Context:
The distances between the points are given as $A B = 30$, $B C = 80$, $C D = 236$, $D E = 86$, $E A = 40$. What is the distance $E C$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a $4 \times 4$ grid, numbers are written in such a way that the sum of the numbers in the neighboring cells (cells that share a side) of each number is equal to 1.

Find the sum of all the numbers in the grid.
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Context:
The three medians of a triangle divide its angles into six smaller angles, among which exactly \( k \) are greater than \( 30^\circ \). What is the maximum possible value of \( k \)?
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Context:
Find all real functions \( f: \mathbb{R} \to \mathbb{R} \) that satisfy the functional equation \[ f(xy+x) + f(y) = f(xy+y) + f(x) \] for all real numbers \( x \) and \( y \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
The edge of cube \( ABCD A_1 B_1 C_1 D_1 \) is 12. Point \( K \) lies on the extension of edge \( BC \) at a distance of 9 from vertex \( C \). Point \( L \) on edge \( AB \) is at a distance of 5 from \( A \). Point \( M \) divides segment \( A_1 C_1 \) in a ratio of 1:3, starting from \( A_1 \). Find the area of the cross-section of the cube by the plane passing through points \( K \), \( L \), and \( M \).
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Context:
Find the limit of the following Riemann sum as $n$ approaches infinity:

$$\lim _{n\to \infty }\frac{1}{n^3}\sum _{k=1}^{2n}\frac{k^2}{8}\arcsin \left(\frac{k}{2n}\right) = L$$
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Context:
Determine the minimum number of keys required to ensure that any 40 out of 50 drivers can operate 40 cars, each with a unique lock.
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Context:
Compute the number of integers \(n \in\{1,2, \ldots, 300\}\) such that \(n\) is the product of two distinct primes, and is also the length of the longest leg of some nondegenerate right triangle with integer side lengths.
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Context:
$ABCD$ is a cyclic quadrilateral with circumcenter $O$ and circumradius $7$. The line $AB$ intersects $CD$ at $E$, and the line $DA$ intersects $CB$ at $F$. Given that $OE = 13$ and $OF = 14$, let $\cos \angle FOE = \frac{p}{q}$, where $p$ and $q$ are coprime integers. Find $p + q$.
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Context:
In the following figure, the bigger wheel has circumference  $12$ m and the inscribed wheel has circumference  $8 $ m.  $P_{1}$  denotes a point on the bigger wheel and  $P_{2}$  denotes a point on the smaller wheel. Initially  $P_{1}$  and  $P_{2}$  coincide as in the figure. Now we roll the wheels on a smooth surface and the smaller wheel also rolls in the bigger wheel smoothly. What distance does the bigger wheel have to roll so that the points will be together again?
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Context:
Determine the measure of the set of all irrational numbers $x$ in the interval $[0,1]$ such that, in its decimal representation, if a $1$ appears, it is immediately followed by a $2$. Assume the decimal representation does not terminate.
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Context:
Let $\boldsymbol{g}$ be an $N$-by-$1$ complex Gaussian random vector with distribution $CN(0, \boldsymbol{I})$. Consider an $N$-by-$M$ complex matrix $\boldsymbol{A}$, whose columns are orthogonal and have unit norm. Denote the $m$th column vector of $\boldsymbol{A}$ by $\boldsymbol{a}_m$. Determine the distribution of the scalar $x$ as $M$ approaches infinity:

$$x = \frac{1}{M} \sum_{m=1}^{M} \boldsymbol{g}^H \boldsymbol{a}_m$$
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Context:
Given a sequence $\{a_n\}$ satisfying $a_1=1$, $a_2=3$, if $|a_{n+1}-a_n|=2^n$ $(n\in\mathbb{N}^*)$, and the sequence $\{a_{2n-1}\}$ is increasing while $\{a_{2n}\}$ is decreasing, then $\lim\limits_{n\to\infty} \frac{a_{2n-1}}{a_{2n}}=$ ______.
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Context:
求一切实数 $p$ ，使得三次方程 $5 x^{3}-5(p+1) x^{2}+(71 p-1) x+1=66 p$ 的三个根均为正整数。
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Context:
Given independent random variables, each with a variance not exceeding 4, determine the number of such variables required so that the probability that the deviation of the arithmetic mean from the arithmetic mean of their expectations is no more than 0.25 exceeds 0.99.
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Context:
Determine the degree of the differential equation \( \left(\frac{d^2y}{dx^2}\right)^{\frac{3}{2}} + \left(\frac{dy}{dx}\right)^{\frac{2}{3}} + y^2 = 0 \) after eliminating the radicals.
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Context:
Compute the number of ways there are to assemble 2 red unit cubes and 25 white unit cubes into a $3 \times 3 \times 3$ cube such that red is visible on exactly 4 faces of the larger cube. (Rotations and reflections are considered distinct.)
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Context:
Compute the limit:
\[ \lim_{n\to\infty} e^{-itn^{1/4}}\left(1-\frac{1}{\sqrt{n}}+\frac{1}{\sqrt{n}}e^{i\frac{t}{n^{1/4}}} \right)^{n} \] where \( t \) is a real number.
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Context:
5. Garland (1 b. 6-11). A string of 100 lights was hung on a Christmas tree in a row. Then the lights started switching according to the following algorithm: all lights turned on, after a second, every second light turned off, after another second, every third light switched: if it was on, it turned off and vice versa. After another second, every fourth light switched, then every fifth light, and so on. After 100 seconds, everything was over. Find the probability that a randomly chosen light is on after this (the lights do not burn out or break).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \(a, b, c\) be positive integers. All the roots of each of the quadratics \(a x^{2}+b x+c, a x^{2}+b x-c, a x^{2}-b x+c, a x^{2}-b x-c\) are integers. Over all triples \((a, b, c)\), find the triple with the third smallest value of \(a+b+c\).
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Context:
Given a sector with a central angle of $\frac{4}{3}\pi$ and an area of $6\pi$, which represents the lateral surface of a cone, calculate the volume of the cone. Express your answer in terms of $\pi$ and include the appropriate units and precision.
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Context:
a 100 meter long train crosses a man standing on the platform in 5 sec . what is the speed of the train ?
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Context:
Let \( x[n] \) denote \( x \) raised to the power of \( x \), repeated \( n \) times. What is the minimum value of \( n \) such that \( 9[9] < 3[n] \)?

(For example, \( 3[2] = 3^3 = 27 \); \( 2[3] = 2^{2^2} = 16 \).)
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Context:
Let $a_0$, $a_1$, ..., $a_6$ be real numbers such that $a_0 + a_1 + ... + a_6 = 1$ and
$$a_0 + a_2 + a_3 + a_4 + a_5 + a_6 =\frac{1}{2}$$
$$a_0 + a_1 + a_3 + a_4 + a_5 + a_6 = \frac{2}{3}$$
$$a_0 + a_1 + a_2 + a_4 + a_5 + a_6 =\frac{7}{8}$$
$$a_0 + a_1 + a_2 + a_3 + a_5 + a_6 =\frac{29}{30}$$
$$a_0 + a_1 + a_2 + a_3 + a_4 + a_6 =\frac{143}{144}$$
$$a_0 + a_1 + a_2 + a_3 + a_4 + a_5 =\frac{839}{840}$$

The value of $a_0$ is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
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Context:
In bridge, a standard 52-card deck is dealt in the usual way to 4 players. By convention, each hand is assigned a number of "points" based on the formula $$4 \times(\# \mathrm{~A} \text { 's })+3 \times(\# \mathrm{~K} \text { 's })+2 \times(\# \mathrm{Q} \text { 's })+1 \times(\# \mathrm{~J} \text { 's })$$ Given that a particular hand has exactly 4 cards that are A, K, Q, or J, find the probability that its point value is 13 or higher.
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Context:
One way to pack a 100 by 100 square with 10000 circles, each of diameter 1, is to put them in 100 rows with 100 circles in each row. If the circles are repacked so that the centers of any three tangent circles form an equilateral triangle, what is the maximum number of additional circles that can be packed?
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Context:
A cylinder is inscribed in a cone, with the height of the cylinder equal to the diameter of the cone's base. The total surface area of the cylinder is equal to the area of the base of the cone. Find the angle between the generatrix (slant height) of the cone and the plane of its base.
Please reason step by step, and put your final answer within \boxed{}.

