Context:
Compute
\[\sum_{1 \le a < b < c} \frac{1}{2^a 3^b 5^c}.\](The sum is taken over all triples $(a,b,c)$ of positive integers such that $1 \le a < b < c.$)The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.
Please reason step by step, and put your final answer within \boxed{}.


Context:
3. Find the largest integer $n$ with $\lambda(n)=12$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Péter took one stamp each of values 1, 2, 3, ..., 37 forints from his stamp collection. He wants to group them so that each group has the same total value of stamps. In how many ways can he do this?
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many numbers \( (x_1 x_2 \ldots x_{1998}) \) are there such that:

\[
(x_1 x_2 \ldots x_{1998}) = 7 \cdot 10^{1996} (x_1 + x_2 + \ldots + x_{1998})
\]

where \( (x_1 x_2 \ldots x_{1998}) \) shows a number with 1998 digits in the decimal system? Provide your answer as an integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Maria must work 8 hours each day, excluding her 45-minute lunch break. If she begins working at 7:25 A.M. and takes her lunch break at noon, what time will her workday end? Express your answer in 12-hour clock format (e.g., 4:10 P.M.).
Please reason step by step, and put your final answer within \boxed{}.


Context:
5.2. A boy presses a vertical rod against a rough horizontal surface with his thumb. Then he gradually tilts the rod, keeping the component of the force directed along the rod applied to its end unchanged. At an angle of inclination of the rod to the horizontal $\alpha=70^{\circ}$, the rod starts to slide along the surface. Determine the coefficient of friction between the surface and the rod, if in the vertical position the normal reaction force of the plane was 21 times the weight of the rod. Round your answer to the hundredths.
Please reason step by step, and put your final answer within \boxed{}.


Context:
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{}.


Context:
The derivative of the function $f(x)$ is given by $f'(x) = x(1 - x)$. Find the interval(s) over which $f(x)$ is monotonically increasing.
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Context:
Given the function $f(x)=x^{2}+ax+1$, if there exists $x\_0$ such that $|f(x\_0)| \leqslant \frac{1}{4}$ and $|f(x\_0+1)| \leqslant \frac{1}{4}$ hold simultaneously, then the range of the real number $a$ is $\_\_\_\_\_\_\_\_\_\_.$
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Context:
Let $k$ be a positive integer such that $1 + \frac{1}{2} + \frac{1}{3} + \ldots + \frac{1}{13} = \frac{k}{13!}$. Find the remainder when $k$ is divided by $7$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Jim starts with $80 in his investment portfolio. Over the next two years, he experiences the following growth rates and transactions:

Year 1:
- Q1: 15% growth
- Q2: 5% growth, Jim adds $28
- Q3: 6% growth, Jim withdraws $10
- Q4: 3% decline

Year 2:
- Q1: 10% growth, Jim adds $40
- Q2: 4% decline, Jim withdraws $20
- Q3: 2% growth, Jim adds $12
- Q4: 7% decline, Jim withdraws $15

Calculate Jim's final portfolio worth after two years from when he started, considering all transactions and growth/decline rates throughout the quarters.
Please reason step by step, and put your final answer within \boxed{}.


Context:
\( \mathrm{n} \) is a positive integer not greater than 100 and not less than 10, and \( \mathrm{n} \) is a multiple of the sum of its digits. How many such \( \mathrm{n} \) are there?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a Pascal distribution with parameters r = 3 and p, where 0 < p < 1, find the maximum value of p that satisfies the condition P(X=6) >= P(X=5). Express your answer as a single numerical value.
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Context:
In the independence test, suppose H0: variable X is unrelated to variable Y. Given the estimated probability p(k^2 >= 10.83) ≈ 0.001, what is the probability that variable X is related to variable Y? Express your answer as a percentage (%).
Please reason step by step, and put your final answer within \boxed{}.


Context:
For each positive integer $n$ , find the number of $n$ -digit positive integers that satisfy both of the following conditions:
$\bullet$ no two consecutive digits are equal, and
$\bullet$ the last digit is a prime.
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Context:
All the altitudes of the pyramid $ABCD$, whose faces are acute-angled triangles, are equal to each other. It is known that $AB=9$, $BC=13$, and the angle $ADC$ is $60^\circ$. Find the edge $BD$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a certain land only one or two - letter words are used. The parliament decided to forbid the use of the seventh letter. The people of this land have lost 129 words because of the prohibition. How many different letters does the local language have?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the sport of diving from a high platform, there is a functional relationship between the athlete's height above the water surface $h$ (m) and the time $t$ (s) after the jump: $h(t)=-4.9t^2+6.5t+10$. Determine the moment when the instantaneous velocity is $0 \text{ m/s}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Mark initially has 13 trees in his backyard. He decides to remove 3 trees due to their old age. Mark then buys 18 new trees from a nursery, but only plants 12 of them immediately, saving the rest for later. If Mark plants an additional 25% more trees than he initially planned, how many trees will he have in his backyard in total?
Please reason step by step, and put your final answer within \boxed{}.


Context:
4. (28th Russian Mathematical Olympiad) Find the smallest positive integer that can be represented as the sum of 2002 positive integers with equal sums of digits, and also as the sum of 2003 positive integers with equal sums of digits.
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Context:
59. Let $a, b$ be positive constants. $x_{1}, x_{2}, \cdots, x_{n}$ are positive real numbers. $n \geqslant 2$ is a positive integer. Find the maximum value of $y=$ $\frac{x_{1} x_{2} \cdots x_{n}}{\left(a+x_{1}\right)\left(x_{1}+x_{2}\right) \cdots\left(x_{n-1}+x_{n}\right)\left(x_{n}+b\right)}$. (1999 Polish Mathematical Olympiad Problem)
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Context:
There are 2 boys and 3 girls, making up a total of 5 students standing in a line. If boy A does not stand at either end, and exactly two of the three girls stand together, find the number of different arrangements.
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Context:
Dima wrote a sequence of 0's and 1's in his notebook. He then noticed that 1 follows 0 sixteen times, 0 follows 1 fifteen times, and 0 follows 01 eight times. How many times does 0 follow 11?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that $\binom{18}{8}=31824$, $\binom{18}{9}=48620$, and $\binom{18}{10}=43758$, calculate $\binom{20}{10}$.
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Context:
Let $A X B Y$ be a cyclic quadrilateral, and let line $A B$ and line $X Y$ intersect at $C$. Suppose $A X \cdot A Y=6, B X \cdot B Y=5$, and $C X \cdot C Y=4$. Compute $A B^{2}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Suppose that $x$ and $y$ are real numbers that satisfy the two equations: $x^{2} + 3xy + y^{2} = 909$ and $3x^{2} + xy + 3y^{2} = 1287$. What is a possible value for $x+y$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the sets $P=\{x|\log_{2}x < -1\}$ and $Q=\{x||x| < 1\}$, determine the set $P\cap Q$. Express your answer in interval notation.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The pairs of numbers $(a, b)$ such that $\left|a+b t+t^{2}\right| \leq 1$ for $0 \leq t \leq 1$ fill a certain region in the $(a, b)$-plane. What is the area of this region?The answer is in the form \frac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A sphere is inscribed in a cone. The ratio of the radius of the circle of tangency of the spherical and conical surfaces to the radius of the base of the cone is $k$. Find the cosine of the angle between the slant height of the cone and the plane of the base.
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Context:
The charge for a single room at hotel P is some percent less than the charge for a single room at hotel R and 10 percent less than the charge for a single room at hotel G. The charge for a single room at hotel R is 12.500000000000004 percent greater than the charge for a single room at hotel G. What is the percentage difference between the charges for a single room at hotel P and hotel R?
Please reason step by step, and put your final answer within \boxed{}.


Context:
As shown in the diagram, \(E, F, G, H\) are the midpoints of the sides \(AB, BC, CD, DA\) of the quadrilateral \(ABCD\). The intersection of \(BH\) and \(DE\) is \(M\), and the intersection of \(BG\) and \(DF\) is \(N\). What is \(\frac{S_{\mathrm{BMND}}}{S_{\mathrm{ABCD}}}\)?
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Context:
A car begins its journey traveling at a constant speed of 80 km/h for the first 120 km. Then, the car stops at a gas station for 15 minutes before resuming its trip at a constant speed of 100 km/h for the next 180 km. Finally, the car makes another stop at a checkpoint for 10 minutes before covering the last 75 km at a constant speed of 120 km/h. Given this information, what is the car's average speed for the entire journey, including stops?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $\alpha$ and $\beta$ be two interior angles of a triangle, and let $\tan \alpha$ and $\tan \beta$ be the two real roots of the equation $x^{2}+p(x+1)+1=0$. Find the measure of the obtuse angle of this triangle.
Please reason step by step, and put your final answer within \boxed{}.


Context:
From the set {0,1,2,3,4,5,6}, two distinct numbers a and b are chosen to form the complex number a+bi. How many imaginary numbers can be formed? Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In an acute triangle $ABC$ with an angle $\angle ACB = 75^\circ$, altitudes $AA_3$ and $BB_3$ intersect the circumscribed circle at points $A_1$ and $B_1$, respectively. On the lines $BC$ and $CA$, select points $A_2$ and $B_2$, respectively, such that the line $B_1B_2$ is parallel to the line $BC$ and the line $A_1A_2$ is parallel to the line $AC$. Let $M$ be the midpoint of the segment $A_2B_2$. Find in degrees the measure of the angle $\angle B_3MA_3$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A train travels from Albany to Syracuse, a distance of 120 miles, at an average rate of 50 miles per hour. The train then continues to Rochester, which is 90 miles from Syracuse, before returning to Albany. On its way to Rochester, the train's average speed is 60 miles per hour. Finally, the train travels back to Albany from Rochester, with the total travel time of the train, including all three legs of the journey, being 9 hours and 15 minutes. What was the average rate of speed of the train on the return trip from Rochester to Albany?
Please reason step by step, and put your final answer within \boxed{}.


Context:
James needs to get a new pair of glasses. His frames cost $200, the lenses cost $500, and he wants to add anti-glare and scratch-resistant coatings. Each coating costs $75, but they're currently running a promotion where if you get both, they will only charge for the more expensive coating. James's insurance covers 70% of the cost of lenses and 50% of the cost of the anti-glare coating, but it doesn't cover the scratch-resistant coating. He has a $50 off coupon that is only valid on frames, a 10% off loyalty discount for being a long-time customer, but this discount is only applicable to the total cost of the lenses and coatings. He also received a $20 cash reward from his last purchase, which he can use on any item. Lastly, there's a local sales tax of 7% that's added after all the discounts have been subtracted. How much will he have to pay in total, after all the discounts have been applied and including the sales tax?
Please reason step by step, and put your final answer within \boxed{}.


