inclusionAI/Ring-lite-rl-data huggingface dataset:
Calculate the integral \( \int_{\Sigma}(2x \, dy\land dz + 2y \, dz\land dx + (x^2+y^2) \, dx\land dy) \), where \( \Sigma \) is the boundary of the region \( D \) defined by \( 0\leq z\leq 1-x^2-y^2 \). The surface \( \Sigma \) is oriented such that the outward normal vectors point away from the region \( D \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\pi}
In triangle \(ABC\), \(\angle ABC = 70^\circ\) and \(\angle ACB = 50^\circ\). On \(AB\), a point \(M\) is selected such that \(\angle MCB = 40^\circ\), and on \(AC\), a point \(N\) is chosen such that \(\angle NBC = 50^\circ\). Find \(\angle NMC\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30^\circ}
8 is 2% of a, and a certain percentage of b is 2. c equals b / a. What is the value of the percentage of b?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.5\%}
A 10000-liter tank, half-full of water, is being filled from a pipe with a flow rate of 1 kiloliter every 2 minutes. At the same time, the tank is losing water from two drains. One drain has a rate of 1 kiloliter every 6 minutes. It takes 60 minutes to fill the tank completely. What is the rate of the other drain?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.25}
Find a vector field \( \mathbf{F} \) such that the line integral \( \oint_C \mathbf{F} \cdot d\mathbf{r} \) is negative, where \( C \) is a circle of radius \( r \) centered at the origin, traced once in the clockwise direction.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-y, x)}
In a road traffic survey, cars passing a marker on a road are counted. The streams of cars in the two directions are modeled as independent Poisson processes with intensities of 2 cars per minute and 3 cars per minute, respectively. Counting stops once 400 cars have passed. Let \( T_{400} \) be the time at which this occurs. Compute the time \( \alpha \) such that \( P(T_{400} \leq \alpha) = 0.90 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{85.13}
A box measuring some inches long by 48 inches wide by 12 inches deep is to be filled entirely with identical cubes. No space is to be left unfilled. The smallest number of cubes that can accomplish this objective is 80. What is the length of the box?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{240}
A statue is being carved by a sculptor. The original piece of marble weighed 250 kg. In the first week, 30 percent is cut away. In the second week, 20 percent of the remainder is cut away. In the third week, 25 percent of the remaining marble is cut away. In the fourth week, the sculptor cuts away 15 percent. In the fifth week, 10 percent of the remainder is cut away. In the sixth week, the sculptor removes another 5 percent of the remaining marble. The statue is finally completed in the seventh week when 3 percent of the remainder is cut away. What is the weight of the final statue?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{74.02}
Evaluate the limit as \( N \to \infty \):
\[
\lim_{{N \to \infty}} \left( \left(\frac{\sin \frac{1}{N}}{\frac{1}{N}}\right)^{N} -1 \right) \cdot \frac{1}{N}.
\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Evaluate the integral \( \int_0^{\infty} e^{-x} \sum_{k=0}^{\infty} \frac{(-1)^k (2015x)^{2k}}{((2k)!)^2} \, dx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\cos(2015)}
In the Cartesian coordinate system $xOy$, let $A(-1,1)$, and let $B$ and $C$ be two points on the graph of the function $y= \frac {1}{x} (x > 0)$, and $\triangle ABC$ is an equilateral triangle. The height of $\triangle ABC$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
There are births in West Northland every 6 hours, deaths every 2 days, and a net immigration every 3 days. Calculate the approximate annual increase in population.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1400}
Solve the complex equation $w^3 = 9\overline{w}$, where $\overline{w}$ denotes the conjugate of $w$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{{0, ±3, ±3i}}
