inclusionAI/Ring-lite-rl-data huggingface dataset:
when 242 is divided by a certain divisor the remainder obtained is 6 . when 698 is divided by the same divisor the remainder obtained is 13 . however , when the sum of the two numbers 242 and 698 is divided by the divisor , the remainder obtained is 5 . what is the value of the divisor ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
An island has  $10$  cities, where some of the possible pairs of cities are connected by roads. A *tour route* is a route starting from a city, passing exactly eight out of the other nine cities exactly once each, and returning to the starting city. (In other words, it is a loop that passes only nine cities instead of all ten cities.) For each city, there exists a tour route that doesn't pass the given city. Find the minimum number of roads on the island.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
Evaluate the integral $$\int_{0}^{2\pi}(1+\cos x)^n\sin nx dx$$ using complex analysis.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Evaluate $$\int_0^\infty \dfrac {\log{x}}{(x^2+1)^2}dx$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\frac{\pi}{4}}
Mr. Grey's house was worth $100,000. During a booming market, he sold the house to Mr. Brown at a profit of 12%. After one year, the market experienced a slight decline, and Mr. Brown invested 5% of the purchasing price on improvements. He then sold the house to his friend Mr. Green with a 4% profit. Mr. Green rented out the house for a year, during which the market faced another decline of 3%. Mr. Green then decided to sell the house to Ms. White, who bought it with a 10% loss. After six months, the market started recovering and experienced an 8% increase in property prices. Ms. White then decided to sell the house back to Mr. Brown with a 5% profit. How much was Mr. Brown's final purchase price after buying the house back from Ms. White?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$121,078.76}
Let $A=\{x|x=sin\frac{2π}{2023}+sin\frac{4π}{2023}+sin\frac{6π}{2023}+…+sin\frac{2kπ}{2023},k∈Z,k＞0\}$. Find the number of distinct elements in set $A$. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1012}
There is a bag containing 8 white balls and 2 red balls. Each time a ball is randomly taken out, and then a white ball is put back into the bag. What is the probability that all red balls have been taken out exactly after the 4th draw?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.0434}
When \((1+x)^{38}\) is expanded in ascending powers of \(x\), \(N_{1}\) of the coefficients leave a remainder of 1 when divided by 3, while \(N_{2}\) of the coefficients leave a remainder of 2 when divided by 3. Find \(N_{1} - N_{2}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
1. Naomi has a broken calculator. All it can do is either add one to the previous answer, or square the previous answer. (It performs the operations correctly.) Naomi starts with 2 on the screen. In how many ways can she obtain an answer of 1000 ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{128}
The structure consists of two weightless rods \(AB\) and \(CD\), a weightless pulley, and a load with mass \(m = 3\) kg, which is suspended at point \(G\). All cords are weightless and inextensible. The rod is connected to the hinge at point \(A\), which allows the rod to rotate within the plane of the drawing. The lengths are \(CG = \frac{1}{4} CD\) and \(AE = \frac{1}{4} AB\), and point \(F\) is the midpoint of rod \(AB\). The acceleration due to gravity is \(g = 10 \ \mathrm{m} / \mathrm{s}^2\). Determine the tension force \(T_{0}\) in the cord attached to rod \(AB\) at point \(E\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15 \, \text{N}}
If we also want to know whether the counterfeit coin is lighter or heavier than the others in the previous problem, what is the maximum number of coins that can be determined with three weighings?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
