Context:
770. It is required to divide the segment $\boldsymbol{A} \boldsymbol{B}$ in half. How should the radii of the arcs described from points $\boldsymbol{A}$ and $\boldsymbol{B}$, as centers, be chosen so that the construction is as accurate as possible?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a T-shaped figure made of four squares and eight identical lettered squares, determine how many of the resulting figures can be folded into a topless cubical box after adding one of the lettered squares. Provide your answer as a single integer.
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Context:
In checking the petty cash, a clerk counts $q$ quarters, $d$ dimes, $n$ nickels, and $c$ cents. Later, he discovers that $x$ of the nickels were counted as quarters and $x$ of the dimes were counted as cents. To correct the total obtained, by how many cents must the clerk adjust the total? The original answer is in the form $kx$, please give the value of $k$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
At a crossroads, if vehicles are not allowed to turn back, calculate the total number of possible driving routes. Express your answer as a single integer.
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Context:
Gary plays the following game with a fair \( n \)-sided die whose faces are labeled with the positive integers between 1 and \( n \), inclusive: if \( n=1 \), he stops; otherwise, he rolls the die and starts over with a \( k \)-sided die, where \( k \) is the number his \( n \)-sided die lands on. (In particular, if he gets \( k=1 \), he will stop rolling the die.) If he starts out with a 6-sided die, what is the expected number of rolls he makes?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find maximal value of positive integer $n$ such that there exists subset of $S=\{1,2,...,2001\}$ with $n$ elements, such that equation $y=2x$ does not have solutions in set $S \times S$
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Context:
How many different collections of 9 letters are there? A letter can appear multiple times in a collection. Two collections are equal if each letter appears the same number of times in both collections.
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Context:
A real estate developer determines the price of a 23-story building before sale using the following method: Since both the first and the top floors are duplex structures, the price of the first floor is $a_1$ yuan/$m^2$, and due to the better view, the top floor's price is $a_2$ yuan/$m^2$. The price of the second floor is $a$ yuan/$m^2$. Starting from the third floor, the price increases by $\dfrac{a}{100}$ yuan/$m^2$ for each floor above the previous one. The average price per square meter for this building is $\boxed{\text{answer}}$ yuan/$m^2$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A regular triangular prism \( A B C A_{1} B_{1} C_{1} \) is inscribed in a sphere, with its base \( A B C \) and lateral edges \( A A_{1}, B B_{1}, C C_{1} \). Segment \( C D \) is the diameter of this sphere, and points \( K \) and \( L \) are the midpoints of edges \( A A_{1} \) and \( A B \) respectively. Find the volume of the prism if \( D L = \sqrt{2} \) and \( D K = \sqrt{3} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
There are 13 students in a class (one of them being the monitor) and 13 seats in the classroom. Every day, the 13 students line up in random order and then enter the classroom one by one. Except for the monitor, each student will randomly choose an unoccupied seat and sit down. The monitor, however, prefers the seat next to the door and chooses it if possible. What is the probability that the monitor can choose his favourite seat?
Please reason step by step, and put your final answer within \boxed{}.


Context:
The radii of two circles are 2 and 4. Their common internal tangents are mutually perpendicular. Find the length of each tangent.
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Context:
The points $(6,12)$ and $(0,-6)$ are connected by a straight line. Another point on this line is:
Please reason step by step, and put your final answer within \boxed{}.


Context:
A cardboard box in the shape of a rectangular parallelopiped is to be enclosed in a cylindrical container with a hemispherical lid. If the total height of the container from the base to the top of the lid is  $60$  centimetres and its base has radius  $30$  centimetres, find the volume of the largest box that can be completely enclosed inside the container with the lid on.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A pizza is cut into 10 pieces. Two of the pieces are each \(\frac{1}{24}\) of the whole pizza, four are each \(\frac{1}{12}\), two are each \(\frac{1}{8}\), and two are each \(\frac{1}{6}\). A group of \(n\) friends share the pizza by distributing all of these pieces. They do not cut any of these pieces. Each of the \(n\) friends receives, in total, an equal fraction of the whole pizza. What is the sum of the values of \(n\) with \(2 \leq n \leq 10\) for which this is not possible?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given Lucy starts with an initial term of 8 in her sequence, where each subsequent term is generated by either doubling the previous term and subtracting 2 if a coin lands on heads, or halving the previous term and subtracting 2 if a coin lands on tails, determine the probability that the fourth term in Lucy's sequence is an integer.
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Context:
What is the formula for determining the index of the diagonal entry in the half-vectorization of a symmetric matrix? Given a symmetric matrix $M$, where $m_{ij}$ are its elements, the half-vectorization $V$ is formed by stacking the lower-triangular elements into a vector. The pattern of the diagonal entries in $V$ is shown as follows:
$$ 1,\; n+1,\; 2n,\; 3n-2,\; 4n-5,\; 5n-9,\; 6n-14, ... $$

What formula relates the index of the $i$th diagonal entry of $M$ in vector $V$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Three marketing executives from a chewing gum company were each in charge of a different product line - standard gum, sugar-free gum, and bubble gum. For their respective product lines, they projected an increase in revenue this year over that of last year by 30 percent, 50 percent, and 40 percent. However, at the end of the year, the revenue for the respective product lines actually decreased from the previous year's revenue by 20 percent, 30 percent, and 25 percent.

If the revenue from last year for standard gum, sugar-free gum, and bubble gum was $100,000, $150,000, and $200,000 respectively, what percent of the total projected revenue for these three product lines is the total actual revenue this year?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Rebecca has four resistors, each with resistance 1 ohm. Every minute, she chooses any two resistors with resistance of \(a\) and \(b\) ohms respectively, and combines them into one by one of the following methods:
- Connect them in series, which produces a resistor with resistance of \(a+b\) ohms;
- Connect them in parallel, which produces a resistor with resistance of \(\frac{a b}{a+b}\) ohms;
- Short-circuit one of the two resistors, which produces a resistor with resistance of either \(a\) or \(b\) ohms.
Suppose that after three minutes, Rebecca has a single resistor with resistance \(R\) ohms. How many possible values are there for \(R\)?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $ABC$ be a triangle such that midpoints of three altitudes are collinear. If the largest side of triangle is $10$, calculate the largest possible area of the triangle. Express your answer as a single integer.
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Context:
Given the circle $x^{2}+y^{2}=m$ and the circle $x^{2}+y^{2}+6x-8y+21=0$, find the range of the real number $m$ for which these two circles intersect.
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Context:
Solve the inequality \(\sin x \cdot \sin 1755 x \cdot \sin 2011 x \geqslant 1\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Define $\varphi^{k}(n)$ as the number of positive integers that are less than or equal to $n / k$ and relatively prime to $n$. Find $\phi^{2001}\left(2002^{2}-1\right)$. (Hint: $\phi(2003)=2002$.)
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Context:
$DEB$ is a chord of a circle such that $DE=3$ and $EB=5$. Let $O$ be the center of the circle. Join $OE$ and extend $OE$ to cut the circle at $C$. Given $EC=1$, find the radius of the circle.
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Context:
The base of pyramid P is an irregular quadrilateral with vertices A, B, C, and D. The side lengths of the quadrilateral are AB = 5, BC = 6, CD = 7, and DA = 8. The angles are ∠ABC = 110°, ∠BCD = 120°, and ∠CDA = 100°. If the height of pyramid P from vertex A is 4, what is the volume of P? Note: You may need to use trigonometry in your calculations.
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Context:
Given that the circumradius of \( \triangle ABC \) is 1, and \( AB \), \( BC \), \( \frac{4}{3} CA \) form a geometric sequence in that order, find the maximum value of \( BC \).
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Context:
Marcella has 27 pairs of shoes, consisting of 12 pairs of sneakers, 10 pairs of sandals, and 5 pairs of boots. The sneakers come in 3 different colors: 5 pairs are red, 4 pairs are blue, and 3 pairs are green. The sandals consist of two styles: 6 pairs have straps while 4 pairs are slip-on. All boots are black. If Marcella loses 9 individual shoes randomly, what is the greatest number of matching pairs she could have left?
Please reason step by step, and put your final answer within \boxed{}.


Context:
The minimum positive period of the function $f(x) = \sin \omega x + \sqrt{3}\cos \omega x + 1$ ($\omega > 0$) is $\pi$. When $x \in [m, n]$, $f(x)$ has at least 5 zeros. The minimum value of $n-m$ is \_\_\_\_\_\_.
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Context:
King Arthur has two equally wise advisors - Merlin and Percival. Each of them finds the right answer to any question with a probability of $p$ or the wrong answer - with a probability of $q=1-p$.

If both advisors say the same thing, the king listens to them. If they say the opposite, the king makes a decision by flipping a coin.

One day, Arthur wondered - would the probability of making correct decisions not decrease if he kept only one advisor and listened to him? So the king summoned the advisors and said:

- It seems to me that the probability of making correct decisions will not decrease if I keep only one advisor and listen to him. If this is true, I will have to dismiss one of you. If this is not true, I will leave everything as it is. Answer me, should I dismiss one of you?
- Who exactly are you planning to dismiss, King Arthur? - the advisors asked.
- If I decide to dismiss one of you, I will make the choice by drawing lots, flipping a coin.

The advisors left to think about the answer. The advisors, I repeat, are equally wise, but not equally honest.

Percival is very honest and will try to give the correct answer, even if it means he will be dismissed. Merlin, honest in all other respects, in this situation decides to give an answer that minimizes the probability of his dismissal. What is the probability that Merlin will be dismissed?
Please reason step by step, and put your final answer within \boxed{}.


