Context:
Find the number of primes $p$ between $100$ and $200$ for which $x^{11}+y^{16}\equiv 2013\pmod p$ has a solution in integers $x$ and $y$.
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Context:
It is half past eight on the clock. What is the angle between the hour and minute hands?
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Context:
I have two arithmetic sequences. The first term of the first sequence is $0$. The second term of the first sequence is the first term of the first sequence plus the first term of the second sequence. Similarly, the third term of the first sequence is the second term of the first sequence plus the second term of the second sequence. If the fifth term of the second sequence is $3$, what is the fifth term of the first sequence?
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Context:
Given that \( n \) is a natural number less than 50, find all \( n \) such that \( 3n + 5 \) and \( 5n + 4 \) have a common divisor greater than 1.
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Context:
On Andover's campus, Graves Hall is $60$ meters west of George Washington Hall, and George Washington Hall is $80$ meters north of Paresky Commons. Jessica wants to walk from Graves Hall to Paresky Commons. If she first walks straight from Graves Hall to George Washington Hall and then walks straight from George Washington Hall to Paresky Commons, it takes her $8$ minutes and $45$ seconds while walking at a constant speed. If she walks with the same speed directly from Graves Hall to Paresky Commons, how much time does she save, in seconds?
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Context:
What is the minimum number of circles required to completely cover a circle with twice the radius?
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Context:
Suppose that $A, B, C, D$ are four points in the plane, and let $Q, R, S, T, U, V$ be the respective midpoints of $A B, A C, A D, B C, B D, C D$. If $Q R=2001, S U=2002, T V=$ 2003, find the distance between the midpoints of $Q U$ and $R V$.
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Context:
求所有次数不超过 $2$ 的整系数多项式 $f$, 使得存在整系数多项式 $g$, 满足
$$
x^{3}-1 \mid f(x) g(x)-1
$$
当x=1时请提供所有满足条件的 $f(x)$ 的绝对值和。
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Context:
Let \( n \) be a positive integer. A sequence of \( n \) positive integers (not necessarily distinct) is called full if it satisfies the following condition: For each positive integer \( k \geq 2 \), if the number \( k \) appears in the sequence, then so does the number \( k-1 \), and moreover, the first occurrence of \( k-1 \) comes before the last occurrence of \( k \). For each \( n \), how many full sequences are there?
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Context:
A train 130 m long passes a man, running at a certain speed in the direction opposite to that of the train, in 6 seconds. The speed of the train is 71.99376049916006 kmph. What is the speed of the man in kmph?
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Context:
Two athletes start running simultaneously—one from point $A$ to point $B$ and the other from point $B$ to point $A$. They run at different but constant speeds and meet at a distance of 300 meters from point $A$. After reaching points $A$ and $B$, they immediately turn around and meet again at a distance of 400 meters from point $B$. Find the length of the segment $AB$.
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Context:
Determine the smallest prime number that does not divide any five-digit number whose digits are in strictly increasing order.
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Context:
Points \( A, B, C, \) and \( D \) lie on a straight line in that order. For a point \( E \) outside the line,

\[ \angle AEB = \angle BEC = \angle CED = 45^\circ. \]

Let \( F \) be the midpoint of segment \( AC \), and \( G \) be the midpoint of segment \( BD \). What is the measure of angle \( FEG \)?
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Context:
The trapezoid $ABCD$ is divided by the line segment $CE$ into a triangle and a parallelogram, as shown in the diagram. Point $F$ is the midpoint of the segment $CE$, the line $DF$ passes through the midpoint of the segment $BE$, and the area of the triangle $CDE$ is $3 \text{ cm}^{2}$.

Determine the area of the trapezoid $ABCD$.
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Context:
The number $2013$ has the property that it includes four consecutive digits ($0$, $1$, $2$, and $3$). Find how many $4$-digit numbers include $4$ consecutive digits.
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Context:
Given a point \( P \) in the plane, let \( S_P \) be the set of points whose distance from \( P \) is irrational. What is the smallest number of such sets whose union is the entire plane?
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Context:
从0,1,2,3,4,5,6,7,8,9这10个数中取3个数，使其和为不小于10的偶数，则不同的取法有种. （1998年全国高中联赛试题）
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Context:
How many permutations of 1, 2, 3, ... , n have each number larger than all the preceding numbers or smaller than all the preceding numbers?
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Context:
A pentagonal prism is used as the base of a new pyramid. One of the seven faces of this pentagonal prism will be chosen as the base of the pyramid. Calculate the maximum value of the sum of the exterior faces, vertices, and edges of the resulting structure after this pyramid is added.
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Context:
Humanity finds 12 habitable planets, of which 6 are Earth-like and 6 are Mars-like. Earth-like planets require 3 units of colonization resources, while Mars-like need 1 unit. If 18 units of colonization resources are available, how many different combinations of planets can be colonized, assuming each planet is unique?
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Context:
Determine the values of k for which the function $f(x)=x^k|x|$ is four times differentiable at the origin.
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Context:
Given a regular $n$-sided prism, where the area of the base is $S$. Two planes intersect all the lateral edges of the prism in such a way that the volume of the part of the prism between the planes is $V$. Find the sum of the lengths of the segments of the lateral edges of the prism that are enclosed between the planes, given that the planes do not have any common points inside the prism.
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Context:
Square $ABCD$ has sides of length 3. Segments $CM$ and $CN$ divide the square's area into three equal parts. How long is segment $CM$?
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Context:
先把一个半径为5，弧长为$6\pi $的扇形卷成一个体积为最大的空心圆锥，再把一个实心的铁球融化为铁水倒入此圆锥内(假设圆锥的侧面不渗漏，且不计损耗)，正好把此空心的圆锥浇铸成了一个体积最大的实心圆锥，则此球的半径为________.
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Context:
What is the difference between 70% of 120 and 6/9 of 150 divided by 80% of 250, when compared to 18% of 180 multiplied by 5/7 of 210?
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Context:
4.	Find the biggest positive integer $n$, lesser thar $2012$, that has the following property:
If $p$ is a prime divisor of $n$, then $p^2 - 1$ is a divisor of $n$.
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Context:
There are three commissions of bureaucrats. It is known for each pair of bureaucrats from different commissions that among the members of the remaining commission, there are exactly 10 bureaucrats who are acquainted with both and exactly 10 bureaucrats who are not acquainted with either. Find the total number of bureaucrats in the commissions.
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Context:
In the diagram, \( AE \) and \( BD \) are straight lines that intersect at \( C \). If \( BD = 16 \), \( AB = 9 \), \( CE = 5 \), and \( DE = 3 \), calculate the length of \( AC \). Express your answer as a whole number.
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Context:
A polynomial \( P(x) \) of degree 10 with a leading coefficient of 1 is given. The graph of \( y = P(x) \) lies entirely above the x-axis. The polynomial \( -P(x) \) was factored into irreducible factors (i.e., polynomials that cannot be represented as the product of two non-constant polynomials). It is known that at \( x = 2020 \), all the resulting irreducible polynomials take the value -3. Find \( P(2020) \).
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Context:
Given a triangle  $ABC$  with angles  $\angle A = 60^{\circ}, \angle B = 75^{\circ}, \angle C = 45^{\circ}$ , let  $H$  be its orthocentre, and  $O$  be its circumcenter. Let  $F$  be the midpoint of side  $AB$ , and  $Q$  be the foot of the perpendicular from  $B$  onto  $AC$ . Denote by  $X$  the intersection point of the lines  $FH$  and  $QO$ . Suppose the ratio of the length of  $FX$  and the circumradius of the triangle is given by  $\dfrac{a + b \sqrt{c}}{d}$ , then find the value of  $1000a + 100b + 10c + d$ .
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Context:
For $n \in \mathbb{N}$, let $P(n)$ denote the product of the digits in $n$ and $S(n)$ denote the sum of the digits in $n$. Consider the set \(A=\{n \in \mathbb{N}: P(n)\) is non-zero, square-free, and \(S(n)\) is a proper divisor of \(P(n)\}\). Find the maximum possible number of digits of the numbers in \(A\).
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Context:
The pentagon PQRST is divided into four triangles with equal perimeters. The triangle PQR is equilateral. PTU, SUT, and RSU are congruent isosceles triangles. Find the ratio of the perimeter of the pentagon PQRST to the perimeter of the triangle PQR. Express your answer as a simplified ratio.
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Context:
Find the smallest positive integer $a$ such that $x^4+a^2$ is not prime for any integer $x$.
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Context:
There is a certain number of boxes in a warehouse that can be divided evenly into 5 equal shipments by boat or some equal shipments by truck. The smallest number of boxes that could be in the warehouse is 120. How many equal shipments can be made by truck?
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Context:
Calculate the following limit: $$\lim_{n \to \infty} \frac{1}{(\ln n)^2} \sum_{i=1}^n \ln \left(\frac{i}{\sqrt[i]{e}}\right).$$
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Context:
Given an infinite grid where each cell is either red or blue, such that in any \(2 \times 3\) rectangle exactly two cells are red, determine how many red cells are in a \(9 \times 11\) rectangle.
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Context:
Given that $\sqrt {3}\sin x+\cos x= \frac {2}{3}$, find the value of $\tan (x+ \frac {7\pi}{6})$.
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Context:
Call a date \( mm/dd/yy \) \textit{multiplicative} if its month number times its day number is a two-digit integer equal to its year expressed as a two-digit year. For example, \( 01/21/21 \), \( 03/07/21 \), and \( 07/03/21 \) are multiplicative. Find the number of dates between January 1, 2022 and December 31, 2030 that are multiplicative.
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Context:
Last year, Nicki dedicated herself to a strict exercise regime comprising of running, swimming, and hiking. She started with 10 miles of running per week for the first three months, increased it to 20 miles per week for the next three, and then 30 miles per week for the last six months. For swimming, she maintained 5 miles per week for the first six months but didn't pursue it during the second half of the year. Additionally, every fourth week was a rest week, where she would only engage in hiking, covering a distance of 15 miles each time. Considering these variations and rest weeks, what is the total distance she covered in all her exercises during the year?
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Context:
From home to work and back. If a person walks to work and takes transport back, it takes him a total of one and a half hours. If he takes transport both ways, the entire journey takes him 30 minutes. How much time will it take for the person to walk both to work and back?
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Context:
This year, February $29$ fell on a Sunday. In which year will February $29$ next fall on a Sunday?
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Context:
Let $d(n)$ denote the number of positive integers that divide $n$, including $1$ and $n$. For example, $d(1)=1,d(2)=2,$ and $d(12)=6$. (This function is known as the divisor function.) Let\[f(n)=\frac{d(n)}{\sqrt [3]n}.\]There is a unique positive integer $N$ such that $f(N)>f(n)$ for all positive integers $n\ne N$. What is the sum of the digits of $N$?
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Context:
In a football tournament, 15 teams participated, each of which played exactly once with all the others. A win awarded 3 points, a draw 1 point, and a loss 0 points.

