inclusionAI/Ring-lite-rl-data huggingface dataset:
The bisector of $\angle BAC$ in $\triangle ABC$ intersects $BC$ in point $L$. The external bisector of $\angle ACB$ intersects $\overrightarrow{BA}$ in point $K$. If the length of $AK$ is equal to the perimeter of $\triangle ACL$, $LB=1$, and $\angle ABC=36^\circ$, find the length of $AC$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
A finite increasing sequence $a_{1}, a_{2}, \ldots, a_{n}$ ($n \geq 3$) of natural numbers is given, such that for all $k \leq n-2$, the equality $a_{k+2}=3 a_{k+1}-2 a_{k}-1$ holds. The sequence must contain the term $a_{k}=2021$. Determine the maximum number of three-digit numbers divisible by 25 that this sequence can contain.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36}
The degrees of polynomials $P$ and $Q$ with real coefficients do not exceed $n$. These polynomials satisfy the identity
\[ P(x) x^{n + 1} + Q(x) (x+1)^{n + 1} = 1. \]
Determine all possible values of $Q \left( - \frac{1}{2} \right)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^n}
Given the polynomial f(x) = x^5 + 2x^3 + 3x^2 + x + 1, apply Horner's method to calculate the value at x = 3. What is the value of v_3? Express your answer as a single numerical value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36}
If 7 boys meet at a reunion and each boy shakes hands with each of the others a certain number of times, then the total number of handshakes is 21. How many times does each boy shake hands with the others?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Two circles touch each other internally at point K. Chord \( AB \) of the larger circle touches the smaller circle at point L, with \( A L = 10 \). Find \( B L \) given that \( A K : B K = 2 : 5 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25}
A loaf of sausage has thin transverse rings drawn on it. If you cut along the red rings, you get 5 pieces, if along the yellow rings, you get 7 pieces, and if along the green rings, you get 11 pieces. How many pieces of sausage will you get if you cut along the rings of all three colors?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21}
In a drawer Sandy has $5$ pairs of socks, each pair a different color.  On Monday Sandy selects two individual socks at random from the $10$ socks in the drawer.  On Tuesday Sandy selects $2$ of the remaining $8$ socks at random and on Wednesday two of the remaining $6$ socks at random.  The probability that Wednesday is the first day Sandy selects matching socks is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers,  Find $m+n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{341}
A class has a group of 7 students, and now select 3 of them to swap seats with each other, while the remaining 4 students keep their seats unchanged. Calculate the number of different ways to adjust their seats.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{70}
ABC is an acute-angled triangle. B' is a point on the perpendicular bisector of AC on the opposite side of AC to B such that angle AB'C = 2∠A. A' and C' are defined similarly (with ∠CA'B = 2∠C, ∠BC'A = 2∠B). The lines AA' and B'C' meet at A". The points B" and C" are defined similarly. Find AA'/A"A' + BB'/B"B' + CC'/C"C'.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Find the maximum real number \( M \) such that for all real numbers \( x \) and \( y \) satisfying \( x + y \geqslant 0 \), the following inequality holds:
$$
\left(x^{2}+y^{2}\right)^{3} \geqslant M\left(x^{3}+y^{3}\right)(xy - x - y).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32}
Find the value of the integral $$\int_{0}^{\infty} \frac{\ln^2 z}{1+z^2} \, dz$$ without using contour integration. Utilize standard special functions, such as the zeta, gamma, or beta functions.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{\pi^3}{8}\)}
Consider the sequence: $x_1 = 19$, $x_2 = 95$, $x_{n+2} = \text{lcm}(x_{n+1}, x_n) + x_n$, for $n > 1$, where $\text{lcm}(a, b)$ means the least common multiple of $a$ and $b$. Find the greatest common divisor of $x_{1995}$ and $x_{1996}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{19}
已知向量$\overrightarrow{a}=(0,2,1)$，$\overrightarrow{b}=(-1,1,2)$，则$\overrightarrow{a}$与$\overrightarrow{b}$的夹角为________
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\arccos \frac{2\sqrt{30}}{15}$}