Context:
If 3 eighths of a number is 141, what will be the approximate value of 32.08% of this number?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral $\int_{\lvert z \rvert = 2} \frac{dz}{z \sin\left( \frac{1}{z} \right)}$ using residue theorem.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A, B, and C are lattice points in the plane. The triangle \( \triangle ABC \) contains exactly one lattice point, \( X \), in its interior. The line \( AX \) meets \( BC \) at \( E \). What is the largest possible value of \( \frac{AX}{XE} \)?
Please reason step by step, and put your final answer within \boxed{}.


Context:
For the curve  $ C: y = \frac {1}{1 + x^2}$ , Let  $ A(\alpha ,\ f(\alpha)),\ B\left( - \frac {1}{\alpha},\ f\left( - \frac {1}{\alpha} \right)\right)\ (\alpha > 0).$  Find the minimum area bounded by the line segments  $ OA,\ OB$  and  $ C,$  where  $ O$  is the origin.

Note that you are not allowed to use the integral formula of  $ \frac {1}{1 + x^2}$  for the problem.
Please reason step by step, and put your final answer within \boxed{}.


Context:
已知 \triangle A B C 为椭圆 \frac{x^{2}}{9}+\frac{y^{2}}{4}=1 的内接三角形，且 A B 过点 P(1,0) ，则 \triangle A B C 的面积的最大值为 \frac{m\sqrt{q}}{n} 的形式，请给出 m + n + q 的值。
Please reason step by step, and put your final answer within \boxed{}.


Context:
A positive number is mistakenly divided by $6$ instead of being multiplied by $6.$ Based on the correct answer, the error thus committed, to the nearest percent, is
Please reason step by step, and put your final answer within \boxed{}.


Context:
A type of bulrush grows 3 feet on the first day and subsequently decreases its growth by half each day; while a type of reed grows 1 foot on the first day and subsequently doubles its height each day. Using the reference data: lg 2 = 0.3010, lg 3 = 0.4771, calculate the number of days until the height of the bulrush and the reed are equal. Round your answer to 0.1.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $C$ be a circle centered at point $O$, and let $P$ be a point in the interior of $C$. Let $Q$ be a point on the circumference of $C$ such that $PQ \perp OP$, and let $D$ be the circle with diameter $PQ$. Consider a circle tangent to $C$ whose circumference passes through point $P$. Let the curve $\Gamma$ be the locus of the centers of all such circles. If the area enclosed by $\Gamma$ is $1/100$ the area of $C$, then what is the ratio of the area of $C$ to the area of $D$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Mary tried to improve her health by changing her diet. At first, she dropped a dozen pounds. Then, she added back some weight, then dropped three times more weight than she initially lost, and finally gained back half of a dozen pounds. She weighed 99 pounds at the start of her change in diet, and her final weight was 81 pounds. What is the ratio of the weight she added back to the weight she initially lost?
Please reason step by step, and put your final answer within \boxed{}.


Context:
设等比数列 $\left\{a_{n}\right\}$ 的前 $n$ 项和为 $S_{n}, S_{n}=2^{n}+r$ ( $r$ 为常数),记 $b_{n}=2\left(1+\log _{2} a_{n}\right)\left(n \in \mathbf{N}^{*}\right)$.若对于任意的正整数 $n$, 都有 $\frac{1+b_{1}}{b_{1}} \cdot \frac{1+b_{2}}{b_{2}} \cdots \frac{1+b_{n}}{b_{n}} \geqslant k \sqrt{n+1}$ 成立, 求实数 $k$ 的最大值的$2\sqrt{2}$倍是多少.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given vectors $\overrightarrow {a}$ and $\overrightarrow {b}$ that satisfy $( \overrightarrow {a}-2 \overrightarrow {b})⊥(3 \overrightarrow {a}+ \overrightarrow {b})$, and $| \overrightarrow {a}|= \frac {1}{2}| \overrightarrow {b}|$, find the angle between vectors $\overrightarrow {a}$ and $\overrightarrow {b}$.
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Context:
Four people were guessing the number, $N$, of jellybeans in a jar. No two guesses were equally close to $N$. The closest guess was 80 jellybeans, the next closest guess was 60 jellybeans, followed by 49 jellybeans, and the furthest guess was 125 jellybeans. Find the sum of all possible values for $N$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a speech contest held in a class, there are a total of $5$ contestants, including $3$ females and $2$ males. If the two males cannot appear consecutively, and female A cannot be the first, what is the number of possible sequences for their appearance?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In triangle $DEF$, $DE = 8$, $EF = 6$, and $FD = 10$.

[asy]
defaultpen(1);

pair D=(0,0), E=(0,6), F=(8,0);
draw(D--E--F--cycle);

label("\(D\)",D,SW);
label("\(E\)",E,N);
label("\(F\)",F,SE);

[/asy]

Point $Q$ is arbitrarily placed inside triangle $DEF$. What is the probability that $Q$ lies closer to $D$ than to either $E$ or $F$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A polynomial $P$ of degree $2015$ satisfies the equation $P(n) = \frac{1}{n^2}$ for $n = 1, 2, \dots, 2016$. Find $\lfloor 2017P(2017) \rfloor$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given an arithmetic sequence $\{a_{n}\}$ with the sum of the first $n$ terms as $S_{n}$, for all $n\in N^{*}$, $S_{n}\geqslant S_{3}$. Find the range of $\frac{{{a_6}}}{{{a_5}}}$.
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Context:
How to compute the sum of \begin{align*} \sum\limits_{k=0}^n \frac{(-1)^k}{k! (2k+1)} \frac{1}{(n-k)!} \end{align*}?
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Context:
A train of length 380 m moving at a speed of 72 kmph takes 91.9926405887529 seconds to cross another train of some length, moving at 36 kmph in the same direction. What is the length of the second train?
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Context:
If the oblique section diagram of a horizontally placed figure is an isosceles trapezoid with a base angle of $45^{\circ}$, legs and upper base all measuring $1$, then the area of the original plane figure is _____.
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Context:
For $a \in \mathbb R,a≠-1$,
$$\lim_{n\to\infty}\frac{1^a+2^a+\cdots +n^a}{(n+1)^{a-1}[(na+1)+(na+2)+(na+3)+\cdots+(na+n)]}=\frac{1}{60}$$
Find the value of $a$.
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Context:
In the geometric sequence {a_n}, it is known that a_1 = 1/5 and a_3 = 5. Find the value of a_2. Express your answer as a numerical value, including the ± symbol if necessary.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a sphere with a radius of 4 and two points \( A \) and \( B \) on its surface, where \( AB = 4\sqrt{2} \), and the center of the sphere is \( O \). If a moving point \( C \) on the sphere satisfies that the dihedral angle \( C-AB-O \) is \( 60^\circ \), find the radius of the circumscribed sphere of the tetrahedron \( OABC \).
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Context:
Given are 100 different positive integers. We call a pair of numbers "good" if the ratio of these numbers is either 2 or 3. What is the maximum number of good pairs that these 100 numbers can form? (A number can be used in several pairs.)
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Context:
11 Let $f(x)=a x^{3}+b x+c$ (where $a, b, c$ are real numbers), when $0 \leqslant x \leqslant 1$, $0 \leqslant f(x) \leqslant 1$. Find the maximum possible value of $b$.
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Context:
设 $a, b \in \mathbb{N}_+$，且满足 $\frac{1}{a}-\frac{1}{b}=\frac{1}{2021}$，则所有正整数对 $(a, b)$ 的个数为多少？
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Context:
Dani wrote the integers from 1 to \( N \). She used the digit 1 fifteen times. She used the digit 2 fourteen times.
What is \( N \) ?
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Context:
In a certain year, a specific date was never a Sunday in any month. Determine this date.
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Context:
Two trains of equal length are running on parallel inclined tracks in the same direction, with Train A moving at 42 km/hr and Train B moving at 36 km/hr. The incline of the tracks exerts an additional constant deceleration on both trains. Train A has a higher weight than Train B, and wind resistance is acting against both trains. Taking into account the deceleration due to the incline, the weight difference between the trains, and wind resistance, the faster Train A still manages to pass the slower Train B in 36 seconds. Determine the length of each train.
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Context:
Given a sphere of radius 1 with four non-coplanar points \( A, B, C, D \), and \( AB = CD = x \), \( BC = DA = y \), \( CA = BD = z \), what is the value of \( x^{2} + y^{2} + z^{2} \)?
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Context:
There are 28 garden gnomes in a yard. Three-fourths of them have red hats, and the rest have blue hats. Half the garden gnomes have big noses instead of small noses. If six gnomes with blue hats have big noses, and the rest of the gnomes are divided equally into those with striped shirts or with polka-dotted shirts, how many garden gnomes with red hats, small noses and striped shirts are present in the yard?
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Context:
在核酸检测中,"k合1"混采核酸检测是指:先将k个人的样本混合在一起进行1次检测，如果这k个人都没有感染新冠病毒，则检测结果为阴性，得到每人的检测结果都为阴性，检测结束；如果这k个人中有人感染新冠病毒，则检测结果为阳性，此时需对每人再进行1次检测，得到每人的检测结果，检测结束.现对100人进行核酸检测，假设其中只有2人感染新冠病毒，并假设每次检测结果准确.
$将这100人随机分成10组，每组10人，且对每组都采用“10合1”混采核酸检测. 已知感染新冠病毒的2人分在同一组的概率为\frac{1}{11}. 设X是检测的总次数，求X的数学期望E(X).$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the smallest possible positive integer \( n \) with the following property: For all positive integers \( x \), \( y \), and \( z \) such that \( x \mid y^{3} \) and \( y \mid z^{3} \) and \( z \mid x^{3} \), it always holds that \( x y z \mid (x+y+z)^{n} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Jeremy has two pieces of wood: one is 90 inches and the other is some length. He wants to cut both pieces of wood into smaller pieces so that all these pieces are the same length. He cuts both pieces of wood into 18-inch pieces. How long is the second piece of wood?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the smallest integer $n > 3$ such that, for each partition of $\{3, 4, \ldots, n\}$ into two sets, at least one of these sets contains three (not necessarily distinct) numbers $a, b, c$ for which $ab = c$.
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Context:
In acute triangle $A B C$, let $D, E$, and $F$ be the feet of the altitudes from $A, B$, and $C$ respectively, and let $L, M$, and $N$ be the midpoints of $B C, C A$, and $A B$, respectively. Lines $D E$ and $N L$ intersect at $X$, lines $D F$ and $L M$ intersect at $Y$, and lines $X Y$ and $B C$ intersect at $Z$. Find $\frac{Z B}{Z C}$ in terms of $A B, A C$, and $B C$.
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Context:
We have two regular pyramids with $2n$ sides $(n \geq 2)$ (base: regular polygon, sides: equal). Their congruent bases are aligned to form a convex polyhedron with triangular faces. What is the maximum number of sides a planar cross-section of the resulting polyhedron can have?
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Context:
Let $ABC$ be a triangle in the $xy$ plane, where $B$ is at the origin $(0,0)$. Extend $BC$ to $D$ such that $BC: CD = 1:1$, extend $CA$ to $E$ such that $CA: AE = 1:2$, and extend $AB$ to $F$ such that $AB: BF = 1:3$. Let $G(32,24)$ be the centroid of the triangle $ABC$ and $K$ be the centroid of the triangle $DEF$. Find the length $GK$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
10. A cylindrical container with a base radius of $1 \mathrm{~cm}$ contains four solid iron balls, each with a radius of $\frac{1}{2} \mathrm{~cm}$. The four balls are pairwise tangent, and the two balls at the bottom are tangent to the bottom of the container. Water is now poured into the container until the water level just covers all the iron balls. The amount of water needed is $\qquad$ $\mathrm{cm}^{3}$.
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Context:
A circle $\omega$ is circumscribed around triangle $ABC$. Tangents to the circle at points $A$ and $B$ intersect at point $K$. Point $M$ is the midpoint of side $AC$. A line passing through point $K$ parallel to $AC$ intersects side $BC$ at point $L$. Find the angle $AML$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that $F(-c,0)$ is the left focus of the hyperbola $\frac{{{x}^{2}}}{{{a}^{2}}}- \frac{{{y}^{2}}}{{{b}^{2}}}=1 (a > 0,b > 0)$, $A$ is the left vertex, and the line perpendicular to $AF$ intersects the hyperbola at points $B$ and $C$. The perpendicular lines $AB$ and $AC$ intersect at point $D$. If the distance from $D$ to the line $BC$ is $3(a+c)$, calculate the eccentricity of the hyperbola. Express your answer as a numerical value.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a tetrahedron \( A B C D \) such that \( A B = C D = 2a \), \( A C = B D = B C = A D = \sqrt{10} \), find the range of values for \( a \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
设实数 x_1,x_2,\cdots,x_{1997} 满足：(1) -\df{1}{\sqrt{3}}\leq x_i\leq\sqrt{3},i=1,2,\cdots,1997；(2) x_1+x_2+\cdots+x_{1997}=-318\sqrt{3}.求 x_1^{12}+x_2^{12}+\cdots+x_{1997}^{12} 的最大值.
Please reason step by step, and put your final answer within \boxed{}.