After the tournament ended, it turned out that some 6 teams each scored at least $N$ points. What is the largest possible integer value for $N$?
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Context:
On the sides \( AB \) and \( CD \) of an inscribed quadrilateral \( ABCD \), which is not a trapezoid, points \( P \) and \( R \) are marked, respectively, such that \( AP:PB = CR:RD \). On the segment \( PR \), a point \( Q \) is chosen such that \( PQ:QR = AB:CD \). Find the ratio of the areas of triangles \( AQD \) and \( BQC \) if it is known that \( AD = x \) and \( BC = y \).
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Context:
A man buys a house for $10,000 and rents it. He puts $12\frac{1}{2}\%$ of each month's rent aside for repairs and upkeep; pays $325 a year taxes and realizes $5\frac{1}{2}\%$ on his investment. The monthly rent (in dollars) is:
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Context:
已知椭圆 $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ 与圆 $x^{2}+y^{2}=b^{2}$, 过椭圆上一点 $M$ 作圆的两切线, 切点分别为 $P, Q$, 直线 $P Q$ 与 $x, y$轴分别交于点 $E, F$, 当$a=2,b=2$时，求 $S_{\triangle E O F}$ 的最小值.
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Context:
8(7/18)-(2(11/13)+1(7/18))
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Context:
Let  $ABC$  be an equilateral triangle with side length  $1$ . Points  $A_1$  and  $A_2$  are chosen on side  $BC$ , points  $B_1$  and  $B_2$  are chosen on side  $CA$ , and points  $C_1$  and  $C_2$  are chosen on side  $AB$  such that  $BA_1<BA_2$ ,  $CB_1<CB_2$ , and  $AC_1<AC_2$ .
Suppose that the three line segments  $B_1C_2$ ,  $C_1A_2$ ,  $A_1B_2$  are concurrent, and the perimeters of triangles  $AB_2C_1$ ,  $BC_2A_1$ , and  $CA_2B_1$  are all equal. Find all possible values of this common perimeter.
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Context:
two trains of length 250 m and 120 m are 50 m apart . they start moving towards each other on parallel tracks , at speeds 64 kmph and 42 kmph . after how much time will the trains meet ?
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Context:
一个正方体的8个顶点可以组成 个非等边三角形.
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Context:
During the weekends, Eli delivers milk in the complex plane. On Saturday, he begins at $z$ and delivers milk to houses located at $z^3,z^5,z^7,\ldots,z^{2013}$ in that order; on Sunday, he begins at $1$ and delivers milk to houses located at $z^2,z^4,z^6,\ldots,z^{2012}$ in that order.  Eli always walks directly (in a straight line) between two houses. If the distance he must travel from his starting point to the last house is $\sqrt{2012}$ on both days, find the real part of $z^2$.
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Context:
Two isosceles triangles are given with equal perimeters. The base of the second triangle is 15% larger than the base of the first, and the leg of the second triangle is 5% smaller than the leg of the first triangle. Find the ratio of the sides of the first triangle.
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Context:
When askes: "What time is it?", father said to a son: "Quarter of time that passed and half of the remaining time gives the exact time". What time was it?
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Context:
How many distinct sets of 8 positive odd integers sum to 20 ?
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Context:
Aileen plays badminton where she and her opponent stand on opposite sides of a net and attempt to bat a birdie back and forth over the net. A player wins a point if their opponent fails to bat the birdie over the net. When Aileen is the server (the first player to try to hit the birdie over the net), she wins a point with probability $\frac{9}{10}$. Each time Aileen successfully bats the birdie over the net, her opponent, independent of all previous hits, returns the birdie with probability $\frac{3}{4}$. Each time Aileen bats the birdie, independent of all previous hits, she returns the birdie with probability $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
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Context:
Find the smallest positive interger number  $n$  such that there exists  $n$  real numbers  $a_1,a_2,\ldots,a_n$  satisfied three conditions as follow:
a.  $a_1+a_2+\cdots+a_n>0$ ;
b.  $a_1^3+a_2^3+\cdots+a_n^3<0$ ;
c.  $a_1^5+a_2^5+\cdots+a_n^5>0$ .
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Context:
Captain Rusczyk tracked down a pirate who had stolen $2345_{6}$ dollars worth of goods from his ship. After winning an epic duel, the Captain demands that the pirate return $41324_{5}$ dollars. How much has the pirate gone in debt due to his two encounters with Rusczyk? Express your answer in base 10.
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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 $\sigma(n)$ represent the number of positive divisors of $n$. Define $s(n)$ as the number of positive divisors of $n+1$ such that for every divisor $a$, $a-1$ is also a divisor of $n$. Find the maximum value of $2s(n) - \sigma(n)$.
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Context:
For how many integers $n \neq 1$ does $(n-1)^3$ divide $n^{2018(n-1)} - 1$?
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Context:
Given that the solution set for the quadratic inequality $ax^2+bx+c<0$ is $(1, 2)$, find the solution set for the quadratic inequality $cx^2+bx+a<0$. Express your answer in interval notation.
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Context:
A sequence of integers has a mode of 32, a mean of 22, a smallest number of 10, and a median of m. If m is replaced by m+10, the new sequence has a mean of 24 and a median of m+10. If m is replaced by m-8, the new sequence has a median of m-4. Find the value of m. Express your answer as a single integer.
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Context:
The line of sight of a circular cone is 1, and the angle it forms with the base is $\theta$. The edge length of the inscribed cube in this cone is $\qquad$
$\qquad$ (4 vertices of the cube are on the base of the cone, and the other 4 vertices are on the side surface of the cone).
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Context:
In a sequence of coin tosses, one can keep a record of instances in which a tail is immediately followed by a head, a head is immediately followed by a head, and etc. We denote these by <tt>TH</tt>, <tt>HH</tt>, and etc. For example, in the sequence <tt>TTTHHTHTTTHHTTH</tt> of 15 coin tosses we observe that there are two <tt>HH</tt>, three <tt>HT</tt>, four <tt>TH</tt>, and five <tt>TT</tt> subsequences. How many different sequences of 15 coin tosses will contain exactly two <tt>HH</tt>, three <tt>HT</tt>, four <tt>TH</tt>, and five <tt>TT</tt> subsequences?
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Context:
Positive integers  $a_0<a_1<\dots<a_n$ , are to be chosen so that  $a_j-a_i$  is not a prime for any  $i,j$  with  $0 \le i <j \le n$ . For each  $n \ge 1$ , determine the smallest possible value of  $a_n$ .
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Context:
Anastasia is taking a walk in the plane, starting from $(1,0)$. Each second, if she is at $(x, y)$, she moves to one of the points $(x-1, y),(x+1, y),(x, y-1)$, and $(x, y+1)$, each with $\frac{1}{4}$ probability. She stops as soon as she hits a point of the form $(k, k)$. What is the probability that $k$ is divisible by 3 when she stops?
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Context:
11.183. A regular triangular prism is described around a sphere, and a sphere is described around it. Find the ratio of the surfaces of these spheres.
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Context:
Call a set of positive integers "conspiratorial" if no three of them are pairwise relatively prime. (A set of integers is "pairwise relatively prime" if no pair of them has a common divisor greater than 1.) What is the largest number of elements in any "conspiratorial" subset of the integers 1 through 16?
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Context:
A pentagon  $ABCDE$  is circumscribed about a circle. The angles at the vertices  $A{}$ ,  $C{}$  and  $E{}$  of the pentagon are equal to  $100^\circ$ . Find the measure of the angle  $\angle ACE$ .
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Context:
For polynomials $f(x)$, $g(x)$ of degree $d = 2^{r-1}+1$, how many multiplications in the field $F$ are required when multiplying $f(x)$ and $g(x)$ using the Karatsuba method?
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Context:
A circle intersects the  $y$ -axis at two points  $(0, a)$  and  $(0, b)$  and is tangent to the line  $x+100y = 100$  at  $(100, 0)$ . Compute the sum of all possible values of  $ab - a - b$ .
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Context:
Riley has 64 cubes with dimensions $1 \times 1 \times 1$. Each cube has its six faces labelled with a 2 on two opposite faces and a 1 on each of its other four faces. The 64 cubes are arranged to build a $4 \times 4 \times 4$ cube. Riley determines the total of the numbers on the outside of the $4 \times 4 \times 4$ cube. How many different possibilities are there for this total?
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Context:
Given a trapezoid \( ABCD \) with \( AD \parallel BC \). It turns out that \( \angle ABD = \angle BCD \). Find the length of segment \( BD \) if \( BC = 36 \) and \( AD = 64 \).
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Context:
Find the center of gravity of the arc of a circle \( x^{2} + y^{2} = R^{2} \), located in the first quadrant, if the density \( \rho = \rho_{0} x y \).
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Context:
There are 11 ones, 22 twos, 33 threes, and 44 fours on the blackboard. The following operation is performed: each time, three different numbers are erased, and two more of the fourth type of number (the one that was not erased) are added. For instance, if one 1, one 2, and one 3 are erased, then two more of 4 are added. After several operations, only three numbers remain on the blackboard, and no further operations can be performed. What is the product of the remaining three numbers?
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Context:
今有茭草一千八百二十束，欲令撒星形垛之，问底子几何？（撒星形的放法是第一层1束，第二层(1+3)束，第三层(1+3+6)束，···，第n层$( 1 + 3 + 6 + 1 0 + \cdots + \frac { n ( n + 1 ) } { 2 } )$束．本题即n层共1820束，求n.)
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Context:
若实数 $x, y$ 满足 $x-4 \sqrt{y}=2 \sqrt{x-y}$, 则 $x$ 的取值范围是 \qquad. 请给出区间端点之和。
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Context:
称平面上的 n 个正方形是"三连通的"，如果(1) 所有正方形都全等；(2) 若两个正方形有公共点 P，则 P 是它们每个的顶点；(3) 每个正方形都恰与其他三个正方形有公共点.问：在 2018\leq n\leq3018 中有多少个正整数 n，使得存在 n 个三连通的正方形？
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Context:
Given a rectangle \(ABCD\). On two sides of the rectangle, different points are chosen: six points on \(AB\) and seven points on \(BC\). How many different triangles can be formed with vertices at the chosen points?
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Context:
给定正整数 $n(n \geqslant 3)$. 对于 $1,2, \cdots, n$的任意一个排列 $P=\left(x_{1}, x_{2}, \cdots, x_{n}\right)$, 若 $i<j<k$, 则称 $x_{j}$ 介于 $x_{i}$ 和 $x_{k}$ 之间（如在排列（ $1,3,2,4$ ）中，3 介于 1 和 4 之间， 4 不介于 1 和 2 之间）。设集合 $S=\left\{P_{1}, P_{2}, \cdots, P_{m}\right\}$的每个元素 $P_{i}$ 是 $1,2, \cdots, n$ 的排列. 已知 $\{1,2, \cdots, n\}$ 的任意三个不同数中都有一个数，它在每个 $P_{i}(1 \leqslant i \leqslant m)$ 中都不介于另外两个数之间. 求 $m$ 的最大值。当 $n = 11$ 时，请给出答案。
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Context:
If x and y are numbers such that (x + 4)(y - 4) equals some number, the smallest possible value of x^2 + y^2 is 16. What is the number that (x + 4)(y - 4) equals?
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Context:
抛物线 $y=\frac{1}{2} x^{2}$ 与直线 $l: y=x+4$ 所围成区域中有一个矩形 $A B C D$, 且点 $A, B$ 在抛物线上, 点 $D$ 在直线 $l$ 上, 其中点 $B$ 在 $y$ 轴右侧, 且 $|A B|=2 t(t>0)$.
当边 $C D$ 在直线 $l$ 上时,求矩形 $A B C D$ 面积的最大值.
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Context:
On the island of Liars and Knights, a circular arrangement is called correct if everyone standing in the circle can say that among his two neighbors there is a representative of his tribe. One day, 2019 natives formed a correct arrangement in a circle. A liar approached them and said: "Now together we can also form a correct arrangement in a circle." How many knights could there have been in the initial arrangement?
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Context:
Problems 8, 9 and 10 use the data found in the accompanying paragraph and figures
Four friends, Art, Roger, Paul and Trisha, bake cookies, and all cookies have the same thickness. The shapes of the cookies differ, as shown.
$\circ$ Art's cookies are trapezoids:

$\circ$ Roger's cookies are rectangles:

$\circ$ Paul's cookies are parallelograms:

$\circ$ Trisha's cookies are triangles:

Each friend uses the same amount of dough, and Art makes exactly $12$ cookies. Art's cookies sell for $60$ cents each. To earn the same amount from a single batch, how much should one of Roger's cookies cost in cents?
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Context:
Ostap Bender organized an elephant distribution to the population in the city of Fux. 28 union members and 37 non-members arrived, and Ostap distributed the elephants equally among all the union members and equally among all the non-members. It turned out that there is only one way to distribute the elephants (so that all elephants are distributed). What is the maximum number of elephants Ostap Bender could have had? (It is assumed that each attendee received at least one elephant.)
Please reason step by step, and put your final answer within \boxed{}.


Context:
For which natural number \( K \) does the expression \(\frac{n^{2}}{1.001^{n}}\) reach its maximum value?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the diagram, \(PW\) is parallel to \(QX\). \(S\) and \(T\) lie on \(QX\), and \(U\) and \(V\) are the points of intersection of \(PW\) with \(SR\) and \(TR\), respectively. If \(\angle SUV = 120^\circ\) and \(\angle VTX = 112^\circ\), calculate the measure of \(\angle URV\). Express your answer in degrees (°).
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Context:
Let \( S(x) \) represent the sum of the digits of the natural number \( x \). Determine the set of \( x \) that satisfy the equation \( x + S(x) + S(S(x)) = 2014 \).
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Context:
Given two subsets $A$ and $B$ of size $k$ in the symmetric group $S_n$, both containing a given integer $j$, how many permutations in $S_n$ take $B$ onto $A$ under the action of $S_n$ induced on the subsets of $[n]$ of size $k$?
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Context:
In a chess festival that is held in a school with  $2017$  students, each pair of students played at most one match versus each other. In the end, it is seen that for any pair of students which have played a match versus each other, at least one of them has played at most  $22$  matches. What is the maximum possible number of matches in this event?
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Context:
Given a function f(x) defined on the real numbers, satisfying f''(x) < f''(-x) for any x in (0, +inf), and non-zero real numbers a and b such that f(a) - f(b) > f(-b) - f(-a), what is the relationship between a^2 and b^2? Express your answer as an inequality.
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Context:
Find the number of 9-digit US social security numbers that do not use the digit 8, using exponential generating functions.
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Context:
A bag contains two red beads and two green beads. You reach into the bag and pull out a bead, replacing it with a red bead regardless of the color you pulled out.  What is the probability that all beads in the bag are red after three such replacements? Express your answer as a common fraction.
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Context:
Shaq sees the numbers  $1$  through  $2017$  written on a chalkboard. He repeatedly chooses three numbers, erases them, and writes one plus their median. (For instance, if he erased  $-2, -1, 0$  he would replace them with  $0$ .) If  $M$  is the maximum possible final value remaining on the board, and if m is the minimum, compute  $M - m$ .
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let  $ABCD$  be a quadrilateral with an inscribed circle, centre  $O$ . Let
\[AO = 5, BO =6, CO = 7, DO = 8.\]
If  $M$  and  $N$  are the midpoints of the diagonals  $AC$  and  $BD$ , determine  $\frac{OM}{ON}$  .
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Context:
Find the explicit Dirichlet density of primes $p$ in $\\mathbb{Z}$ that split completely in the field $\\mathbb{Q}(\\sqrt{m})$ and remain inert in the field $\\mathbb{Q}(2^{1/3})$, where $m$ is a non-zero, square-free integer.
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Context:
Four friends went mushroom picking in the forest. Upon returning, each pair of them counted how many mushrooms they collected in total. They obtained the numbers $6, 7, 9, 9, 11, 12$. How many mushrooms did each friend collect?
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Context:
How many different (non-similar) triangles are there whose angles have an integer number of degrees?
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Context:
一只小虫在正八面体的表面上爬行，每秒从某一个顶点等可能地爬往四个相邻顶点之一。则小虫在第8秒爬回初始位置的概率的256倍为_______。
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Context:
A solid metal sphere with a radius of 20 has a cylindrical hole with a radius of 12 drilled through its center along a diameter. The remaining part is melted and recast into a new solid sphere. Find the radius of the new sphere.
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Context:
Determine all positive integer numbers  $n$  satisfying the following condition:
 the sum of the squares of any  $n$  prime numbers greater than  $3$  is divisible by  $n$ .
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Context:
Given that all elements of set A are real numbers, and it satisfies: if $a \in A$, then $\frac{1+a}{1-a} \in A$. If $a = -3$, please write down all elements of set A \_\_\_\_\_\_ .
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Context:
Let \( M \) be a subset of the set \(\{1, 2, \cdots, 25\}\). Any two disjoint subsets of \( M \) must have different sums of their elements. Find the maximum possible sum of the elements in \( M \).
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Context:
Find the number of all integer-sided isosceles obtuse-angled triangles with perimeter $2008$.
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Context:
Find the largest natural number consisting of distinct digits such that the product of its digits equals 2016.
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Context:
Given an arithmetic sequence where all terms are positive, the sum of the first $n$ terms is $S_{n}$. It is known that $2S_{2}=a_{2}(a_{2}+1)$ and $a_{1}=1$. Find the minimum value of $\frac{2S_{n}+13}{n}$.
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Context:
Let $F = \max_{1 \leq x \leq 3} |x^3 - ax^2 - bx - c|$.  When $a$, $b$, $c$ run over all the real numbers, find the smallest possible value of $F$.
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Context:
A height $A H$ is drawn from the vertex $A$ to the base $BC$ of isosceles triangle $ABC$. A point $P$ is marked on the side $A B$ such that $C P = BC$. The segment $C P$ intersects the height $A H$ at point $Q$. It turns out that the area of triangle $BH Q$ is 4 times smaller than the area of triangle $AP Q$. Find the angles of triangle $ABC$.
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Context:
Find the value of $a \in \mathbb{R}$, where $a \neq 1$, such that \[ \lim_{n \to \infty} \frac{1^a + 2^a + \cdots + n^a}{(n+1)^{a-1} \left[ (na+1) + (na+2) + \cdots + (na+n) \right]} = \frac{1}{60}. \]
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Context:
Let  $p$  be a prime number, and define a sequence by:  $x_i=i$  for  $i=,0,1,2...,p-1$  and  $x_n=x_{n-1}+x_{n-p}$  for  $n \geq p$
Find the remainder when  $x_{p^3}$  is divided by  $p$ .
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Context:
Example 1.7. Find $\int \frac{6 d x}{x(x-1)(x-2)(x-3)}$.
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Context:
In a Craps game, calculate the expected number of rolls needed to get outcome 2, given that outcome 2 occurs before outcome 1.
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Context:
A group of $12$ pirates agree to divide a treasure chest of gold coins among themselves as follows. The $k^{\text{th}}$ pirate to take a share takes $\frac{k}{12}$ of the coins that remain in the chest. The number of coins initially in the chest is the smallest number for which this arrangement will allow each pirate to receive a positive whole number of coins. How many coins does the $12^{\text{th}}$ pirate receive?
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Context:
Six people enter two rooms, with the conditions that: ①each room receives three people; ②each room receives at least one person. How many distribution methods are there for each condition?
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Context:
Calculate the line integral of the vector field \( F(x,y,z) = \left(4x\left(1+\frac{z^2}{8}\right),\frac{\pi}{3}\cos\left(\frac{\pi}{3}y\right),\frac{x^2z}{2}-e^{\frac{z}{3}}\right) \) along the curve \( C \) parametrized by \( \gamma(t) = (e^{t(t-3)}, t, 2t) \). Evaluate \( \int_C F \cdot ds \).
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Context:
Compute the line integral \( \oint_C \left( \frac{-y^3}{(x^2+y^2)^2} \right) dx + \left( \frac{xy^2}{(x^2+y^2)^2} \right) dy \) using Green's theorem, where \( C \) is the boundary of any simply connected region that contains the unit disk centered at the origin.
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Context:
Let \[S = 1 + \frac 18 + \frac{1\cdot 5}{8\cdot 16} + \frac{1\cdot 5\cdot 9}{8\cdot 16\cdot 24} + \cdots + \frac{1\cdot 5\cdot 9\cdots (4k+1)}{8\cdot 16\cdot 24\cdots(8k+8)} + \cdots.\] Find the positive integer $n$ such that $2^n < S^{2007} < 2^{n+1}$.
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Context:
Let $ABCD$ be a parallelogram with area $15$. Points $P$ and $Q$ are the projections of $A$ and $C,$ respectively, onto the line $BD;$ and points $R$ and $S$ are the projections of $B$ and $D,$ respectively, onto the line $AC.$ See the figure, which also shows the relative locations of these points.

