inclusionAI/Ring-lite-rl-data huggingface dataset:
Eight hockey teams are competing against each other in a single round to advance to the semifinals. What is the minimum number of points that guarantees a team advances to the semifinals?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
Determine the real numbers $x$, $y$, $z > 0$ for which
$xyz \leq \min\left\{4(x - \frac{1}{y}), 4(y - \frac{1}{z}), 4(z - \frac{1}{x})\right\}$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ x = y = z = \sqrt{2} }
Select numbers $a$ and $b$ between 0 and 1 independently and at random, and let $c$ be their sum. Let $A$, $B$, and $C$ be the results when $a$, $b$, and $c$, respectively, are rounded to the nearest integer.  What is the probability that $A+B=C$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3}{4}}
In an equilateral triangle \(\triangle ABC\), a point \(P\) is located inside the triangle such that \(\angle PAC = x\), \(\angle PCA = 3x\), and \(\angle PBC = 2x\). Determine the value of \(x\) in degrees.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
A repunit is a positive integer, all of whose digits are 1s. Let $a_{1}<a_{2}<a_{3}<\ldots$ be a list of all the positive integers that can be expressed as the sum of distinct repunits. Compute $a_{111}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1223456}
Find all four-digit numbers that, after removing the middle two digits, are reduced by 120 times.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1200, 2400, 3600, 4800}
A *strip* is the region between two parallel lines.  Let  $A$  and  $B$  be two strips in a plane.  The intersection of strips  $A$  and  $B$  is a parallelogram  $P$ .  Let  $A'$  be a rotation of  $A$  in the plane by  $60^\circ$ .  The intersection of strips  $A'$  and  $B$  is a parallelogram with the same area as  $P$ .  Let  $x^\circ$  be the measure (in degrees) of one interior angle of  $P$ .  What is the greatest possible value of the number  $x$ ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{150^\circ}
A rectangular piece of paper whose length is \(\sqrt{3}\) times the width has area \(A\). The paper is divided into three equal sections along the opposite lengths, and then a dotted line is drawn from the first divider to the second divider on the opposite side as shown. The paper is then folded flat along this dotted line to create a new shape with area \(B\). The ratio of the new shape's area to the original area is given in the form \(\frac{k}{m}\). Please calculate the value of \(k + m\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Evaluate the surface integral \(\iint_{S_1}\mathbf{F}\mathbf{\cdot} \hat{\mathbf n}\ dS\) over the sphere \(S_1\) given by \(x^2 + (y-2)^2 + z^2 = 9\) oriented outwards, where the vector field is \(\mathbf F(x,y, z) = \frac{x\hat{i} + y\hat{j} + z\hat{k}}{[x^2+y^2+z^2]^{3/2}}\). Note that \(\mathbf{F}\) is undefined at the origin, so the divergence theorem cannot be directly applied.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4\pi}
Find the limits of the following sequence, given $a_0 = 1$ and $a_n = \sum_{j=0}^{\lfloor n/2\rfloor} \frac{a_j}{(n-2j)!}$ for $n \ge 1$:

$$\lim_{n\to \infty}\frac{a_n}{n!}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
The train was delayed by \( t \) hours. By increasing the speed by \( m \) km/h, the driver made up for the delay over a section of \( s \) km. Determine the speed the train should have had on this section if there had been no delay.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{t m (4 s + t m)} - t m}{2 t}}
设 $U=\{1,2,3, \cdots, n\}(n \geqslant 3), S$ 是 $U$ 的一个子集. 如果不属于 $S$ 的一个元素出现在 $U$ 的一个排列 $P$ 的某处,并且在该排列 $P$ 中它处在 $S$ 中的两个元素之间, 那么我们称 $S$ 是由排列 $P$ 分裂出来的(例如排列 13542 可分裂出子集 $\{1,2,3\}$, 而不能分裂出子集 $\{3,4,5\}$ ), 试求最大正整数 $m$, 使得对 $U$ 的任意 $m$ 个子集(每个子集至少有 2个元, 至多有 $n-1$ 个元), 都存在 $U$ 的一个排列分裂出所有这 $m$ 个子集.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$n-2$.}
Piazzanos Pizzeria has a buy-1-get-1-free special on triple cheese pizzas, and a buy-2-get-1-free special on meat lovers pizzas. If the standard price for a pizza is $5, it costs $55 to purchase 10 triple cheese and a certain number of meat lovers pizzas using the special pricing. How many meat lovers pizzas were purchased?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
