Context:
Sara has 47 red flowers, 63 yellow flowers, 54 blue flowers, 29 orange flowers, and 36 pink flowers. She wants to make bouquets with the same number of each color flower in each bouquet. What is the greatest number of bouquets she can make?
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the smallest positive integer that satisfies the congruence $5x \equiv 17 \pmod{31}$?
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Context:
There is a very popular race course where runners frequently go for a daily run. Assume that all runners randomly select a start time, a starting position on the course, and a direction to run. Also assume that all runners make exactly one complete circuit of the race course, all runners run at the same speed, and all runners complete the circuit in one hour. Suppose that one afternoon you go for a run on this race course, and you count $300$ runners which you pass going in the opposite direction, although some of those runners you count twice since you pass them twice. What is the expected value of the number of different runners that you pass not counting duplicates?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $A, B, C$ be points in that order along a line, such that $AB = 20$ and $BC = 18$. Let $\omega$ be a circle of nonzero radius centered at $B$, and let $\ell_1$ and $\ell_2$ be tangents to $\omega$ through $A$ and $C$, respectively. Let $K$ be the intersection of $\ell_1$ and $\ell_2$. Let $X$ lie on segment $\overline{KA}$ and $Y$ lie on segment $\overline{KC}$ such that $XY \parallel BC$ and $XY$ is tangent to $\omega$. What is the largest possible integer length for $XY$?
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Context:
On the Cartesian plane $\mathbb{R}^2$, a circle is said to be \textit{nice}
if its center is at the origin $(0,0)$ and it passes through at least one lattice point
(i.e. a point with integer coordinates).
Define the points $A = (20,15)$ and $B = (20,16)$. How many nice circles intersect the open segment $AB$?

For reference, the numbers $601$, $607$, $613$, $617$, $619$, $631$, $641$, $643$, $647$, $653$, $659$, $661$, $673$, $677$, $683$, $691$ are the only prime numbers between $600$ and $700$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the Cartesian coordinate system $xOy$, the hyperbola $C$: $x^{2}- \frac{y^{2}}{b^{2}}=1$ ($b > 0$) has two asymptotes that intersect with the circle $O$: $x^{2}+y^{2}=2$ at four points sequentially named $A$, $B$, $C$, and $D$. If the area of rectangle $ABCD$ is $b$, then the value of $b$ is.
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Context:
What is the maximal number of checkers that can be placed on an $8 \times 8$ checkerboard so that each checker stands on the middle one of three squares in a row diagonally, with exactly one of the other two squares occupied by another checker?
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Context:
Two types of shapes composed of unit squares, each with an area of 3, are placed in an $8 \times 14$ rectangular grid. It is required that there are no common points between any two shapes. What is the maximum number of these two types of shapes that can be placed in the $8 \times 14$ rectangular grid?
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Context:
Let's call the arrangement of $N$ grasshoppers in distinct points on a line $k$-lucky if the grasshoppers, by making the necessary number of moves according to the checkers' rules, can achieve that the sum of pairwise distances between them decreases by at least $k$ times. For which $N \geq 2$ is there an arrangement that is $k$-lucky for all natural $k$? (In the game, one of the grasshoppers jumps to a point symmetrical to it relative to another grasshopper.)
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Context:
Chantal and Jean start hiking from a trailhead toward a fire tower. Jean is wearing a heavy backpack and walks slower. Chantal starts walking at $4$ miles per hour. Halfway to the tower, the trail becomes really steep, and Chantal slows down to $2$ miles per hour. After reaching the tower, she immediately turns around and descends the steep part of the trail at $3$ miles per hour. She meets Jean at the halfway point. What was Jean's average speed, in miles per hour, until they meet?
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Context:
Let \(a \star b = ab - 2\). Compute the remainder when \((((579 \star 569) \star 559) \star \cdots \star 19) \star 9\) is divided by 100.
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Context:
Given a circle of radius $3$ units, find the area of the region consisting of all line segments of length $6$ units that are tangent to the circle at their midpoints.
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Context:
the average of 15 consecutive integers is 25. Then, 14 is deducted from the first consecutive number, 12 is added to the second number, 10 is deducted from the third, 8 is added to the fourth, and so on until the 14th number which has 12 added to it. In the end, the last (15th) number remains unchanged. What is the new average?
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Context:
The trapezoid $ABCD$ has bases with lengths $7$ and $17$ and an area of $120$. Find the difference of the areas of $\triangle ACD$ and $\triangle CDB$.
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Context:
The lengths of the diagonals of a rhombus and the length of its side form a geometric progression. Find the sine of the angle between the side of the rhombus and its longer diagonal, given that it is greater than \( \frac{1}{2} \).
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Context:
The expression \( 1000 \sin 10^{\circ} \cos 20^{\circ} \cos 30^{\circ} \cos 40^{\circ} \) can be simplified as \( a \sin b^{\circ} \), where \( a \) and \( b \) are positive integers with \( 0 < b < 90 \). Find the value of \( 100a + b \).
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Context:
If $2^x = 3^y = 5^z$, and $x, y, z$ are all positive numbers, determine the order of $2x, 3y, 5z$ from smallest to largest.
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Context:
For a positive integer $n$, let $f(n)$ be equal to $n$ if there exists an integer $x$ such that $x^2 - n$ is divisible by $2^{12}$, and let $f(n)$ be $0$ otherwise. Determine the remainder when \( \sum_{n=0}^{2^{12}-1} f(n) \) is divided by $2^{12}$.
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Context:
Given that O is the origin and a perpendicular line l is drawn through point P(2,1) to line OP, find the value of p such that triangle OPQ is a right isosceles triangle, where Q is the midpoint of segment AB on the parabola y^2 = 2px (p > 0). Express your answer as a single number.
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Context:
Let  $ABC$  be an equilateral triangle. Denote by  $D$  the midpoint of  $\overline{BC}$ , and denote the circle with diameter  $\overline{AD}$  by  $\Omega$ . If the region inside  $\Omega$  and outside  $\triangle ABC$  has area  $800\pi-600\sqrt3$ , find the length of  $AB$ .
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Context:
For a convex function $f$ with $L$-Lipschitz continuous gradient, what is the lower bound for the expression $\left( x - y \right)^T {\color{red} A} \left( \nabla f(x) - \nabla f(y)\right)$, where $A$ is a positive semi-definite matrix?
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Context:
A class of 30 students is completing an art project. They are assigned to separate groups and will need to take markers from some boxes of markers. The first group has 10 students who will have 2 markers each and the second group has 15 students who will have 4 markers each. The last group takes the remaining markers and divides them equally, with each student receiving 6 markers. How many boxes of markers are there?
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Context:
Let \( x, y \), and \( z \) be positive integers. If \( |x-y|^{2010} + |z-x|^{2011} = b \) and \( c = |x-y| + |y-z| + |z-x| \), find the value of \( c \).
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Context:
8. (SPA 2) In a test, $3 n$ students participate, who are located in three rows of $n$ students in each. The students leave the test room one by one. If $N_{1}(t), N_{2}(t), N_{3}(t)$ denote the numbers of students in the first, second, and third row respectively at time $t$, find the probability that for each $t$ during the test,
$$
\left|N_{i}(t)-N_{j}(t)\right|<2, \quad i \neq j, \quad i, j=1,2, \ldots .
$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
If $MATH + WITH = GIRLS$, compute the smallest possible value of $GIRLS$. Here $MATH$ and $WITH$ are 4-digit numbers and $GIRLS$ is a 5-digit number (all with nonzero leading digits). Different letters represent different digits.
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Context:
Given that $F$ is the left focus of the hyperbola $C: \frac{{x}^{2}}{{a}^{2}} - \frac{{y}^{2}}{{b}^{2}} = 1 (a > 0, b > 0)$, point $P(0, \sqrt{6}a)$, and the line $PF$ intersects the hyperbola $C$ at exactly one point, then the eccentricity of the hyperbola $C$ is ______.
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Context:
Given a triangle $\triangle ABC$ with sides $a$, $b$, $c$ opposite to angles $A$, $B$, $C$ respectively. If $a=2$, $A= \frac{\pi}{3}$, and $\frac{\sqrt{3}}{2} - \sin(B-C) = \sin 2B$, find the area of $\triangle ABC$.
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Context:
If six geometric means are inserted between $16$ and $11664$, calculate the sixth term in the geometric series.
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Context:
Given the function $y=x^3-3x+d$ has exactly two common points with the x-axis, then $d=\ $
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Context:
Milo can roll downhill on his skateboard at a certain speed compared to his running speed. And Milo's best friend, Cory, can drive his wheelchair at twice the speed that Milo can roll downhill on his skateboard. If Cory always drives his wheelchair at 12 miles per hour, what is the ratio of Milo's downhill skateboarding speed to his running speed?
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Context:
在$\Delta ABC$中，已知$2{{\cos }^{2}}\frac{A}{2}=\frac{\sqrt{3}}{3}\sin A$，若$a=2\sqrt{3}$，则$\Delta ABC$周长的取值范围为__________．
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Context:
We are given sufficiently many stones of the forms of a rectangle $2\times 1$ and square $1\times 1$. Let $n > 3$ be a natural number. In how many ways can one tile a rectangle $3 \times n$ using these stones, so that no two $2 \times 1$ rectangles have a common point, and each of them has the longer side parallel to the shorter side of the big rectangle?
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Context:
A four-digit number \(\overline{abcd} (1 \leqslant a \leqslant 9, 0 \leqslant b, c, d \leqslant 9)\) is called a \(P\) type number if \(a > b, b < c, c > d\). It is called a \(Q\) type number if \(a < b, b > c, c < d\). Let \(N(P)\) and \(N(Q)\) be the number of \(P\) type and \(Q\) type numbers respectively. Find the value of \(N(P) - N(Q)\).
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Context:
In the city park, there are various attractions for kids and their parents. The entrance ticket to the park is $5, but for each attraction, you have to pay separately - such ticket costs $2 for kids and a certain amount for parents. A family with 4 children, their parents, and her grandmother paid $55 for visiting the park and one attraction inside. How much does the ticket for one attraction cost for parents?
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Context:
On the coordinate plane, find the area of the part enclosed by the curve $C: (a+x)y^2=(a-x)x^2\ (x\geq 0)$ for $a>0$.
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Context:
Given that the range of the function $f(x) = \log(5^x + \frac{4}{5^x} + m)$ is $\mathbb{R}$, find the range of values for $m$.
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Context:
Keesha wants to get her hair, nails, and makeup done for prom. She decides to go to three different boutiques for these services. For hair, updos cost $50 and come with a 10% student discount. Manicures cost $30 and have a buy-one-get-one 50% off deal on pedicures, which cost $35 each. Makeup applications cost $40, with a 5% tax applied. Keesha decides to tip each beautician 20%. Calculate the total cost Keesha will pay for these services, taking into account service costs, discounts, taxes, and tips.
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Context:
A five-digit number \(abcde\) satisfies:
\[ a < b, \, b > c > d, \, d < e, \, \text{and} \, a > d, \, b > e. \]

For example, 34 201, 49 412. If the digit order's pattern follows a variation similar to the monotonicity of a sine function over one period, then the five-digit number is said to follow the "sine rule." Find the total number of five-digit numbers that follow the sine rule.

