inclusionAI/Ring-lite-rl-data huggingface dataset:
A magician has a hundred cards, numbered from 1 to 100. He places each of them into one of three boxes - a red box, a white box, and a blue box - in such a way that each box contains at least one card.

A member of the audience selects two of the three boxes and draws one card from each, then announces the sum of the numbers on the drawn cards. Knowing this sum, the magician can tell which is the box from which no card was drawn.

In how many ways can the cards be placed in the boxes so that this trick always works? (Two arrangements are considered different if there is at least one card that ends up in a different box.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
In trapezoid \(ABCD\) (\(AD \parallel BC\)), the angle bisectors of \(\angle DAB\) and \(\angle ABC\) intersect on side \(CD\). Find \(AB\) if \(AD = 5\) and \(BC = 2\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Find the number of ordered pairs of integers \((a, b) \in \{1, 2, \ldots, 35\}^{2}\) (not necessarily distinct) such that \(ax + b\) is a quadratic residue modulo \(x^2 + 1\) and 35.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{225}
A rectangle with vertices (0,0), (100,0), (100,50), (0,50) has a circle with center (75,30) and radius 10 cut out of it. Find the slope of the line through the point (75,30) that divides the remaining area of the rectangle into two shapes of equal area. Express your answer as a simplified fraction.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{5}}
In the number \(2016^{* * * *} 02 *\), each of the 5 asterisks needs to be replaced with any of the digits \(0, 2, 4, 6, 7, 8\) (digits can be repeated) so that the resulting 11-digit number is divisible by 6. In how many ways can this be done?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2160}
A school is holding a round-robin football tournament. Each participating team plays one match against every other team. Winning a match earns 2 points, a draw earns 1 point, and losing a match earns 0 points. It is known that only one team has the highest total points, but it has the fewest number of wins. What is the minimum number of teams participating for this scenario to be possible?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
What is the smallest integer $n > 0$ such that for any integer m in the range $1, 2, 3, ... , 1992$ we can always find an integral multiple of $\frac{1}{n}$ in the open interval $(\frac{m}{1993}, \frac{m+1}{1994})$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3987}
A person is waiting at the $A$ HÉV station. They get bored of waiting and start moving towards the next $B$ HÉV station. When they have traveled $1 / 3$ of the distance between $A$ and $B$, they see a train approaching $A$ station at a speed of $30 \mathrm{~km/h}$. If they run at full speed either towards $A$ or $B$ station, they can just catch the train. What is the maximum speed at which they can run?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
There are $30$ cities in the empire of Euleria. Every week, Martingale City runs a very well-known lottery. $900$ visitors decide to take a trip around the empire, visiting a different city each week in some random order. $3$ of these cities are inhabited by mathematicians, who will talk to all visitors about the laws of statistics. A visitor with this knowledge has a probability of $0$ of buying a lottery ticket, else they have a probability of $0.5$ of buying one. What is the expected number of visitors who will play the Martingale Lottery?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{405}
## 73. Man and Dog.

