Context:
P.G. Zhenoarove

In a box, there are 111 balls: red, blue, green, and white. It is known that if you pull out 100 balls from the box without looking, there will definitely be four balls of different colors among them. What is the smallest number of balls that need to be pulled out without looking to ensure that there are three balls of different colors among them?

#
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Context:
In how many distinct ways can I arrange my six keys on a keychain, if my house key must be exactly opposite my car key and my office key should be adjacent to my house key? For arrangement purposes, two placements are identical if one can be obtained from the other through rotation or flipping the keychain.
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Context:
Point $M$ lies on the leg $AC$ of the right triangle $ABC$ with a right angle at $C$, such that $AM=2$ and $MC=6$. Segment $MH$ is the altitude of triangle $AMB$. Point $D$ is located on the line $MH$ such that the angle $ADB$ is $90^{\circ}$, and points $C$ and $D$ lie on the same side of line $AB$. Find the length of segment $DC$ if the tangent of the angle $ACH$ is $1 / 7$. (16 points)
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Context:
Given the function f(x) = ax^2 - (2a+1)x + a + 1, find the range of real number x such that the inequality f(x) < 0 holds true for all a in [-1,1]. Express your answer as an interval or a range of values.
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Context:
Let $A B C$ be a triangle with $A B=7, B C=9$, and $C A=4$. Let $D$ be the point such that $A B \| C D$ and $C A \| B D$. Let $R$ be a point within triangle $B C D$. Lines $\ell$ and $m$ going through $R$ are parallel to $C A$ and $A B$ respectively. Line $\ell$ meets $A B$ and $B C$ at $P$ and $P^{\prime}$ respectively, and $m$ meets $C A$ and $B C$ at $Q$ and $Q^{\prime}$ respectively. If $S$ denotes the largest possible sum of the areas of triangles $B P P^{\prime}, R P^{\prime} Q^{\prime}$, and $C Q Q^{\prime}$, determine the value of $S^{2}$.
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Context:
As shown in the diagram, a cube with a side length of 12 cm is cut once. The cut is made along \( IJ \) and exits through \( LK \), such that \( AI = DL = 4 \) cm, \( JF = KG = 3 \) cm, and the section \( IJKL \) is a rectangle. The total surface area of the two resulting parts of the cube after the cut is \( \quad \) square centimeters.
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Context:
Find the value of the triple integral $$ \int_1^2\int_{\frac{\pi}{4}}^{\frac{\pi}{2}}\int_0^3 yz\cos(xz) \,dxdydz.$$
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Context:
Given the ellipse \( C: \frac{x^{2}}{25}+\frac{y^{2}}{9}=1 \) and a moving circle \( \Gamma: x^{2}+y^{2}=r^{2}(3<r<5) \). Let \( M \) be a point on the ellipse \( C \) and \( N \) be a point on the moving circle \( \Gamma \), such that the line segment \( MN \) is tangent to both the ellipse \( C \) and the moving circle \( \Gamma \). Find the maximum distance \( |MN| \) between points \( M \) and \( N \).
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Context:
An investor has three investments A, B, and C. The annual interest rates earned by these investments, compounded with varying frequencies, increased by 10 percent from last year to this year. The compounding frequencies for investments A, B, and C are quarterly, monthly, and semi-annually, respectively. If the combined effective annual interest rate earned by the investments this year was 11 percent, what was the combined effective annual interest rate for investments A, B, and C last year?
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Context:
Given the function $f(x) = \log_a(2a-x)$ is increasing on the interval $(0,1)$, find the range of values for $a$.
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Context:
Coins are arranged in a row from left to right. It is known that two of them are fake, they lie next to each other, the left one weighs 9 grams, the right one weighs 11 grams, and all the remaining are genuine and each weighs 10 grams. The coins are weighed on a balance scale, which either shows which of the two sides is heavier, or is in balance indicating the weights on both sides are equal. What is the maximum number \( n \) of coins for which it is possible to find the 11-gram coin in three weighings?
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Context:
What is the square root of 25? Express your answer using the ± symbol to denote both roots.
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Context:
A globe has 17 parallels and 24 meridians. Into how many parts is the surface of the globe divided?
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Context:
Consider the differential equation:

$$ xf''(x)+3x[f'(x)]^2=1-e^{-x}.$$

Let $h(x) = \frac{f'(x)}{x}$. Find the smallest constant $B$ such that $h(x) \leq Bx$ for all $x \geq 0$, given that $f(0) = f'(0) = 0$.
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Context:
What is the smallest value of $k$ for which it is possible to mark $k$ cells on a $9 \times 9$ board such that any placement of a three-cell corner touches at least two marked cells?
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Context:
Suppose $E, I, L, V$ are (not necessarily distinct) nonzero digits in base ten for which the four-digit number $\underline{E} \underline{V} \underline{I} \underline{L}$ is divisible by 73 , and the four-digit number $\underline{V} \underline{I} \underline{L} \underline{E}$ is divisible by 74 . Compute the four-digit number $\underline{L} \underline{I} \underline{V} \underline{E}$.
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Context:
Calculate the angle between the bisectors of the coordinate angles $x O y$ and $y O z$.
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Context:
Bored in an infinitely long class, Evan jots down a fraction whose numerator and denominator are both  $70$ -character strings, as follows:
\[ r = \frac{loooloolloolloololllloloollollolllloollloloolooololooolololooooollllol}
{lolooloolollollolloooooloooloololloolllooollololoooollllooolollloloool}. \]
If  $o=2013$  and  $l=\frac{1}{50}$ , find  $\lceil roll \rceil$ .
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Context:
The mayor commissioned two artists to paint 50 murals around the city. Once the work was completed, Celina was paid $1,000 more than some multiple of the amount Diego got. The mayor paid the two a total of $50,000. What is the multiple of Diego's payment that Celina received?
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Context:
How many subgroups of $S_4$ are isomorphic to the Klein's four group $K_4$?
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Context:
Assume a particle X is situated within a $d$-dimensional hypercube, with each dimension independently and uniformly distributed. The distance measure $D$ is given by $D = \frac{1}{2} \max_{i=1...d} |\frac{1}{2} - X_i|$. What is the probability $P(D > a)$?
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Context:
Given $a_{n}=\log_{(n+1)}(n+2)$ $(n\in\mathbb{N}^*)$, define the product $a_{1}\cdot a_{2}\ldots a_{k}$ as an integer for $k(k\in\mathbb{N}^*)$ called "ideal number", then the sum of all ideal numbers in the interval $[1,2015]$ is ______.
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Context:
The set of real values $a$ such that the equation $x^4-3ax^3+(2a^2+4a)x^2-5a^2x+3a^2$ has exactly two nonreal solutions is the set of real numbers between $x$ and $y,$ where $x<y.$ If $x+y$ can be written as $\tfrac{m}{n}$ for relatively prime positive integers $m,n,$ find $m+n.$
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Context:
$f(x)=\sum_{n=0}^{\infty} 2^{-n} \cos (n x)$, 记 $A=\int_{0}^{2 \pi} f^{2}(x) \mathrm{d} x$, 则 $[10A]$=
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Context:
Given a finite sequence of numbers where the sum of any 3 consecutive terms is negative and the sum of any 4 consecutive terms is positive, find the maximum number of terms in this sequence.
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Context:
A shopkeeper fixes the marked price of an item at a certain percentage above its cost price. The percentage of discount allowed to gain 8% is 16.92307692307692%. What is the percentage above the cost price at which the marked price is fixed?
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Context:
Given a sequence $\{a_{n}\}$ that satisfies the equation: ${a_{n+1}}+{({-1})^n}{a_n}=3n-1$ ($n∈{N^*}$), calculate the sum of the first $60$ terms of the sequence $\{a_{n}\}$.
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Context:
In a 10×10 grid square, the centers of all unit squares are marked (a total of 100 points). What is the minimum number of lines, not parallel to the sides of the square, needed to go through all the marked points?
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Context:
The flag of the United Nations consists of a polar map of the world, with the North Pole as center, extending approximately to $45^{\circ}$ South Latitude. The parallels of latitude are concentric circles with radii proportional to their co-latitudes. Australia is near the periphery of the map and is intersected by the parallel of latitude $30^{\circ} \mathrm{S}$. In the very close vicinity of this parallel, how much are East and West distances exaggerated as compared to North and South distances? The original answer is in the format $m\pi/n\sqrt{j}\sim2.42$, please provide the value of m + n + j.
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Context:
In the spring round of the 2000 City Tournament, high school students in country $N$ were presented with six problems. Each problem was solved by exactly 1000 students, and no two students solved all six problems together. What is the smallest possible number of high school students in country $N$ who participated in the spring round?
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Context:
设实数 $x ,y$ 满足 $\frac{x^{2}}{5}+\frac{y^{2}}{4}=1$, 求 $\sqrt{x^{2}+y^{2}-2 y+1}+\sqrt{x^{2}+y^{2}-2 x+1}$ 的最小值。答案的形式为 $k\sqrt{m} - \sqrt{n}$，请给出k+m+n的值。
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Context:
Given an integer  $n\ge 2$ , compute  $\sum_{\sigma} \textrm{sgn}(\sigma) n^{\ell(\sigma)}$ , where all  $n$ -element permutations are considered, and where  $\ell(\sigma)$  is the number of disjoint cycles in the standard decomposition of  $\sigma$ .
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Context:
In the finals of a beauty contest among giraffes, there were two finalists: the Tall one and the Spotted one. There are 135 voters divided into 5 districts, each district is divided into 9 precincts, and each precinct has 3 voters. Voters in each precinct choose the winner by majority vote; in a district, the giraffe that wins in the majority of precincts is the winner; finally, the giraffe that wins in the majority of districts is declared the winner of the final. The Tall giraffe won. What is the minimum number of voters who could have voted for the Tall giraffe?
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Context:
The number of distinct even numbers that can be formed using the digits 0, 1, 2, and 3.
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Context:
# Task 4.

