example, $3$ is a stepstool number, as $3$ has $2$ divisors and $4$ has $2 + 1 = 3$ divisors. Find the sum of all stepstool numbers less than $300$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{687}
Regular tetrahedron  $ABCD$  has center  $O$  and side length  $1$ . Points  $A'$ ,  $B'$ ,  $C'$ , and  $D'$  are defined by reflecting  $A$ ,  $B$ ,  $C$ , and  $D$  about  $O$ . Compute the volume of the polyhedron with vertices  $ABCDA'B'C'D'$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{-3/2}}
The train arrives at a station randomly between 1:00 PM and 3:00 PM and waits for 30 minutes before departing. If John also arrives randomly at the station within the same time period, what is the probability that he will find the train at the station?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{7}{32}}
How many different solutions are there on the interval $[0, \pi]$ for the equation: $$6\sqrt{2} \sin x \cdot \tan x - 2\sqrt{2} \tan x + 3\sin x - 1 = 0$$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Find the area in the first quadrant bounded by the hyperbola \(x^{2}-y^{2}=1\), the \(x\)-axis, and the line \(3x=4y\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\ln 7}{4}}
As shown in the diagram, \(FGHI\) is a trapezium with side \(GF\) parallel to \(HI\). The lengths of \(FG\) and \(HI\) are 50 and 20 respectively. The point \(J\) is on the side \(FG\) such that the segment \(IJ\) divides the trapezium into two parts of equal area. What is the length of \(FJ\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{35}
Let $\Omega$ be a circle with radius $18$ and let $\mathcal{S}$ be the region inside $\Omega$ that the centroid of $\triangle XYZ$ sweeps through as $X$ varies along all possible points lying outside of $\Omega$, $Y$ varies along all possible points lying on $\Omega$ and $XZ$ is tangent to the circle. Compute the greatest integer less than or equal to the area of $\mathcal{S}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{904}
Let {a_n} be a geometric sequence with the first term 1 and common ratio q, and let S_n be the sum of the first n terms. Define b_n = S_n + 2. Find the value of q such that {b_n} is also a geometric sequence.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3}{2}}
Devin starts out as 65 inches tall and grows 3 inches. The chances of making the junior high basketball team are given by the following formula: 

P(Team) = Base Probability + Height Factor * (h - 66) + Age Factor * (a - 12) + PPG Modifier * p

Here, h is the height in inches, a is the age in years, and p is the percentage of games where the player scores above 10 points.

The Base Probability is 5%. The Height Factor is 8% for every inch above 66 inches but decreases by 1% for each year above 12 years old. The Age Factor is 3% for each year above 12 years old, up to a maximum of 9%. The PPG Modifier is 5% if the player has scored above 10 points in more than 50% of the games.

Devin is a 13-year-old with a height of 68 inches and has scored above 10 points in 40% of his games. What are his chances of making the basketball team?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{22\%}
设 $P$ 为正 2006 边形. 如果 $P$ 的一条对角线的两端将 $P$ 的边界分为两部分, 每部分都包含 $P$ 的奇数条边, 那么该对角线称为 "好边", 规定 $P$ 的每条边均为 "好边".已知 2003 条在 $P$ 内部不相交的对角线将 $P$ 分割成若干三角形. 试问在这种分割之下,最多有多少个有两条 "好边" 的等腰三角形?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1003}
记 $S=\left\{(x, y, z, w) \in \mathbb{Z}^{4} \mid 0 \leq x, y, z, w \leq 18, x^{2}+y^{2} \equiv z^{2}+w^{2}(\bmod 19)\right\}$,则 $S$ 中有 $\qquad$个元素.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7201}
In the crowded garage at Connor's house, there is space for only 50 total vehicles at any given time. At the moment, 20 bicycles, 10 cars, 5 motorcycles, 3 tricycles, and 2 quads are parked inside. However, during the next hour, 7 bicycles will leave, and 4 cars will arrive. Additionally, 3 new motorcycles will enter the garage while another 2 will leave. Keep in mind that 5 bicycles have one wheel missing, 2 cars have one broken wheel each, and a motorcycle has 2 wheels missing. Calculate the total number of wheels in the garage at the end of the hour, considering the incoming and outgoing vehicles.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{102}
Given $f(x) = (1+x)^m + (1+2x)^n$ ($m, n \in \mathbb{N}^*$), the coefficient of $x$ in its expansion is 11. When the coefficient of $x^2$ takes the minimum value, the sum of the coefficients of the odd powers of $x$ in the expansion of $f(x)$ is \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
Find the smallest positive integer $n$ such that the divisors of $n$ can be partitioned into three sets with equal sums.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{120}
Evaluate the following limit: 