Context:
You can arrange 15 balls in the shape of a triangle, but you cannot arrange 96 balls in the shape of a square (missing one ball). Out of how many balls, not exceeding 50, can you arrange them both in the shape of a triangle and a square?
Please reason step by step, and put your final answer within \boxed{}.


Context:
From one point on a circle, two chords with lengths of 9 cm and 17 cm are drawn. Find the radius of the circle if the distance between the midpoints of these chords is 5 cm.
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Context:
In a square, there are $K$ points ($K>2$). What is the smallest number of triangles needed to divide the square so that each triangle contains no more than one point?
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Context:
In a row of 10 chairs, one of which is broken and cannot be used, Mary and James randomly select their seats. What is the probability that they do not sit next to each other?
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Context:
A pyramid of non-negative integers is constructed as follows:

1. The first row consists of only $0$.
2. The second row consists of $1$ and $1$.
3. The $n^{th}$ row (for $n > 2$) is an array of $n$ integers, where the leftmost and rightmost elements are equal to $n - 1$, and the interior numbers are equal to the sum of two adjacent numbers from the $(n - 1)^{th}$ row.

Let $S_n$ be the sum of numbers in the $n^{th}$ row. Determine the remainder when dividing $S_{2018}$ by $2018$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $f:\{1,2,\dots,2019\}\to\{-1,1\}$ be a function, such that for every $k\in\{1,2,\dots,2019\}$, there exists an $\ell\in\{1,2,\dots,2019\}$ such that $$ \sum_{i\in\mathbb{Z}:(\ell-i)(i-k)\geqslant 0} f(i)\leqslant 0. $$ Determine the maximum possible value of $$ \sum_{i\in\mathbb{Z}:1\leqslant i\leqslant 2019} f(i). $$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Sherry is waiting for a train. Every minute, there is a $75 \%$ chance that a train will arrive. However, she is engrossed in her game of sudoku, so even if a train arrives she has a $75 \%$ chance of not noticing it (and hence missing the train). What is the probability that Sherry catches the train in the next five minutes?
Please reason step by step, and put your final answer within \boxed{}.


Context:
The probability of randomly selecting 5 different numbers from the set {1, 2, ..., 20} such that at least two of them are consecutive.
Please reason step by step, and put your final answer within \boxed{}.


Context:
12. Using the digits $1,2,3, \cdots 9$, without repetition, form 3 three-digit numbers such that the sum of the first two equals the third, for example: $245+736=981$. The minimum value of the sum of these three three-digit numbers is $\qquad$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $P(x)=x^2-3x-9$. A real number $x$ is chosen at random from the interval $5\leq x \leq 15$. The probability that $\lfloor \sqrt{P(x)} \rfloor = \sqrt{P(\lfloor x \rfloor )}$ is equal to $\dfrac{\sqrt{a}+\sqrt{b}+\sqrt{c}-d}{e}$, where $a,b,c,d$ and $e$ are positive integers and none of $a,b,$ or $c$ is divisible by the square of a prime. Find $a+b+c+d+e$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the cells of a $3 \times 3$ square are the numbers $1, 2, 3, \ldots, 9$. It is known that any two consecutive numbers are located in adjacent (side-by-side) cells. What number can be in the central cell if the sum of the numbers in the corner cells is $18$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given any \( n \) points \( P_{1}, P_{2}, \cdots, P_{n} \) on a plane, where no three points are collinear, each point \( P_{i} \) (\(1 \leqslant i \leqslant n \)) is colored either red or blue. Let \( S \) be the set of triangles with vertices from \(\{ P_{1}, P_{2}, \cdots, P_{n} \}\) that satisfy the property: for any two line segments \( P_{i} P_{j} \) and \( P_{u} P_{v} \), the number of triangles in \( S \) that have \( P_{i} P_{j} \) as an edge is the same as the number of triangles that have \( P_{u} P_{v} \) as an edge. Find the smallest \( n \) such that \( S \) always contains at least two triangles, each of whose vertices are all the same color.
Please reason step by step, and put your final answer within \boxed{}.


Context:
\(\left(\left(\frac{\sqrt{a}-1}{\sqrt{a}+1}\right)^{-1} \cdot\left(\frac{\sqrt{a}-1}{\sqrt{a}+1}\right)^{1 / 2}-\sqrt{a-1} \cdot(\sqrt{a}+1)^{-1}\right)^{-2} \cdot \frac{1}{a^{2 / 3}+a^{1 / 3}+1}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
How can we calculate the probability that a number $n$ is prime if it passes the Fermat test $x$ times, excluding Carmichael numbers?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the limit of the following expression as $n$ approaches infinity: $\lim_{n \rightarrow \infty} \left( \sin \frac{1}{n} \cdot \sin \frac{2}{n} \cdot \ldots \cdot \sin 1 \right)^{\frac{1}{n}}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Four people went to a bookstore to buy books. Each person bought 4 different books, and each pair of people had exactly 2 books in common. How many different books did these 4 people buy at least?
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Context:
In the Cartesian coordinate system, the slope angle of the line $y=- \frac{ \sqrt{3}}{3}x+1$ is ______.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the rectangular coordinate system, points (8, 0) and a point (x, 0) both lie on circle C. The maximum possible value of the radius of C is 8. What is the value of x?
Please reason step by step, and put your final answer within \boxed{}.


Context:
若复数 $z_{1}, z_{2}$ 满足 $\left|z_{1}\right|=2,\left|z_{2}\right|=3,3 z_{1}-2 z_{2}=\frac{3}{2}-i$, 则 $z_{1} \cdot z_{2}$ 的原始答案是 $-\frac{a}{b} + \frac{c}{d} i$ 的形式，请给出a+b+c+d和的值。
Please reason step by step, and put your final answer within \boxed{}.


Context:
If line $l_1: (2m+1)x - 4y + 3m = 0$ is parallel to line $l_2: x + (m+5)y - 3m = 0$, determine the value of $m$.
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Context:
The expression  $\circ \ 1\  \circ \ 2 \ \circ 3 \ \circ \dots \circ \ 2012$  is written on a blackboard. Catherine places a  $+$  sign or a  $-$  sign into each blank. She then evaluates the expression, and finds the remainder when it is divided by 2012. How many possible values are there for this remainder?
Please reason step by step, and put your final answer within \boxed{}.


Context:
计算 $2+\sqrt{3}$ 和 $2-\sqrt{3}$ 的等比中项是多少？
Please reason step by step, and put your final answer within \boxed{}.


Context:
In triangle $\triangle ABC$, $(3\overrightarrow{AB}+2\overrightarrow{AC})\cdot \overrightarrow{BC}=0$, and for $t\in R$, the minimum value of $|\overrightarrow{BA}-t\overrightarrow{BC}|$ is $\frac{6}{5}|BC|$, then $\angle BAC=$____.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The inclination angle of the line $\sqrt{3}x + y - 1 = 0$ is ______.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given non-negative real numbers $x_{1}, x_{2}, \cdots, x_{n}$ satisfying the inequality $x_{1} + x_{2} + \cdots + x_{n} \leqslant \frac{1}{2}$, find the minimum value of $f\left(x_{1}, x_{2}, \cdots, x_{n}\right) = \prod_{i=1}^{n}\left(1 - x_{i}\right)$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
For $a > 0$, find the minimum value of $\int_{0}^{\frac{1}{a}}(a^{3}+4x-a^{5}x^{2})e^{ax}\ dx.$
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Context:
When you turn on a computer, a certain bit (0/1) in its memory is initialized to the value 0. Suppose that starting from when you turn on your computer the bit is rewritten at random instants, on average λ times per millisecond, and each rewrite results in a 1 independently with probability 𝑝 (or a 0 is written with probability (1 − 𝑝) when a rewrite occurs). What is the probability that after t milliseconds the bit stores the value 0?
Please reason step by step, and put your final answer within \boxed{}.

