inclusionAI/Ring-lite-rl-data huggingface dataset:
Helga went shopping and visited 6 stores. 

At the first store, she tried on 7 pairs of shoes and looked at 4 handbags. 

At the second store, she tried on twice the number of shoes she did at the first store and also checked out 6 more bags than the first store. 

At the third store, she did not try on any shoes, but she did buy a scarf and a pair of earrings. 

At the fourth store, she tried on the same number of shoes as the total number of bags she looked at in the first and second stores combined. 

At the fifth store, she tried on half the number of shoes she did at the fourth store but also checked out 8 bags. 

Finally, at the sixth store, she tried on the square root of the total pairs of shoes she tried on at the second and fifth stores combined, and also looked at 5 more bags than the total number of bags she had checked out previously. 

What is the total number of pairs of shoes and bags Helga tried on or looked at before buying her items?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{95}
设 $A 、 B 、 C$ 分别是复数 $Z_{0}=a \mathrm{i}, Z_{1}=\frac{1}{2}+b \mathrm{i}, Z_{2}=1+c \mathrm{i}$ (其中 $a, b, c$ 都是实数) 对应的不共线的三点. 曲线

$$
Z=Z_{0} \cos ^{4} t+2 Z_{1} \cos ^{2} t \sin ^{2} t+Z_{2} \sin ^{4} t \quad(t \in \mathrm{R})
$$

与 $\triangle A B C$ 中平行于 $A C$ 的中位线只有一个公共点, 求出此点.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\left(\frac{1}{2},\frac{1}{4}(a+2 b+c)\right)$}
求具有下述性质的最小正数 $c$ ：对任意整数 $n \geq 4$,以及集合 $A \subseteq\{1,2, \cdots, n\}$, 若 $|A|>c n$, 则存在函数 $f: A \rightarrow\{1,-1\}$, 满足
$$
\left|\sum_{a \in A} f(a) \cdot a\right| \leq 1
$$
原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Six horizontal lines and five vertical lines are drawn in a plane. If a specific point, say (3, 4), exists in the coordinate plane, in how many ways can four lines be chosen such that a rectangular region enclosing the point (3, 4) is formed?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
Initially, Aslı distributes  $1000$  balls to  $30$  boxes as she wishes. After that, Aslı and Zehra make alternated moves which consists of taking a ball in any wanted box starting with Aslı. One who takes the last ball from any box takes that box to herself. What is the maximum number of boxes can Aslı guarantee to take herself regardless of Zehra's moves?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
Find the greatest value of positive integer $ x$ , such that the number 

  $ A\equal{} 2^{182} \plus{} 4^x \plus{} 8^{700}$ 

 is a perfect square .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2008}
A company hires 4 types of workers with varying levels of expertise and pay rates to work on a project. Type A workers earn Rs. 200 per day, Type B workers earn Rs. 250 per day, Type C workers earn Rs. 300 per day, and Type D workers earn Rs. 350 per day.

A team of 4 men, consisting of one worker of each type, work together 10 hours a day to finish the project and are paid their full rates if they work at least 8 hours a day. For every hour worked beyond 8 hours, they receive overtime pay equal to 1.5 times their standard hourly rate.

Later, the company decides to redistribute the workload. They assign 9 men to the project, with 3 workers of Type A, 2 workers of Type B, 3 workers of Type C, and 1 worker of Type D. This new team works 6 hours a day for the entire week.

Calculate how much the company would have to pay the new team per week including overtime pay if applicable.