Over two days, 50 financiers raised funds to combat a new virus. Each of them made a one-time contribution of a whole number of thousands of rubles, not exceeding 100. Each contribution on the first day did not exceed 50 thousand, while on the second day, it was more than this amount; and no pair of the 50 contributions differed by exactly 50 thousand. What amount was raised?
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Context:
Using the digits 0, 1, 2, 3, and 4, how many even numbers can be formed without repeating any digits?
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Context:
A square and a circle intersect so that each side of the square contains a chord of the circle equal in length to twice the radius of the circle. What is the ratio of the area of the square to the area of the circle? Express your answer as a common fraction in terms of $\pi$.
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Context:
Let $g$ be a function satisfying $g(x^2y) = g(x)/y^2$ for all positive real numbers $x$ and $y$. If $g(800) = 4$, what is the value of $g(7200)$?
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Context:
An isosceles trapezoid \(ABCD\) is circumscribed around a circle. The lateral sides \(AB\) and \(CD\) are tangent to the circle at points \(M\) and \(N\), respectively, and \(K\) is the midpoint of \(AD\). In what ratio does the line \(BK\) divide the segment \(MN\)?
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Context:
Let $ABC$ be a triangle with $AB = 4024$, $AC = 4024$, and $BC = 2012$. The reflection of line $AC$ over line $AB$ meets the circumcircle of $\triangle ABC$ at a point $D \neq A$. Find the length of segment $CD$.
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Context:
John and Roger can finish the work in 24 days if they work together. They worked together for 16 days and then Roger left. After Roger left, a third person, Anna, helped John for 8 days before leaving as well. John finished the remaining work in another 8 days. In how many days can John alone finish the work?
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Context:
The sequence \(\left\{a_{n}\right\}_{n \geq 1}\) is defined by \(a_{n+2}=7 a_{n+1}-a_{n}\) for positive integers \(n\) with initial values \(a_{1}=1\) and \(a_{2}=8\). Another sequence, \(\left\{b_{n}\right\}\), is defined by the rule \(b_{n+2}=3 b_{n+1}-b_{n}\) for positive integers \(n\) together with the values \(b_{1}=1\) and \(b_{2}=2\). Find \(\operatorname{gcd}\left(a_{5000}, b_{501}\right)\).
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Context:
Trevor is a comic book illustrator. In the past three months, he has drawn 220 pages of the comic. The comic he illustrates releases a new issue once a month. The first month, Trevor drew at a speed of 5 pages per day, while in the second and third months, he drew at a speed of 4 pages per day. The first and second months' issues were the same number of pages, but the third month's issue was four pages longer than either of the others. Additionally, each month's issue contained 3 bonus pages created by another artist. How many pages was the first issue of the comic, excluding the bonus pages?
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Context:
Let $N = 99999$. Then $N^3 = \ $
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Context:
Ten football teams played each other exactly once. As a result, each team ended up with exactly $x$ points.

What is the largest possible value of $x$? (A win earns 3 points, a draw earns 1 point, and a loss earns 0 points.)
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Context:
What is the limit of the following expression as $r$ approaches infinity?

$$\lim_{r \to \infty} \frac{\int_0^{\pi/2}\left(\frac{x}{\pi/2}\right)^{r-1}\cos x\,dx}{\int_0^{\pi/2}\left(\frac{x}{\pi/2}\right)^r\cos{x}\,dx}$$
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Context:
Given that the sum of the squares of two numbers $x$ and $y$ is 7, and their sum of cubes is 10, find the maximum value of $x+y$.
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Context:
Sixteen dots are arranged in a 4x4 grid. The distance between any two dots is defined as the minimum number of horizontal and vertical steps along the grid lines needed to travel from one dot to the other. For instance, two adjacent dots are a distance of 1 apart, and two dots at opposite corners of the grid are a distance of 6 apart.

The problem is to find the mean distance between two distinct dots in the grid, expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Determine the value of $m + n$.
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Context:
Find the least upper bound for the set of values \((x_1 x_2 + 2x_2 x_3 + x_3 x_4) / (x_1^2 + x_2^2 + x_3^2 + x_4^2)\), where \(x_i\) are real numbers, not all zero.
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Context:
Given the sample 7, 8, 9, x, y has an average of 8 and a standard deviation of $\sqrt{2}$, the value of xy is \_\_\_\_\_\_.
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Context:
In the city park, there are various attractions for kids and their parents. The entrance ticket to the park is $5, but for each attraction, you have to pay separately - such ticket costs $2 for kids and $4 for parents. A family with a certain number of children, their parents, and her grandmother paid $55 for visiting the park and one attraction inside. How many children were in the family?
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Context:
Meiyangyang is using two types of decorations to decorate a "Christmas tree". The decorations are hung on each vertex of small triangles. Meiyangyang can decorate in any order. If two vertices of a small triangle already have the same decoration, then the third vertex will have the other decoration; otherwise, the third vertex will have the same decoration. How many different designs of the "Christmas tree" are there? (No rotation or flipping is allowed)
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Context:
If the real part and the imaginary part of both $\frac{z}{20}$ and $\frac{20}{z}$ are between 0 and 1, what is the area of the geometric figure formed by such complex numbers $z$?
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Context:
Let $ABCD$ be a parallelogram with an area of 160. The diagonals $AC$ and $BD$ intersect at point $E$. Point $P$ is on the line segment $\overline{AE}$ such that $EC = 4EP$. If the line $DP$ intersects $AB$ at point $F$, find the area of quadrilateral $BFPC$.
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Context:
将一个正方形分割成有限个白色和红色矩形，其边均与正方形的边平行. 在每个白色矩形内，写下它的宽与高的比值；在每个红色矩形内，写下它的高与宽的比值. 我们计算出所有矩形内写的数的总和 x. 如果两种颜色矩形的总面积相同，求 x 的最小可能值.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:
Define a sequence $a_i$ as follows: \( a_1 = 181 \) and for \( i \ge 2 \), \( a_i = a_{i-1}^2 - 1 \) if \( a_{i-1} \) is odd and \( a_i = \frac{a_{i-1}}{2} \) if \( a_{i-1} \) is even. Find the least \( i \) such that \( a_i = 0 \).
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Context:
Four, (50 points) Let $X$ be a 56-element set. Find the smallest positive integer $n$, such that for any 15 subsets of set $X$, if the union of any seven of them has at least $n$ elements, then there must exist three subsets among these 15 such that their intersection is non-empty.
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Context:
Given the function $f(x)=\sin (ωx+ \frac {π}{3})(ω > 0)$, the graph of the function $y=f(x)$ is shifted to the right by $\frac {2}{3}π$ units and overlaps with the original graph. What is the minimum value of $ω$?
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Context:
A sequence is defined recursively as follows: \( t_{1} = 1 \), and for \( n > 1 \):

- If \( n \) is even, \( t_{n} = 1 + t_{\frac{n}{2}} \).
- If \( n \) is odd, \( t_{n} = \frac{1}{t_{n-1}} \).

Given that \( t_{n} = \frac{19}{87} \), find the sum of the digits of \( n \).

(From the 38th American High School Mathematics Examination, 1987)
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Context:
Among all positive integers \( n \) less than 2012, how many \( n \) are there such that \( 2^n - n^2 \) is divisible by 7?
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Context:
A positive integer has exactly $81$ divisors, which are arranged in a $9 \times 9$ table. In this table, for any two numbers in the same row or column, one of them divides the other. Find the maximum possible number of distinct prime divisors of $n$.
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Context:
Let $a$, $b \in \mathbb{R}$, $c \in [0,2\pi)$. If for any real number $x$, the equation $2\sin (3x- \frac {\pi}{3})=a\sin (bx+c)$ holds, determine the number of sets of parameters $(a, b, c)$ that satisfy this condition.
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Context:
At the beginning of the week, a toy car factory, Factory A, made 60 cars on Monday. Factory B, which is twice as efficient as Factory A, also started producing toy cars that day. From Tuesday to Friday, each factory doubled its production rate from the previous day, but on some days, defective products were found.

On Tuesday, Factory A had a 5% defect rate, and on Thursday, Factory B had a 3% defect rate. There were no defective items on the other days. How many non-defective toy cars did the factories make in total during the week?
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Context:
Given the sequence ${a_n}$ satisfying: $a_1=1$, $a_n=a_{n-1}^2+2a_{n-1}$ for $n\geqslant 2$, and let $b_n= \frac {1}{a_{n+1}}+ \frac {1}{a_{n}+2}$ for $n\in\mathbb{N^*}$, find the sum of the first $n$ terms of the sequence ${b_n}$, $S_n=$ ______.
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Context:
In the equilateral triangle \(ABC\), a circle is drawn passing through the center of the triangle and touching side \(BC\) at its midpoint \(D\). A line is drawn from point \(A\) tangent to the circle at point \(E\), such that \(\angle BAE < 30^{\circ}\). Find the area of triangle \(ABE\) if the area of triangle \(ABC\) is \(\frac{10}{4-\sqrt{2}}\).
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Context:
Three equally spaced parallel lines intersect a circle, creating three chords of lengths $38, 38,$ and $34$. What is the distance between two adjacent parallel lines? Provide your answer as an integer.
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Context:
Find the maximum number of colors used in coloring integers  $n$  from  $49$  to  $94$  such that if  $a, b$  (not necessarily different) have the same color but  $c$  has a different color, then  $c$  does not divide  $a+b$ .
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Context:
A rising number, such as $34689$, is a positive integer each digit of which is larger than each of the digits to its left. There are $\binom{9}{5} = 126$ five-digit rising numbers. When these numbers are arranged from smallest to largest, the $97^{\text{th}}$ number in the list does not contain the digit
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Context:
If $f(n)$ is the sum of the digits of $n^2+1$ (where $n \in \mathbb{N}^*$). For example, since $14^2+1=197$, and $1+9+7=17$, thus $f(14)=17$. Let $f_1(n)=f(n)$, $f_2(n)=f(f_1(n))$, ..., $f_{k+1}(n)=f(f_k(n))$, where $k \in \mathbb{N}^*$, then $f_{2005}(8)=$  ．
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Context:
Let \( A_{1}B_{1}C_{1}, A_{2}B_{2}C_{2} \), and \( A_{3}B_{3}C_{3} \) be three triangles in the plane. For \( 1 \leq i \leq 3 \), let \( D_{i}, E_{i} \), and \( F_{i} \) be the midpoints of \( B_{i}C_{i}, A_{i}C_{i} \), and \( A_{i}B_{i} \), respectively. Furthermore, for \( 1 \leq i \leq 3 \) let \( G_{i} \) be the centroid of \( A_{i}B_{i}C_{i} \). Suppose that the areas of the triangles \( A_{1}A_{2}A_{3}, B_{1}B_{2}B_{3}, C_{1}C_{2}C_{3}, D_{1}D_{2}D_{3}, E_{1}E_{2}E_{3} \), and \( F_{1}F_{2}F_{3} \) are 2, 3, 4, 20, 21, and 2020, respectively. Compute the largest possible area of \( G_{1}G_{2}G_{3} \).
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Context:
Consider functions $f$ that satisfy \[|f(x)-f(y)|\leq \frac{1}{2}|x-y|\] for all real numbers $x$ and $y$. Of all such functions that also satisfy the equation $f(300) = f(900)$, find the greatest possible value of
\[f(f(800))-f(f(400))?\]
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Context:
## Task 3 - 100733

Of the students in an 8th grade class, exactly 35 belong to the school choir and exactly 710 belong to the school sports club (SSG). Exactly 25 of the total number of students in this class are members of both the choir and the SSG.