\boxed{\arctan \left( \frac{2(4 + \sqrt{6})}{5} \right)}


Context:
Given vectors $\overrightarrow {m}=(\lambda+1, 1)$ and $\overrightarrow {n}=(4, -2)$. If $\overrightarrow {m} \parallel \overrightarrow {n}$, then $\lambda= \_\_\_\_\_\_$.
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Context:
Yang has the sequence of integers $1,2, \ldots, 2017$. He makes 2016 swaps in order, where a swap changes the positions of two integers in the sequence. His goal is to end with $2,3, \ldots, 2017,1$. How many different sequences of swaps can Yang do to achieve his goal?
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Context:
Star lists the whole numbers $1$ through $50$ once. Emilio copies Star's numbers, but he replaces each occurrence of the digit $2$ by the digit $1$ and each occurrence of the digit $3$ by the digit $2$. Calculate the difference between Star's sum and Emilio's sum.
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Context:
The people in an infinitely long line are numbered $1,2,3,\dots$. Then, each person says either "Karl" or "Lark" independently and at random. Let $S$ be the set of all positive integers $i$ such that people $i$, $i+1$, and $i+2$ all say "Karl," and define $X = \sum_{i \in S} 2^{-i}$. Then the expected value of $X^2$ is $\frac{m}{n}$, where $m$ and $n$ are positive integers with $\gcd(m,n)=1$. Compute $100m+n$.
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Context:
For finite sets  $A,M$  such that  $A \subseteq M \subset \mathbb{Z}^+$ , we define   $$ f_M(A)=\{x\in M \mid x\text{ is divisible by an odd number of elements of }A\}. $$   Given a positive integer  $k$ , we call  $M$  *k-colorable* if it is possible to color the subsets of  $M$  with  $k$  colors so that for any  $A \subseteq M$ , if  $f_M(A)\neq A$  then  $f_M(A)$  and  $A$  have different colors.
Determine the least positive integer  $k$  such that every finite set  $M \subset\mathbb{Z}^+$  is k-colorable.
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Context:
In a regular truncated quadrilateral pyramid, a section is made through the diagonals of the bases, and another section passes through a side of the lower base and the opposite side of the upper base. The angle between the intersecting planes is $\alpha$. Find the ratio of the areas of the sections.
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Context:
The perimeter of a right-angled triangle is 160 cm. If two of the sides are 40 cm and 50 cm, and the triangle is isosceles, find the length of the third side.
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Context:
Find the largest positive integer \(d\) for which there exists a degree $d$ polynomial $P$ with real coefficients such that there are at most $d$ different values among $P(0), P(1), \ldots, P\left(d^{2}-\right.$ $d)$.
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Context:
The lateral edges of a triangular pyramid are pairwise perpendicular, with one of them equal to \(a\) and also equal to the sum of the other two. Find the radius of the sphere that touches the base of the pyramid and its lateral faces' extensions.
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Context:
In an infinite sequence of tosses of a fair coin, what is the expected number of tosses required for the pattern HTTH to appear?
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Context:
Calculate the present value of a continuously varying, continuously paid annuity with payments $(8k + tk)$ made to an account, where $0 \leq t \leq 10$. The interest is credited at a force of interest $\delta_t = \frac{1}{8+t}$. After 10 years, the account's value is $20,000$. Find the value of $k$.
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Context:
Given the equation \( px^2 + y^2 + qz^2 + 2yz + zx + 3xy = 0 \) represents a pair of perpendicular planes, find the value of \( p - q \).
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Context:
Find the greatest integer \( c \) such that the function \( f(x) = \frac{x-1}{c-x^2+1} \) does not take any values in the interval \([-1, -\frac{1}{3}]\).
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Context:
The perimeter of triangle $ABC$ is 1. Circle $\omega$ is tangent to side $BC$, the extension of side $AB$ at point $P$, and the extension of side $AC$ at point $Q$. The line passing through the midpoints of $AB$ and $AC$ intersects the circumcircle of triangle $APQ$ at points $X$ and $Y$. Find the length of segment $XY$.
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Context:
Find the number of pairs  $(n,C)$  of positive integers such that  $C\leq 100$  and  $n^2+n+C$  is a perfect square.
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Context:
Let  $N$  be an odd number,  $N\geq 3$ .  $N$  tennis players take part in a championship. Before starting the championship, a commission puts the players in a row depending on how good they think the players are. During the championship, every player plays with every other player exactly once, and each match has a winner. A match is called *suprising* if the winner was rated lower by the commission. At the end of the tournament, players are arranged in a line based on the number of victories they have achieved. In the event of a tie, the commission's initial order is used to decide which player will be higher.

It turns out that the final order is exactly the same as the commission's initial order. What is the maximal number of suprising matches that could have happened.
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Context:
Given an equilateral triangular prism \( S-ABC \) with height \( SO = 3 \) and a base edge length of 6. A perpendicular is drawn from point \( A \) to the opposite face \( SBC \), with the foot of the perpendicular at \( O' \). On \( AO' \), a point \( P \) is chosen such that \( AP:PO' = 8 \). Find the area of the cross-section passing through point \( P \) and parallel to the base.
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Context:
Two swimmers, at opposite ends of a $90$-foot pool, start to swim the length of the pool, one at the rate of $3$ feet per second, the other at $2$ feet per second. They swim back and forth for $12$ minutes. Allowing no loss of times at the turns, find the number of times they pass each other.
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Context:
The number $5^{867}$ is between $2^{2013}$ and $2^{2014}$.  How many pairs of integers $(m,n)$ are there such that $1\leq m\leq 2012$ and $5^n<2^m<2^{m+2}<5^{n+1}$?
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Context:
In a certain pond, a biologist conducts an experiment over three days to estimate the fish population. On the first day, 80 fish were caught, tagged, and returned to the pond. On the second day, 100 fish were caught, of which 20 were found to have been tagged. On the third day, 120 fish were caught, and 36 of them were found to have been tagged. If the percent of tagged fish caught on each day approximates the percent of tagged fish in the pond, what is the approximate number of fish in the pond?
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Context:
A fair $6$-sided die is repeatedly rolled until an odd number appears. What is the probability that every even number appears at least once before the first occurrence of an odd number? Express your answer in the form \(\frac{k}{m}\), where k and m are integers with no common factors other than 1. Find the value of k + m.
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Context:
Brendan works online and is paid every week. Once he gets the money, he recharges his debit card with half of his pay. In June, he earned $5000 and bought himself a used car. His total amount of remaining money at the end of the month is $1000. What is the worth of the used car he bought?
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Context:
9. The base of the pyramid $\mathrm{TABCD}$ is an isosceles trapezoid $\mathrm{ABCD}$, the midline of which is equal to $5 \sqrt{3}$. The ratio of the areas of the parts of the trapezoid $\mathrm{ABCD}$, into which it is divided by the midline, is $7: 13$. All lateral faces of the pyramid $\mathrm{TABCD}$ are inclined to the base plane at an angle of $30^{\circ}$. Find the volume of the pyramid TAKND, where points $\mathrm{K}$ and $\mathrm{N}$ are the midpoints of edges $\mathrm{TB}$ and $\mathrm{TC}$, respectively, and AD is the larger base of the trapezoid ABCD.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \(ABCD\) be an isosceles trapezoid with \(AB = 1\), \(BC = DA = 5\), and \(CD = 7\). Let \(P\) be the intersection of diagonals \(AC\) and \(BD\), and let \(Q\) be the foot of the altitude from \(D\) to \(BC\). Let \(PQ\) intersect \(AB\) at \(R\). Compute \(\sin \angle RPD\).
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Context:
Determine whether there are any solutions to the functional equation \( f(f(x)-x) = f(f(x)) \) other than the constant solution \( f(x) = k \), where \( k \in \mathbb{R} \).
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Context:
Randomly fill the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9 into a $3 \times 3$ grid, with each small square filled with one number and all numbers being distinct. What is the probability that the sum of the numbers in each row and each column is odd?
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Context:
3. For how many positive integers $n$ less than 200 is $n^{n}$ a cube and $(n+1)^{n+1}$ a square?
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Context:
Calculate the flux of the vector field \( F = g(y,z)i + 8j - (2z+3)k \) through the surface of a curved pipe \( S \), where \( S \) is bounded by two circles: \( C_1 \) with radius 3 in the plane \( y=0 \) and \( C_2 \) with radius 1 in the plane \( z=4 \). The volume of \( S \) is 12. Assume \( g \) has continuous partial derivatives.
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Context:
How many essentially different proportions can be made from the four letters $x, y, z, t$?
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Context:
Call a positive integer  $n\geq 2$  *junk* if there exist two distinct  $n$  digit binary strings  $a_1a_2\cdots a_n$  and  $b_1b_2\cdots b_n$  such that

- $a_1+a_2=b_1+b_2,$
- $a_{i-1}+a_i+a_{i+1}=b_{i-1}+b_i+b_{i+1}$  for all  $2\leq i\leq n-1,$  and
- $a_{n-1}+a_n=b_{n-1}+b_n$ .

Find the number of junk positive integers less than or equal to  $2016$ .

[i]Proposed by Nathan Ramesh
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Context:
Mary's income is 60% more than Tim's income, and Tim's income is 20% less than Juan's income. Mary saves 15% of her income, while Tim and Juan save 12% and 18% of their incomes, respectively. Mary's expenses are 25% of her income, while Tim's expenses are 30% of his income, and Juan spends 20% of his income on rent. Calculate the percentage difference between Mary's and Juan's total savings and expenses combined.
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Context:
A quadrilateral and a pentagon are given on the plane (both may be non-convex), such that no vertex of one lies on the side of the other. What is the maximum possible number of points of intersection of their sides?
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Context:
Natural numbers \(a, b, c\) are chosen such that \(a < b < c\). It is also known that the system of equations \(2x + y = 2019\) and \(y = |x-a| + |x-b| + |x-c|\) has exactly one solution. Find the minimum possible value of \(c\).
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Context:
If 5% more is gained by selling an article for Rs. 350 than by selling it for Rs. 345, what is the cost of the article?
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Context:
Find the maximum value of $ x_{0}$ for which there exists a sequence $ x_{0},x_{1}\cdots ,x_{1995}$ of positive reals with $ x_{0} \equal{} x_{1995}$, such that
\[ x_{i \minus{} 1} \plus{} \frac {2}{x_{i \minus{} 1}} \equal{} 2x_{i} \plus{} \frac {1}{x_{i}},
\]
for all $ i \equal{} 1,\cdots ,1995$.
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Context:
Shapovalov A.V.

Through the origin, lines (including the coordinate axes) are drawn that divide the coordinate plane into angles of $1^{\circ}$.

Find the sum of the abscissas of the points of intersection of these lines with the line $y=100-x$.

#
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Context:
Seven children sit in a circle. The teacher distributes pieces of candy to the children such that:

- Every child gets at least one piece of candy.
- No two children receive the same number of pieces of candy.
- The numbers of candy pieces given to any two adjacent children have a common factor other than 1.
- There is no prime dividing every child's number of candy pieces.

What is the smallest number of pieces of candy that the teacher must have ready for the children?
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Context:
Given a triangular pyramid $P-ABC$, where the volume ${V}_{P-ABC}=\frac{4\sqrt{3}}{3}$, $\angle APC=\frac{\pi }{4}$, $\angle BPC=\frac{\pi }{3}$, $PA\perp AC$, $PB\perp BC$, and the plane $PAC$ is perpendicular to the plane $PBC$, find the volume of the circumscribed sphere of the triangular pyramid $P-ABC$.
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Context:
2018年9月10日，全国教育大会在北京召开，习近平总书记在会上提出“培养德智体美劳全面发展的社会主义建设者和接班人”.某学校贯彻大会精神，为学生开设了一门模具加工课，经过一段时间的学习，拟举行一次模具加工大赛，学生小明、小红打算报名参加大赛.