Remember there are 7 working days in a week, and overtime pay applies only when a worker works more than 8 hours a day.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16,450}
Evaluate the surface integral \(\int_{S}\nabla\times\vec{F}\cdot{\rm d}\vec{S}\), where \(\vec{F}(x,y,z) = y^2\hat{i} + 2xy\hat{j} + xz^2\hat{k}\), and \(S\) is the upward-facing surface of the paraboloid \(z = x^2 + y^2\) in the first quadrant, bounded by the planes \(x=0\), \(y=0\), and \(z=1\). Use Stokes' theorem to find the boundary curve \(C\) and evaluate the integral.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2}{7}}
已知 $A, B$ 分别为椭圆 $C: \frac{x^{2}}{2}+y^{2}=1$ 的左、右顶点, $P$ 是椭圆在第一象限内一点,满足 $|P A|=\lambda|P B|$ 且 $\angle P B A=2 \angle P A B$, 求 $\lambda$ 的值。原始答案的形式为 $\frac{\sqrt{m}}{n}$，请给出 $m + n$ 的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{33}
Given condition p: $(x + 1)^2 > 4$, and condition q: $x > a$, and $\neg p$ is a sufficient but not necessary condition for $\neg q$, determine the range of values for $a$. Express your answer as an inequality involving $a$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a \geq 1}
200 participants arrived at the international table tennis championship. The matches are elimination matches, meaning each match involves two players; the loser is eliminated from the tournament, and the winner remains. Find the maximum possible number of participants who have won at least three matches.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{66}
From noon till midnight, Clever Cat sleeps under the oak tree and from midnight till noon he is awake telling stories. A poster on the tree above him says "Two hours ago, Clever Cat was doing the same thing as he will be doing in one hour's time". For how many hours a day does the poster tell the truth?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
Find the greatest $n$ such that $(z+1)^n = z^n + 1$ has all its non-zero roots in the unitary circumference, e.g. $(\alpha+1)^n = \alpha^n + 1, \alpha \neq 0$ implies $|\alpha| = 1.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ n = 7 }
Let $a_n= \frac {1}{n}\sin \frac {n\pi}{25}$, and $S_n=a_1+a_2+\ldots+a_n$. Find the number of positive terms among $S_1, S_2, \ldots, S_{100}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{100}
The graph of a power function passes through the point $(2, \frac{1}{4})$. The interval where it is monotonically increasing is \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-\infty, 0)}
8. [30] Consider the following two-player game. Player 1 starts with a number, $N$. He then subtracts a proper divisor of $N$ from $N$ and gives the result to player 2 (a proper divisor of $N$ is a positive divisor of $N$ that is not equal to 1 or $N$ ). Player 2 does the same thing with the number she gets from player 1 , and gives the result back to player 1. The two players continue until a player is given a prime number, at which point that player loses. For how many values of $N$ between 2 and 100 inclusive does player 1 have a winning strategy?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{47}
Given the function f(x) = (x-1)<sup>3</sup> - $\frac{1}{x-1}$ and g(x) = -x + m, if the sum of the x-coordinates of the intersection points of their graphs is 2, find the value of m.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Let $A B C$ be a triangle such that $A B=13, B C=14, C A=15$ and let $E, F$ be the feet of the altitudes from $B$ and $C$, respectively. Let the circumcircle of triangle $A E F$ be $\omega$. We draw three lines, tangent to the circumcircle of triangle $A E F$ at $A, E$, and $F$. Compute the area of the triangle these three lines determine.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{462}{5}}
已知 $a_{k}$ 为 $\log _{2} k, \log _{3} k, \cdots, \log _{2018} k$ 中整数项的个数. 则 $\sum\limits_{k=1}^{2018} a_{k}=$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4102}
Given the function $f(x)=f′( \frac {π}{4})\cos x+\sin x$, find the value of $f( \frac {π}{4})$. Express your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Find the interval that contains the solution to the equation $2^{x-1}+x=5$. Express your answer as an interval in standard notation (e.g., (a, b)).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(2, 3)}
Determine the number of ways to assign each integer from $1$ to $11$ one of the letters $L$, $M$, or $T$ such that no two consecutive multiples of any positive integer $n$ are assigned the same letter.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
Define an odd function $f(x)$ on $\mathbb{R}$ that is monotonically increasing on $(0, +\infty)$ and satisfies $f(1) = 0$. Then, the solution set of the inequality $f(x) > 0$ is.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-1, 0) \cup (1, +\infty)}
The highway from Lishan Town to the provincial capital is 189 kilometers long, passing through the county town. It is 54 kilometers from Lishan Town to the county town. At 8:30 AM, a bus left from Lishan Town to the county town and arrived at 9:15 AM. After a 15-minute stop, it continued to the provincial capital and arrived at 11:00 AM. Another bus left from the provincial capital to Lishan Town on the same morning at 8:50 AM, traveling at 60 kilometers per hour. What is the time when the two buses meet?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10:08}
Determine the biggest possible value of $ m$ for which the equation $ 2005x \plus{} 2007y \equal{} m$ has unique solution in natural numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 \cdot 2005 \cdot 2007}
In the quadrilateral \(ABCD\), points \(E\) and \(F\) are on \(BC\) and \(CD\) respectively, with \(\frac{DF}{FC} = 1\) and \(\frac{CE}{EB} = 2\). If the area of \(\triangle ADF\) is \(m\), and the area of the quadrilateral \(AECF\) is \(n\) (with \(n > m\)), then what is the area of the quadrilateral \(ABCD\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3}{2}n + \frac{1}{2}m}
Evaluate the improper integral using the residue theorem: $$\int_{-\infty}^\infty \frac{x\cos\left(-\frac{\pi}{2}x\right)}{x^2-2x+5} \,dx$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\pi e^{-\pi}}
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}
Calculate the area of the spherical triangle defined by the points \((0, 0, 1)\), \((0, 1, 0)\), and \(\left(\dfrac{1}{\sqrt{2}}, 0, \dfrac{1}{\sqrt{2}}\right)\) using the spherical Gauss-Bonnet formula.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{4}}
You are given  $n \ge 2$  distinct positive integers. Let's call a pair of these integers *elegant* if their sum is an integer power of  $2$ . For every  $n$  find the largest possible number of elegant pairs.

