inclusionAI/Ring-lite-rl-data huggingface dataset:
A square in the \( xy \)-plane has area \( A \), and three of its vertices have \( x \)-coordinates 2, 0, and 18 in some order. Find the sum of all possible values of \( A \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1168}
Let \(O\) be the origin. The line \(y = c\) intersects the curve \(y = 2x - 3x^3\) at points \(P\) and \(Q\) in the first quadrant and cuts the y-axis at \(R\). Find \(c\) so that the region \(OPR\) bounded by the y-axis, the line \(y = c\), and the curve has the same area as the region between \(P\) and \(Q\) under the curve and above the line \(y = c\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{4}{9}}
7. If $z$ is a complex number, and $|z|=1$, then the maximum value of $\left|\frac{\sqrt{3} \mathrm{i}-z}{\sqrt{2}-z}\right|$ is

保留源文本的换行和格式，翻译结果如下：

7. If $z$ is a complex number, and $|z|=1$, then the maximum value of $\left|\frac{\sqrt{3} \mathrm{i}-z}{\sqrt{2}-z}\right|$ is
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["\\sqrt{7}+\\sqrt{5}"]}
If a positive integer is equal to the sum of all its factors (including 1 but excluding the number itself), then this number is called a "perfect number". For example, 28 is a "perfect number" because $1 + 2 + 4 + 7 + 14 = 28$. If the sum of all factors of a positive integer (including 1 but excluding the number itself) is one less than the number, then this number is called an "almost perfect number". For example, 8 is an "almost perfect number" because $1 + 2 + 4 = 7$. The fifth "almost perfect number" in ascending order is     .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32}
Assume a particle X in a d-dimensional hypercube. Each dimension is independent of another and the particle's position is distributed uniformly in each. A distance measure of $D = \frac{1}{2} \max\limits_{i\in 1\cdots d} \lvert \frac{1}{2} - X_i\rvert$ is computed. What is $P(D >a)$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1-(4a)^d}
Given that $\tan (3π-x)=2$, find the value of $\frac {2\cos ^{2} \frac {x}{2}-\sin x-1}{\sin x+\cos x}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-3}
To the natural number \( N \), the largest divisor of \( N \) that is less than \( N \) was added, and the result is a power of ten. Find all such \( N \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{75}
We need to find a number which, when multiplied by itself, added to 2, then doubled, added to 3, divided by 5, and finally multiplied by 10, results in 50.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
8. Positive integers $a_{1}, a_{2}, \ldots, a_{7}, b_{1}, b_{2}, \ldots, b_{7}$ satisfy $2 \leq a_{i} \leq 166$ and $a_{i}^{b_{i}} \equiv a_{i+1}^{2}(\bmod 167)$ for each $1 \leq i \leq 7$ (where $a_{8}=a_{1}$ ). Compute the minimum possible value of $b_{1} b_{2} \cdots b_{7}\left(b_{1}+b_{2}+\cdots+b_{7}\right.$ ).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{675}
Laura wishes to cut a shape, made up of nine small squares, into pieces that she can then rearrange to make a $3 \times 3$ square. What is the smallest number of pieces that she needs to cut the shape into so that she can do this? Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
What is the expected number of steps an ant needs to take when performing a random walk on the edges of a cube to reach the diagonally opposite vertex?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
A $3 \times 3$ square is partitioned into $9$ unit squares. Each unit square is painted either white or black with each color being equally likely, chosen independently and at random. The square is then rotated $90^{\circ}$ clockwise about its center, and every white square in a position formerly occupied by a black square is painted black. The colors of all other squares are left unchanged. What is the probability the grid is now entirely black?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{49}{512}}
There is a triangular pyramid $A-BCD$, where the sum of the face angles at each vertex is $180^{\circ}$. The lengths of its three unequal edges are $\sqrt{34}$, $\sqrt{58}$, and $\sqrt{74}$. Find the volume of this pyramid.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{35}
Given an even function f(x) defined on the real numbers, where f(x+2) = f(x), and f(x) = x^2 + 4x + 3 for x in the interval [-3, -2], determine the number of zeros of the function y = f[f(x)] + 1 within the interval [-3, 3]. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
The sum of the positive numbers \(a, b, c,\) and \(d\) does not exceed 4. Find the maximum value of the expression

