inclusionAI/Ring-lite-rl-data huggingface dataset:
There are $10$ birds on the ground. For any $5$ of them, there are at least $4$ birds on a circle. Determine the least possible number of birds on the circle with the most birds.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Gabriela found an encyclopedia with  $2023$  pages, numbered from  $1$  to  $2023$ . She noticed that the pages formed only by even digits have a blue mark, and that every three pages since page two have a red mark. How many pages of the encyclopedia have both colors?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{44}
The sequence  $\{a_{n}\}$  is defined as follows. \[a_{1}=\frac{\pi}{4},\ a_{n}=\int_{0}^{\frac{1}{2}}(\cos \pi x+a_{n-1})\cos \pi x\ dx\ \ (n=2,3,\cdots)\] Find  $\lim_{n\to\infty}a_{n}$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{4(\pi - 1)}}
The lengths of the three sides of $\triangle ABC$ are 4 cm, 7 cm, and $a$ cm. Using the triangle inequality theorem, determine the possible integer values of $a$. Express your answer as a list or range of integers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4, 5, 6, 7, 8, 9, 10}
2015 people sit down at a restaurant. Each person orders a soup with probability $\frac{1}{2}$. Independently, each person orders a salad with probability $\frac{1}{2}$. What is the probability that the number of people who ordered a soup is exactly one more than the number of people who ordered a salad?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["\\frac{\\binom{4030}{2016}}{2^{4030}}"]}
A finite sequence of three-digit integers has the property that the tens and units digits of each term are, respectively, the hundreds and tens digits of the next term, and the tens and units digits of the last term are, respectively, the hundreds and tens digits of the first term. For example, such a sequence might begin with the terms 247, 475, and 756 and end with the term 824. Let $S$ be the sum of all the terms in the sequence. What is the largest prime factor that always divides $S$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{37}
In a clinical trial, only 0.1% of new treatments are actually effective. Given a significance level of 0.05 and a power of 0.8, calculate the proportion of tests that declare a new treatment effective which are actually false positives.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.984}
从正15边形的顶点中选出3个连成钝角三角形，不同的选法有多少种？
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{315}
Suppose two equally strong tennis players play against each other until one player wins three games in a row. The results of each game are independent, and each player will win with probability $\frac{1}{2}$. What is the expected value of the number of games they will play?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
In the four-digit number 4753, three two-digit numbers are formed by successive pairs of digits (47, 75, 53). Exactly two of these two-digit numbers are prime. Find all four-digit numbers in which all four digits are prime, and all three two-digit numbers formed by successive digits are prime.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2373, 3737, 5373, 7373}
Andrea is three times as old as Jim was when Jim was twice as old as he was when the sum of their ages was  $47$ . If Andrea is  $29$  years older than Jim, what is the sum of their ages now?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{79}
In a group of 2009 people, each pair of people shares exactly one common friend. Determine the smallest possible difference between the maximum and minimum number of friends any person in the group can have.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2006}
Determine the values of \( \alpha \in \mathbb{R} \) for which the integral \( \int_0^1 \frac{\sin(\frac{1}{x})}{(\sqrt{x} - x)^\alpha} \, dx \) converges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-\infty, 1)}
Compute the smallest positive integer $M$ such that there exists a positive integer $n$ such that 

[list] [*] $M$ is the sum of the squares of some $n$ consecutive positive integers, and
[*] $2M$ is the sum of the squares of some $2n$ consecutive positive integers.
[/list]

[i]Proposed by Jaedon Whyte[/i]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ 4250 }
In triangle $\triangle ABC$, $\angle C=90^{\circ}$, $\angle B=30^{\circ}$, and $AC=2$. Let $M$ be the midpoint of $AB$. The triangle $\triangle ACM$ is folded along $CM$ such that the distance between points $A$ and $B$ becomes $2\sqrt{2}$. Determine the distance from point $M$ to the plane $ABC$. \(\text{Express your answer in simplest form.}\)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2\sqrt{6}}{3}}
Calculate the flux of the vector field \( \vec{F}(x,y) = (x^2+y^2)^{-1}\begin{bmatrix} x \\ y \end{bmatrix} \) across the circle \( C \) of radius \( a \) centered at the origin, with positive orientation.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\pi}
What is the smallest three-digit number in Pascal's triangle that is also a multiple of 5?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{100}
Find the remainder when the infinite power tower \(3^{3^{3^{\:
\phantom{}^{.^{.^.}}}}}\) with 2020 levels is divided by 46. Use Euler's totient theorem and modular order reduction to solve this problem.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{41}
## Task B-3.7.