Context:
There were 3 candidates in an election. Two candidates received 7636 and 11628 votes respectively. The winning candidate got 49.69230769230769% of the total votes in that election. How many votes did the third candidate receive?
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Context:
Today, Ivan the Confessor prefers continuous functions \(f:[0,1] \rightarrow \mathbb{R}\) satisfying \(f(x)+f(y) \geq|x-y|\) for all pairs \(x, y \in[0,1]\). Find the minimum of \(\int_{0}^{1} f\) over all preferred functions.
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Context:
In trapezoid \(ABCD\), side \(AB\) is perpendicular to the bases \(AD\) and \(BC\). A circle touches side \(AB\) at point \(K\), which lies between points \(A\) and \(B\), has a single common point \(C\) with segment \(BC\), passes through point \(D\), and intersects segment \(AD\) at point \(E\), different from point \(D\). Find the distance from point \(K\) to line \(CD\), given that \(AD=48\) and \(BC=12\).
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Context:
Square $ABCD$ has side length $s$, a circle centered at $E$ has radius $r$, and $r$ and $s$ are both rational. The circle passes through $D$, and $D$ lies on $\overline{BE}$. Point $F$ lies on the circle, on the same side of $\overline{BE}$ as $A$. Segment $AF$ is tangent to the circle, and $AF=\sqrt{9+5\sqrt{2}}$. The original answer is in \frac{k}{m} format, where $k$ and $m$ are integers. Please give the value of $k + m$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Suppose $a_1$, $a_2$, $\ldots$, $a_{10}$ are nonnegative integers such that \[\sum_{k=1}^{10}a_k=15 \text{ and } \sum_{k=1}^{10}ka_k = 80.\] Let $M$ and $m$ denote the maximum and minimum respectively of $\sum_{k=1}^{10}k^2a_k$. Compute $M-m$.
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Context:
Let's define the distance between numbers as the absolute value of their difference. It is known that the sum of the distances from sixteen consecutive natural numbers to a certain number \(a\) is 636, and the sum of the distances from these same sixteen numbers to the number \(a^2\) is 591. Find all possible values of \(a\).
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Context:
Let \( x, y, z, w \) be different positive real numbers such that \( x+\frac{1}{y}=y+\frac{1}{z}=z+\frac{1}{w}=w+\frac{1}{x}=t \). Find \( t \).
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Context:
There are two rows of seats, with 4 seats in the front row and 5 seats in the back row. Now, we need to arrange seating for 2 people, and these 2 people cannot sit next to each other (sitting one in front and one behind is also considered as not adjacent). How many different seating arrangements are there?
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Context:
The teacher plans to give the children the following type of problem. He will tell them that he has in mind a polynomial \( P(x) \) of degree 2017 with integer coefficients, whose leading coefficient is 1. Then he will tell them \( k \) integers \( n_{1}, n_{2}, \ldots, n_{k} \), and separately will provide the value of the expression \( P\left(n_{1}\right) \cdot P\left(n_{2}\right) \cdot \ldots \cdot P\left(n_{k}\right) \). Using this data, the children must find the polynomial that the teacher intended. What is the minimum value of \( k \) such that the polynomial found by the children would necessarily match the one intended by the teacher?
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Context:
Let $a_{0}, a_{1}, \ldots$ and $b_{0}, b_{1}, \ldots$ be geometric sequences with common ratios $r_{a}$ and $r_{b}$, respectively, such that $$\sum_{i=0}^{\infty} a_{i}=\sum_{i=0}^{\infty} b_{i}=1 \quad \text { and } \quad\left(\sum_{i=0}^{\infty} a_{i}^{2}\right)\left(\sum_{i=0}^{\infty} b_{i}^{2}\right)=\sum_{i=0}^{\infty} a_{i} b_{i}$$ Find the smallest real number $c$ such that $a_{0}<c$ must be true.
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Context:
There are 2 prime numbers between a certain number and 85 / 6. What is the lower limit of the range?
Please reason step by step, and put your final answer within \boxed{}.


Context:
On the International Space Station, there was an electronic clock displaying time in the HH:MM format. Due to an electromagnetic storm, the device started malfunctioning, causing each digit on the display to either increase by 1 or decrease by 1. What was the actual time when the storm occurred, if immediately after the storm the clock showed 09:09?
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Context:
Out of 500 participants in a remote math olympiad, exactly 30 did not like the problem conditions, exactly 40 did not like the organization of the event, and exactly 50 did not like the method used to determine the winners. A participant is called "significantly dissatisfied" if they were dissatisfied with at least two out of the three aspects of the olympiad. What is the maximum number of "significantly dissatisfied" participants that could have been at this olympiad?
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Context:
Let O be the circumcenter of △ABC. If there exists a positive real number k such that $\overrightarrow {AO}$ = $\overrightarrow {AB}$ + k $\overrightarrow {AC}$, then the range of k is _______.
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Context:
Consider an infinite sequence $x_1, x_2, \dots$ of positive integers such that, for every integer $n \geq 1$:

- If $x_n$ is even, then $x_{n+1} = \frac{x_n}{2}$.
- If $x_n$ is odd, then $x_{n+1} = \frac{x_n - 1}{2} + 2^{k-1}$, where $2^{k-1} \leq x_n < 2^k$.

Determine the smallest possible value of $x_1$ for which $2020$ is in the sequence.
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Context:
A triangle is inscribed in a circle with radius \( R \). The vertices of the triangle divide the circle into three segments in the ratio \( 2: 5: 17 \). Find the area of the triangle.
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Context:
Let \( S = \{1, 2, \cdots, 98\} \). Find the smallest positive integer \( n \) such that, in any subset of \( S \) with \( n \) elements, it is always possible to select 10 numbers, and no matter how these 10 numbers are evenly divided into two groups, there will always be one number in one group that is relatively prime to the other 4 numbers in the same group, and one number in the other group that is not relatively prime to the other 4 numbers in that group.
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Context:
In a store where all items are priced in whole numbers of rubles, there are two special offers:
1) A customer who buys at least three items at once can select one item for free, provided that its price does not exceed the minimum of the paid items' prices.
2) A customer who buys exactly one item priced at no less than \(N\) rubles receives a 20% discount on their next purchase (which can include any number of items).

A customer, visiting this store for the first time, wants to buy exactly four items with a total price of 1000 rubles, where the cheapest item is priced at no less than 99 rubles. Determine the maximum \(N\) such that the second offer is more beneficial for the customer than the first one.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Problem 2. It is well known that the smallest periods of the functions $\cos x$ and $\sin x$ are $2 \pi$. Find the smallest period of the function $\cos (\sin x)$. If necessary, round your answer to 0.01. For the answer "no period," enter -1.
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Context:
求所有的实数 k，使得 a^3+b^3+c^3+d^3+1\geq k(a+b+c+d) 对任意 a,b,c,d\in [-1,+\infty) 都成立.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:
40 multiplied by 30, plus (12 + 8) multiplied by a certain number, divided by 5 equals 1212. What is the number?
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Context:
10,11

All dihedral angles of a trihedral angle are equal to $60^{\circ}$. Find the angles formed by the edges of this trihedral angle with the planes of the opposite faces.
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Context:
Let $(a_n)_{n=1}^\infty$ be a real sequence such that \[a_n = (n-1)a_1 + (n-2)a_2 + \dots + 2a_{n-2} + a_{n-1}\] for every $n \geq 3$. If $a_{2011} = 2011$ and $a_{2012} = 2012$, find the value of $a_{2013}$.
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Context:
If 5 points are placed in the plane at lattice points (i.e., points $(x, y)$ where $x$ and $y$ are both integers) such that no three are collinear, then there are 10 triangles whose vertices are among these points. What is the minimum possible number of these triangles that have area greater than $\frac{1}{2}$?
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Context:
Let $N$ be the smallest positive integer for which $$x^{2}+x+1 \quad \text { divides } \quad 166-\sum_{d \mid N, d>0} x^{d}$$ Find the remainder when $N$ is divided by 1000.
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Context:
Marcella has 23 pairs of shoes in total, consisting of 6 different colors and 3 distinct sizes. If she loses 9 individual shoes with varying colors and sizes, what is the greatest number of matching pairs, considering both color and size, she could have left?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let X, Y, and Z denote the initial investments made by partners A, B, and C respectively in starting the business. The total profit for one year is $56,700. The profit-sharing ratios for A, B, C, and D are 7:9:10:4, and partner D joins the business 6 months into the year. A, B, and C invested their amounts at the beginning of the year. How much profit did partner C receive in the first year of the business?
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Context:
点集 $S=\left\{(x, y) \mid x \leq 5, y \leq 4, x, y \in \mathrm{~N}^{*}\right\}$, 则由 $S$ 中的点可组成 $\qquad$个不同的三角形。
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Context:
the average student age of a certain class which has 45 students is 14 . if one student aged 15 go to another class and the age of the class teacher is included the average changes to 14.66 . what is the age of class teacher ?
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Context:
Let $\mathrm {Q}$ be the product of the roots of $z^8+z^6+z^4+z^3+z+1=0$ that have a positive imaginary part, and suppose that $\mathrm {Q}=s(\cos{\phi^{\circ}}+i\sin{\phi^{\circ}})$, where $0<s$ and $0\leq \phi <360$. Find $\phi$.
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Context:
Let x and y be complex numbers such that |x| = |y| = 1. Find the maximum value of the following expression:

$$|1+x|+|1+xy|+|1+xy^2|+\cdots+|1+xy^{2013}|-1007|1+y|$$
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Context:
There are 8 people sitting in a row. Now, we want to select 3 people and swap their positions, while the positions of the remaining 5 people remain unchanged. How many different ways can this be done?
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Context:
Given a hollow, sealed (with negligible surface thickness) tetrahedron craft with an edge length of $3 \sqrt {6}$, if a cube that can rotate freely inside it is embedded, the maximum edge length of the cube is \_\_\_\_\_\_.
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Context:
Determine the following limit:

$$\lim_{n\to\infty}\frac{1}{n}\sqrt[n]{(n+1)(n+2)\ldots (n+n)} $$
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Context:
已知 $z=\cos \frac{2}{7} \pi+\mathrm{i} \sin \frac{2}{7} \pi$, 求 $\frac{z}{1+z^{2}}+\frac{z^{4}}{1+z^{3}}+\frac{z^{3}}{1+z^{4}}$ 的值.
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Context:
If a function $f(x)$ defined on $\mathbb{R}$ satisfies $f(x)+f′(x) > 1$ and $f(0)=4$, find the solution set of the inequality $f(x) > \frac {3}{e^{x}}+1$.
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Context:
Every one of the six trucks of a construction company drove for $8$ hours, and they all together spent $720$ litres of oil. How many litres should $9$ trucks spend if every one of them drives for $6$ hours?
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Context:
At the Intergalactic Math Olympiad held in the year 9001, there are 6 problems, and on each problem, you can earn an integer score from 0 to 7. The contestant's score is the product of the scores on the 6 problems, and ties are broken by the sum of the 6 problems. If 2 contestants are still tied after this, their ranks are equal. In this olympiad, there are $8^6 = 262144$ participants, and no two get the same score on every problem. Find the score of the participant whose rank was $7^6 = 117649$.
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Context:
In the interior of the square $ABCD$, the point $P$ is located such that $\angle DCP = \angle CAP = 25^{\circ}$. Find all possible values of $\angle PBA$.
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Context:
求函数

$$
f(x)=\sqrt{15-12 \cos x}+\sqrt{4-2 \sqrt{3} \sin x}+\sqrt{7-4 \sqrt{3} \sin x}+\sqrt{10-4 \sqrt{3} \sin x-6 \cos x}
$$