\boxed{e^{-\lambda t} + (1 - e^{-\lambda t})(1 - p)}


Context:
Given that $\sin(a + \frac{\pi}{4}) = \sqrt{2}(\sin \alpha + 2\cos \alpha)$, determine the value of $\sin 2\alpha$.
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Context:
Suppose that point $D$ lies on side $B C$ of triangle $A B C$ such that $A D$ bisects $\angle B A C$, and let $\ell$ denote the line through $A$ perpendicular to $A D$. If the distances from $B$ and $C$ to $\ell$ are 5 and 6 , respectively, compute $A D$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
When a tree tends to fall, forest rangers often use two wooden sticks to support the tree. This is an application of the triangle's ______ in real life.
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Context:
Determine the greatest positive integer $ n$ such that in three-dimensional space, there exist n points $ P_{1},P_{2},\cdots,P_{n},$ among $ n$ points no three points are collinear, and for arbitary $ 1\leq i < j < k\leq n$, $ P_{i}P_{j}P_{k}$ isn't obtuse triangle.
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the maximum $k$-dimensional volume ($V_{n,k}$) of the projection of an $n$-dimensional cube ($C_n$) with side length 1 onto a $k$-dimensional plane ($P_k$), where $k \leq n$? Specifically, focus on the case where $k = n - 1$. Provide a formula for $V_{n,k}$ when $k = n - 1$.
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Context:
For $\{1, 2, 3, \ldots, n\}$ and each of its non-empty subsets a unique <b>alternating sum</b> is defined as follows. Arrange the numbers in the subset in decreasing order and then, beginning with the largest, alternately add and subtract successive numbers. For example, the alternating sum for $\{1, 2, 3, 6,9\}$ is $9-6+3-2+1=5$ and for $\{5\}$ it is simply $5$. Find the sum of all such alternating sums for $n=7$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
If 1000 were expressed as a sum of at least three distinct powers of 2, what would be the least possible sum of the exponents of these powers?
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many ways to fill the board $ 4\times 4$ by nonnegative integers, such that sum of the numbers of each row and each column is 3?
Please reason step by step, and put your final answer within \boxed{}.


Context:
记$$F(x)=16 \int_{\frac{\pi^{2}}{4}+\cos \left(3 x+\frac{\pi}{2}\right)}^{\frac{25 \pi^{2}}{4}+2 x^{2}} \sin \sqrt{t} d t$$$F^{\prime}(0)$ 是 $F(x)$ 在 $x=0$ 处的导数. 求取整 $\left[F^{\prime}(0)\right]$ 的值.
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Context:
Nelson is having his friend drop his unique bouncy ball from a $12$ foot building, and Nelson will only catch the ball at the peak of its trajectory between bounces. On any given bounce, there is an $80\%$ chance that the next peak occurs at $\frac13$ the height of the previous peak and a $20\%$ chance that the next peak occurs at $3$ times the height of the previous peak (where the first peak is at $12$ feet). If Nelson can only reach $4$ feet into the air and will catch the ball as soon as possible, the probability that Nelson catches the ball after exactly $13$ bounces is $2^a \times 3^b  \times 5^c  \times 7^d  \times 11^e$ for integers $a, b, c, d$, and $e$. Find $|a| + |b| + |c| + |d| + |e|$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the function f(x) = 1/2 * e^(2x) + (a-e) * e^x - a*e + b, where a, b are real numbers and e is the base of the natural logarithm, and it has a maximum value at x = 1, find the range of a. Express your answer as an inequality.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the definite integral:

$$
\int_{\pi / 4}^{\arccos (1 / \sqrt{26})} \frac{d x}{(6 - \tan x) \sin 2 x}
$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
3. A function $g: \mathbb{N} \rightarrow \mathbb{N}$ satisfies the following:
(a) If $m$ is a proper divisor of $n$, then $g(m)<g(n)$.
(b) If $m$ and $n$ are relatively prime and greater than 1 , then
$$
g(m n)=g(m) g(n)+(n+1) g(m)+(m+1) g(n)+m+n \text {. }
$$

Find the least possible value of $g(2016)$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
If the positive solution for the fractional equation $\frac{a}{{x+3}}=\frac{1}{2}$ with respect to $x$ is $a$, then the range of values for $a$ is ____.
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Context:
How many non-congruent isosceles triangles, including equilateral triangles, have positive integer side lengths and a perimeter less than 20?
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Context:
Find the positive integer $k$ such that the roots of the polynomial $x^3 - 15x^2 + kx - 1105$ are three distinct collinear points in the complex plane.
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Context:
In an isosceles trapezoid, one base is \(40 \text{ cm}\) and the other is \(24 \text{ cm}\). The diagonals of this trapezoid are mutually perpendicular. Find its area.
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Context:
Given a sequence ${a_n}$ that satisfies ${a_{n+1}} + {a_n} = (n+1)\cos \frac{n\pi}{2}$ $(n\geqslant 2, n\in \mathbb{N^*})$, and ${S_n}$ is the sum of the first $n$ terms of the sequence. If ${S_{2017}} + m = 1012$ and ${a_1}m > 0$, find the minimum value of $\frac{1}{{a_1}} + \frac{1}{m}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
10. Centipedes and three-headed dragons. In a certain herd of centipedes and three-headed dragons, there are a total of 26 heads and 298 legs. Each centipede has one head. How many legs does a three-headed dragon have?
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Context:
In the convex quadrilateral ABCD, it is known that ∠DAB = ∠BCD = 90°, and ∠CDA : ∠ABC = 2 : 1, AD : CB = 1 : √3. Calculate the measure of ∠BDA in degrees.
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Context:
In a tennis tournament, 512 schoolchildren participate. For a win, 1 point is awarded, for a loss, 0 points. Before each round, pairs are drawn from participants with an equal number of points (those who do not have a pair are awarded a point without playing). The tournament ends as soon as a sole leader is determined. How many schoolchildren will finish the tournament with 6 points?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the surface integral of the second kind:

$$
\iint_{S}\left(x y^{2}+z^{2}\right) d y d z+\left(y z^{2}+x^{2}\right) d z d x+\left(z x^{2}+y^{2}\right) d x d y
$$

where \( S \) is the exterior surface of the upper ( \(z \geq 0 \) ) hemisphere \( x^{2}+ y^{2}+z^{2}=a^{2} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the closed form for a sum involving permutations, given by $nPk + (n-1)Pk + (n-2)Pk + \ldots + kPk$, where $n$ is the total number of items and $k$ is the number of items being permuted? For example, when $n=7$ and $k=2$, the sum is $7P2 + 6P2 + 5P2 + \ldots + 2P2$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The number $a=\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers, has the property that the sum of all real numbers $x$ satisfying
\[\lfloor x \rfloor \cdot \{x\} = a \cdot x^2\]
is $420$, where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$ and $\{x\}=x- \lfloor x \rfloor$ denotes the fractional part of $x$. What is $p+q$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
考虑正 60 边形的 60 个顶点，将其中 10 个染红色， 20 个染黄色，其余 30 个染蓝色. 求最小的正整数 $m$ ，使得总能找到不过任何顶点的 $m$ 条直线将平面分成若干区域，满足每个区域中所有点的颜色相同。
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Context:
All sides of the convex pentagon $ABCDE$ are of equal length, and $\angle A = \angle B = 90^{\circ}$. What is the degree measure of $\angle E$?
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Context:
A two-digit number is given in which the first digit is greater than the second. Its digits were rearranged to obtain a new two-digit number. The difference between the first and the second number turned out to be a perfect square. How many possible two-digit numbers satisfy this condition?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the diagram, square PQRS has side length 40. Points J, K, L, and M are on the sides of PQRS, so that JQ = KR = LS = MP = 10. Line segments JZ, KW, LX, and MY are drawn parallel to the diagonals of the square so that W is on JZ, X is on KW, Y is on LX, and Z is on MY. Find the area of quadrilateral WXYZ.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In Mrs. Dawson's rose garden, there are 30 rows of roses. In each row, there are 50 roses. The roses are planted in such a way that, going from left to right:

- 2/5 of the roses are red until a blue rose appears.
- After a blue rose, 1/4 of the remaining roses are white until two yellow roses appear together.
- After the two yellow roses, 3/8 of the remaining roses are purple until three orange roses appear together.
- The rest of the roses in the row are pink.

How many roses at Mrs. Dawson's rose garden are pink, considering the above color distribution pattern?
Please reason step by step, and put your final answer within \boxed{}.


Context:
已知数列 $\left\{a_{n}\right\}$ 满足 $a_{1}=2, a_{n+1}=2^{a_{n}}(n \in$ $\left.\mathbf{N}^{*}\right)$ ，其中数列 $\left\{b_{n}\right\}: b_{1}=5, b_{n+1}=5^{b_{n}}\left(n \in \mathbf{N}^{*}\right)$ ，则满足 $b_{m}>a_{25}$ 的最小正整数 $m$ 的值是 $\qquad$.
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Context:
In different historical periods, the conversion between "jin" and "liang" was different. The idiom "ban jin ba liang" comes from the 16-based system. For convenience, we assume that in ancient times, 16 liang equaled 1 jin, with each jin being equivalent to 600 grams in today's terms. Currently, 10 liang equals 1 jin, with each jin equivalent to 500 grams today. There is a batch of medicine, with part weighed using the ancient system and the other part using the current system, and it was found that the sum of the number of "jin" is 5 and the sum of the number of "liang" is 68. How many grams is this batch of medicine in total?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the sum of the real part and the imaginary part of the complex number $\frac{i}{1+i}-\frac{1}{2i}$. Express your answer as a single fraction or integer in the form $\boxed{[answer]}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
If $a_n=2n^2+\lambda n+3$ where $\lambda$ is a real constant and $n\in\mathbb{N}^*$, and the sequence $\{a_n\}$ is monotonically increasing, then the range of real $\lambda$ is ______.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Five balls are arranged around a circle. Chris chooses two adjacent balls at random and interchanges them. Then Silva does the same, with her choice of adjacent balls to interchange being independent of Chris's. What is the expected number of balls that occupy their original positions after these two successive transpositions?
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Context:
Given the quadratic equation $x^2 + px + q = 0$, where $p$ and $q$ are positive numbers, and the roots of this equation differ by $2$, find the value of $p$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Simplify $rac{1}{2+rac{2}{3}}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $ABCD$ be a trapezoid such that $AB \parallel CD$ and let $P = AC \cap BD$. Given $AB = 21$, $CD = 7$, $AD = 13$, and the area of trapezoid $[ABCD] = 168$. A line parallel to $AB$ through $P$ intersects the circumcircle of $\triangle BCP$ at $X$. The circumcircles of $\triangle BCP$ and $\triangle APD$ intersect at points $P$ and $Y$. Let $XY \cap BC = Z$. If $\angle ADC$ is obtuse, then $BZ = \frac{a}{b}$, where $a$ and $b$ are coprime positive integers. Compute $a + b$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let  $n$  be a positive integer,  $n \ge 2$ , and put  $\theta=\frac{2\pi}{n}$ . Define points  $P_k=(k,0)$  in the *xy*-plane, for  $k=1,2,\dots,n$ . Let  $R_k$  be the map that rotates the plane counterclockwise by the angle  $\theta$  about the point  $P_k$ . Let  $R$  denote the map obtained by applying in order,  $R_1$ , then  $R_2$ , ..., then  $R_n$ . For an arbitrary point  $(x,y)$ , find and simplify the coordinates of  $R(x,y)$ .
Please reason step by step, and put your final answer within \boxed{}.


