Context:
Concentric circles  $\Omega_1$  and  $\Omega_2$  with radii  $1$  and  $100$ , respectively, are drawn with center  $O$ . Points  $A$  and  $B$  are chosen independently at random on the circumferences of  $\Omega_1$  and  $\Omega_2$ , respectively. Denote by  $\ell$  the tangent line to  $\Omega_1$  passing through  $A$ , and denote by  $P$  the reflection of  $B$  across  $\ell$ . Compute the expected value of  $OP^2$ .
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Context:
In a tournament with $n$ players, where $n < 10$, each player competes once against every other player. Players score 1 point for a win and 0 points for a loss, with no draws occurring. In a particular tournament, only one player finished with an odd number of points and was ranked fourth. Determine if this scenario is possible. If it is possible, calculate the number of wins the player achieved.
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Context:
Given that the two roots of the equation x^2 + (k^2 - 4)x + k - 1 = 0 are opposites of each other, find the value of k. Express your answer as a single numerical value.
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Context:
Let  $n > 1$  be an integer and  $X = \{1, 2, \cdots , n^2 \}$ . If there exist  $x, y$  such that  $x^2\mid y$  in all subsets of  $X$  with  $k$  elements, find the least possible value of  $k$ .
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Context:
已知$z _ { 1 } , z _ { 2 } , z _ { 3 } , \cdots , z _ { 6 }$是2+i的六次方根，而它们的辐角主值分别为$\alpha _ { 1 } , \alpha _ { 2 } , \alpha _ { 3 } , \cdots , \alpha _ { 6 }$ 求$\tan \alpha _ { 1 } \tan \alpha _ { 2 } + \tan \alpha _ { 2 } \tan \alpha _ { 3 } + \cdots + \tan \alpha _ { 6 } \tan \alpha _ { 1 }$的值.
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Context:
Suppose the estimated $20$ billion dollar cost to send a person to the planet Mars is shared equally by the $250$ million people in the U.S. Then each person's share is
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Context:
A roulette wheel can land on any number from 0 to 2007 with equal probability. The roulette wheel is spun repeatedly. Let \( P_{k} \) denote the probability that at some point the sum of the numbers that have come up in all the spins made equals \( k \). Which number is greater: \( P_{2007} \) or \( P_{2008} \)?
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Context:
Alex starts with a rooted tree with one vertex (the root). For a vertex  $v$ , let the size of the subtree of  $v$  be  $S(v)$ . Alex plays a game that lasts nine turns. At each turn, he randomly selects a vertex in the tree, and adds a child vertex to that vertex. After nine turns, he has ten total vertices. Alex selects one of these vertices at random (call the vertex  $v_1$ ). The expected value of  $S(v_1)$  is of the form  $\tfrac{m}{n}$  for relatively prime positive integers  $m, n$ . Find  $m+n$ .**Note:** In a rooted tree, the subtree of  $v$  consists of its indirect or direct descendants (including  $v$  itself).

*Proposed by Yang Liu*
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Context:
Two different cubes of the same size are to be painted, with the color of each face being chosen independently and at random to be either black or white. What is the probability that after they are painted, the cubes can be rotated to be identical in appearance?
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Context:
一城市设计成分为若干方块的矩形形状，n条街道互相平行，另外m条街道与它们相交成直角.警察在城市的街道上（不是十字路口）值勤.每个警察报告从他身旁开过去的汽车的号码、运行方向及通过时间.为了能根据警察提供的数据同时再现沿着闭合线路（即不重复通过同一地段的线路）行驶的任何汽车的路线，至少需要多少警察值勤？
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Context:
The numbers $2^{1989}$ and $5^{1989}$ are written out one after the other (in decimal notation). How many digits are written altogether?
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Context:
For which $n$ can an $n \times n$ grid be divided into one $2 \times 2$ square and some number of strips consisting of five cells, in such a way that the square touches the side of the board?
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Context:
Form a six-digit number using the digits 1, 2, 3, 4, 5, 6 without repetition, requiring that any two adjacent digits have different parities, and 1 and 2 are adjacent. The number of such six-digit numbers is (answer in digits).
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Context:
In how many ways can 10 fillér and 50 fillér coins be placed side by side (with all centers on a straight line) to cover a $1 \mathrm{~m}$ long segment (not more), using at least 50 coins, and considering the order of the two types of coins? (Coins of the same value are not distinguished. The diameter of the 10 fillér is $19 \mathrm{~mm}$, and the diameter of the 50 fillér is $22 \mathrm{~mm}$.)
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Context:
Six positive integers are written on the faces of a cube. Each vertex is labeled with the product of the numbers on the three faces adjacent to that vertex. If the sum of the numbers on the vertices is $1512$, and the sum of the numbers on one pair of opposite faces is $8$, what is the sum of the numbers on all the faces?
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Context:
Given a sequence $\{a_{n}\}$ such that $a_{1}+3a_{2}+\ldots +\left(2n-1\right)a_{n}=\frac{1}{3}-\frac{1}{{2n+3}}$, $n\in N^{*}$, then $a_{1}+a_{2}+\ldots +a_{n}=$____.
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Context:
In the diagram, \( PQR \) is a straight line segment and \( QS = QT \). Also, \( \angle PQS = x^\circ \) and \( \angle TQR = 3x^\circ \). If \( \angle QTS = 76^\circ \), find the value of \( x \).
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Context:
Given that there are \( C \) integers that satisfy the equation \( |x-2| + |x+1| = B \), find the value of \( C \).
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Context:
Find the distance covered by a man walking uphill for 25 minutes at a speed of 8 km/hr and then downhill for 11 minutes at a speed of 12 km/hr. The uphill slope is at an angle of 15 degrees, and the downhill slope is at an angle of 10 degrees.
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Context:
Given that α and β are acute angles, and $\tan \alpha = \frac{2}{t}$, $\tan \beta = \frac{t}{15}$. When $10\tan \alpha + 3\tan \beta$ reaches its minimum value, the value of $\alpha + \beta$ is \_\_\_\_\_\_.
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Context:
Given a positive integer $k$, let $f(k)$ be the sum of the $k$-th powers of the primitive roots of $73$. For how many positive integers $k < 2015$ is $f(k)$ divisible by $73?$
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Context:
What is the least positive integer m such that $m\cdot 2! \cdot 3! \cdot 4! \cdot 5! \ldots 16!$ is a perfect square? Listing out all possibilities of m is not allowed.
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Context:
Two unit-radius spheres intersect in such a way that the resulting 3 parts have equal volume. How far apart are the centers of the spheres?
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Context:
Louis is making himself a velvet suit for a high-end gala. The velvet fabric he chose costs $24 per yard. He purchases a pattern for $15, two spools of silver thread for $3 each, lining fabric at an unknown price per yard, a set of 12 stylish buttons for $14, and 3 yards of exquisite trim for $19 per yard. Additionally, he has a 10% discount coupon specific to fabrics, which he applies only to the velvet and lining fabric. He spent a total of $310.50 after applying the discount. If Louis bought 5 yards of velvet fabric and 4 yards of lining fabric, what is the cost per yard of the lining fabric?
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Context:
For a sequence  $s = (s_1, s_2, . . . , s_n)$ , define  $$ F(s) =\sum^{n-1}_{i=1} (-1)^{i+1}(s_i - s_{i+1})^2. $$  Consider the sequence  $S =\left(2^1, 2^2, . . . , 2^{1000}\right)$ . Let  $R$  be the sum of all  $F(m)$  for all non-empty subsequences  $m$  of  $S$ . Find the remainder when  $R$  is divided by  $1000$ .