的最小值。
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Context:
In an acute triangle $A B C$ with angle $C$ equal to $30^{\circ}$, the altitudes intersect at point $M$. Find the area of triangle $A M B$, if the distances from the center of the circle circumscribed around triangle $A B C$ to sides $B C$ and $A C$ are $\sqrt{2}$ and $\frac{\sqrt{3}}{3}$, respectively.
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Context:
Let \(ABCD\) be a quadrilateral such that \(\widehat{DAB} = 60^\circ\), \(\widehat{ABC} = 90^\circ\), and \(\widehat{BCD} = 120^\circ\). Let \(M\) be the intersection point of the diagonals \([AC]\) and \([BD]\). Given that \(BM = 1\) and \(DM = 2\), calculate the area of quadrilateral \(ABCD\).
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Context:
The sequence \( u_0, u_1, u_2, \ldots \) is defined as follows: \( u_0 = 0 \), \( u_1 = 1 \), and \( u_{n+1} \) is the smallest integer \( > u_n \) such that there is no arithmetic progression \( u_i, u_j, u_{n+1} \) with \( i < j < n+1 \). Find \( u_{100} \).
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Context:
Example 5.25. Calculate $e^{2}$ with an accuracy of 0.01.
Please reason step by step, and put your final answer within \boxed{}.


Context:
## Task 6 - 321236

Let $k_{1}, k_{2}$, and $k_{3}$ be three concentric circles with radii $r_{1}=5, r_{2}=3 \sqrt{2}$, and $r_{3}=1$, respectively.

Determine the maximum possible value for the area of a triangle $A B C$ with the property that $A$ lies on $k_{1}$, $B$ lies on $k_{2}$, and $C$ lies on $k_{3}$.

Hint: It can be assumed as a known fact that among all triangles with the mentioned property, there exists a triangle with the maximum possible area.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \(a\), \(b\), and \(c\) represent the lengths of the sides of a triangle, with \(\alpha\), \(\beta\), and \(\gamma\) being the angles opposite these sides respectively. Given that \(a^2 + b^2 = 2023c^2\), determine the value of
\[
\frac{\cot \alpha}{\cot \beta + \cot \gamma}.
\]
Please reason step by step, and put your final answer within \boxed{}.


Context:
A piece of string is cut in two at a point selected at random. Find the probability that the longer piece is at least x times as large as the shorter piece. Express your answer as a fraction in terms of x.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Emily starts with an empty bucket. Every second, she either adds a stone to the bucket or removes a stone from the bucket, each with probability \(\frac{1}{2}\). If she wants to remove a stone from the bucket and the bucket is currently empty, she merely does nothing for that second (still with probability \(\frac{1}{2}\)). What is the probability that after 2017 seconds her bucket contains exactly 1337 stones?
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Context:
In the cube $ABCD-A_1B_1C_1D_1$, point $P$ is located on edge $AB$. A line $l$ passes through point $P$ such that it makes an angle of $30^\circ$ with both plane $ABCD$ and plane $ABC_1D_1$. Find the number of such lines $l$.
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Context:
Find the fourth term in the Taylor series expansion of \(\cos(\sin x)\) around \(x=\frac{\pi}{2}\).
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Context:
Brian starts at the point  $\left(1,0\right)$  in the plane. Every second, he performs one of two moves: he can move from  $\left(a,b\right)$  to  $\left(a-b,a+b\right)$  or from  $\left(a,b\right)$  to  $\left(2a-b,a+2b\right)$ . How many different paths can he take to end up at  $\left(28,-96\right)$ ?

*2018 CCA Math Bonanza Individual Round #14*
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Context:
A cylindrical water tank, placed horizontally, has an interior length of 15 feet and an interior diameter of 8 feet. If the surface area of the water exposed is 60 square feet, find the depth of the water in the tank.
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Context:
In a school there are \( n \) students, each with a different student number. Each student number is a positive factor of \( 60^{60} \), and the H.C.F. of any two student numbers is not a student number in the school. Find the greatest possible value of \( n \).
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Context:
Let $E$ be the set of all nonnegative continuous functions $f:[0,1]\to \mathbb{R}$ such that \(f(x)\,f(y)\ge |x-y|\) for all \(\{x,y\}\subset [0,1]\). Find the minimum value of \(\int_0^1f(x) \,dx\) for \(f \in E\).
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Context:
Construct a square such that two adjacent vertices lie on a circle with a unit radius, and the side connecting the other two vertices is tangent to the circle. Calculate the sides of the square!
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Context:
It is now 3:00:00 PM, as read on a 12-hour digital clock. In 189 hours, 58 minutes, and 52 seconds, the time will be $X:Y:Z$ on the clock. What is the value of $X + Y + Z$?
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Context:
In a racing tournament with 12 stages and \( n \) participants, each participant receives points \( a_{k} \) based on their place \( k \) in each stage, where \( a_{1}>a_{2}>\ldots>a_{n} \) and all \( a_{k} \) are natural numbers. Determine the smallest \( n \) such that the tournament organizer can choose the points \( a_{1}, \ldots, a_{n} \) so that after the penultimate stage, at least two participants still have a chance to win the tournament. What is the smallest possible value of \( n \)?
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Context:
How many $n$-digit numbers composed only of 0 and 1 have exactly $m$ blocks of "01"?
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Context:
Figure $ABCD$ is a trapezoid with $AB || DC, AB = 5, BC = 3 \sqrt 2, \measuredangle BCD = 45^\circ$, and $\measuredangle CDA = 60^\circ$. The length of $DC$ is
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Context:
Given vectors $\overrightarrow{a}=( \frac{1}{3},	an\alpha)$ and $\overrightarrow{b}=(\cos \alpha)$, where $\overrightarrow{a}$ is parallel to $\overrightarrow{b}$, find the value of $\cos 2\alpha$. Express your answer as a numerical value.
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Context:
A truncated cone is circumscribed around a sphere, with the area of its lower base being a times larger than the area of its upper base. By what factor is the volume of the truncated cone greater than the volume of the sphere?
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\boxed{\frac{a + \sqrt{a} + 1}{2 \sqrt{a}}}


Context:
A total of $2010$ coins are distributed in $5$ boxes. At the beginning the quantities of coins in the boxes are consecutive natural numbers. Martha should choose and take one of the boxes, but before that she can do the following transformation finitely many times: from a box with at least 4 coins she can transfer one coin to each of the other boxes.
What is the maximum number of coins that Martha can take away?
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Context:
Given the tetrahedron \(P Q R S\) with \(P Q = 4\), \(S R = 6\), \(\angle Q R S = \angle P S R = 50^\circ\), and \(\angle Q S R = \angle P R S = 40^\circ\). A sphere is circumscribed around the tetrahedron. Consider the set of all points on this sphere where the sum of the spherical distances from these points to \(P\), \(Q\), \(R\), and \(S\) is no less than \(6\pi\). What is the area of this set? Note that the spherical distance between two points on a sphere is the length of the shortest arc of a great circle connecting these points.
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Context:
Stephen had a 10:00 a.m. appointment 60 km from his home. He averaged 80 km/h for the trip and arrived 20 minutes late for the appointment. At what time did he leave his home? Express your answer in 12-hour clock format (e.g., 9:00 a.m.).
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Context:
Find the number of local extrema of the function $$f(x)=\left(\frac{1-x}{2-x}\right)x^{p-1}~\text{where}~p>1;~x\in[0,1]$$
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Context:
Let  $S$  be a finite set of points in the plane, such that for each  $2$  points  $A$  and  $B$  in  $S$ , the segment  $AB$  is a side of a regular polygon all of whose vertices are contained in  $S$ . Find all possible values for the number of elements of  $S$ .

Proposed by *Viktor Simjanoski, Macedonia*
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Context:
When a positive integer $N$ is fed into a machine, the output is a number calculated according to the rule shown below.

[asy] size(300); defaultpen(linewidth(0.8)+fontsize(13)); real r = 0.05; draw((0.9,0)--(3.5,0),EndArrow(size=7)); filldraw((4,2.5)--(7,2.5)--(7,-2.5)--(4,-2.5)--cycle,gray(0.65)); fill(circle((5.5,1.25),0.8),white); fill(circle((5.5,1.25),0.5),gray(0.65)); fill((4.3,-r)--(6.7,-r)--(6.7,-1-r)--(4.3,-1-r)--cycle,white); fill((4.3,-1.25+r)--(6.7,-1.25+r)--(6.7,-2.25+r)--(4.3,-2.25+r)--cycle,white); fill((4.6,-0.25-r)--(6.4,-0.25-r)--(6.4,-0.75-r)--(4.6,-0.75-r)--cycle,gray(0.65)); fill((4.6,-1.5+r)--(6.4,-1.5+r)--(6.4,-2+r)--(4.6,-2+r)--cycle,gray(0.65)); label("$N$",(0.45,0)); draw((7.5,1.25)--(11.25,1.25),EndArrow(size=7)); draw((7.5,-1.25)--(11.25,-1.25),EndArrow(size=7)); label("if $N$ is even",(9.25,1.25),N); label("if $N$ is odd",(9.25,-1.25),N); label("$\frac N2$",(12,1.25)); label("$3N+1$",(12.6,-1.25)); [/asy]
For example, starting with an input of $N=7,$ the machine will output $3 \cdot 7 +1 = 22.$ Then if the output is repeatedly inserted into the machine five more times, the final output is $26.$\[7 \to 22 \to 11 \to 34 \to 17 \to 52 \to 26\]When the same $6$-step process is applied to a different starting value of $N,$ the final output is $1.$ What is the sum of all such integers $N?$\[N \to \rule{0.5cm}{0.15mm} \to \rule{0.5cm}{0.15mm} \to \rule{0.5cm}{0.15mm} \to \rule{0.5cm}{0.15mm} \to \rule{0.5cm}{0.15mm} \to 1\]
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Context:
Let point O be located inside a regular tetrahedron ABCD, and let G be the centroid of the tetrahedron. The line OG intersects each face of the tetrahedron at points A1, B1, C1, and D1. Find the value of the sum of the ratios A1O to A1G, B1O to B1G, C1O to C1G, and D1O to D1G.
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Context:
The mean of $\frac12 , \frac34$ , and $\frac56$ differs from the mean of $\frac78$ and $\frac{9}{10}$ by $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
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Context:
Given a geometric sequence {a_n} where a_2 * a_6 = 36, find the value of a_4. Express your answer in the form ±n, where n is the absolute value of a_4.
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Context:
In quadrilateral \(ABCD\), the diagonals intersect at point \(O\). It is known that \(S_{ABO} = S_{CDO} = \frac{3}{2}\), \(BC = 3\sqrt{2}\), and \(\cos \angle ADC = \frac{3}{\sqrt{10}}\). Find the minimum area of such a quadrilateral.
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Context:
Given the power function $f(x)=(m-2)x^{m^2-2m}$, find $m$.
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Context:
Given a positive integer $n (n \geq 2)$, select $m$ different numbers from $1, 2, \ldots, 3n$. Among these numbers, there must exist four distinct numbers $a, b, c, d$ such that $a = b + c + d$. Find the minimum value of $m$.
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Context:
The manufacturing cost of a shoe is Rs. 200, and the transportation cost is Rs. 500 for 100 shoes. The selling price is Rs. 246. What is the percentage gain on the selling price?
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Context:
A man walking at a constant rate of 5 miles per hour is passed by a woman traveling in the same direction along the same path at a certain rate. The woman stops to wait for the man 2 minutes after passing him, while the man continues to walk at his constant rate. The woman must wait 4 minutes until the man catches up. What is the woman's traveling rate in miles per hour?
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Context:
How many equivalence relations on the set $S=\{a,b,c,d,e\}$ satisfy the conditions $a \sim b, b \sim c$, and $a \nsim e$?
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Context:
[ Tangent lines and tangent circles ] Described quadrilaterals $\quad]$