Context:
The vertices of  $\triangle ABC$  are labeled in counter-clockwise order, and its sides have lengths  $CA = 2022$ ,  $AB = 2023$ , and  $BC = 2024$ .  Rotate  $B$   $90^\circ$  counter-clockwise about  $A$  to get a point  $B'$ .  Let  $D$  be the orthogonal projection of  $B'$  unto line  $AC$ , and let  $M$  be the midpoint of line segment  $BB'$ .  Then ray  $BM$  intersects the circumcircle of  $\triangle CDM$  at a point  $N \neq M$ .  Compute  $MN$ .
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that $e$ is the base of the natural logarithm, if for any $x \in [0,1]$, there exists a unique $y \in [-1,1]$ such that $x + y^2e^y - a = 0$ holds, then the range of the real number $a$ is _____.
Please reason step by step, and put your final answer within \boxed{}.


Context:
(4) Given a regular tetrahedron $P-ABC$ with the side length of the equilateral triangle base being 1, and the center $O$ of its circumscribed sphere satisfies $\overrightarrow{O A}+\overrightarrow{O B}+\overrightarrow{O C}=\overrightarrow{0}$, then the volume of this regular tetrahedron is $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given \( a, b \in \mathbf{R} \), the circle \( C_{1}: x^{2}+y^{2}-2 x+4 y-b^{2}+5=0 \) and the circle \( C_{2}: x^{2}+y^{2}-2(a-6) x-2 a y +2 a^{2}-12 a+27=0 \) intersect at two distinct points \( A\left(x_{1}, y_{1}\right) \) and \( B\left(x_{2}, y_{2}\right) \), and \( \frac{y_{1}+y_{2}}{x_{1}+x_{2}}+\frac{x_{1}-x_{2}}{y_{1}-y_{2}} = 0 \). Find the value of \( a \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Solve in positive integers the following equation:

$$ \left[ \sqrt{1} \right] + \left[ \sqrt{2} \right] + \left[ \sqrt{3} \right] + \ldots + \left[ \sqrt{x^2 - 2} \right] + \left[ \sqrt{x^2 - 1} \right] = 125, $$

where $[a]$ denotes the integer part of the real number $a$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A positive integer is considered **detestable** if the sum of its digits is a multiple of $11$. Determine how many positive integers below $10,000$ are detestable.
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Context:
When $0.73\overline{864}$ is expressed as a fraction in the form $\frac{y}{999900}$, what is the value of $y$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the smallest possible value of $x+y$ where $x, y \geq 1$ and $x$ and $y$ are integers that satisfy $x^{2}-29y^{2}=1$
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the Generic Math Tournament,  $99$  people participate. One of the participants, Alfred, scores 16th in Algebra, 30th in Combinatorics, and 23rd in Geometry (and does not tie with anyone). The overall ranking is computed by adding the scores from all three tests. Given this information, let  $B$  be the best ranking that Alfred could have achieved, and let  $W$  be the worst ranking that he could have achieved. Compute  $100B+W$ .
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Context:
Nine lines parallel to the base of a triangle divide the other sides each into  $10$  equal segments and the area into  $10$  distinct parts. If the area of the largest of these parts is  $38$, calculate the area of the original triangle.
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Context:
Let $P$ be a point not on line $XZ$ and $Q$ a point on line $XZ$ such that $PQ \perp XZ.$ Meanwhile, $R$ is a point on line $PZ$ such that $SR \perp PZ.$ If $SR = 5,$ $PQ = 6,$ and $XZ = 7,$ then what is the length of $PZ?$
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Context:
Let  $ABC$  be a non-equilateral triangle with integer sides. Let  $D$  and  $E$  be respectively the mid-points of  $BC$  and  $CA$  ; let  $G$  be the centroid of  $\Delta{ABC}$ . Suppose,  $D$ ,  $C$ ,  $E$ ,  $G$  are concyclic. Find the least possible perimeter of  $\Delta{ABC}$ .
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Context:
Danny is sitting on a rectangular box. The area of the front face of the box is half the area of the top face, and the area of the top face is some multiple of the area of the side face. The volume of the box is 192 and the area of the side face of the box is approximately 32. What is the ratio of the area of the top face to the side face?
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Context:
a、b为正整数，满足$\frac { 1 } { a } - \frac { 1 } { b } = \frac { 1 } { 2 0 1 8 }$，则所有正整数对（a，b）的个数为
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Context:
At the New Year's school party in the city of Lzheretsark, 301 students came. Some of them always tell the truth, while the rest always lie. Each of the 200 students said: "If I leave the hall, then among the remaining students, the majority will be liars." Each of the other students stated: "If I leave the hall, then among the remaining students, there will be twice as many liars as those who tell the truth." How many liars were at the party?
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Context:
3. There are 207 different cards with numbers $1,2,3,2^{2}, 3^{2}, \ldots, 2^{103}, 3^{103}$ (each card has exactly one number, and each number appears exactly once). In how many ways can 3 cards be chosen so that the product of the numbers on the chosen cards is a square of an integer divisible by 6?
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Context:
In quadrilateral \( \square ABCD \), \( AB = 4 \), \( BC = 2 \), and \( \angle DAB = 60^\circ \). A point \( P \) on side \( BC \) and a point \( Q \) on the extension of side \( DC \) satisfy \( S_{\triangle BPQ} = \frac{\sqrt{3}}{32} \). Find the minimum value of \( \overrightarrow{AP} \cdot \overrightarrow{BQ} \).
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Context:
Alice, Bob, Chad, and Denise decide to meet for a virtual group project between 1 and 3 PM, but they don't decide on a specific time. Each of the four group members signs on to Zoom at a uniformly random time between 1 and 2 PM, and they stay for 1 hour. The group gets work done whenever at least three members are present. What is the expected number of minutes that the group gets work done?
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Context:
-1200° is an angle in the    quadrant.
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Context:
A large sphere is on a horizontal field on a sunny day. At a certain time, the shadow of the sphere reaches out a distance of $10$ m from the point where the sphere touches the ground. At the same instant, a meter stick (held vertically with one end on the ground) casts a shadow of length $2$ m. What is the radius of the sphere in meters? (Assume the sun's rays are parallel and the meter stick is a line segment.) The original answer is in the form \( k\sqrt{m} - n \). Please give the value of \( k + m + n \).
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Context:
Let \( n \) (\( \geqslant 3 \)) be a positive integer, and \( M \) be an \( n \)-element set. Find the largest positive integer \( k \) that satisfies the following condition:

There exists a family \( \psi \) of \( k \) three-element subsets of \( M \) such that the intersection of any two elements in \( \psi \) (note: the elements of \( \psi \) are all three-element sets) is nonempty.
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Context:
There are 100 participants in an international StarCraft championship. The tournament is single-elimination, meaning in each match, two players compete, the loser is eliminated, and the winner continues. Find the maximum possible number of participants who won exactly two matches.
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Context:
A unit square \(ABCD\) and a circle \(\Gamma\) have the following property: if \(P\) is a point in the plane not contained in the interior of \(\Gamma\), then \(\min (\angle APB, \angle BPC, \angle CPD, \angle DPA) \leq 60^\circ\). The minimum possible area of \(\Gamma\) can be expressed as \(\frac{a \pi}{b}\) for relatively prime positive integers \(a\) and \(b\). Compute \(100a+b\).
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Context:
30. In each of the following 6-digit positive integers: $555555,555333,818811$, 300388, every digit in the number appears at least twice. Find the number of such 6-digit positive integers.
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Context:
 The population of a certain bacteria is known to double every 10 hours. Assuming exponential growth, determine the time that it would take for the bacteria to triple in number.
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Context:
Given a parabola \(C\) with the center of ellipse \(E\) as its focus, the parabola \(C\) passes through the two foci of the ellipse \(E\), and intersects the ellipse \(E\) at exactly three points. Find the eccentricity of the ellipse \(E\).
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Context:
Let $k>0$ and define $$A=-\sqrt{2} \frac{\Gamma(\frac{k+1}{2})}{\Gamma(\frac{k}{2})}$$ and $$\gamma(k)=\frac{(k+1)A-3kA+2A^3}{(k-A^2)^{\frac{3}{2}}}.$$ Where $\Gamma$ is the Gamma function. What is $$\lim_{k \to 0^{+}} \gamma(k)?$$
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Context:
A regular hexagon is inscribed in an equilateral triangle. If the hexagon has an area of 12 square units, find the area of the equilateral triangle. Express your answer in square units as a whole number.
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Context:
A flea is jumping on the vertices of square \(ABCD\), starting from vertex \(A\). With each jump, it moves to an adjacent vertex with a probability of \(\frac{1}{2}\). The flea stops when it reaches the last vertex it has not yet visited. Determine the probability that each vertex will be the last one visited.
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Context:
Find the maximum of the expression
$$
|| \ldots|| x_{1}-x_{2}\left|-x_{3}\right|-\ldots\left|-x_{2023}\right|,
$$
where \( x_{1}, x_{2}, \ldots, x_{2023} \) are distinct natural numbers between 1 and 2023.
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Context:
In the diagram, $\triangle QRS$ is an isosceles right-angled triangle with $QR=SR$ and $\angle QRS=90^{\circ}$. Line segment $PT$ intersects $SQ$ at $U$ and $SR$ at $V$. If $\angle PUQ=\angle RVT=y^{\circ}$, find the value of $y$. Express your answer as a decimal value.
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Context:
38. In a tournament where each pair of teams played each other twice, 4 teams participated. For each win, two points were awarded, for a draw - one, and for a loss - 0. The team that finished in last place scored 5 points. How many points did the team that finished in first place score?
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Context:
How many distinct trees with exactly 7 vertices exist?
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Context:
Each pair  $(x, y)$  of nonnegative integers is assigned number  $f(x, y)$  according the conditions: $f(0, 0) = 0$ ; $f(2x, 2y) = f(2x + 1, 2y + 1) = f(x, y)$ , $f(2x + 1, 2y) = f(2x, 2y + 1) = f(x ,y) + 1$  for  $x, y \ge 0$ .
Let  $n$  be a fixed nonnegative integer and let  $a$ ,  $b$  be nonnegative integers such that  $f(a, b) = n$ . Decide how many numbers satisfy the equation  $f(a, x) + f(b, x) = n$ .
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Context:
Teams A, B, and C need to complete two projects, $A$ and $B$. The workload of project $B$ is $\frac{1}{4}$ more than the workload of project $A$. If teams A, B, and C work alone, they can finish project $A$ in 20 days, 24 days, and 30 days respectively. To complete these two projects simultaneously, team A is assigned to project $A$, and teams B and C work together on project $B$. After a few days, team C joins team A to complete project $A$. How many days did team C and team B work together?
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Context:
The middle school sold 6 more than two times the number of fair tickets as it did tickets to the baseball game. If 25 fair tickets were sold, how many baseball game tickets did the school sell?
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Context:
Given that \(\alpha, \beta \in\left(0, \frac{\pi}{2}\right)\), and
\[
\tan ^{2}(\alpha + \beta) + \cot ^{2}(\alpha + \beta) = 3\alpha - \alpha^{2} - \frac{1}{4}.
\]
Determine \(\beta\).
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Context:
If the function \( f(x) = \frac{(\sqrt{1008} x + \sqrt{1009})^{2} + \sin 2018 x}{2016 x^{2} + 2018} \) has a maximum value of \( M \) and a minimum value of \( m \), then \( M + m \) equals \_\_\_\_.
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Context:
Given that \( 2^{2013} < 5^{867} < 2^{2014} \), how many pairs of integers \((m, n)\) satisfy:
\[ 5^n < 2^m < 2^{m+2} < 5^{n+1} \]
where \( 1 \leq m \leq 2012 \).
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Context:
An auditorium with $20$ rows of seats has $10$ seats in the first row. Each successive row has one more seat than the previous row. If students taking an exam are permitted to sit in any row, but not next to another student in that row, find the maximum number of students that can be seated for an exam.
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Context:
Thirty nine nonzero numbers are written in a row. The sum of any two neighbouring numbers is positive, while the sum of all the numbers is negative. Is the product of all these numbers negative or positive?    (4 points)