[asy] size(350); defaultpen(linewidth(0.8)+fontsize(11)); real theta = aTan(1.25/2); pair A = 2.5*dir(180+theta), B = (3.35,0), C = -A, D = -B, P = foot(A,B,D), Q = -P, R = foot(B,A,C), S = -R; draw(A--B--C--D--A^^B--D^^R--S^^rightanglemark(A,P,D,6)^^rightanglemark(C,Q,D,6)); draw(B--R^^C--Q^^A--P^^D--S,linetype("4 4")); dot("$A$",A,dir(270)); dot("$B$",B,E); dot("$C$",C,N); dot("$D$",D,W); dot("$P$",P,SE); dot("$Q$",Q,NE); dot("$R$",R,N); dot("$S$",S,dir(270)); [/asy]

Suppose $PQ=6$ and $RS=8,$ and let $d$ denote the length of $\overline{BD},$ the longer diagonal of $ABCD.$ Then $d^2$ can be written in the form $m+n\sqrt p,$ where $m,n,$ and $p$ are positive integers and $p$ is not divisible by the square of any prime. Find the value of $m+n+p$.
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Context:
the length of each side of square a is increased by 150 percent to make square b . if the length of the side of square b is increased by 60 percent to make square c , by what percent is the area of square c greater than the sum of the areas of squares a and b ?
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Context:
In a trapezoid with bases $x$ and 5, express the distance between the midpoints of the diagonals as a function of $x$.
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Context:
What is the maximum number of handshakes that could have occurred in a group of 20 people, given that no matter which three people are chosen from the group, there will always be at least two who have not shaken hands?
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Context:
A triangle has sides of length 888, 925, and $x>0$. Find the value of $x$ that minimizes the area of the circle circumscribed about the triangle.
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Context:
Consider the sequence of numbers defined recursively by $t_1=1$ and for $n>1$ by $t_n=1+t_{(n/2)}$ when $n$ is even and by $t_n=\frac{1}{t_{(n-1)}}$ when $n$ is odd. Given that $t_n=\frac{19}{87}$, the sum of the digits of $n$ is
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Context:
A satellite of mass 'm' is revolving around Earth in a circular orbit of radius \( r_0 \) with speed \( v_0 \). At a certain point in its orbit, the direction of motion of the satellite is suddenly changed by an angle \( \theta = \cos^{-1}\left(\frac{3}{5}\right) \), while maintaining the same speed. As a result, the satellite enters an elliptical orbit around Earth. Calculate the ratio of the speed at perigee to the speed at apogee of the new elliptical orbit.
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Context:
Suppose $g: [-1,1] \to [-1,1]$ is a bijective function defined by $g(x)=px^3+qx+r$. Determine the possible values of $g(-1)$. Provide your answer as a set of values.
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Context:
Example 6 The adjacent terms $a_{n}, a_{n-1}$ of the sequence $\left\{a_{n}\right\}$ are the two roots of the equation $x^{2}-c_{n} x+\left(\frac{1}{3}\right)^{n}=0$, and $a_{1}=2$. Find the sum of the first $2 n$ terms of the sequence $\left\{c_{n}\right\}$, $S_{2 n}$.
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Context:
A servant is sent to a spring with two vessels of 7 pints and 11 pints capacity. She needs to bring back exactly 2 pints of water.

What is the minimum number of operations required in this case? An "operation" is defined as filling a vessel, emptying a vessel, or pouring water from one vessel to another.
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Context:
Meghal is playing a game with 2016 rounds $1,2, \cdots, 2016$. In round $n$, two rectangular double-sided mirrors are arranged such that they share a common edge and the angle between the faces is $\frac{2 \pi}{n+2}$. Meghal shoots a laser at these mirrors and her score for the round is the number of points on the two mirrors at which the laser beam touches a mirror. What is the maximum possible score Meghal could have after she finishes the game?
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Context:
A point is randomly thrown onto the segment [3, 8] and let $k$ be the resulting value. Find the probability that the roots of the equation $\left(k^{2}-2 k-3\right) x^{2}+(3 k-5) x+2=0$ satisfy the condition $x_{1} \leq 2 x_{2}$.
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Context:
In Figure 1, \(\angle A = 60^\circ\), \(\angle B = \angle D = 90^\circ\), \(BC = 2\), \(CD = 3\), and \(AB = x\). Find the value of \(x\).
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Context:
If $\sin \alpha = 3 \sin \left( \alpha + \frac{\pi}{6} \right)$, find the value of $\tan \left( \alpha + \frac{\pi}{12} \right)$.
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Context:
Two real numbers $x$ and $y$ are such that $8 y^{4}+4 x^{2} y^{2}+4 x y^{2}+2 x^{3}+2 y^{2}+2 x=x^{2}+1$. Find all possible values of $x+2 y^{2}$.
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Context:
Let $M$ and $N$ be two closed subspaces of a Hilbert space $H$ such that $N \subset M$. Let $P_M$ and $P_N$ be the orthogonal projections onto $M$ and $N$, respectively. Determine the range of the orthogonal projection $P_M - P_N$. \( \text{Provide your answer as a subspace of } H. \)
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Context:
Let \( H \) be a Hilbert space with a complete orthonormal sequence \((i_k)_{k=1}^\infty\). Suppose there exist complex numbers \(a_k \in \mathbb{C}\) for \(k \in \mathbb{N}\), and a bounded linear operator \(T: H \rightarrow H\) such that \(T(i_k) = a_ki_k\), with the sequence \((a_k)\) being bounded. Determine the norm of \(T\), denoted by \(||T||\).
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Context:
Simplify the following sum:

$$\sum_{k=0}^{10}{20\choose k}(1/4)^k (3/4)^{20-k}$$
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Context:
Determine the minimum number of squares that must be colored in a 65x65 grid so that in any configuration of four cells forming an "L" shape, at least one of the squares is colored.
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Context:
Determine the number of positive integer solutions to the equation \(a^2 + b^2 = p_1p_2p_3\), where \(p_1, p_2,\) and \(p_3\) are distinct primes, each congruent to 1 modulo 4.
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Context:
There are 80 cable cars traveling in a circular route between ski resorts A and B, all moving in the same direction. The distance between two adjacent cable cars is the same. Xiaoming, who is sitting in one of the cable cars, encounters an opposing cable car every 15 seconds. How long does it take to travel from A to B?
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Context:
When $x \in (0,1)$, the graph of the function $f(x)=e^{x}-1$ is not below the graph of the function $g(x)=x^{2}-ax$. Determine the range of values for the real number $a$.
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Context:
Let $p,$ $q,$ $r$ be positive real numbers. Find the smallest possible value of
\[4p^3 + 6q^3 + 24r^3 + \frac{8}{3pqr}.\]
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Context:
Evaluate the line integral \( \int_{c} (3a(x^2+y^2)-y^3)dx+3xy(2a-y)dy \) where \( c \) is the upper part of the circle defined by \( y=a+\sqrt{a^2-x^2} \).
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Context:
Evaluate the integral \( \int_{0}^1 \int_{0}^{1} \int_{x^2}^{1}  12xze^{zy^2} \, dy \, dx \, dz \) using a change of variables.
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Context:
A hot air balloon with 200 balloons is blowing up. After about half an hour, while at an altitude of 1500 feet, 1/5 of the total number of balloons in the hot air balloon have blown up. During the following hour, the balloon ascends to an altitude of 3500 feet, causing 30% of the remaining balloons to blow up due to pressure changes. Meanwhile, the wind speed increases, creating a variation in the rate at which balloons blow up. Additionally, balloons inflated at different times show varying durability. Out of the balloons still intact, 10% have double durability and are less likely to blow up in the same conditions. At the end of the second hour, twice the number of balloons that had already blown up, excluding the more durable ones, also blow up, as the wind speed decreases. How many balloons in the hot air balloon remain intact?
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Context:
We have a calculator with two buttons that displays and integer $x$. Pressing the first button replaces $x$ by $\lfloor \frac{x}{2} \rfloor$, and pressing the second button replaces $x$ by $4x+1$. Initially, the calculator displays $0$. How many integers less than or equal to $2014$ can be achieved through a sequence of arbitrary button presses? (It is permitted for the number displayed to exceed 2014 during the sequence. Here, $\lfloor y \rfloor$ denotes the greatest integer less than or equal to the real number $y$).
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Context:
14. Find the area of the plane region $S$ $=\left\{(x, y) \mid x, y \in\left[0, \frac{\pi}{2}\right], \sin ^{2} x-\sin x \sin y+\sin ^{2} y \leqslant \frac{3}{4}\right\}$ is $\qquad$
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Context:
Let $T$ be a totally ordered family of countable sets under set inclusion. If the cofinality of $T$ is uncountable, determine the cardinality of $\bigcup_{E \in T} E$. \(\text{Provide your answer as a cardinality.}\)
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Context:
In an $8 \times 8$ chessboard, how many ways are there to select 56 squares so that all the black squares are selected, and each row and each column has exactly seven squares selected?
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Context:
Determine the spectrum of the operator $S: l^q \to l^q$, where $1 \leq q \leq \infty$, defined by $$ S(x_1, x_2, x_3, x_4, \ldots )=(x_2, -x_1, x_4, -x_3, \ldots). $$
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Context:
Simplify the expression: $$\frac{d}{dx}\left(\int_{-(x^2)}^{x^3}\left(\frac{\sin(t)}{1+\cos(t)}+1\right)dt+\int_{x^3}^{x^2}\left(\frac{\sin(t)}{1+\cos(t)}-1\right)dt\right)$$ for \(-1 \le x \le 1\).
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Context:
Find the value of $a$ such that the derivative $f'(x) = 2xe^{-x} - x^2 e^x$ has a tangent line that passes through the origin, and the function $f(x)$ also passes through the origin.
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Context:
Given that the sequence $\{a_{n}\}$ is a geometric sequence with a common ratio $q\neq 1$, $a_{1}=3$, $3a_{1}$, $2a_{2}$, $a_{3}$ form an arithmetic sequence, and the terms of the sequence $\{a_{n}\}$ are arranged in a certain order as $a_{1}$, $a_{1}$, $a_{2}$, $a_{1}$, $a_{2}$, $a_{3}$, $a_{1}$, $a_{2}$, $a_{3}$, $a_{4}$, $\ldots$, determine the value of the sum $S_{23}$ of the first 23 terms of the new sequence $\{b_{n}\}$.
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Context:
Compute the dimension of the Dolbeault cohomology group \(H^{(0,1)}\left(B(2)-B(1)\right)\), where \(B(r)\) is the ball of radius \(r\) in \(\mathbb{C}^2\).
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Context:
In a random graph with $n$ nodes, where each node is connected to $k$ randomly chosen nodes, determine the probability that the graph is connected if $k = \frac{\log n + c}{n}$, where $c$ is a constant that tends to infinity.
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Context:
Ninety-eight apples who always lie and one banana who always tells the truth are randomly arranged along a line. The first fruit says "One of the first forty fruits is the banana!" The last fruit responds "No, one of the \emph{last} forty fruits is the banana!" The fruit in the middle yells "I'm the banana!" In how many positions could the banana be?
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Context:
Let \( f(n) \) be the integer nearest to \( \sqrt{n} \). Find the value of
\[
\sum_{n=1}^{\infty} \frac{\left(\frac{3}{2}\right)^{f(n)} + \left(\frac{3}{2}\right)^{-f(n)}}{\left(\frac{3}{2}\right)^{n}}
\]
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Context:
4. Let $n$ be a positive integer. Find the smallest positive integer $k$ with the property that for any colouring of the squares of a $2 n \times k$ chessboard with $n$ colours, there are 2 columns and 2 rows such that the 4 squares in their intersections have the same colour.
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Context:
Consider the non-linear differential equation \( y'(t) = y^2 + t \). Determine the maximum number of times the solution \( y(t) \) is differentiable for \( t \in \mathbb{R} \).
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Context:
Determine the number of ways to consider \(\Bbb Q\) as an \(\Bbb R\)-module.
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Context:
The vertices $M$, $N$, $K$ of rectangle $KLMN$ lie on the sides $AB$, $BC$, $CA$ respectively of a regular triangle $ABC$ in such a way that $AM = 2$, $KC = 1$. The vertex $L$ lies outside the triangle. Find the value of $\angle KMN$.
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Context:
A number, when divided by 44, gives 432 as quotient and a certain remainder. When dividing the same number by 34, the remainder is 2. What is the original remainder when dividing by 44?
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Context:
Team A and team B play a series. The first team to win three games wins the series. Before each game, each team is equally likely to win each game, there are no ties, and the outcomes of the individual games are independent. If it turns out that team B won the second game and team A won the series, what is the conditional probability that team B won the first game?
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Context:
Find the formula for the order of the wreath product of two groups, G and H, where H is a permutation group on n elements.
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Context:
During a secret meeting, 20 interns elect their favorite animator. Each votes for two animators. It is known that if you take any two interns, there is always at least one animator for whom they have both voted. What is the minimum number of votes received by the animator who wins the election?
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Context:
The manufacturing cost of a shoe is Rs. 210, and the transportation cost is Rs. 500 for 100 shoes. The selling price is Rs. 258. What is the percentage gain on the selling price?
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Context:
There are $1000$ rooms in a row along a long corridor. Initially, the first room contains $1000$ people, and the remaining rooms are empty. Each minute, the following happens: for each room containing more than one person, someone in that room decides it is too crowded and moves to the next room. All these movements are simultaneous (so nobody moves more than once within a minute). After one hour, how many different rooms will have people in them?
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Context:
Given the parabola with a focus at \((-1, -1)\) and a tangent line at the point \((7, 13)\) described by the equation \(y = 3x - 8\), find the length of the latus rectum of the parabola.
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Context:
Natural numbers \(a, b, c\) are chosen such that \(a < b < c\). It is also known that the system of equations \(2x + y = 2037\) and \(y = |x-a| + |x-b| + |x-c|\) has exactly one solution. Find the minimum possible value of \(c\).
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Context:
After simple interest for two months at $5$% per annum was credited, a Boy Scout Troop had a total of $\textdollar 255.31$ in the Council Treasury. The interest credited was a number of dollars plus the following number of cents
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Context:
An $80 \times 80$ grid is colored orange and black. A square is black if and only if either the square below it or the square to the left of it is black, but not both. If there is no such square, consider it as if it were orange. The only exception is the bottom left square, which is black. Consider the diagonal from the upper left to the lower right. How many black squares does this diagonal have?
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Context:
Point $A$ has two tangents drawn to a circle, touching the circle at points $M$ and $N$, respectively. A secant from the same point $A$ intersects the circle at points $B$ and $C$, and intersects the chord $MN$ at point $P$. Given that $AB:BC = 2:3$, find the ratio $AP:PC$.
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Context:
$100$ chicks are sitting in a circle. The first chick in the circle says the number $1$, then the chick $2$ seats away from the first chick says the number $2$, then the chick $3$ seats away from the chicken that said the number $2$ says the number $3$, and so on. The process will always go clockwise. Some of the chicks in the circle will say more than one number while others might not even say a number at all. The process stops when the $1001$th number is said. How many numbers would the chick that said $1001$ have said by that point (including $1001$)?
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Context:
In a regular triangular prism \(P-ABC\) where all edge lengths are \(1\), points \(L\), \(M\), and \(N\) are the midpoints of edges \(PA\), \(PB\), and \(PC\) respectively. Determine the area of the cross-section of the circumscribed sphere intersected by the plane \(LMN\).
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Context:
Given 2414 cards, each with a unique natural number from 1 to 2414. We need to choose two cards such that the sum of the numbers on them is divisible by 100. In how many ways can this be done?
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Context:
The doll maker crafts one doll in 1 hour and 45 minutes. After making every three dolls, the maker takes a 30-minute break. Ten dolls need to be made for gifts. At what time the following day (specify hours and minutes) will the order be completed if the maker started crafting dolls at 10 AM and also worked throughout the night?
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Context:
In a movie theater, five friends took seats from 1 to 5 (the leftmost seat is number 1). During the movie, Anya left to get popcorn. When she came back, she found that Varya had moved one seat to the right, Galya had moved three seats to the left, and Diana and Ellya had swapped seats, leaving the end seat for Anya. On which seat was Anya sitting before she left?
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Context:
A sequence $a_1, a_2, \ldots$ satisfies $a_1 = \dfrac 52$ and $a_{n + 1} = {a_n}^2 - 2$ for all $n \ge 1.$ Let $M$ be the integer which is closest to $a_{2023}.$ Find the last digit of $M$. Express your answer as a single digit (0-9).
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Context:
In the diagram, \(ABCD\) is a trapezoid with \(BC\) parallel to \(AD\) and \(AB = CD\). Point \(E\) is on \(AD\) such that \(BE\) is perpendicular to \(AD\). Point \(F\) is the point of intersection between \(AC\) and \(BE\). If \(AF = FB\) and \(\angle AFE = 50^\circ\), what is the measure of \(\angle ADC\)?
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Context:
Consider a dice game with an n-sided fair die. On the first roll, rolling an n results in a win, while rolling a 1 results in a loss. For any other result, the player rolls again. On subsequent rolls, the player wins by rolling an n and loses by rolling the current or lower values. What is the probability of winning as the number of sides on the die approaches infinity (n tends to infinity)?
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Context:
Santana has 10 siblings in total. 4 of them have birthdays in March, 1 has a birthday in May, 1 has a birthday in June, 1 has a birthday in October, 1 has a birthday in November, and 2 of them were born in December. Apart from buying each of her siblings a birthday present and a Christmas present, she also buys them a present each for a family holiday celebrated in August. How many more presents does she have to buy in the second half of the year than the first half of the year?
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Context:
Alice chooses three primes \( p, q, r \) independently and uniformly at random from the set of primes of at most 30. She then calculates the roots of \( p x^{2} + q x + r \). What is the probability that at least one of her roots is an integer?
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Context:
A fair coin is flipped eight times in a row. Let $p$ be the probability that there is exactly one pair of consecutive flips that are both heads and exactly one pair of consecutive flips that are both tails. If $p=\frac{a}{b}$, where $a, b$ are relatively prime positive integers, compute $100a+b$.
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Context:
Calculate the arc length of the curve defined by the equation \(6xy - y^4 = 3\) from the point \(\left(\frac{19}{12}, 2\right)\) to the point \(\left(\frac{14}{3}, 3\right)\).
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Context:
The number \(5^{867}\) is between \(2^{2013}\) and \(2^{2014}\). Find how many pairs of integers \((m,n)\) there are such that \(1\leq m\leq 2012\) and
\[5^n<2^m<2^{m+2}<5^{n+1}.\]
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Context:
3.17. A circle of radius $R$ is circumscribed around an isosceles trapezoid with an acute angle $\alpha$ at the base. Find the perimeter of the trapezoid.
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Context:
Given the set $M=\{x\in\mathbb{N}|8-x\in\mathbb{N}\}$, count the number of elements in set $M$. Express your answer as a single integer.
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Context:
集合 $U=\{1,2,3, \cdots, 10\}$ ，则 $U$ 的元素两两互素的三元子集个数为 $\qquad$.
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Context:
Given a group \( G \) with order \( |G| = 231 \), determine the maximum number of subgroups of order 3 that \( G \) can have.
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Context:
Find the inverse Mellin transform of the function \((ix)^{-s}\) for \(x > 0\).
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Context:
Asli will distribute $100$ candies among her brother and $18$ friends of his. Asli splits the friends of her brother into several groups and distributes all the candies into these groups. In each group, the candies are shared in a fair way such that each child in a group takes the same number of candies, and this number is the largest possible. Then, Asli's brother takes the remaining candies of each group. At most, how many candies can Asli's brother have?
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Context:
In a triangle, let the angle bisectors of angles \(\alpha\) and \(\beta\) intersect the circumcircle at points \(D\) and \(E\) respectively. Determine the angles formed by the segment \(DE\) with the angle bisector of angle \(\gamma\).
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Context:
Find the ratio of the volumes obtained by the successive rotation of a parallelogram around each of its two adjacent sides.
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Context:
The steamboat "Rarity" travels for three hours at a constant speed after leaving the city, then drifts with the current for an hour, then travels for three hours at the same speed, and so on. If the steamboat starts its journey in city A and goes to city B, it takes it 10 hours. If it starts in city B and goes to city A, it takes 15 hours. How long would it take to travel from city A to city B on a raft?
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Context:
I4.4 A boy walks from home to school at a speed of $p \mathrm{~km} / \mathrm{h}$ and returns home along the same route at a speed of $3 \mathrm{~km} / \mathrm{h}$. If the average speed for the double journey is $\frac{24}{q} \mathrm{~km} / \mathrm{h}$, find $q$.
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Context:
Given that the lines $l_{1}$: $ax+y+3=0$ and $l_{2}$: $2x+\left(a-1\right)y+a+1=0$ are parallel, find the value of $a$.
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Context:
There were 1238 apples and 374 more pears than apples in the fruit shop. Of these, a few apples were sold, and 276 pears were bought, so the total of apples and pears was 2527. How many apples have been sold?
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Context:
What is the maximum number of sides of a convex polygon that can be divided into right triangles with acute angles measuring 30 and 60 degrees?
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Context:
Given the complex-number sequence $\left\{z_{n}\right\}$ satisfying $z_{1}=1$ and $z_{n+1}=\overline{z_{n}}+1+n \mathrm{i}$ for $n=1,2,3 \cdots$, where $\mathrm{i}$ is the imaginary unit and $\overline{z_{n}}$ represents the complex conjugate of $z_{n}$, find the value of $z_{2015}$.
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Context:
Evaluate the limit \( \lim_{{k \to \infty}} \int_{0}^{\infty} k^{3/2} f(x) e^{-kx} \, dx \), where \( f(x) = \int_{0}^{x} \frac{\sin(t)}{t^{3/2}} \, dt \).
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Context:
Let $A = \{ z \in \mathbb{C} : |z-1| \leq |z-i| \text{ and } |z-(2+2i)| \leq 1 \}$. Find the modulus of $z \in A$ such that the argument of $z$ is minimized. Express your answer as a single value.
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Context:
Given the equation \(x^2 + xy + y^3 = 1\), find the value of \(y'''\) at the point where \(x = 1\).
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Context:
A total amount of Rs. 10,000 is to be distributed amongst p, q, r, and s such that "p" gets twice as that of "q" and "s" gets 4 times as that of "r." Additionally, "q" and "r" are to receive equal amounts, and the amount received by "s" is half of that of "p." What is the difference between the amounts received by s and p?
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Context:
Calculate the integral of the 1-form \( \alpha = dx_1 + dx_2 - dx_3 \) over the image of the map \( G: S^1 \rightarrow S^1 \times S^1 \times S^1 \) given by \( G(q) = (q, q^2, q^3) \).
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Context:
Let \( {\left\{  {a}_{n}\right\}  }_{n = 0}^{\infty },{\left\{  {b}_{n}\right\}  }_{n = 0}^{\infty } \) ,and \( {\left\{  {c}_{n}\right\}  }_{n = 0}^{\infty } \) be sequences of real numbers such that for all \( k \geq  1 \) ,
\( {a}_{k} = \left\lfloor  {\sqrt{2} + \frac{k - 1}{2024}}\right\rfloor   + {a}_{k - 1} \)
\( {b}_{k} + {c}_{k} = 1 \)
 \( {a}_{k - 1}{b}_{k} = {a}_{k}{c}_{k}. \)
Suppose that \( {a}_{0} = 1,{b}_{0} = 2 \) ,and \( {c}_{0} = 3 \) . Given that \( \sqrt{2} \approx  {1.4142} \) ,compute
$\sum_{k=1}^{2024}(a_kb_k-a_{k-1}c_k).$
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Context:
Compare the smallest positive roots of the equations \( \cos(\sin x) = x \) and \( \sin(\cos x) = x \) within the interval \([0, \frac{\pi}{2}]\). Let \(x_1\) be the root of \(\sin(\cos x) = x\) and \(x_2\) be the root of \(\cos(\sin x) = x\). Determine which is greater: \(x_1\) or \(x_2\).
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Context:
On a $4 \times 4$ board, the numbers from $1$ to $16$ are written, one in each box. Andres and Pablo choose four numbers each. Andres chooses the biggest number from each row, and Pablo chooses the biggest number from each column. The same number can be chosen by both. Then, all chosen numbers are removed from the board. What is the greatest possible sum of the numbers left on the board?
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Context:
Let $\triangle ABC$ have side lengths $AB = 12$, $AC = 16$, and $BC = 20$. Inside $\angle BAC$, two circles are positioned, each tangent to rays $\overline{AB}$ and $\overline{AC}$, and the segment $\overline{BC}$. Compute the distance between the centers of these two circles.
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Context:
给定空间中 10 个点, 其中任意四点不在一个平面上, 将某些点之间用线段相连, 若得到的图形中没有三角形也没有空间四边形, 试确定所连线段数目的最大值.
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Context:
Dana creates a "rectangular spiral" by drawing line segments of lengths, in cm, of \(1,1,2,2,3,3,4,4, \ldots\). Dana's pen runs out of ink after the total of all the lengths he has drawn is \(3000 \, \text{cm}\). Calculate the length of the longest line segment that Dana draws. Express your answer as a whole number in cm.
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Context:
Let $[x]$ denote the greatest integer not exceeding $x$, e.g. $[5.6]=5$ and $[3]=3$. How many different integers are there in the sequence $\left[\frac{1^{2}}{2006}\right], \left[\frac{2^{2}}{2006}\right], \left[\frac{3^{2}}{2006}\right], \ldots, \left[\frac{2006^{2}}{2006}\right]$?
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Context:
Find the number of ordered 2012-tuples of integers $\left(x_{1}, x_{2}, \ldots, x_{2012}\right)$, with each integer between 0 and 2011 inclusive, such that the sum $x_{1}+2 x_{2}+3 x_{3}+\cdots+2012 x_{2012}$ is divisible by 2012.
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Context:
Let $f(n)$ be a function defined as $f(n)= 1! +2! +3!+... +n!$. If there exist polynomials $P(x)$ and $Q(x)$ such that $f(n+2)=P(n)f(n+1)+Q(n)f(n)$ for all $n \geq 1$, determine the degrees of the polynomials $P(x)$ and $Q(x)$. Provide your answer as a pair of integers representing the degrees of $P(x)$ and $Q(x)$.
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Context:
Find the number of 7-tuples \( \left( {{n}_{1},\ldots ,{n}_{7}}\right) \) of integers such that
 \( \mathop{\sum }\limits_{{i = 1}}^{7}{n}_{i}^{6} = {96957}. \)
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Context:
Determine the function $f(x)$ that satisfies the equation $f(x^2 + yf(x)) = f(x^2)$ for all real numbers $x$ and $y$.
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Context:
Given the Markov chain with state space \(\{1, 2, 3\}\) and transition matrix \(P = \begin{pmatrix} 0 & \frac{1}{2} & \frac{1}{2} \\ \frac{1}{3} & \frac{1}{2} & \frac{1}{6} \\ 0 & 1 & 0 \end{pmatrix}\), where the initial state \(X_0 = 1\) almost surely, find the probability that the chain returns to state 1 before reaching state 3 for the first time. Let \(u(i)\) denote the probability of visiting state 1 before state 3 when starting from state \(i\).
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Context:
The circles with centers \(O_{1}\) and \(O_{2}\) have a common chord \(AB\), and \(\angle AO_{1}B = 60^\circ\). The ratio of the length of the first circle to the length of the second circle is \(\sqrt{2}\). Find the angle \(\angle AO_{2}B\).
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Context:
Roy's cat eats $\frac{1}{3}$ of a can of cat food every morning and $\frac{1}{4}$ of a can of cat food every evening. Before feeding his cat on Monday morning, Roy opened a box containing $6$ cans of cat food. On what day of the week did the cat finish eating all the cat food in the box?
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Context:
Lance wants to finish reading a 100 page book in 3 days. Yesterday, he started to read 35 pages. Today, he read 5 fewer pages than yesterday. Lance wants to read an even number of pages tomorrow, in increments of 10 pages. How many pages should he read tomorrow to finish the book?
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Context:
设集合 $M=\{1,2, \cdots, 10\}, T$ 为 $M$ 的某些二元子集构成的集合，满足对 $T$ 中任意两个不同的元素 $\{a, b\} 、\{x, y\}$ ，均有
$11 \nmid(a x+b y)(a y+b x)$ .
求 $T$ 的元素个数的最大值.
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Context:
A set $S \subseteq \mathbb{N}$ satisfies the following conditions:

1. If $x, y \in S$ (not necessarily distinct), then $x + y \in S$.
2. If $x$ is an integer and $2x \in S$, then $x \in S$.

Find the number of pairs of integers $(a, b)$ with $1 \le a, b \le 50$ such that if $a, b \in S$ then $S = \mathbb{N}$.
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Context:
A group $H$ of order $p^4$ is non-abelian. If it has $p^3+p^2+p+1$ subgroups of order $p$, how many subgroups of order $p^3$ does it have?
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Context:
Given real numbers \( a_1, a_2, \ldots, a_n \in [0, 2] \) with the condition that \( a_{n+1} = a_1 \), find the maximum value of the expression \(\frac{\sum_{i=1}^{n} a_{1}^{2} a_{i+1} + 8n}{\sum_{i=1}^{n} a_{i}^{2}}\).
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Context:
11. (This question is worth 20 points) In the plane rectangular coordinate system $x O y$, let $A B$ be a chord of the parabola $y^{2}=4 x$ passing through the point $F(1,0)$, and the circumcircle of $\triangle A O B$ intersects the parabola at point $P$ (different from points $O, A, B$). If $P F$ bisects $\angle A P B$, find all possible values of $|P F|$.
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Context:
k是实数，令$f ( x ) = \frac { x ^ { 4 } + k x ^ { 2 } + 1 } { x ^ { 4 } + x ^ { 2 } + 1 }$.若对每三个实数a，b，c（可以相同），存在一个三角形，其三边长为f(a)、f(b)、f(c)，则k的取值范围是 .
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Context:
A palindrome is a nonnegative integer number that reads the same forwards and backwards when written in base $10$ with no leading zeros. A $6$-digit palindrome $n$ is chosen uniformly at random. Find the probability that \(\frac{n}{11}\) is also a palindrome. The original answer is in the form of \(\frac{k}{m}\), where \(\frac{k}{m}\) is a simplified fraction. Please give the value of \(k + m\).
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Context:
There are 3 boys, 3 girls, and 1 teacher taking a photo together. The teacher must be in the middle, and those adjacent to the teacher cannot be boys or girls at the same time. How many possible arrangements are there?
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Context:
Determine the complex polynomial in the form \( p(z) = (z-(a+bi))^n + q \) that has roots at the vertices of a regular hexagon, with one root located at \( 2 + 0i \).
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Context:
Compute the length of the segment tangent from the origin to the circle that passes through the points $(3,4),$ $(6,8),$ and $(5,13).$
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Context:
Find the limit of the sequence \(x_n = \dfrac{\lfloor x \rfloor + \lfloor 3^2x \rfloor + \ldots + \lfloor (2n-1)^2x \rfloor}{n^3}\) as \(n\) approaches infinity.
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Context:
12. The parabolas $y=x^{2}-b x$, $y=-x^{2}+b x$ and the line $y=k x$ can divide the plane into at most
$\qquad$ parts.
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Context:
Evaluate the integral \( \int_{0}^{\infty} \left(e^{-\frac{1}{x^2}}-e^{-\frac{4}{x^2}}\right) dx \) using appropriate substitution and integration techniques.
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Context:
For what value of \(a\) are the graphs of \(y = a^x\) and \(y = \log_a x\) tangent to each other at exactly one point?
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Context:
Five haikus each consist of three lines with syllable counts of five, seven, and five. If you randomly scramble the lines of these five haikus, what is the probability that you end up with five completely good haikus, each with the correct five, seven, five syllable structure? Express this probability as a fraction \( \frac{m}{n} \) in simplest form, where \( m \) and \( n \) are positive integers with \( \text{gcd}(m, n) = 1 \). Find the sum of \( m \) and \( n \).
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Context:
在平面直角坐标系 $x O y$ 中, 椭圆 $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)$ 的右焦点为 $F(c, 0)$,若存在经过点 $F$ 的一条直线 $l$ 交椭圆于 $A, B$ 两点，使得 $O A \perp O B$. 求该椭圆的离心率 $e=\frac{c}{a}$ 的取值范围。该取值范围两端点的和形如\frac{a\sqrt{b}+c}{d}。请求出a+b+c+d的值
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Context:
At the "Lukomorye" station, tickets are sold for one, five, and twenty trips. All tickets cost an integer number of gold coins. Five single-trip tickets are more expensive than one five-trip ticket, and four five-trip tickets are more expensive than one twenty-trip ticket. It turned out that the cheapest way for 33 warriors to travel was to buy tickets for 35 trips, spending 33 gold coins. How much does a five-trip ticket cost?
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Context:
Let $a$, $b$, and $c$ be real numbers satisfying the equations
$$a^3+abc=26$$
$$b^3+abc=78$$
$$c^3-abc=104.$$
Find $a^3+b^3+c^3$.
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Context:
Given real numbers $b$ and $c$, and the function $f(x) = x^2 + bx + c$, the equation $f(f(x)) = 0$ has exactly three different real roots. Find the maximum value of the sum of the roots of $f(x)$.
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Context:
For each of the nine natural numbers \( n, 2n, 3n, \ldots, 9n \), the first digit from the left in its decimal representation was written on the board. \( n \) was chosen so that the number of different digits among the nine written digits is as small as possible. What is this minimum number of different digits?
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Context:
Given seven natural numbers \(a, b, c, a+b-c, a+c-b, b+c-a, a+b+c\), it is known that all of them are distinct prime numbers. Find all possible values of the smallest of these seven numbers.
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Context:
The side \( AB \) of triangle \( ABC \) is equal to \( c \). Point \( M \) is taken on side \( AB \) such that \(\angle CMA = \varphi\).

Find the distance between the orthocenters of triangles \( AMC \) and \( BMC \).
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Context:
Using three colors $\mathbf{R}$, $\mathbf{G}$, and $\mathbf{B}$ to color a $2 \times 5$ table in a way that two squares sharing a common edge must be colored differently. How many different coloring methods are there?
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Context:
We consider 2018 lines on the plane, each pair of which are not parallel and no three of which are concurrent. Let $E$ be the set of their intersection points. We wish to assign a color to each point in $E$ such that any two points on the same line, whose connecting segment does not contain any other point from $E$, have different colors.

How many colors are needed at minimum to achieve such a coloring?
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Context:
In a 5x6 seating arrangement for 30 students, two students, A and B, from the same school cannot be seated adjacent to each other in any direction (front, back, left, right). Calculate the number of different seating arrangements for students A and B. Express your answer as a numerical value.
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Context:
Vasya wrote consecutive natural numbers \( N \), \( N+1 \), \( N+2 \), and \( N+3 \) in rectangles. Under each rectangle, he wrote the sum of the digits of the corresponding number in a circle.

The sum of the numbers in the first two circles turned out to be 200, and the sum of the numbers in the third and fourth circles turned out to be 105. What is the sum of the numbers in the second and third circles?
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Context:
Determine if there exists a natural number that forms a perfect pair with 122, where both the sum and the product of the two numbers are perfect squares.
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Context:
Let \( I \) be the augmentation ideal of the group ring \( \mathbb{Z}/p[\mathbb{Z}/p] \). For a given odd prime \( p \) and an integer \( n \) such that \( 1 \leq n \leq p-1 \), determine the dimension of \( I^n \) as a vector subspace of \( \mathbb{Z}/p[\mathbb{Z}/p] \).
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Context:
The numbers 1, 2, 3, 4, 5, 6, 7 are written in a circle in some order. A number is called "good" if it is equal to the sum of the two numbers written next to it. What is the maximum possible number of "good" numbers among those written?
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Context:
In a $25 \times n$ grid, each square is colored with a color chosen among $8$ different colors. Let $n$ be as minimal as possible such that, independently from the coloration used, it is always possible to select $4$ coloumns and $4$ rows such that the $16$ squares of the interesections are all of the same color. Find the remainder when $n$ is divided by $1000$.

[i]Proposed by [b]FedeX333X[/b][/i]
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Context:
Alex drives from point \( P \) to point \( Q \), then immediately returns to \( P \). Bob drives from point \( Q \) to point \( P \), then immediately returns to \( Q \). They travel on the same road, start at the same time, and each travels at a constant speed. Alex's speed is three times Bob's speed. They meet each other for the first time 15 minutes after the start. How many minutes after the start will they meet each other for the second time? Express your answer as a whole number.
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Context:
In how many ways can we paint $16$ seats in a row, each red or green, in such a way that the number of consecutive seats painted in the same colour is always odd?
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Context:
Determine the interval of values for $x$ such that the inequality \((x^2-5x+4)(y^2+y+1)<2y\) holds for all real values of $y$. What is this interval?
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Context:
Evaluate the double series
$$
\sum_{j=0}^{\infty} \sum_{k=0}^{\infty} 2^{-3 k-j-(k+n)^{2}}
$$The answer is in the form \frac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.
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Context:
For a polynomial $f(x)$ of degree 3 with rational coefficients that touches the x-axis, determine how many of its roots are rational.
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Context:
A cafe has 3 tables and 5 individual counter seats. People enter in groups of size between 1 and 4, inclusive, and groups never share a table. A group of more than 1 will always try to sit at a table, but will sit in counter seats if no tables are available. Conversely, a group of 1 will always try to sit at the counter ﬁrst. One morning, $M$ groups consisting of a total of $N$ people enter and sit down. Then, a single person walks in, and realizes that all the tables and counter seats are occupied by some person or group. What is the minimum possible value of $M + N$?
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Context:
Find the sum of the coefficients of the terms containing $x^4t^2$ in the expansion of $(x^2 + y + 2t + 3k)^{12}$ using the multinomial expansion formula.
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Context:
Evaluate the integral $$\int_0^\pi \log^2\left(\tan\frac{x}{2}\right)dx.$$
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Context:
Determine the number of digits that can be filled in the square to make the product of □4×32 a three-digit number. Provide your answer as a single integer.
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Context:
How many ordered triples of integers $ (a, b, c)$ satisfy\[|a + b| + c = 19\quad\text{and}\quad ab + |c| = 97?\]
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Context:
Given the hyperbola x^2/a^2 - y^2/b^2 = 1 (a > 0, b > 0) with its right focus at F, if a line passing through point F with a slope angle of 60° intersects the right branch of the hyperbola at two points, calculate the range of the eccentricity of this hyperbola. Express your answer in interval notation.
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Context:
12.217. Find the angle between the generatrix and the base of a truncated cone, the total surface area of which is twice the surface area of a sphere inscribed in it.
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Context:
Two natural numbers differ by 2, and the sum of their individual digits is divisible by 4. Find the smallest possible average of these two numbers.
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Context:
Calculate the area of the figure bounded by the closed curve \((x^{2}+y^{2})^{2}=4ay^{3}\), where \(a>0\).
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Context:
Evaluate the limit: \[ \lim\limits_{n\to\infty} \frac{1}{n^2}\sum_{2\leq i < j \leq n} \log_i j \]
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Context:
Find a function \( \beta \) such that \( \beta \circ f \), where \( f \) is the Dirichlet function, equals the characteristic function of the rationals, \( x_\mathbb{Q} \).
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Context:
Every time these two wheels are spun, two numbers are selected by the pointers. What is the probability that the sum of the two selected numbers is even? Express your answer as a fraction in simplest form. The original answer is in the format \frac{k}{m}, please give the value of k + m.
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Context:
The integer sequence \(\left\{a_{n}\right\}\) is defined as \(a_{1}=1\), \(a_{2}=2\), and \(a_{n+2}=5a_{n+1}+a_{n}\). Evaluate the expression \(\left[\frac{a_{2}}{a_{1}}\right] \left\{\left[\frac{a_{3}}{a_{2}}\right\} \left\{\frac{a_{4}}{a_{3}}\right\} \cdots \left\{\left[\left\{\frac{a_{2025}}{a_{2024}}\right\}\left[\frac{a_{2024}}{a_{2}}\right]\right.\right.\right.\).
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Context:
6 / 9 of the population of the country of Venezia lives in Montague province, while the rest lives in Capulet province. In the upcoming election, 80% of Montague residents support Romeo, while 70% of Capulet residents support Juliet; each resident of Venezia supports exactly one of these two candidates. Rounded if necessary to the nearest percent, what is the probability that a Juliet supporter chosen at random resides in Capulet province?
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Context:
A and B run on a circular track at their respective constant speeds. If both start running from the same point in opposite directions, and after their first meeting, B takes another 8 minutes to reach the starting point, given that A takes 6 minutes to complete one lap, how many minutes does B take to complete one lap? Express your answer in minutes as a whole number.
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Context:
Petya has a 3x3 grid. He places pieces in the cells according to the following rules:

- No more than one piece can be placed in each cell;
- A piece can be placed in an empty cell if the total number of pieces already present in the corresponding row and column is even (0 is considered an even number).