Two circles of radii 4 and 3 touch each other externally. Common external tangents $P Q$ and $R S$ are drawn to these circles such that points $P$ and $S$ lie on the circle of the larger radius, and points $Q$ and $R$ lie on the circle of the smaller radius. Find the radius of the circle that touches segments $R S$, $S P$, and $P Q$.
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Context:
Expand and find the sum of the coefficients of the expression $-(2x - 5)(4x + 3(2x - 5))$.
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Context:
Given School A and School B each have 3 teachers signing up for volunteer teaching, with School A having 2 males and 1 female, and School B having 1 male and 2 females, calculate the probability that the selected 2 teachers have the same gender.
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Context:
In the arithmetic sequence $\{a_n\}$, if $\frac{a_{1010}}{a_{1009}} < -1$, and the sum of its first $n$ terms, $S_n$, has a maximum value, then the maximum positive integer value of $n$ for which $S_n > 0$ is ______.
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Context:
The lines \(x = \frac{1}{4}y + a\) and \(y = \frac{1}{4}x + b\) intersect at the point \((1,2)\). The original answer is in \(\frac{k}{m}\) format, please find the value of \(k + m\).
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Context:
There are several rows in the stands of a hockey arena, each row with 168 seats. For the final match, 2016 students from several sports schools were invited as spectators, with no more than 45 from each school. Students from any school must be seated in the same row. What is the minimum number of rows required in the arena to ensure this can always be achieved?
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Context:
Given $m \in \mathbb{R}$, if $f(x) = (m + 1)x^{\frac{2}{3}} + mx + 1$ is an even function, find the interval where $f(x)$ is monotonically increasing.
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Context:
7. (10 points) The Spring Festival is approaching. Starting from January 17, 2011 (Monday), workers in the factory gradually return home for the New Year to reunite with their families. If the number of workers leaving the factory each day is the same, by January 31, there are still 121 workers left in the factory. During this 15-day period, the total workdays (one person working one day counts as 1 workday, and workers do not need to be counted after they leave the factory) recorded in the factory is 2011 workdays, with weekends off and no one absent. How many workers have returned home by January 31? $\qquad$
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Context:
A farmer bought 164 rabbits. If the farmer wanted to fill a certain number of cages with the same number of rabbits and needed to buy 6 more rabbits, how many cages did the farmer want to fill?
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Context:
Using the inequality $1 - \frac{1}{2} + \frac{1}{3} - \ldots + \frac{(-1)^{2k+1}}{2k} \approx \ln 2$ and $1 - \frac{1}{2} + \frac{1}{3} - \ldots + \frac{(-1)^{2k+2}}{2k+1} \leq \ln 2$, determine the minimum number of terms needed to be summed in the series $\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n}$ to approximate $\ln 2$ within an error of $0.001$ without computing the exact value of $\ln 2$. The initial attempt involved comparing the differences for $k=1$, resulting in a difference of $-\frac{1}{6}$. How can this be used to find the required number of terms ($k$)?
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Context:
Hamilton Avenue has eight houses. On one side of the street are the houses numbered $1, 3, 5, 7$ and directly opposite are houses $2, 4, 6, 8$ respectively. An eccentric postman starts deliveries at house $1$ and delivers letters to each of the houses, finally returning to house $1$ for a cup of tea. Throughout the entire journey he must observe the following rules:
1. The numbers of the houses delivered to must follow an odd-even-odd-even pattern throughout.
2. Each house except house $1$ is visited exactly once (house $1$ is visited twice).
3. The postman at no time is allowed to cross the road to the house directly opposite.

How many different delivery sequences are possible?
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Context:
Let \( \triangle ABC \) be a triangle with \( AB = 3 \), \( AC = 8 \), \( BC = 7 \). \( M \) and \( N \) are the midpoints of \( \overline{AB} \) and \( \overline{AC} \), respectively. Point \( T \) is selected on side \( BC \) such that \( AT = TC \). The circumcircles of triangles \( BAT \) and \( MAN \) intersect at \( D \). Compute \( DC \).
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Context:
What is the maximum value of \(\frac{(2^t-3t)t}{4^t}\) for real values of \(t?\) The original answer is in \(\frac{k}{m}\) format, where \(\frac{k}{m}\) is a simplified fraction. Please provide the value of k + m.
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Context:
A train 120 m in length crosses a telegraph post in 6 seconds. What is the speed of the train?
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Context:
In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are respectively $a$, $b$, $c$, and the height from $BC$ is $\frac{a}{2}$. The maximum value of $\frac{c}{b}+\frac{b}{c}$ is \_\_\_\_.
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Context:
Suppose that $f$ is a function from $\mathbb{R}$ to $\mathbb{R}$ such that $$f(x)+f\left(1-\dfrac{1}{x}\right)=\arctan x$$ for all real $x\neq0.$ (As usual, y=arctan $x$ means $-\pi/2<$ $y<\pi/2$ and tan$y=x.)$ Find $$\int_0^1f(x)\:dx.$$ The original answer is in the format \(\frac{m \pi}{n}\). Please give the value of \(m + n\).
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Context:
3rd Centromerican 2001 Problem A2 C and D are points on the circle diameter AB such that ∠AQB = 2 ∠COD. The tangents at C and D meet at P. The circle has radius 1. Find the distance of P from its center.
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Context:
Two kinds of vodka are mixed in the ratio 1 : 2 and 2 : 1 and they are sold fetching the profit 40 % and 20 % respectively. If the vodkas are mixed in equal ratio and the individual profit percent on them are increased by 4 / 3 and 5 / 3 times respectively, what is the profit percentage of the mixture?
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Context:
已知抛物线 $\Gamma: y^{2}=8 x$ 的焦点为 $F$ ，过 $F$ 作一条直线 $l$ 与抛物线 $\Gamma$ 交于 $A 、 B$ 两点，分别过 $A 、 B$ 作抛物线 $\Gamma$ 的切线，与 $y$ 轴交于 $P 、 Q$ 两点. 求四边形 $A P Q B$ 面积的最小值.
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Context:
There are 10 positive even integers less than a certain number that contain digits 5 or 9. What is that number?
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Context:
Given a geometric sequence {a_n} with real terms, and the sum of the first n terms is S_n. If S_{10} = 10 and S_{30} = 70, then find the value of S_{40}. Express your answer as a single numerical value.
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Context:
From an equal quantity of squares with sides 1, 2, and 3, create a square of the smallest possible size.
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Context:
1. [4] Submit an integer $x$ as your answer to this problem. The number of points you receive will be $\max (0,8-|8 x-100|)$. (Non-integer answers will be given 0 points.)
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Context:
Let the function $f(x)$ satisfy $f(x) = f\left(\frac{1}{{x+1}})$. The domain is $D=\left[0,+\infty \right)$, and the range is $A$. If the set $\{y|y=f(x), x\in \left[0,a\right]\}$ can take on all values in $A$, then the range of parameter $a$ is ____.
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Context:
Let set $T = \{1, 2, 3, 4, 5, 6, 7, 8\}$. Find the number of all nonempty subsets $A$ of $T$ such that $3 \mid S(A)$ and $5 \nmid S(A)$, where $S(A)$ is the sum of all the elements in $A$.
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Context:
What digits should replace the asterisks to make the number 454** divisible by 2, 7, and 9?
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Context:
In a line formation of 5 students, including 2 male students and 3 female students, if male student A does not stand at either end and only two of the three female students stand next to each other, then the number of different possible arrangements is ____.
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Context:
11. Stephanie enjoys swimming. She goes for a swim on a particular date if, and only if, the day, month (where January is replaced by '01' through to December by '12') and year are all of the same parity (that is they are all odd, or all are even). On how many days will she go for a swim in the two-year period between January 1st of one year and December 31st of the following year inclusive?
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Context:
Let $1,\alpha_1,\alpha_2,...,\alpha_{10}$ be the roots of the polynomial $x^{11}-1$. It is a fact that there exists a unique polynomial of the form $f(x) = x^{10}+c_9x^9+ \dots + c_1x$ such that each $c_i$ is an integer, $f(0) = f(1) = 0$, and for any $1 \leq i \leq 10$ we have $(f(\alpha_i))^2 = -11$. Find $\left|c_1+2c_2c_9+3c_3c_8+4c_4c_7+5c_5c_6\right|$.
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Context:
9.3. Find the value of the expression $x-\sqrt{2022 x}+2023$,

if $x-\sqrt{\frac{2022}{x}}=2023$.
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Context:
When a plane intersects a cube, what type of regular polygon cannot be formed as a cross-section? Express your answer as the name of the polygon.
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Context:
On the side \( AB \) of the convex quadrilateral \( ABCD \), point \( M \) is chosen such that \( \angle AMD = \angle ADB \) and \( \angle ACM = \angle ABC \). The quantity three times the square of the ratio of the distance from point \( A \) to the line \( CD \) to the distance from point \( C \) to the line \( AD \) is equal to 2, and \( CD = 20 \). Find the radius of the inscribed circle of triangle \( ACD \).
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Context:
Define the sequence of positive integers $\{a_n\}$ as follows:
- Let $a_1=1$, $a_2=3$.
- For each $n>2$, let $a_n$ be the result of expressing $a_{n-1}$ in base $n-1$, then reading the resulting numeral in base $n$, then adding $2$ (in base $n$).