Note: A subsequence is a sequence that can be obtained from another sequence by deleting some non-negative number of values without changing the order.
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Context:
Compute the surface integral $\iint_S F\cdot dS$ where $F=\langle x,y,z^2\rangle$ and $S$ is the unit sphere centered at the origin. The surface $S$ is parameterized by $r(\phi,\theta)=\langle \sin\theta \cos\phi,\sin\theta \sin\phi, \cos\theta\rangle$ for $0\leq\theta\leq\pi$ and $0\leq\phi\leq2\pi$.
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Context:
Find the area of the figure on the coordinate plane bounded by the straight lines $x = 0$, $x = 2$, and the graphs of the functions $y = \sqrt{x^3 + 1}$ and $y = -\sqrt[3]{x^2 + 2x}$.
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Context:
Find the integral $\int \operatorname{cth}^{2} x \, dx$.
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Context:
Let $S(N)$ be the number of 1's in the binary representation of an integer $N$, and let $D(N) = S(N + 1) - S(N)$.  Compute the sum of $D(N)$ over all $N$ such that $1 \le N \le 2017$ and $D(N) < 0$.
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Context:
In a group of 100 cars, some cars do not have air conditioning. If at least 53 cars have racing stripes, the greatest number of cars that could have air conditioning but not racing stripes is 47. How many cars do not have air conditioning?
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Context:
$S$ is the sum of the first 9 terms of an increasing arithmetic sequence $a_{1}, a_{2}, a_{3}, \ldots$, consisting of integers. It is known that $a_{5} a_{18} > S - 4$ and $a_{13} a_{10} < S + 60$. Determine all possible values of $a_{1}$.
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Context:
Find the exact value of $t$ such that $\frac{d}{dt}\int_{t}^{t+1}\sin e^x \ dx=0$.
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Context:
A train of a certain length passes a man, running at 6 kmph in the direction opposite to that of the train, in 6 seconds. The speed of the train is 77.993280537557 kmph. What is the length of the train?
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Context:
There are knights, liars, and followers living on an island; each knows who is who among them. All 2018 islanders were arranged in a row and asked to answer "Yes" or "No" to the question: "Are there more knights on the island than liars?". They answered in turn such that everyone else could hear. Knights told the truth, liars lied. Each follower gave the same answer as the majority of those who answered before them, and if "Yes" and "No" answers were equal, they gave either answer. It turned out that the number of "Yes" answers was exactly 1009. What is the maximum number of followers that could have been among the islanders?
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Context:
The Binomial Expansion is valid for exponents that are not integers. That is, for all real numbers $x,y$ and $r$ with $|x|>|y|$ ,
$(x+y)^r=x^r+rx^{r-1}y+\dfrac{r(r-1)}{2}x^{r-2}y^2+\dfrac{r(r-1)(r-2)}{3!}x^{r-3}y^3 \cdots$
What are the first three digits to the right of the decimal point in the decimal representation of $(10^{2002}+1)^{\frac{10}{7}}$ ?
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Context:
In convex quadrilateral \(ABCD\) with \(AB = 11\) and \(CD = 13\), there is a point \(P\) for which \(\triangle ADP\) and \(\triangle BCP\) are congruent equilateral triangles. Compute the side length of these triangles.
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Context:
Let  $R$  be the set of points  $(x, y)$  such that  $x$  and  $y$  are positive,  $x + y$  is at most 2013, and
\[
  \lceil x \rceil \lfloor y \rfloor = \lfloor x \rfloor \lceil y \rceil.
\]
Compute the area of set  $R$ .  Recall that  $\lfloor a \rfloor$  is the greatest integer that is less than or equal to  $a$ , and  $\lceil a \rceil$  is the least integer that is greater than or equal to  $a$ .
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Context:
A certain percentage of antifreeze solution and 54 % antifreeze solution were combined to give 39 gallons of 58 % antifreeze solution. The amount of the first solution used was 26 gallons. What was the percentage of the first antifreeze solution?
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Context:
The last three digits of \( 1978^n \) and \( 1978^m \) are the same. Find the positive integers \( m \) and \( n \) such that \( m+n \) is minimized (here \( n > m \geq 1 \)).
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Context:
Given that |a| = 7, |b| = 5, and a < b, calculate all possible values of a - b. Express your answer as a list of integers.
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Context:
Let  $ABC$  be a triangle with  $\angle BAC=40^\circ $ ,  $O$  be the center of its circumscribed circle and  $G$  is its centroid. Point  $D$  of line  $BC$  is such that  $CD=AC$  and  $C$  is between  $B$  and  $D$ . If  $AD\parallel OG$ , find  $\angle ACB$ .
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Context:
Suppose \(a\), \(b\), and \(c\) are three distinct numbers randomly chosen from \(1, 2, \cdots, 5\), and \(ab + c\) is odd. Find the number of possible values of \(ab + c\). Express your answer as a single integer.
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Context:
One side of a triangle has length $75$. Of the other two sides, the length of one is double the length of the other. What is the maximum possible area for this triangle?
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Context:
For positive integers $n$, let $c_n$ be the smallest positive integer for which $n^{c_n}-1$ is divisible by $210$, if such a positive integer exists, and $c_n = 0$ otherwise. What is $c_1 + c_2 + \dots + c_{210}$?
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Context:
Find the smallest possible value of $n$ such that $n+2$ people can stand inside or on the border of a regular $n$-gon with side length $6$ feet where each pair of people are at least $6$ feet apart.
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Context:
128 students represent x percent of the boys at a school. If the boys at the school make up 50% of the total school population of x students, what is x?
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Context:
$A, B, G$ are three points on a circle with $\angle A G B=48^\circ$. The chord $AB$ is trisected by points $C$ and $D$ with $C$ closer to $A$. The minor arc $AB$ is trisected by points $E$ and $F$ with $E$ closer to $A$. The lines $EC$ and $FD$ meet at $H$. If $\angle A H B = x^\circ$, find $x$.
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Context:
Evita wants to write the numbers 1 to 8 in the boxes of the grid shown, so that the sums of the numbers in the boxes in each row are equal and the sums of the numbers in the boxes in each column are equal. She has already written numbers 3, 4, and 8, as shown. What number should she write in the shaded box? Express your answer as a single integer.
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Context:
Define a sequence of functions recursively by  $f_1(x) = |x-1|$  and  $f_n(x)=f_{n-1}(|x-n|)$  for integers  $n > 1$ . Find the least value of  $n$  such that the sum of the zeros of  $f_n$  exceeds  $500{,}000$ .
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Context:
Two trains move in the same direction at some speeds. The slower train moves at 32 kmph. A man in the slower train observes that 15 seconds elapse before the faster train completely passes by him. The length of the faster train is 75.006 meters. What is the speed of the faster train?
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Context:
Let $a,$ $b,$ $c$ be nonzero real numbers such that $a + b + c = 0,$ and $ab + ac + bc \neq 0.$  Find all possible values of
\[
\frac{a^7 + b^7 + c^7}{abc (ab + ac + bc)}.
\]
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Context:
Suppose $a$, $b$ and $c$ are positive integers with $a+b+c=2006$, and $a!b!c!=m\cdot 10^n$, where $m$ and $n$ are integers and $m$ is not divisible by $10$. What is the smallest possible value of $n$?
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Context:
A group of 7 fruit baskets contains 9 apples, 15 oranges, 14 bananas, 12 grapes, and 18 strawberries in the first three baskets. The fourth basket has 2 fewer of each fruit than the first basket, the fifth basket has 3 more apples and 5 fewer oranges than the first basket, the sixth basket contains twice as many bananas and half as many grapes as the third basket, and the seventh basket holds 1.5 times the number of strawberries in the second basket and 4 more oranges than the first basket. How many fruits are there in total?
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Context:
In a survey of parents and caregivers, exactly 5/6 of the mothers, 3/4 of the fathers, and 1/2 of the grandparents held full-time jobs. Part-time jobs were held by 1/6 of the mothers, 1/8 of the fathers, and 1/4 of the grandparents. The remaining parents and grandparents in the survey were either unemployed or not in the labor force. If 55 percent of the respondents surveyed were women, 35 percent were men, and 10 percent were grandparents, what percent of the surveyed individuals did not hold a full-time or part-time job?
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Context:
Given an ellipse $C$: $\frac{x^2}{8}+\frac{y^2}{4}=1$ with the left and right foci $F_{1}$ and $F_{2}$ respectively. A line $l$ passing through the right focus $F_{2}$ intersects $C$ at points $A$ and $B$. If the center of the circle passing through points $A$, $B$, and $(-\sqrt{7},0)$ lies on the $y$-axis, then the slope of line $l$ is ______.
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Context:
Misha drew a triangle with a perimeter of 11 and cut it into parts with three straight cuts parallel to the sides, as shown in the figure. The perimeters of the three shaded figures (trapezoids) turned out to be 5, 7, and 9. Find the perimeter of the small triangle that resulted from the cuts.
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Context:
设函数

$$f(\:x\:)=\sqrt{2x^2\:+2x\:+41}\:-\sqrt{2x^2\:+4x\:+4}\:(\:x\:\in\:\mathbf{R}\:)\:.$$
则$f(x)$的最大值为
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Context:
At a meeting of cactus enthusiasts, 80 cactophiles presented their collections, each consisting of cacti of different species. It turned out that no single species of cactus is found in all collections simultaneously, but any 15 people have cacti of the same species. What is the minimum total number of cactus species that can be in all collections?
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Context:
Jane started babysitting when she was 18 years old. Whenever she babysat for a child, that child was no more than half her age at the time. She charged different rates based on the child's age: $5 per hour for children under 2 years old, $7 per hour for children between 2 and 5 years old, and $8 per hour for children older than 5 years. Jane is currently 34 years old. She stopped babysitting for one year when she was 24, then resumed for three years before stopping permanently 6 years ago. During her babysitting career, Jane babysat four children with different ages: Child A was half her age when she started babysitting, Child B was 2 years younger than Child A, Child C was 3 years older than Child B, and Child D was the same age as Child C. Each child was babysat for a different number of hours: 50 hours for Child A, 90 hours for Child B, 130 hours for Child C, and 70 hours for Child D. Calculate the total amount of money Jane earned from babysitting all four children, considering their ages and varying babysitting rates.
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Context:
if 5 a = 3125 , then the value of 5 ( a - 3 ) is :
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Context:
A new mixed-use development has 60 floors, with varying types of residential floors and commercial spaces. The residential floors are divided into three different categories: regular floors, luxury floors, and penthouse floors. There are 25 regular floors, 20 luxury floors, and 10 penthouse floors. The remaining 5 floors are dedicated to commercial spaces, amenities, and uncounted residential units.

On the regular residential floors, there are 14 units on odd-numbered floors and 12 units on even-numbered floors. Luxury floors have a range of 6 to 10 units per floor, with the number of units decreasing as the floor number increases. Penthouse floors have only 2 units each.

There are 3 floors of commercial units with 5 shops each and 2 separate floors for amenities and uncounted residential units. The floor with amenities has 4 uncounted residential units, and the remaining floor has 6 uncounted residential units.

