inclusionAI/Ring-lite-rl-data huggingface dataset:
Parkway Elementary School has 800 seventh-grade students. Out of these, 420 are boys and 380 are girls. These students can participate in a range of extracurricular activities including soccer, basketball, chess club, math club, and glee club. The school allows each student to participate in only one extracurricular activity. There are 320 students participating in soccer, out of which 70% are boys, 280 students participating in basketball, of which 65% are girls, 70 students participating in the chess club that's majorly boys (80%), and the math club has a total of 50 members with an equal gender representation. The remaining students participate in the glee club. 

The school also has an absentee record where 5% of total boys and 8% of total girls were absent throughout the year and were not able to participate in any of the clubs. How many girl students in seventh grade at Parkway Elementary School are not participating in any extracurricular activities, considering the absenteeism as well?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{33}
Given the XOR result X = 0000 0011, determine if it is possible to uniquely identify the original values A and B, where A and B are 7-bit ASCII characters ranging from 32 to 126. Calculate the number of combinations a computer would need to try to find the correct values of A and B.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{94}
Identify the group \((D, \circ)\) of the symmetries of the circle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{O(2)}
Angelo wrote down the word TEAM. He then swapped two adjacent letters around and wrote down the new order of the letters. He proceeded in this way until he obtained the word MATE. What is the minimum number of swaps that Angelo could have used to transform the word "TEAM" into "MATE"? Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
How many ordered triples $(A,B,C)$ of subsets of $\{1,2,...,n\}$ exist such that $A\cap B\cap C = \emptyset$, $A\cap B \neq \emptyset$, and $A\cap C \neq \emptyset$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7^n - 2*6^n + 5^n}
Leap Day, February 29, 2004, occurred on a Sunday. On what day of the week will Leap Day, February 29, 2020, occur?



Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{Saturday 

}
## C2

In a country with $n$ cities, all direct airlines are two-way. There are $r>2014$ routes between pairs of different cities that include no more than one intermediate stop (the direction of each route matters). Find the least possible $n$ and the least possible $r$ for that value of $n$.

Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2016}
Given a continuous function $f: \mathbb{R} \to \mathbb{C}$, if $\int_{\mathbb{R}} |f(x)| \, dm(x) < \infty$, is it true that $\text{ess}\sup |f| < \infty$? Answer "yes" or "no."
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{no}
Determine the cardinality of the superstructure $\mathbb V$ over $\mathbb R$ and its subset $\mathcal{P}_{fin}(\mathbb V)$, where $\mathbb V=\cup_{n<\omega}V_n$ with $V_0=\mathbb R$ and $V_{n+1}=V_n\cup \mathcal{P}(V_n)$. What is the cardinality of these sets?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\beth_\omega}
Let \( S \) be the set of all points(x,y)in the coordinate plane such that \( 0 \leq  x \leq  \frac{\pi }{2} \) and \( 0 \leq  y \leq  \frac{\pi }{2} \) . What is the area of the
 subset of \( S \) for which
 \( {\sin }^{2}x - \sin x\sin y + {\sin }^{2}y \leq  \frac{3}{4}? \)

Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi^2}{6}}
8,9

Point $K$ is located on edge $AD$ of tetrahedron $ABCD$, point $N$ is on the extension of edge $AB$ beyond point $B$, and point $M$ is on the extension of edge $AC$ beyond point $C$, such that $AK:KD=3:1$, $BN=AB$, and $CM:AC=1:3$. Construct the section of the tetrahedron by the plane passing through points $K, M, N$. In what ratio does this plane divide the volume of the tetrahedron?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2:33}
Determine the rank of the field of rational numbers \(\mathbb{Q}\) over the ring of integers \(\mathbb{Z}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
There are 1996 points on a circle, each painted in one of several different colors. The number of points of each color is different. Determine the number of colors and the distribution of points per color that maximizes the number of inscribed polygons that can be formed by selecting one point from each color, ensuring that all vertices of the polygon are of different colors.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{61}
Let \( K = \mathbb{Q} \) and consider the polynomial \( f = X^3 - 3 \). Determine the basis and the degree of the splitting field of \( f \) over \( K \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Determine whether the set $X=\{A \in P : 1 \in A\}$, where $P=\{A \subseteq \mathbb{N} : A \text{ is non-empty, finite, and has an even number of elements}\}$, has a lower bound in the partially ordered set $(P,\subseteq)$. Provide your answer as "yes" or "no".
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{no}
Determine whether the growth in the number of decimal digits of the factorial function is exponential or sub-exponential. For example, 7! = 5040, which has 4 digits.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{sub-exponential}
The equation \( x^{2} + mx + 1 + 2i = 0 \) has real roots. Find the minimum value of the modulus of the complex number \( m \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{2 + 2\sqrt{5}}}
Among all the triangles inscribed in a given circle, find the one for which the sum of the squares of the lengths of the sides is maximized.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9R^2}
Given the set $M=\{x|x^2-5x-6\leq0, x\in \mathbb{N}\}$, if two numbers are randomly selected from $M$ and added together, the sum will be an element of set $N$. How many non-empty proper subsets does $N$ have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2046}
Ana, Bob, and CAO bike at constant rates of $8.6$ meters per second, $6.2$ meters per second, and $5$ meters per second, respectively. They all begin biking at the same time from the northeast corner of a rectangular field whose longer side runs due west. Ana starts biking along the edge of the field, initially heading west, Bob starts biking along the edge of the field, initially heading south, and Cao bikes in a straight line across the field to a point $D$ on the south edge of the field. Cao arrives at point $D$ at the same time that Ana and Bob arrive at $D$ for the first time. The ratio of the field's length to the field's width to the distance from point $D$ to the southeast corner of the field can be represented as $p : q : r$, where $p$, $q$, and $r$ are positive integers with $p$ and $q$ relatively prime. Find $p+q+r$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{61}
$2006$ points are evenly spaced on a circle. Given one point, find the maximum number of points that are less than one radius distance away from that point.


Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{668
}
In a pentagon, all diagonals are drawn. Identify seven angles between two diagonals or between diagonals and sides such that if these angles are equal, the pentagon is regular.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Determine whether the functional equation \[3=\sqrt{1+f(2)\sqrt{1+f(3)\sqrt{1+\cdots}}}\] has a unique solution \(f(x)=x\) for any positive, continuous, and differentiable function \(f(x)\) on \((0,\infty)\) that is not constant.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{f(x) = x}
Determine if the integral \( \int_0^\infty \frac{1-\cos x}{x^2(x^2+1)}\,dx \) can be evaluated without using complex analysis techniques such as complex integration, residues, and Cauchy's Theorem.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{2e}}
Lucy has some spare change. If she were to convert it into quarters and pennies, the minimum number of coins she would need is $66$. If she were to convert it into dimes and pennies, the minimum number of coins she would need is $147$. How much money, in cents, does Lucy have?

Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$210$}
Ed has five identical green marbles, and a large supply of identical red marbles. He arranges the green marbles and some of the red ones in a row and finds that the number of marbles whose right hand neighbor is the same color as themselves is equal to the number of marbles whose right hand neighbor is the other color. An example of such an arrangement is GGRRRGGRG. Let $m$ be the maximum number of red marbles for which such an arrangement is possible, and let $N$ be the number of ways he can arrange the $m+5$ marbles to satisfy the requirement. Find the remainder when $N$ is divided by $1000$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{003}
1. A bear walks one mile south, one mile east, and one mile north, only to find itself where it started. Another bear, more energetic than the first, walks two miles south, two miles east, and two miles north, only to find itself where it started. However, the bears are not white and did not start at the north pole. At most how many miles apart, to the nearest .001 mile, are the two bears' starting points?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["3.477"]}
Suppose that \( \sum_{j=1}^{\infty} a_j \) is a convergent series with \( a_j \geq 0 \). Determine whether it is true that \( \lim_{n \to \infty} S_n = 0 \), where \( S_n = \sum_{j=1}^{n-1} \frac{a_j}{n-j} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Consider a square matrix \(A\) of dimension \(n \times n\) where \(n = 2, 3, 4, \ldots\). The matrix is hollow (all elements on the principal diagonal are zero), symmetric, and all off-diagonal elements are equal to \(a\). Determine if the spectral radius (maximum eigenvalue) of this matrix is equal to \((n-1) \times a\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(n-1)a}
Determine whether \( \biggl\|\frac{vv^T}{v^Tv}\biggr\| = 1 \) for any vector \( v \in \mathbb{R}^n \), where \( \|.\| \) denotes a matrix norm.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine the cardinality of the set \(X = \bigcup_{n \in \mathbb{N}} X_n\), where each set \(X_n\) has infinite cardinality \(\alpha\). Is the cardinality of \(X\) equal to \(\aleph_0 \alpha\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\aleph_0 \alpha}
The first AMC $8$ was given in $1985$ and it has been given annually since that time. Samantha turned $12$ years old the year that she took the seventh AMC $8$. In what year was Samantha born?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$1979$}
Determine whether the limit \( \lim_{x \to \infty} \sum_{n=1}^{x-1} \frac{1}{x-n} \) exists.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Determine if the function \( f: \mathbb{R}^2 \to \mathbb{R} \) is differentiable at the point \((0,0)\):
\[
f(x,y) = \begin{cases} \cfrac{x^n y^m}{x^2 - xy + y^2} & \text{for } (x,y) \ne (0,0) \\ 0 & \text{for } (x,y) = (0,0) \end{cases}
\]
where \( m, n \in \mathbb{N}^* \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n + m > 3}
4. Consider a 20 -sided convex polygon $K$, with vertices $A_{1}, A_{2}, \ldots, A_{20}$ in that order. Find the number of ways in which three sides of $K$ can be chosen so that every pair among them has at least two sides of $K$ between them. (For example $\left(A_{1} A_{2}, A_{4} A_{5}, A_{11} A_{12}\right)$ is an admissible triple while $\left(A_{1} A_{2}, A_{4} A_{5}, A_{19} A_{20}\right)$ is not.)

Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{520}
Let $S$ be the sum of all numbers of the form $a/b,$ where $a$ and $b$ are relatively prime positive divisors of $1000.$ What is the greatest integer that does not exceed $S/10?$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
248
}
Determine whether the following limit is true: \[ \lim_{x\rightarrow\infty}\frac{x}{\pi(x)}-\ln(x) =-1, \] where \( \pi(x) \) is the number of primes not exceeding \( x \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
Given two sequences of vectors \(\{a_n\}\) and \(\{b_n\}\) that converge to vectors \(a\) and \(b\), respectively, determine if the sequence of cross products \(\{a_n \times b_n\}\) converges to \(a \times b\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a \times b}
In Heidi's history class, the only grades that count toward the semester average are the $6$ tests she has already taken and the upcoming final exam. The final exam counts as two tests. Heidi has determined that if she earns $99$ points on the final she will have exactly a $90$-point average for the semester. On average, how many points has Heidi scored on each test prior to the final exam?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{87}
Determine whether the following identity holds for all integers \( n \):
\[ \frac{\sum_{k=0}^{\infty}\frac{F_{n+k}}{k!}}{\sum_{k=0}^{\infty}\frac{F_{n-k}}{k!}}=e, \]
where \( F_n \) denotes the Fibonacci sequence.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e}
In the plane, $2022$ points are chosen such that no three points lie on the same line. Each of the points is coloured red or blue such that each triangle formed by three distinct red points contains at least one blue point.
What is the largest possible number of red points?

Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
1012
}
Determine whether the sum \( \sum_{k=1}^N \frac{(k+2)^3 c_k}{(N+3)^4} \) converges, given that \( \{c_k\} \) is a convergent sequence with limit \( c \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{c}{4}}
Suppose that a polynomial function $p(x)$ crosses the horizontal line $y = 1$ at exactly one point, crosses the horizontal line $y = 3$ at exactly three points, and crosses the horizontal line $y = 5$ at exactly five points ($p$ is not tangent at any point to any of those three horizontal lines). What is the least possible degree of $p$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Determine if the function \( f(x) = \int_{0}^{\infty} t^{x^2+1} e^{-2t} \, dt \) can be expressed as a power series in a neighborhood of zero. If it can, find the radius of convergence of this series.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{2}}
Determine the convergence radius and convergence domain of the complex power series $\sum\limits_{n=0}^\infty \frac{1}{2^{n}} z^{2^n}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Four cubes with edge lengths $1$, $2$, $3$, and $4$ are stacked as shown. Find the length of the portion of $\overline{XY}$ contained in the cube with edge length $3$. The original answer is in the format \(\frac{k\sqrt{n}}{m}\). Please calculate the value of k + m + n.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{41}
Evaluate the limit: $$\lim_{x\to \infty} \left(\cosh^{-1}(x^{3}) + \coth^{-1}(\sqrt{x^{2}+1}) - 3\sinh^{-1}(x)\right)$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2 \ln 2}
In the $4 \times 5$ grid shown, six of the $1 \times 1$ squares are not intersected by either diagonal. When the two diagonals of an $8 \times 10$ grid are drawn, how many of the $1 \times 1$ squares are not intersected by either diagonal?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{48}
Determine the number of continuous functions \( f(x) \) defined on the interval \([0,1]\) that satisfy the following conditions: \( \int_0^1 f(x) \, dx = 1 \), \( \int_0^1 x f(x) \, dx = a \), and \( \int_0^1 x^2 f(x) \, dx = a^2 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Three balls are drawn simultaneously from the urn (as described in Problem 4). Find the probability that all the drawn balls are blue (event $B$).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1/12}
In a fluid dynamics context, suppose you have a piece of wood with a mass of 0.5 kg and densities of different materials as follows: $\rho_{\text{wood}} = 600 \ \mathrm{kg/m^3}$, $\rho_{\text{copper}} = 8900\ \mathrm{kg/m^3}$, and $\rho_{\text{water}} = 1000 \ \mathrm{kg/m^3}$. What is the minimum mass of copper, in kilograms, that must be attached to the wood to make the combined object submerged in water?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.375}
In the diagram, \( \triangle QRS \) is an isosceles right-angled triangle with \( QR = SR \) and \( \angle QRS = 90^{\circ} \). Line segment \( PT \) intersects \( SQ \) at \( U \) and \( SR \) at \( V \). If \( \angle PUQ = \angle RVT = y^{\circ} \), calculate the value of \( y \) in degrees.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{67.5^{\circ}}
Find all integer triplets \((x, y, z)\) such that:
\[ x^{3} + y^{3} + z^{3} - 3xyz = 2003. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(668, 668, 667), (668, 667, 668), (667, 668, 668)}
It's pouring down rain, and the amount of rain hitting point  $(x,y)$  is given by   $$ f(x,y)=|x^3+2x^2y-5xy^2-6y^3|. $$  If you start at the origin  $(0,0)$ , find all the possibilities for  $m$  such that  $y=mx$  is a straight line along which you could walk without any rain falling on you.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1, \frac{1}{2}, -\frac{1}{3}}
In the diagram, square PQRS has side length 40. Points J, K, L, and M are on the sides of PQRS, so that JQ = KR = LS = MP = 10. Line segments JZ, KW, LX, and MY are drawn parallel to the diagonals of the square so that W is on JZ, X is on KW, Y is on LX, and Z is on MY. Find the area of quadrilateral WXYZ. Express your answer in square units as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{200}
Find the limit of the infinite series \( \lim_{n\rightarrow \infty} e^{-n}\displaystyle\sum_{k=0}^{n-1}\frac{n^{k-1}}{k!} \). What is the value of this limit?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Find all integers \( x \) such that \( 2x^2 + x - 6 \) is a positive integral power of a prime positive integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-3, 2, 5}
给定正整数 $m(m \geq 3)$. 设正项等差数列 $\left\{a_{n}\right\}$ 与正项等比数列 $\left\{b_{n}\right\}$ 满足: $\left\{a_{n}\right\}$ 的首项等于 $\left\{b_{n}\right\}$ 的公比, $\left\{b_{n}\right\}$ 的首项等于 $\left\{a_{n}\right\}$ 的公差, 且 $a_{m}=b_{m}$ 。求 $a_{m}$ 的最小值, 并确定当 $a_{m}$ 取到最小值时 $a_{1}$ 与 $b_{1}$ 的比值。请给出当 $m = 3$ 时 $a_{1}$ 与 $b_{1}$ 的比值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Pinocchio correctly solved the example but smudged his notebook.