The base of the pyramid is a right-angled triangle with sides of length 1, $a, a^{2}, a>1$. The apex of the pyramid is orthogonally projected onto the vertex of the right angle of the base. The acute angle opposite the side of length 1 is equal to the angle at which one lateral face is inclined to the base. Calculate the volume of the pyramid.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{6}\sqrt{\frac{\sqrt{5}-1}{2}}}
In $\triangle ABC$, $\angle ABC = 50^{\circ}$ and $\angle ACB = 20^{\circ}$. Point $N$ is inside $\triangle ABC$ such that $\angle NAB = 40^{\circ}$ and $\angle NBC = 30^{\circ}$. Find the measure of $\angle NCB$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10^\circ}
Let \( f: \mathbb{N} \rightarrow \mathbb{N} \) be a function that satisfies

\[ f(1) = 2, \]
\[ f(2) = 1, \]
\[ f(3n) = 3f(n), \]
\[ f(3n + 1) = 3f(n) + 2, \]
\[ f(3n + 2) = 3f(n) + 1. \]

Find how many integers \( n \leq 2014 \) satisfy \( f(n) = 2n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{127}
A rectangle and an isosceles triangle share a common side, which lies along the $x$-axis, with the right vertex of the rectangle and the left vertex of the triangle located at $(15, 0)$. The dimensions of the rectangle are $15$ units wide (along the $x$-axis) and $10$ units high, while the base and height of the triangle are both $10$ units. A segment is drawn from the top left vertex of the rectangle to the top vertex of the triangle. Calculate the area of the triangle formed by this segment and part of the $x$-axis.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{75 \text{ sq. units}}
若实数 $x 、 y$ 满足
$2^{x}+4 x+12=\log _{2}(y-1)^{3}+3 y+12=0$,则 $x+y=$          .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2}
Benjamin walks to work and home five days a week, walks his dog twice a day every day, walks to his best friend’s house once a week, and walks to the convenience store twice a week. Work is six miles away, dog walks are two miles, and his best friend’s house is one mile. He walks 95 miles in a week. How far is the convenience store?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2.5}
Consider all the positive integers  $N$  with the property that all of the divisors of  $N$  can be written as  $p-2$  for some prime number  $p$ . Then, there exists an integer  $m$  such that  $m$  is the maximum possible number of divisors of all
numbers  $N$  with such property. Find the sum of all possible values of  $N$  such that  $N$  has  $m$  divisors.

*Proposed by **FedeX333X***
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{135}
A collection of circles in the upper half-plane, all tangent to the $x$-axis, is constructed in layers as follows. Layer $L_0$ consists of two circles of radii $70^2$ and $73^2$ that are externally tangent. For $k \ge 1$, the circles in $\bigcup_{j=0}^{k-1}L_j$ are ordered according to their points of tangency with the $x$-axis. For every pair of consecutive circles in this order, a new circle is constructed externally tangent to each of the two circles in the pair. Layer $L_k$ consists of the $2^{k-1}$ circles constructed in this way. Let $S=\bigcup_{j=0}^{6}L_j$, and for every circle $C$ denote by $r(C)$ its radius. What is
\[\sum_{C\in S} \frac{1}{\sqrt{r(C)}}?\]
[asy] import olympiad; size(350); defaultpen(linewidth(0.7)); // define a bunch of arrays and starting points pair[] coord = new pair[65]; int[] trav = {32,16,8,4,2,1}; coord[0] = (0,73^2); coord[64] = (2*73*70,70^2); // draw the big circles and the bottom line path arc1 = arc(coord[0],coord[0].y,260,360); path arc2 = arc(coord[64],coord[64].y,175,280); fill((coord[0].x-910,coord[0].y)--arc1--cycle,gray(0.75)); fill((coord[64].x+870,coord[64].y+425)--arc2--cycle,gray(0.75)); draw(arc1^^arc2); draw((-930,0)--(70^2+73^2+850,0)); // We now apply the findCenter function 63 times to get // the location of the centers of all 63 constructed circles. // The complicated array setup ensures that all the circles // will be taken in the right order for(int i = 0;i<=5;i=i+1) { int skip = trav[i]; for(int k=skip;k<=64 - skip; k = k + 2*skip) { pair cent1 = coord[k-skip], cent2 = coord[k+skip]; real r1 = cent1.y, r2 = cent2.y, rn=r1*r2/((sqrt(r1)+sqrt(r2))^2); real shiftx = cent1.x + sqrt(4*r1*rn); coord[k] = (shiftx,rn); } // Draw the remaining 63 circles } for(int i=1;i<=63;i=i+1) { filldraw(circle(coord[i],coord[i].y),gray(0.75)); }[/asy]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{143}{14}}
Simplify the expression: 