                                                       Boris Frenkin
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Context:
Find the number of distinct roots of the equation $\overline{z} + z^2 = i(\overline{z} - z^2)$ in the set of complex numbers, where $\overline{z}$ represents the complex conjugate of $z$.
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Context:
Three cards are dealt at random from a standard deck of 52 cards. What is the probability that the first card is an Ace, the second card is a diamond, and the third card is a Jack?
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Context:
考虑 \ln(1+x+x^2+x^3) 在 x=0 处的泰勒展开，设 x^{2024} 项的系数为 a. 求 [|a|^{-1}].
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Context:
If 125% of j is equal to 25% of k, some percentage of k is equal to 50% of l, and 175% of l is equal to 75% of m, then 20% of m is equal to 700% of 100% of j. What is the percentage of k that is equal to 50% of l?
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Context:
Tom decides to lease a car. He drives 50 miles on Monday, Wednesday, and Friday, and Sunday for the rest of the days he drives 100 miles. He has to pay $.1 per mile he drives. He also has to pay a certain weekly fee. He has to pay $7800 in a year. How much is the weekly fee he has to pay?
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Context:
In the context of Number Theory, determine the explicit value of the sum $\sum_{n=1}^{9999}\frac{1}{(n+1)^{\frac{1}{4}}}$ using a method other than Abel's formula.
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Context:
What is the maximum number of white and black pieces that can be placed on a chessboard so that in each row and each column the number of white pieces is exactly twice the number of black pieces?
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Context:
一个盒中有 9 个正品和 3 个废品, 每次取一个产品, 取出后不再放回。则在取出 2个正品前已取出 3 个废品的概率为 $\qquad$.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
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Context:
设 $\mathbf{M}=\{1,2,3, \cdots, 1995\}, \mathbf{A}$ 是 $\mathbf{M}$ 的子集且满足条件: 当 $x \in \mathbf{A}$ 时, $15 x \notin \mathbf{A}$, 则 $\mathbf{A}$ 中元素的个数最多是
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Context:
定义在$R$上的函数$f\left( x \right)$满足$f\left( 0 \right)=0$，$f\left( x \right)+f\left( 1-x \right)=1$，$f\left( \frac{x}{5} \right)=\frac{1}{2}f\left( x \right)$，且当$0\le {{x}_{1}}<{{x2}}\le 1$时，$f\left( {{x}_{1}} \right)\le f\left( {{x2}} \right)$，则$f\left( \frac{1}{2019} \right)=$_______．
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Context:
A merchant has two types of goods. He marks up the first type of goods by 45% and then offers a discount of 15% on the marked price. For the second type of goods, he marks them up by 75% and then offers a discount of 30% on the marked price. If the merchant sells equal amounts of both types of goods, what is the overall percentage profit for the merchant after the discounts?
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Context:
5. Let's call the arrangement of $N$ grasshoppers on a line at various points of it $k$-successful if, after making the necessary number of moves according to the rules of leapfrog, the grasshoppers can achieve that the sum of pairwise distances between them decreases by at least $k$ times. For which $N \geqslant 2$ does there exist an arrangement that is $k$-successful for all natural $k$? (In leapfrog, on a move, one of the grasshoppers jumps to the point symmetric to it relative to another grasshopper.)

M. Svyatlovsky
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Context:
Let \[ f(N, r) \equiv \prod_{p \leq N} (1 + p^{-r}) \quad 0 < r < 1 \] where $p$ runs through all the primes less than or equal to $N$. What is the order of magnitude of $f(N, r)$ as $N \to \infty$?
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Context:
Susie babysits every day for 3 hours a day at the rate of $10 per hour. She spent 3/10 of the money she earned from last week to buy a make-up set. She then spent 2/5 of her money on her skincare products. How much is left from her earnings last week, in dollars?
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Context:
Determine the number that does not divide the expression $n^{2225} - n^{2005}$ for every integer value of $n$. Express your answer as a single integer.
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Context:
Evaluate the integral $$\iiint_E\sqrt{3x^2+3z^2}\,dV$$ where the solid $E$ is bounded by the paraboloid $y=2x^2+2z^2$ and the plane $y=8$.
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Context:
Arrange six positive integers \(a, b, c, d, e, f\) in a sequence in alphabetical order such that \(a=1\). If any positive integer is greater than 1, then the number that is one less than this integer must appear to its left. For example, \(1,1,2,1,3,2\) meets the requirement; \(1,2,3,1,4,1\) meets the requirement; \(1,2,2,4,3,2\) does not meet the requirement. Find the number of different sequences that meet the requirement.
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Context:
A school is adding 25 rows of seats to the auditorium. The first 10 rows will have 20 seats each, the next 10 rows will have 15 seats each, and the remaining 5 rows will have 10 seats each. Seat prices vary by section: each seat in the first 10 rows costs $60, each seat in the next 10 rows costs $50, and each seat in the remaining rows costs $40. A parent, being a seat manufacturer, offers a tiered discount based on seat types:

- For the $60 seats: 12% discount for each group of 20 seats purchased.
- For the $50 seats: 10% discount for each group of 15 seats purchased, and an extra 3% discount if 30 or more seats are purchased.
- For the $40 seats: 8% discount for every group of 10 seats purchased.

How much will the school pay for the new seats considering this tiered discount system with variable seat prices and discount rates?
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Context:
Given the circle $O: x^2 + y^2 = 1$ and the functions $f(x) = x^3$, $f(x) = \tan(x)$, and $f(x) = x\sin(x)$, how many of these functions have graphs that can bisect the area of circle $O$ based on their symmetry properties? Express your answer as a single integer.
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Context:
Let \( ABC \) be a triangle and \( D, E \), and \( F \) be points located on segments \( AC, BA \), and \( CB \) respectively, such that the segments \( AF, BD \), and \( CE \) concur at a point \( P \) inside the triangle. It is given that \( BP=6, PD=6, PC=9, PE=3 \), and \( AF=20 \). Find the area of triangle \( ABC \).
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Context:
5. In an isosceles trapezoid $M N K L$ with bases $M L$, $N K$ the diagonals are perpendicular to the sides $M N, \quad K L$ and intersect at an angle of $22.5^{\circ}$. Find the height of the trapezoid if the length $N Q=3$, where $Q-$ is the midpoint of the larger base.
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Context:
There are 128 positive integers q with 3 digits that when divided by 7 leave a certain remainder. What is the remainder?
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Context:
Let \( S = \{1, 2, 3, \ldots, 1990\} \), and let \( A \subset S \) with \( |A| = 26 \). If the sum of all elements in \( A \) is a multiple of 5, then \( A \) is called a "good subset" of 26 elements of \( S \). Find the number of "good subsets" of 26 elements of \( S \).
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Context:
已知欧拉常数\gamma=\displaystyle\lim_{n\rightarrow+\infty}\left(1+\df{1}{2}+\df{1}{3}+\cdots+\df{1}{n}-\ln n\right)约等于 0.5772156649. 记 I=\df{100}{\displaystyle\int_0^1\left(\df{1}{x}-\left[\df{1}{x}\right]\right)\dd x}，则 [I]=__________.
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Context:
Let the sequence $\{a_n\}$ have a sum of the first $n$ terms denoted by $S_n$. It is known that $4S_n = 2a_n - n^2 + 7n$ ($n \in \mathbb{N}^*$). Find $a_{11}$.
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Context:
Sandra has 5 pairs of shoes in a drawer, each pair a different color. Every day for 5 days, Sandra takes two shoes out and throws them out the window. If they are the same color, she treats herself to a practice problem from a past HMMT. What is the expected value (average number) of practice problems she gets to do?
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Context:
A 230 m long train running at a certain speed crosses another train running in opposite direction at the speed of 80 km/hr in 9 sec. The length of the other train is 270.04 m. What is the speed of the first train in km/hr?
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Context:
Find the number of solutions to the equation \(x = \left\lfloor \frac{x}{2} \right\rfloor + \left\lfloor \frac{x}{3} \right\rfloor + \left\lfloor \frac{x}{5} \right\rfloor\), where \(\left\lfloor x \right\rfloor\) denotes the greatest integer less than or equal to \(x\).
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Context:
Jonathan eats 2500 calories every day except for Saturday, when he consumes an extra 1000 calories. He burns a certain amount of calories every day. His weekly caloric deficit is 2500 calories. How many calories does Jonathan burn daily?
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Context:
The yearly changes in the population census of a city for five consecutive years are, respectively, 20% increase, 10% increase, 30% decrease, 20% decrease, and 10% increase. Calculate the net change over these five years, to the nearest percent.
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Context:
Given a parabola $y = ax^2 + bx + c$ ($a \neq 0$) with its axis of symmetry on the left side of the y-axis, where $a, b, c \in \{-3, -2, -1, 0, 1, 2, 3\}$. Let the random variable $X$ represent the value of $|a-b|$. Calculate the expected value $E(X)$.
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Context:
There are 10 individuals on a mysterious island, comprising both elves and wizards. Each person has a treasure chest containing a distinct number of gold coins ranging from 1 to 10. When asked about the number of gold coins in their chest, every wizard lies, while every elf tells the truth. If the reported numbers of gold coins are all within the range of 1 to 10 and the total sum of their reported gold coins is 36, then the island must have at least $\qquad$ wizards.
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Context:
In a circle with radius $6$, two chords $PQ$ and $RS$ intersect at a point $T$. Chord $RS$ bisects chord $PQ$ such that $PT = TQ = 8$ and $PQ$ is the only chord from $P$ that $RS$ bisects. It is known that the sine of the central angle subtended by minor arc $PR$ is a rational number. If expressed as a fraction $\frac{p}{q}$ in lowest terms, what is the product $pq$?
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Context:
Given a triangle $\triangle ABC$ with side length $b = 2$ and angle $B = 60^\circ$, there are two possible solutions for the triangle. Determine the range of possible values for side length $a$.
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Context:
How many bitstrings of length 77 are there that start with 010 (i.e., have 010 at positions 1, 2, and 3) or have 101 at positions 2, 3, and 4, or have 010 at positions 3, 4, and 5?
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Context:
(5) The set of integer points on the plane $S=\{(a, b) \mid 1 \leqslant a, b \leqslant 5, a, b \in \mathbf{Z}\}$, $T$ is a set of integer points on the plane, such that for any point $P$ in $S$, there exists a point $Q$ in $T$ different from $P$, such that the line segment $P Q$ has no other integer points except $P$ and $Q$. How many elements does $T$ need to have at minimum? (Provided by Yonggao Chen)
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Context:
In an examination, 35% of the students passed, 25% of the students scored in the "B" grade range, and 481 students failed. How many students appeared for the examination?
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Context:
一个六面体的各个面和一个正八面体的各个面都是边长为 $a$ 的正三角形, 这样两个多面体的内切球的半径之比是一个既约分数 $\frac{m}{n}$ .那么, 积 $m \cdot n$ 是  $\qquad$ .
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Context:
In the bathhouse, a bathtub that can be filled with water up to 10 liters (L) is filled with water for the guests. 3 guests used 1 liter (L) and 500 milliliters (㎖) of water from this bathtub, and 1 guest used 1 liter (L) and 750 milliliters (㎖) of water. If the remaining guests each use 1 liter (L) and 250 milliliters (㎖) of water, how many people can take shower?
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Context:
某市发出车牌号码均由 6 个数字 (从 0 到 9 )组成, 但要求任意 2 个车牌至少有 2 位不同（如车牌 038471 和 030471 不能同时使用）。 求该市最多能发出多少个不同车牌？
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Context:
Let \( P(x) \) be an \( n \)-degree polynomial such that
\[ P(x) = 2^{x}, \quad x = 1, 2, \cdots, n+1. \]