For example:
- $a_2=3_{10}=11_2$, so $a_3=11_3+2_3=6_{10}$.

Express $a_{2013}$ in base $10$.
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Context:
Let \(ABCD\) be a square with side length 5, and let \(E\) be the midpoint of side \(AB\). Let \(P\) and \(Q\) be the feet of perpendiculars from \(B\) and \(D\) to \(CE\), respectively, and let \(R\) be the foot of the perpendicular from \(A\) to \(DQ\). The segments \(CE, BP, DQ\), and \(AR\) partition \(ABCD\) into five regions. What is the median of the areas of these five regions?
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Context:
Determine all integers $ a$ for which the equation $ x^2\plus{}axy\plus{}y^2\equal{}1$ has infinitely many distinct integer solutions $ x,y$.
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Context:
In quadrilateral $ABCD$, $\overrightarrow{AB}=(1,1)$, $\overrightarrow{DC}=(1,1)$, $\frac{\overrightarrow{BA}}{|\overrightarrow{BA}|}+\frac{\overrightarrow{BC}}{|\overrightarrow{BC}|}=\frac{\sqrt{3}\overrightarrow{BD}}{|\overrightarrow{BD}|}$, calculate the area of the quadrilateral.
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Context:
Let \( f(x) \) be an odd function defined on \( \mathbf{R} \) such that \( f(x) = x^2 \) for \( x \geq 0 \). If the inequality \( f(x+a) \geq 2 f(x) \) holds for any \( x \in [a, a+2] \), determine the range of the real number \( a \).
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Context:
When $x \in (0, +\infty)$, the inequality $c^{2}x^{2}-(cx+1)\ln x+cx\geqslant 0$ always holds, then the range of the real number $c$ is ______.
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Context:
On a line, there are blue and red points, with at least 5 red points. It is known that on any segment with endpoints at red points that includes at least one red point inside, there are at least 3 blue points. Additionally, on any segment with endpoints at blue points that includes exactly 2 blue points inside, there are at least 2 red points. What is the maximum number of blue points that can be on a segment with endpoints at red points, not containing other red points?
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Context:
The jury, when preparing versions of the district math olympiad problems for grades $7, 8, 9, 10, 11$, aims to ensure that each version for each grade contains exactly 7 problems, of which exactly 4 do not appear in any other version. What is the maximum number of problems that can be included in the olympiad?
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Context:
十边形内任意三条对角线都不会在其内部交于同一点，问这个十边形所有对角线可以把这个十边形划分为 $\qquad$个区域.
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Context:
The equation
$$
1 - 3 \sin \theta + 5 \cos 3\theta = 0 \quad (0 < \theta < 2\pi)
$$
has how many distinct solutions? Provide your answer as a single integer.
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Context:
A number is supposed to be multiplied by 5 and divided by 2, then squared. However, instead of multiplying the irrational number pi (π) by 5 and dividing it by 2, the number is divided by 5 and multiplied by 2 before being squared. What is the percentage of error obtained in the final result?
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Context:
Given the function $y = \cos\left(x+ \frac {\pi}{5}\right)$, where $x\in\mathbb{R}$, determine the horizontal shift required to obtain this function's graph from the graph of $y=\cos x$.
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Context:
If $3x=2x+6$, then $3x \_\_\_\_\_\_=6.$
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Context:
Given that $x,y,z>0$ and $x\geq max\{y,z\}$, find the minimum value of: $$A=\frac{x}{y}+2\sqrt{1+\frac{y}{z}}+3\sqrt[3]{1+\frac{z}{x}}.$$
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Context:
A necklace has a total of 99 beads. Among them, the first bead is white, the 2nd and 3rd beads are red, the 4th bead is white, the 5th, 6th, 7th, and 8th beads are red, the 9th bead is white, and so on. Determine the total number of red beads on this necklace.
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Context:
30 pikes were released into a pond, and they gradually eat each other. A pike is considered to be full if it has eaten at least three pikes (whether full or hungry). What is the maximum number of pikes that can be full?
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Context:
Let  $P_1$ ,  $P_2$ ,  $P_3$ ,  $P_4$ ,  $P_5$ , and  $P_6$  be six parabolas in the plane, each congruent to the parabola  $y = x^2/16$ .  The vertices of the six parabolas are evenly spaced around a circle.  The parabolas open outward with their axes being extensions of six of the circle's radii.  Parabola  $P_1$  is tangent to  $P_2$ , which is tangent to  $P_3$ , which is tangent to  $P_4$ , which is tangent to  $P_5$ , which is tangent to  $P_6$ , which is tangent to  $P_1$ .  What is the diameter of the circle?
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Context:
11.069. The lateral edge of a regular triangular prism is equal to the height of the base, and the area of the section made through this lateral edge and the height of the base is $Q$. Determine the volume of the prism.
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Context:
In the sequence \(\{a_n\}\), \(a_1 = -1\), \(a_2 = 1\), and \(a_3 = -2\), for all \(n \in \mathbb{N_+}\) it holds that \(a_n a_{n+1} a_{n+2} a_{n+3} = a_n + a_{n+1} + a_{n+2} + a_{n+3}\) and \(a_{n+1} a_{n+2} a_{n+3} \neq 1\). Determine the value of the sum of the first 4321 terms of the sequence \(S_{4321}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
$1 \cdot 411,2,3,4,5,6$ 的排列 $\left(n_{1}, n_{2}, n_{3}, n_{4}, n_{5}, n_{6}\right)$ 具有如下性质: 对于 $i=1,2,3,4,5,\left(n_{1}, n_{2}, \cdots, n_{i}\right)$ 都不是 $(1,2, \cdots, i)$ 的排列. 求这种排列的个数.

The permutation $\left(n_{1}, n_{2}, n_{3}, n_{4}, n_{5}, n_{6}\right)$ of $1, 2, 3, 4, 5, 6$ has the following property: for $i=1,2,3,4,5, \left(n_{1}, n_{2}, \cdots, n_{i}\right)$ is not a permutation of $(1,2, \cdots, i)$. Find the number of such permutations.
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Context:
Let  $P(x)$  be a real quadratic trinomial, so that for all  $x\in \mathbb{R}$  the inequality  $P(x^3+x)\geq P(x^2+1)$  holds. Find the sum of the roots of  $P(x)$ .
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Context:
Given two parallel lines $3x - 2y - 1 = 0$ and $6x + ay + c = 0$, the distance between them is $\frac{2\sqrt{13}}{13}$. Find the value of $\frac{c + 2}{a}$.
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Context:
Three cards have numbers on their front and back sides as follows: 1 and 2, 3 and 4, 5 and 6. If these three cards are placed side by side, how many different three-digit numbers can be formed, given that 6 cannot be used as 9? Express your answer as a single integer.
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Context:
Select 3 people from 5, including A and B, to form a line, and determine the number of arrangements where A is not at the head.
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Context:
Let y = f(x) be a function defined on R with a period of 1. If g(x) = f(x) + 2x and the range of g(x) on the interval [1,2] is [-1,5], calculate the range of the function g(x) on the interval [-2020,2020]. Express your answer as an interval [min, max].
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the sequence $a_1, a_2, \ldots, a_n, \ldots$, insert 3 numbers between each pair of consecutive terms to form a new sequence. Determine the term number of the 69th term of the new sequence.
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Context:
Interior numbers begin in the third row of Pascal's Triangle. The sum of the interior numbers in the fourth row is 6. The sum of the interior numbers of the fifth row is 14. What is the sum of the interior numbers of the seventh row?
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Context:
There are 4 prime numbers between 1 and a certain number that are factors of 210. What is the ending number?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \(\mathbb{N}\) be the set of all positive integers. A function \( f: \mathbb{N} \rightarrow \mathbb{N} \) satisfies \( f(m + n) = f(f(m) + n) \) for all \( m, n \in \mathbb{N} \), and \( f(6) = 2 \). Also, no two of the values \( f(6), f(9), f(12) \), and \( f(15) \) coincide. How many three-digit positive integers \( n \) satisfy \( f(n) = f(2005) \) ?
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Context:
A four-digit natural number $M$, where the digits in each place are not $0$, we take its hundreds digit as the tens digit and the tens digit as the units digit to form a new two-digit number. If this two-digit number is greater than the sum of the thousands digit and units digit of $M$, then we call this number $M$ a "heart's desire number"; if this two-digit number can also be divided by the sum of the thousands digit and units digit of $M$, then we call this number $M$ not only a "heart's desire" but also a "desire fulfilled". ["Heart's desire, desire fulfilled" comes from "Analects of Confucius. On Governance", meaning that what is desired in the heart becomes wishes, and all wishes can be fulfilled.] For example, $M=3456$, since $45 \gt 3+6$, and $45\div \left(3+6\right)=5$, $3456$ is not only a "heart's desire" but also a "desire fulfilled". Now there is a four-digit natural number $M=1000a+100b+10c+d$, where $1\leqslant a\leqslant 9$, $1\leqslant b\leqslant 9$, $1\leqslant c\leqslant 9$, $1\leqslant d\leqslant 9$, $a$, $b$, $c$, $d$ are all integers, and $c \gt d$. If $M$ is not only a "heart's desire" but also a "desire fulfilled", where $\frac{{10b+c}}{{a+d}}=11$, let $F\left(M\right)=10\left(a+b\right)+3c$. If $F\left(M\right)$ can be divided by $7$, then the maximum value of the natural number $M$ that meets the conditions is ____.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the inverse Laplace transform of the following function:
$$\sum_{k=0}^\infty \frac{(-1)^k (s+1)^{2k}}{ (2k)!}$$
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Context:
Given the hyperbola $\dfrac {x^{2}}{a^{2}}- \dfrac {y^{2}}{b^{2}}=1$ ($a > 0$, $b > 0$) with its left and right foci being $F_{1}$ and $F_{2}$ respectively, a line passing through point $F_{2}$ intersects the right branch of the hyperbola at two distinct points $M$ and $N$. If $\triangle MNF_{1}$ is an equilateral triangle, then the eccentricity of this hyperbola is ______.
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Context:
A trapezoid field is uniformly planted with wheat. The trapezoid has one pair of parallel sides measuring 80 m and 160 m, respectively, with the longer side on the bottom. The other two non-parallel sides each measure 120 m. The angle between a slanted side and the longer base is $45^\circ$. At harvest, the wheat at any point in the field is brought to the nearest point on the field's perimeter. What fraction of the crop is brought to the longest side?
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Context:
In △ABC, the sides opposite to angles A, B, and C are a, b, and c, respectively. If (2b-c)cosA=acosC, calculate the value of angle A in degrees. Express your answer in the format \boxed{[answer]°}, where [answer] is the numerical value of the angle.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Three noncollinear points and a line $\ell$ are given in the plane. Suppose no two of the points lie on a line parallel to $\ell$ (or $\ell$ itself). There are exactly $n$ lines perpendicular to $\ell$ with the following property: the three circles with centers at the given points and tangent to the line all concur at some point. Find all possible values of $n$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Define the binary operation $a \Delta b = ab + a - 1$. Compute the following expression:
\[
10 \Delta (10 \Delta (10 \Delta (10 \Delta (10 \Delta (10 \Delta (10 \Delta (10 \Delta (10 \Delta 10))))))))
\]
where $10$ is written $10$ times.
Please reason step by step, and put your final answer within \boxed{}.