Calculate what fraction of the total number of students in this class are neither in the choir nor in the SSG!
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Context:
If the line $2ax - by + 2 = 0$ ($a, b \in \mathbb{R}$) always bisects the circumference of the circle $(x+1)^2 + (y-2)^2 = 4$, what is the maximum value of $ab$?
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Context:
A rectangular grazing area is to be fenced off on three sides using part of a 100 meter rock wall as the fourth side. The area to be fenced is 36 m by 60 m. Fence posts are to be placed every 12 meters along the fence, including the two posts where the fence meets the rock wall. What is the minimum number of fence posts required to fence this area? Express your answer as a whole number.
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Context:
There are \( N \geq 5 \) natural numbers written on the board. It is known that the sum of all the numbers is 80, and the sum of any five of them is not more than 19. What is the smallest possible value of \( N \)?
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Context:
Determine an expression for the area of $\triangle QCA$ in terms of $p$, where the coordinates of point $A$ are now $(3, 12)$. [asy]
size(5cm);defaultpen(fontsize(9));
pair o = (0, 0); pair q = (0, 12); pair b = (12, 0);
pair a = (3, 12); pair t = (3, 0); pair c = (0, 9);

draw((-2, 0)--(15, 0), Arrow);
draw((0, -2)--(0, 15), Arrow);
draw(q--a--b);
draw(a--c--b);

label("$Q(0, 12)$", q, W);
label("$A(3, 12)$", a, NE);
label("$B(12, 0)$", b, S);
label("$O(0, 0)$", o, SW);
label("$x$", (15, 0), E);
label("$y$", (0, 15), N);
label("$C(0, p)$", c, W);
[/asy]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that the right focus of the ellipse $\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1(a>b>0)$ is $F(\sqrt{6},0)$, a line $l$ passing through $F$ intersects the ellipse at points $A$ and $B$. If the midpoint of chord $AB$ has coordinates $(\frac{\sqrt{6}}{3},-1)$, calculate the area of the ellipse.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find a formula for the following series:

$$ \sum_{i=1}^m {m \choose i} i! S(n,i)$$

Where $S(n,m)$ is the Stirling numbers of the second kind.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Two diameters and one radius are drawn in a circle of radius 1, dividing the circle into 5 sectors. The largest possible area of the smallest sector can be expressed as \(\frac{a}{b} \pi\), where \(a, b\) are relatively prime positive integers. Compute \(100a + b\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
A semicircle is inscribed in a triangle, with the semicircle touching the base and the diameter (with its ends on the lateral sides of the triangle) parallel to the base. Find the radius of the semicircle if the base of the triangle is $a$ and the height is $h$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Solomiya wrote the numbers $1, 2, \ldots, 2024$ on the board. In one move, she can erase any two numbers $a, b$ from the board and write the sum $a+b$ instead of each of them. After some time, all the numbers on the board became equal. What is the minimum number of moves Solomiya could make to achieve this?
Please reason step by step, and put your final answer within \boxed{}.


Context:
The Gnollish language consists of 3 words, "splargh", "glumph", and "amr". In a sentence, "splargh" cannot come directly before "glumph", and "amr" cannot come directly after "glumph". How many valid 3-word sentences are there in Gnollish?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let f(x) be an odd function defined on R with f(2) = 0, and xf''(x) + f(x) < 0 holds true for all x > 0. Determine the solution set of the inequality xf(x) > 0 in interval notation.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the area of the region enclosed by the curve $y=x^{2}-1$ and the lines $x=2$, $y=0$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $\mathbf{p}$ be the projection of vector $\mathbf{v}$ onto vector $\mathbf{u},$ and let $\mathbf{q}$ be the projection of $\mathbf{p}$ onto $\mathbf{u}.$ If $\frac{\|\mathbf{p}\|}{\|\mathbf{v}\|} = \frac{3}{4},$ then find $\frac{\|\mathbf{q}\|}{\|\mathbf{u}\|}.$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Suppose that $G$ is a group generated by elements $A$ and $B$, that is, every element of $G$ can be written as a finite "word" $A^{n_1}B^{n_2}A^{n_3}\cdots B^{n_k}$, where $n_{1},\cdots, n_k$ are any integers, and $A^0=B^0=1$ as usual. Also, suppose that $A^4=B^7= ABA^{-1}B=1$, $A^2\neq1$, and $B\neq1$. How many elements of $G$ are of the form $C^2$ with $C$ in $G$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
The number of edges of a convex polyhedron is 99. What is the maximum number of edges a plane, not passing through its vertices, can intersect?
Please reason step by step, and put your final answer within \boxed{}.


Context:
510 athletes participate in an arm wrestling tournament. For a victory, 1 point is awarded, for a defeat - 0 points. If a winner initially had fewer points than the opponent, they additionally receive one point from the loser. In each round, participants meet whose number of points differs by no more than 1. The tournament ends as soon as a sole leader is determined. What is the minimum number of rounds that need to be held?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In an exam, there are 4 multiple-choice questions, each with 3 possible answers. A group of students takes the exam, and it is found that for any 3 students, there is at least 1 question for which their answers are all different. What is the maximum number of students that could have taken the exam?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that the base- $17$  integer  $\overline{8323a02421_{17}}$  (where a is a base- $17$  digit) is divisible by  $\overline{16_{10}}$ , find  $a$ . Express your answer in base  $10$ .
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the ratio of length $AD$ to width $AB$ of the rectangle is $4:3$ and $AB$ is 40 inches, determine the ratio of the area of the rectangle to the combined area of the semicircles.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given three equilateral triangles $\triangle HOW$, $\triangle BOW$, and $\triangle DAH$ in a plane:
- $WO = 7$
- $AH = 2$

It is also given that the points $D$, $A$, and $B$ are collinear in that order.

Find the length of $BA$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many ways are there to arrange the letters of the word $\text{B}_1\text{A}_1\text{N}_1\text{A}_2\text{N}_2\text{A}_3\text{B}_2$, where three A's, two N's, and two B's are all considered different within each letter group but identical between groups?
Please reason step by step, and put your final answer within \boxed{}.


Context:
On a seven-day trip, Carrie drove 135 miles the first day, 124 miles more the second day, 159 miles the third day, 189 miles the fourth day, 210 miles the fifth day, 156 miles the sixth day, and 240 miles the seventh day. If her phone had a charging capacity that lasted for 106 miles initially, but increased by an increment of 15 miles every day, how many times did she charge her phone for the whole trip?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let us consider all rectangles with sides of length $a$ and $b$, both of which are whole numbers. Do more of these rectangles have a perimeter of $2000$ or a perimeter of $2002$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A five-digit number is called irreducible if it cannot be expressed as a product of two three-digit numbers.

What is the greatest number of consecutive irreducible five-digit numbers?
Please reason step by step, and put your final answer within \boxed{}.


Context:
The generatrix of a cone forms an angle $\alpha$ with the plane of its base, $\cos \alpha=\frac{1}{4}$. A sphere is inscribed in the cone, and a plane is drawn through the circle of contact between the sphere and the lateral surface of the cone. The volume of the part of the cone enclosed between this plane and the base plane of the cone is 37. Find the volume of the remaining part of the cone.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The Red Sox play the Yankees in a best-of-seven series that ends as soon as one team wins four games. Suppose that the probability that the Red Sox win Game \( n \) is \(\frac{n-1}{6}\). What is the probability that the Red Sox will win the series?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Points $D$ and $E$ are marked on the sides $AC$ and $BC$ of triangle $ABC$ respectively, such that $AD = EC$. It turns out that $BD = ED$ and $\angle BDC = \angle DEB$. Find the length of the segment $AC$, given that $AB = 7$ and $BE = 2$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
For how many positive integers $n \le 500$ is $n!$ divisible by $2^{n-2}$?

[i]Proposed by Eugene Chen[/i]
Please reason step by step, and put your final answer within \boxed{}.


Context:
设 $f(x)=\frac{1}{x^{3}-x}$, 求满足不等式 $f(2)+f(3)+\cdots+f(n)>\frac{499}{2020}$ 的最小正整数 $n$. 请给出$n$的值。
Please reason step by step, and put your final answer within \boxed{}.


Context:
We select a real number $\alpha$ uniformly and at random from the interval $(0,500)$. Define
\[ S = \frac{1}{\alpha} \sum_{m=1}^{1000} \sum_{n=m}^{1000} \left\lfloor \frac{m+\alpha}{n} \right\rfloor. \]
Let $p$ denote the probability that $S \ge 1200$. Compute $1000p$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider an infinite grid of equilateral triangles. Each edge (that is, each side of a small triangle) is colored one of \( N \) colors. The coloring is done in such a way that any path between any two nonadjacent vertices consists of edges with at least two different colors. What is the smallest possible value of \( N \)?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that $x=x_{1}$ and $x=x_{2}$ are respectively the local minimum and maximum points of the function $f\left(x\right)=2a^{x}-ex^{2}$ where $a \gt 0$ and $a\neq 1$. If $x_{1} \lt x_{2}$, then the range of values for $a$ is $\underline{\hspace{1cm}}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A 480 m long train is running at a speed of 55 km/hr. It crossed a platform of a certain length in 71.99424046076314 seconds. What is the length of the platform?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Rationalize the denominator of $\frac{5}{4\sqrt{7} + 3\sqrt{13}}$ and express your answer in the form $\frac{A\sqrt{B} + C\sqrt{D}}{E}$ where $B < D$, the fraction is in lowest terms, and all radicals are in simplest radical form. What is $A+B+C+D+E$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
设整数 n>4，(x+2\sqrt{y}-1)^n 的展开式中 x^{n-4} 与 xy 两项的系数相等，则 n 的值为________.
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the smallest possible area of a right-angled triangle whose hypotenuse contains the point \( M(1, 3) \) and whose legs lie on the lines \( y = x \) and \( y = -x \)?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A school organizes a trip for teachers and students. If they rent several 48-seat buses, they can fill them exactly. If they rent 64-seat buses, they can rent one less bus, and there will be one bus that is not fully occupied but is more than half full. Knowing that renting a 48-seat bus costs 260 yuan per bus and renting a 64-seat bus costs 300 yuan per bus, which type of bus is more cost-effective to rent?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A positive integer $n$ is called *nice* if it has at least 3 proper divisors and it is equal to the sum of its three largest proper divisors. For example, $6$ is *nice* because its largest three proper divisors are $3, 2, 1$ and $6 = 3 + 2 + 1$. Find the number of *nice* integers not greater than $3000$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
11. In the Kingdom of Fantasy, the way of representing numbers is quite unique. If a number or a part of a number consists of consecutive digits, it must be written in another form. For example, what we write as 12345, they must write as $1 ※ 5$; what we write as 134569, they must write as $13 ※ 69$; what we write as 7543210, they must write as $75 ※ 0$. Of course, there are also troublesome times, such as 98, which they must write as $9 ※ 8$. The following calculation problem is a math question from the Kingdom of Fantasy. Please calculate the correct result (expressed in the normal way).
$$
20 ※ 13 + 20 ※ 3 + 2 ※ 0 ※ 13 =
$$
$\qquad$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a positive integer $n$ with prime factorization $p_1^{e_1}p_2^{e_2}... p_k^{e_k}$ , we define $f(n)$ to be  $\sum^k_{i=1}p_ie_i$. In other words, $f(n)$ is the sum of the prime divisors of $n$, counted with multiplicities. Let $M$ be the largest odd integer such that $f(M) = 2023$, and $m$ the smallest odd integer so that $f(m) = 2023$. Suppose that $\frac{M}{m}$ equals $p_1^{e_1}p_2^{e_2}... p_l^{e_l}$ , where the $e_i$ are all nonzero integers and the $p_i$ are primes. Find $\left| \sum^l_{i=1} (p_i + e_i) \right|$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
在平面直角坐标系 $x O y$, 点集 $K=\{(x, y) \mid x, y \in-1,0,1\}$ 。在 $K$ 中随机取出三个点,则这三个点两两之间的距离均不超过 2 的概率是 $\qquad$.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
Please reason step by step, and put your final answer within \boxed{}.


