inclusionAI/Ring-lite-rl-data huggingface dataset:
Stringing 6 red balls, 1 green ball, and 8 yellow balls into a necklace, how many possible different types can appear (balls of the same color are not distinguished)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1519}
At a certain pizzeria, there are five different toppings available, and a pizza can be ordered with any (possibly empty) subset of them on it. Determine the number of ways to order an unordered pair of pizzas such that at most one topping appears on both pizzas and at least one topping appears on neither.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{271}
How many squares whose sides are parallel to the axes and whose vertices have coordinates that are integers lie entirely within the region bounded by the line $y=\pi x$, the line $y=-0.1$ and the line $x=5.1?$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50}
According to the directions on a packet of smoothie mix, 1 3-ounce packet of smoothie mix is to be combined with 15 ounces of water to make a smoothie. To prepare 150 smoothies of a certain size, 180 3-ounce packets of smoothie mix are required. What is the size of each smoothie in ounces?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21.6}
For transporting cargo from one place to another, a certain number of trucks with the same capacity were required. Due to road conditions, each truck had to carry 0.5 tons less than originally planned, thus 4 additional trucks of the same capacity were required. The mass of the transported cargo was at least 55 tons but did not exceed 64 tons. How many tons of cargo were transported by each truck?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2.5 \text{ tons}}
Person A arrives between 7:00 and 8:00, while person B arrives between 7:20 and 7:50. The one who arrives first waits for the other for 10 minutes, after which they leave. Calculate the probability that the two people will meet.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{3}}
What fraction of the area of triangle $ABC$ is occupied by those points $K$ which lie inside the triangle and for which the point-similarity with center $K$ and ratio $-1/2$ maps $A$ and $B$ to internal points of the triangle, but maps $C$ to an external point?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{9}}
A regular  $17$ -gon with vertices  $V_1, V_2, . . . , V_{17}$  and sides of length  $3$  has a point  $ P$  on  $V_1V_2$  such that  $V_1P = 1$ . A chord that stretches from  $V_1$  to  $V_2$  containing  $ P$  is rotated within the interior of the heptadecagon around  $V_2$  such that the chord now stretches from  $V_2$  to  $V_3$ . The chord then hinges around  $V_3$ , then  $V_4$ , and so on, continuing until  $ P$  is back at its original position. Find the total length traced by  $ P$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4\pi}
Let $S$ be a real number. It is known that however we choose several numbers from the interval $(0, 1]$ with sum equal to $S$, these numbers can be separated into two subsets with the following property: The sum of the numbers in one of the subsets doesn’t exceed 1 and the sum of the numbers in the other subset doesn’t exceed 5. 

Find the greatest possible value of $S$. 
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5.5}
There are four tiles marked X, and three tiles marked O. The seven tiles are randomly arranged in a row. What is the probability that the two outermost positions in the arrangement are occupied by X and the middle one by O?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{6}{35}}
知曲线 $C$ 上任意一点, 到定点 $A(1,0)$ 与定直线 $x=4$ 的距离之和等于 5 , 对于给定的点 $B(b, 0)$, 在曲线上恰有三对不同的点关于点 $B$ 对称, 求 $b$ 的取值范围.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ $b$ 的取值范围 $\frac{5}{2}<b<4$.}
Calculate the volumes of solids formed by the rotation of regions bounded by the graphs of the functions around the y-axis.