A regular tetrahedron with four equilateral triangular faces has a sphere inscribed within it and another sphere circumscribed about it. Each of the four faces of the tetrahedron is tangent to a unique external sphere which is also tangent to the circumscribed sphere, but now these external spheres have radii larger than those in the original setup. Assume new radii are 50% larger than the radius of the inscribed sphere. A point $P$ is selected at random inside the circumscribed sphere. Compute the probability that $P$ lies inside one of these external spheres.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.5}
Given the function $f(x)=\ln(e^{x}+e^{-x})+x^{2}$, find the range of $x$ for which $f(x) > f(2x-1)$ holds.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left(\frac{1}{3},1\right)}
The square  $BCDE$  is inscribed in circle  $\omega$  with center  $O$ .  Point  $A$  is the reflection of  $O$  over  $B$ . A "hook" is drawn consisting of segment  $AB$  and the major arc  $\widehat{BE}$  of  $\omega$  (passing through  $C$  and  $D$ ). Assume  $BCDE$  has area  $200$ .  To the nearest integer, what is the length of the hook?

*Proposed by Evan Chen*
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{67}
Annie was given a pack of crayons. The pack contained some crayons. She already had a box of 36 crayons in her locker. Her friend Bobby gave her half the amount she already had in her locker. She decided to give her sister Mary 25 crayons. How many crayons were in the pack she was given?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{29}
The equation $x^5-2x^4-1=0$ has five complex roots $r_1,r_2,r_3,r_4,r_5$.  Find the value of \[\dfrac1{r_1^8}+\dfrac1{r_2^8}+\dfrac1{r_3^8}+\dfrac1{r_4^8}+\dfrac1{r_5^8}.\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
The surface area of a spherical cap is $n$ times larger than the lateral surface area of the cone that can be inscribed in the corresponding spherical slice. What is the ratio of the height of the slice to the diameter of the sphere?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{n^2 - 1}{n^2}}
Let $$f(x)= \lim_{n \to \infty}\dfrac{3^n(\overbrace{\sin(\sin(...\sin(x)))}^{\text{n times}}+(\sqrt 2 \cos x+2)^n+2^n\cos x}{3^n+\sin x(\sqrt 2\cos x+2)^n}.$$ If $l = \lim\limits_{x\to \frac{\pi}{4}^+}f(x)$ and $m = \lim\limits_{x\to \frac{\pi}{4}^-}f(x)$, find the value of $l^2+m^2$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
The real numbers \( x_1, x_2, \ldots, x_{1997} \) have a sum of \(-318\sqrt{3}\) and satisfy \(-\frac{1}{\sqrt{3}} \le x_i \le \sqrt{3}\). What is the maximum possible value for the sum of their 12th powers?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{189548}
In $\triangle ABC$ in the adjoining figure, $AD$ and $AE$ trisect $\angle BAC$. The lengths of $BD$, $DE$ and $EC$ are $2$, $3$, and $6$, respectively. The length of the shortest side of $\triangle ABC$ is
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\sqrt{10}}
Find the value of $$\lim_{x\to-\infty} \frac{\int_0^x e^{t^2} dt}{\sqrt{-\pi x}}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Given points $A(1,m)$ and $B(1,2\sqrt{5}-m)$, if there exists only one point $P$ on the circle $C:x^{2}+y^{2}+2x=0$ such that $PA\perp PB$, then one possible value of the real number $m$ is _____. (Provide one value that satisfies the condition)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{5}+2}
At an interview, ten people were given a test consisting of several questions. It is known that any group of five people together answered all the questions (i.e., for each question, at least one of the five gave the correct answer), but any group of four did not. What is the minimum number of questions this could have been?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{210}
Given a convex quadrilateral $ABCD$ inscribed in a circle with diameter $1$. Inside $ABCD$, there exists a point $M$ such that the sum of its squared distances to the vertices is $2$, i.e., $|MA|^2 + |MB|^2 + |MC|^2 + |MD|^2 = 2$. Determine the area of $ABCD$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Reese joined a Pinterest group where members contributed an average of some pins per day. The group owner deleted older pins at the rate of 5 pins per week per person. The group has 20 people and the total number of pins is 1000. After Reese had been a member for a month, there were 6,600 pins. How many pins did members contribute per day on average?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Task 2. (10 points) A circle touches the extensions of two sides $A B$ and $A D$ of square $A B C D$ with a side length of 4 cm. Two tangents are drawn from point $C$ to this circle. Find the radius of the circle if the angle between the tangents is $60^{\circ}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4(\sqrt{2}+1)}