Context:
When $p = \sum\limits_{k=1}^{6} k \ln{k}$, the number $e^p$ is an integer.  What is the largest power of 2 that is a factor of $e^p$ ?
Please reason step by step, and put your final answer within \boxed{}.


Context:
8,9

The acute angle of an isosceles trapezoid is $75^{\circ}$. Lines passing through the ends of one of the bases of the trapezoid, parallel to the opposite lateral sides, intersect on the circle circumscribed around the trapezoid. Find the ratio of the bases of the trapezoid.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Select several numbers from $1, 2, 3, \cdots, 9, 10$ so that every number among $1, 2, 3, \cdots, 19, 20$ can be expressed as either one of the selected numbers or the sum of two selected numbers (which can be the same). What is the minimum number of selections needed?
Please reason step by step, and put your final answer within \boxed{}.


Context:
On the edge of a dihedral angle of $120^{\circ}$, a segment of length $c$ is taken, and from its ends, perpendiculars to it are drawn, lying in different faces of the given dihedral angle and having lengths $a$ and $b$. Find the length of the segment of the line connecting the ends of these perpendiculars.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a function $f(x)$ with domain $\mathbb{R}$ and its derivative $f'(x)$, it satisfies $f'(x) - 2f(x) > 4$. If $f(0) = -1$, then the solution set of the inequality $f(x) + 2 > e^{2x}$ is ______.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $b_n$ be the number obtained by writing the integers 1 to $n$ from left to right in reverse order. For example, $b_4 = 4321$ and $b_{12} = 121110987654321$. For $1 \le k \le 150$, how many $b_k$ are divisible by 9?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the expression $\dfrac{\sqrt[6]{5}}{\sqrt[4]{5}}$. What power of 5 does this expression represent?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Three swimmers have to swim a 50-meter lane in the pool, immediately turn back and return to the starting point. The first swimmer starts first, the second swimmer starts 5 seconds later, and the third swimmer starts another 5 seconds after the second. At some point in time, before reaching the end of the lane, all three swimmers were equidistant from the start. The third swimmer, having reached the end of the lane and turning back, met the second swimmer 4 meters from the end of the lane, and the first swimmer 7 meters from the end of the lane. Find the speed of the third swimmer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A certain clock rings 2 notes at quarter past the hour, 4 notes at half past, and 6 notes at 3-quarters past. On the hour, it rings some notes plus an additional number of notes equal to whatever hour it is. The clock will ring 103 notes from 1:00 p.m. through 5:00 p.m., including the rings at 1:00 and 5:00. How many base notes does the clock ring on the hour, before adding the additional notes equal to the hour?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the maximum value of the expression $\left|e^{i\theta}-a\right|+\left|e^{i\theta}+a\right|$ for $0\le\theta\le2\pi$ and $a\in\mathbb{R}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a convex quadrilateral with area \( S \). A point inside it is chosen and reflected symmetrically with respect to the midpoints of its sides. This results in four vertices of a new quadrilateral. Find the area of the new quadrilateral.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The perimeter of a square is equal to the radius of a circle. The area of the square is 784.3155595568603 sq cm. What is the area of the circle?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Angle $2023^{\circ}$ lies in the ______ quadrant.
Please reason step by step, and put your final answer within \boxed{}.


Context:
已知 $x, y, z$ 均为正整数, 且 $\frac{x(y+1)}{x-1}, \frac{y(z+1)}{y-1}, \frac{z(x+1)}{z-1}$ 均为正整数, 则 $x y z$ 的最大值和最小值之和为 $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The domain of the function $y=\sin x$ is $[a,b]$, and its range is $\left[-1, \frac{1}{2}\right]$. Find the maximum value of $b-a$. Express your answer as a single value in radians, simplified if possible.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given 2005 distinct numbers  $a_1,\,a_2,\dots,a_{2005}$ . By one question, we may take three different indices  $1\le i<j<k\le 2005$  and find out the set of numbers  $\{a_i,\,a_j,\,a_k\}$  (unordered, of course).  Find the minimal number of questions, which are necessary to find out all numbers  $a_i$ .
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the function
$$
f(x)=\left(1-x^{2}\right)\left(x^{2}+b x+c\right) \text{ for } x \in [-1, 1].
$$

Let $\mid f(x) \mid$ have a maximum value of $M(b, c)$. As $b$ and $c$ vary, find the minimum value of $M(b, c)$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Eleven girls and \( n \) boys went to the forest to pick mushrooms. Together, they collected \( n^{2} + 9n - 2 \) mushrooms, and each of them collected an equal number of mushrooms.

Who were more in number: boys or girls?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In triangle $ABC,$ $D,$ $E,$ and $F$ are points on sides $\overline{BC},$ $\overline{AC},$ and $\overline{AB},$ respectively, so that $BD:DC = CE:EA = AF:FB = 1:2.$

[asy]
unitsize(0.8 cm);

pair A, B, C, D, E, F, P, Q, R;

A = (2,5);
B = (0,0);
C = (7,0);
D = interp(B,C,1/3);
E = interp(C,A,1/3);
F = interp(A,B,1/3);
P = extension(A,D,C,F);
Q = extension(A,D,B,E);
R = extension(B,E,C,F);

fill(P--Q--R--cycle,gray(0.7));
draw(A--B--C--cycle);
draw(A--D);
draw(B--E);
draw(C--F);

label("$A$", A, N);
label("$B$", B, SW);
label("$C$", C, SE);
label("$D$", D, S);
label("$E$", E, NE);
label("$F$", F, W);
label("$P$", P, NE);
label("$Q$", Q, NW);
label("$R$", R, S);
[/asy]

Line segments $\overline{AD},$ $\overline{BE},$ and $\overline{CF}$ intersect at $P,$ $Q,$ and $R,$ as shown above.  Compute $\frac{[PQR]}{[ABC]}.$
Please reason step by step, and put your final answer within \boxed{}.


Context:
The instantaneous rate of change of carbon-14 content is $-\frac{\ln2}{20}$ (becquerel/year) given that at $t=5730$. Using the formula $M(t) = M_0 \cdot 2^{-\frac{t}{5730}}$, determine $M(2865)$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the polar coordinate system, $P$ is a moving point on the curve $\rho=12\sin \theta$, and $Q$ is a moving point on the curve $\rho=12\cos (\theta- \frac {\pi}{6})$. Find the maximum value of $PQ$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A carpenter worked alone for 1 day on a job. He and another carpenter completed the job in 2 more days. It would have taken the second carpenter 5 days to do the complete job working alone. How many more days would it have taken the first carpenter to finish the job alone?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $n$ be a positive integer. All numbers $m$ which are coprime to $n$ satisfy $m^6 \equiv 1 \pmod{n}$. Find the maximum possible value of $n$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
On the first day, $2^n$ students played a knockout table tennis tournament: first, two played, then the winner played against the third, the winner of that pair played against the fourth, and so on, until the last student played (draws are not possible in table tennis).

On the second day, the same students competed for a cup: they randomly paired up and played in pairs, the losers were eliminated, and the winners were randomly paired up again and played in pairs, and so on. It turned out that the sets of pairs that played on the first and second days were the same (possibly with different winners). Find the largest possible value of $n$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In an acute scalene triangle  $ABC$ , points  $D,E,F$  lie on sides  $BC, CA, AB$ , respectively, such that  $AD \perp BC, BE \perp CA, CF \perp AB$ . Altitudes  $AD, BE, CF$  meet at orthocenter  $H$ . Points  $P$  and  $Q$  lie on segment  $EF$  such that  $AP \perp EF$  and  $HQ \perp EF$ . Lines  $DP$  and  $QH$  intersect at point  $R$ . Compute  $HQ/HR$ .

*Proposed by Zuming Feng*
Please reason step by step, and put your final answer within \boxed{}.


Context:
The telephone company wants to add an area code composed of 2 letters to every phone number. In order to do so, the company chose a special sign language containing 124 different signs. The company used some of the signs fully and two remained unused. The company can create 488 additional area codes if they use all 124 signs. How many signs were used fully by the company?
Please reason step by step, and put your final answer within \boxed{}.


Context:
When two perpendicular lines are rotated about each other (out of their common plane), what is the new angle between them?

More formally: Let $\ell$ and $m$ be two perpendicular lines, and let their common plane be $P$. Rotate $\ell$ about $m$ out of plane $P$ by angle $\theta$, and rotate $m$ about $\ell$ out of plane $P$ by angle $\phi$. Now $\ell'$ and $m'$ have common plane $Q$. What is the angle $\psi$ between $\ell'$ and $m'$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Suppose hops, skips and jumps are specific units of length. If $b$ hops equals $c$ skips, $d$ jumps equals $e$ hops, and $f$ jumps equals $g$ meters, then one meter equals how many skips?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Two players play on an $8 \times 8$ chessboard. The player who starts the game makes the first move - placing a knight on the board. Then they take turns moving it (according to the usual rules), and it is not allowed to place the knight on a square where it has already been. The player who has no legal moves loses. Who wins with correct play - the starting player or their partner?
Please reason step by step, and put your final answer within \boxed{}.