$参考数据 \left(\text{其中}t_i=\frac{1}{x_i}\right)：$

| $\sum \limits^{7}_{{i=1}}t_iy_i$ | $\overline{t}$ | $\sum \limits^{7}_{{i=1}}t_i^2-7\overline{t}^2$ |
| ------ | ------ | ------ |
| 1845   | 0.37   | 0.55   |

$参考公式：对于一组数据(u1,v1)，(u2,v2)，\ldots ，(un,vn)，其线性回归直线v=\alpha +\beta u的斜率和截距的最小二乘估计公式分别为 \hat{\beta } = \frac{\sum \limits^{n}_{{i=1}}u_iv_i-n\overline{u} \cdot \overline{v}}{\sum \limits^{n}_{{i=1}}u^2_i-n\overline{u}^2} ， \hat{\alpha } = \overline{v} - \hat{\beta } \cdot  \overline{u}.$
$小明和小红拟先举行一次模拟赛，每局比赛各加工一个模具，先加工完成模具的人获胜.两人约定先胜4局者赢得比赛.若小明每局获胜的概率为\frac{3}{5}，已知在前3局中小明胜2局，小红胜1局.若每局不存在平局，请你估计小明最终赢得比赛的概率.$
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Context:
Find the angle within the range of -360° to 0° that has the same terminal side as the angle 1250°.
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Context:
If one takes 6 different letters from "CN Dream" and arranges them in a row, how many different permutations are there that include the letter combination "ea" (in the same order)? Express your answer as a single integer.
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Context:
How many digits in the product of the two eight-digit numbers 11111111 and 99999999 are odd?
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Context:
12 is 4% of a, and a certain percentage of b is 4. c equals b / a. What is the value of the percentage of b?
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Context:
Philatelist Andrey decided to distribute all his stamps equally into 3 envelopes, but there was one extra stamp left. When he distributed them equally into 5 envelopes, there were 3 stamps left; finally, when he distributed them equally into 7 envelopes, 5 stamps were left over. How many stamps does Andrey have in total, given that he recently bought an additional album that can hold 150 stamps since his old album was no longer sufficient?
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Context:
During the summer, Mike mowed lawns for 20 days. He earned 5 dollars for each lawn mowed, and on average, he mowed 2 lawns every weekday and 3 lawns on each weekend day. Mike spent 24 dollars on new mower blades and an additional 15 dollars on gasoline. How many 5 dollar games could he buy with the money he had left?
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Context:
What is the value of the sum
\[
  \sum_z \frac{1}{{\left|1 - z\right|}^2} \, ,
\]where $z$ ranges over all 7 solutions (real and nonreal) of the equation $z^7 = -1$?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:
Let $x_1, x_2, x_3$ be the solutions to the equation $(x - 13)(x - 33)(x - 37) = 1337$. Find the value of
\[
\sum_{i=1}^3 \left[(x_i - 13)^3 + (x_i - 33)^3 + (x_i - 37)^3\right].
\]
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Context:
The sum of the areas of all triangles whose vertices are also vertices of a $1$ by $1$ by $1$ cube is $m + \sqrt{n} + \sqrt{p},$ where $m, n,$ and $p$ are integers. Find $m + n + p.$
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Context:
For each positive integer $n$ let $a_n$ be the least positive integer multiple of $23$ such that $a_n \equiv 1 \pmod{2^n}.$ Find the number of positive integers $n$ less than or equal to $1000$ that satisfy $a_n = a_{n+1}.$
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Context:
A train passes a platform in 34 seconds. The same train passes a man standing on the platform in 20 seconds. The speed of the train is some km/hr, and the length of the platform is 210.0168 meters. What is the speed of the train?
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Context:
设 $a=1+10^{-4}$ 。在 $2023 \times 2023$ 的方格表的每个小方格中填入区间 $[1, a]$ 中的一个实数。设第 $i$ 行的总和为 $x_{i}$ ，第 $i$ 列的总和为 $y_{i}$ ， $1 \leq i \leq 2023$ 。 用$a$表示$\frac{y_{1} y_{2} \cdots y_{2023}}{x_{1} x_{2} \cdots x_{2023}}$ 的最大值为$\frac{(m a+n)^{p}}{p^{p} a^{m}}$，求$m+n+p$的值。
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Context:
Xonathan Jue goes to the casino with exactly $1000. Each week, he has a $\frac{1}{3}$ chance of breaking even and a $\frac{2}{3}$ chance of losing $500. Evaluate the expected amount of weeks before he loses everything.
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Context:
Find the minimum positive integer value of the expression (x + y^2)(x^2 - y)/(xy), where x and y are distinct positive integers.
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Context:
Given a circle with equation $x^2 + y^2 - 4x - 4y - 10 = 0$, there are at least three distinct points on the circle whose distances to the line $l : ax + by = 0$ are $2\sqrt{2}$. Determine the range of possible values for the angle of inclination of line $l$.
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Context:
Determine the number of homomorphisms from a group $G$ of order $2^n(2m+1)$ to an abelian group of order $2^s$, where $0 \leq s \leq n$.
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Context:
A palindrome is a nonnegative integer number that reads the same forwards and backwards when written in base 10 with no leading zeros. A 6-digit palindrome $n$ is chosen uniformly at random. What is the probability that $\frac{n}{11}$ is also a palindrome?
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Context:
6. Find the largest possible number of integers that can be selected from the set $\{1,2,3, \ldots, 100\}$ such that no two of them differ by 2 or 5.
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Context:
Inscribe a rectangle of base  $b$  and height  $h$  in a circle of radius one, and inscribe an isosceles triangle in the region of the circle cut off by one base of the rectangle (with that side as the base of the triangle). For what value of  $h$  do the rectangle and triangle have the same area?
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Context:
I have 6 friends and during a vacation I met them during several dinners. I found that I dined with all the 6 exactly on 1 day; with every 5 of them on 2 days; with every 4 of them on 3 days; with every 3 of them on 4 days; with every 2 of them on 5 days. Further every friend was present at 7 dinners and every friend was absent at 7 dinners. How many dinners did I have alone?
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Context:
From the natural numbers between 1 and 99, 50 different numbers were chosen such that no two of them sum to 100 and no two of them sum to 99. Which numbers were chosen?
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Context:
There are $2022$ equally spaced points on a circular track $\gamma$ with a circumference of $2022$. The points are labeled $A_1, A_2, \ldots, A_{2022}$ in some order, each label used once. Initially, Bunbun the Bunny begins at $A_1$. She hops along $\gamma$ from $A_1$ to $A_2$, then from $A_2$ to $A_3$, until she reaches $A_{2022}$, after which she hops back to $A_1$. When hopping from $P$ to $Q$, she always hops along the shorter of the two arcs $\widehat{PQ}$ of $\gamma$; if $\overline{PQ}$ is a diameter of $\gamma$, she moves along either semicircle.

Determine the maximal possible sum of the lengths of the $2022$ arcs which Bunbun traveled, over all possible labelings of the $2022$ points.
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Context:
Call a set of integers <i>spacy</i> if it contains no more than one out of any three consecutive integers. How many subsets of $\{1,2,3,\ldots,12\},$ including the empty set, are spacy?
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Context:
Determine the maximum length of a repeating loop that can be obtained by repeatedly applying the function \( f(x) \) to any natural number \( x \).
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Context:
How many different positive three-digit integers can be formed using only the digits in the set $\{4, 4, 5, 6, 6, 7, 7\}$, with no digit used more times than it appears in the set?
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Context:
Find the total derivative of \( u(x,z) = \ln\left(\frac{x^\alpha + z^\alpha}{z^\alpha}\right) \) with respect to \( x \), given that \( z = h(x) = (ax^4 + b)^{1/3} \) and \( u'(x,z) = 0 \).
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Context:
Find the angle between the tangents to the curve \( y = \frac{x^{2} \sqrt{3}}{24} \) that pass through the point \( M(4, -2 \sqrt{3}) \).
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Context:
Sir Bedevir will only participate in a jousting tournament if he knows that he has at least a $1 / 2$ probability of winning. In any given match, the probability of the opponents winning is proportional to their combat skills. Bedevir's combat skill is $1$, while the combat skill of the $n$-th opponent is $\frac{1}{2^{n+1}-1}$. How many knights must have registered for the tournament if, after careful calculation, Sir Bedevir decided to participate?
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Context:
Counting from the right end, what is the $2500$th digit of $10000!$?
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Context:
In the expansion of \((1+x^3+x^5)^{20}\), how many distinct terms are present? Express the powers of \(x\) as \(P = 3b + 5c\) with \(a + b + c = 20\). Determine the number of distinct terms in the expansion.
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Context:
Find the expected value of $g(X) = (X+1)f(X+2) - Xf(X+1)$ for a Poisson random variable $X$ with mean $\lambda$.
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Context:
Evaluate the integral \(\displaystyle\oint_C\left(\frac{1}{\sin(z)-1}\right)dz\) where \(C\) is a circle with radius 5, oriented clockwise.
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Context:
Let $x_1, x_2, \dots, x_{100}$ be real numbers such that $|x_1| = 63$ and $|x_{n+1}| = |x_n + 1|$ for $n = 1, 2, \dots, 99$. Find the largest possible value of $(-x_1 - x_2 - \cdots - x_{100})$.
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Context:
Evaluate the integral \( \oint_{|z|=4} \frac{e^{1 /(z-1)}}{z-2} \, dz \) using Cauchy's Integral Formula.
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Context:
For positive integers $N$ and $k$ define $N$ to be $k$-nice if there exists a positive integer $a$ such that $a^k$ has exactly $N$ positive divisors. Determine the quantity of positive integers smaller than $1500$ that are neither $9$-nice nor $10$-nice.
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Context:
If line $l$ passes through the origin and $(-1, 1)$, then the size of the inclination angle of line $l$ is __________.
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Context:
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $a=2 \sqrt {3}$, $c=2 \sqrt {2}$, and $1+ \frac {\tan A}{\tan B}= \frac {2c}{b}$, find $\angle C$.
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Context:
Given three independent and identically distributed (i.i.d.) uniform random variables \( a \), \( b \), and \( c \) with \( a, b, c \sim \text{Unif}(0,1) \), what is the probability that \( c \) is the maximum if \( a+b < 1 \)?
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Context:
Note that there are exactly three ways to write the integer  $4$  as a sum of positive odd integers where the order of the summands matters:
\begin{align*}
1+1+1+1&=4,
1+3&=4,
3+1&=4.
\end{align*}
Let  $f(n)$  be the number of ways to write a natural number  $n$  as a sum of positive odd integers where the order of the summands matters.  Find the remainder when  $f(2008)$  is divided by  $100$ .
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Context:
A starts walking from a place at a uniform speed of 4 kmph in a particular direction. After some time, B starts from the same place and walks in the same direction as A at a uniform speed of 4.555555555555555 kmph and overtakes A after 1 hour 48 minutes. How long after A started walking did B start walking?
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Context:
One hundred points labeled 1 to 100 are arranged in a $10 \times 10$ grid such that adjacent points are one unit apart. The labels are increasing left to right, top to bottom (so the first row has labels 1 to 10 , the second row has labels 11 to 20, and so on). Convex polygon $\mathcal{P}$ has the property that every point with a label divisible by 7 is either on the boundary or in the interior of $\mathcal{P}$. Compute the smallest possible area of $\mathcal{P}$.
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Context:
Consider the sequence of numbers defined recursively by $t_1=1$ and for $n>1$ by $t_n=1+t_{\left(\frac{n}{2}\right)}$ when $n$ is even and by $t_n=\frac{1}{t_{(n-1)}}$ when $n$ is odd. Given that $t_n=\frac{19}{87}$, find the sum of the digits of $n$.
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Context:
In a graduating class, there are three different salary ranges: low income, middle income, and high income. The difference between the highest and lowest salaries in each range is $100,000. The median salary for low income earners is $50,000 higher than the lowest salary, while the median salary for middle and high income earners is $40,000 and $30,000 higher than the lowest salary in their respective ranges. The average salary across all income groups is $20,000 higher than the overall median salary. Furthermore, the number of high income earners is twice the number of low income earners, and the number of middle income earners is equal to the sum of low and high income earners. What is the minimum number of students, e, in the class?
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Context:
Let $n$ be a positive integer greater than $4$ such that the decimal representation of $n!$ ends in $k$ zeros and the decimal representation of $(2n)!$ ends in $3k$ zeros. Let $s$ denote the sum of the four least possible values of $n$. What is the sum of the digits of $s$?
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Context:
5. For the Day of the Russian Flag, the seller decided to decorate the store window with 10 horizontal strips of fabric in three colors. At the same time, he follows two conditions:
1) strips of the same color should not hang next to each other;
2) each blue strip must hang between a white and a red one.