- Walking the dog, - a mathematician friend once told me, - gives me plenty of food for thought. Once, for example, my dog, after waiting for me to go out, looked to see which way I was going to head, and when I started down the path, he raced to the end of it. Then he returned to me, ran to the end of the path again, and returned, and so on, doing this 4 times in total. All this time he moved at a constant speed, and when he ran to me for the last time, he covered the remaining distance of 81 m. Measuring the distance from my door to the end of the path later, I found it to be 625 m. At what speed did my dog run, if I was walking at a speed of 4 km/h?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
We plotted the graph of the function \( f(x) = \frac{1}{x} \) in the coordinate system. How should we choose the new, still equal units on the axes, if we want the curve to become the graph of the function \( g(x) = \frac{2}{x} \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{2}}{2}}
The triangle \(PQR\) is isosceles with \(PR = QR\). Angle \(PRQ = 90^\circ\) and length \(PQ = 2 \text{ cm}\). Two arcs of radius \(1 \text{ cm}\) are drawn inside triangle \(PQR\). One arc has its center at \(P\) and intersects \(PR\) and \(PQ\). The other arc has its center at \(Q\) and intersects \(QR\) and \(PQ\). What is the area of the shaded region, in \(\text{cm}^2\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1 - \frac{\pi}{4}}
The number of the members of a club is more than 20 and less than 50. When a certain number of people sit at a table, other people are exactly divided into 6-people groups or 7-people groups. If the members are divided into 8-people groups, 7 people will be left. How many people sit at the first table?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
The businessman Boris Mikhailovich's Lexus navigator reports how much longer it will take to reach the destination if traveling at the current average speed. Boris Mikhailovich left his house for his country house. Halfway, the navigator reported that it would take 1 more hour to reach the destination. At that moment, a tractor driven by Vasya entered the road right in front of the Lexus, making it impossible to overtake. After covering half of the remaining distance, the navigator reported that there were 2 hours left to travel. How many hours after this will the businessman reach his country house, assuming he never overtakes the tractor? (The tractor's speed is constant.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5 \text{ hours}}
In cyclic quadrilateral $ABCD$ with $AB = AD = 49$ and $AC = 73$, let $I$ and $J$ denote the incenters of triangles $ABD$ and $CBD$. If diagonal $\overline{BD}$ bisects $\overline{IJ}$, find the length of $IJ$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{28}{5}\sqrt{69}}
Rectangle  $ABCD$  has side lengths  $AB = 6\sqrt3$  and  $BC = 8\sqrt3$ . The probability that a randomly chosen point inside the rectangle is closer to the diagonal  $\overline{AC}$  than to the outside of the rectangle 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{17}
A group of students decided to collect as many paise from each member of the group as the product of the number of members and the group's average height in centimeters. If the total collection amounts to Rs. 3625.25, the number of members in the group is:

(Note: 1 Indian Rupee (Rs.) = 100 paise)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
The odd function $y=f(x)$ has a domain of $\mathbb{R}$, and when $x \geq 0$, $f(x) = 2x - x^2$. If the range of the function $y=f(x)$, where $x \in [a, b]$, is $[\frac{1}{b}, \frac{1}{a}]$, then the minimum value of $b$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
What is the expected number of additional coin flips needed to reach a state where the number of heads equals the number of tails, given that the first flip lands on heads? Assume the coin is unfair, with the probability of heads being $p$, where $p < \frac{1}{2}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{1}{1-2p}$}
Given a trapezoid \(ABCD\) and a point \(M\) on the side \(AB\) such that \(DM \perp AB\). It is found that \(MC = CD\). Find the length of the upper base \(BC\), if \(AD = d\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{d}{2}}
12. A small ball with a radius of 1 can move freely in all directions inside a regular tetrahedron container with an internal edge length of $4 \sqrt{6}$. The area of the container's inner wall that the ball can never touch is $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{72\sqrt{3}}
A line with a slope of 1 passes through the focus of the parabola $x^2=2py$ ($p>0$) and intersects the parabola at points A and B. The orthogonal projections of A and B onto the x-axis are D and C, respectively. If the area of trapezoid ABCD is $12\sqrt{2}$, then $p=$ \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Along the southern shore of a boundless sea stretches an archipelago of an infinite number of islands. The islands are connected by an endless chain of bridges, and each island is connected by a bridge to the shore. In the event of a strong earthquake, each bridge independently has a probability $p=0.5$ of being destroyed. What is the probability that after a strong earthquake, one can travel from the first island to the shore using the remaining intact bridges?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2/3}
Find the number of ordered triples of nonnegative integers \((a, b, c)\) that satisfy
\[
(a b+1)(b c+1)(c a+1)=84.
\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Let $n$ be a positive integer. Find the number of pairs $P,Q$ of polynomials with real coefficients such that $$(P(X))^2+(Q(X))^2=X^{2n}+1$$ and $\deg P>\deg Q.$ Provide the value when $n=3$. The answer should be an integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
Given that 4 empty mineral water bottles can be exchanged for 1 bottle of mineral water, and there are 16 empty mineral water bottles, what is the maximum number of bottles of mineral water that can be drunk if no money is paid? Express your answer as a whole number of bottles.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
If 5 points are placed in the plane at lattice points (i.e. points $(x, y)$ where $x$ and $y$ are both integers) such that no three are collinear, then there are 10 triangles whose vertices are among these points. What is the minimum possible number of these triangles that have area greater than $1 / 2$ ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Points \( K \) and \( M \) are located on sides \( AB \) and \( BC \) of triangle \( ABC \), with \( BK : KA = 1 : 4 \) and \( BM : MC = 3 : 2 \). Lines \( MK \) and \( AC \) intersect at point \( N \).