Context:
When \( x \in [1, 2017] \), find the minimum value of
$$
f(x) = \sum_{i=1}^{2017} i |x - i|
$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
A projectile is fired with an initial speed $v$ from the ground at an angle between $0^\circ$ and $90^\circ$ to the horizontal. The trajectory of the projectile can be described by the parametric equations
\[
x = vt \cos \theta, \quad y = vt \sin \theta - \frac{1}{2} gt^2,
\]
where $t$ is the time, $g$ is the acceleration due to gravity, and $\theta$ varies from $0^\circ$ to $90^\circ$. As $\theta$ varies within this range, the highest points of the projectile paths trace out a curve. Calculate the area enclosed by this curve, which can be expressed as $c \cdot \frac{v^4}{g^2}$. Find the value of $c$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The cash realized on selling a 14% stock is Rs. 108.25, with the brokerage being 1/4%. What is the net amount received after deducting the brokerage?
Please reason step by step, and put your final answer within \boxed{}.


Context:
There are 20 rooms, with some lights on and some lights off. The people in these rooms want to have their lights in the same state as the majority of the other rooms. Starting with the first room, if the majority of the remaining 19 rooms have their lights on, the person will turn their light on; otherwise, they will turn their light off. Initially, there are 10 rooms with lights on and 10 rooms with lights off, and the light in the first room is on. After everyone has had their turn, how many rooms will have their lights off?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Eight friends, Aerith, Bob, Chebyshev, Descartes, Euler, Fermat, Gauss, and Hilbert, bought tickets for adjacent seats at the opera. However, when they arrived, they mixed up their seats:

- Bob sat in his assigned seat.
- Chebyshev sat two seats to the right of Gauss' assigned seat.
- Descartes sat one seat to the left of Fermat's assigned seat.
- Euler sat four seats to the left of Hilbert's assigned seat.
- Fermat sat five seats to the right of Descartes' assigned seat.
- Gauss sat one seat to the right of Euler's assigned seat.
- Hilbert sat three seats to the left of Aerith's assigned seat.

In whose seat did Aerith sit?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that points P1 and P2 are two adjacent centers of symmetry for the curve $y= \sqrt {2}\sin ωx-\cos ωx$ $(x\in\mathbb{R})$, if the tangents to the curve at points P1 and P2 are perpendicular to each other, determine the value of ω.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the maximum value that the expression \(a e k - a f h + b f g - b d k + c d h - c e g\) can take, given that each of the numbers \(a, b, c, d, e, f, g, h, k\) equals \(\pm 1\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
给定 1991 个集合, 其中每个集合中都恰好有 45 个元素, 并且每两个集合的并集中都恰好有 89 个元素. 试问, 这 1991 个集合的并集中含有多少个元素?
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Context:
A license plate in a certain state consists of 5 digits, not necessarily distinct, and 3 letters, with the condition that at least one of the letters must be a vowel (A, E, I, O, U). These letters do not need to be next to each other but must be in a sequence. How many distinct license plates are possible if the digits and the letter block can appear in any order?
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Context:
Last year, 10% of the net income from our school's ball was allocated to clubs for purchases, and the remaining part covered the rental cost of the sports field. This year, we cannot sell more tickets, and the rental cost remains the same, so increasing the share for the clubs can only be achieved by raising the ticket price. By what percentage should the ticket price be increased to make the clubs' share 20%?
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Context:
We call a number greater than  $25$ , *semi-prime* if it is the sum of some two different prime numbers. What is the greatest number of consecutive natural numbers that can be *semi-prime*?
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Context:
When $11^4$ is written out in base 10, the sum of its digits is $16=2^4$. What is the largest base $b$ such that the base-$b$ digits of $11^4$ do not add up to $2^4$?  (Note: here, $11^4$ in base $b$ means that the base-$b$ number $11$ is raised to the fourth power.)
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Context:
Given that July 1, 1997, was a Tuesday, determine the day of the week for June 9, 1898. Express your answer as the full day of the week (e.g., Monday, Tuesday, etc.).
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Context:
Let \( a \) and \( b \) be positive integers such that \( 79 \mid (a + 77b) \) and \( 77 \mid (a + 79b) \). Find the smallest possible value of \( a + b \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find all prime numbers \( p \) and \( q \) such that \( p q \) divides \( 2^{p} + 2^{q} \).
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Context:
Jacob uses the following procedure to write down a sequence of numbers. First, he chooses the first term to be \(6\). To generate each succeeding term, he flips a fair coin. If it comes up heads, he doubles the previous term and subtracts \(1\). If it comes up tails, he takes half of the previous term and subtracts \(1\). What is the probability that the fourth term in Jacob's sequence is an integer? The original answer is in \(\frac{k}{m}\) format, please give the value of \(k + m\).
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Context:
Let $ABC$ be a scalene triangle with inradius $1$ and exradii $r_A$, $r_B$, and $r_C$ such that
\[ 20\left(r_B^2r_C^2+r_C^2r_A^2+r_A^2r_B^2\right)=19\left(r_Ar_Br_C\right)^2. \]
If
\[ \tan\frac{A}{2}+\tan\frac{B}{2}+\tan\frac{C}{2}=2.019, \]
then the area of $\triangle{ABC}$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $100m+n$.
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Context:
Points  $A_1, A_2, \ldots, A_{2022}$  are chosen on a plane so that no three of them are collinear. Consider all angles  $A_iA_jA_k$  for distinct points  $A_i, A_j, A_k$ . What largest possible number of these angles can be equal to  $90^\circ$ ?

*Proposed by Anton Trygub*
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Context:
Given an isosceles triangle $\triangle ABC$ where $AB = BC$. A point $K$ is inside $\triangle ABC$ such that $CK = AB = BC$ and $\angle KAC = 30^\circ$. Find $\angle AKB$.
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Context:
The sequence $\{a_n\}_{n\geq 1}$ is defined by the recurrence relation $a_{n+2} = 7a_{n+1} - a_n$ for positive integers $n$, with initial values $a_1 = 1$ and $a_2 = 8$. Another sequence, $\{b_n\}$, is defined by the recurrence relation $b_{n+2} = 3b_{n+1} - b_n$ for positive integers $n$, with initial values $b_1 = 1$ and $b_2 = 2$. Find $\gcd(a_{5000}, b_{501})$.
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Context:
Let $N$ be the set $\{1, 2, \dots, 2018\}$. For each subset $A$ of $N$ with exactly $1009$ elements, define
\[f(A) = \sum\limits_{i \in A} i \sum\limits_{j \in N, j \notin A} j.\]
If $\mathbb{E}[f(A)]$ is the expected value of $f(A)$ as $A$ ranges over all the possible subsets of $N$ with exactly $1009$ elements, find the remainder when the sum of the distinct prime factors of $\mathbb{E}[f(A)]$ is divided by $1000$.
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Context:
A volleyball net is in the shape of a rectangle with dimensions of $50 \times 600$ cells.

What is the maximum number of strings that can be cut so that the net does not fall apart into pieces?
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Context:
Find the mass percentage of Ca in Calcium oxide
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Context:
How many ways are there to split the integers $1$ through $14$ into $7$ pairs such that in each pair, the greater number is at least $2$ times the lesser number?
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Context:
Three planets orbit a star circularly in the same plane. Each moves in the same direction and moves at constant speed. Their periods are $60, 84$, and $140$ years. The three planets and the star are currently collinear . What is the fewest number of years from now that they will all be collinear again?
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Context:
We have four containers. The first three contain water, while the fourth is empty. The second container holds twice as much water as the first, and the third holds twice as much water as the second. We transfer half of the water from the first container, one-third of the water from the second container, and one-quarter of the water from the third container into the fourth container. Now, there are 26 liters of water in the fourth container. How much water is there in total in all the containers?
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Context:
Two congruent right circular cones each with base radius $3$ and height $8$ have the axes of symmetry that intersect at right angles at a point in the interior of the cones a distance $3$ from the base of each cone. A sphere with radius $r$ lies within both cones. The maximum possible value of $r^2$ is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
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Context:
Danil has 6 cards with letters, from which he managed to form the word WNMWNM, depicted in the image. Note that this word has the remarkable property that if you turn it 180 degrees, it becomes the same. How many such words can Danil form using all 6 cards at once?
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Context:
On the angle bisector of angle \(B A C\) of triangle \(A B C\), a point \(M\) is marked, and on the extension of side \(A B\) beyond point \(A\), a point \(N\) is marked such that \(A C = A M = 1\) and \(\angle A N M = \angle C N M\). Find the length of segment \(A N\).
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Context:
$设函数f(x)=x-x^3e^{ax+b}，曲线y=f(x)在点(1， f(1))处的切线方程为y=-x+1.$
$求f(x)的极值点个数.$
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Context:
The integers $1, 2, \ldots, n$ are written in order on a long slip of paper. The slip is then cut into five pieces, so that each piece consists of some (nonempty) consecutive set of integers. The averages of the numbers on the five slips are $1234$, $345$, $128$, $19$, and $9.5$ in some order. Compute $n$.
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Context:
Two trains 135 meters and another of unknown length are running in opposite directions, one at the rate of 80 km and the other at the rate of 65 kmph. They will be completely clear of each other from the moment they meet in 7.447680047665153 seconds. How long is the second train?
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Context:
In a certain math competition, there are 6 questions, each worth 7 points (each question's score is an integer between 0 and 7). The total score of each participant is the product of the scores of the 6 questions. If two participants have the same total score, the sum of the scores for the 6 questions is calculated to determine the ranking. If still tied, they are considered to have the same rank. In this competition, there are $8^{6}=262144$ participants, and there are no ties. The score of the participant ranked $7^{6}=117649$ is $\qquad$ points.
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Context:
The integers $1$ through $1000$ are arranged on the circumference of a circle in natural order. Starting with $1$, every fifteenth number (i.e., $1, 16, 31, \ldots$) is marked. The marking continues until an already marked number is reached. How many numbers will remain unmarked?
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Context:
Given a triangular prism $ABC-A_1B_1C_1$, draw lines through the midpoints of any two edges. How many of these lines are parallel to the plane $ABBA_1$?
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Context:
As shown in the figure, in the right triangular prism $ABC-A_{1}B_{1}C_{1}$, $\angle ACB=90^\circ$, $BC=CC_1=2$, and $AC=4\sqrt{2}$. Point $P$ is on $BC_1$. Find the minimum value of $CP + PA_1$.
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Context:
William is biking from his home to his school and back, using the same route. When he travels to school, there is an initial  $20^\circ$  incline for  $0.5$  kilometers, a flat area for  $2$  kilometers, and a  $20^\circ$  decline for  $1$  kilometer. If William travels at  $8$  kilometers per hour during uphill  $20^\circ$  sections,  $16$  kilometers per hours during flat sections, and  $20$  kilometers per hour during downhill  $20^\circ$  sections, find the closest integer to the number of minutes it take William to get to school and back.