Vasya has $n{}$ candies of several types, where $n>145$. It is known that for any group of at least 145 candies, there is a type of candy which appears exactly 10 times. Find the largest possible value of $n{}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
160
}
Evaluate the improper integral: $$\int_{0}^{\infty} \frac{\sin x}{x^{3/2}} \, dx.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{2\pi}}
Determine the maximum value of the sum
\[S = \sum_{n=1}^\infty \frac{n}{2^n} (a_1 a_2 \cdots a_n)^{1/n}\]
over all sequences $a_1, a_2, a_3, \cdots$ of nonnegative real numbers satisfying
\[\sum_{k=1}^\infty a_k = 1.\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2/3}
Natural numbers \( a, b, c \) are chosen such that \( a < b < c \). Given that the system of equations \( 2x + y = 2033 \) and \( y = |x-a| + |x-b| + |x-c| \) has exactly one solution, find the minimum possible value of \( c \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1017}
Solve for \( n \) in the equation: \[ \frac{365!}{365^{n}(365-n)!} = 0.5 \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{23}
In trapezoid \(ABCD\), the base \(AD\) is equal to 16, the sum of the side \(AB\) and the diagonal \(BD\) is equal to 40, and the angle \(CBD\) is \(60^\circ\). The ratio of the areas of triangles \(ABO\) and \(BOC\), where \(O\) is the intersection point of the diagonals, is 2. Find the area of the trapezoid.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{126\sqrt{3}}
Let $S$ be a set of intervals defined recursively as follows: Initially, $[1,1000]$ is the only interval in $S$. If $l \neq r$ and $[l, r] \in S$, then both $\left[l,\left\lfloor\frac{l+r}{2}\right\rfloor\right],\left[\left\lfloor\frac{l+r}{2}\right\rfloor+1, r\right] \in S$. An integer $i$ is chosen uniformly at random from the range $[1,1000]$. What is the expected number of intervals in $S$ which contain $i$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10.976}
In our number system, the base is ten. If the base were changed to four, you would count as follows: $1, 2, 3, 10, 11, 12, 13, 20, 21, 22, 23, 30, \ldots$ What would be the twentieth number in this system?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{110}
Let $K/F$ be a finite-dimensional field extension with $[K:F]=n$. Consider the collection of $F$-linear transformations $\mathcal{L}_F(K)$ from $K$ to $K$. This collection forms an algebra over $F$ and is isomorphic to the $n \times n$ matrices over $F$, making it an $n^2$-dimensional vector space over $F$. Determine the dimension of $\mathcal{L}_F(K)$ as a vector space over $K$. \( \text{Express your answer as a single number.} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n}
Given that \(9^{-1} \equiv 90 \pmod{101}\), find \(81^{-1} \pmod{101}\), as a residue modulo 101. (Give an answer between 0 and 100, inclusive.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
Xiao Wu, Xiao Yi, Xiao Jie, Xiao Kuai, and Xiao Le are standing in a row. If Xiao Yi does not appear in the first or last position, and among Xiao Wu, Xiao Jie, and Xiao Le, exactly two are adjacent, how many different arrangements satisfy these conditions?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{48}
Determine the interval of convergence for the power series representation of \( f(x) = \sqrt{x} \) centered at \( x = 16 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[0, 32]}
Let $g: \mathbb{R} \to \mathbb{R}$ be a three-times differentiable function that satisfies $$g(x^3)g''(x)=g'(x)g'(x^3)$$ for all $x \in \mathbb{R}$. If $g(1) = 2$ and $g'''(1) = 12$, determine the value of $g'(1) + g''(1)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
For complex numbers n and q, if m = | | n – 3 | – 2 | q | + | | q - n | + 3 | , for how many values of n is m a purely real number, m = 5 ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
What is the limit of the following sum as $n$ approaches infinity?