Calculate the total number of residential and commercial units in the entire mixed-use development, considering the varying number of units per floor type and the uncounted residential units on the specified floors.
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Context:
If we increase one leg of a right triangle by 5 and decrease the other leg by 5, the area of the triangle increases by 5. How does the area of the square drawn on the hypotenuse change in the meantime?
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Context:
Each cell of an infinite table (infinite in all directions) is colored with one of  $n$  given colors. All six cells of any  $2\times 3$  (or  $3 \times 2$ ) rectangle have different colors. Find the smallest possible value of  $n$ .
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Context:
Let $m$ be a real number where $m > 0$. If for any $x \in (1, +\infty)$, the inequality $2e^{2mx} - \frac{ln x}{m} ≥ 0$ always holds, then find the minimum value of the real number $m$.
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Context:
If you increase a number  $X$  by  $20\%,$  you get  $Y.$  By what percent must you decrease  $Y$  to get  $X?$
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Context:
A candy store has $100$ pieces of candy to give away. When you get to the store, there are five people in front of you, numbered from $1$ to $5$. The $i$th person in line considers the set of positive integers congruent to $i$ modulo $5$ which are at most the number of pieces of candy remaining. If this set is empty, then they take no candy. Otherwise, they pick an element of this set and take that many pieces of candy. For example, the first person in line will pick an integer from the set $\{1,6,\dots,96\}$ and take that many pieces of candy. How many ways can the first five people take their share of candy so that after they are done there are at least $35$ pieces of candy remaining?
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Context:
Find the greatest common divisor in $\mathbb{Z}[i]$ of $11+7i$ and $18-i$ using the Euclidean algorithm or ideal membership tests.
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Context:
将等差数列 $1,5,9,13, \ldots, 2017$ 排成一个大数 $15913 \ldots 2017$, 则该数被 9 除的余数为多少？
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Context:
Let \( A = \{1,2,3,4,5\} \). How many functions \( f: \mathscr{P}(A) \backslash \{\varnothing\} \rightarrow A \) are there that satisfy \( f(B) \in B \) and \( f(B \cup C) \in \{f(B), f(C)\} \) for all non-empty subsets \( B \) and \( C \) of \( A \)?
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Context:
The natural numbers \(a, b, c\) are chosen such that \(a < b < c\). It is also known that the system of equations \(2x + y = 2023\) and \(y = |x - a| + |x - b| + |x - c|\) has exactly one solution. Find the minimum possible value of \(c\).
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Context:
An  $m \times n$  chessboard where  $m \le n$  has several black squares such that no two rows have the same pattern. Determine the largest integer  $k$  such that we can always color  $k$  columns red while still no two rows have the same pattern.
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Context:
Suppose \(S\) cans of soda can be purchased from a vending machine for \(Q\) quarters. Find the expression that describes the number of cans of soda that can be purchased for \(D\) dollars, where \(1\) dollar is worth \(4\) quarters. The original answer is in \(\frac{kDS}{mQ}\) format, please give the value of k + m + n.
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Context:
Point \(P\) is inside an equilateral \(\triangle ABC\) such that the measures of \(\angle APB, \angle BPC, \angle CPA\) are in the ratio 5:6:7. Determine the ratio of the measures of the angles of the triangle formed by \(PA, PB, PC\) (in increasing order). Express your answer as a ratio of three numbers, separated by colons (e.g., a:b:c).
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Context:
Let  $ABC$  be a triangle (right in  $B$ ) inscribed in a semi-circumference of diameter  $AC=10$ . Determine the distance of the vertice  $B$  to the side  $AC$  if the median corresponding to the hypotenuse is the geometric mean of the sides of the triangle.
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Context:
Find the sum of all values of $a$ such that there are positive integers $a$ and $b$ satisfying $(a - b)\sqrt{ab} = 2016$.
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Context:
On the lateral side $CD$ of trapezoid $ABCD (AD \parallel BC)$, a point $M$ is marked. From vertex $A$, a perpendicular $AH$ is dropped to the segment $BM$. It is found that $AD = HD$. Find the length of segment $AD$ if it is known that $BC = 16$, $CM = 8$, $MD = 9$.
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Context:
If  $x$  is a real number, let  $\lfloor x \rfloor$  be the greatest integer that is less than or equal to  $x$ .  If  $n$  is a positive integer, let  $S(n)$  be defined by
\[
  S(n)
    = \left\lfloor \frac{n}{10^{\lfloor \log n \rfloor}} \right\rfloor
        + 10 \left( n - 10^{\lfloor \log n \rfloor}
               \cdot \left\lfloor \frac{n}{10^{\lfloor \log n \rfloor}} \right\rfloor
             \right) \, .
\]
(All the logarithms are base 10.)  How many integers  $n$  from 1 to 2011 (inclusive) satisfy  $S(S(n)) = n$ ?
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Context:
In a notebook, a triangular grid is drawn (see the figure). Tanya placed integers at the grid nodes. We define two numbers as close if they are located at adjacent grid nodes. It is known that:

- The sum of all ten numbers is 43.
- The sum of any three numbers, such that any two of them are close, is 11.

What is the value of the central number?
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Context:
As shown in the figure, \( C \) and \( D \) are two points on segment \( AB \), and \( M \) and \( N \) are the midpoints of \( AC \) and \( DB \) respectively. If \( AB = 10 \) and \( CD = 2 \), then find the length of \( MN \). Express your answer as a whole number.
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Context:
## Problem Statement

Calculate the definite integral:

$$
\int_{\frac{\pi}{3}}^{\frac{\pi}{2}} \frac{\cos x \, dx}{1+\sin x-\cos x}
$$
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Context:
Ten distinct positive real numbers are given and the sum of each pair is written (So 45 sums). Between these sums there are 5 equal numbers. If we calculate product of each pair, find the biggest number  $k$  such that there may be  $k$  equal numbers between them.
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Context:
What is the general method to find the number of lines that are both tangent and normal to a given curve, and how many such lines exist for the curve $y^2=x^4$?
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Context:
Given a function $f$ for which \[f(x) = f(398 - x) = f(2158 - x) = f(3214 - x)\] holds for all real $x,$ what is the largest number of different values that can appear in the list $f(0),f(1),f(2),\ldots,f(999)?$
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Context:
有1000盏灯和1000个开关，每个开关同时控制着所有号码是它的倍数的灯，开始所有灯都是亮着的，现在拉2,3,5号三个开关，那么亮着的灯有＿个．
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Context:
In the diagram, six squares form a \( 2 \times 3 \) grid. The middle square in the top row is marked with an \( R \). Each of the five remaining squares is to be marked with an \( R \), \( S \), or \( T \). In how many ways can the grid be completed so that it includes at least one pair of squares side-by-side in the same row or same column that contain the same letter?
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Context:
An editor is working on a new translation of a classic novel. The novel has a total of 480 pages. Each day, the editor translates an average of 12 pages. However, she also spends time reviewing her translations, which takes an additional 3 pages worth of time each day. If she works 5 days a week, how many weeks will it take her to complete the translation and review of the entire novel?
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Context:
In summation notation, what is the expected value of the squared difference between the sample mean and each individual observation, assuming the observations are iid normally distributed with $N(\mu , \sigma ^2)$?
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Context:
In triangle \(ABC\), medians \(AL\) and \(BM\) intersect at point \(K\). Vertex \(C\) lies on the circle that passes through points \(K\), \(L\), and \(M\). The length of side \(AB\) is \(a\). Find the length of median \(CN\).
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Context:
Evaluate $$\int \ln(2x+3)\mathrm{d}x$$
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\boxed{\left(x+\frac{3}{2}\right)\ln(2x+3) - x + C}


Context:
计算下列实变函数定积分．

$\int _ { 0 } ^ { \infty } \frac { x ^ { 2 } + 1 } { x ^ { 6 } + 1 } d x ,$
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Context:
The current value of a piece of machinery is $40,000. Its value depreciates at a rate of 5% per annum. Additionally, there are annual maintenance costs of $2000 that also increase by 3% every year due to inflation. Calculate the total net worth of the machine after 2 years considering both depreciation and maintenance costs.
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Context:
Points \( M \) and \( K \) are marked on the sides of angle \( ABC \) such that angles \( \angle BMC \) and \( \angle BKA \) are equal, \( BM = BK \), \( AB = 15 \), \( BK = 8 \), and \( CM = 9 \).

Find the perimeter of triangle \( COK \), where \( O \) is the intersection point of lines \( AK \) and \( CM \).
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Context:
Five line segments are drawn inside a rectangle, creating a decagon (10-sided polygon). Find the sum of the six marked angles. Express your answer in degrees as a whole number.
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Context:
A hotel has one triple room, one double room, and one single room available. Three adults and two children come to stay, and the children cannot stay in a room by themselves (they must be accompanied by an adult). If all three rooms are occupied, there are \_\_\_\_\_\_\_\_\_ different arrangements for their accommodation.
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Context:
设有三个函数，第一个是 $y=\varphi(x)$ ，它的反函数就是第二个函数，而第三个函数的图像与第二个函数的图像关于直线 $x+y=0$ 对称。求第三个函数的表达式。原始答案的形式为 $y = k\varphi(mx)$，请给出 $k + m$ 的值。
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Context:
On a table, there are three cones standing on their bases, touching each other. The heights of the cones are the same, and the radii of their bases are \(2r\), \(3r\), and \(10r\). A ball with a radius of 2 is placed on the table, touching all three cones. It turns out that the center of the ball is equidistant from all points of contact with the cones. Find \(r\).
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Context:
Given that \( A \), \( B \), and \( C \) are any three non-collinear points on a plane, and point \( O \) is inside \( \triangle ABC \) such that:
\[
\angle AOB = \angle BOC = \angle COA = 120^\circ.
\]
Find the maximum value of \( \frac{OA + OB + OC}{AB + BC + CA} \).
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Context:
For a nonnegative integer $n$ and a strictly increasing sequence of real numbers $t_0, t_1, \ldots, t_n$, let $f(t)$ be the corresponding real-valued function defined for $t \geq t_0$ by the following properties:
(a) $f(t)$ is continuous for $t \geq t_0$, and is twice differentiable for all $t>t_0$ other than $t_1, \ldots, t_n$;
(b) $f\left(t_0\right)=1 / 2$;
(c) $\lim _{t \rightarrow t_k^{+}} f^{\prime}(t)=0$ for $0 \leq k \leq n$;
(d) For $0 \leq k \leq n-1$, we have $f^{\prime \prime}(t)=k+1$ when $t_k<t<t_{k+1}$, and $f^{\prime \prime}(t)=n+1$ when $t>t_n$.
Considering all choices of $n$ and $t_0, t_1, \ldots, t_n$ such that $t_k \geq t_{k-1}+1$ for $1 \leq k \leq n$, what is the least possible value of $T$ for which $f\left(t_0+T\right)=2023$?
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Context:
Given \( k \geq 1 \). Find the maximum real number \( \lambda \) such that for any positive real numbers \( x_i > 0 \) \((i=1,2,\ldots,n)\) satisfying \( \sum_{i=1}^{n} \frac{1}{x_{i}} = 2016 \), the following inequality holds:
$$
\lambda \sum_{i=1}^{n} \frac{x_{i}}{1+x_{i}} \leq \left[\sum_{i=1}^{n} \frac{1}{x_{i}^{k}\left(1+x_{i}\right)}\right]\left(\sum_{i=1}^{n} x_{i}^{k}\right).
$$
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Context:
The letters L, K, R, F, O and the digits 1, 7, 8, 9 are "cycled" separately and put together in a numbered list. Determine the line number on which LKRFO 1789 will appear for the first time.
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Context:
In a hockey championship, teams receive a certain number of points when they win and 1 point when they tie. In one season, a team won a championship with 60 points. They won 12 games more than they tied. How many points does a team receive when they win?
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Context:
What is the minimum number of acute-angled triangles that must be arranged without gaps and overlaps to form an isosceles triangle with a $120^\circ$ vertex angle?
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Context:
The deputies in a parliament were split into $10$ fractions. According to regulations, no fraction may consist of less than five people, and no two fractions may have the same number of members. After the vacation, the fractions disintegrated and several new fractions arose instead. Besides, some deputies became independent. It turned out that no two deputies that were in the same fraction before the vacation entered the same fraction after the vacation.