*Proposed by William Yue*
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Context:
Nancy's ex owes her child support. He's supposed to pay 30% of his income each year. For the first 3 years, he made $30,000/year, but he also received annual bonuses that varied as follows: $2,000 in the first year, $3,000 in the second year, and $4,000 in the third year. After the first 3 years, he got a 20% raise on his base salary and made $36,000/year for the next 4 years, but he also received year-end bonuses in this period: $5,000 in the fourth year, $6,000 in the fifth year, $7,000 in the sixth year, and $8,000 in the seventh year. During the sixth year, his child support obligation was adjusted to 25% of his income. If he's only ever paid a total of $1,200, how much does he owe Nancy?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the value of the integral $\int_{0}^{π} \frac{1}{({3+cosθ})^2}dθ$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the center of gravity of a plate bounded by the lines \( x=0 \), \( y=0 \), \( \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 \), \( y \geqslant 0 \), \( x \geqslant 0 \), given that the surface density \( P = x y \).
Please reason step by step, and put your final answer within \boxed{}.

\boxed{\left( \frac{8a}{15}, \frac{8b}{15} \right)}


Context:
Using "学" and "习" to represent two different digits, the four-digit number "学学学学" and "羽习习习习" have a product that is a seven-digit number. And the units digit and the million's place digit of the product are the same as the digit represented by "学". How many different two-digit numbers can "学习" represent?
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Context:
Let $x$ and $y$ be real numbers such that
\[4x^2 + 8xy + 5y^2 = 1.\]Let $m$ and $M$ be the minimum and maximum values of $2x^2 + 3xy + 2y^2,$ respectively.  Find the product $mM.$The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.
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Context:
Find the coefficient of $a^5b^5c^5d^6$ in the expansion of the expression $(bcd + acd + abd + abc)^7$.
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Context:
There are 5 pots of chrysanthemums: 2 pots of yellow chrysanthemums, 2 pots of white chrysanthemums, and 1 pot of red chrysanthemums. When arranging them in a row, the 2 pots of yellow chrysanthemums must be adjacent to each other, and the 2 pots of white chrysanthemums must not be next to each other. Find the number of different arrangements for these 5 pots of flowers. Express your answer as a whole number.
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Context:
Determine the smallest positive integer $m$ such that $529^n + m \cdot 132^n$ is divisible by $262417$ for all odd positive integers $n$.
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Context:
Given two non-zero vectors $\overrightarrow {a}$ and $\overrightarrow {b}$ that satisfy $| \overrightarrow {a} + \overrightarrow {b} | = | \overrightarrow {a} - \overrightarrow {b} | = \sqrt {2} | \overrightarrow {a} |$, find the angle between vector $\overrightarrow {b}$ and $\overrightarrow {a} + \overrightarrow {b}$.
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Context:
7. A sphere is circumscribed around the tetrahedron $ABCD$, and another sphere with radius 1 is tangent to the plane $ABC$ and the two spheres are internally tangent at point $D$. Given $AD=3, \cos \angle BAC=\frac{4}{5}, \cos \angle BAD=\cos \angle CAD=\frac{1}{\sqrt{2}}$, then the volume of the tetrahedron $ABCD$ is . $\qquad$
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Context:
Calculate the definite integral:

$$
\int_{0}^{\frac{\pi}{2}} \frac{\cos x \, dx}{5+4 \cos x}
$$
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Context:
Given a group with the numbers $-3, 0, 5, 8, 11, 13$, and the following rules: the largest isn't first, and it must be within the first four places, the smallest isn't last, and it must be within the last four places, and the median isn't in the first or last position, determine the average of the first and last numbers.
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Context:
In a convex quadrilateral \(ABCD\), it is known that \(\angle ACB = 25^\circ\), \(\angle ACD = 40^\circ\), and \(\angle BAD = 115^\circ\). Find the angle \(\angle ADB\).
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Context:
Find the indefinite integral of $$\int \tan (x) e^ {(\tan (x))} \, dx$$
Please reason step by step, and put your final answer within \boxed{}.

\boxed{\tan(x) e^{\tan(x)} - e^{\tan(x)} + C}


Context:
Given that the point $(\frac{1}{2}, 2)$ is on the graph of the power function $y = f(x)$ and the point $(-2, \frac{1}{4})$ is on the graph of the power function $y = g(x)$, find the value of $f(2) + g(-1)$.
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Context:
If $\alpha$ is an angle in the third quadrant, then $180°-\alpha$ is an angle in the \_\_\_\_\_\_ quadrant.
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Context:
What is the formula for the sum of the form $\frac{\alpha_1 + \alpha_2 + \ldots + \alpha_k}{k}$, where $(\alpha_1, \alpha_2, \ldots, \alpha_k)$ are positive integers satisfying $1 \leq \alpha_1 < \alpha_2 < \ldots < \alpha_k \leq n$?
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Context:
Given a square \(ABCD\) with side length 2, \(E\) is the midpoint of \(AB\). The square is folded along lines \(EC\) and \(ED\) so that \(AE\) coincides with \(BE\), and point \(A\) coincides with point \(B\), named point \(O\). Calculate the volume of the tetrahedron \(O-CDE\).
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Context:
Two joggers each run at their own constant speed and in opposite directions from one another around an oval track. They meet every 36 seconds. The first jogger completes one lap of the track in a time that, when measured in seconds, is a number (not necessarily an integer) between 80 and 100. The second jogger completes one lap of the track in a time, t seconds, where t is a positive integer. What is the product of the smallest and largest possible integer values of t?
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Context:
Let  $T$  be the set of ordered triples  $(x,y,z)$ , where  $x,y,z$  are integers with  $0\leq x,y,z\leq9$ . Players  $A$  and  $B$  play the following guessing game. Player  $A$  chooses a triple  $(x,y,z)$  in  $T$ , and Player  $B$  has to discover  $A$ *'s* triple in as few moves as possible.  A *move* consists of the following:  $B$  gives  $A$  a triple  $(a,b,c)$  in  $T$ , and  $A$  replies by giving  $B$  the number  $\left|x+y-a-b\right |+\left|y+z-b-c\right|+\left|z+x-c-a\right|$ . Find the minimum number of moves that  $B$  needs to be sure of determining  $A$ *'s*  triple.
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Context:
If \[x+\sqrt{x^2-1}+\frac{1}{x-\sqrt{x^2-1}}=20,\] then \[x^2+\sqrt{x^4-1}+\frac{1}{x^2+\sqrt{x^4-1}}=\]
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Context:
If the coefficients $A$ and $B$ of the line $Ax+By=0$ can take different values from the set $\{0, 2, 3, 4, 5, 6\}$, the number of these equations representing different lines is.
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Context:
A number should have been multiplied by 5, but instead, it was processed in a different way. The percentage of error obtained is 96%. What operation was performed on the number instead of multiplication?
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Context:
Calculate the limit $$ \lim_{n\to \infty}\left(\dfrac{f\left(a+\frac{1}{n}\right)}{f(a)}\right)^{1/n}, $$ given a function $f:\mathbb{R}\to \mathbb{R}$ that is differentiable at $a\in \mathbb{R}$ and satisfies $f(a) > 0$.
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Context:
Candace is trying to decide whether to wear her old shoes or new high-tech shoes for a hike. The high-tech shoes will help Candace walk twice as fast, but they're not broken in yet, so she'll get 1 blister after every 2 hours she spends walking in them. Each blister slows Candance down by 2 miles per hour. If Candance walks 6 miles per hour in the old shoes and plans to hike for 4 hours, how many miles per hour can she go in the new shoes?
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Context:
Let the set \( T = \left\{ x_{1}, x_{2}, \cdots, x_{10} \right\} \) contain five-element subsets such that any two elements of \( T \) appear together in at most two of these subsets. What is the maximum number of such subsets?
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Context:
In the convex pentagon ABCDE, all sides are of equal length, and angles A and B are right angles (90 degrees). Calculate the degree measure of angle E. Express your answer in degrees.
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Context:
Let \( S_{n} = 1 + \frac{1}{2} + \cdots + \frac{1}{n} \) for \( n = 1, 2, \cdots \). Find the smallest positive integer \( n \) such that \( S_{n} > 10 \).
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Context:
3. Initially, the numbers $1, 2, 3, \cdots, 2013$ are written on the blackboard. In each step, you can erase two numbers $a, b$ from the blackboard and write down $a+b$. What is the minimum number of steps required to ensure that the sum of any remaining numbers on the blackboard is not equal to 2014?
(Lin Chang provided the problem)
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Context:
Write a negative number greater than $-\frac{1}{2}$ ______.
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Context:
In a cube, a line determined by two vertices and a plane containing four vertices can form an "orthogonal line-plane pair". How many such pairs are possible in the cube? Express your answer as a whole number.
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Context:
Consider all the real sequences $x_0,x_1,\cdots,x_{100}$ satisfying the following two requirements:
(1)$x_0=0$;
(2)For any integer $i,1\leq i\leq 100$,we have $1\leq x_i-x_{i-1}\leq 2$.
Find the greatest positive integer $k\leq 100$,so that for any sequence $x_0,x_1,\cdots,x_{100}$ like this,we have
\[x_k+x_{k+1}+\cdots+x_{100}\geq x_0+x_1+\cdots+x_{k-1}.\]
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Context:
Given that the sequence {a_n} is an increasing arithmetic sequence, and S_n is the sum of the first n terms of the sequence {a_n}. Let's denote the first term of the sequence as a_1 and the common difference as d.