In how many ways can he do this?
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Context:
Given a parabola with the equation $x^{2}=2py$ ($p > 0$), where the distance between the vertex and focus is equal to the diameter of a circle with an area of $4π$. A line $L$ passing through the point $(-1,0)$ intersects the parabola at only one point. Determine the distance from the focus to line $L$.
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Context:
设 A 是正交矩阵（即 AA^T=A^TA=I），且 A 的每个矩阵元均为 \df{1}{2} 或 -\df{1}{2}. 则这样的 A 的个数为__________.
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Context:
设 I=\displaystyle\int_0^{10}\dd x\int_0^5e^{\max\{x^2,4y^2\}}\ \dd y，则 [10\ln I]=__________.
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Context:
12. What is the sum of all possible  $\left(\begin{array}{l}i  j\end{array}\right)$  subject to the restrictions that  $i \geq 10, j \geq 0$ , and  $i+j \leq 20$  ? Count different  $i, j$  that yield the same value separately - for example, count both  $\left(\begin{array}{c}10  1\end{array}\right)$  and  $\left(\begin{array}{c}10  9\end{array}\right)$ .
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Context:
If the integral of $\int \frac{2\sin x}{2\cos x+3}dx$ is $\ln (2\cos x+3)+c$, what is the integral of $\int \frac{4\sin x}{4\cos x+6}dx$?
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Context:
平面上同时和三直线 y=\df{3}{4}x, y=-\df{4}{3}(x-5), y=0 相切的所有圆的半径的乘积为__________.
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Context:
Evariste has drawn twelve triangles such that two consecutive triangles share exactly one edge.

Sophie colors every triangle side in red, green, or blue. Among the $3^{24}$ possible colorings, how many have the property that every triangle has one edge of each color?
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Context:
Given that the solution set of the inequality in terms of $x$, $\left(a-2\right)x^{2}+\left(2a-1\right)x+6 \gt 0$, is $M$. If $3\in M$ and $5\notin M$, then the range of real number $a$ is ______.
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Context:
David, Hikmet, Jack, Marta, Rand, and Todd were in a $12$-person race with $6$ other people. Rand finished $6$ places ahead of Hikmet. Marta finished $1$ place behind Jack. David finished $2$ places behind Hikmet. Jack finished $2$ places behind Todd. Todd finished $1$ place behind Rand. Marta finished in $6$th place. Who finished in $8$th place? Please provide the position of the first letter of their name in the alphabet.
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Context:
Two trains travel in opposite directions at a certain speed and 45 kmph, and a man sitting in the slower train passes the faster train in 4 seconds. The length of the faster train is 90.0072 meters. What is the speed of the slower train?
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Context:
Given a polynomial $$ P(x)=x^{3}-x^{2}-x+a, a \in R $$ where $$ x_{1} \neq x_{2} \neq x_{3}$$ are the roots of P(x). Find the value of $$ {P(x_{1})-x_{1}^{3}\over P'(x_{1})}+{P(x_{2})-x_{2}^{3}\over P'(x_{2})}+{P(x_{3})-x_{3}^{3}\over P'(x_{3})}$$
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Context:
Let triangle \(ABC\) have \(AB=5\), \(BC=6\), and \(AC=7\), with circumcenter \(O\). Extend ray \(AB\) to point \(D\) such that \(BD=5\), and extend ray \(BC\) to point \(E\) such that \(OD=OE\). Find \(CE\).
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Context:
The product of three successive numbers is 1,059,460. Which is the smallest of the three numbers?
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Context:
In a country, there are 4 southern cities and 5 northern cities. Transportation connects each southern city with each northern city in both directions; there is no transportation between other pairs of cities. A traveler starts from a specific southern city and plans to visit all the southern cities once and return to the starting point. How many different routes for such a trip exist?
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Context:
The numbers which contain only even digits in their decimal representations are written in ascending order such that \[2,4,6,8,20,22,24,26,28,40,42,\dots\] What is the  $2014^{\text{th}}$  number in that sequence?
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Context:
2. Using the six digits $0,1,2,3,4,5$ to form a four-digit even number without repeating digits, and arrange these even numbers in ascending order, then the 99th number is $\qquad$ .
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Context:
Given the sample 7, 5, x, 3, 4 has an average of 5, the variance of this sample is \_\_\_\_\_\_.
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Context:
In triangle $ABC$ with centroid $G$ and circumcircle $\omega$, line $\overline{AG}$ intersects $BC$ at $D$ and $\omega$ at $P$. Given that $GD = DP = 3$, and $GC = 4$, find $AB^2$.
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Context:
6. Point $K$ lies on side $A B$ of triangle $A B C$ with an angle of $120^{\circ}$ at vertex $C$. Circles are inscribed in triangles $A K C$ and $B K C$ with centers $O$ and $Q$ respectively. Find the radius of the circumcircle of triangle $O Q C$, if $O K=6, K Q=7$.
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Context:
Compute the quadratic variation of the Ornstein-Uhlenbeck process.
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Context:
Let \( f: \mathbb{Z}_{>0} \rightarrow \mathbb{Z} \) be a function with the following properties:

(i) \( f(1)=0 \),

(ii) \( f(p)=1 \) for all prime numbers \( p \),

(iii) \( f(xy)=y f(x)+x f(y) \) for all \( x, y \in \mathbb{Z}_{>0} \).

Determine the smallest integer \( n \geq 2015 \) that satisfies \( f(n)=n \).

