inclusionAI/Ring-lite-rl-data huggingface dataset:
Giraldo wrote five distinct natural numbers on the vertices of a pentagon. Next, he wrote on each side of the pentagon the least common multiple (LCM) of the numbers written on the two vertices of that side. He noticed that the five numbers written on the sides were equal. What is the smallest number Giraldo could have written on the sides?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
Recall that a palindrome is a number that reads the same forward and backward. Find the greatest integer less than $1000$ that is a palindrome both when written in base ten and when written in base eight, such as $292 = 444_{eight}.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{585}
Starting from a certain number, the digit 5 will be written 300 times when listing the integers up to 1000. What is the starting number of this range?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{500}
Abby, Bernardo, Carl, and Debra play a game in which each of them starts with four coins. The game consists of four rounds. In each round, four balls are placed in an urn---one green, one red, and two white. The players each draw a ball at random without replacement. Whoever gets the green ball gives one coin to whoever gets the red ball. What is the probability that, at the end of the fourth round, each of the players has four coins?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{96}}
You, your friend, and two strangers are sitting at a table. A standard $52$-card deck is randomly dealt into $4$ piles of $13$ cards each, and each person at the table takes a pile. You look through your hand and see that you have one ace. Compute the probability that your friend’s hand contains the three remaining aces.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{22}{703}}
Let $P(x) = x + 1$ and $Q(x) = x^2 + 1$. Consider all sequences $\langle (x_k, y_k) \rangle_{k \in \mathbb{N}}$ such that $(x_1, y_1) = (1, 3)$ and $(x_{k+1}, y_{k+1})$ is either $(P(x_k), Q(y_k))$ or $(Q(x_k), P(y_k))$ for each $k$. We say that a positive integer $n$ is nice if $x_n = y_n$ holds in at least one of these sequences. Find all nice numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
The roots of a monic cubic polynomial  $p$  are positive real numbers forming a geometric sequence. Suppose that the sum of the roots is equal to  $10$ . Under these conditions, the largest possible value of  $|p(-1)|$  can be written as  $\frac{m}{n}$ , where  $m$ ,  $n$  are relatively prime integers. Find  $m + n$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2224}
4. In a coffee shop, 55 Indians and Turks met, each drinking tea or coffee. All Indians tell the truth when drinking tea and lie when drinking 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 coffee shop are drinking tea?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Point \(P\) is inside an equilateral \(\triangle ABC\) such that the measures of \(\angle APB, \angle BPC, \angle CPA\) are in the ratio 5:6:7. Determine the ratio of the measures of the angles of the triangle formed by \(PA, PB, PC\) (in increasing order). Express your answer as a ratio of three numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2:3:4}
If the domains of the functions \( f(x) \) and \( g(x) \) are both the set of non-negative real numbers, and for any \( x \geq 0 \), \( f(x) \cdot g(x) = \min \{ f(x), g(x) \} \), given \( f(x) = 3 - x \) and \( g(x) = \sqrt{2x + 5} \), then the maximum value of \( f(x) \cdot g(x) \) is ______ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 \sqrt{3} - 1}
Azar and Carl play a game of tic-tac-toe. Azar places an in $X$ one of the boxes in a $3$-by-$3$ array of boxes, then Carl places an $O$ in one of the remaining boxes. After that, Azar places an $X$ in one of the remaining boxes, and so on until all boxes are filled or one of the players has of their symbols in a row--horizontal, vertical, or diagonal--whichever comes first, in which case that player wins the game. Suppose the players make their moves at random, rather than trying to follow a rational strategy, and that Carl wins the game when he places his third $O$. How many ways can the board look after the game is over?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$148$}
Given the function $y=(m^2-3)x^{2m}$ is a power function, find the value of $m$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pm2}