Let \( k=-\frac{1}{2}+\frac{\sqrt{3}}{2} \mathrm{i} \). In the complex plane, the vertices of \(\triangle ABC\) correspond to the complex numbers \( z_{1}, z_{2}, z_{3} \) which satisfy the equation
\[ z_{1}+k z_{2}+k^{2}\left(2 z_{3}-z_{1}\right)=0 \text {. } \]

Find the radian measure of the smallest interior angle of this triangle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\pi}{6}}
Given the curve $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 (a \geqslant b > 0)$ passes through the point $(\sqrt{3}, 1)$, find the area of the region formed by all points on these curves that satisfy $y \geqslant 1$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{4\pi}{3} - \sqrt{3}}
There are two people wearing red clothes, two people wearing yellow clothes, and one person wearing blue clothes. When arranging these five people in a row, people wearing the same color cannot be adjacent. Calculate the total number of different arrangements.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{48}
求值: \(\lim _{n \rightarrow+\infty}\left(\sum_{k=1}^{n} \arctan \frac{2}{k^{2}}\right) = x\pi\)，其中答案为\(\frac{k}{m}\)的形式，请计算k+m的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
A pair of dice is rolled twice, and the coordinates of point $P(m,n)$ are determined by the sequence of the dots $m$ and $n$ obtained. The probability that point $P$ lies outside the circle $x^2+y^2=17$ is $\_\_\_\_\_\_\_.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{13}{18}}
Two circles have radii $2$ and $3$, and the distance between their centers is $10$. Let $E$ be the intersection of their two common external tangents, and $I$ be the intersection of their two common internal tangents. Compute $EI$.

A common external tangent is a tangent line to two circles such that the circles are on the same side of the line, while a common internal tangent is a tangent line to two circles such that the circles are on opposite sides of the line.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
In the coordinate plane, a rectangle is drawn with vertices at $(34,0),(41,0),(34,9),(41,9)$. Find the smallest value of the parameter $a$ for which the line $y=ax$ divides this rectangle into two parts such that the area of one part is twice the area of the other. If the answer is not an integer, write it as a decimal fraction.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.08}
Charles Dickens' 200th anniversary of birth was celebrated on February 7, 2012, which was a Tuesday. A year is a leap year if it is divisible by 400 or divisible by 4 but not by 100. Using this rule, calculate the day of the week on which Charles Dickens was born. Provide your answer as the full day of the week.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{Friday}
In a distant future, humans have built 6 cities on Mars, evenly distributed on a circle. If a straight high-speed track is laid between any two cities, and a station is set up at each intersection of the tracks, with each of the 6 cities also having a station, then a total of $\qquad$ stations will be set up.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{19}
If the proposition $p$: "There exists $x_0 \in \mathbb{R}$, such that $2^{x_0}-2 \leqslant a^2-3a$" is a false proposition, then the range of values for the real number $a$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[1,2]}
Xiao Wang and Xiao Li have to process the same number of identical parts and start working at the same time. It is known that Xiao Wang processes 15 parts per hour and must rest for 1 hour after every 2 hours of work. Xiao Li works continuously, processing 12 parts per hour. Both of them finish at the exact same time. How many parts did Xiao Wang process?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60}
Determine the fixed point through which the graph of the function $f(x)=\log_{a}(2x-1)+1$ (where $a > 0$ and $a \neq 1$) always passes.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(1,1)}