Behind each blot is hidden the same digit different from zero. Find this digit.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
In chess, there are two types of minor pieces, the bishop and the knight. A bishop may move along a diagonal, as long as there are no pieces obstructing its path. A knight may jump to any lattice square $\sqrt{5}$ away as long as it isn't occupied.

One day, a bishop and a knight were on squares in the same row of an infinite chessboard, when a huge meteor storm occurred, placing a meteor in each square on the chessboard independently and randomly with probability $p$. Neither the bishop nor the knight were hit, but their movement may have been obstructed by the meteors.

The value of $p$ that would make the expected number of valid squares that the bishop can move to and the number of squares that the knight can move to equal can be expressed as $\frac{a}{b}$ for relatively prime positive integers $a, b$. Compute $100a + b$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{102}
In the future, MIT has attracted so many students that its buildings have become skyscrapers. Ben and Jerry decide to go ziplining together. Ben starts at the top of the Green Building, and ziplines to the bottom of the Stata Center. After waiting $a$ seconds, Jerry starts at the top of the Stata Center, and ziplines to the bottom of the Green Building. The Green Building is 160 meters tall, the Stata Center is 90 meters tall, and the two buildings are 120 meters apart. Furthermore, both zipline at 10 meters per second. Given that Ben and Jerry meet at the point where the two ziplines cross, compute $100 a$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{740}
There are 1997 odd numbers whose sum equals their product. Among them, three numbers are not 1 but are three different prime numbers. What are these three prime numbers?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5, 7, 59}
The graph of the function \( f(x) = x^4 + 4x^3 - 16x^2 + 6x - 5 \) has a common tangent line at \( x = p \) and \( x = q \). Find the product \( pq \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-10}
In Zuminglish, words consist only of the letters $M, O,$ and $P$. $O$ is a vowel, while $M$ and $P$ are consonants. A valid Zuminglish word must have at least two consonants between any two $O's$. Determine the number of valid 7-letter words in Zuminglish and find the remainder when this number is divided by $1000$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{912}
已知平面上有 $100$ 条直线，用 $T$ 表示由这些直线中的某三条直线围成的直角三角形的集合。求 $|T|$ 的最大值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{62500}
Find all triangles whose side lengths are consecutive integers, and one of whose angles is twice another.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4, 5, 6}
At a school, all 60 students play on at least one of three teams: Basketball, Soccer, and Mathletics. 8 students play all three sports, half the students play basketball, and the ratio of the size of the math team to the size of the basketball team to the size of the soccer team is $4:3:2$. How many students at the school play on exactly two teams?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
设a,b,c是△ABC的三边,P: ${{a}^{2}}={{b2}}+{{c}^{2}}$, Q:方程x$^{2}$ +2ax+b$^{2}$ = 0与方程x$^{2}$ +2cx－b$^{2}$ = 0有公共根. 则P是Q的_____.(填：充分不必要条件,必要而不充分条件,充要条件,既不充分也不必要条件)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{充要条件}