In triangle $ABC$, angle $C$ is $135^{\circ}$. A square with center $O$ is constructed on side $AB$ outside the triangle. Find $OC$ if $AB = 6$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3\sqrt{2}}
设 $A, B$ 分别是 $x$ 轴, $y$ 轴上的动点, 若以 $A B$ 为直径的圆 $C$ 与直线 $2 x+y-4=0$ 相切, 则圆 $C$ 面积的最小值为 $\frac{k}{m}\pi$，请计算k+m的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Cathy has $5$ red cards and $5$ green cards. She shuffles the $10$ cards and lays out $5$ of the cards in a row in a random order. She will be happy if and only if all the red cards laid out are adjacent and all the green cards laid out are adjacent. For example, card orders $RRGGG, GGGGR$, or $RRRRR$ will make Cathy happy, but $RRRGR$ will not. The probability that Kathy will be happy 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{157}
求最小的正整数 $m$, 使得对任意整数 $A, B$, 存在多项式 $f(x)$, 满足 $f(-2)=$ $A, f(-3)=B$, 且 $f(x)$ 的各项系数都是不超过 $m$ 的非负整数.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Steve has a bank account that earns 10% interest every year for the first three years, and then the interest rate changes to 8% for the next two years. He initially deposits $100 in it. Then, he makes annual deposits of $10 for the first two years and increases his annual deposits to $15 for the remaining three years. How much money is in his bank account after five years with the changing interest rates and deposit amounts?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$230.89}
Given that the four vertices of the quadrilateral $MNPQ$ are on the graph of the function $f(x)=\log_{\frac{1}{2}} \frac{ax+1}{x+b}$, and it satisfies $\overrightarrow{MN}= \overrightarrow{QP}$, where $M(3,-1)$, $N\left( \frac{5}{3},-2\right)$, then the area of the quadrilateral $MNPQ$ is \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{26}{3}}
Given that the positive integers \( a_{1}, a_{2}, a_{3}, a_{4} \) satisfy the condition that in any circular arrangement of these four numbers, there exist two adjacent elements which are not coprime. Find the maximum number of ordered triplets \((i, j, k)\) such that \( i, j, k \in {1, 2, 3, 4} \), \( i \neq j \), \( j \neq k \), \( k \neq i \), and \( (a_{i}, a_{j})^2 \mid a_{k} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
Let point $P$ be any point on edge $BC$ of a regular tetrahedron $ABCD$. Find the range of the angle formed by lines $AP$ and $DC$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ \left[\frac{\pi}{3}, \frac{\pi}{2}\right] }
甲，乙、丙三人轮流值日，从周一至周六，每人值班两天，若甲不值周一，乙不值周六，则可排出的不同值日表有 $\qquad$种。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{42}
对于任何集合 $S$ ，用 $|S|$ 表示集合 $S$中的元素个数，用 $n(S)$ 表示集合 $S$ 的子集个数。若 $A 、 B 、 C$ 为三个有限集，且满足
(1) $|A|=|B|=2016$;
(2) $n(A)+n(B)+n(C)=n(A \cup B \cup C)$ 。
则 $|A \cap B \cap C|$ 的最大值为 $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2015}
A school selects a sample from 90 students numbered sequentially from 01, 02, ..., 90 using systematic sampling. Given that the adjacent group numbers in the sample are 14 and 23, what is the student number from the fourth group in the sample?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32}
The ratio of a geometric sequence is an integer. We know that there is a term in the sequence which is equal to the sum of some other terms of the sequence. What can the ratio of the sequence be?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
In a chess festival that is held in a school with $2017$ students, each pair of students played at most one match versus each other. In the end, it is seen that for any pair of students which have played a match versus each other, at least one of them has played at most $22$ matches. What is the maximum possible number of matches in this event?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{43890}
Natural numbers \(a, b, c\) are chosen such that \(a < b < c\). It is also known that the system of equations \(2x + y = 2029\) and \(y = |x - a| + |x - b| + |x - c|\) has exactly one solution. Find the minimum possible value of \(c\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1015}
Determine which positive integers n have the following property: For all integers m that are relatively prime to n, there exists a permutation $\pi:\{1,2,\ldots,n\}\to\{1,2,\ldots,n\}$ such that $\pi(\pi(k))\equiv mk(\bmod n)$ for all $k\in\{1,2,\ldots,n\}$. Provide the sum of all such positive integers n.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