In an isosceles triangle, the lengths of two sides are $3cm$ and $7cm$ respectively. The perimeter of the triangle is ______ $cm$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{17}
Three red beads, two white beads, and one blue bead are placed in line in random order. What is the probability that no two neighboring beads are the same color? Express your answer in the form \(\frac{k}{m}\) where the fraction is in its simplest form, and find the value of \(k + m\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Five numbers were written on a board. Then these numbers were erased and their pairwise sums were written down:

$$
4, 8, 10, 12, 14, 18, 19, 21, 25, 29
$$

What were the five numbers originally written on the board?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1, 3, 7, 11, 18}
A hurricane is approaching the southern coast of Texas, and a rancher is planning to move some head of cattle 60 miles to higher ground to protect them from possible inland flooding that might occur. His animal transport truck holds 20 head of cattle. Traveling at 60 miles per hour, it took him 40 hours to transport all of his cattle to higher ground. How many head of cattle did the rancher have to move?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{400}
Given the equations \( 2^x + 4x + 12 = 0 \) and \( \log_2((y-1)^3) + 3y + 12 = 0 \), where \( x \) and \( y \) are real numbers, find the value of \( x + y \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2}
Determine the condition on the integer $k$ such that the norm of the Gaussian integer $p (1+i)^{-k}$ is between 1 and 2, where $p$ is a prime Gaussian integer with a norm greater than 2.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^k < N(p) < 2^{k+1}}
Three identical square sheets of paper each with side length $6$ are stacked on top of each other. The middle sheet is rotated clockwise $30^\circ$ about its center and the top sheet is rotated clockwise $60^\circ$ about its center, resulting in the $24$-sided polygon shown in the figure below. The area of this polygon can be expressed in the form $a-b\sqrt{c}$, where $a$, $b$, and $c$ are positive integers, and $c$ is not divisible by the square of any prime. Find the value of $a+b+c$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{147}
In each of the $9$ small circles of the following figure, write positive integers less than $10$, without repetitions. Additionally, it is true that the sum of the $5$ numbers located around each one of the $3$ circles is always equal to $S$. Find the largest possible value of $S$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{28}
On the eve of the 2010 Guangzhou Asian Games, a 12-person tour group took a commemorative photo near a venue of the Asian Games. They initially stood in a formation with 4 people in the front row and 8 people in the back row. Now, the photographer plans to keep the order of the front row unchanged, and move 2 people from the back row to the front row, ensuring that the 2 moved people are not adjacent in the front row. The number of different ways to adjust their positions is _____. (Answer in numerals)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{560}
In △ABC, a, b, and c are the sides opposite to angles A, B, and C respectively. If c = 2b and the area of △ABC is 1, find the minimum value of a.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{3}}
Quadrilateral $PQRS$ is a square. A circle with center $S$ has arc $PXC$. A circle with center $R$ has arc $PYC$. If $PQ = 3$ cm, what is the total number of square centimeters in the football-shaped area of regions II and III combined? Express your answer as a decimal to the nearest tenth.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5.1}