Context:
5. Given a positive integer $n$ greater than 2004, fill the numbers $1,2,3, \cdots, n^{2}$ into the squares of an $n \times n$ chessboard (consisting of $n$ rows and $n$ columns) such that each square contains exactly one number. If a square contains a number that is greater than the numbers in at least 2004 squares in its row and at least 2004 squares in its column, then this square is called a “good square”. Find the maximum number of “good squares” on the chessboard.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Timur and Alexander are counting the trees growing around the house. They move in the same direction but start counting from different trees. How many trees are growing around the house if the tree that Timur counted as the 12th, Alexander counted as the 33rd, and the tree that Timur counted as the 105th, Alexander counted as the 8th?
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Context:
There has been a successive increase of some percentage in the price of gas from the previous month, and then a 20% increase. A driver should reduce gas consumption by 35.89743589743589% so that the expenditure does not change. What was the percentage increase in the price of gas in the first month?
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Context:
Given a set of consecutive positive integers $I=\{1, 2, 3, \ldots, 238\}$, if $T$ is a subset of $I$ and satisfies the condition: if $x \in T$, then $7x \notin T$, find the maximum number of elements in the set $T$. Express your answer as a whole number.
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Context:
What is the sum of the first 1234 terms of the sequence where the number of 2s between consecutive 1s increases by 1 each time?
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Context:
A pond has a rare diatom present that is randomly distributed in its surface water with a rate of \( \lambda = 3 \) cells/ml. If three 1.5 ml samples of surface water are independently taken, what is the probability that exactly two of these three samples will contain four or more diatoms?
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Context:
To be eligible for an algebra class a student must have an average of at least $83\%$ over all four quarters in his or her pre-algebra class. If Fisher had an $82\%$, $77\%$, and $75\%$ in the first three quarters, what is the minimum score he must earn in the 4th quarter to move on to algebra?
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Context:
Find the asymptotic bounds in the form of \(\Theta(\cdot)\) for the recurrence relation: \[T(n) = T\left(\frac{n}{\log n}\right) + \log \log n\]
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Context:
1. A column of scouts has a length of $1 \mathrm{~km}$ and is moving uniformly. The courier, who is at the front of the column, runs to the end of the column, delivers the message, and then returns to the front of the column. During this time, the column travels a distance of $1 \mathrm{~km}$. How far did the courier travel?
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Context:
Four identical isosceles triangles $A W B, B X C, C Y D$, and $D Z E$ are arranged with points $A, B, C, D$, and $E$ lying on the same straight line. A new triangle is formed with sides the same lengths as $A X, A Y,$ and $A Z$. If $A Z = A E$, what is the largest integer value of $x$ such that the area of this new triangle is less than 2004?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the Galois group of the polynomial $T^3-2$ over the finite field $\mathbb{F}_5$. Provide your answer as a group description or notation.
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Context:
On the Richter scale, which measures the total amount of energy released during an earthquake, a reading of x - 1 indicates one-tenth the released energy as is indicated by a reading of x. The frequency at which an earthquake occurs is represented by f(x) for a given Richter scale value x. Earthquake A occurred at location M with a Richter scale reading of 5, while earthquake B occurred at location N with a reading of 3. If the frequency of earthquake events at location M is 2 times the frequency of those at N, find the ratio of the energy release of earthquake A to earthquake B in terms of f(x).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Stacy and Heather are some miles apart and walk towards each other along the same route. Stacy walks at a constant rate that is 1 mile per hour faster than Heather's constant rate of 5 miles/hour. Heather starts her journey 24 minutes after Stacy. How far apart were Stacy and Heather initially if Heather has walked 5.7272727272727275 miles when the two meet?
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Context:
In trapezoid \(ABCD\) with bases \(AD\) and \(BC\), the angle \(\angle BCD\) is \(120^\circ\). A circle with radius 1 is inscribed in this angle, passing through points \(A\), \(B\), and \(D\). Find the area of triangle \(ABD\).
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Context:
If the chord cut by the line $ax-by+2=0$ on the circle $x^{2}+y^{2}+2x-4y+1=0$ is $4$, find the minimum value of $\dfrac{2}{a}+\dfrac{3}{b}$.
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Context:
Given that $i$ is the imaginary unit, and the complex number $z = (x^2 - 1) + (x + 1)i$ is purely imaginary. Find the value(s) of the real number $x$.
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Context:
Eastern rattlesnakes have 6 segments in their tails, while Western rattlesnakes have 8 segments. Additionally, Southern rattlesnakes have 7 segments, and Northern rattlesnakes have 9 segments in their tails. Calculate the average percentage difference in tail sizes of the Eastern, Southern, and Northern rattlesnakes compared to the Western rattlesnake's tail size.
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Context:
If the distance from a point A on the parabola $C: y^2 = 4x$ to the focus of the parabola is 3, and $O$ is the origin, then the slope of line $OA$ is ______.
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Context:
There is a pack of 27 distinct cards, and each card has three values on it. The first value is a shape from $\{\Delta,\square,\odot\}$; the second value is a letter from $\{A,B,C\}$; and the third value is a number from $\{1,2,3\}$.

In how many ways can we choose an unordered set of 3 cards from the pack, so that no two of the chosen cards have two matching values?

For example, we can choose $\{\Delta A1, \Delta B2, \odot C3\}$, but we cannot choose $\{\Delta A1, \square B2, \Delta C1\}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Among the coordinates $(x,y)$ where $1 \leq x,y \leq 101$, choose some points such that there do not exist $4$ points which form an isosceles trapezium with its base parallel to either the $x$ or $y$ axis (including rectangles). Find the maximum number of coordinate points that can be chosen.
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Context:
Find the least positive integer $n$ such that
$\frac 1{\sin 45^\circ\sin 46^\circ}+\frac 1{\sin 47^\circ\sin 48^\circ}+\cdots+\frac 1{\sin 133^\circ\sin 134^\circ}=\frac 1{\sin n^\circ}.$
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Context:
For \(0 < m < 10\), let \(S(m, n)\) be the set of all positive integers with \(n\) 1s, \(n\) 2s, \(n\) 3s, \ldots, \(n\) \(m\)s. For a positive integer \(N\), let \(d(N)\) be the sum of the absolute differences between all pairs of adjacent digits. For example, \(d(122313) = 1 + 0 + 1 + 2 + 2 = 6\). Find the mean value of \(d(N)\) for \(N\) in \(S(m, n)\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that the number of rabbits in a farm increases such that the difference between the populations in year $n+2$ and year $n$ is directly proportional to the population in year $n+1$, and the populations in the years $2001$, $2002$, and $2004$ were $50$, $80$, and $170$, respectively, determine the population in $2003$.
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Context:
A point $P$ is chosen at random in the interior of equilateral triangle $ABC$. What is the probability that $\triangle ABP$ has a greater area than each of $\triangle ACP$ and $\triangle BCP$?The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.
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Context:
A space probe is to be launched from a space station 200 miles above Earth. Determine its escape velocity in miles/s. Take Earth's radius to be 3960 miles.
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Context:
A container is already filled with water. There are three lead balls: large, medium, and small. The first time, the small ball is submerged in the water; the second time, the small ball is removed, and the medium ball is submerged in the water; the third time, the medium ball is removed, and the large ball is submerged in the water. It is known that the water spilled the first time is 3 times the water spilled the second time, and the water spilled the third time is 3 times the water spilled the first time. Find the ratio of the volumes of the three balls.
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Context:
Let \(a_n = \frac{1}{n} \sin \frac{n\pi}{25}\) and \(S_n = a_1 + a_2 + \cdots + a_n\). Calculate the number of positive terms among \(S_1, S_2, \cdots, S_{100}\). Express your answer as a single integer.
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Context:
Calculate the area enclosed by the curve defined by the equation $$\lfloor x + y\rfloor + \lfloor x - y\rfloor = 5$$ for $x\ge y$ and $x, y \ge 0$, where $\lfloor z\rfloor$ represents the greatest integer less than or equal to $z$.
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Context:
A number, when divided by the sum of a certain number and 445, gives 2 times their difference as quotient and 25 as remainder. The number is 220025. What is the first number?
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Context:
Given a regular pentagon \(ABCDE\). Point \(K\) is marked on side \(AE\), and point \(L\) is marked on side \(CD\). It is known that \(\angle LAE + \angle KCD = 108^\circ\) and the ratio \(AK : KE = 3 : 7\). Find the ratio \(CL : AB\).