An elliptic curve $E: y^2 = x^3 + bx + c$ is defined over the integers modulo $221 = 13 \cdot 17$. Points $P$ and $Q$ on $E$ satisfy $11P = \mathcal{O}$ and $7Q = \mathcal{O}$, where $\mathcal{O}$ is the point at infinity. Determine the number of points on $E$, denoted as $\sharp E$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{231}
在平面直角坐标系内, $M \in\left\{(x, y) \left\lvert\, \frac{x^{2}}{200}+\frac{y^{2}}{8} \leq 1\right.\right\}, A(2,1)$, 若 $\Delta O M A$ 的面积不超过 3, 则满足条件的整点 $M$ 个数为 $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{65}
Linda was gathering eggs from her chicken coops. When she was done, she had three times as many white eggs as brown eggs, and she had an equal number of small, medium, and large eggs. Additionally, she collected 6 green eggs which were all medium-sized. On her way back to the house, she stumbled, dropping her egg basket. Upon inspecting the damage, she found that all 20 of the brown eggs survived - 10 large, 5 small, and 5 medium, but now she only had 35 eggs left. How many eggs did Linda break?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{51}
Given that the circumcircle of $\triangle ABC$ has a radius of $1$, with its center at $O$, and $3 \overrightarrow{OA} + 4 \overrightarrow{OB} + 5 \overrightarrow{OC} = \overrightarrow{0}$, find the measure of $\angle AOB$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\pi}{2}}
Compute the sum of all positive real numbers \(x \leq 5\) satisfying \(x=\frac{\left\lceil x^{2}\right\rceil+\lceil x\rceil \cdot\lfloor x\rfloor}{\lceil x\rceil+\lfloor x\rfloor}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{85}
If \( f \) is a function such that \( f(a+b) = \frac{1}{f(a)} + \frac{1}{f(b)} \), find all possible values of \( f(2011) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pm \sqrt{2}}
In an $n \times 6$ rectangular grid with $6n$ unit squares, each unit square is filled with either a 0 or a 1. A filling method is called an "N-type filling" if it ensures that there is no rectangular subgrid whose four corners have the same number. Otherwise, the filling method is called a "Y-type filling." Find the smallest positive integer $n$ such that no matter how the numbers are filled, the filling method will always be a "Y-type filling."
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
The 30 edges of a regular icosahedron are distinguished by labeling them $1,2,…,30$. How many different ways are there to paint each edge red, white, or blue such that each of the 20 triangular faces of the icosahedron has two edges of the same color and a third edge of a different color? The standard answer format is $a^{b} c^{d}$. Please give the value of a + b + c + d.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{35}
In $\triangle ABC$, if $\angle B=30^\circ$, $AB=2 \sqrt {3}$, $AC=2$, find the area of $\triangle ABC$\_\_\_\_\_\_．
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\sqrt {3}}
在空间直角坐标系 $x y z$ 中，先以 z 轴为旋转轴旋转 $\frac{\pi}{3}$ ，再以 x 轴为旋转轴旋转 $\frac{\pi}{3}$ ，两次操作后等效于绕某一轴旋转，旋转角为 $\theta(\theta \in(0, \pi])$ ，则
$\left[\tan ^{2} \theta\right]=$ $\qquad$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{63}
The average score of 60 students is 72. After disqualifying two students whose scores are 85 and 90, calculate the new average score for the remaining class.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{71.47}
In how many ways can the number 1500 be represented as a product of three natural numbers? Variations where the factors are the same but differ in order are considered identical.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32}
对于非负整数，函数f定义如下：

$f ( 0 ) = 0 , f ( x ) = f ( [ \frac { x } { 1 0 } ] ) + [ \lg \frac { 1 0 } { x - 1 0 [ \frac { x - 1 } { 1 0 } ] } ] ( x > 0 )$.问：f(x)在$0\leq x\leq 1996$中取最大值时，x的值是多少？
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1111}
Let the universal set be U = {1,2,3,4,5,6,7,8,9,10}. Given the conditions: 