若不小于 1 的实数 $t$ ：对任意 $a, b \in[-1, t]$, 总存在 $c, d \in[-1, t]$, 使得 $(a+c)(b+d)=1$，则实数 $t$ 满足$\frac{m}{n} \leq t \leq p$，求$m+n+p$的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Three clever monkeys divide a pile of bananas. The first monkey takes some bananas from the pile, keeps three-fourths of them, and divides the rest equally between the other two. The second monkey takes some bananas from the pile, keeps one-fourth of them, and divides the rest equally between the other two. The third monkey takes the remaining bananas from the pile, keeps one-twelfth of them, and divides the rest equally between the other two. Given that each monkey receives a whole number of bananas whenever the bananas are divided, and the numbers of bananas the first, second, and third monkeys have at the end of the process are in the ratio $3: 2: 1,$what is the least possible total for the number of bananas?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{408}
The people in an infinitely long line are numbered $1,2,3,\dots$. Then, each person says either ”Karl“ or ”Lark” independently and at random. Let $S$ be the set of all positive integers $i$ such that people $i$, $i+1$, and $i+2$ all say ``Karl, and define $X = \sum\limits_{i \in S} 2^{-i}$. Then the expected value of $X^2$ is $\frac{m}{n}$, where $m$ and $n$ are positive integers with $\gcd(m,n)=1$. Compute $100m+n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
1028
}
Given the fractional equation $\frac{6}{x-2}-1=\frac{ax}{2-x}$, find the value of $a$ that allows the equation to have a positive root. Express your answer as a single numerical value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-3}
Let \[ p(x,y) = a_0 + a_1x + a_2y + a_3x^2 + a_4xy + a_5y^2 + a_6x^3 + a_7x^2y + a_8xy^2 + a_9y^3. \] Suppose that \( p(0,0) = p(1,0) = p(-1,0) = p(0,1) = p(0,-1) = p(1,1) = p(1,-1) = p(2,2) = 0. \) There is a point \( \left(\frac{a}{c},\frac{b}{c}\right) \) for which \( p\left(\frac{a}{c},\frac{b}{c}\right) = 0 \) for all such polynomials, where \( a, b, \) and \( c \) are positive integers, \( a \) and \( c \) are relatively prime, and \( c > 1 \). Find \( a + b + c \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40}
Compute the partial derivative of the first derivative of \( x(t) \) with respect to \( x(t) \), represented as \[ \frac{\partial\left(\frac{d(x(t))}{dt}\right)}{\partial x(t)} \].
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Grant has worked as a freelance math worker for the last three months. The first month he made $350. The second month he made $50 more than double he made the first month. The third month he made some multiple of the sum of the first two months. Grant made a total of $5500 in his first three months. What is the relationship between the third month's earnings and the sum of the first two months' earnings?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4:1}
Starting at $12:00:00$ AM on January $1,$ $2022,$ after $13!$ seconds it will be $y$ years (including leap years) and $d$ days later, where $d < 365.$ Find $y + d.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{317}
Peter has $2022$ pieces of magnetic railroad cars, which are of two types: some have the front with north and the rear with south magnetic polarity, and some have the rear with north and the rear with south magnetic polarity (on these railroad cars the front and the rear can be distinguished). Peter wants to decide whether there is the same number of both types of cars. He can try to fit together two cars in one try. What is the least number of tries needed?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2021}
Evaluate the integral of the function \( f(z) = \exp(1/z) \) over the unit circle in the complex plane.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\pi i}
设 $a_{n}$ 是与 $\sqrt{\frac{n}{2}}$ 的差的绝对值最小的整数, $b_{n}$ 是与 $\sqrt{2 n}$ 的差的绝对值最小的整数. 记 $\left\{\frac{1}{a_{n}}\right\}$ 的前 $n$ 项和为 $S_{n},\left\{\frac{1}{b_{n}}\right\}$ 的前 $n$ 项和为 $T_{n}$ 。则 $2 T_{100}-S_{100}$ 的值为 $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Vasya wrote a note on a piece of paper, folded it into quarters, and labeled the top with "MAME". Then he unfolded the note, wrote something else on it, folded the note along the creases randomly (not necessarily as before), and left it on the table with a random side facing up. Find the probability that the inscription "MAME" is still on top.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1/8}