Find \( P(n+3) \).
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Context:
Find the number of ordered integer solutions (a, b, c) that satisfy the following system of equations:
|a+b| + c = 19 and ab + |c| = 97.
Express your answer as a single integer.
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Context:
The sum of the digits of the result of the expression $\underbrace{99 \cdots 99}_{2021 \text{ digits}} \times \underbrace{99 \cdots 99}_{2020 \text{ digits}}$ is $\qquad$
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Context:
$n$ 个棋手参加象棋比赛，每两个棋手比赛一局. 规定：胜者得 1 分，负者得 0 分，平局各得 0.5 分。如果赛后发现任何 $m$ 个棋手中都有一个棋手胜了其余 $m-1$ 个棋手，也有一个棋手输给了其余 $m-1$ 个棋手，就称此赛况具有性质 $P(m)$ 。对给定的 $m(m \geqslant 4)$ ，求 $n$ 的最小值 $f(m)$ ，使得对具有性质 $P(m)$ 的任何赛况，都有所有 $n$ 名棋手的得分各不相同。请提供当m=10时，f(m)的值。
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Context:
A merchant placed on display some dresses, each with a marked price. He then posted a sign "$1/3$ off on these dresses." The cost of the dresses was $3/4$ of the price at which he actually sold them. Then the ratio of the cost to the marked price was:
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Context:
Find the maximum value of the following expression:
$$
|\cdots||| x_{1}-x_{2}\left|-x_{3}\right|-x_{4}\left|-\cdots-x_{1990}\right|,
$$

where \( x_{1}, x_{2}, \cdots, x_{1990} \) are distinct natural numbers from 1 to 1990.
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Context:
Represent the number 1000 as a sum of the maximum possible number of natural numbers, the sums of the digits of which are pairwise distinct.
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Context:
设 $M=\{1,2, \cdots, 20\}, A_1, A_2, \cdots A_n$ 是 $M$ 的互不相同的非空子集, 当 $i \neq j$ 时, $A_i \cap A_j$ 至多有两个元, 求 $n$ 的最大值.
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Context:
If the sum of the units digit, tens digit, and hundreds digit of a four-digit number $M$ is $12$, then this four-digit number $M$ is called an "eternal number." By swapping the thousands digit and hundreds digit of the "eternal number" $M$, and swapping the tens digit and units digit, we obtain a new four-digit number $N$. Let $F(M)=\frac{{M-N}}{9}$. If the difference between the hundreds digit and units digit of an "eternal number" $M$ is exactly the thousands digit, and $\frac{{F(M)}}{9}$ is an integer, then the maximum value of $F(M)$ is ____.
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Context:
How many $5$-digit numbers $N$ (in base $10$) contain no digits greater than $3$ and satisfy the equality $\gcd(N,15)=\gcd(N,20)=1$? (The leading digit of $N$ cannot be zero.)
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Context:
Triangle $ABC$ has $\angle A = 90^\circ$, $\angle C = 30^\circ$, and $AC = 12$. Let the circumcircle of this triangle be $W$. Define $D$ to be the point on arc $BC$ not containing $A$ so that $\angle CAD = 60^\circ$. Define points $E$ and $F$ to be the feet of the perpendiculars from $D$ to lines $AB$ and $AC$, respectively. Let $J$ be the intersection of line $EF$ with $W$, where $J$ is on the minor arc $AC$. The line $DF$ intersects $W$ at $H$ other than $D$. The area of the triangle $FHJ$ is in the form $\frac{a}{b}(\sqrt{c}-\sqrt{d})$ for positive integers $a,b,c,d$, where $a,b$ are relatively prime, and the sum of $a,b,c,d$ is minimal. Find $a+b+c+d$.
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Context:
On an island of Liars and Knights, a circular arrangement is called correct if each person in the circle can say that among their two neighbors, there is at least one member of their tribe. One day, 2019 natives formed a correct circle. A liar approached them and said, "Now we too can form a correct circle." How many knights could have been in the initial arrangement?
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Context:
Given a right triangle $ABC$. On the extension of the hypotenuse $BC$, a point $D$ is chosen such that the line $AD$ is tangent to the circumcircle $\omega$ of triangle $ABC$. The line $AC$ intersects the circumcircle of triangle $ABD$ at point $E$. It turns out that the angle bisector of $\angle ADE$ is tangent to the circle $\omega$. In what ratio does point $C$ divide segment $AE$?
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Context:
Kayla draws three triangles on a sheet of paper. What is the maximum possible number of regions, including the exterior region, that the paper can be divided into by the sides of the triangles?
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Context:
Given that $F\_1$ and $F\_2$ are the left and right foci of the hyperbola $(x^{2}- \dfrac {y^{2}}{24}=1)$, and $P$ is a point on the hyperbola, find the area of $\triangle F\_1PF\_2$.
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Context:
12. B6 (FIN 3) Four distinct circles $C, C_{1}, C_{2}, C_{3}$ and a line $L$ are given in the plane such that $C$ and $L$ are disjoint and each of the circles $C_{1}, C_{2}, C_{3}$ touches the other two, as well as $C$ and $L$. Assuming the radius of $C$ to be 1 , determine the distance between its center and $L$.
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Context:
A TV set worth Rs. 35,000 is to be paid in 36 installments of Rs. 2,300 each. If the rate of interest is 9% per annum compounded monthly, and the first installment is paid at the time of purchase, with an additional processing fee of Rs. 1,000 added to the last installment, what will be the value of the last installment covering the interest, principal, and processing fee?
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Context:
Given the circle O: $x^2+y^2=4$, and the equation of line $l$ is $x+y=m$. If there are exactly three points on circle O which are at a distance of 1 from line $l$, then the real number $m$ equals \_\_\_\_\_.
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Context:
A three-digit number has different digits in each position. By writing a 2 to the left of this three-digit number, we get a four-digit number; and by writing a 2 to the right of this three-digit number, we get another four-digit number. The difference between these two four-digit numbers is 945. What is this three-digit number?
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Context:
Let $d_1 = a^2 + 2^a + a \cdot 2^{(a+1)/2} + a^3$ and $d_2 = a^2 + 2^a - a \cdot 2^{(a+1)/2} + a^3$. If $1 \le a \le 300$, how many integral values of $a$ are there such that $d_1 \cdot d_2$ is a multiple of $3$?
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Context:
How many pairs of positive integers $(a,b)$ are there such that the roots of the polynomial $x^2-ax-b$ are not greater than $5$? Provide the number of such pairs.
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Context:
$39$ students participated in a math competition. The exam consisted of $6$ problems and each problem was worth $1$ point for a correct solution and $0$ points for an incorrect solution. For any $3$ students, there is at most $1$ problem that was not solved by any of the three. Let $B$ be the sum of all of the scores of the $39$ students. Find the smallest possible value of $B$.
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Context:
Let $\triangle ABC$ be a triangle with $AB=65$, $BC=70$, and $CA=75$. A semicircle $\Gamma$ with diameter $\overline{BC}$ is constructed outside the triangle. Suppose there exists a circle $\omega$ tangent to $AB$ and $AC$ and furthermore internally tangent to $\Gamma$ at a point $X$. The length $AX$ can be written in the form $m\sqrt{n}$ where $m$ and $n$ are positive integers with $n$ not divisible by the square of any prime. Find $m+n$.
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Context:
Find the minimum value of the expression \((\sqrt{2(1+\cos 2x)} - \sqrt{3-\sqrt{2}} \sin x + 1) \cdot (3 + 2\sqrt{7-\sqrt{2}} \cos y - \cos 2y)\). If the answer is not an integer, round it to the nearest whole number.
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Context:
A metallic sheet with a thickness of 0.5 m is of rectangular shape with dimensions 48 m x 38 m. From each of its corners, a square is cut off so as to make an open box. If the length of the square is 8 m, taking into account the material thickness, determine the volume of the open box (in m³).
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Context:
An isosceles trapezoid has two rows of small identical isosceles triangles drawn along its top and bottom bases. Each small isosceles triangle has an area of 1. Find the area of the entire isosceles trapezoid.
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Context:
What is the largest finite number of roots that the equation

$$
\left|x-a_{1}\right| + \cdots + |x-a_{50}| = \left|x-b_{1}\right| + \cdots + |x-b_{50}|
$$