(Gerhard J. Woeginger)
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Context:
The lateral surface of a cone is unfolded into a sector with a radius of 3 and a central angle of $\frac{2\pi}{3}$. The height of this cone is \_\_\_\_\_.
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Context:
A cube-shaped container with an edge length of $12 \mathrm{~cm}$ was filled with liquid up to $\frac{5}{8}$ of its volume, and then slightly tilted along one of its edges. The figure shows the cross-section of the container and the liquid level inside. We know that the length of segment $LC$ is exactly twice the length of segment $KB$. Determine the length of segment $LC$.
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Context:
Let $[x]$ denote the greatest integer less than or equal to $x$, and define $a_{k}=\left[\frac{2009}{k}\right]$ for $k=1, 2, \ldots, 100$. Determine how many distinct integers are there among these 100 integers.
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Context:
Find the missing digit in the number $3^{37}$ when expressed in base 10 using modular arithmetic.
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Context:
Determine the values of the parameter \( a \) for which the equation \( f(x) = p(x) \) has exactly one solution, where \( f(x) = \left|\frac{2 x^{3}-5 x^{2}-2 x+5}{(1.5 x-3)^{2}-(0.5 x-2)^{2}}\right| \) and \( p(x) = |2 x+5|+a \). If there is more than one such value of \( a \), find the sum of these values.
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Context:
Given the function \( f(x) = \sin \omega x + \sin 2x \), where \( \omega \in \mathbf{N}_{+} \) and \( \omega \leq 2023 \), determine the number of possible values of the constant \( \omega \) such that \( f(x) < 2 \) always holds.
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Context:
What is the largest $n$ for which the numbers $1,2, \ldots, 14$ can be colored in red and blue so that for any number $k=1,2, \ldots, n$, there are a pair of blue numbers and a pair of red numbers, each pair having a difference equal to $k$?
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Context:
Find the largest constant \( K > 0 \) such that for any triangle with sides \( a, b, c \) and semiperimeter \( s \), the inequality \( l_a^2 + l_b^2 + l_c^2 > K s^2 \) holds, where \( l_a, l_b, l_c \) are the lengths of the angle bisectors of the triangle.
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Context:
A house worth $9000 is sold by Mr. A to Mr. B at a 10% loss. Mr. B sells the house back to Mr. A at a 10% gain. Calculate Mr. A's net loss from the two transactions. Express your answer in dollars as a whole number.
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Context:
A train 200 m in length crosses a telegraph post in 8 seconds. What is the speed of the train?
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Context:
Find the sum of the decimal digits of the number $(10^{2021} + 2021)^2$?
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Context:
I have two cents and Bill has $n$ cents. Bill wants to buy some pencils, which come in two different packages. One package of pencils costs 6 cents for 7 pencils, and the other package of pencils costs a dime for a dozen pencils (i.e. 10 cents for 12 pencils). Bill notes that he can spend all $n$ of his cents on some combination of pencil packages to get $P$ pencils. However, if I give my two cents to Bill, he then notes that he can instead spend all $n+2$ of his cents on some combination of pencil packages to get fewer than $P$ pencils. What is the smallest value of $n$ for which this is possible?
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Context:
Given an ellipse $C:\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b>0)$ with its left vertex $A$, left focus $F$, and a vertical line passing through the right focus of $C$ intersecting the $x$-axis at point $P$. When the eccentricity of the ellipse $C$ is $\frac{1}{5}$, the maximum value of $\sin \angle APF$ is ______.
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Context:
What is the negation of the proposition "∃x＞0, 3x＜x2"? Express your answer as a proposition.
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Context:
If you are responsible for making digitally symmetric license plates that start with an 8 or 9 as the first digit following the letter "C" and have five digits in total, calculate the maximum number of such plates that can be made. Express your answer as a single integer.
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Context:
The sides and vertices of a pentagon are labeled with the numbers $1$ through $10$. Each side of the pentagon must have the same sum of numbers. What is the smallest possible value of this sum?
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Context:
An urn contains 100 balls, some of which are white and some are black. Let $p$ be the probability of drawing a white ball. You draw one ball at a time, replacing it after each draw. After 100 draws, you observe that all 100 balls drawn are white. What is the value of $p$?
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Context:
In the triangle \(ABC\), points \(K\), \(L\), and \(M\) are taken on sides \(AB\), \(BC\), and \(AD\) respectively. It is known that \(AK = 5\), \(KB = 3\), \(BL = 2\), \(LC = 7\), \(CM = 1\), and \(MA = 6\). Find the distance from point \(M\) to the midpoint of \(KL\).
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Context:
a man goes into a barbershop and gets a $ 15 haircut . he pays the barber with a $ 20 bill , but the barber does n ' t have change because it is an especially slow day . the barber goes to the flower shop next to his own and trades the $ 20 bill for a $ 10 dollar bill and two $ 5 bills . then he gives the man his $ 5 in change . later the lady from the flower shop confronts the barber and tells him that the $ 20 bill he gave her was counterfeit . the barber agrees and he gives the lady a different $ 20 bill . later that day he tries to figure out how much money he lost . what did he lose ?
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Context:
Given that \( \int_{1}^{2} e^{x^2} \, \mathrm{d}x = a \), find the value of \( \int_{e}^{e^4} \sqrt{\ln x} \, \mathrm{d}x \).
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Context:
Find the smallest ordinal \( \beta \) such that \( \omega + \beta = \beta \).
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Context:
For $1\leq n\leq 2016$, how many integers $n$ satisfying the condition: the reminder divided by $20$ is smaller than the one divided by $16$.
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Context:
In a methodology class, I'm studying sample size determination. When reviewing a past exam question, I encountered an issue. I used the formula $n = \frac{z^2 \cdot Var}{error^2}$ to calculate the sample size, but I'm unsure how to find the expected proportion without knowing the mean. Consider a scenario where we have a group of 10,000 students, and we want to determine the percentage who would recommend their school. We aim for a 99% confidence level with an error margin of 4 percentage points. Given that the minimum sample size is 373, what is the expected percentage of students who would recommend their school?
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Context:
Observe the following equations: $a=c=\sqrt{6}$, $a^2+b^2=3$, $a^4+b^4=7$, $a^5+b^5=11$. Recognizing that each term is equal to the sum of the previous two terms, calculate the value of $a^{10}+b^{10}$. Express your answer as a single integer.
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Context:
Compute the determinant of the linear transformation $T:V\to V$ given by $T(X) = AX + XA$, where $$A = \begin{pmatrix}1&1&0\\0&2&0\\0&0&-1 \end{pmatrix}.$$
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Context:
Find the least value of  $k$  such that for all  $a,b,c,d \in \mathbb{R}$  the inequality
\[ \begin{array}  c \sqrt{(a^2+1)(b^2+1)(c^2+1)} +\sqrt{(b^2+1)(c^2+1)(d^2+1)} +\sqrt{(c^2+1)(d^2+1)(a^2+1)} +\sqrt{(d^2+1)(a^2+1)(b^2+1)}   \  \ge 2( ab+bc+cd+da+ac+bd)-k  \end{array}\]
holds.
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Context:
Determine the maximal number of members, or an upper bound, in a family of ordered set partitions $C_1, \dots, C_m$ of $[n]$ with exactly $k$ blocks, such that for each $j=1, \dots, k$, the union $B_{1j} \cup \cdots \cup B_{mj}$ is a partition of $[n]$. Consider the case $k=2$ specifically.
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Context:
Determine the smallest positive integer \( n \) such that for all positive integers \( x, y, \) and \( z \) with \( x \mid y^{3} \), \( y \mid z^{3} \), and \( z \mid x^{3} \), it is always true that \( x y z \mid (x+y+z)^{n} \).
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Context:
From the numbers 1, 2, 3,..., 7, select several numbers such that the sum of the even numbers is equal to the sum of the odd numbers. How many ways are there to satisfy this condition? Express your answer as a single integer.
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Context:
Bill bought 13 notebooks, 26 pens, and 19 markers for 25 dollars. Paula bought 27 notebooks, 18 pens, and 31 markers for 31 dollars. How many dollars would it cost Greg to buy 24 notebooks, 120 pens, and 52 markers?
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Context:
将函数 \(y=\sqrt{4+6 x-x^{2}}-2(x \in[0,6])\) 的图象逆时针方向旋转 \(\theta(0 \leqslant \theta \leqslant \alpha)\), 得到曲线 \(C\). 若对于每一个旋转角 \(\theta\)，曲线 \(C\) 都是一个函数的图象，则求 \(\alpha\) 的最大值。原始答案的形式为 \(\arctan \frac{k}{m}\)，请给出k+m的值。
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Context:
The curve $y=\frac{1}{2}{x^2}-2$ has a slope of $\frac{1}{2}$ at the point $(1$,$-\frac{3}{2})$. Find the angle of inclination of the tangent line at this point.
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Context:
Two points $A$ and $B$ are selected independently and uniformly at random along the perimeter of a unit square with vertices at $(0,0)$, $(1,0)$, $(0,1)$, and $(1,1)$. The probability that the $y$-coordinate of $A$ is strictly greater than the $y$-coordinate of $B$ can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute $100m+n$.
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Context:
How many distinct positive integers can be expressed in the form $ABCD - DCBA$, where $ABCD$ and $DCBA$ are 4-digit positive integers? Here, $A$, $B$, $C$, and $D$ are digits, possibly equal.

**Clarification:** $A$ and $D$ cannot be zero because otherwise $ABCD$ or $DCBA$ wouldn't be a true 4-digit integer.
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Context:
A car tire with a certain diameter will make approximately 672.1628045157456 revolutions if the car travels 1/2 mile. What is the diameter of the car tire?
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Context:
As shown in the diagram, \( D \), \( E \), and \( F \) are points on the sides \( BC \), \( CA \), and \( AB \) of \(\triangle ABC\), respectively, and \( AD \), \( BE \), \( CF \) intersect at point \( G \). Given that the areas of \(\triangle BDG\), \(\triangle CDG\), and \(\triangle AEG\) are 8, 6, and 14 respectively, find the area of \(\triangle ABC\). Express your answer as a whole number.
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Context:
For each set of five integers $S= \{a_1, a_2, a_3, a_4, a_5\}$, let $P_S$ be the product of all differences between two of the elements, namely:

$$P_S = (a_5-a_1)(a_4-a_1)(a_3-a_1)(a_2-a_1)(a_5-a_2)(a_4-a_2)(a_3-a_2)(a_5-a_3)(a_4-a_3)(a_5-a_4)$$

Determine the greatest integer $n$ such that given any set $S$ of five integers, $n$ divides $P_S$.
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Context:
已知圆锥面 $x^{2}+y^{2}=\frac{z^{2}}{3}$, 记沿该圆锥面从 $P(-\sqrt{3}, 3,6)$ 到 $Q(\sqrt{3}, 0,3)$ 的曲线长度的最小为值为 I, 则 $[10 I]=$ $\qquad$.
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Context:
To assess the shooting level of a university shooting club, an analysis group used stratified sampling to select the shooting scores of $6$ senior members and $2$ new members for analysis. After calculation, the sample mean of the shooting scores of the $6$ senior members is $8$ (unit: rings), with a variance of $\frac{5}{3}$ (unit: rings$^{2}$). The shooting scores of the $2$ new members are $3$ rings and $5$ rings, respectively. What is the variance of the shooting scores of these $8$ members?
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Context:
The store sells 20 items, each priced uniquely in whole numbers from 1 to 20 rubles. The store has a promotion: when purchasing any 5 items, one of them is given for free, and the customer can choose which one. Vlad wants to buy all 20 items in this store, paying as little as possible. How many rubles does he need? (Each of the 20 items is sold in a single unit.)
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Context:
Flora the frog starts at $0$ on the number line and makes a sequence of jumps to the right. In any one jump, independent of previous jumps, Flora leaps a positive integer distance $m$ with probability $\frac{1}{2^m}$. Find the probability that Flora will eventually land at $10$. The original answer is in \frac{k}{m} format, please give the value of k + m.
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Context:
Five balls are arranged around a circle. Chris chooses two adjacent balls at random and interchanges them. Then Silva does the same, with her choice of adjacent balls to interchange being independent of Chris's. If the expected number of balls that occupy their original positions after these two successive transpositions is \( x \), find the value of \( 10x \).
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Context:
The Fibonacci sequence $F_{1}, F_{2}, F_{3}, \ldots$ is defined by $F_{1}=F_{2}=1$ and $F_{n+2}=F_{n+1}+F_{n}$. Find the least positive integer $t$ such that for all $n>0, F_{n}=F_{n+t}$.
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Context:
Ana has \(22\) coins. She can take from her friends either \(6\) coins or \(18\) coins, or she can give \(12\) coins to her friends. She can perform these operations as many times as she wants. Find the least number of coins Ana can have.
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Context:
In how many ways can the nine digits from 1 to 9 be placed in a 3 × 3 grid so that the sums of the rows and the sums of the columns are all equal?
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Context:
In triangle \(ABC\) with \(BC = 4\) and \(AB = 2\sqrt{19}\), it is known that the center of the circle passing through the midpoints of the sides of the triangle lies on the bisector of angle \(C\). Find \(AC\).
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Context:
若方程 $|x-1|=k(x-2 a)+a$ 对任意实数 $k$ 都有解, 则实数 $a$ 的取值范围是
Please reason step by step, and put your final answer within \boxed{}.