$$\lim_{x \to 0} x \tan \left(ax + \arctan\left(\frac{b}{x}\right)\right)$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{b}{1 - ab}}
Calculate the volumes of the bodies formed by the rotation of the figures bounded by the graphs of the functions about the $y$-axis.
$$
y = \arccos \frac{x}{3}, \quad y = \arccos x, \quad y = 0
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\pi^2}
4. Find the minimum value of the function $f(x)=\sum_{i=1}^{2011}|i x-1|$.

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Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{592043}{711}}
Sansa is a famous artist, she can draw a portrait and sell it according to its size. She sells an 8-inch portrait for $5, and a 16-inch portrait for a certain multiple of the price of the 8-inch portrait. If she sells three 8-inch portraits and five 16-inch portraits per day, she earns $195 every 3 days. What is the ratio of the price of the 16-inch portrait to the price of the 8-inch portrait?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2:1}
A wealthy king has his blacksmith fashion him a large cup, whose inside is a cone of height 9 inches and base diameter 6 inches. At one of his many feasts, he orders the mug to be filled to the brim with cranberry juice. For each positive integer $n$, the king stirs his drink vigorously and takes a sip such that the height of fluid left in his cup after the sip goes down by $\frac{1}{n^{2}}$ inches. Shortly afterwards, while the king is distracted, the court jester adds pure Soylent to the cup until it's once again full. The king takes sips precisely every minute, and his first sip is exactly one minute after the feast begins. As time progresses, the amount of juice consumed by the king (in cubic inches) approaches a number $r$. Find $r$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{216 \pi^{3}-2187 \sqrt{3}}{8 \pi^{2}}}
A novel is recorded onto compact discs, taking a total of 505 minutes to read aloud. Each disc can hold up to 53 minutes of reading. Assuming the smallest possible number of discs is used and each disc contains the same length of reading, calculate the number of minutes of reading each disc will contain.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50.5}
Rachel has two indistinguishable tokens, and places them on the first and second square of a $1 \times 6$ grid of squares. She can move the pieces in two ways: If a token has a free square in front of it, then she can move this token one square to the right. If the square immediately to the right of a token is occupied by the other token, then she can 'leapfrog' the first token; she moves the first token two squares to the right, over the other token, so that it is on the square immediately to the right of the other token. If a token reaches the 6th square, then it cannot move forward any more, and Rachel must move the other one until it reaches the 5th square. How many different sequences of moves for the tokens can Rachel make so that the two tokens end up on the 5th square and the 6th square?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{42}
Richard has an infinite row of empty boxes labeled $1, 2, 3, \ldots$ and an infinite supply of balls. Each minute, Richard finds the smallest positive integer $k$ such that box $k$ is empty. Then, Richard puts a ball into box $k$, and if $k \geq 3$, he removes one ball from each of boxes $1,2,\ldots,k-2$. Find the smallest positive integer $n$ such that after $n$ minutes, both boxes $9$ and $10$ have at least one ball in them.