A trader cheats both his supplier and customer by using faulty weights. When he buys from the supplier, he takes 10% more than the indicated weight. When he sells to his customer, he gives the customer a weight such that some percentage of that is added to the weight, the weight claimed by the trader is obtained. If he charges the cost price of the weight that he claims, his profit percentage is 43. What is the percentage by which the trader adds weight when selling to the customer?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30\%}
A root of unity is a complex number that is a solution to $z^{n}=1$ for some positive integer $n$. Determine the number of roots of unity that are also roots of $z^{2}+a z+b=0$ for some integers $a$ and $b$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Define a $\emph{smart integer}$ as an even integer that is greater than 20, less than 120, and such that the sum of its digits is 10. What fraction of all smart integers is divisible by 25? Express your answer as a common fraction.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Let  $m, n, a, k$  be positive integers and  $k>1$  such that the equality   $$ 5^m+63n+49=a^k $$  holds. Find the minimum value of  $k$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
How many parallelograms with sides of lengths 1 and 2 and angles of $60^{\circ}$ and $120^{\circ}$ can be placed inside a regular hexagon with a side length of 3?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
How many 9-digit numbers are there in the decimal system that are divisible by 11 and include each digit from 1 to 9 exactly once?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{31680}
7. In $\square A B C D$, $A B=4, B C=2, \angle D A B$ $=60^{\circ}$, a point $P$ on side $B C$ and a point $Q$ on the extension of $D C$ satisfy $S_{\triangle B P Q}=\frac{\sqrt{3}}{32}$. Then the minimum value of $\overrightarrow{A P} \cdot \overrightarrow{B Q}$ is $\qquad$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{97}{16}}
Chidi is a small business owner in Rivers State, Nigeria, who runs a local grocery store. Every week, he buys 120 bags of rice at ₦8,500 per bag and 200 crates of eggs at ₦1,500 per crate. If Chidi aims to make a profit of 20% on the rice and 25% on the eggs, what should be the total selling price for all the rice and eggs for that week?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{₦1599000}
How many points does one have to place on a unit square to guarantee that two of them are strictly less than 1/2 unit apart?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
In a café, 55 Indians and Turks met, each of whom drank either tea or coffee. All Indians tell the truth when they drink tea and lie when they drink coffee, while all Turks do the opposite. When asked "Are you drinking coffee?" 44 people answered "yes", when asked "Are you a Turk?" 33 people answered "yes", and 22 people agreed with the statement "It is raining outside." How many Indians in the café are drinking tea?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
During a certain two-week period, 48 percent of the movies rented from a video store were comedies, 16 percent were action movies, and the remaining movies rented consisted of dramas, thrillers, and sci-fi movies. If the number of rented dramas was 3 times the number of action movies, the number of rented thrillers was 2 times the number of rented dramas, and the number of rented sci-fi movies was equal to the number of rented action movies, and there were a action movies rented, then how many comedies, in terms of a, were rented during that two-week period?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(40/3)a}
How many distinct permutation cycles can be expressed for the elements $(a, b, c, d)$, considering cycles to be different if a cyclic shift results in a new cycle? For instance, $(a, b, c, d)$ and $(d, a, b, c)$ are considered different.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
已知$M$是抛物线${{x}^{2}}=4y$上一点，$F$为其焦点，点$A$在圆$C:{{\left( x+1 \right)2}}+{{\left( y-6 \right)}^{2}}=1$上，则$\left| MA \right|+\left| MF \right|$的最小值是__________．
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
A table can seat 6 people. Two tables joined together can seat 10 people. Three tables joined together can seat 14 people. Following this pattern, if 10 tables are arranged in two rows with 5 tables in each row, how many people can sit?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{44}
Find the largest positive integer $n$ such that the following equations have integer solutions in $x, y_1, y_2, \ldots, y_n$:
\[
(x+1)^2 + y_1^2 = (x+2)^2 + y_2^2 = \ldots = (x+n)^2 + y_n^2.