Given that S_14 = 14 and S_9 = 18 - λa_10, calculate the value of λ. Express your answer as a single integer.
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Context:
Given a trapezoid \(ABCD\) with bases \(AB\) and \(CD\), and angles \(\angle C = 30^\circ\) and \(\angle D = 80^\circ\). Find \(\angle ACB\), given that \(DB\) is the bisector of \(\angle D\).
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Context:
What is the maximum number of natural numbers that can be written in a row such that the sum of any three consecutive numbers is even, and the sum of any four consecutive numbers is odd?
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Context:
Assume that the targeted respondents belong to three different demographics: demographic A, demographic B, and demographic C. The response rates for these demographics are 70%, 60%, and 50%, respectively. The population proportion of each demographic is 40%, 35%, and 25%. In order to receive 210 responses, what is the minimum number of questionnaires that should be mailed out, taking into consideration the different demographics and their respective response rates?
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Context:
全体正整数的集合可以分成两个互不相交的正整数子集

{f(1),f(2),⋯,f(n),⋯},{g(1),g(2),⋯,g(n),⋯}

其中 f(1)<f(2)<⋯<f(n)<⋯

g(1)<g(2)<⋯<g(n)<⋯

且有 g(n)=f(f(n))+1,n≥1

求f(240).
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Context:
$已知数列a_n的奇数项是公差为d_1的等差数列，偶数项是公差为d_2的等差数列，S_n是数列a_n的前n项和，a_1=1,a_2=2.$
$若S_4=11,a_{10}=2a_{7}，求a_{12}；$
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Context:
The quiz participants were asked four questions: 90 participants answered the first question correctly, 50 answered the second correctly, 40 answered the third correctly, and 20 answered the fourth correctly. No one was able to answer more than two questions correctly. What is the minimum number of participants in the quiz under these conditions?
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Context:
Compute the following limit: $$\lim_{x \to \infty}\dfrac {f^{-1}(8x)-f^{-1}(x)}{x^{1/3}},$$ where $f: \mathbb R \to \mathbb R$ is defined as $f(x)=8x^3+3x$, and it is known that $f$ is continuous, strictly increasing, and satisfies $\lim _{x\to \infty}f(x)=\infty$ and $\lim_{x \to -\infty}f(x)=-\infty$.
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Context:
8-1. Find all common points of the graphs

$$
y=8 \cos \pi x \cdot \cos ^{2} 2 \pi x \cdot \cos 4 \pi x \quad \text { and } \quad y=\cos 9 \pi x
$$

with abscissas belonging to the segment $x \in[0 ; 1]$. In your answer, specify the sum of the abscissas of the found points.
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Context:
A bag of potatoes weighs 36 kg, which is 1.5 times the weight of a bag of carrots. Two bags of tomatoes weigh an additional 16 pounds. Convert the weights of the bags of potatoes and carrots to pounds and combine them with the weight of the tomatoes to find the total weight of all the produce in pounds. How much do the bags of potatoes, carrots, and tomatoes weigh together?
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Context:
In trapezoid ABCD, AB is parallel to DC, angle ABC = 90 degrees, and E is the midpoint of leg AD. If EC = sqrt(13) and AB + BC + CD = 2*sqrt(26), then calculate the measure of angle BCE in degrees.
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Context:
Consider a fair coin and a fair 6-sided die.  The die begins with the number 1 face up.  A *step* starts with a toss of the coin: if the coin comes out heads, we roll the die; otherwise (if the coin comes out tails), we do nothing else in this step.  After 5 such steps, what is the probability that the number 1 is face up on the die?
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Context:
The equation \( x^{2} + a x - 2 = 0 \) has two distinct roots \( x_{1} \) and \( x_{2} \); moreover,

\[ x_{1}^{3} + \frac{22}{x_{2}} = x_{2}^{3} + \frac{22}{x_{1}} \]

Find all possible values of \( a \).
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Context:
已知存在实数 $r$, 使得圆周 $x^{2}+y^{2}=r^{2}$ 上恰好有 $n$ 个整点, 则 $n$ 可以等于（）。请提供所有满足条件的n值之和。
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Context:
105 A cube of wood with an edge length of 3 meters, has a square hole with a side length of 1 meter drilled through the center of each face, all the way through, with the edges of the hole parallel to the corresponding edges of the cube. The total surface area (including the inside) of this object is $\qquad$ square meters.
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Context:
In the sequence $\left\{a_{n}\right\}$, where $a_{1}=1$, the sum of the first $n$ terms is $S_{n}$. For any integer $n \geqslant 2$, the sequence $S_{n-1}-8, a_{n}, 8 S_{n}-4$ is an arithmetic sequence. Find $\lim _{n \rightarrow \infty}\left(a_{n}+S_{n}\right)$.
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Context:
已知双曲线$\Gamma:\frac{x^{2}}{3}-y^{2}=1$，点$A_{1}(2,0)$，$A_{2}(3,0)$，过点$A_{1}$的直线与双曲线交于点$B_{1}、C_{1}$，过点$A_{2}$的直线与双曲线交于点$B_{2}、C_{2}$，且$k_{B_{1}C_{1}}k_{B_{2}C_{2}}=\frac{1}{2}$。分别以$B_{1}C_{1}$、$B_{2}C_{2}$为直径的圆交于点$X、Y$。求原点到直线$XY$的最大距离。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:
Points  $M$ ,  $N$ ,  $P$  are selected on sides  $\overline{AB}$ ,  $\overline{AC}$ ,  $\overline{BC}$ , respectively, of triangle  $ABC$ .  Find the area of triangle  $MNP$  given that  $AM=MB=BP=15$  and  $AN=NC=CP=25$ .

*Proposed by Evan Chen*
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Context:
Given the parabola equation \( y = ax^2 + bx + c \) (\( a \neq 0 \)), there is an inscribed circle with radius \( \frac{1}{|a|} \) touching the parabola, denoted as \( \odot C_{1} \). A circle \( \odot C_{2} \) is drawn, which is externally tangent to \( \odot C_{1} \) and also touches the parabola internally. Then, a circle \( \odot C_{3} \) is drawn, which is externally tangent to \( \odot C_{2} \) and also touches the parabola internally, and so on, forming a sequence of circles \( C_{1}, C_{2}, C_{3}, \cdots \). Find the sum of the areas of the first \( n \) circles.
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Context:
Sheela deposits some amount in a bank savings account. If this is 32% of her monthly income, and her monthly income is Rs. 11875, how much did she deposit in the bank savings account?
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Context:
Shift the graph of the function $f(x) = \sin x + \sqrt{3}\cos x$ to the right by $\varphi$ ($\varphi > 0$) units. If the resulting graph is symmetric about the origin, then the minimum value of $\varphi$ is ______.
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Context:
Determine how many ordered pairs of positive integers $(x, y)$ where $x < y$, such that the harmonic mean of $x$ and $y$ is equal to $24^{10}$.
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Context:
Given that the angle between vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ is $\frac{π}{3}$, and $\overrightarrow{a}⋅\overrightarrow{b}=3$, vector $\overrightarrow{c}$ satisfies $\overrightarrow{c}=λ\overrightarrow{a}+({1-λ})\overrightarrow{b}$ ($0＜λ＜1$), and $\overrightarrow{a}⋅\overrightarrow{c}=\overrightarrow{b}⋅\overrightarrow{c}$. Let $x=\frac{{\overrightarrow{c}⋅\overrightarrow{a}}}{{|{\overrightarrow{a}}|}}$ and $y=\frac{{\overrightarrow{c}⋅\overrightarrow{b}}}{{|{\overrightarrow{b}}|}}$. Find the maximum value of $x^{2}+y^{2}-xy$.
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Context:
Given a real number $t$ with $0 < t < 1$, define the real-valued function $f(t, \theta) = \sum^{\infty}_{n=-\infty} t^{|n|}\omega^n$, where $\omega = e^{i \theta} = \cos  \theta + i\sin  \theta$. For $\theta \in [0, 2\pi)$, the polar curve $r(\theta) = f(t, \theta)$ traces out an ellipse $E_t$ with a horizontal major axis whose left focus is at the origin. Let $A(t)$ be the area of the ellipse $E_t$. Let $A\left( \frac12 \right) = \frac{a\pi}{b}$ , where $a, b$ are relatively prime positive integers. Find $100a +b$ .
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Context:
Given $x \gt 0$, $y \gt 0$, and $x+2y+\frac{2}{x}+\frac{1}{y}=6$, there exists a real number $m$ such that $m\geqslant x+2y$ always holds. Find the minimum value of $m$.
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Context:
II Let $a, d \geqslant 0, b, c>0$ and $b+c \geqslant a+d$. Find the minimum value of $\frac{b}{c+d}+\frac{c}{a+b}$.
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Context:
Calculate the volume of the solid $D$ formed by the union of all line segments connecting the origin $(0,0,0)$ to points $(x,y,z)$ on the surface of the sphere $x^2 + y^2 + z^2 = 1$ where $0 \leq z \leq \frac{1}{\sqrt{2}}$.
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Context:
John adopts a dog from a shelter and decides to take care of all the dog's necessities. The dog ends up having health problems, which requires 3 vet appointments during the year, costing $400 each. In addition, John spends an average of $50 per month on medication, $30 per month on pet food, and $15 per month on play toys. Every 3 months, John also spends $60 on grooming services. Furthermore, the dog suffers from an irregular health issue twice a year that costs an additional $200 each time.

After the first vet appointment, John pays $100 for annual pet insurance that covers 80% of the subsequent vet visits, 50% of the medication costs, and 25% of the irregular health issue expenses. Calculate the total amount John paid for the dog-related expenses during the year, taking into account the insurance coverage.
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Context:
Amaya scored 20 marks fewer in Maths than she scored in Arts. She also got some marks more in Social Studies than she got in Music. She scored 70 in Music and scored 1/10 less in Maths. What's the total number of marks she scored in all the subjects if the difference between Social Studies and Music marks is 10?
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Context:
设曲线$y = \sqrt { 2 x - x ^ { 2 } }$与x轴所围成的区域为D，向区域D内随机投一点，该点落在D内任一小区域的概率只与该小区域的面积成比例，则该点落入区域$\{ ( x , y ) \in D \vert x ^ { 2 } + y ^ { 2 } < 2 \}$内的概率为
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Context:
What is the largest positive integer $n$ for which there is a unique integer $k$ such that $\frac{3}{7} < \frac{n}{n + k} < \frac{8}{19}$?
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Context:
The set
$$
A=\{\sqrt[n]{n} \mid n \in \mathbf{N} \text{ and } 1 \leq n \leq 2020\}
$$

has the largest element as $\qquad$ .
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Context:
Elena intends to buy 7 binders priced at $\textdollar 3$ each. Coincidentally, a store offers a 25% discount the next day and an additional $\textdollar 5$ rebate for purchases over $\textdollar 20$. Calculate the amount Elena could save by making her purchase on the day of the discount.
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Context:
Given the set $M=\{1, 2, 3, 4, 5\}$, for its non-empty subset $A$, if we multiply each element $k$ in $A$ by $(-1)^k$ and then sum them up, this sum is called the "Element Characteristic Sum" of $A$. For example: the "Element Characteristic Sum" of $A=\{4\}$ is $(-1)^k \times 4 = 4$, and for $A=\{1, 2, 5\}$, it is $(-1)^1 \times 1 + (-1)^2 \times 2 + (-1)^5 \times 5 = -4$. Then, the total sum of the "Element Characteristic Sums" of all non-empty subsets of set $M$ equals \_\_\_\_\_\_.
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Context:
In a convex quadrilateral \(ABCD\), the segment \(CM\), connecting vertex \(C\) with point \(M\) located on side \(AD\), intersects diagonal \(BD\) at point \(K\). It is known that \(CK : KM = 2 : 1\), \(CD : DK = 5 : 3\), and \(\angle ABD + \angle ACD = 180^\circ\). Find the ratio of side \(AB\) to diagonal \(AC\).
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Context:
A public pool opened for the summer. A total of 246 people came swimming over the first 3 days it was open. On the first day, 79 came to swim. On the second day, some more people swam than on the third day, when 120 people swam. How many more people swam on the second day than on the third day?
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Context:
A circle is constructed on one of the equal sides of an isosceles triangle as its diameter, dividing the other equal side into segments of length \(a\) and \(b\).