$$
\sqrt[4]{2 a^{2}+a^{2} b} + \sqrt[4]{2 b^{2}+b^{2} c} + \sqrt[4]{2 c^{2}+c^{2} d} + \sqrt[4]{2 d^{2}+d^{2} a}
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4 \sqrt[4]{3}}
Matt has some baseball cards worth $6 each. He trades two of them to Jane in exchange for 3 $2 cards and 1 $9 card. He makes a profit of $3. How many baseball cards did Matt have initially?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
由 $1,2, \cdots, 9$ 这九个正整数构成的所有圆排列中, 任意相邻两数之积均不超过 60 的圆排列的个数为 $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21600}
在平面直角坐标系内, 有四个定点 $A(-3,0), B(1,-1), C(0,3), D(-1,3)$及一个动点 $P$ ，则 $|P A|+|P B|+|P C|+|P D|$ 的最小值为原始答案是 a\sqrt{b}+c\sqrt{d} 格式，请给出 a+b+c+d的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Karen is constantly trying to stop the raccoons from getting in her trash. The first lock she tries stalls them for 5 minutes. The next lock stalls them for 3 minutes less than a certain multiple of the first lock's time. When Karen tries both locks at once, it stalled the raccoons for five times as long as the second lock alone, which took 60 minutes. What is the multiple of the first lock's time that the second lock is compared to?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Given the sets $A=\{x|x=2n-1,n\in\mathbb{N}^*\}$ and $B=\{x|x=2^n,n\in\mathbb{N}^*\}$. Arrange all elements of $A\cup B$ in ascending order to form a sequence $\{a_n\}$. Let $S_n$ denote the sum of the first $n$ terms of the sequence $\{a_n\}$. Find the smallest value of $n$ such that $S_n > 12a_{n+1}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{27}
In a five-candidate election, the winning candidate received 48% of the votes, the runner-up candidate received 34% of the votes, and the remaining candidates received a combined 15% of the votes. Considering that 3% of the votes were spoiled or invalid, the winner won the election by 2,112 votes. Find the number of votes casted for the winning candidate.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7,241}
Given the ellipse equation \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) (\( a > b > 0 \)), let \( AB \) be the major axis of length \(2a\). Suppose \( AB \) is divided into 100 equal parts, and perpendiculars are drawn through each division point to \( AB \), intersecting the upper half of the ellipse at \( P_1, P_2, \dots, P_{99} \). Let \( F_1 \) be the left focus of the ellipse. Find the value of \( |F_1A| + |F_1P_1| + |F_1P_2| + \cdots + |F_1P_{99}| + |F_1B| \). Express your answer as a multiple of \(a\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{101a}
Observe the following numbers: $1,-\frac{1}{3},\frac{2}{5},-\frac{2}{7},\frac{1}{3},-\frac{3}{11}$, $\ldots$, according to this pattern, the $2023$rd number is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1012}{4045}}
Petya's favorite TV game is called "Lottery on the Couch." During the game, viewers can send SMS messages with three-digit numbers containing only the digits 1, 2, 3, and 4. At the end of the game, the host announces a three-digit number, also consisting only of these digits. An SMS is considered a winning one if the number in it differs from the host's number by no more than one digit (for example, if the host announces the number 423, then messages 443 and 123 are winning, but 243 and 224 are not).

Petya wants to send as few messages as possible so that at least one is guaranteed to be winning. How many SMS messages does he need to send?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
There are a total of 1,800 fish swimming westward, 3,200 swimming eastward, 500 swimming north, and 2,300 fish swimming south. Fishers catch the following proportions of the fish based on their swimming direction: 

- westward: a/b
- eastward: c/d
- southward: e/f

Find the number of fish left in the sea after the fishers catch some fish, if a=3, b=4, c=2, d=5, e=1, and f=3.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4,403}
10.8. Given a natural number $n \geqslant 2$. Consider all colorings of the cells of an $n \times n$ board in $k$ colors such that each cell is painted in exactly one color, and all $k$ colors are used. For what smallest $k$ will any such coloring contain four cells painted in four different colors, located at the intersection of two rows and two columns?