1. A is a subset of U.
2. If x is in A, then 2x is not in A.
3. If x is not in A, then 2x is not not in A.
Calculate the number of sets A that satisfy all three conditions simultaneously.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32}
In a certain candy store, 22% of the customers are caught sampling the candy and are charged a small fine. However, 10% of the customers who sample the candy are not caught. Additionally, 15% of the customers who are caught sampling candy are repeat offenders, and 35% of the customers who sample the candy end up buying additional items after sampling. What is the total percent of all customers who sample candy, including those who are caught, not caught, repeat offenders, and those who buy additional items?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24.44\%,}
Given a real number \(a\) such that there exists exactly one square whose four vertices lie on the curve \(y = x^3 + a x\). Determine the side length of this square.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt[4]{72}}
Find the number of positive integers $n \le 600$ whose value can be uniquely determined when the values of $\left\lfloor \frac n4\right\rfloor$, $\left\lfloor\frac n5\right\rfloor$, and $\left\lfloor\frac n6\right\rfloor$ are given, where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to the real number $x$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{80}
In a geometric sequence of real numbers, the sum of the first $2$ terms is $7$, and the sum of the first $6$ terms is $91$. The sum of the first $4$ terms is
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$28$}
In $\triangle ABC$, $H$ is the orthocenter, $BC = a$, the circumradius of $\triangle ABC$ is $R$, and $a = 2AH - 2R$. Find the value of $\sin A$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sin A = \frac{3}{5}}
In triangle $ABC$, $AB=10$, $BC=12$ and $CA=14$. Point $G$ is on $\overline{AB}$, $H$ is on $\overline{BC}$, and $I$ is on $\overline{CA}$. Let $AG=s\cdot AB$, $BH=t\cdot BC$, and $CI=u\cdot CA$, where $s$, $t$, and $u$ are positive and satisfy $s+t+u=3/4$ and $s^2+t^2+u^2=3/7$. The ratio of the area of triangle $GHI$ to the area of triangle $ABC$ can be written in the form $x/y$, where $x$ and $y$ are relatively prime positive integers. Find $x+y$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{295}
Find the greatest natural number  $n$  such there exist natural numbers  $x_{1}, x_{2}, \ldots, x_{n}$  and natural  $a_{1}< a_{2}< \ldots < a_{n-1}$  satisfying the following equations for  $i =1,2,\ldots,n-1$ : \[x_{1}x_{2}\ldots x_{n}= 1980 \quad \text{and}\quad x_{i}+\frac{1980}{x_{i}}= a_{i}.\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
8 men, working a certain number of hours a day can complete a work in 24 days. If 12 men work the same number of hours a day, they can complete the same work in 16 days. How many hours a day were the 8 men working?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
If the maximum value and the minimum value of the function \( f(x) = \frac{a + \sin x}{2 + \cos x} + b \tan x \) sum up to 4, then find \( a + b \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
已知${{2}^{2x}}-{{25}}={{2}^{x+2}}$，则$\lg \left( {{x2}}+1 \right)=$__________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$1$}
In the expansion of $(x-2{y}^{3}){(x+\frac{1}{y})}^{5}$, the coefficient of $x^{2}$ term is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-20}
令 $z_{k}=\cos \frac{2 k-2}{5} \pi+i \sin \frac{2 k-2}{5} \pi(k=1,2,3,4,5)$ ，若 $a_{i}=\prod_{i \neq j}\left(z_{i}-z_{j}\right)$ ，请计算 $a_{1} a_{2} a_{3} a_{4} a_{5}$ 的值，并将结果表示为一个整数。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3125}
A calculator treats angles as radians. It initially displays 1. What is the largest value that can be achieved by pressing the buttons `cos` or `sin` a total of 2001 times? (So you might press `cos` five times, then `sin` six times, and so on with a total of 2001 presses.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
13.364. The base of the power was increased by $k$ times, and the exponent was decreased by the same factor, as a result of which the power itself did not change. Find the base of the power that has such a property.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt[k-1]{k}}
Compute the integral $$\int_{|z|=1} \frac{z^7}{10z^8-5z^4+2z^2+iz+1}dz$$ given that the polynomial in the denominator has all its zeros inside the open unit disc.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$$\frac{2\pi i}{10}$$}
Given that $\alpha$ and $\beta$ are two interior angles of an oblique triangle, if $\frac{{\cos \alpha - \sin \alpha}}{{\cos \alpha + \sin \alpha}} = \cos 2\beta$, then the minimum value of $\tan \alpha + \tan \beta$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{1}{4}}
Evaluate the integral $\int_{\gamma}\frac{dz}{z^2+3}$ where $\gamma$ is the upper half of the unit circle $|z|=1$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Any six points are taken inside or on a rectangle with dimensions $2 \times 1$. Let $b$ be the smallest possible number with the property that it is always possible to select one pair of points from these six such that the distance between them is equal to or less than $b$. Determine the value of $b$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{5}}{2}}
To estimate the lower probability that the average burning duration of the selected 200 electric lamps differs from the average burning duration of the entire batch in absolute value by less than 5 hours, given that the standard deviation of the burning duration of lamps in each box is less than 7 hours.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.9902}
For positive real numbers \(a\), \(b\), and \(c\), compute the maximum value of
\[
\frac{abc(a + b + c + ab)}{(a + b)^3 (b + c)^3}.
