inclusionAI/Ring-lite-rl-data huggingface dataset:
Find the value of x. (47% of some number - 36% of 1412) + 63 = x. If x is 3, what is the number?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{953.87}
Points $A, B, Q, D,$ and $C$ lie on the circle shown. The measures of arcs $\widehat{BQ}$ and $\widehat{QD}$ are $42^\circ$ and $38^\circ$, respectively. Find the sum of the measures of angles $P$ and $Q$ in degrees.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40}
If the product of a and b is 240, the average (arithmetic mean) of b and c is 60, and the average of c and d is 90, what is the value of d - a, given that a, b, c, and d are all positive integers?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{116}
A function \( g \) is ever more than a function \( h \) if, for all real numbers \( x \), we have \( g(x) \geq h(x) \). Consider all quadratic functions \( f(x) \) such that \( f(1)=16 \) and \( f(x) \) is ever more than both \( (x+3)^{2} \) and \( x^{2}+9 \). Across all such quadratic functions \( f \), compute the minimum value of \( f(0) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{21}{2}}
Find the asymptotic bounds for the recurrence relation \(T(n)=T(n/4+n^{1/3})+n^{1/2}\) given that \(T(n)=1\) for \(n\le 5\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\Theta(\sqrt{n})}
The ones digit of the product of four consecutive positive integers is 4. If this product is greater than 1000, what is the sum of the four smallest such integers?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
In the figure below, $ADE$ is a triangle with $\angle AED = 120^\circ$, and $B$ and $C$ are points on side $AD$ such that $BCE$ is an equilateral triangle. If $AB = 4$ cm, $CD = 16$ cm, and $BC = x$ cm, find the value of $x$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
In the diagram, points P, Q, and R lie on a circle with center O and radius 12, and point S lies on OR. If angle POR = 135 degrees, calculate the exact area of trapezoid OPQS.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{108}
In the figure, letters represent different digits within the range 1-9. Adjacent cells can form 24 two-digit numbers (e.g., $\overline{\mathrm{AB}}, \overline{\mathrm{BA}}, \overline{\mathrm{GD}}$), and three digits in the same row or column can form 12 three-digit numbers in sequence (e.g., $\overline{\mathrm{ABC}}, \overline{\mathrm{BEH}}, \overline{\mathrm{HEB}}$). Among these 36 numbers, the maximum number of composite numbers is $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{33}
Find all positive integers  $k$ , for which the product of some consecutive  $k$  positive integers ends with  $k$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1, 2, 4}
There are  $1,000,000$  piles of  $1996$  coins in each of them, and in one pile there are only fake coins, and in all the others - only real ones. What is the smallest weighing number that can be used to determine a heap containing counterfeit coins if the scales used have one bowl and allow weighing as much weight as desired with an accuracy of one gram, and it is also known that each counterfeit coin weighs  $9$  grams, and each real coin weighs  $10$  grams?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
A starship enters an extraordinary meteor shower. Some of the meteors travel along a straight line at the same speed, equally spaced. Another group of meteors travels similarly along another straight line, parallel to the first, with the same speed but in the opposite direction, also equally spaced. The ship travels parallel to these lines. Astronaut Gavrila recorded that every 7 seconds the ship encounters meteors coming towards it, and every 13 seconds it overtakes meteors traveling in the same direction. He wondered how often the meteors would pass by if the ship were stationary. He thought that he should take the arithmetic mean of the two given times. Is Gavrila correct? If so, write down this arithmetic mean. If not, indicate the correct time in seconds, rounded to the nearest tenth.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9.1}
Let \( n \) be a positive integer and \( \mathbf{A} \) an \( n \)-by-\( n \) matrix with integer entries such that \( k = \big|\det(\mathbf{A})\big| \neq 0 \). Determine the number of \( n \)-by-\( 1 \) column vectors \( \mathbf{x} \in \mathbb{Q}^n \) with entries in the interval \([0,1)\) that satisfy \( \mathbf{A} \mathbf{x} = \mathbf{b} \) for some \( n \)-by-\( 1 \) column vector \( \mathbf{b} \in \mathbb{Z}^n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{k}
Given a positive integer $n>1$. In the cells of an $n\times n$ board, marbles are placed one by one. Initially there are no marbles on the board. A marble could be placed in a free cell neighboring (by side) with at least two cells which are still free. Find the greatest possible number of marbles that could be placed on the board according to these rules.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ n^2 - n }