$$\lim_{n\to\infty} \sum_{k=0}^{\left\lfloor \frac{n}{2} \right\rfloor} \binom{n}{2k} \left(4^{-k} \binom{2k}{k}\right)^{\frac{2n}{\log_2{n}}}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Let \( R \) be a semicircle with diameter \( XY \). A trapezoid \( ABCD \) in which \( AB \) is parallel to \( CD \) is circumscribed about \( R \) such that \( AB \) contains \( XY \). If \( AD = 4 \), \( CD = 5 \), and \( BC = 6 \), determine \( AB \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
By selling some pencils for a rupee, a man loses 35%. In order to gain 30%, he should sell 10 pencils for a rupee. How many pencils was he selling for a rupee initially?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
1.48 In the expression $x_{1}: x_{2}: \cdots: x_{n}$, use parentheses to indicate the order of operations, and the result can be written in fractional form:
$$\frac{x_{i_{1}} x_{i_{2}} \cdots x_{i_{k}}}{x_{j_{j_{1}}}^{x_{j_{2}}} \cdots x_{j_{n-k}}}$$
(At the same time, each letter in $x_{1}, x_{2}, \cdots, x_{n}$ may appear in the numerator or in the denominator.) How many different fractions can be obtained by adding parentheses in all possible ways?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{n-2}}
Determine the Galois group of the polynomial \( p(x) = x^4 - 2x^2 - 2 \) over the field \( \mathbb{F}_5 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{Z}/4\mathbb{Z}}
In $\triangle ABC$, $\angle A = 60^{\circ}$ and $AB > AC$. Point $O$ is the circumcenter. The two altitudes $BE$ and $CF$ intersect at point $H$. Points $M$ and $N$ are on segments $BH$ and $HF$ respectively, and satisfy $BM = CN$. Find the value of $\frac{MH + NH}{OH}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{3}}
Let \(A_{1}, A_{2}, A_{3}, A_{4}, A_{5}\), and \(A_{6}\) be six points on a circle in this order such that \(\overline{A_{1} A_{2}}=\overline{A_{2} A_{3}}\), \(\widehat{A_{3} A_{4}}=\widehat{A_{4} A_{5}}\), and \(\widehat{A_{5} A_{6}}=\widehat{A_{6} A_{1}}\), where \(\widehat{A_{1} A_{2}}\) denotes the arc length of the arc \(A_{1} A_{2}\) etc. It is also known that \(\angle A_{1} A_{3} A_{5}=72^{\circ}\). Find the size of \(\angle A_{4} A_{6} A_{2}\) in degrees.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{54}
Determine the side length of the smallest square sheet of paper required to wrap a unit cube without cutting or stretching the paper.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1 + \sqrt{2}}
Determine the coefficient of $z$ in the Laurent series expansion of $\frac{e^z}{z-1}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2}
In a certain company, there are three departments: IT, HR, and Marketing. Employee attendance at the annual company picnic varies depending on the department, as shown below:

IT Department:
- 25% of the men and 60% of the women attended the picnic.

HR Department:
- 30% of the men and 50% of the women attended the picnic.

Marketing Department:
- 10% of the men and 45% of the women attended the picnic.

The percentage composition of employees in each department is as follows:
- IT Department: 40% men and 25% women
- HR Department: 30% men and 20% women
- Marketing Department: 30% men and 55% women

What percentage of all the employees attended the annual company picnic, considering the departmental break-up and gender proportions?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{71.75\%}
a and b together have $1210. If 1/3 of a's amount is equal to 1/4 of b's amount, how much amount does b have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$691.43}
Given that $a$ and $b$ are positive real numbers satisfying $$\frac {1}{a+b}+ \frac {1}{a-b}=1$$, find the minimum value of $3a+2b$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3 + \sqrt {5}}
For every non-negative integer \( n \), a polynomial \( p_n(x) \) exists such that \( \int x^n e^x \,dx = p_n(x) e^x + C \). Determine the floor value of \( 1000 \cdot L \), where \( L \) is the limit defined as \( L = \lim_{n \to \infty} \frac{|p_n(2)|}{n!} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{135}
Let $ m$ be a positive odd integer, $ m > 2.$ Find the smallest positive integer $ n$ such that $ 2^{1989}$ divides $ m^n - 1.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$ n =  2^{1988}$}