In a language, there are three letters - Ш, У, and Я. A word is defined as a sequence of 100 letters, exactly 40 of which are vowels (У or Я), and the remaining 60 are the letter Ш. What is the maximum number of words that can be chosen so that for any two selected words, there is at least one position among the hundred where different vowels stand simultaneously?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{40}}
There are $100$ lightbulbs $B_1, B_2, \ldots, B_{100}$ spaced evenly around a circle in this order. Additionally, there are $100$ switches $S_1, S_2, \ldots, S_{100}$ such that for all $1 \leq i \leq 100$, switch $S_i$ toggles the states of lights $B_{i-1}$ and $B_{i+1}$ (where $B_{101} = B_1$ and $B_0 = B_{100}$). Suppose David chooses whether to flick each switch with probability $\frac{1}{2}$. What is the expected number of lightbulbs which are on at the end of this process given that not all lightbulbs are off?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50}
For which natural numbers \( n \) is the inequality

$$
\sin n \alpha + \sin n \beta + \sin n \gamma < 0
$$

true for any angles \( \alpha, \beta, \gamma \) that are the angles of an acute triangle?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Find the number of different quadruples $(a, b, c, d)$ of positive integers such that $ab =cd = a + b + c + d - 3$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
We draw a radius of a circle. We draw a second radius $23$ degrees clockwise from the first radius. We draw a third radius $23$ degrees clockwise from the second. This continues until we have drawn $40$ radii each $23$ degrees clockwise from the one before it. What is the measure in degrees of the smallest angle between any two of these $40$ radii?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
A finite sequence of three-digit integers has the property that the tens and units digits of each term are, respectively, the hundreds and tens digits of the next term, and the tens and units digits of the last term are, respectively, the hundreds and tens digits of the first term. For example, such a sequence might begin with the terms $247, 475$, and $756$ and end with the term $824$. Let $S$ be the sum of all the terms in the sequence. Find the largest prime factor that always divides $S$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{37}
Given $\lg 2=a$, $\lg 3=b$, then $\log _{6}15=$______(express the result using $a$ and $b$).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{b + 1 - a}{a + b}}
8. The function $y=f(t)$ is such that the sum of the roots of the equation $f(\sin x)=0$ on the interval $[3 \pi / 2, 2 \pi]$ is $33 \pi$, and the sum of the roots of the equation $f(\cos x)=0$ on the interval $[\pi, 3 \pi / 2]$ is $23 \pi$. What is the sum of the roots of the second equation on the interval $[\pi / 2, \pi]$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{17\pi}
A set of 3 red balls, 2 white balls, and 2 blue balls is arranged randomly. What is the probability of having 2 adjacent white balls or 2 adjacent blue balls in this arrangement? Express your answer as a fraction.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{10}{21}}
Simplify and solve for the value of the expression \u221a[3]{5+2\u221a{13}} + \u221a[3]{5-2\u221a{13}}. Express your answer as a single numerical value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
8. In a circle, three chords $A A_{1}, B B_{1}, C C_{1}$ intersect at one point. The angular measures of the arcs $A C_{1}, A B, C A_{1}$, and $A_{1} B_{1}$ are $150^{\circ}, 30^{\circ}, 60^{\circ}$, and $30^{\circ}$, respectively. Find the angular measure of the arc $B_{1} C_{1}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60}
In a rectangle that is sub-divided into 3 identical squares of side length 1, if \(\alpha^{\circ} = \angle ABD + \angle ACD\), find the value of \(\alpha\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{45}
Three runners move along a circular track at equal constant speeds. When two runners meet, they instantly turn around and start running in the opposite direction.

At a certain moment, the first runner met the second runner. After 20 minutes, the second runner met the third runner for the first time. Another half hour later, the third runner met the first runner for the first time.