Find the smallest possible number of independent deputies after the vacation.
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Context:
Determine the minimum of the following function defined in the interval $45^{\circ}<x<90^{\circ}$:

$$
y=\tan x+\frac{\tan x}{\sin \left(2 x-90^{\circ}\right)}
$$
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Context:
What is the volume of a regular quadrangular pyramid, given its base area Q and lateral area S?
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Context:
I was given the following exercise. Let $X$ be a random variable that takes non-negative natural number values such that $E[X]=1,E[X^2]=2,E[X^3]=5$. Find the minimum value of $P(X=0)$ using the Taylor expansion of the probability generating function at $z=1$.
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Context:
How many distinct trees with exactly 7 vertices are there? A tree in graph theory refers to a connected graph without cycles, which can be simply understood as connecting n vertices with n-1 edges. Provide your answer as a single integer.
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Context:
Compute
\[\tan 5^\circ + \tan 25^\circ + \tan 45^\circ + \dots + \tan 165^\circ.\]
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Context:
Given that the function y=f(x) is an even function, when x>0, f(x)=ln x. If M=f(-π), N=f(e), K=f(2), what is the relationship among M, N, and K? Express your answer as an inequality using the symbols >, <, and =.
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Context:
11 将集合$\{ 2 ^ { x } + 2 ^ { y } + 2 ^ { z } \vert x , y , z \in N ,$ $x$<$y$<$z$}中的数从小到大排列，第100个是（用数字作答）．
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Context:
Several students are competing in a series of three races. A student earns 5 points for winning a race, 3 points for finishing second, and 1 point for finishing third. There are no ties. What is the smallest number of points that a student must earn in the three races to be guaranteed of earning more points than any other student?
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Context:
How many ways can a schedule of 4 mathematics courses - algebra, geometry, number theory, and calculus - be created in an 8-period day if exactly one pair of these courses can be taken in consecutive periods, and the other courses must not be consecutive?
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Context:
For every integer n ≥ 3, the function g(n) is defined as the product of all the odd integers from 1 to n, inclusive. The difference between g(106) and g(some number) is 105. What is that some number?
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Context:
There are two values of \(a\) for which the equation \(4x^2 + ax + 8x + 9 = 0\) has only one solution for \(x\). The discriminant of the quadratic equation can be expressed in the form \(m + n = 0\). Please find the sum of these values of \(m\) and \(n\).
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Context:
Given that \( m \) and \( n \) are two distinct positive integers and the last four digits of \( 2019^{m} \) and \( 2019^{n} \) are the same, find the minimum value of \( m+n \).
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Context:
Define the function $f(x) = \lfloor x \rfloor + \lfloor \sqrt{x} \rfloor + \lfloor \sqrt{\sqrt{x}} \rfloor$ for all positive real numbers $x$. How many integers from $1$ to $2023$ inclusive are in the range of $f(x)$? Note that $\lfloor x \rfloor$ is known as the $\textit{floor}$ function, which returns the greatest integer less than or equal to $x$.
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Context:
Let $f(x)=-4^{x}+2^{x+1}-1$, $g(x)=\log_{10} (ax^{2}-4x+1)$. If for any $x_{1}\in \mathbb{R}$, there exists $x_{2}\in \mathbb{R}$ such that $f(x_{1})=g(x_{2})$, then the range of real number $a$ is ______.
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Context:
Given a positive integer $n$, let $M(n)$ be the largest integer $m$ such that \[ \binom{m}{n-1} > \binom{m-1}{n}. \] Evaluate \[ \lim_{n \to \infty} \frac{M(n)}{n}. \]
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Context:
On a $10 \times 10$ chessboard, several knights are placed. In any $2 \times 2$ square, there is at least one knight. What is the smallest number of cells these knights can threaten? (Note: A knight does not threaten the square on which it stands, but it does threaten the squares on which other knights are standing.)
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Context:
Amongst the seven numbers \( 3624, 36024, 360924, 3609924, 36099924, 360999924, \) and \( 3609999924 \), there are \( n \) of them that are divisible by 38. Find the value of \( n \).
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Context:
Freddy the Frog is situated at 1 on an infinitely long number line. On day $n$, where $n \ge 1$, Freddy can choose to hop 1 step to the right, stay where he is, or hop $k$ steps to the left, where $k$ is an integer at most $n+1$. After day 5, how many sequences of moves are there such that Freddy has landed on at least one negative number?
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Context:
Let $a=\sqrt{17}$ and $b=i \sqrt{19}$, where $i=\sqrt{-1}$. Find the maximum possible value of the ratio $|a-z| /|b-z|$ over all complex numbers $z$ of magnitude 1 (i.e. over the unit circle $|z|=1$ ).
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Context:
7. A digital clock displays hours, minutes, and seconds using two digits each (e.g., 10:09:18). Between 05:00:00 and 22:59:59 on the same day, the probability that all six digits on the clock are different is
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Context:
Determine the value of $s$ for which the function $v(x, y) = x^4 − sxy^3 + 10xy − 10x$ satisfies Laplace's Equation.
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Context:
Given that the sum of two prime numbers is $102$ and one of the prime numbers is greater than $30$, calculate the product of these two prime numbers.
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Context:
Let $ABCDE$ be an equiangular convex pentagon of perimeter $1$. The pairwise intersections of the lines that extend the sides of the pentagon determine a five-pointed star polygon. Let $s$ be the perimeter of this star. What is the difference between the maximum and the minimum possible values of $s$?
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Context:
将函数 $y=\sqrt{4+6 x-x^{2}}-2(x \in[0, 6])$ 的图象绕坐标原点逆时针方向旋转角 $\theta(0 \leqslant \theta \leqslant \alpha)$，得到曲线 $C$ 。若对于每一个旋转角 $\theta$ ，曲线 $C$ 都是一个函数的图象，则 $\alpha$ 的最大值为多少？答案的形式为$\arctan \frac{a}{b}$，请给出$a+b$的值。
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Context:
8. (5 points) The founder of a noble family received a plot of land. Each man in the family, upon dying, divided the land he inherited equally among his sons. If he had no sons, the land went to the state. No other members of the family gained or lost any land in any other way. In total, there were 150 people in the family. What is the smallest fraction of the original plot of land that any member of the family could have received?
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Context:
设a，b，c是互不相等的实数，则$( a - b ) ^ { 2 }$.$( b - c ) ^ { 2 }$. $( c - a ) ^ { 2 }$ 中不超过$\frac { 1 } { 2 } ( a ^ { 2 } + b ^ { 2 } + c ^ { 2 } )$的最少有几个？
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Context:
In the triangular pyramid \(ABCD\) with base \(ABC\), the lateral edges are pairwise perpendicular, \(DA = DB = 5, DC = 1\). A ray of light is emitted from a point on the base. After reflecting exactly once from each lateral face (the ray does not reflect from the edges), the ray hits a point on the pyramid's base. What is the minimum distance the ray could travel?
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Context:
In the figure, there are two identical regular hexagons inside a parallelogram. Calculate the fraction of the area of the parallelogram that the shaded portion occupies. Express your answer as a simplified fraction.
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Context:
Two nonadjacent vertices of a rectangle are $(4,3)$ and $(-4,-3),$ and the other two vertices have integer coordinates. How many rectangles satisfy these conditions?
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Context:
Find the number of integers $n$ for which $\sqrt{\frac{(2020 - n)^2}{2020 - n^2}}$ is a real number.
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Context:
Jeff makes 100 donuts a day for 365 days in a year. Jeff indulges in his creations, eating 4% of donuts each day. His friends Chris, Sam, and Emily drop by every month, consuming 15, 20, and 25 donuts respectively each time. He also donates 10% of his monthly donuts to a local charity and gives his neighbor 6% of his donuts at the end of the year. If 50 donuts can fit into one large box, how many such boxes can Jeff fill with his remaining donuts at the end of the year?
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Context:
A thousand points form the vertices of a convex polygon with 1000 sides. Inside this polygon, there are another 500 points placed such that no three of these 500 points are collinear. The polygon is triangulated in such a way that all of these 1500 points are vertices of the triangles, and none of the triangles have any other vertices. How many triangles result from this triangulation?
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Context:
将前 12 个正整数构成的集合 $M=\{1,2, \cdots, 12\}$ 中的元素分成 $M_{1}, M_{2}, M_{3}, M_{4}$ 四个三元子集, 使得每个三元子集中的三数都满足: 其中一数等于另外两数之和, 试求不同的分法的种数.
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Context:
Given the number 2550, calculate the sum of its prime factors.
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Context:
An acute isosceles triangle, $ABC$, is inscribed in a circle. Through $B$ and $C$, tangents to the circle are drawn, meeting at point $D$. If $\angle ABC = \angle ACB = 3 \angle D$ and $\angle BAC = k \pi$ in radians, then find $k$.
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Context:
We roll a fair die consecutively until the sum of the numbers obtained, \( S \), exceeds 100. What is the most probable value of \( S \)?
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Context:
Consider the integer \[N = 9 + 99 + 999 + 9999 + \cdots + \underbrace{99\ldots 99}_\text{321 digits}.\]Find the sum of the digits of $N$.
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Context:
Two lines with slopes $\dfrac{1}{3}$ and $3$ intersect at $(3,3)$. Find the area of the triangle enclosed by these two lines and the line $x+y=12$.
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Context:
A certain amount of money is divided amongst a, b, and c so that a may get 2/3 as much as b and c together. B gets a certain fraction as much as a and c together. The share of a is $80, and the total amount of money is $200. What is the ratio of b's share to the combined share of a and c?
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Context:
Given the vector
$$
\boldsymbol{a}_{k}=\left(\cos \frac{k \pi}{6}, \sin \frac{k \pi}{6}+\cos \frac{k \pi}{6}\right),
$$
where $k=0,1,\cdots,12$, find the value of $\sum_{k=0}^{11} \boldsymbol{a}_{k} \cdot \boldsymbol{a}_{k+1}$.
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Context:
Given a mixture of AlCl3, Al2(SO4)3, and Al(OH)3, determine the mass percentage of Al in the entire mixture if the mass of each compound is 50 g, 70 g, and 40 g, respectively.
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Context:
Given the function $y=f(x)$, let $A(x\_1,y\_1)$ and $B(x\_2,y\_2)$ be two distinct points on its graph with corresponding tangent line slopes $k\_A$ and $k\_B$. Define $K(A,B)=\frac{|{K}\_{A}-{K}\_{B}|}{|AB|}$ (where $|AB|$ is the length of the line segment $AB$) as the "approximate curvature" of the curve $y=f(x)$ between points $A$ and $B$. For the curve $y=\frac{1}{x}$, let $A\left(a,\frac{1}{a}\right)$ and $B\left(\frac{1}{a},a\right)$ with $a>0$ and $a\neq 1$. If $m\cdot K(A,B)>1$ holds true, find the range of values for the real number $m$.
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Context:
Determine the least real number $k$ such that the inequality
\[
\left(\frac{2a}{a-b}\right)^2 + \left(\frac{2b}{b-c}\right)^2 + \left(\frac{2c}{c-a}\right)^2 + k \geq 4\left(\frac{2a}{a-b} + \frac{2b}{b-c} + \frac{2c}{c-a}\right)
\]
holds for all real numbers $a, b, c$.
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Context:
Given a function $y = f(x)$ defined on $\mathbb{R}$ that is an even function satisfying $f(2 + x) = f(2 - x)$, with at least one zero in the interval $(0, 4)$ and $f(0) = 0$, determine the minimum number of zeros the function $y = f(x)$ has in the interval $(-8, 10]$. Express your answer as a single integer.
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Context:
The product of two positive numbers is equal to $50$ times their sum and $75$ times their difference.  Find their sum.
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Context:
A car starts from point \( O \) and moves along a straight road with a constant speed \( v \). At the same time, a person on a bicycle, located at a distance \( a \) from point \( O \) and \( b \) from the road, begins to ride with the intention of delivering a letter to the car driver. What is the minimum speed the cyclist must maintain to achieve this goal?
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Context:
In the following right-angled triangle ABC, ∠C = BC = 1 and DEF is an arc of a circle with center A. Suppose the shaded areas BDE and CEF are equal and AD = x/√π. Calculate the value of x.
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Context:
According to the Gregorian calendar, a total of 97 leap days must be inserted over 400 consecutive years. After how many years will there be a 1-day discrepancy between the Gregorian calendar and the "exact" calendar if a year has a length of 365 days, 5 hours, 48 minutes, and 46 seconds?
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Context:
A person crosses a 2,500 m long street in 8 minutes. What is their speed in miles per hour?
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Context:
从 $[0,10]$ 中任取一个数 $x$, 从 $[0,6]$ 中任取一个数 $y$, 计算使得
$$
|x-5|+|y-3| \leqslant 4
$$
的概率。原始答案的形式为\frac{k}{m}，请给出k+m的值。
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Context:
In a competition with a total of 4 multiple-choice questions, each question has three choices: $A, B,$ and $C$. For any three contestants participating, there is at least one question for which their answers are all different. What is the maximum number of participants allowed in the competition?
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Context:
8.6. What is the maximum number of sides a coin can have as a projection of a convex polyhedron with $n$ faces?
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Context:
A person having contracted conjunctivitis infects a total of 144 people after two rounds of infection. Determine the average number of people each infected person infects in each round.
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Context:
A *short-sighted* rook is a rook that beats all squares in the same column and in the same row for which he can not go more than  $60$ -steps.
What is the maximal amount of short-sighted rooks that don't beat each other that can be put on a  $100\times 100$  chessboard.
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Context:
Given the cone $z^2=x^2+y^2$, $z\geq 0$ and the cylinder $z^2+y^2=64$, find the surface area of the section of the cone inside the cylinder.
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Context:
There are some positive factors of 15 and another number, and these are also the common factors in these numbers. The total number of these factors is 3. What is the second number?
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Context:
In a warehouse, transporting out 30 tons of goods is recorded as -30 tons. Therefore, transporting in -20 tons of goods represents ______.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $ABCD$ be a convex quadrilateral with $AC = 20$, $BC = 12$, and $BD = 17$. If $\angle CAB = 80^{\circ}$ and $\angle DBA = 70^{\circ}$, then find the area of $ABCD$.
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Context:
A cylinder has a volume of $72\pi$ $\text{cm}^3$. Calculate the combined volume of a cone with the same height as the cylinder and a hemisphere that fits exactly on the top of the cylinder's open end. The radius of the hemisphere is equal to the radius of the cylinder and the cone. Express your answer in terms of $\pi$.
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Context:
Two semicircles of diameters 2 and 4 are placed such that the smaller is inside the larger one and their diameters are along the same line. A square of side 1 is inscribed in the larger semicircle such that one of its sides is along the diameter of the semicircles. Calculate the area of the lune formed by the space in the smaller semicircle that is outside the larger semicircle and subtract the area covered by the square.
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Context:
The café has enough chairs to seat $310_5$ people. If $3$ people are supposed to sit at one table, how many tables does the café have?
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Context:
Given the quadratic equation $ax^{2}+bx+c=0$ with $a > 0$ and $b, c \in \mathbb{R}$, and the roots of the equation lying in the interval $(0, 2)$, determine the minimum value of the real number $a$ given that $25a+10b+4c \geqslant 4$ for $c \geqslant 1$.
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Context:
11 people were standing in line under the rain, each holding an umbrella. They stood so close together that the umbrellas touched each other. Once the rain stopped, people closed their umbrellas and maintained a distance of 50 cm between each other. By how many times did the length of the queue decrease? Assume people can be considered as points and umbrellas as circles with a radius of 50 cm.
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Context:
(5 points) Out of the eight playing cards which are Hearts of 2, 3, 4, 5 and Clubs of 2, 3, 4, 5, four cards are drawn and placed in a row. If the sum of the numbers on the drawn cards is 14, then the total number of different arrangements is ___ (answer in numerals).
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Context:
the radius of a semi circle is 7 cm then its perimeter is ?
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Context:
58 flags were planted at equal intervals on both sides of the 191.8 meters (m) long road. If flags are planted at both the beginning and end of the road, find the distance between the flags in meters (m).
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Context:
a trader cheats both his supplier and customer by using faulty weights . when he buys from the supplier , he takes 10 % more than the indicated weight . when he sells to his customer , he gives the customer a weight such that 50 % of that is added to the weight , the weight claimed by the trader is obtained . if he charges the cost price of the weight that he claims , find his profit percentage .
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Context:
Mona plays a multiplayer video game on the weekends. She joins groups with four other players to complete objectives in the game. The past weekend, she joined 9 groups and grouped with 33 unique players. In one of the groups, she had grouped with a certain number of players before that weekend, and another group included one person she had grouped with before. How many players had Mona grouped with before in the first group?
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Context:
What is the sum of the interior numbers of the eighth row of Pascal's Triangle?
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Context:
Let $P(x)=x^3+ax^2+bx+c$ where $a,b,c$ are positive real numbers. If $P(1)\geq 2$ and $P(3)\leq 31$, find how many integers $P(4)$ can take.
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Context:
3. (3 points) Anya, Vanya, Danya, Sanya, and Tanya were collecting apples. It turned out that each of them collected a whole percentage of the total number of collected apples, and all these numbers were different and greater than zero. Then Tanya, who collected the most apples, ate her apples. After that, it turned out that each of the children still had a whole percentage, but now of the remaining number of apples. What is the minimum number of apples that could have been collected?
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Context:
A circle with a radius of \(1 + \sqrt{2}\) is circumscribed around an isosceles right triangle. Find the radius of a circle that is tangent to the legs of this triangle and internally tangent to the circumscribed circle.
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Context:
Let $P$ be the set of positive integers that are prime numbers. Find the number of subsets of $P$ that have the property that the sum of their elements is $34$, such as $\{3, 31\}$.
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Context:
For some positive integer $n > m$, it turns out that $n$ can be represented as a sum of $2021$ non-negative integer powers of $m$, and also as a sum of $2021$ non-negative integer powers of $m+1$. Find the maximal value of the positive integer $m$.
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Context:
求 $\sum\limits_{i=1}^{2024}\left[\frac{19^{i}}{20}\right]$ 模 $7$ 的余数.
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Context:
For how many rational numbers $p$ is the area of the triangle formed by the intercepts and vertex of $f(x) = -x^2 + 4px - p + 1$ an integer?
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Context:
Given that $k \in \mathbb{Z}$, if the curve $x^{2}+y^{2}=k^{2}$ has no intersection points with the curve $xy=k$, then $k=$ _____ .
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Context:
Given are sheets and the numbers  $00, 01, \ldots, 99$  are written on them. We must put them in boxes  $000, 001, \ldots, 999$  so that the number on the sheet is the number on the box with one digit erased. What is the minimum number of boxes we need in order to put all the sheets?
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Context:
Given $x$, $y$, and $a \in R^*$, and when $x + 2y = 1$, the minimum value of $\frac{3}{x} + \frac{a}{y}$ is $6\sqrt{3}$. Then, calculate the minimum value of $3x + ay$ when $\frac{1}{x} + \frac{2}{y} = 1$.
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Context:
In the binomial expansion of (x^2 - $\frac {3}{x}$)^n, the binomial coefficient of the fifth term is the maximum and unique. Find the value of $C_{ n }^{ 0 }$ - $\frac {1}{2}$ $C_{ n }^{ 1 }$ + $\frac {1}{4}$ $C_{ n }^{ 2 }$ - ... + (-1)^n $\frac {1}{2}$ $C_{ n }^{ n }$.
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Context:
a train 600 m long can cross an over brdge of 100 m long in 70 sec and then find the speed of the train ?
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Context:
Find the range of the function
\[f(x) = \sqrt{x^2 - 10x + 34} - \sqrt{x^2 + 4}.\]
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Context:
Given the proposition ∀x > 1, x² + x + 1 > 0, what is the negation of this proposition? Express your answer as a logical statement.
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Context:
If the complex number \( z \) satisfies \( |z| = 2 \), then the maximum value of \( \frac{\left|z^{2}-z+1\right|}{|2z-1-\sqrt{3}i|} \) is _______.
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Context:
Find the biggest real number  $C$ , such that for every different positive real numbers  $a_1,a_2...a_{2019}$  that satisfy inequality : $\frac{a_1}{|a_2-a_3|} + \frac{a_2}{|a_3-a_4|} + ... + \frac{a_{2019}}{|a_1-a_2|} > C$
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Context:
In $\triangle ABC,$ $AB=20$, $AC=24$, and $BC=18$. Points $D,E,$ and $F$ are on sides $\overline{AB},$ $\overline{BC},$ and $\overline{AC},$ respectively. Angle $\angle BAC = 60^\circ$ and $\overline{DE}$ and $\overline{EF}$ are parallel to $\overline{AC}$ and $\overline{AB},$ respectively. What is the perimeter of parallelogram $ADEF$?
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Context:
已知奇素数 $x 、 y 、 z$ 满足
$$
x\left|\left(y^{5}+1\right), y\right|\left(z^{5}+1\right), z \mid\left(x^{5}+1\right)
$$
求乘积 $x y z$ 的最小值。
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Context:
$已知f(x) = (x-1)e^{x} + \frac{1}{2}ax^{2}.$
$若f(x)有2个不同零点，求a的取值范围.$
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Context:
The trains. At Würzltown station, an old lady looked out of the window and shouted:
- Conductor! How long is the trip from here to Madville?
- All trains take 5 hours in either direction, ma'am, he replied.
- And how many trains will I encounter on my way?