(D. Khramov)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2n}
We call a pair $(a,b)$ of positive integers a "pupusa" if $a < 391$ and \(\text{lcm}(a,b) > \text{lcm}(a,391)\). Find the minimum value of $b$ across all "pupusa" pairs.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
In a test with 84 questions, if you get 58 out of 84 questions correct, what is your percentage of correct answers?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{69\%}
Joel collected a bin of old toys to donate. He was able to collect some stuffed animals, 42 action figures, 2 board games, and 13 puzzles. His sister gave him some of her old toys from her closet, and then Joel added twice as many toys from his own closet as his sister did from hers. In all, Joel was able to donate 108 toys, and 22 of the donated toys were Joel’s. How many stuffed animals did Joel collect?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
40% of major airline companies equip their planes with wireless internet access, 70% of major airlines offer passengers free on-board snacks, 60% provide in-flight entertainment, and 50% have premium seating options. What is the greatest possible percentage of major airline companies that offer wireless internet, free on-board snacks, in-flight entertainment, and premium seating options?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40\%}
Find the minimal value of integer n that guarantees that among n sets, there exist at least three sets such that any of them does not include any other, or there exist at least three sets such that any two of them include the other. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
In a group of 89 students, some are taking history, and 32 are taking statistics. If 59 students are taking history or statistics or both, and 27 students are taking history but not statistics, how many students are taking history?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{54}
Nadine collected different colored pebbles. She has 20 white pebbles, half as many red pebbles as white pebbles, and one-third as many blue pebbles as red pebbles. She also has 5 more green pebbles than blue pebbles. How many pebbles does she have in all?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{41}
What is the area of the region \( \mathcal{R} \) formed by all points \( L(r, t) \) with \( 0 \leq r \leq 10 \) and \( 0 \leq t \leq 10 \) such that the area of triangle \( JKL \) is less than or equal to 10, where \( J(2,7) \) and \( K(5,3) \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{313}{4}}
Given $sin(\theta+\frac{π}{12})=\frac{1}{3}$, and $\theta∈(\frac{π}{2},π)$, find $cos(\theta+\frac{π}{3})$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\frac{\sqrt{2}+4}{6}}
Regular hexagon $ABCDEF$ has vertices $A$ and $C$ at $(0,0)$ and $(7,1)$ respectively. What is the area of the hexagon? The answer should be in the form $k\sqrt{m}$, where k and m are integers. Please find the value of k + m.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{28}
Given the ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 (a>b>0)\), let \(A\) be the left vertex, \(F(c, 0)\) be the right focus, and \(2b\), \(a\), and \(c\) form a geometric progression. A line through the point \(F\) intersects the ellipse at points \(M\) and \(N\). The lines \(AM\) and \(AN\) intersect the right directrix at points \(P\) and \(Q\) respectively. Determine the size of the angle \(\angle PFQ\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{90^\circ}
Elfs and trolls are seated at a round table, 60 creatures in total. Trolls always lie, and all elfs always speak the truth, except when they make a little mistake. 
Everybody claims to sit between an elf and a troll, but exactly two elfs made a mistake! How many trolls are there at this table?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
已知点$P(3,a)$,若圆$O:{{x}^{2}}+{{y2}}=4$上存在点$A$,使得线段$PA$的中点也在圆$O$上,则$a$的取值范围是__________．
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$[-3\sqrt{3},3\sqrt{3}]$}
Let $AB$ and $BC$ be two consecutive sides of a regular polygon with 9 vertices inscribed in a circle with center $O$. Let $M$ be the midpoint of $AB$ and $N$ be the midpoint of the radius perpendicular to $BC$. Find the measure of the angle $\angle OMN$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
The members of a distinguished committee were choosing a president, and each member gave one vote to one of the $27$ candidates. For each candidate, the exact percentage of votes the candidate got was smaller by at least $1$ than the number of votes for that candidate. What was the smallest possible number of members of the committee?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{134}
14. (15 points) In the four idioms “虚有其表”, “表里如一”, “一见如故”, “故弄玄虚”, each Chinese character represents one of 11 consecutive non-zero natural numbers. The same character represents the same number, and different characters represent different numbers. Additionally, “表” > “一” > “故” > “如” > “虚”, and the sum of the numbers represented by the four characters in each idiom is 21. What is the largest number that “弄” can represent?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
James has 20 pairs of red socks, half as many black socks, and twice as many white socks as red and black combined. Additionally, James has the square root of the total number of red, black, and white sock pairs combined in green sock pairs. How many total socks does he have combined?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{198}
设正数列 $x_{1}, x_{2}, \cdots, x_{n}, \cdots$ 满足
$(8 x_{2}-7 x_{1}) x_{1}^{7}=8$
及 $x_{k+1} x_{k-1}-x_{k}^{2}=\frac{x_{k-1}^{8}-x_{k}^{8}}{(x_{k} x_{k-1})^{7}}(k \geqslant 2)$ 。
求正实数 $a$, 使得当 $x_{1}>a$ 时, 有单调性 $x_{1}>x_{2}>\cdots>x_{n}>\cdots$ ；当 $0<x_{1}<a$ 时，不具有单调性。标准答案格式为k^{\frac{m}{n}}，请给出k+m+n的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{17}
B4. The infinite sequence of numbers