Compute the sum of all 2-digit prime numbers  $p$  such that there exists a prime number  $q$  for which  $100q + p$  is a perfect square.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{179}
Find the least natural number $n$, such that the following inequality holds:$\sqrt{\dfrac{n-2011}{2012}}-\sqrt{\dfrac{n-2012}{2011}}<\sqrt[3]{\dfrac{n-2013}{2011}}-\sqrt[3]{\dfrac{n-2011}{2013}}$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
4024
}
What is the range of the function \( y = \arcsin[\sin x] + \arccos[\cos x] \) (for \( x \in [0, 2\pi) \) and where \([x]\) denotes the greatest integer less than or equal to \( x \))?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{{0, π/2, π}}
Among 100 points in the plane, no three are collinear, and exactly 4026 pairs are connected by line segments. Each point is randomly assigned a distinct integer from 1 to 100. Find the expected value of the number of segments that join two points whose labels differ by at least 50.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1037}
The number $12ab9876543$ is divisible by $101$, where $a, b$ represent digits between $0$ and $9$. What is $10a + b$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{58}
Determine the number of non-invertible elements in the multiplicative group $(\mathbb{Z}/p^n\Bbb{Z})^\times$, where $p$ is a prime number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{p^{n-1}}
Calculate the moment of inertia about the x-axis for the region bounded by the curves \( x = 2y - y^2 \) and \( x = y^2 \), given that the density at any point \((x, y)\) is \( y + 1 \). Use double integration to find your answer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{6}}
Find the volume bounded by the surfaces \(x^2+z=1\) and \(y^2+z=1\) in the first octant.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
Compute the Cauchy principal value of the integral \( \int_{-\infty}^{\infty} \frac{x}{(x^{2}+4)(x-1)} \, dx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2\pi}{5}}
How can we calculate $|M|^2$ for the complex expression $M=\frac{e^{2(1+{\rm i})l}-e^{-2(1+{\rm i})l}}{e^{2(1+{\rm i})x}-e^{-2(1+{\rm i})x}}$, where $l$ and $x$ are real numbers?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\cosh(4l)-\cos(4l)}{\cosh(4x)-\cos(4x)}}
Rectangle $ABCD$, pictured below, shares $50\%$ of its area with square $EFGH$. Square $EFGH$ shares $20\%$ of its area with rectangle $ABCD$. What is $\frac{AB}{AD}$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Find the maximum number $E$ such that the following holds: there is an edge-colored graph with 60 vertices and $E$ edges, with each edge colored either red or blue, such that in that coloring, there is no monochromatic cycles of length 3 and no monochromatic cycles of length 5 .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1350}
Deanna has $44$ pieces of candy. What is the largest number of kids whom she can give an equal number of candies such that every kid gets at least $3$ pieces?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
What is the $111^{\text{st}}$ smallest positive integer which does not have $3$ and $4$ in its base-$5$ representation?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{755}
Determine the value of $-1+2+3+4-5-6-7-8-9+...+10000$, where the signs change after each perfect square.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1000000}
Determine the radius of convergence of the power series \( \sum_{n=0}^{\infty} \left(\sqrt{4^n + 3^n} + (-1)^n \sqrt{4^n - 3^n}\right)x^n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
Find the value of $\sin(x)$ that maximizes the expression $(5+3\sin(x))^2 (7-3\sin(x))^3$. What is this value of $\sin(x)$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{1}{15}}
Determine the number of surjective ring homomorphisms from \(\mathbb{Z}[i]\) to \(\mathbb{F}_{11^2}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Compute the Lie bracket of the vector fields \(\xi = \left(\frac{x}{\sqrt{x^2 + y^2}}, \frac{y}{\sqrt{x^2+y^2}} \right)\) and \(\eta = (-y, x)\) in \(\mathbb{R}^2\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(0, 0)}
Consider $h+1$ chess boards. Number the squares of each board from 1 to 64 such that when the perimeters of any two boards are aligned in any possible manner, no two squares in the same position have the same number. What is the maximum value of $h$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Two robots, A and B, start from the same point on a circular track with a circumference of 8 meters and move in opposite directions. Robot A walks 5 meters per minute, while Robot B walks 3 meters per minute. When they meet head-on, Robot B turns around. If Robot A catches up with Robot B from behind, Robot A turns around. Determine the distance from the starting point to the location where they meet for the 10th time. (Catching up from behind is also considered a meeting.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Given the hyperbola $C$: $\frac{x^{2}}{4} - \frac{y^{2}}{b^{2}}=1 (b > 0)$, the distance from its focus to the asymptote is $3$. Find the length of the conjugate axis of the hyperbola $C$. Express your answer as a numerical value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