*Proposed by Oleksiy Masalitin*
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ n-1 }
Find the minimum value of \((s+5-3|\cos t|)^{2}+(s-2|\sin t|)^{2}\) as \(s\) and \(t\) range over all real numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Roberto is raising chickens for eggs. He buys 4 chickens for $20 each. They cost $1 in total a week to feed and each produces 3 eggs a week that will last forever. He used to buy 1 dozen eggs a week and spent a certain amount per dozen. After 81 weeks, the chickens will be cheaper than buying his eggs. How much did he spend per dozen eggs before buying the chickens?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$1.99}
Find the value of $$\lim_{n\to \infty} \left[\left(1-\frac{1}{n}\right)^n-\left(1-\frac{1}{n}\right)\right]^{-n}$$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
在黑板上写有97个数：$\frac { 4 8 } { k } ( k = 1 ,2 , \cdots , 9 7 )$，每次可任意选择两个数$a$，$b$，将它们擦去，用$2ab-a-b+1$代替，这样经过96次操作后，黑板上还剩下一个数，问：这个数是多少？
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac { 1 } { 2 }$}
8. In the阶梯教室 (tiered classroom) installed with connected desks, 6 people sit in one row, and candidates can only exit the examination room from the two ends of the desk. The time when candidates finish their exams varies. If a candidate sitting inside needs to submit the paper first, they will disturb others. The number of people disturbing others to submit their papers in a row is regarded as a random variable $X$, then the mathematical expectation of $X$ equals $\qquad$ (Su Chun provided the question)

Note: "阶梯教室" is translated as "tiered classroom" to maintain the meaning in English.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{21}{10}}
a scale 7 ft . 6 inches long is divided into 5 equal parts . find the length of each part .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
The base of the quadrangular pyramid \( M A B C D \) is a parallelogram \( A B C D \). Given that \( \overline{D K} = \overline{K M} \) and \(\overline{B P} = 0.25 \overline{B M}\), the point \( X \) is the intersection of the line \( M C \) and the plane \( A K P \). Find the ratio \( M X: X C \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3 : 4}
Perpendiculars $AP$ and $AK$ are drawn from vertex $A$ to the angle bisectors of the external angles $\angle ABC$ and $\angle ACB$ of triangle $ABC$. Find the length of segment $PK$ if the perimeter of triangle $ABC$ is $p$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{p}{2}}
Evaluate the expression \( z = \sqrt{i} - \sqrt{-i} \) and find its value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{i\sqrt{2}}
Determine all pairs ( $a, p$ ) of positive integers with $p$ prime such that $p^{a}+a^{4}$ is a perfect square. Please provide the sum of all values of $a$ and $p$ for the pairs that satisfy the condition.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
Find all the rational roots of
\[2x^3 - 9x^2 - 32x - 21 = 0.\]Enter all the rational roots, separated by commas.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\frac{3}{2}, -1, 7}
Compute
$\begin{aligned} \lim_{A\to+\infty}\frac{1}{A}\int_{1}^{A}A^{\frac{1}{x}}dx. \end{aligned}$