\boxed{$\frac{1}{3} \leqslant a \leqslant 1$}


Context:
Given that for all $x\_1 \in (0, 2]$, there exists an $x\_2 \in [1, 2]$ such that the inequality $4x\_1 \ln x\_1 - x\_1^2 + 3 + 4x\_1 x\_2^2 + 8a x\_1 x\_2 - 16x\_1 \geq 0$ holds true, find the range of values for the real number $a$.
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Context:
Using the vertices of a cube as vertices, how many triangular pyramids can you form?
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Context:
Determine the remainder when
\[ \sum_{i=0}^{2015} \left\lfloor \frac{2^i}{25} \right\rfloor \]
is divided by 100, where \( \lfloor x \rfloor \) denotes the largest integer not greater than \( x \).
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Context:
If \( x_{i}=\frac{i}{101} \), then calculate the value of \( S=\sum_{i=0}^{101} \frac{x_{i}^{3}}{3 x_{i}^{2}-3 x_{i}+1} \). Express your answer as a single integer.
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Context:
Four numbers w, x, y, z satisfy w < x < y < z. Each of the six possible pairs of distinct numbers has a different sum. The four smallest sums are 1, 2, 3, and 4. Find the sum of all possible values of z. Express your answer as a single value, which can be a fraction or integer.
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Context:
What is the unit digit of the expression
$$
\left(\frac{\sqrt{(a-2)(|a-1|)}+\sqrt{(a-2)(1-|a|)}}{1+\frac{1}{1-a}}+\frac{5a+1}{1-a}\right)^{1988},
$$
where a is a real number?
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Context:
Given the plane point set \( A = \{ (x, y) \mid x = 2 \sin \alpha + 2 \sin \beta, y = 2 \cos \alpha + 2 \cos \beta \} \) and \( B = \{ (x, y) \mid \sin (x + y) \cos (x + y) \geq 0 \} \), find the area of the region represented by \( A \cap B \).
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Context:
Find the value of the limit \( \lim_{a\searrow 0}\sum_{n=1}^\infty\frac{(-1)^{n-1}}{n^a} \).
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Context:
14.5. 14 ** A storybook contains 30 stories, with lengths of $1, 2, \cdots, 30$ pages respectively. Starting from the first page of the book, stories are published, and each subsequent story begins on a new page. Question: What is the maximum number of stories that can start on an odd-numbered page?
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Context:
Let \( X_1, X_2, \ldots, X_{50} \) be a random sample of size 50 from a distribution with probability density function \( f(x) = \frac{1}{x^2} \) for \( 1 < x < \infty \). Approximate the probability that at least 20 of these random variables have values greater than 3.
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Context:
Find the least positive integer $N$ such that the only values of $n$ for which $1 + N \cdot 2^n$ is prime are multiples of $12$.
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Context:
Determine the minimum number of terminals, \( f(n) \), required in a 3-connector network to ensure that it contains a windmill with \( n \) blades.
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Context:
求满足如下条件的最小正整数 \(n\)：在 \(\odot O\) 的圆周上任取 \(n\) 个点 \(A_1, A_2, \cdots, A_n\)，则在 \(C_n^2\) 个角 \(\angle A_iOA_j\) \((1 \leq i < j \leq n)\) 中，至少有 2007 个不超过 \(120^\circ\)。
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Context:
Assign weights to the edges of a complete graph on $n$ vertices such that the sum of the weights is 1 and the weight of the minimum spanning tree is maximized. What is the weight of the minimum spanning tree after this assignment?
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Context:
Determine the sign of the expression \( S_n + T_n - \frac{1}{2} \), where \( S_n = \frac{1}{n}\sum_{k=1}^n \left(\frac{k}{n}\right)^3 \) and \( T_n = \frac{1}{n}\sum_{k=1}^n \left(\frac{k-1}{n}\right)^3 \).
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Context:
Determine the dimension of the real vector space $S_N$ consisting of $3 \times 3$ matrices $M$ such that $NM = 0$, where $N = \begin{bmatrix} 1 & 1 & -1 \\ 0 & 1 & 0 \\ 1 & 0 & -1 \end{bmatrix}$. \( \text{Provide your answer as a single integer.} \)
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Context:
In a restaurant, the profit is some percentage of the cost. If the cost increases by 12% but the selling price remains constant, the profit is approximately 53.333333333333336% of the selling price. What is the original percentage of the profit with respect to the cost?
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Context:
Into how many regions do the x-axis and the graphs of \( y = 2 - x^2 \) and \( y = x^2 - 1 \) split the plane?
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Context:
Given the ellipse: $\dfrac{x^2}{9} + y^2 = 1$, a line passing through the left focus $F$ with an inclination angle of $\dfrac{\pi}{6}$ intersects the ellipse at points $A$ and $B$. The length of chord $AB$ is equal to ______.
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Context:
The volume of the tetrahedron \( ZABC \) is 5. A plane passes through the midpoint \( K \) of edge \( ZA \) and the midpoint \( P \) of edge \( BC \), and intersects edge \( ZC \) at point \( M \) such that \( ZM : MC = 2 : 3 \). If the distance from vertex \( A \) to the plane is 1, find the area of the cross-section formed by the plane.
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Context:
Two students, A and B, are playing a chess match using a "best of five" format (the first player to win three games wins the match). Based on their past performance, the probability of A winning each game is $p$ $(0 < p < 1)$, and the outcomes of each game are independent. If the probability of A winning with a score of $3:1$ is not higher than the probability of A winning with a score of $3:2$, then the possible range of values for $p$ is ______.
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Context:
设正三棱柱 $S-A B C$ 的底面边长为 3，侧棱长为 2。求侧棱 $S A$ 与底面 $A B C$ 所成的角的大小。如果答案是 $x^\circ$，那么求 $x$ 的值。
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Context:
In triangle $ABC$, $AB=AC$ and $D$ is a point on $\overline{AC}$ so that $\overline{BD}$ bisects angle $ABC$. If $BD=BC$, what is the measure, in degrees, of angle $A$?
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Context:
Determine the twenty-fifth number in a sequence obtained by counting in base 5.
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Context:
The numbers \(1, 2, 3, \ldots, 10\) are written in some order around a circle. Peter computed the sums of all 10 triples of neighboring numbers and wrote the smallest of these sums on the board. What is the maximum possible number that could have been written on the board?
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Context:
A display screen has a row of 4 adjacent holes, each of which can display 0 or 1. If each time it displays two of the holes, but two adjacent holes cannot be displayed at the same time, the total number of signals that the display screen can show is ______.
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Context:
In the quadrilateral \(ABCD\), it is known that \(DO = 4\), \(BC = 5\), \(\angle ABD = 45^\circ\), where \(O\) is the point of intersection of the diagonals. Find \(BO\) if the area of the quadrilateral \(ABCD\) is \(\frac{1}{2}(AB \cdot CD + BC \cdot AD)\).
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Context:
给定正整数 $n 、 k$, 其中, $n$ 为偶数, $k \geqslant 2$,且 $n>4 k$. 在圆周上有 $n$ 个点, 若 $\frac{n}{2}$ 条在圆内互不相交的弦的端点恰为这 $n$ 个点，则称这些弦为一个"匹配".求最大的整数 $m$ ，满足对任意一个匹配，均存在圆周上连续 $k$ 个点，使得这个匹配中至少有 $m$ 条弦的所有端点均在这 $k$ 个点中. 请给出当 $k = 5$ 时，最大整数 $m$ 的值。
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Context:
3.20. In the planes of the faces of the tetrahedron \(ABCD\), points \(A_1, B_1, C_1\), and \(D_1\) are taken such that the lines \(AA_1, BB_1, CC_1\), and \(DD_1\) are parallel. Find the ratio of the volumes of the tetrahedra \(ABCD\) and \(A_1B_1C_1D_1\).

## § 4. Calculation of Volume
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Context:
What is the measure of $\angle X Z Y$ if $M$ is the midpoint of $Y Z$, $\angle X M Z=30^{\circ}$, and $\angle X Y Z=15^{\circ}$?
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Context:
In the geometric sequence {a_n}, it is known that (a_2 = 1/3) and (a_8 = 27). Determine the value of a_5. Express your answer as a numerical value, including the ± symbol if necessary.
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Context:
The number 5.6 may be expressed uniquely (ignoring order) as a product $\underline{a} \cdot \underline{b} \times \underline{c} . \underline{d}$ for digits $a, b, c, d$ all nonzero. Compute $\underline{a} \cdot \underline{b}+\underline{c} . \underline{d}$.
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Context:
Arnold paid $5 for 1 dozen apples and 1 bunch of bananas. Tony paid a certain amount for twice the number of apples Arnold bought and 1 bunch of bananas. How much does a bunch of bananas cost?
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Context:
A right triangular prism has exactly one inscribed sphere (the sphere is tangent to the two base faces and three lateral faces of the prism) and one circumscribed sphere (the sphere passes through the 6 vertices of the prism). The ratio of the surface area of the inscribed sphere to that of the circumscribed sphere is ______.
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Context:
Point \( K \) lies on side \( BC \) of parallelogram \( ABCD \), and point \( M \) lies on its side \( AD \). Segments \( CM \) and \( DK \) intersect at point \( L \), and segments \( AK \) and \( BM \) intersect at point \( N \). Find the maximum value of the ratio of the areas of quadrilateral \( KLMN \) to \( ABCD \).
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Context:
In triangle $AHI$, which is equilateral, lines $\overline{BC}$, $\overline{DE}$, and $\overline{FG}$ are all parallel to $\overline{HI}$. The lengths satisfy $AB = BD = DF = FH$. However, the point $F$ on line segment $AH$ is such that $AF$ is half of $AH$. Determine the ratio of the area of trapezoid $FGIH$ to the area of triangle $AHI$.
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Context:
If $\sin^{4}{\alpha} + \cos^{4}{\alpha} = 1$, then the value of $\sin{\alpha} + \cos{\alpha}$ is \_\_\_\_\_\_.
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Context:
A *T-tetromino* is formed by adjoining three unit squares to form a  $1 \times 3$  rectangle, and adjoining on top of the middle square a fourth unit square.
Determine the least number of unit squares that must be removed from a  $202 \times 202$  grid so that it can be tiled using T-tetrominoes.
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Context:
In a year that has  $365$  days, what is the maximum number of "Tuesday the  $13$ th" there can be?