\[
\frac{\sin 2 \alpha + \cos 2 \alpha - \cos 6 \alpha - \sin 6 \alpha}{\sin 4 \alpha + 2 \sin^2 2 \alpha - 1}
\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 \sin 2\alpha}
Suppose that the polynomial $p(x) = x^{2010} \pm x^{2009} \pm \ldots \pm x \pm 1$ does not have a real root. What is the maximum number of coefficients that can be $-1$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1005}
Our class got 125 books from the library. Then we gave 42 books away to a different class, and later added 68 books from another source. After that, we decided to donate 31 more books to the school's book drive. How many books from the library does our class have left?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{52}
Determine the number of integer pairs \((x, y)\) such that \((|x| - 2)^2 + (|y| - 2)^2 < 5\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{48}
ΔABC是边长为1的正三角形，PAL平面ABC，且$P A = \frac { \sqrt { 6 } } { 4 } , A$点关于平面PBC的对称点为$A ^ { \prime }$，则直线$A ^ { \prime } C$与AB所成角的余弦值是 .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
A triangle has sides \(a, b, c\). Semicircles are constructed outwardly on each of its sides as diameters, forming a figure \(F\) composed of the triangle and the three semicircles. Find the diameter of \(F\) (the diameter of a set in the plane is the greatest distance between its points).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{a+b+c}{2}}
Given that when $-1 < x < 0$, the value of the quadratic function $y = x^2 - 4mx + 3$ is always greater than 1, find the range of values for $m$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{m > -\frac{3}{4}}
Given \(0 < b < 1\) and \(0 < a < \frac{\pi}{4}\), determine the correct ordering of the three numbers defined as follows: 
\[ x = \sin a^{\log_{b} (b^{\sin a})}, \]
\[ y = (\cos a)^{\log_{6} (b^{\cos a})}, \]
\[ z = (\sin a)^{\log_{b} (\cos a)}. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{x < z < y}
Two trains of different lengths run at the speed of 60 km/hr and 40 km/hr respectively in opposite directions on parallel tracks. One train is 300 m long, and they take 17.998560115190788 seconds to cross each other. How long is the other train?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{200}
Rafael works 10 hours on Monday and 8 hours on Tuesday on his delivery job. He must work a total of 40 hours per week. However, there is a 5-day work week constraint, meaning that he can only work a maximum of 8 hours per day from Wednesday to Friday. Given that he is paid $20 per hour for the first 8 hours of work each day and $30 per hour for any overtime (hours worked after the first 8 hours), calculate his total pay for the week, taking into account a 10% tax deduction on his earnings.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$738}
Evaluate the integral: $$\int_{0}^{1}\dfrac{x^4+1}{x^6+1}\,dx.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{3}}
Find the maximum value of \( \sum_{i=1}^n \frac{1}{|x-p_i|} \) for \( 0 \le x \le 1 \), where \( 0 \le p_i \le 1 \) for \( i = 1, 2, \dots, n. \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Determine the condition for repeated roots in the fourth-order polynomial equation: \( x^4 + ax^3 + bx^2 + cx + d = 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\Delta = 0}
A man can buy a new house in 4 years with all of his earnings. He spends the same amount on household expenses in some years and on savings in 12 years. He needs 24 years to buy the house. How many years does it take for him to spend the same amount on household expenses?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
In the ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 \ (a>b>0)\), let the left focus be \(F\), the right vertex be \(A\), and the endpoint on the upper side of the minor axis be \(B\). If the eccentricity of this ellipse is \(\frac{\sqrt{5}-1}{2}\), then find \(\angle ABF\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{90^{\circ}}
Given $m\in R$, let point $A$ be on the moving line $x+my=0$ and point $B$ be on the moving line $mx-y-m+3=0$. These two lines intersect at point $P(x,y)$. Find the maximum value of $|PA|\cdot|PB|$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Petya is thinking of a four-digit number of the form \( \overline{20 * *} \).