$$
0,1,2,2,1,-1,-2,-1,1,3, \ldots
$$

satisfies the following rule. For each quadruple of consecutive numbers $\ldots, a, b, c, d, \ldots$ in the sequence, it always holds that $d$ is equal to $c$ minus the smallest of the two numbers $a$ and $b$. Thus, the ninth number in the sequence is equal to $-1-(-2)=1$ and the tenth number is equal to $1-(-2)=3$. Calculate the 100th number in the sequence.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2187}
How many times must a die be rolled so that the probability of the inequality

\[ \left| \frac{m}{n} - \frac{1}{6} \right| \leq 0.01 \]

is at least as great as the probability of the opposite inequality, where \( m \) is the number of times a specific face appears in \( n \) rolls of the die?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{632}
The average (arithmetic mean) of the 5 positive integers k, m, r, s, and t is 10, and k < m < r < s < t. The greatest possible value of the median of the 5 integers is 13. What is the value of t?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
Anita plays the following single-player game: She is given a circle in the plane. The center of this circle and some point on the circle are designated “known points”. Now she makes a series of moves, each of which takes one of the following forms:
(i) She draws a line (infinite in both directions) between two “known points”; or
(ii) She draws a circle whose center is a “known point” and which intersects another “known point”.
Once she makes a move, all intersections between her new line/circle and existing lines/circles become “known points”, unless the new/line circle is identical to an existing one. In other words, Anita is making a ruler-and-compass construction, starting from a circle.
What is the smallest number of moves that Anita can use to construct a drawing containing an equilateral triangle inscribed in the original circle?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
If the circle $x^{2}+y^{2}=4$ is externally tangent to the circle $x^{2}+y^{2}-2mx+m^{2}-1=0$, then the real number $m=$ \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pm3}
Let $b_n$ be the number obtained by writing the integers $1$ to $n$ from left to right, and then reversing the sequence. For example, $b_4 = 43211234$ and $b_{12} = 121110987654321123456789101112$. For $1 \le k \le 100$, how many $b_k$ are divisible by 9?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{22}
已知幂函数$y=\frac{1}{{{x}^{2-m-{{m2}}}}}$在第二象限内为$x$的减函数，求$m$的最大负整数值.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$-3$}
Find the smallest possible sum of a set of prime numbers that use each of the nine nonzero digits (1, 2, 3, 4, 5, 6, 7, 8, 9) exactly once. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{207}
In the triangular prism $P-ABC$, the vertex $P$ projects orthogonally onto the circumcenter $O$ of the base $\triangle ABC$, and the midpoint of $PO$ is $M$. Construct a cross-section $\alpha$ through $AM$ parallel to $BC$, and denote $\angle PAM$ as $\theta_1$. The acute dihedral angle between $\alpha$ and the base plane $ABC$ is denoted as $\theta_2$. When $\theta_1$ is maximized, find the value of $\tan \theta_2$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["\\frac{\\sqrt{2}}{2}"]}