From a \(6 \times 6\) checkered square, gray triangles were cut out. What is the area of the remaining figure? The side length of each cell is 1 cm. Give the answer in square centimeters.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{27}
Two sides of a regular \( n \)-gon are extended to meet at a \( 28^\circ \) angle. What is the smallest possible value for \( n \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{45}
Determine the order of the normalizer of the subgroup \( H = \langle (1,2)(3,4),(1,3)(2,4) \rangle \) in the alternating group \( A_5 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
In $\Delta XYZ$, $\overline{MN} \parallel \overline{XY}$, $XM = 5$ cm, $MY = 8$ cm, and $NZ = 9$ cm. What is the length of $\overline{YZ}$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{23.4}
Evaluate the integral: $$\int_{0}^{1} \left( 1 + \sin^2 \left( \frac{1}{x} \right) + \frac{1}{x^2} \cos^2 \left( \frac{1}{x} \right) - \frac{1}{x} \sin \left( \frac{2}{x} \right) \right) dx$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Van Helsing gets paid by the town to remove all the vampires and werewolves. He gets $5 per vampire and $10 per werewolf. He removes half the vampires and removes 8 werewolves. There were 4 times as many werewolves as vampires. How much did he earn in total?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$85}
The numbers from 1 to 8 are placed at the vertices of a cube such that the sum of the numbers at any three vertices lying on the same face is at least 10. What is the minimum possible sum of the numbers at the vertices of one face?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
Given the equation \( z - c = \rho\frac{1 + it}{1 - it} \), where \( c \) is a complex number, \( \rho \) is a real number, and \( t \) is a real parameter, find the relationship between \( z \), \( c \), \( \rho \), and \( t \). Assume \( z = x + iy \) and \( c = a + ib \), where \( x \), \( y \), \( a \), and \( b \) are real numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{|z - c| = \rho}
Evaluate the limit: \[ \lim_{n\to\infty}\int_0^1 n\ln\left(1+\frac{1}{n\sqrt{x}}\right)\, dx \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Calculate the area of the region enclosed by the curves $xy=10$, $xy=16$, $xy^4=3$, and $xy^4=9$. Provide your answer as a single numerical value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 \ln 3}
The cook, engineer, and sailor have a conversation about their ages:

- Cook: "When I was as old as the sailor is now, I was twice as old as him."
- Engineer: "I am only 4 years older than the sailor."
- Sailor: "Only the cook's age is an odd number, and the least common multiple of our three ages is the captain's year of birth."

Determine the captain's year of birth.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1938}
A father tells his son: "Today is both of our birthdays, and you are exactly 2 times younger than me. This is the eighth time in your life that you have been younger than me by an integer multiple." If the father is no older than 75 years, how old is the son?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
Albert has a very large bag of candies and he wants to share all of it with his friends. At first, he splits the candies evenly amongst his $20$ friends and himself and he finds that there are five left over. Ante arrives, and they redistribute the candies evenly again. This time, there are three left over. If the bag contains over $500$ candies, what is the fewest number of candies the bag can contain?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{509}
Quadrilateral \(ABCD\) is inscribed in a circle. At point \(C\), a tangent \(\ell\) to this circle is drawn. Circle \(\omega\) passes through points \(A\) and \(B\) and touches the line \(\ell\) at point \(P\). Line \(PB\) intersects segment \(CD\) at point \(Q\). Find the ratio \(\frac{BC}{CQ}\), given that \(B\) is tangent to circle \(\omega\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
A magician has one hundred cards numbered 1 to 100. He puts them into three boxes, a red one, a white one and a blue one, so that each box contains at least one card. A member of the audience draws two cards from two different boxes and announces the sum of numbers on those cards. Given this information, the magician locates the box from which no card has been drawn. 