$$
y=\arcsin \frac{x}{5}, y=\arcsin x, y=\frac{\pi}{2}
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6 \pi^2}
Wendy eats sushi for lunch. She wants to eat six pieces of sushi arranged in a $2 \times 3$ rectangular grid, but sushi is sticky, and Wendy can only eat a piece if it is adjacent to (not counting diagonally) at most two other pieces. In how many orders can Wendy eat the six pieces of sushi, assuming that the pieces of sushi are distinguishable?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{360}
Suppose the function \( y= \left| \log_{2} \frac{x}{2} \right| \) has a domain of \([m, n]\) and a range of \([0,2]\). What is the minimum length of the interval \([m, n]\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3}{2}}
Given the universal set U = {0,1,2,3,4}, set M = {0,2,3}, and the complement of N in U, ∁_U N = {1,2,4}, find the intersection of M and N, M ∩ N. Express your answer as a set of elements enclosed in curly brackets, with elements separated by commas.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{{0,3}}
Consider the signal $x(t)=e^{-t}u(t)$ where $u(t)=\mathbb{1}(t\geq0)$, i.e. the Heaviside function. Find the signal $y(t)=x(t)*x(-t)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{1}{2}e^{-|t|}\)}
Let $M$ be the intersection of diagonals of the convex quadrilateral $ABCD$, where $m(\widehat{AMB})=60^\circ$. Let the points $O_1$, $O_2$, $O_3$, $O_4$ be the circumcenters of the triangles $ABM$, $BCM$, $CDM$, $DAM$, respectively. Calculate the ratio of the area of quadrilateral $ABCD$ to the area of quadrilateral $O_1O_2O_3O_4$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3/2}
Given $x < y < 0$, compare the magnitude of $(x^2+y^2)(x-y)$ with $(x^2-y^2)(x+y)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(x^2 + y^2)(x - y) > (x^2 - y^2)(x + y)}
The coefficient of $x^{4}y^{2}$ in the expansion of $({x-\frac{1}{x}+y})^8$ is ______ (expressed as a constant).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-168}
Last year, Manfred received 26 paychecks. Each of his first 6 paychecks was $750. After that, Manfred received a 5% pay raise for his next 10 paychecks. Then, Manfred received another 3% pay raise for his remaining paychecks. Also, Manfred had a one-time bonus of $250 added to his 10th paycheck and a medical deductible of $120 taken from his 20th paycheck. Calculate the average (arithmetic mean) amount of his paychecks for the year, to the nearest dollar, after considering federal tax deductions at a rate of 15% and the other mentioned adjustments.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$674}
Given $f(x) = x^5 - 2x^3 + 3x^2 - x + 1$, calculate the value of $f(x)$ at $x=3$ using the Horner's method. The value of $v_3$ is \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
To calculate $41^2$, Tom mentally computes $40^2$ and adds a number. To find $39^2$, Tom subtracts a number from $40^2$. What number does he add to calculate $41^2$ and subtract to calculate $39^2$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{79}
A circle centered at \( O_1 \) with a radius of 2, inscribed in the triangle \( ABC \), touches side \( BC \) at point \( D \). A second circle centered at \( O_2 \) with a radius of 4 touches the extensions of sides \( AB \) and \( AC \), as well as side \( BC \) at point \( E \). Find the area of the quadrilateral \( O_1 D O_2 E \) if the measure of angle \( ACB \) is \( 120^\circ \). (20 points)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{30}{\sqrt{3}}}
11. Let $n$ be a given natural number, $n \geqslant 3$, and for $n$ given real numbers $a_{1}, a_{2}, \cdots, a_{n}$, denote the minimum value of $\left|a_{i}-a_{j}\right|(1 \leqslant$ $i<j \leqslant n)$ as $m$. Find the maximum value of $m$ under the condition $\sum_{i=1}^{n} a_{i}^{2}=1$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{\frac{12}{n(n^{2}-1)}}}
10. The sum $\sum_{k=1}^{2020} k \cos \left(\frac{4 k \pi}{4041}\right)$ can be written in the form

$$
\frac{a \cos \left(\frac{p \pi}{q}\right)-b}{c \sin ^{2}\left(\frac{p \pi}{q}\right)}
$$

where $a, b, c$ are relatively prime positive integers and $p, q$ are relatively prime positive integers where $p<q$. Determine $a+b+c+p+q$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["4049"]}
Let  $S$  be the set of all ordered triples  $(p,q,r)$  of prime numbers for which at least one rational number  $x$  satisfies  $px^2+qx+r=0.$  Which primes appear in seven or more elements of  $S?$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
$\left[\begin{array}{ll}[\text { Classical combinatorics (other) })]\end{array}\right]$