Note: The months of April, June, September and November have  $30$  days each, February has  $28$  and all others have  $31$  days.
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Context:
A company has 200 employees randomly assigned numbers from 1 to 200, and they are evenly divided into 40 groups. If employee number 22 is selected from the 5th group, find the number of the employee to be selected from the 10th group.
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Context:
Analyzing the intersection of $y = x^3 - 6x + 2$ and $y = m$ where $-10 < m < 10$, define $L(m)$ as the smallest $x$ coordinate of their intersection points. Calculate the function $r = \frac{L(-m) - L(m)}{m}$ as $m$ approaches zero. Find the limit of this function.
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Context:
Determine the set of real numbers  $\alpha$  that can be expressed in the form \[\alpha=\sum_{n=0}^{\infty}\frac{x_{n+1}}{x_n^3}\]
where  $x_0,x_1,x_2,\dots$   is an increasing sequence of real numbers with  $x_0=1$ .
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Context:
At most how many rooks can be placed on an $8 \times 8$ chessboard so that each one is attacked by at most one other?
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Context:
Taequan is playing a dice game where he rolls two, fair, 6-sided dice twice. If he gets the same roll on his second try as his first, he wins. He rolled an 8 on his first try. What is the probability that he will win with his second roll?
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Context:
11. In the Cartesian coordinate system, points $A, B, C$ lie on the hyperbola $x y=1$, satisfying that $\triangle A B C$ is an isosceles right triangle. Find the minimum value of the area of $\triangle A B C$.
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Context:
In the regular hexagon $ABCDEF$, a point $X$ is marked on the diagonal $AD$ such that $\angle AEX = 65^\circ$. What is the degree measure of the angle $\angle XCD$?
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Context:
Given non-zero vectors $\mathbf{a}$, $\mathbf{b}$, and $\mathbf{c}$ satisfy $\mathbf{a} + \mathbf{b} + \mathbf{c} = 0$, the angle between vector $\mathbf{a}$ and $\mathbf{b}$ is $60^\circ$, and $|\mathbf{a}| = |\mathbf{b}| = 1$, then the angle between vector $\mathbf{a}$ and $\mathbf{c}$ is _______.
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Context:
在 1,2,\cdots,10 中随机选出三个不同的数，它们两两互素的概率为__________.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:
Inside triangle \(ABC\), a point \(O\) is chosen such that \(\angle ABO = \angle CAO\), \(\angle BAO = \angle BCO\), and \(\angle BOC = 90^{\circ}\). Find the ratio \(AC : OC\).
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Context:
A wooden cube has edges of length $3$ meters. Square holes, of side one meter, centered in each face are cut through to the opposite face. The edges of the holes are parallel to the edges of the cube. The entire surface area including the inside, in square meters, is
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Context:
Determine the radius of convergence for the following power series: C) $\sum_{n=1}^\infty \frac{z^{2^n}}{n!}$
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Context:
Tanya wrote numbers in the form $n^7 - 1$ for $n = 2, 3, \ldots$ and noticed that for $n = 8$, she obtained a number divisible by $337$. For what minimal $n$ did she get a number divisible by $2022$?
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Context:
There are 400 tons of cargo stored in a warehouse, each with a weight that is a multiple of a quintal and does not exceed 10 tons. It is known that any two pieces of cargo have different weights. What is the minimum number of trips a 10-ton truck must make to guarantee the transportation of all this cargo from the warehouse?
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Context:
Given points A(3,2,0), B(0,4,0), and C(3,0,2), calculate a normal vector of the plane ABC in the format (x, y, z).
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Context:
A square grid of size \(n \times n\) is initially colored in a black and white checkerboard pattern, with at least one of the corner cells being black. In one move, it is allowed to recolor all four cells in a \(2 \times 2\) square according to the following rule: each white cell is recolored black, each black cell is recolored green, and each green cell is recolored white. For which \(n\) is it possible, after a number of moves, to achieve a checkerboard coloring where the black and white colors have swapped places?
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Context:
The bisectors of the angles  $A$  and  $B$  of the triangle  $ABC$  intersect the sides  $BC$  and  $AC$  at points  $D$  and  $E$ . It is known that  $AE + BD = AB$ . Find the angle  $\angle C$ .
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Context:
If a square is divided into acute-angled triangles, what is the minimum number of parts that can be created?
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Context:
A passenger train departs from point $A$ towards point $B$ at a speed of $60 \mathrm{~km/h}$. A high-speed train, which departs later, must catch up with the passenger train at point $B$ and travels at a speed of $120 \mathrm{~km/h}$. After completing $\frac{2}{3}$ of its journey, the passenger train continues at half its original speed. The high-speed train catches up with the passenger train 80 kilometers before point $B$. How far is point $B$ from point $A$?
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Context:
Consider triangle \(ABC\) with angles \(\angle BAC = 24^\circ\) and \(\angle ACB = 28^\circ\). Point \(D\) is constructed such that \(AB \parallel CD\), \(AD = BC\), and \(AD\) and \(BC\) are not parallel. Similarly, point \(E\) is constructed such that \(AE \parallel BC\), \(AB = CE\), and \(AB\) and \(CE\) are not parallel. Lines \(DE\) and \(AC\) intersect at point \(P\). Determine angle \(\angle CPE\) (in degrees).
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Context:
A row of ants comprised of some ants. A man walked upon it and all except 42 were crushed beneath his foot. There are 42 ants left alive. How many ants were there initially in the row?
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Context:
A person (referred to as "A") and four colleagues each have a private car with license plates ending in 0, 0, 2, 1, and 5, respectively. To comply with the local traffic restrictions from the 5th to the 9th of a certain month (cars with an odd-numbered last digit of the license plate are allowed on odd days, and those with an even number on even days), the five people discussed carpooling. Each day they can choose any car that meets the requirements, but car "A" can be used at most 1 day. Calculate the total number of different car usage plans.
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Context:
If \(\frac{(a-b)(b-c)(c-a)}{(a+b)(b+c)(c+a)}=\frac{2004}{2005}\), find \(\frac{a}{a+b}+\frac{b}{b+c}+\frac{c}{c+a}\).
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Context:
The probability that Class A will be assigned exactly 2 of the 8 awards, with each of the 4 classes (A, B, C, and D) receiving at least 1 award is $\qquad$ .
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Context:
Let \( OP \) be the diameter of the circle \( \Omega \), and let \( \omega \) be a circle with center at point \( P \) and a radius smaller than that of \( \Omega \). The circles \( \Omega \) and \( \omega \) intersect at points \( C \) and \( D \). A chord \( OB \) of the circle \( \Omega \) intersects the second circle at point \( A \). Find the length of the segment \( AB \) if \( BD \cdot BC = 5 \).
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Context:
Compute the sum of all positive integers $n<2048$ such that $n$ has an even number of 1's in its binary representation.
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Context:
Find the resolvent of the Volterra integral equation with the kernel $K(x, t) \equiv 1$.
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Context:
Let $W = \ldots x_{-1}x_0x_1x_2 \ldots$ be an infinite periodic word consisting of only the letters $a$ and $b$. The minimal period of $W$ is $2^{2016}$. Say that a word $U$ appears in $W$ if there are indices $k \le \ell$ such that $U = x_kx_{k+1} \ldots x_{\ell}$. A word $U$ is called special if $Ua, Ub, aU, bU$ all appear in $W$. (The empty word is considered special.) You are given that there are no special words of length greater than 2015.

Let $N$ be the minimum possible number of special words. Find the remainder when $N$ is divided by 1000.
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Context:
Teacher Wang took 48 students from class 3 (5) for a spring outing. Each minibus can accommodate up to 8 passengers. How many minibuses does Teacher Wang need to rent?
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Context:
Consider a hyperbola given by the equation $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 (a > 0, b > 0)$. The distance from the point on the hyperbola with an x-coordinate of $\frac{3a}{2}$ to the right focus is greater than its distance to the left directrix. Determine the range of acute angles formed by the two asymptotes of the hyperbola.
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Context:
Let $p$ be a prime number and let $G$ be a group of order $p^4$ such that $|Z(G)|=p^2$. Determine the number of conjugacy classes of $G$.
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Context:
Given a circle $(x-3)^2+(y+1)^2=4$ with a moving point $P$ on it, and another moving point $Q$ on the line $x=-3$, find the minimum value of the distance $|PQ|$.
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Context:
设 $A, B$ 为平面直角坐标系 $x O y$ 中的曲线 $x y=1(x, y>0)$ 上的两点, 向量 $\vec{m}=(1,|O A|)$ ，则数量积 $\vec{m} \cdot \overrightarrow{O B}$ 的最小值是多少。原始答案是 $a \sqrt[4]{b}$ 的形式，请给出$a + b$的值。
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Context:
On a $11 \times 11$ grid, 22 cells are marked such that there are exactly two marked cells in each row and each column. Two configurations of marked cells are considered equivalent if one can be transformed into the other by permuting rows and columns any number of times. How many distinct configurations of marked cells exist, considering these equivalence conditions?
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Context:
On each side of a unit square, an equilateral triangle of side length 1 is constructed. On each new side of each equilateral triangle, another equilateral triangle of side length 1 is constructed. The interiors of the square and the 12 triangles have no points in common. Let $R$ be the region formed by the union of the square and all the triangles, and $S$ be the smallest convex polygon that contains $R$. What is the area of the region that is inside $S$ but outside $R$?
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Context:
In an acute triangle \(ABC\), heights \(CH\) and \(AH_1\) are drawn. It is known that \(AC = 2\), and the area of the circle circumscribed around triangle \(HBH_1\) is \(\pi / 3\). Find the angle between height \(CH\) and side \(BC\).
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Context:
A rectangle is divided into four smaller rectangles, labelled W, X, Y, and Z. The perimeters of rectangles W, X, and Y are 2, 3, and 5, respectively. What is the perimeter of rectangle Z?
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Context:
\( \triangle ABC \) is an isosceles triangle with \( AB = AC \). Point \( P \) is located inside \( \triangle ABC \) such that \( \angle BCP = 30^\circ \), \( \angle APB = 150^\circ \), and \( \angle CAP = 39^\circ \). Find \( \angle BAP \).
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Context:
Find the number of strictly increasing sequences of nonnegative integers with the following properties:

1. The first term is $0$ and the last term is $12$. Therefore, the sequence has at least two terms.
2. Among any two consecutive terms, exactly one of them is even.
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Context:
Given a proposition related to positive integers, if the proposition holds true when $n = k$ ($k \in \mathbb{N}^+$), then it can be inferred that the proposition also holds true when $n = k + 1$. Now, it is known that the proposition does not hold true when $n = 5$. What is the value of $n$ for which we can infer that the proposition does not hold true, based on the given information?
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Context:
Find the flux of the vector field \(\mathbf{a} = y^2 \mathbf{j} + z \mathbf{k}\) through the part of the surface \(z = x^2 + y^2\), cut off by the plane \(z=2\). The normal vector is taken to be outward with respect to the region bounded by the paraboloid.
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Context:
Let
$$
A=\lim _{n \rightarrow \infty} \sum_{i=0}^{2016}(-1)^{i} \cdot \frac{\binom{n}{i}\binom{n}{i+2}}{\binom{n}{i+1}^{2}}
$$

Find the largest integer less than or equal to \(\frac{1}{A}\).