can have, where \( a_{1}, a_{2}, \dots, a_{50}, b_{1}, b_{2}, \dots, b_{50} \) are distinct numbers?
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Context:
Karen is constantly trying to stop the raccoons from getting in her trash. The first lock she tries stalls them for 5 minutes. The next lock stalls them for 3 minutes less than three times as long as the first lock. When Karen tries both locks at once, it stalled the raccoons for five times as long as the second lock alone. However, the raccoons are adapting to the locks, and with each attempt, they reduce the stall time of the first lock by 10% and the second lock by 15%. On the raccoons' third attempt to break through the locks, how long does it take them to get through both locks combined?
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Context:
Find a positive integer that is divisible by 15 and has a square root between 33 and 33.5.
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Context:
Given $a < 0$, and the terminal side of angle $\alpha$ passes through point $P(-3a, 4a)$, then $\sin\alpha + 2\cos\alpha = \_\_\_\_\_\_$.
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Context:
设方程(1+2+\cdots+k)-3(1+2+\cdots+t)=1的正整数解 (k,t) 按 k+t 从小到大排列为 (k_1,t_1), (k_2,t_2), \cdots. 求 k_6.
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Context:
Let \(\omega=-\tfrac{1}{2}+\tfrac{1}{2}i\sqrt{3}.\) Let \(S\) denote all points in the complex plane of the form \(a+b\omega+c\omega^2,\) where \(0\leq a \leq 1,0\leq b\leq 1,\) and \(0\leq c\leq 1.\) Find the area of \(S.\) The original answer is in the form \(\frac{k}{m}\sqrt{n}\), please give the value of k + m + n.
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Context:
In covering a distance of 30 km, Arun takes 2 hours more than Anil. If Arun changes his speed, then he would take 1 hour less than Anil. Arun's speed is 5 km/h. What is the ratio of Arun's new speed to his original speed?
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Context:
Let \( A \) be a set containing only positive integers, and for any elements \( x \) and \( y \) in \( A \), \(|x-y| \geq \frac{x y}{30}\). Determine at most how many elements \( A \) may contain.
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Context:
Find the value of the sum $$\sum_{k=0}^n \frac{(-1)^k}{k!} \left(\sum_{m=0}^n \binom nm m^k\right)$$ where $n$ is a positive integer.
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Context:
数列 $\left\{a_{n}\right\}$ 满足: $a_{0}=\sqrt{6}, a_{n+1}=\left[a_{n}\right]+\frac{1}{\left\{a_{n}\right\}}$ (其中 $\left[a_{n}\right]$ 和 $\left\{a_{n}\right\}$ 分别表示实数 $a_{n}$ 的整数部分与小数部分), 则 $a_{2020}=a+\sqrt{b}$，求$a+b$的值。
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Context:
设等差数列$\left\{ {{a}_{n}} \right\}$的前$n$项和为${{Sn}}$，且满足${{S}_{2017}}>0$，${{S2018}}<0$对任意$n\in {{N}^{*}}$，都有$\left| {{a}_{n}} \right|\ge \left| {{am}} \right|$，则$m$的值为__________．
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Context:
Find the number of positive integers less than $2018$ that are divisible by $6$ but are not divisible by at least one of the numbers $4$ or $9$.
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Context:
Let \( S = \{1, 2, \ldots, 98\} \). Find the smallest natural number \( n \) such that in any \( n \)-element subset of \( S \), it is always possible to select 10 numbers, and no matter how these 10 numbers are divided into two groups of five, there will always be a number in one group that is coprime with the other four numbers in the same group, and a number in the other group that is not coprime with the other four numbers in that group.
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Context:
In trapezoid $ABCD$, where $AD \parallel BC$ and $\angle ABC + \angle CDA = 270^{\circ}$, compute $AB^2$. Given that $AB \cdot \tan(\angle BCD) = 20$ and $CD = 13$.
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Context:
Let \( n \) and \( k \) be positive integers such that \( k \geq n \) and \( k - n \) is an even number. We are given \( 2n \) lamps numbered from 1 to \( 2n \), each of which can be either on or off. Initially, all the lamps are off. We are considering a sequence of steps, where each step consists of changing the state of one lamp (from on to off or from off to on).

Let \( N \) be the number of sequences of \( k \) steps that result in the lamps numbered from 1 to \( n \) being on and the lamps numbered from \( n+1 \) to \( 2n \) being off.

Let \( M \) be the number of sequences of \( k \) steps that result in the lamps numbered from 1 to \( n \) being on and the lamps numbered from \( n+1 \) to \( 2n \) being off, and no lamps numbered from \( n+1 \) to \( 2n \) are ever turned on at any point in the sequence.

Determine the value of the ratio \( N / M \).
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Context:
A baker bakes 5 loaves of bread an hour in one oven. He has 4 ovens. From Monday to Friday, he bakes for some hours, but on Saturday and Sunday, he only bakes for 2 hours. The baker bakes 1740 loaves of bread in 3 weeks. How many hours does the baker bake from Monday to Friday?
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Context:
Let \( C \) be the circle with radius 1 and center at the origin. A point \( P \) is chosen at random on the circumference of \( C \), and another point \( Q \) is chosen at random in the interior of \( C \). What is the probability that the rectangle with diagonal \( PQ \), and sides parallel to the x-axis and y-axis, lies entirely inside (or on) \( C \)?
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Context:
The perimeter of triangle \( \mathrm{ABC} \) is 1. A circle touches side \( \mathrm{AB} \) at point \( P \) and the extension of side \( \mathrm{AC} \) at point \( Q \). A line passing through the midpoints of sides \( \mathrm{AB} \) and \( \mathrm{AC} \) intersects the circumcircle of triangle \( \mathrm{APQ} \) at points \( X \) and \( Y \). Find the length of segment \( X Y \).
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Context:
Bob's password consists of a positive single-digit number followed by a letter and another positive single-digit number. What is the probability that Bob's password consists of an even single-digit number followed by a vowel (from A, E, I, O, U) and a number greater than 5?
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Context:
Given 10 positive integers, the sums of any 9 of them take exactly 9 different values: 86, 87, 88, 89, 90, 91, 93, 94, 95. After arranging these 10 positive integers in descending order, find the sum of the 3rd and the 7th numbers.
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Context:
A subscriber forgot the last digit of a phone number and therefore dials it randomly. What is the probability that they will have to dial the number no more than three times?
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Context:
A ticket contains six digits \(a, b, c, d, e, f\). This ticket is said to be "lucky" if \(a + b + c = d + e + f\). How many lucky tickets are there (including the ticket 000000)?
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Context:
将函数 $y=\sqrt{4+6 x-x^{2}}-2(x \in[0,6])$ 的图象逆时针方向旋转 $\theta(0 \leqslant \theta \leqslant \alpha)$, 得到曲线 $C$. 若对于每一个旋转角 $\theta$ ，曲线 $C$ 都是一个函数的图象，则 $\alpha$ 的最大值为多少？答案的形式为\arctan \frac{k}{m}，请给出k+m的值。
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Context:
Calculate the integral of the function $f(z) = \tan z$ around a rectangle in the complex plane with vertices at $0$, $2\pi$, $2\pi + 0.5i$, and $0.5i$. The rectangle extends from $0$ to $2\pi$ along the real axis and from $-0.5i$ to $0.5i$ along the imaginary axis.
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Context:
On a table, there are three cones standing on their bases, touching each other. The radii of their bases are 10, 15, and 15. A truncated cone is placed on the table with its smaller base down, which has a common slant height with each of the other cones. Find the area of the smaller base of the truncated cone.
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Context:
In a regular octagon inscribed in a circle of radius 1, if a point X is chosen on one of the diagonals such that it divides the diagonal in a 3:1 ratio, what is the absolute value of the product of the distances from X to all the vertices of the octagon?
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Context:
Each term of a geometric sequence $\left\{a_{n}\right\}$ is a real number, and the sum of the first $n$ terms is $S_{n}$. If $S_{10} = 10$ and $S_{30} = 70$, then find $S_{4n}$.
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Context:
已知正整数 n 的所有正因数排列为：1=d_1<d_2<d_3<\cdots，则在 1, 2, 3, \cdots, 2024 中使得 d_{10}=88 的所有数之和为__________.
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Context:
Calculate: \(\left[\left(11 \frac{1}{9}-3 \frac{2}{5} \times 1 \frac{2}{17}\right)-8 \frac{2}{5} \div 3.6\right] \div 2 \frac{6}{25}\).
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Context:
Given the hyperbola equation x^2/16 - y^2/9 = 1 and point P(4, 4), count the number of lines that pass through point P and intersect the hyperbola at only one point. Express your answer as a single integer.
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Context:
Evaluate the following limit without using L'Hopital's rule: $$\lim_{x\to0} \frac{\ln(1+\sin^2(x))\cdot\cot(\ln^2(1+x))}{x^2}$$
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Context:
Find the order of the element $34 + \langle 18 \rangle$ in the factor group $\mathbb Z_{72} /\langle 18 \rangle$.
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Context:
Let $P$ be a square pyramid whose base consists of the four vertices $(0, 0, 0), (3, 0, 0), (3, 3, 0)$, and $(0, 3, 0)$, and whose apex is the point $(1, 1, 3)$. Let $Q$ be a square pyramid whose base is the same as the base of $P$, and whose apex is the point $(2, 2, 3)$. Find the volume of the intersection of the interiors of $P$ and $Q$.
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Context:
During intervals, students played table tennis. Any two students played against each other no more than one game. At the end of the week, it turned out that Petya played half, Kolya played a third, and Vasya played a fifth of all the games played during the week. How many games could have been played during the week if it is known that Vasya did not play with either Petya or Kolya?
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Context:
In a single-round-robin tournament, 10 chess players are participating. What is the minimum number of rounds after which a sole winner can be determined prematurely? (In each round, the participants are paired.

Win - 1 point, draw - 0.5 points, loss - 0).

#
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Context:
James earns $20 an hour while working at his main job. He earns 20% less while working his second job. He works 30 hours at his main job, during which he spends 5 hours working overtime at a rate of 1.5 times his regular pay. James works 15 hours at his second job, as it is half the hours of his main job. He also has a side gig on weekends where he earns a fixed amount of $100 per day.

At the end of the week, James uses some tax deductions worth $200, which reduces his taxable income. The IRS tax rate for James' income is 18% and the state tax rate is 5%. In order to calculate James' weekly take-home pay, consider his earnings from the main job, including overtime, his second job, side gig income, tax deductions, and the applicable taxes. How much does James take home per week?
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Context:
We have $2012$ sticks with integer length, and sum of length is $n$. We need to have sticks with lengths $1,2,....,2012$. For it we can break some sticks ( for example from stick with length $6$ we can get $1$ and $4$).
For what minimal $n$ it is always possible?
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