This seemingly odd question puzzled the conductor, but he readily answered:
- Trains from Würzltown to Madville and from Madville to Würzltown depart at five minutes past the hour, every hour, with an exact interval of one hour.

The old lady prompted one of her compartment neighbors to find the answer to her question.

How many trains will she encounter on her way?
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Context:
A six-digit number is formed by the digits 1, 2, 3, 4, with two pairs of repeating digits, where one pair of repeating digits is not adjacent, and the other pair is adjacent. Calculate the number of such six-digit numbers.
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Context:
Real nonzero numbers $x, y, z$ satisfy $x + y + z = 0$. Furthermore, it is given that:
\[
A = \frac{x}{y} + \frac{y}{z} + \frac{z}{x} = \frac{x}{z} + \frac{z}{y} + \frac{y}{x} + 1
\]
Determine the value of $A$.
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Context:
For positive integers $n$ and $k$, let $\mho(n,k)$ be the number of distinct prime divisors of $n$ that are at least $k$. For example, $\mho(90, 3)=2$, since the only prime factors of $90$ that are at least $3$ are $3$ and $5$. Find the closest integer to
\[\sum_{n=1}^\infty \sum_{k=1}^\infty \frac{\mho(n,k)}{3^{n+k-7}}.\]

[i]Proposed by Daniel Zhu.[/i]
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Context:
Suppose  $n$  is a perfect square. Consider the set of all numbers which is the product of two numbers, not necessarily distinct, both of which are at least  $n$ . Express the  $n-$ th smallest number in this set in terms of  $n$ .
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Context:
In a single-round chess tournament with 10 players, how many rounds are needed at the minimum for a clear winner to emerge ahead of time? (In each round, participants are paired up. A win awards 1 point, a draw awards 0.5 points, and a loss awards 0 points).
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Context:
Given $m \neq n$, $x = m^4 - m^3n$, $y = nm^3 - n^4$, compare the size of $x$ and $y$.
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Context:
A sequence of positive integers is to be placed in each box. The product of any four adjacent integers is always 120. Given the sequence:

|   |   | 2 |   |   | 4 |   | x |   |   | 3 |   |   |

Find the value of x. Express your answer as a single integer.
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Context:
Inside a semicircle with a radius of 12, there is a circle with a radius of 6 and a small semicircle, all pairwise tangent to each other, as shown in the diagram. Find the radius of the small semicircle.
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Context:
A man with mass \(m\) jumps off a high bridge with a bungee cord attached to his ankles. The man falls through a maximum distance \(H\), at which point the bungee cord brings him to a momentary rest before he bounces back up. The bungee cord is perfectly elastic, obeying Hooke's force law with a spring constant \(k\), and stretches from an original length of \(L_0\) to a final length \(L = L_0 + h\). The maximum tension in the bungee cord is four times the weight of the man.

Determine the spring constant \(k\). The original answer is in the form \(\frac{kmg}{nH}\). Please provide the value of \(k+n\).
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Context:
已知复数$z _ { 1 } 、 z _ { 2 }$满足$\vert z _ { 1 } \cdot z _ { 2 } \vert = 3 , z _ { 1 } + z _ { 2 } = 2 i$，求$\vert z _ { 1 } \vert$的最大值及最小值
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\boxed{$\left|z_1\right|$ 的最大值为 3 , 最小值为 1 .}


Context:
How many single-digit odd numbers and two-digit even numbers are there?
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Context:
6. In square $A B C D$, there is a point $M$, point $P$ is on $A B$, point $Q$ is on $C D$, and $\angle A P M=150^{\circ}$, $\angle M Q C=35^{\circ}$, then $\angle P M Q=$ $\qquad$ $\circ$.
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Context:
Given the vector $\overrightarrow{a}=(2,l)$ and the conditions $\overrightarrow{a} \cdot \overrightarrow{b}=10$ and $|\overrightarrow{a}+\overrightarrow{b}|=5 \sqrt {2}$, find $|\overrightarrow{b}|$.
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Context:
There are 2 boys and 3 girls, a total of 5 students standing in a row. If boy A does not stand at either end, and exactly 2 of the 3 girls stand next to each other, then the number of different arrangements is
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Context:
A square has a 6x6 grid, where every third square in each row following a checkerboard pattern is shaded. What percent of the six-by-six square is shaded?
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Context:
Let point $P$ lie on the curve $y= \frac {1}{2}e^{x}$, and point $Q$ lie on the curve $y=\ln (2x)$. Find the minimum value of $|PQ|$.
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Context:
The factors of $x^4+64$ are:
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Context:
If the positive numbers $a$ and $b$ satisfy the equation $2+\log_{2}a=3+\log_{3}b=\log_{6}(a+b)$, find the value of $\frac{1}{a}+\frac{1}{b}$.
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Context:
Let $a, b$ be two real numbers such that \(\sqrt[3]{a} - \sqrt[3]{b} = 10\) and \(ab = \left( \frac{8 - a - b}{6}\right)^3\). Find \(a - b\).
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Context:
Determine the value of $f_{1991}(2^{1990})$ where for a given positive integer $k$, the function $f_1(k)$ is defined as the square of the sum of its digits. The sequence is recursively defined as $f_{n+1}(k) = f_1(f_n(k))$.
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Context:
A box contains 3 black balls and 4 white balls. Balls are drawn randomly one at a time without replacement. The process continues until all balls of one color have been drawn. Calculate the probability that the last ball drawn is a black ball. Express your answer as a simplified fraction.
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Context:
There is a mathematical operator "$\odot$" that satisfies the following equations: $2 \odot 4=8, 4 \odot 6=14, 5 \odot 3=13, 8 \odot 7=23$. According to this rule, what is $9 \odot 3=?$
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Context:
2. (15 points) The bisectors of the obtuse angles at the base of a trapezoid intersect on the other base of the trapezoid. Find the area of the trapezoid if its height is 12 cm, and the lengths of the bisectors are 15 cm and $13 \mathrm{~cm}$.
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Context:
Compute the exact value of the following limit: $$\lim_{n \to \infty} \frac{{n^2 \choose n}}{{n^2+n-1 \choose n}}.$$
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Context:
Let  $\Omega$  be a circle with radius  $18$  and let  $\mathcal{S}$  be the region inside  $\Omega$  that the centroid of  $\triangle XYZ$  sweeps through as  $X$  varies along all possible points lying outside of  $\Omega$ ,  $Y$  varies along all possible points lying on  $\Omega$  and  $XZ$  is tangent to the circle. Compute the greatest integer less than or equal to the area of  $\mathcal{S}$ .
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Context:
A square with a side of 100 was cut into two equal rectangles. They were joined together as shown in the picture. Find the perimeter of the resulting figure.
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Context:
Bag A initially contains 3 white balls and 5 black balls, while bag B initially contains 4 white balls and 6 black balls. A ball is randomly taken from bag A and put into bag B. After thoroughly mixing, a ball is then randomly taken from bag B and put back into bag A. Find the probability that the number of white balls in bag A does not decrease. Express your answer as a fraction.
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Context:
$n$ consecutive positive integers are arranged in a row (not necessarily in order) such that the sum of any three successive integers in the row is divisible by the leftmost number in the triple. Determine the largest possible value of $n$ if the last number in the row is odd.
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Context:
The number $a=\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers, has the property that the sum of all real numbers $x$ satisfying \[\lfloor x \rfloor \cdot \{x\} = a \cdot x^2\] is $420$, where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$ and $\{x\}=x- \lfloor x \rfloor$ denotes the fractional part of $x$. What is $p+q$?
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Context:
The ratio of the number of games won to the number of games lost (no ties) by the Middle School Middies is $11/4$. To the nearest whole percent, what percent of its games did the team lose?
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Context:
Define the annoyingness of a permutation of the first \( n \) integers to be the minimum number of copies of the permutation that are needed to be placed next to each other so that the subsequence \( 1, 2, \ldots, n \) appears. For instance, the annoyingness of \( 3, 2, 1 \) is 3, and the annoyingness of \( 1, 3, 4, 2 \) is 2. A random permutation of \( 1, 2, \ldots, 2022 \) is selected. Compute the expected value of the annoyingness of this permutation.
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Context:
Origami. Mitya plans to fold a square sheet of paper \(ABCD\). Mitya calls a fold beautiful if side \(AB\) intersects side \(CD\) and the four resulting right triangles are congruent.

Before this, Vanya chooses a random point \(F\) on the sheet. Find the probability that Mitya can make a beautiful fold passing through point \(F\).
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Context:
The weight of cows of the Red Gorbatov breed is a random variable \(X\) distributed according to the normal law with a mean \(a = 470 \, \text{kg}\) and a standard deviation \(\sigma = 30 \, \text{kg}\). What is the probability that two out of three randomly selected cows will have a weight more than 470 kg and less than 530 kg?
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Context:
A function f(n) has the property that f(f(n)) + f(n) = 2n + 3, and f(0) = 1. If f(2015) = 2016, what is the value of n for which we are finding f(n)?
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Context:
A primary school conducted a height survey. For students with heights not exceeding 130 cm, there are 99 students with an average height of 122 cm. For students with heights not less than 160 cm, there are 72 students with an average height of 163 cm. The average height of students with heights exceeding 130 cm is 155 cm. The average height of students with heights below 160 cm is 148 cm. How many students are there in total?
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Context:
Suppose that the number $\sqrt{7200} - 61$ can be expressed in the form $(\sqrt a - b)^3,$ where $a$ and $b$ are positive integers. Find $a+b.$
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Context:
On the board, we write a series of $n$ numbers, where $n \geq 40$, and each one of them is equal to either $1$ or $-1$, such that the following conditions both hold:

1. The sum of every $40$ consecutive numbers is equal to $0$.
2. The sum of every $42$ consecutive numbers is not equal to $0$.