In a sphere with a radius of 1, the chord AB has a length of 1 and is positioned at a 60° angle to the diameter CD. Given that AC = \( \sqrt{2} \) and AC < BC, determine the length of the segment BD.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Lucca bought 1,500 pieces of bread. On the first day, he ate 3/8 of the pieces of bread. On the second day, he ate 7/10 of the remaining pieces. On the third day, he ate 1/6 of what was left, and on the fourth day, he ate 4/9 of the remaining pieces. On the fifth day, he ate 5/18 of the pieces left. Calculate how many pieces of bread are remaining for Lucca after the fifth day.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{94}
Evaluate the line integral \( \int_{C} -\frac{y}{x^2 + y^2} \, dx + \frac{x}{x^2 + y^2} \, dy \), where \( C \) is the positively oriented boundary of the region enclosed by the curves \( x^2 + y^2 = 1 \) and \( |x| + |y| = 5 \). Note that the vector field \( F(x,y)=\left(-\frac{y}{x^2 + y^2} , \frac{x}{x^2+y^2}\right) \) is undefined at \((0,0)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Alicia has 20 gumballs. Pedro has that many gumballs plus an additional number of gumballs equal to a certain multiple of the number Alicia has. They put their gumballs in a bowl, and later Pedro takes out 40% of the gumballs to eat. There are 60 gumballs remaining in the bowl. What is the ratio of the additional gumballs Pedro has to the number of gumballs Alicia has?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3:1}
Given $0 \leq a-b \leq 1$ and $1 \leq a+b \leq 4$, find the value of $8a+2002b$ when $a-2b$ reaches its maximum value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
How many ways are there to list the numbers 1 to 10 in some order such that every number is either greater or smaller than all the numbers before it?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{512}
Let $O$ be the origin, and $F$ be the right focus of the ellipse $C: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ where $a > b > 0$. The line $l$ passing through $F$ intersects the ellipse $C$ at points $A$ and $B$. Two points $P$ and $Q$ on the ellipse satisfy
$$
\overrightarrow{O P}+\overrightarrow{O A}+\overrightarrow{O B}=\overrightarrow{O P}+\overrightarrow{O Q}=0,
$$
and the points $P, A, Q,$ and $B$ are concyclic. Find the eccentricity of the ellipse $C$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{2}}{2}}
Find the residue of the function \( f(z) = \frac{\cot(z)\coth(z)}{z^3} \) at \( z = 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{7}{45}}
Inside the circle $\omega$ are two intersecting circles $\omega_{1}$ and $\omega_{2}$, which intersect at points $K$ and $L$, and are tangent to the circle $\omega$ at points $M$ and $N$. It turns out that the points $K$, $M$, and $N$ are collinear. Find the radius of the circle $\omega$, given that the radii of the circles $\omega_{1}$ and $\omega_{2}$ are 3 and 5, respectively.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Determine if there is a known bound for the VC-dimension of the set of binary classifiers \(\mathcal{H} = \{\boldsymbol{x}\mapsto\boldsymbol{1}\hspace{-0.9ex}\mathrm{1}(\boldsymbol{h}^\mathsf{T}\boldsymbol{x}\ge 0)\mid  \boldsymbol{h} \in \mathcal{C} \}\), where \(\mathcal{C}\) is a pointed, salient, full-dimensional cone in \(\mathbb{R}^d\), that is significantly better than the VC-dimension of the set of all linear classifiers in \(\mathbb{R}^d\) (i.e., \(d\)).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{d}
Two weeks ago, Vartan spent 25 percent of his wages on recreation. Last week, his wages were 5 percent less than two weeks ago, and he spent 35 percent of his wages on recreation. This week, his wages are 15 percent less than last week's wages, and he spent 45 percent of his wages on recreation. The amount he spends on recreation this week is what percent of the amount he spent on recreation two weeks ago?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{145.35}
A tetrahedron has five edges of length 3 and a circumradius of 2. What is the length of the sixth edge?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6\sqrt{\frac{3}{7}}}
In the triangle \(ABC\), angle \(C\) is obtuse, \(D\) is the intersection point of the line \(DB\) perpendicular to \(AB\) and the line \(DC\) perpendicular to \(AC\). The altitude of triangle \(ADC\) drawn from vertex \(C\) intersects side \(AB\) at point \(M\). Given that \(AM = a\) and \(MB = b\), find \(AC\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{a(a+b)}}
Determine the minimum value of \( n \) such that it is possible to create at least 100 ordered lists of length 5 using numbers from 1 to \( n \), with the condition that any two lists share at most one common element at the same index.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