Note: Please disregard any references or examples provided within the original problem if they are not part of the actual problem statement.
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Context:
Find the number of ordered triples of nonnegative integers $(a, b, c)$ that satisfy
\[(ab + 1)(bc + 1)(ca + 1) = 84.\]
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Context:
Find the sum of the digits of the cube of a number consisting of three ones and some number of zeros.
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Context:
We place flower pots with masses $1 \mathrm{~kg}$, $2 \mathrm{~kg}$, and $3 \mathrm{~kg}$ at the corners of an equilateral weightless triangular board. Where should we suspend the board so that the flower pots do not fall off?
Please reason step by step, and put your final answer within \boxed{}.

\boxed{\left( \frac{7}{12}, \frac{\sqrt{3}}{4} \right)}


Context:
Find the remainder when \( 14^{100} \) is divided by 45.
Please reason step by step, and put your final answer within \boxed{}.


Context:
已知 x,y,z 是 3 个大于等于 1 的实数，那么\left(\df{\sqrt{x^2(y-1)^2+y^2}}{xy}+\df{\sqrt{y^2(z-1)^2+z^2}}{yz}+\df{\sqrt{z^2(x-1)^2+x^2}}{xz}\right)^2的最小值写成最简分数后的分子分母之和是__________.
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Context:
14. In triangle $A B C$ with orthocenter $H, A B=13, B C=21$ and $C A=20$. The perpendicular bisector of $C H$ meets $B C$ at $P$ and lines $P H$ and $A B$ meet at $Q$. The line through $Q$ perpendicular to $P Q$ meets $A H$ at $X$. The length of $A X$ can be written in the form $p / q$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.
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Context:
In rectangle \(ABCD\), \(AD = a\) and \(AB = b\) (with \(b > a\)). The rectangle is folded such that point \(A\) coincides with point \(C\), creating fold line \(MN\). Then, the half-plane \(D A M N\) is opened to form a dihedral angle of \(57^\circ\) with the half-plane \(M N C B\). What is the angle between line \(AC\) and line \(MN\)?
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Context:
Calculate the limit:

$$\lim_{x\to 0}\frac{ \ln^{5}(x+1)-\ln(x^{5}+1)}{x^{6}}$$
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Context:
How many ways are there to arrange the numbers \(\{1,2,3,4,5,6,7,8\}\) in a circle so that every two adjacent elements are relatively prime? Consider rotations and reflections of the same arrangement to be indistinguishable.
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Context:
Suppose $ABCD$ is a trapezoid with $AB \parallel CD$ and $AB \perp BC$. Let $X$ be a point on segment $\overline{AD}$ such that $AD$ bisects $\angle BXC$ externally, and denote $Y$ as the intersection of $AC$ and $BD$. If $AB = 10$ and $CD = 15$, compute the maximum possible value of $XY$.
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Context:
It is known that odd prime numbers $x, y, z$ satisfy the following conditions:

- $x \mid (y^5 + 1)$
- $y \mid (z^5 + 1)$
- $z \mid (x^5 + 1)$

Find the minimum value of the product $xyz$.
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Context:
Yuri has been adopting puppies for two and a half months. In the first week, he adopted 20 puppies. In the second week, he adopted 2/5 times as many puppies as the first week. In the third week, he adopted 3/8 times as many puppies as he did in the second week. In the fourth week, he adopted twice the number of puppies he adopted in the second week. During the fifth week, he adopted ten more puppies than on the first week. In the sixth week, he adopted 5 less than the twice the number of puppies he adopted in the third week. In the seventh week, he adopted puppies at an exponential rate, doubling the number of puppies he adopted in the previous week. After half of the eighth week, Yuri adopted 5/6 times the number of puppies he adopted in the whole seventh week. How many puppies does Yuri have now?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A group of 6 students decided to make *study groups* and *service activity groups* according to the following principle:

Each group must have exactly 3 members. For any pair of students, there are same number of study groups and service activity groups that both of the students are members.

Supposing there are at least one group and no three students belong to the same study group and service activity group, find the minimum number of groups.
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Context:
Point \( D \) lies on the extension of side \( AC \) of triangle \( ABC \), whose area is \( S \); point \( A \) is between \( D \) and \( C \). Let \( O \) be the centroid of triangle \( ABC \). It is known that the area of triangle \( DOC \) is \( S_l \). Express the area of triangle \( DOB \) in terms of \( S \) and \( S_l \).
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Context:
Three distinct real numbers form (in some order) a 3-term arithmetic sequence, and also form (in possibly a different order) a 3-term geometric sequence.  Compute the greatest possible value of the common ratio of this geometric sequence.
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Context:
Let $S$ be the set of positive integers $k$ such that the two parabolas\[y=x^2-k~~\text{and}~~x=2(y-20)^2-k\]intersect in four distinct points, and these four points lie on a circle with radius at most $21$. Find the sum of the least element of $S$ and the greatest element of $S$.
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Context:
Evaluate the limit without using L'Hôpital's rule, Taylor series, or differentiation (or integration): $$\lim_{x \to 0} \frac{\ln\left[\dfrac{(1-3x)(1+x)^3}{(1+3x)(1-x)^3}\right]}{x^3}$$
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Context:
一颗质地均匀的正方体骰子，六个面上分别标有点数 $1,2,3,4,5,6$. 随机地抛掷该骰子三次 (各次抛掷结果相互独立), 所得的点数依次为 $a_{1}, a_{2}, a_{3}$, 则事件 " $\left|a_{1}-a_{2}\right|+\left|a_{2}-a_{3}\right|+\left|a_{3}-a_{1}\right|=6$ " 发生的概率为 $\qquad$。原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值
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Context:
There are 408 boys and some girls in a school which are to be divided into equal sections of either boys or girls alone. The total number of sections thus formed is 26. How many girls are there in the school?
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Context:
Jeremy buys 30 watermelons. Each week, he eats a certain number of watermelons and gives away some to his family and neighbor. The number of watermelons Jeremy eats per week fluctuates as follows: he eats 3 watermelons the first week, 4 the second week, 5 the third week, and then the pattern repeats. He gives 2 watermelons to his dad on the first week, and the number increases by 1 each week. He gives 1 watermelon to his sister and 1 to his neighbor on the first week, and the numbers increase by 2 for his sister and decrease by 1 for his neighbor each week. How many weeks will the watermelons last?
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Context:
The solutions to the equation \((z+6)^8=81\) are connected in the complex plane to form a convex regular polygon, three of whose vertices are labeled \(A, B,\) and \(C\). What is the least possible area of \(\triangle ABC\)? The original answer is in the form \(\frac{k}{m}\sqrt{n}-\frac{k}{m}\), please give the value of \(k + m + n\).
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Context:
Add parentheses: $a-b+c=a-\_\_\_\_\_\_.$
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Context:
8. What is the maximum number of parts into which the surface of a sphere (sphere) can be divided by 3, 4, and, generally, $n$ circles?
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Context:
On the Cartesian plane \(\mathbb{R}^{2}\), a circle is said to be nice if its center is at the origin \((0,0)\) and it passes through at least one lattice point (i.e. a point with integer coordinates). Define the points \(A=(20,15)\) and \(B=(20,16)\). How many nice circles intersect the open segment \(AB\)?
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Context:
Reimu has 2019 coins \( C_{0}, C_{1}, \ldots, C_{2018} \), one of which is fake. These coins look identical, making each coin equally likely to be fake. She uses a machine that takes any two coins and picks one that is not fake. If both coins are not fake, the machine selects one uniformly at random. For \( i = 1, 2, \ldots, 1009 \), she puts \( C_{0} \) and \( C_{i} \) into the machine once, and the machine picks \( C_{i} \). What is the probability that \( C_{0} \) is fake?
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Context:
Evaluating the expression $\left(\left((3+1)^{-1} \cdot 2\right)^{-1} \cdot 2\right)^{-1} \cdot 2$.
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Context:
In triangle \( ABC \) with \( AB = 8 \) and \( AC = 10 \), the incenter \( I \) is reflected across side \( AB \) to point \( X \) and across side \( AC \) to point \( Y \). Given that segment \( XY \) bisects \( AI \), compute \( BC^2 \). (The incenter \( I \) is the center of the inscribed circle of triangle \( ABC \).)
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Context:
Given the power function $y=(m^2-5m-5)x^{2m+1}$ is a decreasing function on $(0, +\infty)$, then the real number $m=$ .
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Context:
A survey of $150$ teachers determined the following:
- $90$ had high blood pressure
- $60$ had heart trouble
- $50$ had diabetes
- $30$ had both high blood pressure and heart trouble
- $20$ had both high blood pressure and diabetes
- $10$ had both heart trouble and diabetes
- $5$ had all three conditions