Consider a $5 \times 5$ grid, consisting of 25 white cells. Vincent wants to color some cells in red so that every $2 \times 3$ or $3 \times 2$ rectangle on the grid contains at least one colored cell. What is the minimum number of cells he needs to color?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
9. Arrange $1,2, \cdots, 7$ in a row such that the sum of the first term, the sum of the first two terms, the sum of the first three terms, ... up to the sum of all seven terms are not divisible by 3. Then there are $\qquad$ different arrangements.
(Thailand)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{360}
Let $\phi = \frac{1+\sqrt{5}}{2}$. A base-$\phi$ number $(a_n a_{n-1} \ldots a_1 a_0)_{\phi}$, where $0 \le a_n, a_{n-1}, \ldots, a_0 \le 1$ are integers, is defined by 
\[
(a_n a_{n-1} \ldots a_1 a_0)_{\phi} = a_n \cdot \phi^n + a_{n-1} \cdot \phi^{n-1} + \ldots + a_1 \cdot \phi^1 + a_0.
\]
Compute the number of base-$\phi$ numbers $(b_j b_{j-1} \ldots b_1 b_0)_\phi$ which satisfy $b_j \ne 0$ and
\[
(b_j b_{j-1} \ldots b_1 b_0)_\phi = \underbrace{(100 \ldots 100)_\phi}_{\text{Twenty} \ 100's}.
\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1048576}
A lady grows cauliflower in her garden that is in the shape of a square. Each cauliflower takes a certain amount of area in her garden. This year, she has increased her output by 401 cauliflower when compared to last year. She produced 40401 cauliflowers this year. How many square feet of area does each cauliflower take in her garden?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Let $ABCD$ be a convex quadrilateral with $BA = BC$ and $DA = DC$. Let $E$ and $F$ be the midpoints of $BC$ and $CD$ respectively, and let $BF$ and $DE$ intersect at $G$. If the area of $CEGF$ is $50$, what is the area of $ABGD$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{200}
$15 \cdot 2$ In the set $\{1000,1001, \cdots, 2000\}$, how many pairs of consecutive integers can be added without carrying?

(10th American Invitational Mathematics Examination, 1992)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{156}
A solid right prism $ABCDEF$ has a height of $16,$ as shown. Also, its bases are equilateral triangles with side length $12.$ Points $X,$ $Y,$ and $Z$ are the midpoints of edges $AC,$ $BC,$ and $DC,$ respectively. A part of the prism above is sliced off with a straight cut through points $X,$ $Y,$ and $Z.$ Determine the surface area of solid $CXYZ,$ the part that was sliced off.  [asy]
pair A, B, C, D, E, F, X, Y, Z;
A=(0,0);
B=(12,0);
C=(6,-6);
D=(6,-22);
E=(0,-16);
F=(12,-16);
X=(A+C)/2;
Y=(B+C)/2;
Z=(C+D)/2;
draw(A--B--C--A--E--D--F--B--C--D);
draw(X--Y--Z--X, dashed);
label("$A$", A, NW);
label("$B$", B, NE);
label("$C$", C, N);
label("$D$", D, S);
label("$E$", E, SW);
label("$F$", F, SE);
label("$X$", X, SW);
label("$Y$", Y, SE);
label("$Z$", Z, SE);
label("12", (A+B)/2, dir(90));
label("16", (B+F)/2, dir(0));
[/asy]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{48+9\sqrt{3}+3\sqrt{91}}
Two pipes can fill a tank in 18 minutes and 20 minutes. An outlet pipe can empty the tank in some minutes. If all the pipes are opened when the tank is empty, it will take 0.08333333333333333 minutes to fill the tank. How many minutes does it take for the outlet pipe to empty the tank?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{45}