\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{16}}
Aunt Gemma has six dogs with different sizes and appetites. She feeds them three times a day. The first dog consumes 250 grams of food, the second dog eats 350 grams, the third dog requires 450 grams, the fourth dog needs 550 grams, the fifth dog consumes 300 grams, and the sixth dog requires 400 grams per meal. If Aunt Gemma bought 2 sacks of dog food, each weighing 50 kilograms, and feeds the dogs their specified amounts at each meal, how many days will those sacks of food last?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
Let  $I$  be the incenter of  $\triangle ABC$ , and  $O$  be the excenter corresponding to  $B$ . If  $|BI|=12$ ,  $|IO|=18$ , and  $|BC|=15$ , then what is  $|AB|$ ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
记 M(100,\bR) 是所有 100 阶实方阵的集合，设 S=\{A\in M(100,\bR)\ |\ A^4+A^2=0\}. 称 A,B\in S 是相似的，如果存在可逆的 100 阶实方阵 P 使得 P^{-1}AP=B. 求 S 中互不相似的矩阵个数的最大值.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1326}
Let positive real numbers $x$ and $y$ satisfy $2x+y=1$. The minimum value of $\frac{8}{{x+1}}+\frac{1}{y}$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{25}{3}}
Given the function y = |log2(x)| with a domain of [1/m, n] (m, n are positive integers) and a range of [0, 2], calculate the number of integer pairs (m, n) that satisfy these conditions.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Let $ABC$ be a triangle with $AB = 26$, $BC = 51$, and $CA = 73$. Let $O$ be an arbitrary point in the interior of $\triangle ABC$. Lines $\ell_1$, $\ell_2$, and $\ell_3$ pass through $O$ and are parallel to $\overline{AB}$, $\overline{BC}$, and $\overline{CA}$, respectively. The intersections of $\ell_1$, $\ell_2$, and $\ell_3$ with the sides of $\triangle ABC$ form a hexagon whose area is $A$. Compute the minimum value of $A$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{280}
在$\triangle{ABC}$中，中线$\overline{AD}$和$\overline{BE}$在$G$点相交，且$\triangle{AGE}$是等边三角形。那么$\cos(C)$可以写成$\frac{m\sqrt{p}}{n}$的形式，其中$m$和$n$是互质的正整数，$p$是一个不被任何素数的平方整除的正整数。请计算$m+n+p$的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{44}
Given real numbers \( a, b, \) and \( c \) that satisfy
\[ f(x) = a \cos x + b \cos 2x + c \cos 3x \geq -1 \]
for any real number \( x \). What is the maximum value of \( a + b + c \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Initially, all edges of the  $K_{2024}$  are painted with  $13$  different colors. If there exist  $k$  colors such that the subgraph constructed by the edges which are colored with these  $k$  colors is connected no matter how the initial coloring was, find the minimum value of  $k$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
在边长为 8 的正方形 $A B C D$ 中, $M$ 为边 $B C$ 的中点, $N$ 为边 $D A$ 上一点, 且 $D N=3 N A$. 若对于常数 $m$ ，在正方形 $A B C D$ 的边上恰有六个不同的点 $P$ ，使得 $\overrightarrow{P M} \cdot \overrightarrow{P N}=m$ ，则实数 $m$ 的取值范围是 $($用不等式表示$)$，形如$a < m < b$，请给出$a + b$的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
If $x+y+z=30$, $3x+y-z=50$, and $x$, $y$, $z$ are non-negative real numbers, then the range of $M=5x+4y+2z$ is.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{120 \leq M \leq 130}
Two distinct squares of the 8 by 8 chessboard $C$ are said to be adjacent if they have a vertex or side in common. Also, $g$ is called a $C$-gap if for every numbering of the squares of $C$ with all the integers $1,2, \ldots, 64$ there exist two adjacent squares whose numbers differ by at least $g$. Determine the largest $C$-gap $g$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
A rhombus, where the shorter diagonal is equal to its side of length 1, rotates around a line passing through the end of the longer diagonal and perpendicular to this diagonal. Find the volume of the resulting solid of revolution.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3 \pi}{2}}
在数列 $\left\{a_1, a_2, \cdots\right\}=\{1,3,3,3,5,5,5,5,5, \cdots\}$ 中每个奇数 $k$ 出现 $k$ 次, 设有整数 $p, q, r$存在, 对所有正整数 $n$, 满足 $a_n=p[\sqrt{n+q}]+r$, 其中 $[x]$ 表示不大于 $x$ 的最大整数, 求 $p+q+$ $r$ 的值.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Given that the function $f$ satisfies the equation $f(xy) = f(x) + f(y)$ for any positive $x$ and $y$, and that $f\left(\frac{1007}{1}\right) = 1$, find the value of $f(2007)$. 