How many ways are there to put the cards in the three boxes so that the trick works?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
The ten smallest positive odd numbers \( 1, 3, \cdots, 19 \) are arranged in a circle. Let \( m \) be the maximum value of the sum of any one of the numbers and its two adjacent numbers. Calculate the minimum value of \( m \). Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{33}
From the numbers 1, 2, …, 9, select $n$ numbers, among which there must be several numbers (at least one, or all of them) whose sum is divisible by 10. Find the minimum value of $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
100 thimbles are arranged in a circle, and a coin is hidden under one of them. In one move, you can flip four thimbles to check if the coin is under one of them, then return them to their original position. After each move, the coin moves to a neighboring thimble. What is the minimum number of moves required to guarantee finding the coin?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50}
Determine the values of \( n \) for which a cube with edge length \( n \) can be composed of white and black unit cubes such that each white cube shares a face with exactly three black cubes, and each black cube shares a face with exactly three white cubes.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Given a monotonically increasing sequence, $\{a_n\}$, satisfying $a_n + a_{n+1} + a_{n+2} = 3n - 6$, and $a_2 = \frac {1}{2}a_1$, determine the range of possible values for $a_1$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left(-\frac {12}{5}, -\frac {3}{2}\right)}
Jessica had 7 roses in the vase. She went to her flower garden, which had a total of 84 roses, and cut some roses. However, her friend Sarah, who initially gave her 15 roses, wanted 10 of the roses from the garden back. After giving Sarah her roses, Jessica put the remaining roses into the vase. There are now 37 roses in the vase. How many roses did Jessica cut from her garden?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40}
In triangle \( ABC \), \( AB = 33 \), \( AC = 21 \), and \( BC = m \), where \( m \) is a positive integer. If point \( D \) can be found on \( AB \) and point \( E \) can be found on \( AC \) such that \( AD = DE = EC = n \), where \( n \) is a positive integer, what must the value of \( m \) be?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
In a convex 13-gon, all diagonals are drawn. They divide it into polygons. Consider the polygon with the largest number of sides among them. What is the greatest number of sides that it can have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
Four points \( B, A, E, L \) are on a straight line. \( G \) is a point off the line such that \(\angle B A G = 120^\circ\) and \(\angle G E L = 80^\circ\). If the reflex angle at \( G \) is \( x^\circ \), then \( x \) equals:
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{340}
A fair coin is tossed. If the first toss shows heads, it is tossed again: if the second toss shows heads, record 0; if the second toss shows tails, record 1. If the first toss shows tails, record any random real number in the closed interval [0, 1]. This process is repeated two independent times, obtaining two real numbers x and y. What is the probability that |x-y| > 1/2?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{7}{16}$}
Calculate the Gaussian curvature $K$ for the set $S = \{(x,y) \in \mathbb{R}^2 \mid x^2 + y^2 < 1\}$, given that the first fundamental form is $ds^2 = \frac{4}{(1-x^2-y^2)^2}(dx^2 + dy^2)$. The coefficients of the first fundamental form are $E = G = \frac{4}{(1-x^2-y^2)^2}$ and $F = 0$. Express your answer in terms of $x$ and $y$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
Find the limit of $P_{n-t}(n)$ as $n$ approaches infinity, where $P_{k}(n)$ is the number of partitions of $n$ into $k$ parts and $t$ is a fixed non-negative integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{p(t)}
Given a pentagon $ABCDE$ with the following properties:

1. Angles $\angle A$ and $\angle C$ are right angles.
2. The side lengths are $|AB| = 4$, $|BC| = 5$, $|CD| = 10$, and $|DE| = 6$.
3. Point $C'$ is the reflection of point $C$ across line $BD$ and lies on line segment $AE$.

Find the measure of angle $\angle E$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{90}
The squares in a  $7\times7$  grid are colored one of two colors: green and purple. The coloring has the property that no green square is directly above or to the right of a purple square. Find the total number of ways this can be done.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3432}
Calculate the Fourier coefficient $\hat{s}(-1)$ for the 1-periodic signal $s : \mathbb{R}/\mathbb{Z} \to \mathbb{C}$ defined by the equation $s(t) = (2\cos(\pi t))^{16}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11440}
Some of the 20 metal cubes, which are identical in size and appearance, are made of aluminum, and the rest are made of duralumin (which is heavier). How can you determine the number of duralumin cubes using 11 weighings on a balance scale without weights?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
A woman carried water from the well to the kitchen in a straight cylindrical bucket. The bucket had a hole on the side at a height of \(\frac{1}{4}\) of the bucket's height from the bottom, causing half of the water to leak out on her way from the well to the kitchen. After she fixed this hole, the bottom of the bucket developed a hole, and \(\frac{2}{3}\) of the water leaked out on the same journey. She fixed this hole too, but later both plugs failed, and by the time she was 1 meter away from the kitchen, only \(\frac{1}{40}\) of the water was left in the bucket. How far was the well from the kitchen? Assume water leaks evenly over time from each hole.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{140}{23}}
Can you use the four basic arithmetic operations (addition, subtraction, multiplication, division) and parentheses to write the number 2016 using the digits 1, 2, 3, 4, 5, 6, 7, 8, 9 in sequence?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2016}
The work team was working at a rate fast enough to process  $1250$  items in ten hours. But after working for six hours, the team was given an additional  $150$  items to process. By what percent does the team need to increase its rate so that it can still complete its work within the ten hours?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30\%}
Fill the numbers $10, 15, 20, 30, 40,$ and $60$ into the circles in the figure, so that the product of the 3 numbers at the vertices of each of the three smaller triangles A, B, and C are all equal. What is the maximum value of this equal product?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18000}
Evaluate the limit: $$\lim_{x \rightarrow \infty} \frac{\int_0^x\frac{\sin(t)dt}{t}-x}{x^3}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}