In triangle $\mathrm{V} A B C$, where $A, B, C$ represent the three interior angles, find the minimum value of $\frac{1}{\sin ^{2} A} + \frac{1}{\sin ^{2} B} + \frac{4}{1 + \sin C}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16 - 8\sqrt{2}}
Given Elmer’s new car provides a 70% better fuel efficiency and uses a type of fuel that is 35% more expensive per liter, calculate the percentage by which Elmer will save money on fuel costs if he uses his new car for his journey.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20.6\%}
Mason is three times younger than Sydney and Sydney is some years younger than Mason's father. If Mason is 20 years old, how old is his father?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60}
In convex quadrilateral ABCD, $\angle BAD = \angle BCD = 90^{\circ}$, and BC = CD. Let E be the intersection of diagonals AC and BD. Given that $\angle AED = 123^{\circ}$, find the degree measure of $\angle ABD$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$78$}
In triangle  $ABC$ , let  $M$  be the midpoint of  $BC$ ,  $H$  be the orthocenter, and  $O$  be the circumcenter. Let  $N$  be the reflection of  $M$  over  $H$ . Suppose that  $OA = ON = 11$  and  $OH = 7.$  Compute  $BC^2$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{288}
Let $r$ and $s$ be positive integers. Derive a formula for the number of ordered quadruples $(a,b,c,d)$ of positive integers such that
$$3^r \cdot 7^s = \text{lcm}(a,b,c)= \text{lcm}(a,b,d)=\text{lcm}(a,c,d)=\text{lcm}(b,c,d),$$
depending only on $r$ and $s.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(6r^2 + 4r + 1)(6s^2 + 4s + 1)}
The average age of a group of men is increased by years when two of them, whose ages are 21 years and 23 years, are replaced by two new men with an average age of 32. How many men are in the group?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
Integers $a$, $b$, $c$, and $d$, not necessarily distinct, are chosen independently and at random from 0 to 2007, inclusive. What is the probability that $ad-bc$ is even?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{5}{8}}
In the quadrilateral \( ABCD \), \( AB = a \), and the ratio of the measures of the four angles \( A, B, C, D \) is \( 3:7:4:10 \). Additionally, \(\angle CDB = 30^{\circ}\). Find the length of \( BC \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{2}}{3} a}
Given a rectangle \(A B C D\). A circle intersects the side \(A B\) at points \(K\), \(L\), and \(N\). Find the length of the segment \(M N\) if \(A K = 10\), \(K L = 17\), and \(D N = 7\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{23}
A couple, both phenotypically normal, have a son who suffers from both hemophilia (an X-linked recessive condition) and phenylketonuria (an autosomal recessive condition). If they have another daughter, what is the probability that she will be phenotypically normal? Express your answer as a simplified fraction.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3}{4}}
3 设2005条线段首尾相连，组成封闭的折线，且折线的任何两段都不在同一条直线上，那么，此折线自身相交的交点最多有多少个？
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2007005}
Two circles have a radius of $9$, and one circle has a radius of $7$. Each circle is externally tangent to the other two circles, and each circle is internally tangent to two sides of an isosceles triangle, as shown in the figure. The sine of the base angle of the triangle 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{7}
The figure shows a shape consisting of a regular hexagon of side \( 18 \, \text{cm} \), six triangles, and six squares. The outer perimeter of the shape is \( P \, \text{cm} \). What is the value of \( P \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{216}
Let  $P$  and  $Q$  be the midpoints of sides  $AB$  and  $BC,$  respectively, of  $\triangle ABC.$  Suppose  $\angle A = 30^{\circ}$  and  $\angle PQC = 110^{\circ}.$  Find  $\angle B$  in degrees.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{80^\circ}
Let $A_1B_1C_1$, $A_2B_2C_2$, and $A_3B_3C_3$ be three triangles in the plane. For $1 \le i \le 3$, let $D_i$, $E_i$, and $F_i$ be the midpoints of $B_iC_i$, $A_iC_i$, and $A_iB_i$, respectively. Furthermore, for $1 \le i \le 3$ let $G_i$ be the centroid of $A_iB_iC_i$.