How many minutes does it take for one runner to complete the entire track?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{100 \text{ minutes}}
The number of elements in a finite set $P$ is denoted as $\text{card}(P)$. It is known that $\text{card}(M) = 10$, $A \subseteq M$, $B \subseteq M$, $A \cap B = \emptyset$, and $\text{card}(A) = 2$, $\text{card}(B) = 3$. If the set $X$ satisfies $A \subseteq X \subseteq M$, then the number of such sets $X$ is ____. (Answer with a number)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{256}
Let \( f(x) \) and \( g(x) \) be an odd function and an even function, respectively, defined on \(\mathbf{R}\). When \( x<0 \), \( f^{\prime}(x) g(x) + f(x) g^{\prime}(x) > 0 \), and \( g(3) = 0 \). Determine the solution set for the inequality \( f(x) \cdot g(x) < 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-\infty, -3) \cup (0, 3)}
We say that a real number $V$ is good if there exist two closed convex subsets $X, Y$ of the unit cube in $\mathbb{R}^{3}$, with volume $V$ each, such that for each of the three coordinate planes (that is, the planes spanned by any two of the three coordinate axes), the projections of $X$ and $Y$ onto that plane are disjoint. Find $\sup \{V \mid V$ is good $\}$. The original answer is in \frac{m}{n} format, please give the value of m + n.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Let $F$ be the focus of the parabola $C_1$: $y^2=2px$ ($p > 0$), and point $A$ be a common point of the parabola and an asymptote of the hyperbola $C_2$: $\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2}=1$ ($a > 0, b > 0$). If line $AF$ is perpendicular to the x-axis, calculate the eccentricity of the hyperbola.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{5}}
Harry walked along the beach and collected 34 sea stars, 21 seashells, 29 snails, and 17 crabs. During his walk, 5 of the sea stars, 3 of the seashells, and 4 of the snails reproduced, resulting in twice their original number. However, he lost some of the items and had 105 items left at the end of his walk. How many sea creatures did he lose in all?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
The sum of a set is defined as the sum of all its elements. Let $S$ be a set composed of positive integers not exceeding 15, such that the sums of any two disjoint subsets of $S$ are not equal. Among all sets with this property, find the set $S$ whose sum is the maximum. Calculate the sum of this set $S$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{61}
Calculate the area of the part of the sphere $x^2 + y^2 + z^2 = 2$ that lies inside the cylinder $x^2 + y^2 = 1$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4\pi(2 - \sqrt{2})}
If the interior angles \( A, B, C \) of triangle \( \triangle ABC \) satisfy \(\sin A = \cos B = \tan C\), find the value of \(\cos^3 A + \cos^2 A - \cos A\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{2}}
What is the prime factorization of the complex number $6+12i$ in the Gaussian integers $\Bbb Z[i]$? In other words, express $6+12i$ as a product of irreducible elements in $\Bbb Z[i]$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3 \times (1+i) \times (1-i) \times (1+2i)}
For $0\leq a\leq 2$, find the minimum value of $\int_0^2 \left|\frac{1}{1+e^x}-\frac{1}{1+e^a}\right|\ dx.$

[i]2010 Kyoto Institute of Technology entrance exam/Textile e.t.c.[/i]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ \ln \left( \frac{2+2e^2}{1+2e+e^2} \right) }
Forty-five percent of the members of a swim club have passed the lifesaving test, while thirty percent have passed an advanced diving test. Among the members who have not passed either test, 10 have taken the preparatory course for both tests, 6 have taken only the lifesaving course, 8 have taken only the advanced diving course, and 40 have not taken any courses. Additionally, it is known that 20% of the members who have passed the lifesaving test have also passed the advanced diving test. Find the total number of members in the swim club.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{188}
Let $A$, $B$, and $C$ be three points on the edge of a circular chord such that $B$ is due west of $C$ and $\triangle ABC$ is an equilateral triangle with each side measuring $86$ meters. A boy swam from $A$ directly towards $B$. After covering a distance of $x$ meters, he turned and swam westward, reaching the shore after covering a distance of $y$ meters. If $x$ and $y$ are both positive integers, determine $y$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