Find the ratio \( AC : CN \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5:1}
An enterprise produces a certain product over the year in amounts $x_{1}$ for January, $x_{2}$ for February, $\ldots, x_{12}$ for December. The average production of the product from the beginning of the year is calculated as follows:

$x_{1}=x_{1}, \quad \bar{x}_{2}=\frac{1}{2}\left(x_{1}+x_{2}\right), \quad \bar{x}_{3}=\frac{1}{3}\left(x_{1}+x_{2}+x_{3}\right), \quad \ldots, \quad \bar{x}_{12}=\frac{1}{12}\left(x_{1}+x_{2}+\ldots+x_{12}\right)$.

It is known that $\bar{x}_{k}<x_{k}$ for $k$ from 2 to 6 and $\bar{x}_{k}>x_{k}$ for $k$ from 7 to 12. In which month was the average production of the product from the beginning of the year the highest?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{June}
The incircle of $\triangle ABC$ is tangent to sides $\overline{BC}$, $\overline{AC}$, and $\overline{AB}$ at $D$, $E$, and $F$, respectively. Point $G$ is the intersection of lines $AC$ and $DF$. The sides of $\triangle ABC$ have lengths $AB = 73$, $BC = 123$, and $AC = 120$. Find the length $EG$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{119}
How many homomorphisms exist between the cyclic group $\mathbb{Z}_3$ and the symmetric group $S_4$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
In triangle \(ABC\), the median \(AM\) is drawn. Circle \(\gamma\) passes through point \(A\), is tangent to line \(BC\) at point \(M\), and intersects sides \(AB\) and \(AC\) at points \(D\) and \(E\) respectively. On the arc \(AD\) that does not contain \(E\), a point \(F\) is chosen such that \(\angle BFE = 72^\circ\). It is known that \(\angle DEF = \angle ABC\). Find the angle \(\angle CME\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36^\circ}
Let $R$, $S$, and $T$ be squares that have vertices at lattice points (i.e., points whose coordinates are both integers) in the coordinate plane, together with their interiors. The bottom edge of each square is on the $x$-axis. The left edge of $R$ and the right edge of $S$ are on the $y$-axis, and $R$ contains $\frac{9}{4}$ as many lattice points as does $S$. The top two vertices of $T$ are in $R \cup S$, and $T$ contains $\frac{1}{4}$ of the lattice points contained in $R \cup S.$ See the figure (not drawn to scale).
[asy] size(8cm); label(scale(.8)*"$y$", (0,60), N); label(scale(.8)*"$x$", (60,0), E); filldraw((0,0)--(55,0)--(55,55)--(0,55)--cycle, yellow+orange+white+white); label(scale(1.3)*"$R$", (55/2,55/2)); filldraw((0,0)--(0,28)--(-28,28)--(-28,0)--cycle, green+white+white); label(scale(1.3)*"$S$",(-14,14)); filldraw((-10,0)--(15,0)--(15,25)--(-10,25)--cycle, red+white+white); label(scale(1.3)*"$T$",(3.5,25/2)); draw((0,-10)--(0,60),EndArrow()); draw((-34,0)--(60,0),EndArrow()); [/asy]
The fraction of lattice points in $S$ that are in $S \cap T$ is $27$ times the fraction of lattice points in $R$ that are in $R \cap T$. What is the minimum possible value of the edge length of $R$ plus the edge length of $S$ plus the edge length of $T$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{337}
Find the number of positive integers \( x \), where \( x \neq 9 \), such that
\[
\log _{\frac{x}{9}}\left(\frac{x^{2}}{3}\right)<6+\log _{3}\left(\frac{9}{x}\right) .
\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{223}
The line $y=ax+1$ intersects the curve $x^{2}+y^{2}+bx-y=1$ at two points, and these two points are symmetrical with respect to the line $x+y=0$. Find the value of $a+b$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Given a sequence $\{a_n\}$ with the general term formula $a_n=n^2+bn$, if the sequence $\{a_n\}$ is monotonically increasing, then the range of values for the real number $b$ is
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-3,+\infty)}
Take a clay sphere of radius 13, and drill a circular hole of radius 5 through its center. Take the remaining "bead" and mold it into a new sphere. What is this sphere's radius?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
A rigid board with a mass of \(m\) and a length of \(l = 24\) meters partially lies on the edge of a horizontal surface, hanging off by two-thirds of its length. To prevent the board from falling, a stone with a mass of \(2m\) is placed at its very edge. How far from the stone can a person with a mass of \(m\) walk on the board? Neglect the dimensions of the stone and the person compared to the dimensions of the board.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