Proposed by Jan Šustek, University of Ostrava
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
1
}
Given that the odd function $f(x)$ is also a periodic function, and the smallest positive period of $f(x)$ is $\pi$, when $x \in \left(0, \frac{\pi}{2}\right)$, $f(x) = 2\sin x$. Find the value of $f\left(\frac{11\pi}{6}\)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
What is $(a^3+b^3)\div(a^2-ab+b^2+c)$ for $a=7$, $b=6$, and $c=1$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{559}{44}}
已知在三棱锥 $D-A B C$ 中, $\angle A C B=\angle A B D=90^{\circ}, C A=C B, \angle B A D=30^{\circ}$.若点 $C$ 在平面 $A B D$ 上的射影恰好在 $A D$ 上, 则二面角 $C-A B-D$ 的平面角的正弦值大小为 \frac{\sqrt{m}}{n}，请给出m + n的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Let $g(x)$ be a function piecewise defined as \[g(x) = \left\{
\begin{array}{cl}
-x & x\le 0, \\
2x-41 & x>0.
\end{array}
\right.\] If $a$ is negative, find $a$ so that $g(g(g(10.5)))=g(g(g(a)))$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-30.5}
In a city, there are 10,000 bicycles with all possible numbers from 1 to 10,000. What is the probability that the number of the first bicycle encountered does not contain the digit 8?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.6561}
The weather forecast says that in the next three days, the probability of rain each day is $0.4$. Someone used a computer to generate random integers between $0$ and $9$, using $1, 2, 3, 4$ to represent rain, and $5, 6, 7, 8, 9, 0$ to represent no rain. They generated $3$ random numbers as a group, producing $20$ groups of random numbers as follows: $027556488730113537989907966191925271932812458569683431257393$. Based on this, the probability of having at least two rainy days in these three days is approximately \_\_\_\_\_\_
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.35}
Given that f(x) is an even function defined on R, and when x < 0, f(x) + x * f'(x) < 0, and f(-4) = 0, find the solution set of the inequality x * f(x) > 0. Express your answer as an interval or union of intervals.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-\infty,-4)\cup(0,4)}
What is the genus of the field extension $K = \mathbb{C}(T)(\sqrt{T^n + 1})$ over $\mathbb{C}(T)$, where $n$ is an even integer greater than or equal to 2?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{n - 2}{2}\)}
Determine the number of terms required in the alternating series representation of \( \pi \), given by \( \pi = 4\sum_{k=1}^\infty \frac{(-1)^{k+1}}{2k-1} \), to approximate \( \pi \) accurately up to 10 decimal places.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40000000000}
What is the expected value of the number of balls remaining in an urn containing $a$ red and $a$ blue balls if balls are drawn without replacement until all blue balls are drawn?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{a}{a+1}\)}
Place the numbers 1, 2, 3, 4, 5, 6, 7, and 8 on the eight vertices of a cube such that the sum of any three numbers on a face is at least 10. Find the minimum sum of the four numbers on any face.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
Given a matrix \( A \in \mathbb{R}^{5 \times 5} \) and a scalar \( \alpha \in \mathbb{R} \), consider the linear system \( Ax = \begin{pmatrix} 1 \\ -1 \\ a^2+a \\ 1 \\ -1 \end{pmatrix} \). The general solution is given by \( \begin{pmatrix} t+u+1 \\ s+t-u \\ s-t+1 \\ -s-t+u \\ -s-u-1 \end{pmatrix} \). Additionally, the vectors \( \begin{pmatrix} 1 \\ 0 \\ 1 \\ 0 \\ a^2-3 \end{pmatrix} \) and \( \begin{pmatrix} 0 \\ 1 \\ 1 \\ -1 \\ a^2-2 \end{pmatrix} \) are in the column space of \( A \). It is known that \( \alpha = 0 \) or \( \alpha = -1 \) satisfies these conditions. Determine which value of \( \alpha \) is still valid if \( A^2 = 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
The segment \( AB = 8 \) intersects the plane \( \alpha \) at an angle of \( 30^\circ \) and is divided by this plane in the ratio \( 1:3 \). Find the radius of the sphere passing through points \( A \) and \( B \) and intersecting the plane \( \alpha \) in a circle of the smallest radius.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\sqrt{7}}