On a $11 \times 11$ grid, 22 cells are marked such that exactly two cells are marked in each row and each column. Two arrangements of marked cells are equivalent if, by swapping any number of columns among themselves and any number of rows among themselves, one arrangement can be transformed into the other. How many non-equivalent arrangements of marked cells exist?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
For positive integers $n$, let $h(n)$ return the smallest positive integer $k$ such that $\frac{1}{k}$ has exactly $n$ digits after the decimal point, and $k$ is divisible by 3. How many positive integer divisors does $h(2010)$ have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4022}
Given $x>0$, $y>0$, $a=x+y$, $b= \sqrt{x^{2}-xy+y^{2}}$, $c=\lambda \sqrt{xy}$, if $a$, $b$, $c$ can be the lengths of the sides of a triangle, then the range of the positive real number $\lambda$ is \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1<\lambda<3}
Let $p$ be a prime. We arrange the numbers in ${\{1,2,\ldots ,p^2} \}$ as a $p \times p$ matrix $A = ( a_{ij} )$. Next we can select any row or column and add $1$ to every number in it, or subtract $1$ from every number in it. We call the arrangement [i]good[/i] if we can change every number of the matrix to $0$ in a finite number of such moves. How many good arrangements are there?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2(p!)^2}
Given an ellipse x^2/36 + y^2/b^2 = 1 (6 > b > 0) and any point P on the ellipse other than the vertices, it satisfies |OP + OF| = 7, where point F is the left focus of the ellipse, and point O is the origin of the coordinate system. Find the distance |PF|. Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Compute the number of non-empty subsets $S$ of $\{-3, -2, -1, 0, 1, 2, 3\}$ with the following property: for any $k \ge 1$ distinct elements $a_1, \dots, a_k \in S$, we have $a_1 + \dots + a_k \neq 0$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
A rectangle is divided into three smaller congruent rectangles. Each smaller rectangle is similar to the large rectangle. Find the ratio of the length of a longer side to that of a shorter side in each of the four rectangles. Express your answer as a simplified ratio (e.g., x:y).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{3} : 1}
Find the probability that the reflected Brownian motion process $Y(t)$ hits $a$ at or after time $t$, given that it hits $a$ before time $t$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
For how many integers n, where 0 <= n < 2013, is the polynomial n^4 + 2n^3 - 20n^2 + 2n - 21 congruent to 0 modulo 2013? Provide your answer as a single integer representing the count of integers n.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
A **Beaver-number** is a positive 5-digit integer whose digit sum is divisible by 17. Call a pair of **Beaver-numbers** differing by exactly $1$ a **Beaver-pair**. The smaller number in a **Beaver-pair** is called an **MIT Beaver**, while the larger number is called a **CIT Beaver**. Find the positive difference between the largest and smallest **CIT Beavers** (over all **Beaver-pairs**).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{79200}
For real numbers $x$, let \[P(x)=1+\cos(x)+i \sin(x)-\cos(2x)-i \sin(2x)+\cos(3x)+i \sin(3x)\] where $i=\sqrt{-1}$. Find the number of values of $x$ with $0 \leq x<2 \pi$ such that $P(x)=0$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
If you multiply a number by 2 instead of multiply it by 20 and add 3, you get 22. How much is it when you calculate correctly?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{223}
In a city with 10 parallel streets and 10 streets crossing them at right angles, what is the minimum number of turns that a closed bus route passing through all intersections can have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
Let $m$ and $n$ be positive integers such that
$$(m^3 -27)(n^3- 27) = 27(m^2n^2 + 27):$$
Find the maximum possible value of $m^3 + n^3$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1792}
Find the smallest value of \( n \) such that given any \( n \) distinct integers, it is always possible to find 4 different integers \( a, b, c, d \) such that \( a + b \equiv c + d \mod 20 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Evaluate the integral $$\oint_{\verts{z}\ =\ 3} {\dd z \over \root{z^{2} - 1}}$$ using a branch of the function that is analytic outside the interval $[-1.1]$ and positive on $(1,\infty)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2πi}
Gunner has a box of mittens with 20 fewer pairs than a box of plugs. He decides to put 30 more pairs of plugs into the box of plugs. If there are now 400 plugs, how many pairs of mittens are in the box of mittens?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{150}