2. Given positive numbers $x, y, z$ satisfy $\left\{\begin{array}{l}x(y+z-x)=6 \\ y(z+x-y)=8 \\ z(x+y-z)=10\end{array}\right.$, then $3(x+y+z)^{2}=$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{242}
If \( p \) and \( q \) are positive integers and \(\frac{2008}{2009} < \frac{p}{q} < \frac{2009}{2010} \), what is the minimum value of \( p \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4017}
The set \( S \) consists of 9 distinct positive integers. The average of the two smallest integers in \( S \) is 5. The average of the two largest integers in \( S \) is 22. What is the greatest possible average of all of the integers of \( S \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
A square with a side length of $1$ is divided into one triangle and three trapezoids by joining the center of the square to points on each side. These points divide each side into segments such that the length from a vertex to the point is $\frac{1}{4}$ and from the point to the center of the side is $\frac{3}{4}$. If each section (triangle and trapezoids) has an equal area, find the length of the longer parallel side of the trapezoids.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3}{4}}
Let \( f(n) \) be the number of ways to write \( n \) as a sum of powers of 2, where the order of the summation is considered. For example, \( f(4) = 6 \) because 4 can be written as \( 4 \), \( 2 + 2 \), \( 2 + 1 + 1 \), \( 1 + 2 + 1 \), \( 1 + 1 + 2 \), and \( 1 + 1 + 1 + 1 \). Find the smallest \( n \) greater than 2013 for which \( f(n) \) is odd.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2047}
In triangle \( \triangle ABC \), \(\cos A\), \(\sin A\), and \(\tan B\) form a geometric sequence with a common ratio of \(\frac{3}{4}\). What is \(\cot C\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\frac{53}{96}}
Given the sequence $\left\{a_{n}\right\}$ defined by \(a_{1}=\frac{2}{3}\), \(a_{n+1}=a_{n}^{2}+a_{n-1}^{2}+\cdots+a_{1}^{2}\) (where \(n \in \mathbf{N}^{*}\)), find the smallest value of \(M\) such that for any \(n \in \mathbf{N}^{*}\), the inequality \(\frac{1}{a_{1}+1}+\frac{1}{a_{2}+1}+\cdots+\frac{1}{a_{n}+1}<M\) always holds true.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{57}{20}}
A coin is altered so that the probability that it lands on heads is less than \(\frac{1}{2}\) and when the coin is flipped four times, the probability of an equal number of heads and tails is \(\frac{1}{6}\). What is the probability that the coin lands on heads? Express your answer in the form \(\frac{k-\sqrt{m}}{n}\), where \(k\), \(m\), and \(n\) are integers, and give the value of \(k+m+n\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
A classroom has $30$ seats arranged into $5$ rows of $6$ seats. Thirty students of distinct heights come to class every day, each sitting in a random seat. The teacher stands in front of all the rows, and if any student seated in front of you (in the same column) is taller than you, then the teacher cannot notice that you are playing games on your phone. What is the expected number of students who can safely play games on their phone?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{163}{10}}
There are 100 houses in a row on a street. A painter comes and paints every house red. Then, another painter comes and paints every third house (starting with house number 3) blue. Another painter comes and paints every fifth house red (even if it is already red), then another painter paints every seventh house blue, and so forth, alternating between red and blue, until 50 painters have been by. After this is finished, how many houses will be red?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{52}
The equation $2x^3+5x-2=0$ has a unique real root $r$, and there exists a unique strictly increasing sequence of positive integers $a_n$, such that $$\frac {2}{5}= \lim_{n\rightarrow \infty }(r^{a_{1}}+r^{a_{2}}+r^{a_{3}}+\cdots +r^{a_{n}})$$ holds. Then $a_n=\_\_\_\_\_\_\_\_\_\_\_\_$．($n\in\mathbb{N}^*$)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3n-2}
If the purchase price of a certain product decreases by 10% while the selling price remains the same, then the profit margin (Profit Margin = $\frac{\text {Sales Price - Purchase Price}}{\text {Purchase Price}} \times 100 \%$) can increase by 12%. What was the original profit margin of this product?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8 \%}
The 62 interns from the Animath internship each take one ice cream ball from 5 flavors offered. We know that