The following decimal approximation might be useful: \(0.6931 < \ln (2) < 0.6932\), where \(\ln\) denotes the natural logarithm function.
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Context:
Consider a line of \( n \) soldiers. They are to be divided into several non-empty units, and a commander is chosen for each unit. Determine the number of ways this can be done.
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Context:
In a $9 \times 9$ grid, each cell contains one beetle. Upon hearing a whistle, each beetle moves diagonally to an adjacent cell. Determine the minimum possible number of empty cells after the movement.
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Context:
11.142. In a regular triangular prism, a plane is drawn through the side of the lower base and the opposite vertex of the upper base, forming an angle of $45^{\circ}$ with the plane of the lower base. The area of the section is $S$. Find the volume of the prism.
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Context:
Points \( A,B,C \) lie on a circle \( \omega \) such that \( {BC} \) is a diameter. \( {AB} \) is extended past \( B \) to point \( {B}^{\prime } \) and \( {AC} \) is extende
 past \( C \) to point \( {C}^{\prime } \) such that \( \overline{{B}^{\prime }{C}^{\prime }}\parallel {BC} \) and tangent to \( \omega \) at point \( D \) . If \( {B}^{\prime }D = 4 \) and \( {C}^{\prime }D = 6 \) , compute \( {BC} \) .
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Context:
Determine the Löwenheim number of the extended First-Order Logic $L(Q^c)$, which includes Chang's quantifier $Q^c$. Provide your answer as a single number.
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Context:
Find the derivative of the trace function Tr($A$) with respect to the matrix $B$ in the space $M(n)$ of $n \times n$ real matrices.
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Context:
Jenny wants to read 5 books in the next 15 days. She can read 60 words per minute, but she takes a 15-minute break after every hour of continuous reading. If the first book has 12,000 words, the second book has 18,000 words, the third book has 24,000 words, the fourth book has 15,000 words, and the fifth book has 21,000 words, how many minutes per day, on average, should she spend reading (including break time)?
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Context:
Determine the \(\hat{r}\) component of the following integral in cylindrical coordinates without performing the integration:

$$\int \limits_0^{2\pi} \int \limits_0^a \frac{r^2}{(r^2+h^2)^{3/2}} \hat{r} \, dr \, d\theta$$
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Context:
There are 4 stacks of coins on a table. The first stack has 9 coins, the second has 7, the third has 5, and the fourth has 10. In one move, you are allowed to add one coin to three different stacks. What is the minimum number of moves required so that all the stacks have an equal number of coins?
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Context:
Calculate \( \lim_{a \rightarrow 0^+} \int_{D_a} \frac{|x|}{y(1+y^2)} \, dx \, dy \) where \( D_a=\{(x,y) \in \mathbb{R}^2: \frac{x^2}{2} \le y \le 2x^2 ; a \le \sqrt{x^2+y^2} \le \frac{1}{a} \} \) without using polar coordinates.
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Context:
A train of a certain length completely passes an electric pole in 8.999280057595392 seconds, if it travels at a speed of 80 kmph. How long is the train?
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Context:
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and $M$ is the midpoint of $BC$ with $BM = 2$. $AM = c - b$. Find the maximum area of $\triangle ABC$.
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Context:
Find the remainder when $5^{5^{5^5}}$ is divided by 500.
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Context:
A car and a truck start traveling towards each other simultaneously from points $A$ and $B$, respectively. It is known that the car's speed is twice the speed of the truck. The car arrives at point $C$ at 8:30, and the truck arrives at point $C$ at 15:00 on the same day. Both vehicles continue moving without stopping at point $C$. Determine the time at which the car and the truck meet.
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Context:
On a board, the 2014 positive integers from 1 to 2014 are written. The allowed operation is to choose two numbers \( a \) and \( b \), erase them, and write in their place the numbers \( \text{gcd}(a, b) \) (greatest common divisor) and \( \text{lcm}(a, b) \) (least common multiple). This operation can be performed with any two numbers on the board, including numbers that resulted from previous operations. Determine the largest number of 1's that we can leave on the board.
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Context:
In $\triangle ABC$, $\angle ACB = 90^{\circ}$, $CD$ and $CE$ are the heights to the base $AB$ and the angle bisector of $\angle C$, respectively. If $\triangle CED \sim \triangle ABC$, then find the measure of $\angle ECD$ in degrees.
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Context:
The two brothers, without waiting for the bus, decided to walk to the next stop. After passing $\frac{1}{3}$ of the way, they looked back and saw a bus approaching the stop. One of the brothers ran backwards, and the other ran forward at the same speed. It turned out that everyone ran to their stop exactly at the moment when the bus approached it. Find the speed of the brothers, if the bus speed is $30$ km/h, neglecting the bus stop time.
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Context:
A rectangle was cut into three rectangles, two of which have dimensions 9 m x 12 m and 10 m x 15 m. What is the maximum possible area of the original rectangle? Express your answer in square meters.
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Context:
In a round-robin chess tournament organized at a school, every two students played one match against each other. Determine the minimum possible number of students in the school given the following conditions:

- Each girl student has at least 21 wins in matches against boy students.
- Each boy student has at least 12 wins in matches against girl students.
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Context:
Let $\sigma$ be a random permutation of $\{0, 1, \ldots, 6\}$. Let $L(\sigma)$ be the length of the longest initial monotonic consecutive subsequence of $\sigma$ not containing $0$; for example, \[L(\underline{2,3,4},6,5,1,0) = 3,\ L(\underline{3,2},4,5,6,1,0) = 2,\ L(0,1,2,3,4,5,6) = 0.\] If the expected value of $L(\sigma)$ can be written as $\frac mn$, where $m$ and $n$ are relatively prime positive integers, then find $m + n$.
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Context:
There are 300 points in space. Four planes $A$, $B$, $C$, and $D$ each have the property that they split the 300 points into two equal sets. (No plane contains one of the 300 points.) What is the maximum number of points that can be found inside the tetrahedron whose faces are on $A$, $B$, $C$, and $D$?
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Context:
In triangle $ABC$, $AB=\sqrt{30}$, $AC=\sqrt{6}$, and $BC=\sqrt{15}$. There is a point $D$ for which $\overline{AD}$ bisects $\overline{BC}$, and $\angle ADB$ is a right angle. Find the value of the ratio
\[\dfrac{[ADB]}{[ABC]}.\]In other words, compute the ratio of the area of triangle $ADB$ to the area of triangle $ABC.$
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Context:
Two identical cars are traveling in the same direction. The speed of one is $36 \kappa \mu / h$, and the other is catching up with a speed of $54 \mathrm{kм} / h$. It is known that the reaction time of the driver in the rear car to the stop signals of the preceding car is 2 seconds. What should be the distance between the cars to avoid a collision if the first driver suddenly brakes? For a car of this make, the braking distance is 40 meters from a speed of $72 \kappa м / h$.
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Context:
一只蚂蚁在正方体 $A B C D-E F G H$ 的顶点 $A$ 处，每次等概率地爬行到相邻三个顶点中的一个，那么六次爬行之后回到顶点 $A$ 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:
Determine the radius $r$ of a circle inscribed within three mutually externally tangent circles of radii $a = 5$, $b = 10$, and $c = 20$ using the formula:
\[
\frac{1}{r} = \frac{1}{a} + \frac{1}{b} + \frac{1}{c} + 2 \sqrt{\frac{1}{ab} + \frac{1}{ac} + \frac{1}{bc}}.
\]
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Context:
When travelling from London to Edinburgh by train, you pass a sign saying 'Edinburgh 200 miles'. Then, \(3 \frac{1}{2}\) miles later, you pass another sign saying 'Half way between London and Edinburgh'. Calculate the total distance in miles from London to Edinburgh.
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Context:
How many ordered triples of nonzero integers $(a, b, c)$ satisfy the equation $2abc = a + b + c + 4$?
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Context:
Four, (50 points) In an $n \times n$ grid, fill each cell with one of the numbers 1 to $n^{2}$. If no matter how you fill it, there must be two adjacent cells where the difference between the two numbers is at least 1011, find the minimum value of $n$.

---

The translation preserves the original text's formatting and structure.
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Context:
设三角形的三边长分别是整数l，m，n，其中l>m>n.已知$\{ \frac { 3 ^ {l } } { 1 0 ^ { 4 } } \} = \{ \frac { 3 ^ { m } } { 1 0 ^ { 4 } } \} =$ $\{ \frac { 3 ^ { n } } { 1 0 ^ { 4 } } \}$，这里{x}=x-[x]，而[x]表示不超过x的最大整数．求这种三角形周长的最小值．
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Context:
Ten numbers are written around a circle with their sum equal to 100. It is known that the sum of each triplet of consecutive numbers is at least 29. Identify the smallest number \( A \) such that, in any such set of numbers, each number does not exceed \( A \).
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Context:
In trapezoid \(ABCD \) with \(AB \parallel CD\) and \(AB + CD = b\), the diagonals \(AC\) and \(BD\) are related by \(5AC = 3BD\). Find the area of the trapezoid if \(\angle BAC = 2 \angle DBA\).
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Context:
【Question 25】
From 49 balls labeled with numbers 1 to 49, select several balls and arrange them in a circle so that the product of the numbers on any two adjacent balls is less than 100. How many balls can be selected at most?
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Context:
On a tram line, trams depart regularly every 10 minutes throughout the day. A tram takes one hour to travel from one end of the line to the other. A passenger gets on a tram at one end stop and travels to the final stop; having nothing else to do, they look out the window and count the trams coming from the opposite direction on the same route. How many trams will they count during the journey?
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Context:
Let $(a_n)_{n \geq 0}$ be the sequence of integers defined recursively by $a_0 = 0$, $a_1 = 1$, and $a_{n+2} = 4a_{n+1} + a_n$ for $n \geq 0$. Find the common divisors of $a_{1986}$ and $a_{6891}$.
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Context:
For which integer values of \( a \) is the polynomial \( x^{13} + x + 90 \) divisible by \( x^2 - x + a \)?
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