Suppose that the areas of the triangles $A_1A_2A_3$, $B_1B_2B_3$, $C_1C_2C_3$, $D_1D_2D_3$, $E_1E_2E_3$, and $F_1F_2F_3$ are $2$, $3$, $4$, $20$, $21$, and $2020$, respectively. Compute the largest possible area of $G_1G_2G_3$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{917}
How can we determine the number of non-isomorphic semi-direct products of a cyclic group of order 5 with a cyclic group of order 31?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
For each positive integer $n$, find the number of $n$-digit positive integers that satisfy both of the following conditions:
$\bullet$ no two consecutive digits are equal, and
$\bullet$ the last digit is a prime.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2 \times 9^n - 2 \times (-1)^n}{5}}
Find all solutions to \( (m^2 + n)(m + n^2) = (m - n)^3 \), where \( m \) and \( n \) are non-zero integers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-1, -1), (8, -10), (9, -6), (9, -21)}
Calculate $\sin 18^{\circ}$, using only the first two terms of series (3), and estimate the resulting error.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.3090}
In Figure 1, the square \(PQRS\) is inscribed in \(\triangle ABC\). The areas of \(\triangle APQ\), \(\triangle PBS\), and \(\triangle QRC\) are 4, 4, and 12 respectively. If the area of the square is \(c\), find the value of \(c\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
Let $n=2^{31}3^{19}.$ How many positive integer divisors of $n^2$ are less than $n$ but do not divide $n$ ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{589}
Two opposite sides of a rectangle are each divided into $n$ congruent segments, and the endpoints of one segment are joined to the center to form triangle $A$. The other sides are each divided into $m$ congruent segments, and the endpoints of one of these segments are joined to the center to form triangle $B$. [See figure for $n=5, m=7$.] What is the ratio of the area of triangle $A$ to the area of triangle $B$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["\\frac{m}{n}"]}
Given the circle equation $2x^2+2y^2+6x-4y-3=0$, find the coordinates of the center and the radius. Express the center coordinates as an ordered pair and the radius as a simplified radical.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left(-\frac{3}{2}, 1\right), \frac{\sqrt{19}}{2}}
There are three table tennis teams, and members from different teams will play a match against each other, but members from the same team will not play against each other. There are a total of 10 players in the three teams, and 31 matches were played in total. How many players are there in each team?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2, 3, 5}
A list of positive integers is called good if the maximum element of the list appears exactly once. A sublist is a list formed by one or more consecutive elements of a list. For example, the list $10,34,34,22,30,22$ the sublist $22,30,22$ is good and $10,34,34,22$ is not. A list is very good if all its sublists are good. Find the minimum value of $k$ such that there exists a very good list of length $2019$ with $k$ different values on it.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
Five line segments are drawn inside a rectangle, creating a decagon. Find the sum of the six marked angles. Express your answer in degrees as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1080^\circ}
Consider a round table on which  $2014$  people are seated. Suppose that the person at the head of the table receives a giant plate containing all the food for supper. He then serves himself and passes the plate either right or left with equal probability. Each person, upon receiving the plate, will serve himself if necessary and similarly pass the plate either left or right with equal probability. Compute the probability that you are served last if you are seated  $2$  seats away from the person at the head of the table.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1/2013}
The radii of the excircles of a triangle are \( r_{a}, r_{b}, \) and \( r_{c} \), and the radius of the circumcircle is \( R \). Determine the angles of the triangle if

$$
r_{a}+r_{b}=3 R \quad \text { and } \quad r_{b}+r_{c}=2 R.
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30^\circ, 60^\circ, 90^\circ}
If w is 40 percent less than q, q is 40 percent less than y, z is 46 percent less than y, and x is 30 percent more than w, then z is greater than x by what percent of w?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
There are three cards, each with a number from the set $\{1,2, \ldots, 10\}$. These three cards were shuffled and distributed to three people, who recorded the numbers on their respective cards. The cards were then collected and the process was repeated several times. After some repetitions, each of the three people summed their records. Knowing that the obtained sums were 13, 15, and 23, what were the numbers on the cards?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3, 5, 9}