P.J. starts with \(m=500\) and chooses a positive integer \(n\) with \(1 \leq n \leq 499\). He applies the following algorithm to \(m\) and \(n\): P.J. sets \(r\) equal to the remainder when \(m\) is divided by \(n\). If \(r=0\), P.J. sets \(s=0\). If \(r>0\), P.J. sets \(s\) equal to the remainder when \(n\) is divided by \(r\). If \(s=0\), P.J. sets \(t=0\). If \(s>0\), P.J. sets \(t\) equal to the remainder when \(r\) is divided by \(s\). For how many of the positive integers \(n\) with \(1 \leq n \leq 499\) does P.J.'s algorithm give \(1 \leq r \leq 15\) and \(2 \leq s \leq 9\) and \(t=0\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
A papaya tree will grow 2 feet in the first year. In the second year, it will grow some percentage more than the first year. In the third year, the tree will grow 50% more than in the second year. In the fourth year, it will grow twice as much as the third year. In the fifth year, it will grow half as much as the fourth year. When the tree is 5 years old, it is 23 feet tall. What is the percentage increase in growth from the first year to the second year?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50\%}
对于平面 $\alpha$, 若多面体的各个顶点到平面 $\alpha$ 的距离均相等, 则称平面 $\alpha$ 为多面体的“中位面”.
四面体有多少个互不相同的中位面?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$7$}
Find a positive constant \( c \) such that the normalized sum \( \frac{S_n}{n^c} \) converges in distribution to a random variable \( A \), where \( S_n = X_1 + X_2 + \ldots + X_n \) and the characteristic function of the iid random variables \( X_i \) is given by \( \phi(t) = 1 - \sqrt{|t|(2-|t|)} \) for \( t \in [-1,1] \) and \( \phi(t) = 0 \) elsewhere.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Determine the norm \( \|\psi\| \) of the functional \( \psi: L^1[-1,1] \to \mathbb{C} \) defined by \( \psi(g)=\int_{-1}^{0}x^2g(1-x)\,\mathrm{d}x. \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Petya and Vasya play the following game. Petya conceives a polynomial $P(x)$ having integer coefficients. On each move, Vasya pays him a ruble, and calls an integer $a$ of his choice, which has not yet been called by him. Petya has to reply with the number of distinct integer solutions of the equation $P(x)=a$. The game continues until Petya is forced to repeat an answer. What minimal amount of rubles must Vasya pay in order to win?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
4
}
Determine the number of ideals of order 8 in the factor ring \( R = \mathbb{F}_2[x, y]/\langle x^2, y^2 \rangle \) without using the brute force method.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Xiaoming's mother went to the store to buy soap. The unit prices of brand $A$ and brand $B$ soaps are 6 yuan and 9 yuan respectively. Xiaoming's mother can buy 2 more pieces of brand $A$ soap than brand $B$ soap with all the money she has, and there is no money left. How much money did Xiaoming's mother bring?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36 \text{ yuan}}
Let $p$ and $q$ be real numbers, and suppose that the roots of the equation \[x^3 - 10x^2 + px - q = 0\] are three distinct positive integers. Compute $p + q.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{37}
Compute the integral \( \int_{-\infty}^{\infty} [F(x+10) - F(x)] \, dx \), where \( F \) is a distribution function.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
We have 21 pieces of type $\Gamma$ (each formed by three small squares). We are allowed to place them on an $8 \times 8$ chessboard (without overlapping, so that each piece covers exactly three squares). An arrangement is said to be maximal if no additional piece can be added while following this rule. What is the smallest $k$ such that there exists a maximal arrangement of $k$ pieces of type $\Gamma$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
In a convex quadrilateral \(ABCD\), the midpoint of side \(AD\) is marked as point \(M\). Segments \(BM\) and \(AC\) intersect at point \(O\). It is known that \(\angle ABM = 55^\circ\), \(\angle AMB = 70^\circ\), \(\angle BOC = 80^\circ\), and \(\angle ADC = 60^\circ\). How many degrees is \(\angle BCA\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{35}