\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
The incircle $\Gamma$ of a scalene triangle $ABC$ touches $BC$ at $D$, $CA$ at $E$, and $AB$ at $F$. Let $r_A$ be the radius of the circle inside $\triangle ABC$ which is tangent to $\Gamma$ and the sides $AB$ and $AC$. Define $r_B$ and $r_C$ similarly. If $r_A = 16$, $r_B = 25$, and $r_C = 36$, determine the radius of $\Gamma$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{74}
Given a positive integer  $n$ , suppose that  $P(x,y)$  is a real polynomial such that 
\[P(x,y)=\frac{1}{1+x+y} \hspace{0.5cm} \text{for all  $x,y\in\{0,1,2,\dots,n\}$ } \] What is the minimum degree of  $P$ ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2n}
The side of an equilateral triangle is $a$. Determine the area of the part of the triangle that lies outside a circle with radius $a / 3$, whose center coincides with the center of the triangle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{a^2 (3\sqrt{3} - \pi)}{18}}
Let  $A_1$ ,  $\ldots$ ,  $A_{2022}$  be the vertices of a regular  $2022$ -gon in the plane. Alice and Bob play a game. Alice secretly chooses a line and colors all points in the plane on one side of the line blue, and all points on the other side of the line red. Points on the line are colored blue, so every point in the plane is either red or blue. (Bob cannot see the colors of the points.)

In each round, Bob chooses a point in the plane (not necessarily among  $A_1, \ldots, A_{2022}$ ) and Alice responds truthfully with the color of that point. What is the smallest number  $Q$  for which Bob has a strategy to always determine the colors of points  $A_1, \ldots, A_{2022}$  in  $Q$  rounds?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{22}
A shopping mall's main staircase from the 1st floor to the 2nd floor consists of 15 steps. Each step has a height of 16 centimeters and a depth of 26 centimeters. The width of the staircase is 3 meters. If the cost of carpeting is 80 yuan per square meter, how much will it cost to buy the carpet needed for the staircase from the 1st floor to the 2nd floor?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1512}
On their school trip, Lisa, Robert, Claire, James, and Emily took photos at various locations. Lisa took 3 times as many photos as Claire, whereas Robert took 5/4 of the number of photos that Claire took. James, on the other hand, took 250% of the photos Robert took, and Emily took the average of the photos that Lisa and Claire took. Added to this, Paul joined them recently but he took only half the photos Emily took. Given that the combined total number of photos taken by all six students is 840, how many photos has Claire taken?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{74}
I am preparing for a French exam, which includes recalling vocabulary. There are 800 words in total on the exam, and the exam grade is based on the percentage of these words that I recall correctly. If I randomly guess the words I do not learn, my guessing has a 5% chance of being correct. What is the least number of words I should learn to ensure I score at least $90\%$ on the exam?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{716}
The cost price of some books is equal to the selling price of 16 books. The gain or loss percentage is 50%. How many books were bought at the cost price?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
The fractional equation $\dfrac{x-5}{x+2}=\dfrac{m}{x+2}$ has a root, determine the value of $m$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-7}
From the numbers 1, 2, 3, 5, 7, 8, two numbers are randomly selected and added together. Among the different sums that can be obtained, let the number of sums that are multiples of 2 be $a$, and the number of sums that are multiples of 3 be $b$. Then, the median of the sample 6, $a$, $b$, 9 is ____.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5.5}
The graph of the function $y=x^2$ ($x>0$) has a tangent line at the point $(a_k, a_k^2)$, which intersects the x-axis at point with an x-coordinate of $a_k + b_k$. Here, $b_k$ is a positive integer. If $a_1=16$, find the sum $a_1 + a_3 + a_5$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{336}