A jar contains 8 red balls and 2 blue balls. Every minute, a ball is randomly removed. The probability that there exists a time during this process where there are more blue balls than red balls in the jar can be expressed as \(\frac{a}{b}\) for relatively prime integers \(a\) and \(b\). Compute \(100a + b\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{209}
What is the minimal polynomial of $\alpha = \frac{1-\sqrt{5}}{2}$ over the finite field $\mathbb{Z}_5$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{x-3}
Unconventional dice are to be designed such that the six faces are marked with numbers from $1$ to $6$, with $1$ and $2$ appearing on opposite faces. Furthermore, each face is colored either red or yellow, with opposite faces always of the same color. Two dice are considered to have the same design if one of them can be rotated to obtain a dice that has the same numbers and colors on the corresponding faces as the other one. Find the number of distinct dice that can be designed.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{48}
\( \triangle ABC \) has \( AB = 4 \) and \( AC = 6 \). Let point \( D \) be on line \( AB \) so that \( A \) is between \( B \) and \( D \). Let the angle bisector of \( \angle BAC \) intersect line \( BC \) at \( E \), and let the angle bisector of \( \angle DAC \) intersect line \( BC \) at \( F \). Given that \( AE = AF \), find the square of the circumcircle's radius' length.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
Karen's class fund contains only $10 bills and another type of bills, which amount to $120. The number of $10 bills is twice as many as the other type of bills. They have 3 of these other bills in their fund. What is the value of these other bills?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$40}
Let  $\triangle ABC$  be a triangle with  $AB = 7$ ,  $AC = 8$ , and  $BC = 3$ . Let  $P_1$  and  $P_2$  be two distinct points on line  $AC$  ( $A, P_1, C, P_2$  appear in that order on the line) and  $Q_1$  and  $Q_2$  be two distinct points on line  $AB$  ( $A, Q_1, B, Q_2$  appear in that order on the line) such that  $BQ_1 = P_1Q_1 = P_1C$  and  $BQ_2 = P_2Q_2 = P_2C$ . Find the distance between the circumcenters of  $BP_1P_2$  and  $CQ_1Q_2$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
The union of sets \( A \) and \( B \), \( A \cup B = \{a_1, a_2, a_3\} \). When \( A \neq B \), pairs \((A, B)\) and \((B, A)\) are considered different. How many such pairs \((A, B)\) are there?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{27}
Given the sets $A={x|x^2-ax+a^2-19=0}$, $B={x|x^2-5x+6=0}$, and $C={x|x^2+2x-8=0}$, if $\emptyset \subset (A\cap B)$ and $A\cap C=\emptyset$ hold simultaneously, find the value of the real number $a$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2}
Among the natural numbers less than 5000, how many numbers are there that are divisible by 11 and have a digit sum of 13?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
Given the points P1(1,1), P2(1,2), P3(1/2, 1/2), and P4(2,2), determine how many of these points are intersections of an exponential function and a logarithmic function. Express your answer as a single numerical value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
On the hypotenuse \(AB\) of the right triangle \(ABC\), a square \(ABDE\) is constructed on the side of the plane that does not include the triangle \(ABC\). Find the distance from the vertex \(C\) of the right angle of the triangle to the center of the square, given that \(BC = a\) and \(AC = b\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{\frac{a^2 + b^2}{2}}}
Find a basis for $\mathbb{Q}(\sqrt{3}+\sqrt{7})$ over $\mathbb{Q}(\sqrt{7})$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{{1, \sqrt{3}}}
A square with a side length of 36 cm was cut into three rectangles in such a way that the areas of all three rectangles are equal and any two rectangles have a common section of the boundary. What is the total length (in cm) of the made cuts?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60 \text{ cm}}