What percent of the teachers surveyed had none of the conditions?
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Context:
In a certain production lot, a percentage of the toys are red and the remaining toys are green. Half of the toys are small and half are large. 10 percent of the toys are red and small, and 40 toys are green and large. There are 60 red and large toys. What is the percentage of red toys in the production lot?
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Context:
Find the smallest real number \( a \) such that for any non-negative real numbers \( x \), \( y \), and \( z \) that sum to 1, the following inequality holds:
$$
a(x^{2} + y^{2} + z^{2}) + xyz \geq \frac{a}{3} + \frac{1}{27}.
$$
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Context:
In a regular pyramid \( PABC \), the side of the base \( ABC \) is equal to \( a \), and the lateral edge is \( 2a \). Points \( P, B \), and \( C \) lie on the lateral surface of a cone with vertex at point \( A \). Find the angle at the vertex of the axial cross-section of the cone.
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Context:
Sally sold 27 Pokemon cards. Dan gave her some new Pokemon cards. Sally bought 20 Pokemon cards. Now, Sally has 34 Pokemon cards. How many Pokemon cards did Dan give her?
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Context:
Peter has a collection of 80 books. He has read 70% of them, his brother has read 35% of them, their friend Sarah has read 40% of the books, and their cousin Alex has read 22% of the collection. How many more of these books has Peter read than the combined total of his brother, Sarah, and Alex?
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Context:
In a large population, a certain characteristic has a distribution that is symmetric about the mean m, with a skewness of γ₁ and a kurtosis of γ₂. If 36 percent of the distribution lies within one standard deviation d₁ of the mean, and 60 percent lies within two standard deviations d₂ from the mean, what percent of the distribution is less than m + d₁, given a skewness of γ₁ = 0.5 and a kurtosis of γ₂ = 3?
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Context:
Calculate the minimum number of digits to the right of the decimal point needed to express the fraction $\frac{987654321}{2^{30} \cdot 5^6 \cdot 3}$.
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Context:
Triangle  $ABC$  has  $BC=1$  and  $AC=2$ . What is the maximum possible value of  $\hat{A}$ .
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Context:
Smith has a small booth where he sells lottery tickets. Customers arrive according to a Poisson process of rate $\lambda$= 1 per minute. Smith will close the shop when the gap between the arrival of two consecutive customers exceeds $b$ minutes. Let $Y$ denote the time until Smith closes the shop. Find $E(Y)$.
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Context:
Select some numbers from \(1, 2, 3, \cdots, 9, 10\) such that each number from \(1, 2, 3, \cdots, 19, 20\) is equal to the sum of one or two of the selected numbers (they can be the same). Find the minimum number of selected numbers needed.
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Context:
In Smithtown, the ratio of right-handed people to left-handed people is 3 to 1 and the ratio of men to women is 3 to 2. If the number of right-handed men is maximized, what percentage of all the people in Smithtown are left-handed women?
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Context:
Form a five-digit number without repeating digits using the numbers 0, 1, 2, 3, 4. How many of these numbers are even and have the digits 1 and 2 adjacent to each other? (Answer with a number).
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Context:
The sum of the first $n$ terms of an arithmetic sequence $\{a\_n\}$ is $S\_n$, where the first term $a\_1 > 0$ and the common difference $d < 0$. For any $n \in \mathbb{N}^*$, there exists $k \in \mathbb{N}^*$ such that $a\_k = S\_n$. Find the minimum value of $k - 2n$.
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Context:
Given that a ( a - 4 ) = some number and b ( b - 4 ) = the same number , where a ≠ b , and a + b = 4. What is that number?
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Context:
5. Žan decided to assign a single-digit number to each two-digit number, and he would do so only by multiplying the digits. For the numbers 91 and 66, he wrote:

$$
\begin{aligned}
& 91 \xrightarrow{9 \cdot 1} 9 \\
& 66 \xrightarrow{6 \cdot 6} 36 \xrightarrow{3 \cdot 6} 18 \xrightarrow{1 \cdot 8} 8
\end{aligned}
$$

How many two-digit numbers did he assign the number 0 to?

## 51st Mathematical Competition for High School Students in Slovenia

Optional Competition, April 4, 2007

## Problems for 3rd Year Students
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Context:
Given the quadratic function $f(x) = x^2 - ax + 4$, if $f(x+1)$ is an even function, find the value of the real number $a$. Express your answer as a single real number.
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Context:
Two adjacent faces of a tetrahedron, which are isosceles right triangles with a hypotenuse of 2, form a dihedral angle of 60 degrees. The tetrahedron is rotated around the common edge of these faces. Find the maximum area of the projection of the rotating tetrahedron onto the plane containing the given edge.
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Context:
Let $f(n) = \frac{n}{3}$ if $n$ is divisible by $3$, and $f(n) = 4n - 10$ otherwise. Find the sum of all positive integers $c$ such that $f^5(c) = 2$. (Here, $f^5(x)$ means $f(f(f(f(f(x)))))$.)
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Context:
Given the irrational number e = 2.71828182845..., define the function f(n) = k, where the domain is the set of positive integers, and k is the n-th digit after the decimal point of e. Given f(0) = 2, determine the range of f(f(f(n))). Express your answer as a set of integers.
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Context:
Let $N$ be the greatest positive integer that can be expressed using all seven Roman numerals $I$, $V$, $X$, $L$, $C$, $D$, and $M$ exactly once each, and let $n$ be the least positive integer that can be expressed using these numerals exactly once each. Find $N - n$. Note that the arrangement $CM$ is never used in a number along with the numeral $D$.
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Context:
If the power function $y=(m^2-4m+1)x^{m^2-2m-3}$ is an increasing function on $(0,+\infty)$, then the value of the real number $m$ is $\_\_\_\_\_\_$.
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Context:
Let the set \( \mathrm{S} = \{1, 2, 3, \ldots, 10\} \). The subset \( \mathrm{A} \) of \( \mathrm{S} \) satisfies \( \mathrm{A} \cap \{1, 2, 3\} \neq \emptyset \) and \( \mathrm{A} \cup \{4, 5, 6\} \neq \mathrm{S} \). Find the number of such subsets \( \mathrm{A} \).
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Context:
On a plot of land, there are 10,000 trees planted in a grid with 100 rows and 100 columns, forming a neat square grid. What is the maximum number of trees that can be cut down such that from the position of any remaining tree, no other tree is obscured? You can assume the trees are thin enough that this rule is strictly followed.
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Context:
As shown in the figure, \( C \) and \( D \) are two points on segment \( AB \), and \( M \) and \( N \) are the midpoints of \( AC \) and \( DB \) respectively. If \( AB = 10 \) and \( CD = 2 \), then calculate the length of \( MN \). Express your answer as a whole number.
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Context:
In triangle $ABC$, the sides opposite to angles $A$, $B$, $C$ are respectively $a$, $b$, $c$. If $2b\cos B = a\cos C + c\cos A$, then $B=$ __________.
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Context:
162 pluses and 144 minuses are placed in a $30 \times 30$ table such that each row and each column contains at most 17 signs. (No cell contains more than one sign.) For every plus, count the number of minuses in its row, and for every minus, count the number of pluses in its column. Find the maximum of the sum of these numbers.
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Context:
Let \( S = \left\{ A = \left(a_{1}, a_{2}, \cdots, a_{8}\right) \mid a_{i} = 0 \text{ or } 1, i = 1,2, \cdots, 8 \right\} \). For any two elements \( A = \left(a_{1}, a_{2}, \cdots, a_{8}\right) \) and \( B = \left(b_{1}, b_{2}, \cdots, b_{8}\right) \) in \( S \), define \( d(A, B) = \sum_{i=1}^{8} \left| a_{i} - b_{i} \right| \), which is called the distance between \( A \) and \( B \). What is the maximum number of elements that can be chosen from \( S \) such that the distance between any two of them is at least \( 5 \)?
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Context:
A trapezoid has side lengths $24$, $25$, $26$, and $27$ in some order. Find its area.
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Context:
A recipe that makes $5$ servings of hot chocolate requires $2$ squares of chocolate, $\frac{1}{4}$ cup sugar, $1$ cup water and $4$ cups milk. Jordan has $5$ squares of chocolate, $2$ cups of sugar, lots of water, and $7$ cups of milk. If he maintains the same ratio of ingredients, what is the greatest number of servings of hot chocolate he can make?
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Context:
In a tetrahedron \(ABCD\), \(AB = AC = AD = BC = 1\), \(BD = \sqrt{3}\), and \(CD = \sqrt{2}\). Find the angle between \(AD\) and \(BC\).
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Context:
John scores 93 on this year's AHSME. Had the old scoring system still been in effect, he would score only 84 for the same answers.
How many questions does he leave unanswered?
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Context:
Given \( f(x) = |1 - 2x| \) for \( x \in [0, 1] \), find the number of solutions to the equation \( f(f(f(x))) = \frac{1}{2} x \).
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Context:
If each edge of a cube is increased by different percentages - 10% for edge A, 15% for edge B, and 20% for edge C, calculate the percentage increase in the surface area of the transformed rectangular prism.
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Context:
A quadrilateral is given, where two of its sides have lengths of $1$ and $4$. Additionally, a diagonal of length $2$ divides the quadrilateral into two isosceles triangles. Determine the perimeter of this quadrilateral.
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Context:
Compute the integral $$\int_0^1 \frac{\arctan x}{(1+x)^2}dx$$
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Context:
Find the limit of $$\lim_{x\to0}\frac{(1+x)^{\sin x}-1}{x}$$ using L'Hopital's rule.
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Context:
Seven cards numbered $1$ through $7$ are to be lined up in a row. Find the number of arrangements of these seven cards where one of the cards can be removed, leaving the remaining six cards in either ascending or descending order.
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Context:
Vasya thought of a four-digit number and wrote down the product of each pair of its adjacent digits on the board. After that, he erased one product, and the numbers 20 and 21 remained on the board. What is the smallest number Vasya could have in mind?
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Context:
In a group of 150 people who own pets, the following information is given:
- 30 people own only dogs
- 25 people own only cats
- 10 people own only birds
- 7 people own only snakes
- 15 people own both cats and dogs
- 12 people own both birds and dogs
- 8 people own both birds and cats
- 3 people own both snakes and dogs
- 4 people own both snakes and cats
- 2 people own both snakes and birds
- 5 people own cats, dogs, and snakes
- 4 people own cats, dogs, and birds
- 6 people own cats, birds, and snakes
- 9 people own dogs, birds, and snakes
- 10 people own all four pets