What is the maximum number of pieces Petya can place?
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Context:
Compute the limit: \[ \lim_{n\to\infty}\left(\frac{a_{n+1}}{\sqrt[n+1]{b_{n+1}}}-\frac{a_n}{\sqrt[n]{b_n}}\right), \] given that \((a_n)_{n\geq 1}\) and \((b_n)_{n\geq 1}\) are positive real sequences satisfying \[ \lim_{n\to\infty}\frac{a_{n+1}-a_n}{n}=a\in \mathbb{R}_{>0}, \quad \lim_{n\to\infty}\frac{b_{n+1}}{nb_n}=b\in \mathbb{R}_{>0}. \]
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Context:
Person A and Person B start from points $A$ and $B$ respectively and walk towards each other. The first time they meet, it is 100 meters away from point $B$. After meeting, Person A doubles their speed. Person A reaches point $B$, turns around immediately, and catches up with Person B when Person B is 50 meters away from point $A$. How long is the distance between points $A$ and $B$?
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Context:
Consider the transformation $T$ given by the equations: $$x=u+v,\;\;\;\;y=v-u^2.$$ A triangle $Q$ in the $(u,v)$ plane has vertices at $(0,0)$, $(2,0)$, and $(0,2)$. The image of $Q$ under $T$ is the region $D$ in the $(x,y)$ plane. Compute the area of $D$ using a double integral over $Q$. The Jacobian of the transformation is $2u$. Calculate the area of $D$.
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Context:
Evaluate the contour integral: $$I=-2i\oint_C\frac{dz}{z^2(2-i)+6z+(2+i)}$$ where $C$ is the unit circle in the complex plane. Use the residue theorem to find the value of $I$.
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Context:
A fluid is poured down through the cone $z = \sqrt{x^2+y^2}$ for $x^2+y^2 \leq 1$, where the vector field describing the fluid flow is $f(x,y,z) = (0,0,-1)$. Determine the flux of the fluid through the cone.
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Context:
a man sells a car to his friend at 11 % loss . if the friend sells it for rs . 54000 and gains 20 % , the original c . p . of the car was :
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Context:
What is the least possible sum of two positive integers $a$ and $b$ where $a \cdot b = 10!$?
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Context:
Two lines are perpendicular and intersect at point $O$. Points $A$ and $B$ move along these two lines at a constant speed. When $A$ is at point $O$, $B$ is 500 yards away from point $O$. After 2 minutes, both points $A$ and $B$ are equidistant from $O$. After another 8 minutes, they are still equidistant from $O$. Find the ratio of the speed of $A$ to the speed of $B$.
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Context:
In rectangle $ABCD$, diagonal $AC$ is intersected by the angle bisector from $B$ at point $B'$ and by the angle bisector from $D$ at point $D'$. Similarly, diagonal $BD$ is intersected by the angle bisector from $A$ at point $A'$ and by the angle bisector from $C$ at point $C'$. The area of quadrilateral $A'B'C'D'$ is $\frac{9}{16}$ of the area of rectangle $ABCD$. Determine the ratio of the longer side to the shorter side of rectangle $ABCD$.
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Context:
Determine if the sequence \( \{Y_n\} \) defined by \( Y_n = \frac{1}{n^3} \sum_{k=1}^n \frac{X_k^2}{k} \), where \( X_k \) are independent uniform random variables on the interval \((0,k)\), converges almost surely. If it does, find its limit.
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Context:
Determine the least possible value of \( f(1998) \), where \( f \) is a function from the set \(\mathbb{N}\) of positive integers into itself such that for all \( m, n \in \mathbb{N} \),
\[ f\left(n^{2} f(m)\right) = m [f(n)]^{2}. \]
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Context:
Determine the degree of the field extension $K=\mathbb{Q}(\alpha\zeta)$ over $\mathbb{Q}$, where $\alpha =2^{1/5}$ and $\zeta=e^{2i\pi/5}$. Provide your answer as a single integer.
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Context:
Four people each input different positive integers into a series of three black boxes. Each black box follows these rules: if the input is odd, the output is \(4k+1\); if the input is even, the output is \(k \div 2\). All four final outputs are the same and minimized. If Sheep Daddy's input is the largest, what is his input value?
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Context:
Determine the number of partial order relations for a set with 6 elements.
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Context:
Five points are chosen uniformly at random on a segment of length 1. What is the expected distance between the closest pair of points?
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Context:
A merchant has two types of goods. For the first type of goods, which costs him $30 per unit, he marks up the price by 45% and then offers a discount of 15% on the marked price. For the second type of goods, which costs him $60 per unit, he marks them up by 75% and then offers a discount of 30% on the marked price. He sells 100 units of the first type and 50 units of the second type. Additionally, the merchant has to pay an 8% sales tax on the final selling price for each good. Calculate the overall percentage profit for the merchant after the discounts and taxes.
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Context:
Three boys and three girls are lined up for a photo. Boy A is next to boy B, and exactly two girls are next to each other. Calculate the total number of different ways they can be arranged.
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Context:
Given that the real numbers \(a\) and \(b\) satisfy the equations \(1003^{a} + 1004^{b} = 2006^{b}\) and \(997^{a} + 1009^{b} = 2007^{a}\), determine the relationship between \(a\) and \(b\).
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Context:
What is the minimum number of Sportloto (6 out of 49) cards you need to buy to ensure that at least one number is guessed in at least one of them?
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Context:
What is the least number $N$ of 4-digits numbers compiled from digits $1,2,3,4,5,6,7,8$ you need to choose, that for any two different digits, both of this digits are in At least in one of chosen $N$ numbers?
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Context:
A solid has a triangular base with sides of lengths $s$, $s$, $s \sqrt{2}$. Two opposite vertices of the triangle extend vertically upward by a height $h$ where $h = 3s$. Given $s = 2\sqrt{2}$, what is the volume of this solid?
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Context:
Let \( S \) be a set of intervals defined recursively as follows:
- Initially, \([1,1000]\) is the only interval in \( S \).
- If \( l \neq r \) and \([l, r] \in S\), then both \(\left[l,\left\lfloor\frac{l+r}{2}\right\rfloor\right]\) and \(\left[\left\lfloor\frac{l+r}{2}\right\rfloor+1, r\right]\) are in \( S \).

(Note that \( S \) can contain intervals such as \([1,1]\), which contain a single integer.)

An integer \( i \) is chosen uniformly at random from the range \([1,1000]\). What is the expected number of intervals in \( S \) which contain \( i \)?
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Context:
Minimize the function \( f(x,y,z) = x^2 + 2y^2 + 3z^2 - xy + 2xz \) subject to the constraints: \( x + y - z = 5 \) and \( x - y + z = 3 \).
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Context:
Determine the angle at which a ball should be struck on a circular billiard table, such that it returns to its original position on the diameter after bouncing twice off the table's edge.
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Context:
Determine the coordinates of all points on the plane that are equidistant from the points of intersection of the parabolas given by the equations \( y = 2x^2 - 1 \) and \( x = 4y^2 - 2 \).
Please reason step by step, and put your final answer within \boxed{}.

\boxed{\left( \dfrac{1}{8}, \dfrac{1}{4} \right)}


Context:
Find the number of positive integers less than $2000$ that are neither $5$-nice nor $6$-nice.
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Context:
If x gets 25% more than y and y gets some percentage more than z, the share of z out of Rs. 370 is Rs. 100. What is the percentage by which y gets more than z?
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Context:
Equilateral triangles $ACB'$ and $BDC'$ are drawn on the diagonals of a convex quadrilateral $ABCD$ so that $B$ and $B'$ are on the same side of $AC$, and $C$ and $C'$ are on the same sides of $BD$. Find $\angle BAD + \angle CDA$ if $B'C' = AB+CD$.
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Context:
What is the smallest number which when diminished by some number, is divisible by 8, 14, 26 and 28 and the result is 746?
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Context:
Let $\alpha$ and $\beta$ be positive integers such that $\frac{43}{197} < \frac{\alpha}{\beta} < \frac{17}{77}$. Find the minimum possible value of $\beta$.
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Context:
Determine the leading order computation cost for solving the system of equations $BM = A$, where $B$ is an $n \times n$ banded matrix with bandwidth $k$, and $A$ is a diagonal matrix.
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Context:
If a complex number $z$ satisfies the equation $z^2-4z= |z|^2+ \frac{16}{|z|^3}$, and it is known that $z$ is neither purely real nor purely imaginary, then what is the value of $|z|^2$?
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Context:
Find the sum of the infinite series where the numerators are Fibonacci numbers and the denominators are powers of 2, starting with 1 and doubling each time.
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Context:
Evaluate the integral $\int_{0}^{\infty}{\dd x \over \pars{1 + x^{2}}\pars{1 + x^{\beta}}}$, where $\beta$ is a positive constant.
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