We denote by $S_n$ the sum of the $n$ numbers on the board. Find the maximum possible value of $S_n$ for all possible values of $n$.
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Context:
Let $a$ and $b$ be two non-perpendicular skew lines. Planes $\alpha$ and $\beta$ are constructed passing through lines $a$ and $b$ respectively. Consider the following four situations: $b \parallel \alpha$, $b \perpendicular \alpha$, $\alpha \parallel \beta$, $\alpha \perpendicular \beta$. How many of these situations are possible? Express your answer as a single integer.
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Context:
Given that \(a, b, x, y\) are all positive numbers, \(n\) is a positive integer, and \(x + y = 1\), find the minimum value of \(\frac{a}{x^n} + \frac{b}{y^n}\).
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Context:
Compute the value of the expression:
\[ 2(1 + 4(1 + 4(1 + 4(1 + 4(1 + 4(1 + 4(1 + 4(1 + 4(1 + 4))))))))) \]
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Context:
In triangle \(ABC\), \(AB = 32\), \(AC = 35\), and \(BC = x\). What is the smallest positive integer \(x\) such that \(1 + \cos^2 A\), \(\cos^2 B\), and \(\cos^2 C\) form the sides of a non-degenerate triangle?
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Context:
设 A 是有限集，且 |A|\geq2. A_1,A_2,\cdots,A_n 是 A 的子集，满足：(1) |A_1|=|A_2|=\cdots=|A_n|=k, k>\df{|A|}{2}；(2) 对任意 a,b\in A，存在 1\leq r<s<t\leq n，使得 a,b\in A_r\cap A_s\cap A_t；(3) 对任意 1\leq i<j\leq n，|A_i\cap A_j|\leq3.求当 k 取最大值时 n 的值.
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Context:
Given a solid $\Omega$ which is the larger part obtained by cutting a sphere $O$ with radius $4$ by a plane $\alpha$, and $\triangle ABC$ is an inscribed triangle of the circular section $O'$ with $\angle A=90^{\circ}$. Point $P$ is a moving point on the solid $\Omega$, and the projection of $P$ on the circle $O'$ lies on the circumference of $O'$. Given $OO'=1$, the maximum volume of the tetrahedron $P-ABC$ is \_\_\_\_\_\_
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Context:
How many elements in $S_8$ are conjugate to $(12)(345)$?
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Context:
Compute the smallest positive integer \( n \) for which
\[
\sqrt{100+\sqrt{n}} + \sqrt{100-\sqrt{n}}
\]
is an integer.
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Context:
Let $a$ be a number with $n$ digits ($n > 1$). A number $b$ with $2n$ digits is obtained by writing two copies of $a$ one after the other. If $\frac{b}{a^2}$ is an integer $k$, find the possible values of $k$.
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Context:
The sides of a triangle, as well as the inradius \( r \) and the radii of the excircles \( r_1, r_2, r_3 \), are integers, and the radii are even. Determine the sides if

\[ r r_{1} r_{2} + r r_{2} r_{3} + r r_{3} r_{1} + r_{1} r_{2} r_{3} = r r_{1} r_{2} r_{3} \]
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Context:
在平面直角坐标系内, 有四个定点 $A(-3,0), B(1,-1), C(0,3), D(-1,3)$及一个动点 $P$ ，则 $|P A|+|P B|+|P C|+|P D|$ 的最小值为a\sqrt{b}+c\sqrt{d}的形式。请给出该形式中所有系数的和。
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Context:
Find all values of \( a \) such that for each \( x \), the largest of the two numbers \( x^{3}+3x+a-9 \) and \( a+2^{5-x}-3^{x-1} \) is positive.
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Context:
From an external point \(A\), a tangent \(AB\) and a secant \(ACD\) are drawn to a circle. Find the area of triangle \(CBD\), given that the ratio \(AC : AB = 2 : 3\) and the area of triangle \(ABC\) is 20.
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Context:
Let \( W \) be the hypercube \(\left\{\left(x_{1}, x_{2}, x_{3}, x_{4}\right) \mid 0 \leq x_{1}, x_{2}, x_{3}, x_{4} \leq 1\right\}\). The intersection of \( W \) and a hyperplane parallel to \( x_{1}+x_{2}+x_{3}+x_{4}=0 \) is a non-degenerate 3-dimensional polyhedron. What is the maximum number of faces of this polyhedron?
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Context:
In a cube with edge length 1, named ABCD-A1B1C1D1, if point P is a point on an edge, then the number of points P that satisfy |PA|+|PC1|=2 is \_\_\_\_\_\_.
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Context:
Let the set $A=\{x|y= \sqrt {x^{2}-4x+3}\}$, and $B=\{y|y=x+ \frac {m}{x}(m > 0),x\in\complement_{\mathbb{R}}A\}$, if $2 \sqrt {m}\in B$, then the range of $m$ is \_\_\_\_\_\_.
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Context:
Evaluate the following double integral:
$$\int_F \arctan\left(\frac{x}{y}\right)\ dx\ dy,$$

where \(F\) is the region in the \(xy\)-plane bounded by the curves \(x^2+y^2=1\), \(x^2+y^2=4\), \(y=-|x|\), and \(y=|x|\).
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Context:
There is a product $a_{1} \cdot a_{2} \cdot \cdots \cdot a_{100}$ written on the blackboard, where $a_{1}, a_{2}, \cdots, a_{100}$ are positive integers. If one of the multiplication symbols is changed to an addition symbol (while keeping the rest multiplication symbols), it is found that 32 out of the 99 resultant sums are even. How many even numbers can there be at most among $a_{1}, a_{2}, \cdots, a_{100}$?
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Context:
Find the number of ordered tuples  $\left(C,A,M,B\right)$  of non-negative integers such that \[C!+C!+A!+M!=B!\]

*2019 CCA Math Bonanza Team Round #4*
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Context:
Karen is considering her winter coat options. One coat costs $300 and will last for 15 years. The other coat costs $120 and will last for a certain number of years. Karen would save $120 over 30 years by buying the more expensive coat. How many years will the less expensive coat last?
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Context:
Given that the graph of a power function passes through the points $(2,16)$ and $(\frac{1}{2},m)$, find the value of $m$.
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Context:
On the radius \(A O\) of a circle with center \(O\), a point \(M\) is chosen. Points \(B\) and \(C\) are chosen on the circle on one side of \(A O\) such that \(\angle A M B = \angle O M C = \alpha\). Find the length of \(B C\) if the radius of the circle is 12 and \(\cos \alpha = \frac{3}{4}\).
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Context:
If \( M = \left\{(x, y) \mid \tan \pi y + \sin^2 \pi x = 0\right\} \) and \( N = \left\{(x, y) \mid x^2 + y^2 \leqslant 2\right\} \), what is the number of elements in \( M \cap N \)?
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Context:
For a positive integer $n$, let $d(n)$ be the number of positive divisors of $n$. What is the smallest positive integer $n$ such that
\[\sum_{t \mid n} d(t)^3\]is divisible by $35$?
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Context:
Larry and Diane start $100$ miles apart along a straight road. Starting at the same time, Larry and Diane
drive their cars toward each other. Diane drives at a constant rate of 30 miles per hour. To make it
interesting, at the beginning of each 10 mile stretch, if the two drivers have not met, Larry ﬂips a fair coin.
If the coin comes up heads, Larry drives the next 10 miles at 20 miles per hour. If the coin comes up tails,
Larry drives the next 10 miles at 60 miles per hour. Larry and Diane stop driving when they meet. The expected number of times that Larry ﬂips the coin is $\frac{m}{n}$ , where m and n are relatively prime positive integers. Find $m + n.$
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Context:
In a regular hexagonal pyramid, the center of the circumscribed sphere lies on the surface of the inscribed sphere. Find the ratio of the radius of the circumscribed sphere to the radius of the inscribed sphere.
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Context:
Let $D$ be a regular ten-sided polygon with edges of length 1. A triangle $T$ is defined by choosing three vertices of $D$ and connecting them with edges. How many different (non-congruent) triangles $T$ can be formed?
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Context:
Determine the number of pairs $(a, b)$, where $1 \le a \le b \le 100$ are positive integers, so that $\frac{a^3+b^3}{a^2+b^2}$ is an integer.
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Context:
Given a regular square pyramid where all edges are of length 2, calculate the angle formed between a lateral edge and the base. Express your answer in degrees, including the degree symbol (°).
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Context:
A pad of paper comes with 60 sheets. Evelyn uses a pad of paper writing notes at work for a certain period. She takes Monday and Friday off from work. She uses 12 sheets of paper per day at work. How often does she use a pad of paper?
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Context:
Let $p$ be a prime with $p \equiv 3\pmod{4}$. Find the remainder of $$\sum_{i=1}^{p-1}\Big(\frac{i^8+1}{p}\Big)$$ divided by $p$.
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Context:
The alphabet in its natural order $\text{ABCDEFGHIJKLMNOPQRSTUVWXYZ}$ is $T_0$. We apply a permutation to $T_0$ to get $T_1$, which is $\text{JQOWIPANTZRCVMYEGSHUFDKBLX}$. If we apply the same permutation to $T_1$, we get $T_2$, which is $\text{ZGYKTEJMUXSODVLIAHNFPWRQCB}$. We continually apply this permutation to each $T_m$ to get $T_{m+1}$. Find the smallest positive integer $n$ such that $T_n = T_0$.
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Context:
In triangle $ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. If $a$, $b$, and $c$ form an arithmetic sequence, and $\sin A$, $\sin B$, $\sin C$ form a geometric sequence, then the angle $B=$ ______.
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Context:
A total of $2010$ coins are distributed in $5$ boxes. Initially, the quantities of coins in the boxes are consecutive natural numbers. Martha should choose and take one of the boxes, but before that, she can perform the following transformation any number of times: from a box with at least $4$ coins, she can transfer one coin to each of the other boxes.