Find the supremum of the fraction \( \frac{|c_1^\top z + b_1|}{c_2^\top z+b_2} \) with respect to \( z \in S = \{x \in \mathbb{R}^n: c_2^\top x + b_2 > 0 \} \), where \( c_1, c_2 \in \mathbb{R}^n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{+\infty}
Let \(I\) be the center of the inscribed circle of triangle \(ABC\), and let \(A^{\prime}, B^{\prime}\), and \(C^{\prime}\) be the symmetric points of \(I\) with respect to lines \(BC\), \(CA\), and \(AB\) respectively. The circumcircle of \(A^{\prime} B^{\prime} C^{\prime}\) passes through \(B\). Find \(\widehat{ABC}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60^\circ}
There are five concentric circles \(\Gamma_{0}, \Gamma_{1}, \Gamma_{2}, \Gamma_{3}, \Gamma_{4}\) whose radii form a geometric sequence with a common ratio \(q\). Find the maximum value of \(q\) such that a closed polyline \(A_{0} A_{1} A_{2} A_{3} A_{4}\) can be drawn, where each segment has equal length and the point \(A_{i} (i=0,1, \ldots, 4)\) is on the circle \(\Gamma_{i}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{5} + 1}{2}}
有八个盒子，每个盒子中有 6 个球，每个球均被染成 n 种颜色中的一种. 已知同一个盒子中的球均不同色，且任意两个盒子中的球至多有一种相同颜色，求 n 的最小可能值.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{23}
Calculate the maximum value of the summation \( \sum_{k=1}^{n} \left(\frac{(-1)^k}{k!} \int_{0}^{k} 3x^2 \, dx\right) \) for a given positive integer \( n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
求使得s$\sin x = \frac { x } { 1 0 0 }$成立的实数x的个数.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{63}
If $x, y, z, p, q, r$ are real numbers such that:
\[ \frac{1}{x+p} + \frac{1}{y+p} + \frac{1}{z+p} = \frac{1}{p} \]
\[ \frac{1}{x+q} + \frac{1}{y+q} + \frac{1}{z+q} = \frac{1}{q} \]
\[ \frac{1}{x+r} + \frac{1}{y+r} + \frac{1}{z+r} = \frac{1}{r} \]
Find the numerical value of $\frac{1}{p} + \frac{1}{q} + \frac{1}{r}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Find the result of the inverse Fourier transform of the complex conjugate of a function in the Fourier domain.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{f^*(-t)}
In $\triangle{ABC}$, $\angle{C} = 90^{\circ}$ and $AB = 12$. Squares $ABXY$ and $ACWZ$ are constructed outside of the triangle. The points $X, Y, Z$, and $W$ lie on a circle. What is the perimeter of the triangle?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$12+12\sqrt{2}$}
Suppose an article has a cost price that is 60% of the marked price for the first half of the quantity and 70% of the marked price for the second half of the quantity. The store offers four successive discounts of 10%, 8%, 5%, and 3% during a sale. Calculate the net gain percentage after accounting for all these factors and applying all the discounts.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{17.38\%}
Given the set $U={1,2,3,4,5}$, select 4 numbers from set $U$ to form a four-digit number with no repeating digits. This four-digit number should be greater than $2345$ and less than $4351$. The number of four-digit numbers that meet these conditions is \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{54}
Ten points in the plane are given, with no three collinear . Four distinct segments joining pairs of these points are chosen at random, all such segments being equally likely. The probability that some three of the segments form a triangle whose vertices are among the ten given points is $m/n,$ 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{489}
What operation should be applied to 49 and then squared to get the result of 49?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{49}
What is the largest power of 2 by which the number \(10^{10} - 2^{10}\) is divisible?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
What is the probability that in a group of 13 people, at least two were born in September? (For simplicity, assume that each month is equally likely.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.296}
Altitudes \( BE \) and \( CF \) of acute triangle \( ABC \) intersect at \( H \). Suppose that the altitudes of triangle \( EHF \) concur on line \( BC \). If \( AB=3 \) and \( AC=4 \), then \( BC^2=\frac{a}{b} \), where \(a\) and \(b\) are relatively prime positive integers. Compute \(100a+b\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{33725}
Two mathematics teachers administer a geometry test, assessing the ability to solve problems and knowledge of theory for each 10th-grade student. The first teacher spends 5 and 7 minutes per student, and the second teacher spends 3 and 4 minutes per student. What is the minimum time needed to assess 25 students?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{110}