inclusionAI/Ring-lite-rl-data huggingface dataset:
In an abandoned chemistry lab Gerome found a two-pan balance scale and three 1-gram weights, three
5-gram weights, and three 50-gram weights. By placing one pile of chemicals and as many weights as
necessary on the pans of the scale, Gerome can measure out various amounts of the chemicals in the pile.
Find the number of diﬀerent positive weights of chemicals that Gerome could measure.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{129}
Let $\alpha$ and $\beta$ be the roots of the polynomials $x^2-2$ and $x^2+2$, respectively, over an algebraic closure of $\mathbb{F}_{2023}$. Determine the degree of $\alpha+\beta$ over $\mathbb{F}_{2023}$: is it $2$ or $4$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Find the integers \( x,y,z \) for which
 \( \frac{1}{x + \frac{1}{y + \frac{1}{z}}} = \frac{7}{17} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$(x, y, z) = (2, 2, 3)$
}
Find the integrating factor for the differential equation: \((3x+\frac{6}{y})\mathrm{d}x+(\frac{x^2}{y}+\frac{3y}{x})\mathrm{d}y=0\). Determine if it is possible to find a common integrating factor for the entire equation.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{xy}
Three spheres have radii $144$, $225$, and $400$, are pairwise externally tangent to each other, and are all tangent to the same plane at points $A$, $B$, and $C$. Compute the area of triangle $ABC$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{86400}
In a new diagram, $A$ is the center of a circle with radii $AB=AC=8$. The sector $BOC$ is shaded except for a triangle $ABC$ within it, where $B$ and $C$ lie on the circle. If the central angle of $BOC$ is $240^\circ$, what is the perimeter of the shaded region?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16 + \frac{32}{3}\pi}
The social activist and the neighborhood watch leader are planning a meeting to discuss strategies to reduce crime in the area. They have invited 15 community members, and each member will bring 2 friends. If the meeting space can hold a maximum of 50 people, how many additional people can the activist and the leader invite without exceeding the capacity?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Derive the gradient of \( \beta = x^TA^Tx \) with respect to \( x \), where \( x \) is an \( n \times 1 \) vector and \( A \) is an \( n \times n \) matrix that does not depend on \( x \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(A + A^\top)x}
Chelsea goes to La Verde's at MIT and buys 100 coconuts, each weighing 4 pounds, and 100 honeydews, each weighing 5 pounds. She wants to distribute them among \( n \) bags, so that each bag contains at most 13 pounds of fruit. What is the minimum \( n \) for which this is possible?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{75}
There are $20$ balls in a bag. $a$ of them are red, $b$ of them are white, and $c$ of them are black. It is known that

$ \bullet$ if we double the white balls, the probability of drawing one red ball is $\dfrac 1{25}$ less than the probability of drawing one red ball at the beginning,
and

$ \bullet$ if we remove all red balls, the probability of drawing one white ball is $\dfrac 1{16}$ more than the probability of drawing one white ball at the beginning.

Find $a,b,c$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$a=4$, $b=5$, $c=11$
}
Determine if the integral \( \int_0^\infty \frac{\log(x)}{(1+x^2)^2} \, dx \) can be evaluated without using complex analysis methods.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{\pi}{4}}
At the beginning of a trip, the mileage odometer read $ 56200$ miles. The driver filled the gas tank with $ 6$ gallons of gasoline. During the trip, the driver filled his tank again with $ 12$ gallons of gasoline when the odometer read $ 56560$. At the end of the trip, the driver filled the tank again with $ 20$ gallons of gasoline. The odometer read $ 57060$. To the nearest tenth, what was the car's average miles-per-gallon for the entire trip?

Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
26.9

}
Evaluate the limit: $$\lim_{x \to \frac{\pi}{2}} \left| \frac{1}{(1+\sin x)\ln(\sin x)} + \frac{1}{\cos^2x} \right| $$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{4}}
甲、乙、丙三个数的比是4,5：2,3：1,2，甲数比丙数大18，则甲、乙的平均数是多少．
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{204}{11}}
Determine the expression for the fourth mixed partial derivative of the probability generating function \( Q(x,y) \) in terms of the expected values of \( X \) and \( Y \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{E[X(X-1)Y(Y-1)]}
A set $S$ of natural numbers is called good, if for each element $x \in  S, x$ does not divide the sum of the remaining numbers in $S$. Find the maximal possible number of elements of a good set which is a subset of the set $A = \{1,2, 3,  ...,63\}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
61
}
In unit square \( {ABCD} \) ,diagonals \( \overline{AC} \) and \( \overline{BD} \) intersect at \( E \) . Let \( M \) be the midpoint of \( \overline{CD} \) ,with \( \overline{AM} \) intersecting \( \overline{BD} \) at \( F \) and \( \overline{BM} \) intersecting \( \overline{AC} \) at \( G \) . Find the area of quadrilateral \( {MFEG} \) .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$\dfrac{1}{12}$}
Determine the sign of the integral \( \iint_{|x|+|y| \leq 1} \ln(x^{2}+y^{2}) \,dx\,dy \) without explicitly computing it.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{negative}
Given a group $G$ and two subgroups $H$ and $K$ such that $H, K \subseteq G$, if for a fixed element $k \in K$, the conjugate $kHk^{-1}$ is a subset of $H$, does it necessarily follow that $kHk^{-1} = H$? Answer "yes" or "no."
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{no}
For a sequence of distinct real numbers $b_1, b_2, \ldots, b_k$ in the interval $[1, 2023]$, determine the minimum value of $k$ such that there must exist indices $i$ and $j$ with $i \neq j$ satisfying $0 < b_i - b_j < 1 + 3\sqrt[3]{b_i \cdot b_j}$. What is this minimum value of $k$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
We call a time on a $12$ hour digital clock nice if the sum of the minutes digits is equal to the hour. For example, $10:55$, $3:12$ and $5:05$ are nice times. How many nice times occur during the course of one day? (We do not consider times of the form $00:\text{XX}$.)

Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{112}
Task 3. Let $n \geq 2$ be a positive integer. Each cell of an $n \times n$ board is colored red or blue. We place dominoes on the board, each covering two cells. We call a domino plain if it lies on two red or two blue cells, and colorful if it lies on one red and one blue cell. Find the largest positive integer $k$ with the following property: no matter how the red/blue coloring of the board is done, it is always possible to place $k$ non-overlapping dominoes on the board that are either all plain or all colorful.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\lfloor\frac{n^{2}}{4}\rfloor}
Determine the values of $x$ for which the matrix \( M = \begin{bmatrix} x & 0 \\ 0 & -\frac{1}{x} \end{bmatrix} \) cannot be expressed as a product of two involutory real matrices.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{x \neq \pm 1}
聪聪计划做一个底面半径5cm，高12cm的圆柱形笔筒，全用人造革做，要用人造革多少cm**2．
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{455.3}
Determine the number of conjugacy classes in the quaternion group $Q$ and find the size of each class.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
A certain speed in m/s is equivalent to 1.1 kilometers per hour. What is the speed in m/s?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.31}
For every real number $x$, let $[x]$ be the greatest integer which is less than or equal to $x$. If the postal rate for first class mail is six cents for every ounce or portion thereof, then the cost in cents of first-class postage on a letter weighing $W$ ounces is always
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-6[-W]}
Let \( T: \mathbb{R}^4 \to \mathbb{R}^4 \) be a linear map such that the null space of \( T \) is \( \{(x,y,w,z) \in \mathbb{R}^4 : x+y+w+z=0\} \) and the rank of \( (T-4I_4) \) is 3. If the minimal polynomial of \( T \) is \( x(x-4)^{\alpha} \), find the value of \( \alpha \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Indecisive Andy starts out at the midpoint of the 1-unit-long segment $\overline{H T}$. He flips 2010 coins. On each flip, if the coin is heads, he moves halfway towards endpoint $H$, and if the coin is tails, he moves halfway towards endpoint $T$. After his 2010 moves, what is the expected distance between Andy and the midpoint of $\overline{H T}$ ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{4}}
How many $4$-digit numbers (without leading zeros) contain only $2,0,1,8$ as digits? Digits can be used more than once.

Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{192}
Find the number of non-singular $3 \times 3$ matrices with exactly four entries as $1$ and all other entries as $0$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36}
In a single-elimination tournament consisting of $2^{9}=512$ teams, there is a strict ordering on the skill levels of the teams, but Joy does not know that ordering. The teams are randomly put into a bracket and they play out the tournament, with the better team always beating the worse team. Joy is then given the results of all 511 matches and must create a list of teams such that she can guarantee that the third-best team is on the list. What is the minimum possible length of Joy's list?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{45}
Given that the function $y=\log_{2}(ax-1)$ is monotonically increasing in the interval $(1,2)$, the range of values for $a$ is.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[1, +\infty)}
Consider a sample median $Y_n$ from $n$ independent and identically distributed samples with a population median $m$ and a probability density function $f$. Given the central limit theorem for the sample median:
\[ \sqrt{n}(Y_n - m) \xrightarrow{d} N(0, [2f(m)]^{-2}), \]
as $n \to \infty$, determine if the variances converge such that:
\[ nE[(Y_n-m)^2] \to [2f(m)]^{-2}. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[2f(m)]^{-2}}
On each side of an equilateral triangle with side length $n$ units, where $n$ is an integer, $1 \leq n \leq 100$, consider $n-1$ points that divide the side into $n$ equal segments. Through these points, draw lines parallel to the sides of the triangle, obtaining a net of equilateral triangles of side length one unit. On each of the vertices of these small triangles, place a coin head up. Two coins are said to be adjacent if the distance between them is 1 unit. A move consists of flipping over any three mutually adjacent coins. Find the number of values of $n$ for which it is possible to turn all coins tail up after a finite number of moves.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{67}
Compute the intersection form for a smooth subvariety $\iota:X=V_+(f)\subset\mathbb{P}^2\times\mathbb{P}^1$, where $f\in H^0(\mathcal{O}(1,2))$. Determine the value of the intersection $\iota^*\mathcal{O}(2,3).\iota^*\mathcal{O}(2,3)$. Provide your answer as a single value or expression.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
Determine the signature (n+, n-, no) of the symmetric bilinear form \( s(f,g) = \int_{-1}^{1} f(x)g(x)(35x^4-30x^2+3)dx \) on the real vector space \( V = \mathbb{R}[X]_{\leq3} \) of polynomials of degree \( \leq 3 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(2, 1, 1)}
Rhombus $ABCD$ has side length $2$ and $\angle B = 120$°. Region $R$ consists of all points inside the rhombus that are closer to vertex $B$ than any of the other three vertices. What is the area of $R$?


Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{${\dfrac{2\sqrt{3}}{3}}$}
Let  $a=1+10^{-4}$ . Consider some  $2023\times 2023$  matrix with each entry a real in  $[1,a]$ . Let  $x_i$  be the sum of the elements of the  $i$ -th row and  $y_i$  be the sum of the elements of the  $i$ -th column for each integer  $i\in [1,n]$ . Find the maximum possible value of  $\dfrac{y_1y_2\cdots y_{2023}}{x_1x_2\cdots x_{2023}}$  (the answer may be expressed in terms of  $a$ ).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Compute the limit: $$ \lim _{n \to \infty} \frac{1}{n}\sum_{k=0}^{n-1}\left[\frac{k+1}{n}-\frac{1}{2}\right]\log(2n-2k-1) .$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{1}{4}}
In a diagram with 10 different natural numbers labeled as \(a, b, c, \ldots, k\), each number pointed to by an arrow equals the sum of the two numbers at the starting points of those arrows. Determine the minimum possible value of \(d\) in this arrangement.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Determine whether the integral \( \int_\pi^{+\infty}\frac{\sin t}{t}\mathrm dt \) is positive, negative, or zero.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{Negative}
The segments of two lines, enclosed between two parallel planes, are in the ratio of \( 5:9 \), and the acute angles between these lines and one of the planes are in the ratio of \( 2:1 \), respectively. Find the cosine of the smaller angle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.9}
Evaluate the limit of the function \( f(x) = \frac{\tan(\pi x)}{2x - 1} + \frac{2}{\pi(2x - 1)^2} \) as \( x \to \frac{1}{2} \). Determine whether the limit exists or if the function diverges at this point.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{6}}
设 $a_{1}, a_{2}, a_{3}, a_{4}$ 为四个有理数, 使得: $\left\{a_{i} a_{j} \mid 1 \leq i<j \leq 4\right\}=\left\{-24,-2,-\frac{3}{2},-\frac{1}{8}, 1,3\right\}$, 求 $a_{1}+a_{2}+a_{3}+a_{4}$ 的值。原始答案为\pm \frac{m}{n}格式，请给出m + n的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
A marksman shoots at a target located $d$ meters away. An observer standing $a$ meters from the marksman and $b$ meters from the target (as shown in Figure 13.10) hears the sound of the shot and the sound of the bullet hitting the target simultaneously. Determine the speed of the bullet, given that the speed of sound is $v$ meters per second.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{dv}{a - b}}
The capacity of a tank is 32 gallons. If a company bought 728 gallons of oil, how much oil is in the tank that is not full?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
Determine the number of equivalence relations on the set \( \{1,2,3,4,5,6,7\} \) that include the pairs \( \{(6,4),(4,7),(3,3),(5,1)\} \) but exclude the pairs \( \{(1,2),(6,1),(2,4)\} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Consider the endomorphism \( \phi: C_b(\mathbb{R}) \to C_b(\mathbb{R}) \) defined by \( \phi(f)(x) = x^2 f(x) \). Determine whether this endomorphism has an infinite norm, where the norm of a function \( f \) is defined as \( \|f\| = \sup_{x \in \mathbb{R}} |f(x)| \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
已知双曲线$\frac{{{x}^{2}}}{{{a2}}}-\frac{{{y}^{2}}}{{{b2}}}=1\left( a>0,b>0 \right)$的左、右焦点分别为${{F}_{1}}$，${{F2}}$，点$A$是双曲线左支上的一点，若直线$A{{F}_{1}}$与直线$y=\frac{b}{a}x$平行且$\Delta A{{F1}}{{F}_{2}}$的周长为$9a$，则双曲线的离心率为______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
The numbers $1, 2, \dots ,N$ are arranged in a circle where $N \geq  2$. If each number shares a common digit with each of its neighbours in decimal representation, what is the least possible value of $N$?


Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
29
}
A tram ticket costs $1$ Tug, which is equivalent to $100$ tugriks. There are $20$ passengers, each having only coins in denominations of $2$ and $5$ tugriks. The conductor starts with no money at all. Despite this, all passengers manage to pay the fare and receive the necessary change. What is the smallest total number of passengers that the tram could have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
In a convex 2011-gon, what is the maximum number of diagonals that can be drawn such that each newly drawn diagonal intersects at most one of the previously drawn diagonals at an interior point?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4016}
A function $f$ satisfies, for all nonnegative integers $x$ and $y$: - $f(0, x)=f(x, 0)=x$ - If $x \geq y \geq 0, f(x, y)=f(x-y, y)+1$ - If $y \geq x \geq 0, f(x, y)=f(x, y-x)+1$ Find the maximum value of $f$ over $0 \leq x, y \leq 100$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{101}
Given the function \( f(x) = \sin x + \cos x \), you are allowed to write the derivative of any previously written function, as well as the sum and product of any two previously written functions. This process can be repeated multiple times. Determine the constant \( c \) such that a function equal to \( c \) for all real \( x \) appears on the board.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Two players, Bilion and Trilion, play a game where they take turns placing $1, $2, $5, or $10 bills into a pile, starting with Bilion. The game ends when the pile totals exactly $1,000,000. The player who makes the last move wins the $1,000,000 in the pile. If a move causes the pile to exceed $1,000,000, the player who made that move loses, and the other player wins the amount in the pile. Assuming both players play optimally, how many dollars will the winning player gain?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1000000}
Determine the isomorphism class of the quotient group \( \mathbb{Z}^3 / M \), where \( M \) is the subgroup of \( \mathbb{Z}^3 \) generated by the vectors \((13,9,2), (29,21,5), (2,2,2)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{Z}_{16}}
Find the inverse function of \( f(x) = \frac{\ln(x)}{x} \) for \( x > e \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ -\frac{W_{-1}(-x)}{x} }
Determine the value of \( x \) such that the Taylor series expansion \( e^{\left(\frac{d}{dx}\right)} = 1 + \frac{d}{dx} + \frac{1}{2} \left(\frac{d}{dx}\right)^2 + \cdots \) applied to a function \( f(x) \) at \( x = 0 \) results in \( f(x) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Express the rotation of a set of $N$ points $p_1, \\dots, p_N$ with $p_1$ stationary, as a rotation matrix. Given $N-1$ vectors $\vec{v}_i$ from $p_1$ to $p_i$ and their new positions $p'_i$ with vectors $\vec{v}'_i$, express the rotation matrix using the data ($p, p', \vec{v}, \vec{v}'$).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{R = V U^\top}
Evaluate the contour integral \( \int_C(z^2+4iz)\sin\left(\frac{3}{z+2i}\right)\,dz \) where \(C\) is the circle \(|z| = 3\) oriented in the counterclockwise direction.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15\pi i}
Calculate the acute angle between the surfaces given by the equations $xy^2z = 3x + z^2$ and $3x^2 - y^2 + 2z = 1$ at the point $(1, -2, 1)$. Express your answer in degrees or radians.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{80^\circ}
Evaluate the integral \( \int_0^a \int_0^a \frac{dx \, dy}{(x^2+y^2+a^2)^{\frac{3}{2}}} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{6a}}
Two integers $a$ and $b$ are randomly chosen from the set $\{1, 2, 13, 17, 19, 87, 115, 121\}$, with $a > b$. What is the expected value of the number of factors of $ab$?

Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{171}{28}$}
Let the operation $\#$ be defined as $\#(a, b, c) = b^2 - 4ac$, for all real numbers $a, b$, and $c$. Define a new operation $\oplus$ by $\oplus(a, b, c, d) = \#(a, b + d, c) - \#(a, b, c)$. What is the value of $\oplus(2, 4, 1, 3)$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{33}
Determine if the quotient module \( \mathbb{Z}^3 / \mathbb{Z} \cdot (2,0,-1) \) is a free module. If it is free, find its rank.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Ryan initially had 120 balls. There were three times as many red balls as blue balls, and the number of green balls was equal to the sum of red and blue balls. The rest of the balls were yellow. However, 10 balls have been lost, so Ryan now has 110 balls in total. If there are 15 blue balls, how many yellow balls are there after taking into account the lost balls?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
In a metric space, determine whether the limit of an open ball centered at a point x with radius r, as r approaches 0, is equal to the set {x} or the empty set.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\{x\}}
Rectangle $ABCD$ is the base of pyramid $PABCD$. If $AB = 10$, $BC = 5$, and the height of the pyramid, $PA$, while not perpendicular to the plane of $ABCD$, ends at the center of rectangle $ABCD$ and is twice the length of $BC$. What is the volume of $PABCD$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{500}{3}}
Let triangle $\vartriangle ABC$ be acute, and let point $M$ be the midpoint of $\overline{BC}$. Let $E$ be on line segment $\overline{AB}$ such that $\overline{AE} \perp \overline{EC}$. Then, suppose $T$ is a point on the other side of $\overleftrightarrow{BC}$ as $A$ is such that $\angle BTM = \angle ABC$ and $\angle TCA = \angle BMT$. If $AT = 14$, $AM = 9,$ and $\frac{AE}{AC} =\frac27$ , compute $BC$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6\sqrt{5}}
Suppose that a triangle whose sides are of integer lengths is inscribed in a circle of diameter $6.25$. Find the sides of the triangle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
The sides of the triangle are 5, 5, and 6.
}
A point \( P \) is chosen uniformly at random inside a square of side length 2. If \( P_{1}, P_{2}, P_{3}, \) and \( P_{4} \) are the reflections of \( P \) over each of the four sides of the square, find the expected value of the area of quadrilateral \( P_{1} P_{2} P_{3} P_{4} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}