What is the maximum number of coins that Martha can take away?
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Context:
From a circular log with radius \( R \), it is required to cut out a rectangular beam of maximum strength. It is known that the strength of the beam is directly proportional to the product of its width and the square of its height. What should the dimensions of the beam be to maximize its strength?
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Context:
Let $C\subseteq\mathbb{P}^{3}$ be the conic of equations $$ C=V(X_{3}, X_{0}X_{2}-X_{1}^{2})=\{(t_{0}^{2}:t_{0}t_{1}:t_{1}^{2}:0)\in\mathbb{P}^{3}:(t_{0}:t_{1})\in\mathbb{P}^{1}\}. $$ Determine the dimension of the set of surfaces $S\subseteq\mathbb{P}^{3}$ of degree $d$ (that we identify with $\mathbb{P}^{\binom{d+3}{3}-1}$) such that $C\subseteq S$.
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Context:
Let \( a, b, c \) be prime numbers such that \( a^5 \) divides \( b^2 - c \), and \( b + c \) is a perfect square. Find the minimum value of \( abc \).
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Context:
On graph paper, two right triangles are drawn. Find the sum of the angles BCA and \(\mathrm{B}_{1} \mathrm{C}_{1} \mathrm{~A}_{1}\).
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Context:
# Task 8.2

For a natural number $N$, all its divisors were listed, and then the sum of digits for each of these divisors was calculated. It turned out that among these sums, all numbers from 1 to 9 were found. Find the smallest value of $\mathrm{N}$.

## Number of points 7
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Context:
Each of the twelve letters in ``MATHEMATICS'' is written on its own square tile and placed in a bag. What is the probability that a tile randomly selected from the bag will have a letter on it that is in the word ``CALM''? Express your answer as a common fraction.
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Context:
Masha came up with the number \( A \), and Pasha came up with the number \( B \). It turned out that \( A + B = 2020 \), and the fraction \( \frac{A}{B} \) is less than \( \frac{1}{4} \). What is the maximum value that the fraction \( \frac{A}{B} \) can take?
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Context:
A light flashes in one of three different colors: red, green, and blue. Every  $3$  seconds, it flashes green. Every  $5$  seconds, it flashes red. Every  $7$  seconds, it flashes blue. If it is supposed to flash in two colors at once, it flashes the more infrequent color. How many times has the light flashed green after  $671$  seconds?
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Context:
设正整数$n \geq 3 , ( a _ { 1 } , a _ { 2 } , \cdots , a _ { n } )$是任意$n$元不同的实数，且其和是正数.若它的一个排列$( b _ { 1 } , b _ { 2 } , \cdots , b _ { n } )$满足对于任意的$k = 1 ,2 , \cdots , n ,$均有$b _ { 1 } + b _ { 2 } + \cdots + b _ { k } > 0$，则这个排列称为好的.求好的排列至少有多少个？
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Context:
For positive reals $p$ and $q$, define the [i]remainder[/i] when $p$ and $q$ as the smallest nonnegative real $r$ such that $\tfrac{p-r}{q}$ is an integer. For an ordered pair $(a, b)$ of positive integers, let $r_1$ and $r_2$ be the remainder when $a\sqrt{2} + b\sqrt{3}$ is divided by $\sqrt{2}$ and $\sqrt{3}$ respectively. Find the number of pairs $(a, b)$ such that $a, b \le 20$ and $r_1 + r_2 = \sqrt{2}$.
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Context:
Tegwen has the same number of brothers as she has sisters. Each one of her brothers has 50% more sisters than brothers. Using algebraic relationships, calculate the total number of children in Tegwen's family. Express your answer as a whole number.
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Context:
Let $x_1$ and $x_2$ be positive real numbers and define, for $n>2$:$$\displaystyle x_{n+1}=\sum_{k=1}^{n}\sqrt[n]{x_k}$$ Evaluate $\displaystyle \lim_{n\to\infty}\frac{x_n-n}{\ln n}$
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Context:
Patrick buys season passes for each of his 4 children to visit the local amusement park as often as they want. The passes have different prices based on the children's ages: $100 for the oldest, $90 for the second oldest, $80 for the third oldest, and $70 for the youngest. The oldest child went to the park 35 times, the second oldest went 25 times, the third oldest went 20 times, and the youngest went 15 times. Calculate the overall average cost per trip to the park for all children combined.
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Context:
S 是集合 \{1,2,\cdots,2023\} 的子集，满足任意两个元素的平方和不是 9 的倍数，则 |S| 的最大值是__________.（这里 |S| 表示 S 的元素个数）
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Context:
Find the number of $7$-tuples $(n_1,\ldots,n_7)$ of integers such that \[\sum_{i=1}^7 n_i^6=96957.\]
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Context:
Let $c$ and $d$ be real numbers such that
\[\frac{c}{d} + \frac{c}{d^3} + \frac{c}{d^6} + \dots = 9.\]
Find
\[\frac{c}{c + 2d} + \frac{c}{(c + 2d)^2} + \frac{c}{(c + 2d)^3} + \dotsb.\]
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Context:
10. There is a cup filled with a $16 \%$ saltwater solution. There are large, medium, and small iron balls, with their volume ratios being $10: 4: 3$. First, the small ball is submerged in the saltwater cup, causing $10 \%$ of the saltwater to overflow, and then the small ball is removed; next, the medium ball is submerged in the saltwater cup and then removed; then the large ball is submerged in the saltwater cup and removed; finally, pure water is added to the cup until it is full. What is the concentration of the saltwater in the cup at this point? (Round to one decimal place)
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Context:
Trains A and B start simultaneously from stations A and B, respectively, and travel towards each other. The speed ratio of Train A to Train B is 3:2. Station C is located between stations A and B. Train A reaches station C at 9 AM and Train B reaches station C at 7 PM. Determine the time at which trains A and B meet.
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Context:
Determine the largest natural number such that all its decimal digits are different and it is divisible by each of its digits.
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Context:
Xiaohua has a number of apples that he wants to distribute into several baskets. If he puts 4 apples in each basket, he will have 20 apples left undistributed. If he puts 8 apples in each basket, there will be one basket that does not have enough apples. Calculate the original number of apples Xiaohua had. Express your answer as a whole number.
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Context:
The points $A = (3,-4,2),$ $B = (5,-8,5),$ $C = (4,-3,0),$ and $D = (6,-7,3)$ in space form a flat quadrilateral.  Find the area of this quadrilateral.The answer is in the form k\sqrt{m}+n,. Please provide the value of k + m + n.
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Context:
Solve the following quadratic equation and find the positive difference between its solutions:
$$x^2 - 5x + m = 13 + (x+5)$$
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Context:
9 people decided to split the restaurant bill evenly . if the bill was $ 514.16 dollars , how much money did they 1 cent is the smallest unit ?
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Context:
Two numbers \(a\) and \(b\) with \(0 \leq a \leq 1\) and \(0 \leq b \leq 1\) are chosen at random. The number \(c\) is defined by \(c=2a + 2b\). The numbers \(a\), \(b\), and \(c\) are each rounded to the nearest integer to give \(A\), \(B\), and \(C\), respectively. Find the probability that \(2A + 2B = C\). Express your answer as a simplified fraction.
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Context:
B2. An integer $n$ is called a combi-number if every pair of different digits from all possible digits $0 \mathrm{t} / \mathrm{m} 9$ appear next to each other at least once in the number. Thus, in a combi-number, the digits 3 and 5 appear next to each other somewhere. It does not matter whether they appear in the order 35 or 53. We agree that a combi-number does not start with the digit 0.

What is the smallest number of digits a combi-number can consist of?
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Context:
A box contains $2$ red marbles, $2$ green marbles, and $2$ yellow marbles. Carol takes $2$ marbles from the box at random; then Claudia takes $2$ of the remaining marbles at random; and then Cheryl takes the last $2$ marbles. What is the probability that Cheryl gets $2$ marbles of the same color? The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.
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Context:
In a right-angled triangle, the angle bisector of the right angle and the median corresponding to the hypotenuse form a $16^{\circ}$ angle with each other. What are the angles of the triangle?
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Context:
What is the mass percentage of N in Dinitrogen pentoxide?
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Context:
Given that $a$, $b$, $c$ are the sides opposite to the angles $A$, $B$, $C$ respectively, with $a=2$, and $\frac{\sin A-\sin B}{\sin C}=\frac{c-b}{2+b}$, the maximum area of $\Delta ABC$ is ______.
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Context:
In a class of 20 students, all but 4 of the students put their names on a typed assignment. If the teacher randomly guesses, what is the probability that she correctly guesses which paper belongs to each of the four remaining students? Express your answer as a common fraction.
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Context:
以正方体的顶点为顶点的锐角三角形有 个.
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Context:
Integer  $n>2$  is given. Find the biggest integer  $d$ , for which holds, that from any set  $S$  consisting of  $n$  integers, we can find three different (but not necesarilly disjoint) nonempty subsets, such that sum of elements of each of them is divisible by  $d$ .
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Context:
Compare: $\tan \left( - \frac {13\pi}{7} \right)$ ______ $\tan \left( - \frac {15\pi}{8} \right)$.
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Context:
The product of the two $99$-digit numbers
$303,030,303,...,030,303$ and $505,050,505,...,050,505$
has thousands digit $A$ and units digit $B$. What is the sum of $A$ and $B$?
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Context:
Square $ABCD$ and rectangle $BEFG$ are placed as shown in the figure. $AG = CE = 2$ cm. What is the difference in the area between square $ABCD$ and rectangle $BEFG$ in square centimeters?
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Context:
Let  $\mathcal{P}$  be a parallelepiped with side lengths  $x$ ,  $y$ , and  $z$ .  Suppose that the four space diagonals of  $\mathcal{P}$  have lengths  $15$ ,  $17$ ,  $21$ , and  $23$ .  Compute  $x^2+y^2+z^2$ .
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Context:
A capricious mathematician writes a book with pages numbered from  $2$  to  $400$ . The pages are to be read in the following order. Take the last unread page ( $400$ ), then read (in the usual order) all pages which are not relatively prime to it and which have not been read before. Repeat until all pages are read. So, the order would be  $2, 4, 5, ... , 400, 3, 7, 9, ... , 399, ...$ . What is the last page to be read?
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