How many total snakes are owned by the people in this group?
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Context:
设整数数列 $a_{1}, a_{2}, \cdots, a_{10}$ 满足 $a_{10}=3 a_{1}, a_{2}+a_{8}=2 a_{5}$, 且 $a_{i+1} \in\left\{1+a_{i}, 2+a_{i}\right\}, i=$ $1,2, \cdots, 9$, 则这样的数列的个数为 $\qquad$.
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Context:
设有三个函数，第一个是 $y=\varphi(x)$ ，它的反函数就是第二个函数，而第三个函数的图像与第二个函数的图像关于直线 $x+y=0$ 对称。第三个函数的表达式可以写成 $ay=b\varphi(cx)$ 的形式，请给出$a+b+c$的值。
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Context:
A bag contains 4 red, 3 blue, and 6 yellow marbles. One marble is drawn and removed from the bag but is only considered in the new count if it is yellow. What is the probability, expressed as a fraction, of then drawing one marble which is either red or blue from the updated contents of the bag?
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Context:
已知整数 $a, b, c, d$ 满足 $a+b+c+d=6$ ，则 $a b+a c+a d+b c+b d+c d$ 的正整数取值个数为 $\qquad$.
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Context:
Given an ellipse $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1 (a>b>0)$ with two vertices A(a, 0) and B(0, b). Draw perpendicular lines to AB through A and B, intersecting the ellipse at distinct points C and D. If 2|BD| = 3|AC|, what is the eccentricity of the ellipse?
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Context:
Given two equiangular polygons $P_1$ and $P_2$ with different numbers of sides; each angle of $P_1$ is $x$ degrees and each angle of $P_2$ is $kx$ degrees, where $k$ is an integer greater than $1$. The number of possibilities for the pair $(x, k)$ is:
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Context:
对于有理数加群 (\bQ,+)，记所有分母不超过 10 有理数对应的象构成的集合是 A. 设包含 A 的最小群是 G，则 G 的阶为__________.
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Context:
A refrigerator's price was increased by 10% twice consecutively, and then by 5% once. Calculate the percentage by which the price should be reduced to restore the original price. Express your answer as an integer percentage.
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Context:
In an isosceles triangle \(ABC\), the angle at the vertex \(B\) is \(20^\circ\). Points \(D\) and \(K\) are taken on the sides \(AB\) and \(BC\) respectively such that \(\angle KAC = 50^\circ\) and \(\angle DCA = 60^\circ\). Calculate \(\angle CDK\).
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Context:
Several energy-saving devices with a total weight of 120 kg were delivered to the factory. It is known that the total weight of the three lightest devices is 31 kg, and the total weight of the three heaviest devices is 41 kg. How many energy-saving devices were delivered to the factory if the weights of any two devices are different?
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Context:
All arrangements of letters $VNNWHTAAIE$ are listed in lexicographic (dictionary) order. If $AAEHINNTVW$ is the first entry, what entry number is $VANNAWHITE$?
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Context:
How many multiples of 15 are between 25 and 225?
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Context:
How many perfect squares are between 100 and 400?
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Context:
A cistern is two-thirds full of water. Pipe A can fill the remaining part in some minutes and Pipe B in 8 minutes. Once the cistern is emptied, they take 14.4 minutes to fill it together completely. How many minutes does Pipe A take to fill the remaining part?
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Context:
Find the minimum value of the distance $|PQ|$ where point P lies on the curve $y=\frac{1}{2}e^x$, and point Q lies on the curve $y=\ln(2x)$.
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Context:
The ends of a segment are marked with two ones. In the middle between them, their sum, the number 2, is written. Then in the middle between every two adjacent numbers, again their sum is written, and so on for 1973 times. How many times will the number 1973 be written?
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Context:
Given 5 points \( A, B, C, D, E \) on a plane, with no three points being collinear. How many different ways can one connect these points with 4 segments such that each point is an endpoint of at least one segment?
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Context:
设 $A=\{1,2,3,4,5,6\}, B=\{7,8,9, \cdots, n\}$, 在 $A$ 中任取 3 个数, 在 $B$ 中取 2个数,组成含有 5 个元素的集合 $A_i(i=1,2,3, \cdots, 20)$, 使得 $\left|A_i \cap A_j\right| \leqslant 2,1 \leqslant i<j \leqslant 20$.求 $n$ 的最小值.
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Context:
On the hypotenuse \( BC \) of a right triangle \( ABC \), points \( D \) and \( E \) are marked such that \( AD \perp BC \) and \( AD = DE \). On the side \( AC \), a point \( F \) is marked such that \( EF \perp BC \). Find the angle \( ABF \).
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Context:
Given in rectangle ABCD, AB=8, BC=6, folding the rectangle along AC to form a dihedral angle B-AC-D, the surface area of the circumscribed sphere of the tetrahedron ABCD is.
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Context:
On the extension of the side \(AC\) of triangle \(ABC\) beyond point \(A\), there is a point \(T\) such that \(\angle BAC = 2 \angle BTC\). Find the area of triangle \(ABC\), given that \(AB = AC\), \(BT = 70\), and \(AT = 37\).
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Context:
Evaluate the improper integral $$\int_0^{\frac{\pi}{2}} \cos(2nx) \log(\sin(x)) \, dx$$ for $n > 1$, considering its connection to Fourier series and utilizing the identity for $\log(\cos(x))$.
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Context:
Determine the number of distinct ring homomorphisms $\psi$ $:$ $\Bbb Z$$_1$$_2$ $\rightarrow$ $\Bbb Z$$_3$. Let $\Bbb Z$$_1$$_2$ $=$ $<x>$ and $\Bbb Z$$_3$ $=$ $<y>$.
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Context:
The domain of the function $y=\sin x$ is $[a,b]$, and its range is $\left[-1, \frac{1}{2}\right]$. Calculate the maximum possible value of $b-a$. Express your answer as a numerical value.
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Context:
Consider a polynomial  $P(x) \in \mathbb{R}[x]$ , with degree  $2023$ , such that  $P(\sin^2(x))+P(\cos^2(x)) =1$  for all  $x \in \mathbb{R}$ . If the sum of all roots of  $P$  is equal to  $\dfrac{p}{q}$  with  $p, q$  coprime, then what is the product  $pq$ ?
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Context:
Rotating an isosceles trapezoid around its axis of symmetry forms a solid of revolution which is ____.
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Context:
Given the parabola $y^{2}=2px (p > 0)$, its focus $F$ forms a line $l$ with an inclined angle of $60^{\circ}$ intersecting the parabola at points $A$ and $B$, with $|AF| > |BF|$. Calculate the value of $\frac{|AF|}{|BF|}$. Provide your answer as a single number.
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Context:
Let \( d \) be a positive divisor of 2015. Then the maximum value of the unit digit of \( d^{\frac{2005}{d}} \) is \(\qquad\).
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Context:
If $n$ is an odd number, find the remainder when ${7^{n}}+C_{n}^{1}{{7}^{n-1}}+C_{n}^{2}{{7}^{n-2}}+ \cdots +C_{n}^{n-1}7$ is divided by $9$.
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Context:
The area of the enclosed figure formed by the line $x+y-2=0$, the curve $y=x^3$, and the x-axis is \_\_\_\_\_\_.
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Context:
Alice and Bob play a game together as a team on a  $100 \times 100$  board with all unit squares initially white. Alice sets up the game by coloring exactly  $k$  of the unit squares red at the beginning. After that, a legal move for Bob is to choose a row or column with at least  $10$  red squares and color all of the remaining squares in it red. What is the
smallest  $k$  such that Alice can set up a game in such a way that Bob can color the entire board red after finitely many moves?

Proposed by *Nikola Velov, Macedonia*
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Context:
A gardener is planning to plant a row of 20 trees. There are two types of trees to choose from: maple trees and sycamore trees. The number of trees between any two maple trees (excluding the two maple trees) cannot be equal to 3. What is the maximum number of maple trees that can be planted among these 20 trees?
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Context:
Two neon signs are turned on at the same time. Both signs blink as they are turned on. One sign blinks every 9 seconds. The other sign blinks every few seconds. They will blink together again in 45 seconds. How often does the second sign blink?
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Context:
The real numbers \( x, y, z \) satisfy the equations \( x + y + z = 2 \) and \( xy + yz + zx = 1 \). Find the maximum possible value of \( x - y \).
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Context:
Find the smallest constant $K$ such that for all real numbers $x_1, x_2,\ldots,x_6$ the inequality $$(x_1 + x_2 + ...+ x_6)^2 \leq K \cdot (x_1 (x_2 + x_3) + x_2 (x_3 + x_4) + \ldots+ x_6 (x_1 + x_2))$$ holds.
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Context:
8. Let set $S$ consist of all positive integers not exceeding $2^{40}$, and whose binary representation contains exactly two 1s. If a number is randomly selected from $S$, the probability that this number is divisible by 9 is $\qquad$ .
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Context:
a and b can do a work in 8 days, b and c can do the same work in some days. a, b, and c together can finish it in 6 days. a and c together can finish the work in 8 days. In how many days can b and c do the work?
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Context:
Let $f(n)$ and $g(n)$ be polynomials of degree 2014 such that $f(n)+(-1)^{n} g(n)=2^{n}$ for $n=1,2, \ldots, 4030$. Find the coefficient of $x^{2014}$ in $g(x)$.
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Context:
Find the range of values of $I=\int _{-1}^{1} f(x)g(x) dx$ where $f$ and $g$ are functions defined on $[-1,1]$ that satisfy the following conditions:

* $f(x)f(-x)=1$ for all $x$
* $g(x)$ is an even function
* $\int _{-1}^{1}g(x)= 1$
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Context:
Using the digits 1, 2, 3, 4, 5, 6, form a six-digit number without repeating any digit, such that any two adjacent digits have different parity, and 1 and 2 are adjacent. Calculate the number of such six-digit numbers.
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Context:
对正整数 $n$ 及实数 $x(0 \leqslant x<n)$, 定义
$$
f(n, x)=(1-\{x\}) \cdot C_{n}^{[x]}+\{x\} \cdot C_{n}^{[x]+1},
$$
其中 $[x]$ 表示不超过实数 $x$ 的最大整数， $\{x\}=x-[x]$ .若整数 $m, n \geqslant 2$ 满足
$$
f\left(m, \frac{1}{n}\right)+f\left(m, \frac{2}{n}\right)+\cdots+f\left(m, \frac{m n-1}{n}\right)=123,
$$
求 $f\left(n, \frac{1}{m}\right)+f\left(n, \frac{2}{m}\right)+\cdots+f\left(n, \frac{m n-1}{m}\right)$ 的值.
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Context:
若 $\triangle A B C$ 的三边长分别是 $2,3,4$, 其外心为 $O$, 求 $\overrightarrow{O A} \cdot \overrightarrow{A B}+\overrightarrow{O B} \cdot \overrightarrow{B C}+$ $\overrightarrow{O C} \cdot \overrightarrow{C A}$ 的值。答案的形式为\frac{k}{m}，请给出k+m的值。
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Context:
已知正三棱柱$ABC-{{A}_{1}}{{B1}}{{C}_{1}}$的所有棱长为2，点$M,N$分别在侧面$AB{{B1}}{{A}_{1}}$和$AC{{C1}}{{A}_{1}}$内，$B{{C1}}$与${{B}_{1}}C$交于点$P$，则$\Delta MNP$周长的最小值为_______．
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Context:
Calculate the Cesàro sum of the series $\sum\limits_{n = 1}^\infty {\cos (2n)}$.
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Context:
设复数 $z_{1}, z_{2}$ 满足 $\operatorname{Re}\left(z_{1}\right)>0, \operatorname{Re}\left(z_{2}\right)>0$, 且 $\operatorname{Re}\left(z_{1}^{2}\right)=\operatorname{Re}\left(z_{2}^{2}\right)=2$, 其中 $\operatorname{Re}(z)$ 表示复数 $z$ 的实部.
求 $\left|z_{1}+2\right|+\left|\overline{z_{2}}+2\right|-\left|\overline{z_{1}}-z_{2}\right|$ 的最小值.
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Context:
A journalist visits a company with $\mathrm{N}$ people. He knows that there is a person $\mathrm{Z}$ in this company who knows everyone else, but no one knows him. The journalist can ask each member of the company the question: "Do you know such-and-such person?" Find the smallest number of questions required to definitively identify $\mathrm{Z}$. (Everyone answers questions truthfully. One person can be asked multiple questions.)
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Context:
If the degree measures of the angles of a triangle are in the ratio $3:3:4$, and if the answer is $x^\circ$, what is the value of $x$?
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Context:
On Sunday, Xiaopang found that his alarm clock was broken when he woke up. He estimated the time and set the alarm clock to 7:00 AM. Then he walked to the science museum and saw the electronic clock on the rooftop indicating 8:50 AM. He toured the science museum for an hour and a half and then returned home at the same speed. When he got home, he saw the alarm clock indicating 11:50 AM. What is the correct time that the alarm clock should be set to?
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Context:
one fourth of a solution that was 14 % salt by weight was replaced by a second solution resulting in a solution that was 16 percent sugar by weight . the second solution was what percent salt by weight ?
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Context:
已知$m>0$，$m\ne \frac{\sqrt{17}-1}{2}$，直线${{l}_{1}}:y=m$与函数$y=\left| {{\log 4}}x \right|$的图像从左至右相交于点$A,B$，直线${{l}_{2}}:y=\frac{4}{m+1}$与函数$y=\left| {{\log 4}}x \right|$的图像从左至右相交于点$C,D$，记线段$AC$和$BD$在$x$轴上的投影长度分别为$a,b$，当$m$变化时，$\frac{b}{a}$的最小值是__________.
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Context:
We have 10 springs, each originally $0.5 \mathrm{~m}$ long with a spring constant of $200 \mathrm{~N}/\mathrm{m}$. A mass of $2 \mathrm{~kg}$ is hung on each spring, and the springs, along with the masses, are hung in a series. What is the length of the resulting chain? (Neglect the mass of the springs.)
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Context:
There are  $2012$  backgammon checkers (stones, pieces) with one side is black and the other side is white.
These checkers are arranged into a line such that no two consequtive checkers are in same color. At each move, we are chosing two checkers. And we are turning upside down of the two checkers and all of the checkers between the two. At least how many moves are required to make all checkers same color?
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Context:
Calculate the exponential of the vector field ${\bf w} = y^2\partial_y$ using the formula $\exp(\varepsilon{\bf w}) = \sum_{n \ge 0} \frac{\varepsilon^n}{n!}{\bf w}^n$.
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Context:
Given the functional equation $$f(x+\frac{1}{y}) + f(x-\frac{1}{y}) = 2f(x)\cdot f(\frac{1}{y}) \quad \text{for all } x,y\in\mathbb{R}-\{0\},$$ with the condition $f(0) = \frac{1}{2}$, determine the value of $f(4)$.
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Context:
An ant moves on the following lattice, beginning at the dot labeled $A$. Each minute he moves to one of the dots neighboring the dot he was at, choosing from among its neighbors at random. What is the probability that after 5 minutes he is at the dot labeled $B$? [asy]
draw((-2,0)--(2,0));
draw((0,-2)--(0,2));
draw((1,1)--(1,-1)--(-1,-1)--(-1,1)--cycle);
dot((0,0)); dot((1,0)); dot((2,0)); dot((-1,0)); dot((-2,0)); dot((0,1)); dot((0,2)); dot((0,-1)); dot((0,-2)); dot((1,1)); dot((1,-1)); dot((-1,-1)); dot((-1,1));
label("$A$",(0,0),SW);
label("$B$",(0,1),NE);
[/asy]The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.
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Context:
A watch was sold at a loss of 9%. If it was sold for Rs. 220 more, there would have been a gain of 4%. What is the cost price of the watch?
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Context:
Let  $S$  be a subset of the natural numbers such that  $0\in S$ , and for all  $n\in\mathbb N$ , if  $n$  is in  $S$ , then both  $2n+1$  and  $3n+2$  are in  $S$ .  What is the smallest number of elements  $S$  can have in the range  $\{0,1,\ldots, 2019\}$ ?
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Context:
设 $M=\{1,2, \cdots, 1995\}, A \subseteq M$, 且当 $x \in A$ 时, $19 x \bar{\in} A$. 求 $|A|$ 的最大值.
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Context:
Let $\alpha$ denote $\cos^{-1}(\tfrac{2}{3})$. The recursive sequence $a_0, a_1, a_2, \ldots$ satisfies $a_0 = 1$ and, for all positive integers $n$,
$$a_n = \dfrac{\cos(n\alpha) - (a_1a_{n-1} + \cdots + a_{n-1}a_1)}{2a_0}.$$
Suppose that the series
$$\sum_{k=0}^\infty\dfrac{a_k}{2^k}$$
can be expressed uniquely as $\tfrac{p\sqrt{q}}{r}$, where $p$ and $r$ are coprime positive integers and $q$ is not divisible by the square of any prime. Find the value of $p+q+r$.
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Context:
Let $ABCD$ be a trapezoid with $AB||CD, AB=11, BC=5, CD=19,$ and $DA=7$. Bisectors of $\angle A$ and $\angle D$ meet at $P$, and bisectors of $\angle B$ and $\angle C$ meet at $Q$. If the area of hexagon $ABQCDP$ is $x\sqrt{3}$, what is the value of $x$?
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Context:
Given a function $y = f(x)$ defined on $\mathbb{R}$ that is an even function satisfying $f(2 + x) = f(2 - x)$, and given that the function $y = f(x)$ has at least one zero in the interval $(0, 4)$ and $f(0) = 0$, find the minimum number of zeros of the function $y = f(x)$ in the interval $(-8, 10]$. Express your answer as a single integer.
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Context:
A manager schedules a consultation session at a café with two assistant managers but forgets to set a specific time. All three aim to arrive randomly between 2:00 and 5:00 p.m. The manager, upon arriving, will leave if both assistant managers are not present. Each assistant manager is prepared to wait for 1.5 hours for the other to arrive, after which they will leave if the other has not yet arrived. Determine the probability that the consultation session successfully takes place.
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Context:
Given that $a_{1}, a_{2}, \cdots, a_{10}$ are ten different positive integers satisfying  the equation $\left|a_{i+1}-a_{i}\right|=2 \text { or } 3$, where $i=1,2, \cdots, 10$, with the condition $a_{11}=a_{1}$, determine the maximum value of $M-m$, where $M$ is the maximum number among $a_{1}, a_{2}, \cdots, a_{10}$ and $m$ is the minimum number among $a_{1}, a_{2}, \cdots, a_{10}$.
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Context:
Five points are located on a line. When the ten distances between pairs of points are listed from smallest to largest, the list reads: 2, 4, 5, 7, 8, k, 13, 15, 17, 19. Calculate the value of the missing distance k.
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Context:
Imagine a dodecahedron (a polyhedron with 12 pentagonal faces) and an ant starts at one of the top vertices. The ant will randomly walk to one of three adjacent vertices, denoted as vertex A. From vertex A, the ant then walks to one of another three randomly selected adjacent vertices, signified as vertex B. Calculate the probability that vertex B is one of the five bottom vertices of the dodecahedron. Express your answer as a common fraction.
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Context:
Find the smallest nomial of this sequence that  $a_1=1993^{1994^{1995}}$  and
\[ a_{n+1}=\begin{cases}\frac{a_n}{2}&\text{if  $n$  is even}a_n+7 &\text{if  $n$  is odd.} \end{cases} \]
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Context:
集合 $\mathbf{A}=\{0,1,2, \cdots, 9\}, \mathbf{B}_1, \mathbf{B}_2, \cdots, \mathbf{B}_k$ 为 $\mathbf{A}$ 的非空子集, 且 $\left|\mathbf{B}_i \cap \mathbf{B}_j\right| \leqslant 2(1 \leqslant i<j \leqslant k)$, 求 $k$ 的最大值.
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Context:
Determine how many positive integer multiples of $2002$ can be represented in the form $10^{j} - 10^{i}$, where $i$ and $j$ are integers and $0 \leq i < j \leq 150$.
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Context:
Let $G$ be a $2k$-regular tripartite graph with $V(G) = X_1 \cup X_2 \cup X_3$ where $|X_i| = 2m+1$ for $i=1,2,3$. Determine the minimum number of matchings into which $G$ can be decomposed.
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Context:
Ten distinct positive real numbers are given and the sum of each pair is written (So 45 sums). Between these sums there are 5 equal numbers. If we calculate product of each pair, find the biggest number $k$ such that there may be $k$ equal numbers between them.
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Context:
There were some roses in the vase. Jessica threw away 34 roses from the vase and cut some more new roses from her flower garden to put in the vase. There are now 15 roses in the vase, and she threw away 6 more roses than those she cut from her garden. How many roses were initially in the vase?
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Context:
All sides of the convex pentagon $ABCDE$ are of equal length, and $\angle A= \angle B = 90^\circ.$ If the answer is $x^\circ$, what is the degree measure of $\angle E?$
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Context:
The function $y=(m^2-m-1)x^{m^2-3m-3}$ is a power function, and it is an increasing function on the interval $(0, +\infty)$. Find the value of $m$.
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Context:
Let the function \( f(x) = \frac{\sin x}{2 + \cos x} \). If for any \( x^3 \geq 0 \), \( f(x) \leq ax \) holds, find the range of the real number \( a \).
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\boxed{\left[ \frac{1}{3}, +\infty \right)}