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\log_{1007} 2007}
Find the orthogonal projection of the vector $(1, 1, 1)$ onto the null space of the matrix $A$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(0, 0, 0)}
Two unbiased coins are tossed repeatedly. Determine the expected number of tosses required until the last five outcomes of both coins are identical. What is this expected number of tosses?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{62}
Find the smallest integer \( n \) such that \(|\sin x - (x - \frac{x^3}{6} + \frac{x^5}{120} - \frac{x^7}{5040} + \frac{x^9}{9!} - \frac{x^{11}}{11!})| < |\cos(c)/13!|\), for some \( c \in (-0.1, 0.1) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
An insurance policy covers two employees of a company, reimbursing no more than one loss per employee per year. The policy reimburses the full amount of the loss up to a company-wide maximum of $8000. The probability of an employee incurring a loss is 40%, independent of the other employee. Each loss amount is uniformly distributed between $1000 and $5000. Given that one employee has incurred a loss exceeding $2000, calculate the probability that the total losses will exceed the reimbursements.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2}{51}}
Find the generating function for the sequence \( B(n) \) defined by the recurrence relation: \[ B(n) = \sum_{a_1 + \cdots + a_k = n, a_i \geq 1} \frac{1}{k!} \binom{n}{a_1 \cdots a_k} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e^{e^x - 1}}
Given a square truth table of size $n^2$, an algorithm takes two arbitrary rows as input ($a$ and $b$ such that $a < b$) and performs a fixed number of logical operations ($s$) on them. The result is a series of Boolean values. To perform the above operations on all combinations of $a$ and $b$ requires $sn^2$ steps, which means the time complexity is $O(n^2)$. Now consider the case in which the result of each iteration as described above returns some unknown number of TRUE values $c$ such that $c < b - a - 1$ and exactly $c$ additional operations must be performed before starting over with a new $a$ and $b$. Determine the time complexity in the worst case.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{O(n^3)}
Evaluate the integral \( \int_{-\infty}^{\infty} (f(x+1) - f(x)) \, dx \) given that \( \lim_{x \to \infty} f(x) = 2 \) and \( \lim_{x \to -\infty} f(x) = 1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Find a number that is divisible by all integers less than or equal to half of itself, but not divisible by any integer greater than half of itself.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
In triangle ABC, where AB = AC and \( \angle ACB = 72^\circ \), point D is located inside the triangle such that \( \angle DBC = 42^\circ \) and \( \angle DCB = 54^\circ \). If line AD intersects BC at point E, determine the measure of \( \angle AEC \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{96^\circ}
Find the minimum value of the function \(f(x)=(\sin(x)+\cos(x)+\tan(x)+\cot(x)+\sec(x)+\csc(x))^2\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9 - 4\sqrt{2}}
Find the value of \( a \) such that the four planes given by the equations:
\[
\begin{cases}
α_1 :& x + 2ay + az = a^2 \\
α_2 :& x + ay + az = a \\
α_3 :& x + a^2 z = a^3 \\
α_4 :& ax + ay + a^2 z = a \\
\end{cases}
\]
do not have any point in common.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a \neq 0, 1, 2}