- the malabar-flavored ice cream was at least as popular as the tabasco-flavored ice cream;
- the number of students who chose cactus-flavored ice cream is a multiple of 6;
- at most 5 students tasted lemon basil ice cream;
- at most one student opted for foie gras ice cream.

When Igor places the order with the ice cream vendor, he communicates, in alphabetical order of the flavors, the number of students who chose each flavor. How many different lists of numbers could he have pronounced?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2016}
A triangle has sides of length 888, 925, and \( x > 0 \). Find the value of \( x \) that minimizes the area of the circle circumscribed about the triangle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{259}
Find the maximum value of the following expression: 
\[
\left| \cdots \left| \left| x_1 - x_2 \right| - x_3\right| - \cdots - x_{1990} \right|
\]
where \( x_1, x_2, \cdots, x_{1990} \) are distinct natural numbers from 1 to 1990.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1989}
Triangle A's base is some percent greater than the base of Triangle B, and A's height is 20% less than the height of Triangle B. The area of Triangle A is 0.25% less than the area of Triangle B. What is the percentage by which Triangle A's base is greater than Triangle B's base?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24.69\%}
数列 $\left\{a_{n}\right\}$ 满足: $a_{1}=\frac{5}{2}, a_{n+1}=a_{n}^{2}-2$，求 $\left[a_{2023}\right]$ 除以 7 的余数。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
We draw the diagonals of the convex quadrilateral $ABCD$, then find the centroids of the 4 triangles formed. What fraction of the area of quadrilateral $ABCD$ is the area of the quadrilateral determined by the 4 centroids?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2}{9}}
Leo and Smilla find  $2020$  gold nuggets with masses  $1,2,\dots,2020$  gram, which they distribute to a red and a blue treasure chest according to the following rule:

First, Leo chooses one of the chests and tells its colour to Smilla. Then Smilla chooses one of the not yet distributed nuggets and puts it into this chest.

This is repeated until all the nuggets are distributed. Finally, Smilla chooses one of the chests and wins all the nuggets from this chest.

How many gram of gold can Smilla make sure to win?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1021110}
The lateral sides of a right trapezoid are 10 and 8. The diagonal of the trapezoid, drawn from the vertex of the acute angle, bisects this angle. Find the area of the trapezoid.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{104}