Solve the equation: $$\begin{aligned}|x+1|-|x|+3|x-1|-2|x-2|=x+2.\end{aligned}$$ Please provide the sum of the minimum and maximum solutions.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Consider the recurrence system for an integer $k$ where $1\le k\le \frac{p-1}{2}$, given by:
\[ c_k = \frac{1}{p}\binom{p}{k} - c_{k-1}, \]
with the initial condition $c_0 = 0$. Here, $p$ is an odd prime. The goal is to find a closed-form solution for $c_k$ in terms of elementary functions. It has been observed that $c_k$ can be expressed as a sum:
\[ c_k = \sum_{i=0}^{k-1} \frac{(-1)^i}{p} \binom{p}{k-i}. \]
The task is to either simplify this sum into a closed-form expression or directly solve the recurrence system to obtain a closed-form solution. What is the closed-form solution for $c_k$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\binom{p-1}{k} - (-1)^k}{p}}
In 2021, a first-class postage stamp cost 85 pence and a second-class postage stamp cost 66 pence. What is the smallest total number of stamps that should be purchased in order to spend an exact number of pounds, given that at least one of each type of stamp must be bought? Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Gilbert is an enthusiastic gardener who loves growing herbs in his yard to use in his cooking. At the beginning of the spring, he planted eight different types of herbs: three basil bushes, one parsley plant, two types of mint, one rosemary plant, and one thyme plant. Each herb has a different growth rate: basil doubles every two weeks, parsley grows steadily with one new plant each week, mint triples each month, rosemary and thyme stay the same.

Halfway through spring, which was six weeks long, one of the basil plants bloomed flowers and dropped seeds, resulting in an extra basil plant in his garden. Unfortunately, during the same week, weather damage ruined all the mint plants, so they didn't grow any further for the rest of the season. Also, in the fifth week, an infestation of caterpillars devoured two of the parsley plants. By the time spring ended, how many herb plants did Gilbert have in his garden?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{35}
Masha wrote on the board, in ascending order, all the natural divisors of a certain number \( N \) (the first divisor is 1, and the largest divisor is the number \( N \) itself). It turned out that the third-to-last divisor is 21 times larger than the second divisor. What is the largest possible value of \( N \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{441}
Each unit square of a $4 \times 4$ square grid is colored either red, green, or blue. Over all possible colorings of the grid, what is the maximum possible number of L-trominos that contain exactly one square of each color?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
In the Cartesian coordinate system, the vertices of rhombus $OABC$ are $O(0,0)$ and $A(1,1)$, and $\overrightarrow{OA} \cdot \overrightarrow{OC} = 1$. Then, $\overrightarrow{AB} \cdot \overrightarrow{AC}$ equals \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
A fruit stand is selling apples for $2 each, bananas for $1 each, and oranges for $3 each. Emmy has $200 while Gerry has $100. They notice that if they buy 10 apples, they get a 20% discount on the apple price. Both Emmy and Gerry want to buy a mix of these three fruits within their budgets, and they want to ensure they have at least 5 of each fruit. Their goal is to maximize the number of apples they buy, considering the discount and the minimum quantity constraint for other fruits. How many apples can Emmy and Gerry buy altogether at most?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{160}
The expression $\lfloor x\rfloor$ denotes the greatest integer less than or equal to $x$. Find the value of $$\left\lfloor\frac{2002!}{2001!+2000!+1999!+\cdots+1!}\right\rfloor.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2000}
At the bus stop near Absent-minded Scientist's house, buses of two routes stop: No. 152 and No. 251. Both go to the metro station. The interval between buses No. 152 is exactly 5 minutes, and the interval between buses No. 251 is exactly 7 minutes. The intervals are strictly adhered to, but these two routes are not coordinated between each other and their schedules do not depend on each other. At a completely random moment in time, the Absent-minded Scientist comes to the bus stop and gets on the first bus that arrives to get to the metro. What is the probability that the Scientist will get on bus No. 251?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{5}{14}}
Let $m$ and $n$ be two odd integers with $m, n \geqslant 3$. In an $m \times n$ grid, each small square is colored either blue or red. We denote $A$ as the number of rows where blue squares are in the majority, and $B$ as the number of columns where red squares are in the majority.