A regular pentagon is a pentagon with all sides equal and all angles equal.
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Context:
On a table, there are $100$ red and $k$ white buckets, all initially empty. In each move, a red and a white bucket are selected, and an equal amount of water is added to both. After a certain number of moves, no bucket is empty, and for every pair of buckets that were selected together at least once, the amount of water in these buckets is the same. Find all possible values of $k$.
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Context:
Let $x, y, z$ be positive real numbers that satisfy at least one of the inequalities: $2xy > 1$, $yz > 1$. Find the least possible value of:
$$xy^3z^2 + \frac{4z}{x} - 8yz - \frac{4}{yz}.$$
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Context:
在ΔABC中，已知AC=3,sinC=ksinA(k≥2)，则ΔABC的面积的最大值为
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Context:
There are  $5$  accents in French, each applicable to only specific letters as follows:

- The cédille: ç
- The accent aigu: é
- The accent circonflexe: â, ê, î, ô, û
- The accent grave: à, è, ù
- The accent tréma: ë, ö, ü

Cédric needs to write down a phrase in French. He knows that there are  $3$  words in the phrase and that the letters appear in the order: \[cesontoiseaux.\] He does not remember what the words are and which letters have what accents in the phrase. If  $n$  is the number of possible phrases that he could write down, then determine the number of distinct primes in the prime factorization of  $n$ .
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Context:
Find the number of ordered pairs of positive integers $(x, y)$ with $x, y \leq 2020$ such that $3 x^{2}+10 x y+3 y^{2}$ is the power of some prime.
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Context:
How many three-digit positive integers $N$ satisfy the following properties?
The number $N$ is divisible by $7$.
The number formed by reversing the digits of $N$ is divisible by $5$.
Provide the number of such integers.
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Context:
Given the function $f(x)= \frac{1}{3}|x^{3}|- \frac{a}{2}x^{2}+(3-a)|x|+b$ with six distinct monotonic intervals, determine the range of values for $a$ .
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Context:
The center of sphere $\alpha$ lies on the surface of sphere $\beta$. The ratio of the surface area of sphere $\beta$ that is inside sphere $\alpha$ to the entire surface area of sphere $\alpha$ is $1 / 5$. Find the ratio of the radii of spheres $\alpha$ and $\beta$.
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Context:
Let $k$ be a positive integer. Each cell of a $3 k \times 3 k$ grid is colored either red or blue. A triple of cells $\left(C_{1}, C_{2}, C_{3}\right)$ is called a good cell group if $C_{1}$ and $C_{2}$ are in the same row, $C_{2}$ and $C_{3}$ are in the same column, and $C_{1}, C_{3}$ are colored blue, while $C_{2}$ is colored red. Find the maximum possible number of good cell groups.
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Context:
How many ways are there to place 31 knights in the cells of an $8 \times 8$ unit grid so that no two attack one another?
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Context:
In \(\triangle ABC\), \(DC = 2BD\), \(\angle ABC = 45^\circ\), and \(\angle ADC = 60^\circ\). Find \(\angle ACB\) in degrees.
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Context:
For which values of \( y \) does the equation \( x^{2}+2 y^{2}+8 z^{2}-2 x y z-9=0 \) have no solutions for any \( z \)?
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Context:
Let point $P$ be a moving point on the parabola $x^2=4y$, and the distance from point $P$ to the $x$-axis is $d$. Point $P_1$ is a moving point on the circle $(x-2)^2+(y+1)^2=1$. When $d+|PP_1|$ is minimized, the coordinates of point $P$ are ______.
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Context:
Let $\alpha \in \left(0, \frac{\pi}{4}\right)$, if $\tan\left(\alpha + \frac{\pi}{4}\right) = 2\cos2\alpha$, then $\alpha = $ ______.
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Context:
Let $ABC$ be a triangle and $D$, $E$, and $F$ be the midpoints of sides $BC$, $AC$, and $BC$ respectively. The medians $AD$ and $BE$ are perpendicular, with $AD = 12$ and $BE = 9$. What is the value of $CF$?
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Context:
In a rectangular parallelepiped, the sides of the base are \( a \) and \( b \), and the acute angle between them is \( 60^\circ \). The larger diagonal of the base is equal to the smaller diagonal of the parallelepiped. Find the volume of the parallelepiped.
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Context:
You are given a rectangular playing field of size  $13 \times 2$  and any number of dominoes of sizes  $2\times 1$  and  $3\times 1$ . The playing field should be seamless with such dominoes and without overlapping, with no domino protruding beyond the playing field may. Furthermore, all dominoes must be aligned in the same way, i. e. their long sides must be parallel to each other. How many such coverings are possible?

(Walther Janous)
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Context:
Given two circles described by the equations $x^2 + y^2 = m$ and $x^2 + y^2 + 6x - 8y + 21 = 0$, find the range of values for the real number $m$ such that the two circles intersect.
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Context:
A paper square with an area equal to four times the area of a cell is placed on a sheet of grid paper. What is the minimum number of grid points that this square can cover? (A grid point is a point where the lines of the grid intersect; if a grid point lies on the edge of the square, it is considered covered.)
Please reason step by step, and put your final answer within \boxed{}.


Context:
8. Fill $n^{2}(n \geqslant 3)$ integers into an $n \times n$ grid, with one number per cell, such that the product of the numbers in each row and each column is 30. Then there are $\qquad$ ways to fill the grid.
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Context:
After a gymnastics meet, each gymnast shook hands once with every gymnast on every team (except herself). Afterwards, a coach came down and only shook hands with each gymnast from her own team. There were a total of 281 handshakes. What is the fewest number of handshakes the coach could have participated in?
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Context:
Petya wants to create an unusual die, which should have the shape of a cube, with dots drawn on the faces (different numbers of dots on different faces). Additionally, on each pair of adjacent faces, the number of dots must differ by at least two (it is allowed to have more than six dots on some faces). How many dots in total are needed for this?
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Context:
In right triangle \( \triangle ACB \), \(\angle C = 90^{\circ}\), points \(D\) and \(E\) lie on \(BC\) and \(CA\) respectively, and \(\frac{BD}{AC} = \frac{AE}{CD} = K\). Let \(BE\) intersect \(AD\) at \(O\). If \(\angle BOD = 75^{\circ}\), what is \(K\)?
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Context:
An old water pump can fill up a large trough in 600 sec, but a second, newer pump takes only 200 sec to fill the trough. A third water pump can fill the same trough in 400 sec. How long would it take to fill the trough using all three pumps at the same time?
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Context:
Let \( a, b, c, d \) be real numbers defined by

$$
a=\sqrt{4-\sqrt{5-a}}, \quad b=\sqrt{4+\sqrt{5-b}}, \quad c=\sqrt{4-\sqrt{5+c}}, \quad d=\sqrt{4+\sqrt{5+d}}
$$

Calculate their product.
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Context:
定义在 \bR 上的奇函数 f(x) 同时是以 2 为周期的周期函数，并且在 x\in(0,1) 时，f(x)=\ln(x^2-2x+\frac{3}{2}). 则函数 f(x) 在 [0,10] 上的零点个数为__________.（李纪琛供题）
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Context:
某品牌儿童服装每件售价60元，不征收附加税时，每年销售80万件；若征收附加税，即每销售值100元征收R元（叫做税率R%），则每年的销售量将减少$$\frac{20}{3}$$ R 万件。若在此项经营中，每年征收附加税不少于128万元。问：税率R应怎样确定？
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Context:
5. Place eight balls, each with a radius of 1, in two layers inside a cylinder such that each ball is tangent to its four adjacent balls, and to one base and the side of the cylinder. Then the height of the cylinder is $\qquad$ .
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Context:
The points \((2,3)\) and \((3, 7)\) lie on a circle whose center is on the \(x\)-axis. What is the radius of the circle?
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Context:
Sasha wrote the numbers $7, 8, 9, \ldots, 17$ on the board and then erased one or more of them. It turned out that the remaining numbers on the board cannot be divided into several groups such that the sums of the numbers in the groups are equal. What is the maximum value that the sum of the remaining numbers on the board can have?
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Context:
Determine the value of the following expression:

$$
X = \sqrt{x \sqrt{y \sqrt{z \sqrt{x \sqrt{y \sqrt{z \cdots}}}}}}
$$
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Context:
You are given the numbers 1, 2, 3, 4, 5, 6, 7, 8 to be placed at the eight vertices of a cube, such that the sum of any three numbers on each face of the cube is at least 10. Find the minimum possible sum of the four numbers on any face.
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Context:
a , b , c , d and e are 5 consecutive points on a straight line . if bc = 3 cd , de = 8 , a certain distance is given and ac = 11 , the length of ae is 21. What is the length of ab?
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Context:
在平面上有一点 O，过点 O 引出 2007 条不同的直线 $l_1, l_2, l_3, \ldots, l_{2007}$，则可形成多少对以 O 为顶点的对顶角？
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Context:
Consider the cubes whose vertices lie on the surface of a given cube. Which one is the smallest among them?
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Context:
Given $p$: $\frac{1}{2} \leq x \leq 1$, $q$: $(x - a)(x - a - 1) > 0$, if $p$ is a necessary but not sufficient condition for $\lnot q$, then the range of values for the real number $a$ is $\_\_\_\_\_\_\_\_$.
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Context:
A set of tiles numbered $1$ through $100$ is modified repeatedly by the following operation: remove all tiles numbered with a perfect square, and renumber the remaining tiles consecutively starting with $1$. Find the number of times the operation must be performed to reduce the number of tiles in the set to one.
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Context:
Let $p$ and $q$ be real numbers, and suppose that the roots of the equation \[x^3 - 8x^2 + px - q = 0\] are three distinct positive integers. Compute $p + q.$
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Context:
Let $S$ be the set of integers $n > 1$ for which $\tfrac1n = 0.d_1d_2d_3d_4\ldots$, an infinite decimal that has the property that $d_i = d_{i+12}$ for all positive integers $i$. Given that $9901$ is prime, how many positive integers are in $S$? (The $d_i$ are digits.)
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Context:
It takes 30 days to fill a laboratory dish with bacteria. The bacteria grows at a certain rate each day, and it took 25 days for the bacteria to fill 1/32 of the dish. What is the growth rate of the bacteria each day?
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Context:
Given the sequence ${a_n}$ where ${a_{n+1} + (-1)^n a_n = 2n - 1}$, calculate the sum of the first 16 terms of the sequence. Express your answer as a single numerical value.
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Context:
To measure the height of tower AB on the opposite bank of a river, a point C is chosen on the bank such that the base of the tower A is exactly to the west of point C. At this point, the elevation angle to the top of the tower B is measured to be 45°. Then, moving from point C in a direction 30° north of east for 30 meters to reach point D, the elevation angle to the top of the tower B from D is measured to be 30°. The height of tower AB is \_\_\_\_\_\_ meters.
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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 $\frac{1}{A}$.
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Context:
The Garfield Super Winners play $100$ games of foosball, where teams score a non-negative integer number of points, and the team with more points after ten minutes wins (if both teams have the same number of points, it is a draw). Suppose that the Garfield Super Winners score an average of $7$ points per game but allow an average of $8$ points per game. Given that the Garfield Super Winners never won or lost by more than $10$, what is the largest possible number of games that they could win?
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Context:
Convert $BD4_{16}$ to base 4.
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Context:
If a line is perpendicular to a plane, then this line and the plane form a "perpendicular line-plane pair". In a cube, the number of "perpendicular line-plane pairs" formed by a line determined by two vertices and a plane containing four vertices is _________.
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Context:
If the range of the function $f(x)$ on $[m, n] (m < n)$ is exactly $[m, n]$, then $f(x)$ is called an "equal value mapping interval" of the function. Among the following functions: $① y = x^2 - 1$; $② y = 2 + \log_2 x$; $③ y = 2^x - 1$; $④ y = \frac{1}{x - 1}$. The number of functions that have a unique "equal value mapping interval" is $\_\_\_\_\_\_$.
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Context:
As shown in the figure, points $C$ and $D$ divide $AB$ into three equal parts. At 8:00, Person A starts walking uniformly from $A$ to $B$. At 8:12, Person B starts walking uniformly from $B$ to $A$. After a few more minutes, Person C starts walking uniformly from $B$ to $A$. When Person A and Person B meet at point $C$, Person C just reaches point $D$. When Person A and Person C meet at 8:30, Person B just reaches $A$. At what time does Person C start?
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Context:
Out of all parallelograms with a given area, find the one where the longest diagonal is minimized.
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Context:
When young fishermen were asked how many fish each of them caught, the first one replied, "I caught half the number of fish that my friend caught, plus 10 fish." The second one said, "And I caught as many as my friend, plus 20 fish." How many fish did the fishermen catch?
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Context:
Find the remainder when $5^{2021}$ is divided by $17$.
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Context:
$已知椭圆C: \frac{x^2}{a^2} + \frac{y^2}{b^2}=1 (a  >  b > 0), F_1，F_2 为C的左，右焦点，点P \left(1,-\frac{3}{2}\right) 为椭圆上一点，且|PF_1|+|PF_2|=4. 过P作两直线与椭圆C相交于相异的两点A,B,直线PA、PB的倾斜角互补，直线AB与x，y轴正半轴相交.$
求直线AB的斜率.
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Context:
Baker made some cakes and 153 pastries. He sold 8 pastries and 97 cakes. Baker sold 89 more cakes than pastries. How many cakes did he make?
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Context:
Two cyclists are moving towards each other at a certain speed. They are now 50 miles apart. At this instance, a fly starts from one cyclist and moves towards the other, moving to and fro until the two cyclists meet each other. The fly is moving at 15 miles/hour, and the total distance covered by the fly is 37.5 miles. What is the speed of the cyclists?
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Context:
On dividing 635 by a prime number, the quotient is a multiple of 7 and the remainder is 11. Find the divisor.
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Context:
Jia, Yi, Bing, Ding, and Wu sit around a circular table to play cards. Jia has a fixed seat. If Yi and Ding cannot sit next to each other, how many different seating arrangements are possible?
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Context:
In the expansion of $\left(x+1\right)\left(2x+1\right)\cdots \left(nx+1\right)$, where $n$ is a positive integer, find the coefficient of the linear term in $x$. Express your answer as a combination formula (e.g., $C_n^k$).
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Context:
Calculate

$$
\sqrt{1+2 \sqrt{1+3 \sqrt{1+\ldots+2017 \sqrt{1+2018 \cdot 2020}}}}
$$
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Context:
The sum of the first $n$ terms of an arithmetic sequence $\{a_n\}$ is $S_n$ ($n=1,2,3,\ldots$). When the first term $a_1$ and the common difference $d$ change, if $a_5+a_8+a_{11}$ is a constant, then among $S_n$ ($n=1,2,3,\ldots$), the constant value is.
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Context:
Omar made a list of all the arithmetic progressions of positive integer numbers such that the difference is equal to $2$ and the sum of its terms is $200$. How many progressions does Omar's list have?
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Context:
Let \( 0 < x < \frac{\pi}{2} \). Determine the range of the function \( y = \sin x - 2 \cos x + \frac{32}{125 \sin x (1 - \cos x)} \).
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\boxed{\left[ \frac{2}{5}, +\infty \right)}


Context:
若 $a, b, c$ 和 $\left(a-\frac{1}{b}\right)\left(b-\frac{1}{c}\right)\left(c-\frac{1}{a}\right)$ 均为正整数，求 $2 a+3 b+5 c$ 的最大值和最小值之差。
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Context:
A sequence $a_1,a_2,\dots$ of non-negative integers is defined by the rule $a_{n+2}=|a_{n+1}-a_n|$ for $n\geq 1$. If $a_1=999$, $a_2<999$ and $a_{2006}=1$, how many different values of $a_2$ are possible?
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Context:
Given several numbers, one of them,  $a$ , is chosen and replaced by the three numbers  $\frac{a}{3}, \frac{a}{3}, \frac{a}{3}$ . This process is repeated with the new set of numbers, and so on. Originally, there are  $1000$  ones, and we apply the process several times. A number  $m$  is called *good* if there are  $m$  or more numbers that are the same after each iteration, no matter how many or what operations are performed. Find the largest possible good number.
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Context:
Find any five consecutive natural numbers less than 100, whose product is divisible by 2014.
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Context:
If 30 sheep eat 45 bags of grass in 40 days, with the fastest sheep eating at twice the rate of the slowest sheep, in how many days will one slowest sheep eat one bag of grass?
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Context:
Let $a, b, c$ be positive integers such that $a, b, c, a+b-c, a+c-b, b+c-a, a+b+c$ are 7 distinct primes. The sum of two of $a, b, c$ is 800. If $d$ is the difference between the largest prime and the smallest prime among these 7 primes, find the maximum value of $d$.
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Context:
Let \(X_1, X_2, \ldots, X_n, \ldots\) be independent random variables, where each \(X_n\) is uniformly distributed on \([0, n]\). Find a sequence \((a_n)\) such that \(\frac{1}{a_n} \sum_{k=1}^n X_k^2 \to 1\) in probability.
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Context:
A unit cube has vertices $P_1,P_2,P_3,P_4,P_1',P_2',P_3',$ and $P_4'$. Vertices $P_2$, $P_3$, and $P_4$ are adjacent to $P_1$, and for $1\le i\le 4,$ vertices $P_i$ and $P_i'$ are opposite to each other. A regular octahedron has one vertex in each of the segments $\overline{P_1P_2}$, $\overline{P_1P_3}$, $\overline{P_1P_4}$, $\overline{P_1'P_2'}$, $\overline{P_1'P_3'}$, and $\overline{P_1'P_4'}$.  Find the side length of the octahedron.

[asy]
import three;

size(5cm);
triple eye = (-4, -8, 3);
currentprojection = perspective(eye);

triple[] P = {(1, -1, -1), (-1, -1, -1), (-1, 1, -1), (-1, -1, 1), (1, -1, -1)}; // P[0] = P[4] for convenience
triple[] Pp = {-P[0], -P[1], -P[2], -P[3], -P[4]};

// draw octahedron
triple pt(int k){ return (3*P[k] + P[1])/4; }
triple ptp(int k){ return (3*Pp[k] + Pp[1])/4; }
draw(pt(2)--pt(3)--pt(4)--cycle, gray(0.6));
draw(ptp(2)--pt(3)--ptp(4)--cycle, gray(0.6));
draw(ptp(2)--pt(4), gray(0.6));
draw(pt(2)--ptp(4), gray(0.6));
draw(pt(4)--ptp(3)--pt(2), gray(0.6) + linetype("4 4"));
draw(ptp(4)--ptp(3)--ptp(2), gray(0.6) + linetype("4 4"));

// draw cube
for(int i = 0; i < 4; ++i){
draw(P[1]--P[i]); draw(Pp[1]--Pp[i]);
for(int j = 0; j < 4; ++j){
if(i == 1 || j == 1 || i == j) continue;
draw(P[i]--Pp[j]); draw(Pp[i]--P[j]);
}
dot(P[i]); dot(Pp[i]);
dot(pt(i)); dot(ptp(i));
}

label("$P_1$", P[1], dir(P[1]));
label("$P_2$", P[2], dir(P[2]));
label("$P_3$", P[3], dir(-45));
label("$P_4$", P[4], dir(P[4]));
label("$P'_1$", Pp[1], dir(Pp[1]));
label("$P'_2$", Pp[2], dir(Pp[2]));
label("$P'_3$", Pp[3], dir(-100));
label("$P'_4$", Pp[4], dir(Pp[4]));
[/asy]
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Context:
Find the number of 12-cycles in the tent map.
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Context:
A convex polyhedron is bounded by 4 regular hexagonal faces and 4 regular triangular faces. At each vertex of the polyhedron, 2 hexagons and 1 triangle meet. What is the volume of the polyhedron if the length of its edges is one unit?
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Context:
Using only pennies, nickels, dimes, and quarters, calculate the smallest number of coins Freddie would need so he could pay any amount of money less than a dollar. Express your answer as a single integer.
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Context:
Calculate the definite integral:

$$
\int_{0}^{\pi} 2^{4} \cdot \sin^{2}\left(\frac{x}{2}\right) \cos^{6}\left(\frac{x}{2}\right) \, dx
$$
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Context:
If \( N=\frac{1987^{4(\sqrt{3}+2)^{1987}+1988}}{1987^{(\sqrt{3}+2)^{1988}}+(\sqrt{3}+2)^{1988}} \), what is the last digit of \( N \)?
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Context:
In a certain football competition, a victory is worth 3 points, a draw is worth some points, and a defeat is worth 0 points. Each team plays 20 matches. If a team scored 8 points after 5 games, it has to win at least 9 of the remaining matches to reach the 40-point mark by the end of the tournament. How many points is a draw worth?
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Context:
An organization has $30$ employees, $20$ of whom have a brand A computer while the other $10$ have a brand B computer. For security, the computers can only be connected to each other and only by cables. The cables can only connect a brand A computer to a brand B computer. Employees can communicate with each other if their computers are directly connected by a cable or by relaying messages through a series of connected computers. Initially, no computer is connected to any other. A technician arbitrarily selects one computer of each brand and installs a cable between them, provided there is not already a cable between that pair. The technician stops once every employee can communicate with each other. What is the maximum possible number of cables used?
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