Vasya consecutively checks whether the number chosen by Petya is divisible by 1, 3, 5, 7, 9, 11. If the number is divisible, Vasya pays Petya 1, 3, 5, 7, 9, or 11 rubles respectively. For example, for the number 2000, Vasya would pay Petya \(1+5=6\) rubles.

What is the maximum number of rubles Petya can receive?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{31}
Given a four-digit number where the digits are distinct non-negative integers and their sum is 12, find the number of such four-digit numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{174}
In the rectangular coordinate plane, the point \((x, y)\) corresponds one-to-one with the complex number \(x + yi\). If points \(A\) and \(B\) correspond to the complex numbers \(z\) and \(z^{-1} \, (z \notin \mathbb{R})\) respectively, then the complex number corresponding to the intersection of the line \(AB\) with the \(x\)-axis (expressed in terms of \(z\)) is:
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{z + \bar{z}}{1 + z \bar{z}}}
Given linearly independent vectors $\left\{a,b,c\right\}\in \mathbb{R}^3$, find the value of $\lambda $ so that the dimension of the subspace generated by the vectors: $2a-3b,\:\:\left(\lambda -1\right)b-2c,\:\:3c-a,\:\:\lambda c-b$ is 2.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
已知抛物线 $\Omega: y=x^{2}$, 动线段 $A B$ 在直线 $y=\sqrt{3} x-3$ 上 ( $B$ 在 $A$ 右侧), 且 $|A B|=2 \sqrt{3}$. 过 $A$ 作 $\Omega$的切线, 取左边的切点为 $M$. 过 $B$ 作 $\Omega$ 的切线, 取右边的切点为 $N$. 当 $M N / / A B$ 时, 求点 $A$ 的横坐标.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Given a sequence ${a_n}$ with its first $n$ terms sum $S_n$, it satisfies $S_n=(-1)^{n}a_{n}+ \frac {1}{2^{n}}$. Let $T_n$ be the sum of the first $n$ terms of ${S_n}$. Find $T_{2017}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac {1}{3}[1-( \frac {1}{2})^{2016}]}
已知数列 \{x_n\} 满足：x_1=a, x_2=b, x_{n+2}=3x_{n+1}+2x_n. 若存在整数 k\geq3，使得 x_k=2019，则这样的有序正整数对 (a,b) 的个数为__________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{370}
In a convex pentagon \(ABCDE\), \(AB = BC\), \(CD = DE\), \(\angle ABC = 100^\circ\), \(\angle CDE = 80^\circ\), and \(BD^2 = \frac{100}{\sin 100^\circ}\). Find the area of the pentagon.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50}
Given $a=\sin 11^\circ$, $b=\cos 10^\circ$, and $c=\sin 168^\circ$, find the relationship between the magnitudes of $a$, $b$, and $c$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a < c < b}
Find the limit of the following integral as n approaches infinity:

$$I_n = \int_0^3 \underbrace{\sin\left(\frac{\pi}{3} \sin\left(\frac{\pi}{3} \sin \left(\frac{\pi}{3} \dots \sin\left(\frac{\pi}{3} x\right) \dots \right)\right)\right)}_\text{n times of sines}dx$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{3}{2}$}
The average age of 15 men is increased by years when one of them whose age is a certain number of years and another whose age is 23 years are replaced by two new men. The average age of the two new men is 37. What is the age of the first man who was replaced?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21}
Assume that each airplane engine has a failure rate of $1-P$ during flight, and whether each engine fails is independent. If at least $50\%$ of the engines are functioning normally, the airplane can successfully fly. To make a 4-engine airplane safer than a 2-engine airplane, what is the range of the value of $P$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(2/3, 1)}
The taxi fare in Metropolis City is $3.00 for the first $\frac{3}{4}$ mile and additional mileage charged at the rate $0.30 for each additional 0.1 mile. You plan to give the driver a $3 tip. Calculate the number of miles you can ride for $15.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3.75}
What is the minimum number of factors that need to be removed from the number 99! (99! is the product of all numbers from 1 to 99) so that the product of the remaining factors ends in 2?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
Let $ABCD$ be a convex quadrilateral with $AB=5$, $BC=6$, $CD=7$, and $DA=8$. Let $M$, $P$, $N$, $Q$ be the midpoints of sides $AB$, $BC$, $CD$, and $DA$ respectively. Compute $MN^2 - PQ^2$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
Given six people, including A, B, and C, are arranged in a row for a photo, and A and B are not on the same side as C, calculate the total number of different arrangements.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{240}
Leticia has a $9 \times 9$ board. She says that two squares are "friends" if they share a side, if they are at opposite ends of the same row, or if they are at opposite ends of the same column. Every square has $4$ friends on the board. Leticia will paint every square one of three colors: green, blue, or red. In each square, a number will be written based on the following rules:

- If the square is green, write the number of red friends plus twice the number of blue friends.
- If the square is red, write the number of blue friends plus twice the number of green friends.
- If the square is blue, write the number of green friends plus twice the number of red friends.

Considering that Leticia can choose the coloring of the squares on the board, find the maximum possible value she can obtain when she sums the numbers in all the squares.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{486}
化简$C _ { n } ^ { 0 } C _ { n } ^ { 1 } + C _ { n } ^ { 1 } C _ { n } ^ { 2 } + \cdots + C _ { n } ^ { n - 1 } C _ { n } ^ { n } =$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$C _ { 2 n } ^ { n - 1 }$ }
Consider a sequence defined as $500, x, 500 - x, \ldots$ where each term of the sequence after the second one is obtained by subtracting the previous term from the term before it. The sequence terminates as soon as a negative term appears. Determine the positive integer $x$ that leads to the longest sequence.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{309}
Among the 99 positive integers from 1 to 99, find the minimum value of k such that any selection of k numbers must include at least two numbers a and b (a!= b) satisfying 1/2 <= b/a <= 2.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
ерешин D.A.

Once, Rabbit was in a hurry to meet with Donkey Eeyore, but Winnie-the-Pooh and Piglet unexpectedly came to him. Being well-mannered, Rabbit offered his guests some refreshments. Pooh tied a napkin around Piglet's mouth and single-handedly ate 10 pots of honey and 22 cans of condensed milk, with each pot of honey taking him 2 minutes and each can of milk taking 1 minute. Upon learning that there was no more sweet food in the house, Pooh said goodbye and took Piglet away. Rabbit sadly thought that he wouldn't have been late for the meeting with the donkey if Pooh had shared with Piglet. Knowing that Piglet eats a pot of honey in 5 minutes and a can of milk in 3 minutes, Rabbit calculated the minimum time it would take for the guests to finish his supplies. What is this time? (A can of milk and a pot of honey can be divided into any parts.)

#
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
## Problem Statement

Calculate the indefinite integral:

$$
\int \frac{1-\sqrt{x}}{\sqrt{x} \cdot(x+1)} d x
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\arctan\sqrt{x}-\ln|x+1|+C}
The diagram shows a smaller rectangle made from three squares, each of area \(25 \ \mathrm{cm}^{2}\), inside a larger rectangle. Two of the vertices of the smaller rectangle lie on the mid-points of the shorter sides of the larger rectangle. The other two vertices of the smaller rectangle lie on the other two sides of the larger rectangle. What is the area, in \(\mathrm{cm}^{2}\), of the larger rectangle?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{150}
In $\triangle ABC$, if $\sin^{2} B + \sqrt{2} \sin B \sin C = \sin^{2} A - \sin^{2} C$, find the value of $A$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3\pi}{4}}