Context:
In trapezoid \(ABCD\) with \(BC \parallel AD\), it is known that \(AD = 3 \cdot BC\). A line intersects the non-parallel sides of the trapezoid at points \(M\) and \(N\) such that \(AM:MB = 3:5\) and \(CN:ND = 2:7\). Find the ratio of the areas of quadrilaterals \(MBCN\) and \(AMND\).
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Context:
Let  $ABC$  be a triangle with  $\angle BAC = 90^o$  and  $D$  be the point on the side  $BC$  such that  $AD \perp BC$ . Let $ r, r_1$ , and  $r_2$  be the inradii of triangles  $ABC, ABD$ , and  $ACD$ , respectively. If  $r, r_1$ , and  $r_2$  are positive integers and one of them is  $5$ , find the largest possible value of    $r+r_1+ r_2$ .
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Context:
Problem 8.6. In a $3 \times 3$ table, real numbers are arranged. It turns out that the product of the numbers in any row and any column is 10, and the product of the numbers in any $2 \times 2$ square is 3. Find the number in the central cell.
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Context:
In the group stage of the World Cup, each group has 4 teams that play in a round-robin tournament. For each match, the winning team gets 3 points, the losing team gets 0 points, and in the event of a draw, both teams get 1 point each. After the group stage, the two teams with the highest total points advance to the next round. If the total points are the same, the teams are then ranked by goal difference. What is the minimum number of points a team must score to ensure advancement?
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Context:
There exists an $x \in (0, +\infty)$ such that the inequality $mx^2 + 2x + m > 0$ holds is a false statement, then the range of values for $m$ is ___.
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Context:
The maximum number of students among whom 781 pens and some pencils can be distributed in such a way that each student gets the same number of pens and same number of pencils is 71. How many pencils are there?
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Context:
Let $f(x)$ and $g(x)$ be odd and even functions defined on $\mathbb{R}$, respectively. If for $x < 0$, $f'(x)g(x) + f(x)g'(x) > 0$ and $g(-3) = 0$, then the solution set for $f(x)g(x) < 0$ is.
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Context:
In a group of 80 children, there are 35 happy, 15 sad, 10 confused, 5 excited, and 15 neither happy nor sad. There are 45 boys and 35 girls. If there are 8 happy boys, 7 sad girls, 4 confused boys and 3 excited girls, how many boys are neither happy nor sad?
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Context:
Triangle \(\triangle P N R\) has side lengths \(P N=20, N R=18\), and \(P R=19\). Consider a point \(A\) on \(P N\). \(\triangle N R A\) is rotated about \(R\) to \(\triangle N^{\prime} R A^{\prime}\) so that \(R, N^{\prime}\), and \(P\) lie on the same line and \(A A^{\prime}\) is perpendicular to \(P R\). Find \(\frac{P A}{A N}\).
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Context:
In the quadrilateral $ABCD$, the angles $B$ and $D$ are right . The diagonal $AC$ forms with the side $AB$ the angle of $40^o$, as well with side $AD$ an angle of $30^o$. Find the acute angle between the  diagonals $AC$ and $BD$.
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Context:
A thin beam of light falls normally on a flat parallel glass plate. Behind the plate, at some distance from it, there is an ideal mirror (its reflection coefficient is equal to one). The plane of the mirror is parallel to the plate. It is known that the intensity of the beam passing through this system is 16 times less than the intensity of the incident beam. The reflection coefficient at the glass-air boundary is considered constant regardless of the direction of the beam. Absorption and scattering of light in both air and glass are negligible. Find the reflection coefficient at the glass-air boundary under these conditions. (10 points)
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Context:
若 $\triangle A B C$ 的三个内角 $A, B, C$ 满足 $\cos A=\sin B=2 \tan \frac{C}{2}$, 则 $\sin A+\cos A$ $+2 \tan A$ 的值为 $\qquad$.
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Context:
In a cricket match, 5 batsmen A, B, C, D, and E scored an average of 36 runs, and all of them scored runs only in multiples of 4. D scored 12 more than E; E scored 8 fewer than A; B scored as many as D and E combined; and B and C scored 107 between them. Moreover, A is the opening batsman and goes on to score the maximum runs among them. How many runs did E score?
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Context:
In the diagram, \( PQ \) is perpendicular to \( QR \), \( QR \) is perpendicular to \( RS \), and \( RS \) is perpendicular to \( ST \). If \( PQ=4 \), \( QR=8 \), \( RS=8 \), and \( ST=3 \), then the distance from \( P \) to \( T \) is
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Context:
How many pairs of positive integers $(x, y)$ satisfy the equation $\sqrt{xy} - 71\sqrt{x} + 30 = 0$? Express your answer as a single integer.
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Context:
Given the sample 7, 8, 9, x, y has an average of 8, and xy=60, then the standard deviation of this sample is \_\_\_\_\_\_.
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Context:
A cuboctahedron is a polyhedron whose faces are squares and equilateral triangles such that two squares and two triangles alternate around each vertex. What is the volume of a cuboctahedron of side length 1?
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Context:
A sequence consists of the digits $122333444455555 \ldots$ such that each positive integer $n$ is repeated $n$ times, in increasing order. Find the sum of the 4501st and 4052nd digits of this sequence.
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Context:
Given the function y=cos(2x- π/3), derive the horizontal shift required to transform the graph of the function y=sin 2x.
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Context:
Given the sequence $\left\{a_{n}\right\}$ defined by $a_{0} = \frac{1}{2}$ and $a_{n+1} = a_{n} + \frac{a_{n}^{2}}{2023}$ for $n=0,1,2,\ldots$, find the integer $k$ such that $a_{k} < 1 < a_{k+1}$.
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Context:
Jindra collects dice, all of the same size. Yesterday he found a box in which he started stacking the dice. He managed to fully cover the square base with one layer of dice. He similarly stacked five more layers, but he ran out of dice halfway through the next layer. Today, Jindra received 18 more dice from his grandmother, which were exactly the amount he needed to complete this layer.

How many dice did Jindra have yesterday?
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Context:
Teacher Li and three students (Xiao Ma, Xiao Lu, and Xiao Zhou) depart from school one after another and walk the same route to a cinema. The walking speed of the three students is equal, and Teacher Li's walking speed is 1.5 times that of the students. Currently, Teacher Li is 235 meters from the school, Xiao Ma is 87 meters from the school, Xiao Lu is 59 meters from the school, and Xiao Zhou is 26 meters from the school. After they walk a certain number of meters, the distance of Teacher Li from the school will be equal to the sum of the distances of the three students from the school. Find the number of meters they need to walk for this condition to be true.
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Context:
Calculate the value of $3^{12} \cdot 3^3$ and express it as some integer raised to the third power.
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Context:
Compute  $\left\lceil\displaystyle\sum_{k=2018}^{\infty}\frac{2019!-2018!}{k!}\right\rceil$ . (The notation  $\left\lceil x\right\rceil$  denotes the least integer  $n$  such that  $n\geq x$ .)

*Proposed by Tristan Shin*
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Context:
Example 10 Let real numbers $x, y, z, w$ satisfy
$$
\left\{\begin{array}{l}
\frac{x^{2}}{2^{2}-1^{2}}+\frac{y^{2}}{2^{2}-3^{2}}+\frac{z^{2}}{2^{2}-5^{2}}+\frac{w^{2}}{2^{2}-7^{2}}=1, \\
\frac{x^{2}}{4^{2}-1^{2}}+\frac{y^{2}}{4^{2}-3^{2}}+\frac{z^{2}}{4^{2}-5^{2}}+\frac{w^{2}}{4^{2}-7^{2}}=1, \\
\frac{x^{2}}{6^{2}-1^{2}}+\frac{y^{2}}{6^{2}-3^{2}}+\frac{z^{2}}{6^{2}-5^{2}}+\frac{w^{2}}{6^{2}-7^{2}}=1, \\
\frac{x^{2}}{8^{2}-1^{2}}+\frac{y^{2}}{8^{2}-3^{2}}+\frac{z^{2}}{8^{2}-5^{2}}+\frac{w^{2}}{8^{2}-7^{2}}=1 .
\end{array}\right.
$$

Try to find the value of $x^{2}+y^{2}+z^{2}+w^{2}$.
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Context:
Suppose a biased coin gives head with probability $\frac{2}{3}$. The coin is tossed repeatedly. If it shows heads, then player $A$ rolls a fair die; otherwise, player $B$ rolls the same die. The process ends when one of the players gets a $6$, and that player is declared the winner.\n\nIf the probability that $A$ will win is given by $\frac{m}{n}$, where $m$ and $n$ are coprime, then what is the value of $m^2n$?
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Context:
Let \( ABCDEF \) be a regular hexagon. A frog starts at vertex \( A \) and can jump to one of the two adjacent vertices randomly on each jump. If the frog reaches point \( D \) within 5 jumps, it stops jumping. If the frog cannot reach \( D \) within 5 jumps, it stops jumping after completing 5 jumps. Determine the total number of different jumping sequences possible from the start until it stops.
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Context:
定义在集合 $\left\{x \in \mathbf{Z}_{+} \mid 1 \leqslant x \leqslant 12\right\}$ 上的函数 $f(x)$ 满足
$$
|f(x+1)-f(x)|=1(x=1,2, \cdots, 11),
$$
且 $f(1) 、 f(6) 、 f(12)$ 成等比数列。若 $f(1)$ $=1$ ，则满足条件的不同函数 $f(x)$ 的个数为
$\qquad$。
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Context:
Use $\max\{a, b\}$ to represent the maximum value between $a$ and $b$. Suppose the function $f(x)=\max\{-4x^{3}+kx-1, \ln x\}$ (where $x > 0$) has three zeros, then the range of the real number $k$ is ______.
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Context:
Let $T_1$ be a triangle with side lengths $2011, 2012,$ and $2013$. For $n \ge 1$, if $T_n = \triangle ABC$ and $D, E,$ and $F$ are the points of tangency of the incircle of $\triangle ABC$ to the sides $AB, BC$, and $AC,$ respectively, then $T_{n+1}$ is a triangle with side lengths $AD, BE,$ and $CF,$ if it exists. What is the perimeter of the last triangle in the sequence $( T_n )$?
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Context:
Compute \( \sum_{k=1}^{420} \gcd(k,420) \).
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Context:
In a factory, some machines make 10 cell phones each minute. It would take 10 machines to make 50 cell phones a minute. How many cell phones do these machines make each minute?
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Context:
Start by writing the integers $1, 2, 4, 6$ on the blackboard. At each step, write the smallest positive integer $n$ that satisfies both of the following properties on the board.
[list]
[*] $n$ is larger than any integer on the board currently.
[*] $n$ cannot be written as the sum of $2$ distinct integers on the board.
[/list]
Find the $100$-th integer that you write on the board. Recall that at the beginning, there are already $4$ integers on the board.
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Context:
Given a sequence $\{a\_n\}$ with the sum of its first $n$ terms denoted as $S\_n$, where $S\_n=n^2+2a|n-2016|$ $(a > 0, n \in \mathbb{N}^+)$, find the maximum value of $a$ that ensures $a\_n \leq a\_{n+1}$ always holds true.
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Context:
There are \( n \) parking spots along a road. Each of the \( n \) drivers drives their car to their favorite spot to park. If their favorite spot is already occupied, they will park in the nearest available spot down the road. If their favorite spot and all the spots beyond it are occupied, they will drive away without parking. How many different arrays \(\left(a_{1}, a_{2}, \cdots, a_{n}\right)\) are there such that no parking spot is left empty? Here, \( a_{i} \) represents the favorite spot of the \( i \)-th driver, and \( a_{1}, a_{2}, \cdots, a_{n} \) are not necessarily distinct.
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Context:
In a square \(ABCD\) with a side length of 2, \(E\) is the midpoint of \(AB\). The figure is folded along segments \(EC\) and \(ED\) so that segments \(EA\) and \(EB\) coincide. The resulting tetrahedron is \(CDEA\) (with point \(B\) coinciding with point \(A\)). What is the volume of this tetrahedron?
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Context:
正整数 $a, b, c, x, y, z$ 满足: $a x=b+c, b y=c+a, c z=a+b$ ，求 $x y z$ 的可能值的个数。
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Context:
In a right triangle $DEF$ where leg $DE = 30$ and leg $EF = 40$, determine the number of line segments with integer length that can be drawn from vertex $E$ to a point on hypotenuse $\overline{DF}$.
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Context:
A number is called *6-composite* if it has exactly 6 composite factors. What is the 6th smallest 6-composite number? (A number is *composite* if it has a factor not equal to 1 or itself. In particular, 1 is not composite.)