Find the base of the triangle.
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Context:
Given the sequence $\{a_n\}$ with general term formula $a_n = -2n^2 + \lambda n$ ($n \in \mathbb{N}^*, \lambda \in \mathbb{R}$), find the range of values for $\lambda$ such that the sequence is decreasing. Express your answer as an interval in standard interval notation.
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Context:
Find the smallest \( a \in \mathbf{N}^{*} \) such that the following equation has real roots:
$$
\cos ^{2} \pi(a-x)-2 \cos \pi(a-x)+\cos \frac{3 \pi x}{2 a} \cdot \cos \left(\frac{\pi x}{2 a}+\frac{\pi}{3}\right)+2=0 .
$$
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Context:
Given an infinite sequence of positive real numbers $\left\{x_{n}\right\}$ satisfying $x_{0}=1$ and $x_{i} \geq x_{i+1}$ for all $i \in \mathbb{N}$, such a sequence is called a "good sequence." Find the smallest constant $c$ such that there exists a "good sequence" $\left\{x_{n}\right\}$ for which the inequality $\sum_{i=0}^{n} \frac{x_{i}^{2}}{x_{i+1}} \leq c$ holds for any positive integer $n$.
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Context:
Given $a \gt 0$, $b \gt 0$, and $9^{a}=3^{1-b}$, find the minimum value of $\frac{1}{81a}+\frac{2}{81b}+ab$.
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Context:
Mike cycled $15$ laps in $57$ minutes. Assume he cycled at a constant speed throughout. Approximately how many laps did he complete in the first $27$ minutes?
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Context:
Daisy is a poodle puppy who loves to play with her dog toys. She often loses them in various ways, and her owner needs to replace them. On Monday, Daisy played with 5 dog toys. On Tuesday, Daisy had only 60% of her dog toys left after losing some, and her owner went to the store and got her 3 more. On Wednesday, Daisy lost 50% of her old and new dog toys, so her owner went to the store and bought her 5 more. If Daisy's owner found all the lost dog toys, including the new dog toys, how many dog toys would Daisy have now?
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Context:
Find the coefficient of $x^{90}$ in the expansion of
\[(x - 1)(x^2 - 2)(x^3 - 3) \dotsm (x^{12} - 12)(x^{13} - 13).\]
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Context:
Through the points \( A(0 ; 14) \) and \( B(0 ; 4) \) two parallel lines are drawn. The first line, passing through point \( A \), intersects the hyperbola \( y = \frac{1}{x} \) at points \( K \) and \( L \). The second line, passing through point \( B \), intersects the hyperbola \( y = \frac{1}{x} \) at points \( M \) and \( N \).

What is the value of \( \frac{A L - A K}{B N - B M} \)?
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Context:
t = { -3, 0, 2, 3, 4, 5, 7, 9 } b = { -2, 4, 5, 6, 7, 8, 10, 12 } Two integers will be randomly selected from the sets above, one integer from set t and one integer from set b. What is the probability that the sum of the two integers will equal 9?
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Context:
Given the ellipse \(\frac{x^{2}}{9} + \frac{y^{2}}{5} = 1\), the right focus is \(F\), and \(P\) is a point on the ellipse. Point \(A\) is at \((0, 2 \sqrt{3})\). When the perimeter of \(\triangle APF\) is maximized, what is the area of \(\triangle APF\)?
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Context:
There are $20$ geese numbered $1-20$ standing in a line. The even-numbered geese are standing at the front in the order $2,4,\dots,20,$ where $2$ is at the front of the line. Then the odd-numbered geese are standing behind them in the order $1,3,5,\dots,19,$ where $19$ is at the end of the line. The geese want to rearrange themselves in order, so that they are ordered $1,2,\dots,20$ (1 is at the front), and they do this by successively swapping two adjacent geese. What is the minimum number of swaps required to achieve this formation?
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Context:
Given an infinite cone, the measure of its unfolding's angle is equal to $\alpha$. A curve on the cone is represented on any unfolding by the union of line segments. Find the greatest integer $n$ such that $n\cdot \alpha < 180^{\circ }$. Please provide the integer value directly.
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Context:
In the regular triangular prism $ABC-A_1B_1C_1$, where $AB = AA_1$, determine the point $E$ on $BB_1$ such that the dihedral angle between the planes $A_1EC$ and $A_1B_1C_1$ is $45^\circ$.
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\boxed{E \text{ is the midpoint of } B B_1}


Context:
Solve the equation $x^2 + 14x = 72$. The positive solution has the form $\sqrt{c} - d$ for positive natural numbers $c$ and $d$. What is $c + d$?
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Context:
正三棱雉 $P-A B C$ 中, $A B=1, A P=2$, 过 $A B$ 的平面 $\alpha$ 将其体积平分, 则棱 $P C$与平面 $\alpha$ 所成角的余弦值为。原始答案的形式为\frac{m\sqrt{n}}{k}，请给出m + n + k的值。
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Context:
已知正数 a,b,c 满足 a+b+c=13. 则 \df{a^2+b^3+c^4+2019}{10b+123c+26} 的最小值为__________.
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Context:
In the addition below, the same letters represent the same digit, and different letters represent different digits. Find the number $A B C D E$.

$A B C D E$ $B C D E$ $C D E$ $D E$ $A A A A A$
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Context:
Determine the smallest positive integer $n \geq 3$ for which $$A \equiv 2^{10 n}\left(\bmod 2^{170}\right)$$ where $A$ denotes the result when the numbers $2^{10}, 2^{20}, \ldots, 2^{10 n}$ are written in decimal notation and concatenated (for example, if $n=2$ we have $A=10241048576$).
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Context:
Find all positive integers $ k$ with the following property: There exists an integer $ a$ so that $ (a\plus{}k)^{3}\minus{}a^{3}$ is a multiple of $ 2007$.
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Context:
If (a + b + e) = 7, (b + c + f) = 10, (c + d + g) = 6, and (e + f + g) = 9, what is the value of (a + d + g)?
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Context:
求最小的实数 k，使得对任意两两不等的实数 a,b,c，均有\left(\df{2a}{a-b}\right)^2+\left(\df{2b}{b-c}\right)^2+\left(\df{2c}{c-a}\right)^2+k\geq4\left(\df{2a}{a-b}+\df{2b}{b-c}+\df{2c}{c-a}\right).
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Context:
2. Two circles touch each other externally and each touches a larger circle internally. The radius of one is half, and that of the other is one-third of the radius of the largest circle. Find the ratio of the length of the segment of the common internal tangent to the smaller circles, contained within the largest, to its diameter.

#
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Context:
The landlord of an apartment building needs to purchase enough digits to label all of the apartments from 100 through 125 on the first floor and 200 through 225 on the second floor. The digits can only be purchased in a package that contains one of each digit 0 through 9. How many packages must the landlord purchase?
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Context:
What integer value will satisfy the equation $$ 14^2 \times 35^2 = 10^2 \times (M - 10)^2 \ ? $$
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Context:
The large cube shown is made up of $27$ identical sized smaller cubes. For each face of the large cube, the opposite face is shaded the same way. Find the total number of smaller cubes that must have at least one face shaded.
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Context:
5. In the country of Lemonia, coins in circulation have denominations of $2^{n}, 2^{n-1} \cdot 3, 2^{n-2} \cdot 3^{2}$, $2^{n-3} \cdot 3^{3}, \ldots, 2 \cdot 3^{n-1}, 3^{n}$ piastres, where $n$ is a natural number. A resident of the country went to the bank without any cash. What is the largest amount that the bank will not be able to give him?
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Context:
Find the positive value of $k$ such that the equation $4x^3 + 9x^2 + kx + 4 = 0$ has exactly one real solution in $x$.
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Context:
Given the function $f(x)=x^{3}+ax^{2}+2x-a^{2}$, if for any $x_{1}$, $x_{2}\in \left[1,2\right]$, where $x_{1} \lt x_{2}$, the inequality $\frac{{f({{x_1}})-f({{x_2}})}}{{{x_1}-{x_2}}}＜2$ holds, then the maximum value of the real number $a$ is ____.
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Context:
King Qi and Tian Ji are having a horse race. Tian Ji's top horse is better than King Qi's middle horse but worse than King Qi's top horse. Tian Ji's middle horse is better than King Qi's lower horse but worse than King Qi's middle horse. Tian Ji's lower horse is worse than King Qi's lower horse. Now, if a horse is randomly selected from both sides for a race, what is the probability of Tian Ji winning? Express your answer as a simplified fraction in the form $\boxed{\frac{a}{b}}$.
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Context:
Suppose $X$ and $Y$ are digits in base $d > 8$ such that $\overline{XY}_d + \overline{XX}_d = 234_d$. Find $X_d - Y_d$ in base $d$.
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Context:
已知实数 $x_{1}, x_{2}, \cdots, x_{2018}$ 满足：对任意的 $1 \leqslant i<j \leqslant 2018$, 均有 $x_{i}+x_{j} \geqslant(-1)^{i+j}$. 求 $\sum_{i=1}^{2018} i x_{i}$ 的最小值.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:
Given that $A$, $B$, and $P$ are three distinct points on the hyperbola ${x^2}-\frac{{y^2}}{4}=1$, and they satisfy $\overrightarrow{PA}+\overrightarrow{PB}=2\overrightarrow{PO}$ (where $O$ is the origin), the slopes of lines $PA$ and $PB$ are denoted as $m$ and $n$ respectively. Find the minimum value of ${m^2}+\frac{{n^2}}{9}$.
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Context:
If the fractional equation in terms of $x$, $\frac{3}{x-4}+\frac{x+m}{4-x}=1$, has a root, then the value of $m$ is ____.
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Context:
If $a$ and $b$ are elements of the set ${ 1,2,3,4,5,6 }$ and $|a-b| \leqslant 1$, calculate the probability that any two people playing this game form a "friendly pair".
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