What is the largest possible value of $A + B$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{m + n - 2}
Given vectors $\vec{a}$ and $\vec{b}$ satisfy $|\vec{a}|=2$, $|\vec{b}|=1$, and $|\vec{a}-2\vec{b}|=2\sqrt{3}$, then the angle between $\vec{a}$ and $\vec{b}$ is __________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2\pi}{3}}
Convex quadrilateral $ABCD$ satisfies the following conditions: \( \angle{CAB} = \angle{ADB} = 30^{\circ} \), \( \angle{ABD} = 77^{\circ} \), \( BC = CD \), and \( \angle{BCD} = n^{\circ} \) for some positive integer \( n \). Compute \( n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{68}
Given that point \( P \) lies on the hyperbola \(\frac{x^{2}}{16} - \frac{y^{2}}{9} = 1\), and the distance from \( P \) to the right directrix of this hyperbola is the arithmetic mean of the distances from \( P \) to the two foci of this hyperbola, find the x-coordinate of \( P \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\frac{64}{5}}
Let $ABC$ be a triangle with $AB = 6$, $AC = 8$, $BC = 7$. Let $H$ be the orthocenter of $\triangle ABC$. Let $D \neq H$ be a point on $\overline{AH}$ such that $\angle HBD = \frac{3}{2} \angle CAB + \frac{1}{2} \angle ABC - \frac{1}{2} \angle BCA$. Find $DH$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
设数列$\left\{ {{a}_{n}} \right\}$满足${{a1}}=\frac{2}{3}$，且对任意的$n\in {{N}^{*}}$，满足${{a}_{n+2}}-{{an}}\le {{2n}}$，${{a}_{n+4}}-{{an}}\ge 5\times {{2}^{n}}$,则${{a}_{2017}}=$_________
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{{{2}^{2017}}}{3}$}
Given a sequence $\{a_{n}\}$ that satisfies $a_{n+1}+(-1)^{n}a_{n}=2n-1$, then the sum of the first $40$ terms of $\{a_{n}\}$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{820}
Six lines are drawn in the plane. Determine the maximum number of points through which at least three lines pass.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
In the arithmetic sequence {a_n}, a_5 > 0 and a_4 + a_7 < 0. Find the term S_n in the sum of the first n terms of {a_n} that corresponds to the maximum sum. Express your answer in the form S_n, specifying the value of n.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{S_5}
On an $11 × 11$ checker-board 22 squares are marked so that exactly two of the marked squares lie in each column and each row. Two arrangements of marked squares are considered equivalent if in any number of permutations of the columns and/or (independent) permutations of rows one arrangement can be obtained from the other. How many nonequivalent arrangements of marked squares are there?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
The average of 20, 40, and 60 is 5 more than the average of some numbers and 35. If the other numbers are 60 and a third number, what is the first number?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
A $8 \times 8$ board is given. Seven out of $64$ unit squares are painted black. Suppose that there exists a positive $k$ such that no matter which squares are black, there exists a rectangle (with sides parallel to the sides of the board) with area $k$ containing no black squares. Find the maximum value of $k$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Let $a$ and $b$ be real constants such that
\[x^4 + ax^3 + 3x^2 + bx + 1 \ge 0\]for all real numbers $x.$  Find the largest possible value of $a^2 + b^2.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40}