*Ray Li.*
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Context:
The sequence \( a_{0}, a_{1}, \dots \) is defined as follows:

\[ a_{0} = 1995, \]
\[ a_{n} = n a_{n-1} + 1, \quad \text{for } n \geq 1. \]

What is the remainder when \( a_{2000} \) is divided by 9?
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Context:
已知直线 $L: x+y-9=0$ 和圆 $M: 2 x^{2}+2 y^{2}-8 x-8 y-1=0$, 点 $A$ 在直线 $L$ 上, $B, C$ 为圆 $M$ 上两点, 在 $\triangle A B C$ 中, $\angle B A C=45^{\circ}, A B$ 过圆心 $M$,则点 $A$ 横坐标范围为 $\qquad$. 请给出该范围的两个端点之和.
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Context:
Which are more: three-digit numbers where all digits have the same parity (all even or all odd), or three-digit numbers where adjacent digits have different parity?
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Context:
Given proposition p: The function y = kx + 1 is an increasing function on R, and proposition q: The function y = x^2 - 2x + 3 has a maximum value of 3 and a minimum value of 2 on the interval [0, k]. If p ∧ q is a false proposition, and p ∨ q is a true proposition, find the range of values for k.
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Context:
The power function $f(x)=(m^{2}+2m-2)x^{m}$ is a decreasing function on $(0,+\infty)$. Find the value of the real number $m$.
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Context:
If six people decide to come to a basketball game, but three of them are only 2/5 sure that they will stay for the entire time (the other three are sure they'll stay the whole time), what is the probability that at the end, at least 5 people stayed the entire time?The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.
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Context:
In parallelogram $A B C D$ with angle $A$ equal to $60^{\circ}$, the bisector of angle $B$ intersects side $C D$ at point $E$. A circle with radius $R$ is inscribed in triangle $E C B$. Another circle is inscribed in trapezoid $A B E D$. Find the distance between the centers of these circles.
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Context:
Given the angles of a triangle and side $c$, express the height corresponding to side $c$.
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\boxed{\frac{c \sin \alpha \sin \beta}{\sin \gamma}}


Context:
one day , raviraj left home and cycled 20 km southwards , turned right and cycled 10 km and turned right and cycled 20 km and turned left and cycled 20 km . how many kilometres will he have to cycle to reach his home straight ?
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Context:
Consider the sum $\overline{a b} + \overline{ c d e}$, where each of the letters is a distinct digit between $1$ and $5$. How many values are possible for this sum?
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Context:
Let $z$ be a complex number satisfying $(z+\tfrac{1}{z})(z+\tfrac{1}{z}+1)=1$. Evaluate $(3z^{100}+\tfrac{2}{z^{100}}+1)(z^{100}+\tfrac{2}{z^{100}}+3)$.
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Context:
已知椭圆$\frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1 ( a > b > 0 )$，过椭圆上一点M作圆$x ^ { 2 } + y ^ { 2 } = b ^ { 2 }$的两条切线，切点为P，Q，设直线P与x轴、y轴分别交于点E、F，求$\triangle EOF$面积的最小值.
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Context:
On a very long narrow highway where overtaking is impossible, $n$ cars are driving in random order, each at its preferred speed. If a fast car catches up to a slow car, the fast car has to slow down and drive at the same speed as the slow car. As a result, the cars form groups. Find the expected number of "lonely" cars, i.e., groups consisting of a single car.
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Context:
Let $ABCDEFGHIJ$ be a regular 10-sided polygon inscribed in a circle with center $O$ and radius $5$. The diagonals $AD$ and $BE$ intersect at point $P$, and the diagonals $AH$ and $BI$ intersect at point $Q$. Calculate the length of segment $PQ$.
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Context:
Find the number of ordered quadruples of positive integers \((a,b,c,d)\) such that \(a,b,c,\) and \(d\) are all (not necessarily distinct) factors of \(30\) and \(abcd > 900\).
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Context:
Mr. Jones teaches algebra. He has a whiteboard with a pre-drawn coordinate grid that runs from $-10$ to $10$ in both the $x$ and $y$ coordinates. Consequently, when he illustrates the graph of a quadratic, he likes to use a quadratic with the following properties:

1. The quadratic sends integers to integers.
2. The quadratic has distinct integer roots both between $-10$ and $10$, inclusive.
3. The vertex of the quadratic has integer $x$ and $y$ coordinates both between $-10$ and $10$, inclusive.

How many quadratics are there with these properties?
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Context:
Tim has a bag of some orange-flavoured sweets and Peter has a bag of 44 grape-flavoured sweets. They have to divide up the sweets into small trays with equal number of sweets; each tray containing either orange-flavoured or grape-flavoured sweets only. If the largest possible number of sweets in each tray without any remainder is 4, how many orange-flavoured sweets does Tim have?
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Context:
Given condition $p: x^2 + 2x > 3$, and condition $q: x > a$, and $\neg p$ is a sufficient but not necessary condition for $\neg q$, determine the range of possible values for $a$.
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Context:
On Fedor's bookshelf, there are volumes of works by various authors. When a friend borrowed two volumes of Pushkin, Fedor noticed that he had fully read at least half of the remaining books. After the friend returned the two volumes of Pushkin and borrowed two volumes of Lermontov, Fedor realized that he had fully read less than one-third of the remaining volumes. What is the maximum number of volumes that could have been on Fedor's shelf?
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Context:
Evaluate the integral: $$\int_{0}^{2\pi} \frac{\cos^2(mx)}{\cos^2(x)} \mathrm dx$$ where $m$ is a positive integer.
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Context:
A flagpole is 12 feet tall. It breaks, folding over at a certain point, such that what was the tip of the flagpole is now dangling two feet above the ground. The flagpole broke 7 feet from the base. At what point did the flagpole break in relation to its height?
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Context:
Let $ABC$ be an equilateral triangle. Extend side $\overline{AB}$ beyond $B$ to a point $B'$ so that $BB'=3 \cdot AB$. Similarly, extend side $\overline{BC}$ beyond $C$ to a point $C'$ so that $CC'=3 \cdot BC$, and extend side $\overline{CA}$ beyond $A$ to a point $A'$ so that $AA'=3 \cdot CA$. What is the ratio of the area of $\triangle A'B'C'$ to the area of $\triangle ABC$?
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Context:
Points \( K \) and \( M \) are the midpoints of the edges \( AB \) and \( AC \) of the triangular pyramid \( ABCD \) with a base area \( p \). Find the area of face \( BCD \), given that the cross-section \( DKM \) has an area \( q \), and the base of the pyramid's height falls on the intersection of the medians of the base triangle \( ABC \).
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Context:
The Knave of Hearts tells only the truth on Mondays, Tuesdays, Wednesdays, and Thursdays. He tells only lies on all the other days. The Knave of Diamonds tells only the truth on Fridays, Saturdays, Sundays, and Mondays. He tells only lies on all the other days. On one day last week, they both said, "Yesterday I told lies." On which day of the week did they make this statement? Provide your answer as the full day of the week.
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Context:
A mathematical demonstration showed that there were distinct positive integers such that $97^4 + 84^4 + 27^4 + 3^4 = m^4$. Calculate the value of $m$.
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Context:
Given that $\alpha$ and $\beta$ are both acute angles, and $\cos(\alpha+\beta)=\sin\alpha\sin\beta$, then $\tan \alpha=$\_\_\_\_\_\_.
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Context:
What is the maximum number of sides of a convex polygon that has no two adjacent obtuse angles?
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Context:
Let \( D \) be a point inside \( \triangle ABC \) such that \( \angle BAD = \angle BCD \) and \( \angle BDC = 90^\circ \). If \( AB = 5 \), \( BC = 6 \), and \( M \) is the midpoint of \( AC \), find the length of \( DM \).
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Context:
On a computer keyboard, the key for the digit 1 is not working. For example, if you try to type the number 1231234, only the number 23234 will actually print.

Sasha tried to type an 8-digit number, but only 202020 was printed. How many 8-digit numbers satisfy this condition?
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Context:
On an 8x8 board, chips are placed in such a way that each chip lies in a row or column that contains only one chip. What is the maximum possible number of chips on the board?
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Context:
On a given circle, six points $A$, $B$, $C$, $D$, $E$, and $F$ are chosen at random, independently and uniformly with respect to arc length. Determine the probability that the two triangles $ABC$ and $DEF$ are disjoint, i.e., have no common points.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:
Triangle \(ABC\) is given with \(AB = 13\), \(BC = 14\), and \(CA = 15\). Let \(E\) and \(F\) be the feet of the altitudes from \(B\) and \(C\), respectively. Let \(G\) be the foot of the altitude from \(A\) in triangle \(AFE\). Find \(AG\).
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Context:
A right triangle with legs each measuring 7 cells was drawn on graph paper. Then all the lines of the grid inside the triangle were outlined. What is the maximum number of triangles that can be found in this drawing?
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Context:
Given a bivariate standard Gaussian distribution $(A, B) \sim \mathbb{N}(0, I_2)$ on $\mathbb{R}^2$, where $I_2$ is the $2 \times 2$ identity matrix, find the expected value of the maximum between $2.5A + B$ and $A + 2.5B$:

$$\mathbb{E}\left[\max(2.5A + B, A + 2.5B)\right]$$

The initial attempt to express this as a conditional expectation did not simplify the calculation:

$$\mathbb{E}(2.5A + B | A > B) \cdot \mathbb{P}(A > B) + \mathbb{E}(A + 2.5B | A < B) \cdot \mathbb{P}(A < B)$$
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Context:
The circle inscribed in the isosceles triangle \( ABC \) with base \( AB \) touches the side \( BC \) at point \( M \). Segment \( AM \) intersects the circle at point \( N \). Find \(\frac{MN}{AN}\) if \(\frac{AB}{BC} = k\).
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Context:
Thirty girls - 13 in red dresses and 17 in blue dresses - were dancing in a circle around a Christmas tree. Afterwards, each of them was asked if her neighbor to the right was in a blue dress. It turned out that only those girls who stood between girls in dresses of the same color answered correctly. How many girls could have answered affirmatively?
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Context:
Find the maximum real number \( k \) such that for any positive numbers \( a \) and \( b \), the following inequality holds:
$$
(a+b)(ab+1)(b+1) \geqslant k \, ab^2.
$$
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Context:
Seven dwarfs lined up by height, starting with the tallest. The first (i.e., the tallest) said: "My height is 60 cm." The second said: "My height is 61 cm." Then in order: "My height is 62 cm," "My height is 63 cm," "My height is 64 cm," "My height is 65 cm," and finally the shortest said: "My height is 66 cm." What is the largest number of dwarfs that could be telling the truth?
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Context:
In the acute triangle \(ABC\), the altitudes \(AA_1\) and \(BB_1\) are drawn. The bisector of the exterior angle at vertex \(C\) intersects lines \(AB\) and \(A_1B_1\) at points \(L\) and \(K\) respectively. It turns out that \(CL = 2CK\). Find the angle \(C\).
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Context:
In triangle \(ABC\), a median \(AM\) is drawn. Circle \(\alpha\) passes through point \(A\), touches line \(BC\) at point \(M\), and intersects sides \(AB\) and \(AC\) at points \(D\) and \(E\) respectively. On arc \(AD\) that does not contain point \(E\), a point \(F\) is chosen such that \(\angle BFE = 72^\circ\). It is found that \(\angle DEF = \angle ABC\). Find \(\angle CME\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the limit of the function:
$$\lim _{x \rightarrow 0}(\cos \pi x)^{\frac{1}{x \cdot \sin \pi x}}$$
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Context:
We call a number antitriangular if it can be expressed in the form \(\frac{2}{n(n+1)}\) for some natural number \(n\). For how many numbers \(k\) (where \(1000 \leq k \leq 2000\)) can the number 1 be expressed as the sum of \(k\) antitriangular numbers (not necessarily distinct)?
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Context:
设复数 \(z\) 满足 \(|3z-7i|=3\)，求 \(\left|\frac{z^{2}-2z+2}{z-1+i}\right|\) 的最大值。答案的形式为 \(\frac{k}{m}\)，请给出k+m的值。
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Context:
Let \( a \) and \( b \) be real numbers, and consider the function \( f(x) = x^{3} + a x^{2} + b x \). If there exist three real numbers \( x_{1}, x_{2}, x_{3} \) such that \( x_{1} + 1 \leqslant x_{2} \leqslant x_{3} - 1 \), and \( f(x_{1}) = f(x_{2}) = f(x_{3}) \), find the minimum value of \( |a| + 2|b| \).
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Context:
设函数 $f(x)=a \ln x+\frac{2 a}{x}-\frac{e^{x}}{x^{2}}$.
若 $a \leq e$, 求 $f(x)$ 的最大值.
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Context:
For a numerical sequence $\left\{x_{n}\right\}$, in which all terms are distinct starting from $n \geq 2$, the relationship $x_{n} = \frac{x_{n-1} + 98 x_{n} + x_{n+1}}{100}$ holds. Find $\sqrt{\frac{x_{2023} - x_{1}}{2022} \cdot \frac{2021}{x_{2023} - x_{2}}} + 2021$.
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Context:
Determine the monotonically increasing interval of the function $y = \ln(4-x) + \ln(2+x)$. Express your answer as an interval in standard interval notation (e.g., (a, b)).
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Context:
From the 4040 integers ranging from -2020 to 2019, three numbers are randomly chosen and multiplied together. Let the smallest possible product be $m$ and the largest possible product be $n$. What is the value of $\frac{m}{n}$? Provide the answer in simplest fraction form.
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Context:
How to calculate the value of the sum $$\sum_{k = 0}^{\infty} \frac{(-1)^k}{k} \sin(2k)$$?
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Context:
There are six piles of apples, and the number of apples in each pile forms an arithmetic sequence. Junjun selects one of the piles and takes out 150 apples, then distributes these to the remaining 5 piles with 10, 20, 30, 40, and 50 apples respectively. After the distribution, Junjun magically finds that the number of apples in these 5 piles correspond to the 2, 3, 4, 5, and 6 times the number of apples in the pile he selected. How many apples are there in total in the six piles?
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Context:
Let $p(x)=x^{2}-x+1$. Let $\alpha$ be a root of $p(p(p(p(x))))$. Find the value of $(p(\alpha)-1) p(\alpha) p(p(\alpha)) p(p(p(\alpha)))$
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Context:
Given the sequence \(\left\{a_{n}\right\}\) satisfying \(a_{1}=a > 2\), \(a_{2017} = 2017\), and for any positive integer \(n\), \(a_{n+1} = a_{n}^{2} - 2\). Compute \(\left\lfloor \frac{\sqrt{a-2}}{10^{6}} a_{1} a_{2} \cdots a_{2017} \right\rfloor\), where \(\lfloor x \rfloor\) denotes the greatest integer not exceeding the real number \(x\).
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Context:
Let \( \triangle ABC \) be a triangle with \( \angle A = 18^{\circ} \) and \( \angle B = 36^{\circ} \). Let \( M \) be the midpoint of \( AB \), \( D \) a point on ray \( CM \) such that \( AB = AD \), \( E \) a point on ray \( BC \) such that \( AB = BE \), and \( F \) a point on ray \( AC \) such that \( AB = AF \). Find \( \angle FDE \).
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Context:
已知点集 $A=\left\{(x, y) \left\lvert\,(x-3)^{2}+(y-4)^{2} \leqslant\left(\frac{5}{2}\right)^{2}\right.\right\}$,
$B=\left\{(x, y) \left\lvert\,(x-4)^{2}+(y-5)^{2}>\left(\frac{5}{2}\right)^{2}\right.\right\}$. 则点集 $A \cap B$ 中的整点（即横、纵坐标均为整数的点）的个数是 $\qquad$.
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Context:
A right circular cone is sliced into five pieces of equal height by planes parallel to its base. Determine the ratio of the volume of the second-largest piece to the volume of the largest piece.
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Context:
Travis has to babysit the terrible Thompson triplets. Knowing that they love big numbers, Travis devises a counting game for them. First Tadd will say the number $1$, then Todd must say the next two numbers ($2$ and $3$), then Tucker must say the next three numbers  ($4$, $5$, $6$), then Tadd must say the next four numbers ($7$, $8$, $9$, $10$), and the process continues to rotate through the three children in order, each saying one more number than the previous child did, until the number $10,000$ is reached. What is the $2019$th number said by Tadd?
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Context:
Bob's Rice ID number has six digits, each a number from 1 to 9, and any digit can be used any number of times. The ID number satisfies the following property: the first two digits is a number divisible by 2, the first three digits is a number divisible by 3, etc. so that the ID number itself is divisible by 6. One ID number that satisfies this condition is 123252. How many different possibilities are there for Bob's ID number?
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Context:
How many rectangles can be formed where each vertex is a point on a 4x4 grid of equally spaced points?
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Context:
Evaluate the product $\frac{1}{3} \cdot \frac{9}{1} \cdot \frac{1}{27} \cdot \frac{81}{1} \dotsm \frac{1}{6561} \cdot \frac{19683}{1}$.
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Context:
Given the ellipse $\dfrac {x^{2}}{a^{2}}+ \dfrac {y^{2}}{b^{2}}=1(a > b > 0)$, $F_{1}$ and $F_{2}$ are the left and right foci of the ellipse, respectively. A line passing through $F_{1}$ intersects the ellipse at points $A(x_{1},y_{1})$ and $B(x_{2},y_{2})$. If the radius of the incircle of $\triangle ABF_{2}$ is $1$, $|F_{1}F_{2}|=2$, and $|y_{1}-y_{2}|=3$, then the eccentricity of the ellipse is \_\_\_\_\_\_.
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Context:
Given a quadrilateral $ABCD$ inscribed in a circle with side $AB$ extended beyond $B$ to point $E$, if $\measuredangle BAD=92^\circ$ and $\measuredangle ADC=68^\circ$, find $\measuredangle EBC$.
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Context:
Given a line $l$ passes through point $P(0, -2)$, and it always intersects with the line segment $AB$ where $A(1, -1)$ and $B(2, -4)$ are endpoints, find the range of values for the inclination angle of line $l$.
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Context:
There are 84 positive integer solutions for an equation involving x, y, z, and t. The equation is x + y + z + t = 10. What is the value of t in one of the solutions?
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Context:
Let \( f \) be a function taking positive integers to positive integers, such that:

(i) \( f \) is increasing (\( f(n + 1) > f(n) \) for all positive integers \( n \))
(ii) \( f(mn) = f(m) f(n) \) for all positive integers \( m \) and \( n \)
(iii) if \( m \neq n \) and \( m^n = n^m \), then \( f(m) = n \) or \( f(n) = m \)

Find all possible values of \( f(60) \).
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Context:
What is the value of \(b\) such that the graph of the equation \[ 3x^2 + 9y^2 - 12x + 27y = b\] represents a non-degenerate ellipse?
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Context:
There are six lines in the plane. No two of them are parallel, and no point lies on more than three lines. What is the minimum possible number of points that lie on at least two lines?
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Context:
The vending machine fails to drop a snack at a certain rate. One in ten times, it will accidentally drop two snacks. The rest of the time, it drops the purchased snack. If thirty people have used the vending machine once each to purchase a snack, and the vending machine has dropped 28 snacks, what is the rate at which the vending machine fails to drop a snack?
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Context:
A right triangle $ABC$ is inscribed in the circular base of a cone. If two of the side lengths of $ABC$ are $3$ and $4$, and the distance from the vertex of the cone to any point on the circumference of the base is $3$, then the minimum possible volume of the cone can be written as $\frac{m\pi\sqrt{n}}{p}$, where $m$, $n$, and $p$ are positive integers, $m$ and $p$ are relatively prime, and $n$ is squarefree. Find $m + n + p$.
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Context:
1. For all positive real numbers $a, b, c, d$ we have $\left(\frac{a^{3}}{a^{3}+15 b c d}\right)^{\frac{1}{2}} \geq \frac{a^{x}}{a^{x}+b^{x}+c^{x}+d^{x}}$, then $x=$
Please reason step by step, and put your final answer within \boxed{}.


Context:
A sum was put at simple interest at a certain rate for 10 years. Had it been put at a 5% higher rate, it would have fetched Rs. 200 more. What was the sum?
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Context:
On a plane, there is an invisible rabbit hiding on a lattice point. We want to place $n$ hunters on some lattice points to catch the rabbit. In each turn, the following steps occur in order:

1. The rabbit jumps to an adjacent lattice point in the top, bottom, left, or right direction.
2. Each hunter moves to an adjacent lattice point in the top, bottom, left, or right direction (each hunter can move in a different direction). Then, they shoot a bullet which travels $\frac{334111214}{334111213}i$ units in the direction they chose.

If a bullet hits the rabbit, then it is caught. Find the smallest number $n$ such that the rabbit would definitely be caught in a finite number of turns.
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Context:
As a special treat, Georgia makes muffins and brings them to her students on the first day of every month. Her muffin recipe only makes 6 muffins. Georgia makes 36 batches of muffins in 9 months. How many students does Georgia have?
Please reason step by step, and put your final answer within \boxed{}.