[i]Proposed by [b]vvluo[/b] & [b]richy[/b][/i]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{89}
Determine the number of integers  $a$  with  $1\leq a\leq 1007$  and the property that both  $a$  and  $a+1$  are quadratic residues mod  $1009$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{251}
Find the greatest positive real number $M$ such that for all positive real sequence $(a_n)$ and for all real number $m < M$, it is possible to find some index $n \ge 1$ that satisfies the inequality $a_1 + a_2 + a_3 + ...+ a_n +a_{n+1} > m a_n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ M = 4 }
Let $\zeta_7= e^{i2\pi/7}$ be the $7$th root of unity. Find the quadratic subfield of $\mathbb Q(\zeta_7)$ which can be expressed in the form $\mathbb Q(\sqrt D)$, where $D$ is an integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-7}
An engineer designed a ball so that when it was dropped, it rose with each bounce exactly one-half as high as it had fallen. The engineer dropped the ball from a certain height and caught it after it had traveled 45 meters. The ball bounced 4 times. From what height did the engineer drop the ball?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
30 volleyball teams have participated in a league. Any two teams have played a match with each other exactly once. At the end of the league, a match is called **unusual** if the winner of the match has fewer wins than the loser. A team is called **astonishing** if all its matches are **unusual** matches. Find the maximum number of **astonishing** teams.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Given the recursive relationship $(2n+3)a_{n+1}-(2n+5)a_n=(2n+3)(2n+5)\lg (1+ \frac{1}{n} )$ and the initial condition $a_1=5$, find the 2017th term of the sequence $\left\{ \frac{a_n}{2n+3} \right\}$. Express your answer as a single value, including any necessary logarithmic terms.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1+\lg 2017}
On sides \( BC \) and \( AC \) of triangle \( ABC \), points \( D \) and \( E \) are chosen respectively such that \( \angle BAD = 50^\circ \) and \( \angle ABE = 30^\circ \). Find \( \angle BED \) if \( \angle ABC = \angle ACB = 50^\circ \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40}
A lighthouse is located 38° north by east of a ship. Determine the direction of the ship from the lighthouse, including the degree value. Express your answer in the format "Direction by Direction X°" (e.g. South by west X°).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{South by west 38°}
Given the set M = {-1, 0, 1} and N as the set of natural numbers, find the intersection M ∩ N. Express your answer as a set, with elements separated by commas.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{{0, 1}}
Evaluate the following limit:
$$L=\lim_{n\to\infty}\int_0^1\frac{\int_0^t(\frac{x^{n+1}-1}{x-1})\mathrm dx}{t}\mathrm dt $$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$$\frac{\pi^2}{6}$$}
Suppose rectangle $F O L K$ and square $L O R E$ are on the plane such that $R L=12$ and $R K=11$. Compute the product of all possible areas of triangle $R K L$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{414}
Given four distinct real numbers \( a, b, c, d \) such that \(\frac{a}{b} + \frac{b}{c} + \frac{c}{d} + \frac{d}{a} = 4\) and \( ac = bd \), find the maximum value of \(\frac{a}{c} + \frac{b}{d} + \frac{c}{a} + \frac{d}{b} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-12}
If an integer $n > 8$ is a solution of the equation $x^2 - ax + b = 0$ and the representation of $a$ in the base-$n$ number system is $18$, what is the base-$n$ representation of $b$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{80}
Given the function $f(x)=2x-\sin x$, if the positive real numbers $a$ and $b$ satisfy $f(a)+f(2b-1)=0$, then the minimum value of $\dfrac {1}{a}+ \dfrac {4}{b}$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9+4 \sqrt {2}}
Determine \( q(x) \) if the graph of \( \frac{x^3-2x^2-5x+6}{q(x)} \) has vertical asymptotes at \( 3 \) and \( -3 \), no horizontal asymptote, and \( q(2) = 18 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{q(x) = -\frac{18}{5}x^2 + \frac{162}{5}}
If I choose four cards from a standard $52$-card deck, without replacement, what is the probability that I will end up with one card from each suit, in a sequential order (e.g., clubs, diamonds, hearts, spades)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2197}{499800}}
Given that in a convex quadrilateral $ABCD$, $\angle BAC=30^\circ, \angle ACD=40^\circ,\angle ADB=50^\circ,\angle CBD=60^\circ$ and $\angle ABC+\angle ADC>180^\circ.$, if $\angle A=\angle C=a^\circ, \angle B=b^\circ, \angle D=c^\circ.$, find the value of $a+b+c$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{290}
On a plane, there are a finite number of polygons. If any two of these polygons have a line passing through the origin that intersects both of them, then these polygons are said to be properly placed. Find the smallest natural number \( m \) such that for any set of properly placed polygons, we can draw \( m \) lines passing through the origin such that each of these polygons intersects with at least one of these \( m \) lines.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
A marketing firm determined that, of 160 households surveyed, 80 used neither brand A nor brand B soap. 60 used only brand A soap and for every household that used both brands of soap, some used only brand B soap. 5 of the 200 households surveyed used both brands of soap. What is the ratio of households that used only brand B soap to those that used both brands?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11:1}
The graph of the function $f(x)=\sin(\omega x+\varphi)$, where $(\omega>0, |\varphi|<\frac{\pi}{2})$, passes through the point $(0,-\frac{1}{2})$. Find the minimum value of $\omega$ if the graph of this function is shifted to the right by $\frac{\pi}{3}$ units and becomes symmetric about the origin.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{5}{2}}
Given the line $y=x+\sqrt{6}$, the circle $(O)$: $x^2+y^2=5$, and the ellipse $(E)$: $\frac{y^2}{a^2}+\frac{x^2}{b^2}=1$ $(b > 0)$ with an eccentricity of $e=\frac{\sqrt{3}}{3}$. The length of the chord intercepted by line $(l)$ on circle $(O)$ is equal to the length of the major axis of the ellipse. Find the product of the slopes of the two tangent lines to ellipse $(E)$ passing through any point $P$ on circle $(O)$, if the tangent lines exist.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
Let $ABCD$ be an isosceles trapezoid with $AD \parallel BC$ and $BC > AD$ such that the distance between the incenters of $\triangle ABC$ and $\triangle DBC$ is $16$. If the perimeters of $ABCD$ and $ABC$ are $120$ and $114$ respectively, then the area of $ABCD$ can be written as $m\sqrt{n}$, where $m$ and $n$ are positive integers with $n$ not divisible by the square of any prime. Find $100m+n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7114}
A farmer bought $749$ sheep. He sold $700$ of them for the price paid for the $749$ sheep. The remaining $49$ sheep were sold at the same price per head as the other $700$. Based on the cost, the percent gain on the entire transaction is:
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$7$}
Let $E\subseteq\mathbb{R}$ be a Borel measurable set with $m(E)=0$ and $f(x)=x^{2}$. Determine if $m(f(E))=0$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
5. In a regular $1000-$gon, all diagonals are drawn. What is the maximum number of diagonals that can be selected such that among any three of the selected diagonals, at least two have the same length?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2000}
Calculate the line integral of the vector field \( \mathbf{F}(x,y,z) = \left(\frac{-y}{x^2+z^2}, \frac{x}{x^2+z^2}, \frac{z}{x^2+z^2}\right) \) over the curve \( \gamma \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\pi}
Let $ABC$ be a triangle such that $AB = 7$, and let the angle bisector of $\angle BAC$ intersect line $BC$ at $D$. If there exist points $E$ and $F$ on sides $AC$ and $BC$, respectively, such that lines $AD$ and $EF$ are parallel and divide triangle $ABC$ into three parts of equal area, determine the number of possible integral values for $BC$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
4. For any integer $n(n \geqslant 4)$, consider $m$ subsets $A_{1}, A_{2}, \cdots, A_{m}$ of the set $\{1,2, \cdots, n\}$, where $A_{1}$ has one element, $A_{2}$ has two elements, $\cdots \cdots A_{m}$ has $m$ elements, and no subset is contained in another subset. Find the maximum value of $m$.


Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n-2}
$A,B,C$ are the three angles in a triangle such that
$2\sin B\sin (A+B)-\cos A=1$,
$2\sin C\sin (B+C)-\cos B=0$
find the three angles.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$A=120^\circ, B=C=30^\circ$}
In the game of Twister, a spinner randomly selects either an arm or a leg, and also selects one of four colors, one of which is red, each with equal probability, and players have to move the appropriate body part to the appropriately colored spot on the ground. There are four players. Each player spins once, and makes the move the spinner selects. What is the probability that in these four spins, there will be exactly two moves to a red spot, and the body part selected to move will be an arm exactly $3$ times?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{27}{512}}
Ralph has a cylinder with height $15$ and volume $\frac{960}{\pi}$ . What is the longest distance (staying on the surface) between two points of the cylinder?

Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{17}
Let $G$ be a $30$-regular graph. Determine whether $G$ must contain a path of length at least $20$. Answer with 'yes' or 'no'.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{yes}
Determine whether the series \(\sum_{k=1}^\infty \frac{1}{4^k(2k)}\binom{2k}{k}\) converges to \(\ln 2\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\ln 2}
A frog starts at $(1, 1)$. Every second, if the frog is at point $(x, y)$, it moves to $(x + 1, y)$ with probability $\frac{x}{x+y}$ and moves to $(x, y + 1)$ with probability $\frac{y}{x+y}$ . The frog stops moving when its $y$ coordinate is $10$. Suppose the probability that when the frog stops its $x$-coordinate is strictly less than $16$, is given by $m/n$ where $m, n$ are positive integers that are relatively prime. Find $m + n.$

Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
There are 144 bottle caps in a box. Marvin takes 63 bottle caps and gives half of them to his friend John. Then, Marvin's cousin adds 56 more bottle caps to the box. How many bottle caps are in the box now?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{137}