Context:
Find the approximate interval where the zero of the function $f(x) = \ln x - \frac{3}{x}$ is located. Express your answer as an interval in the format (a, b).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a volleyball tournament for the Euro-African cup, there were nine more teams from Europe than from Africa. Each pair of teams played exactly once and the Europeans teams won precisely nine times as many matches as the African teams, overall. What is the maximum number of matches that a single African team might have won?
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Context:
There are 3 kids waiting for the swings and twice as many kids waiting for the slide. Each kid waits 2 minutes for the swings and a certain amount of seconds for the slide. The shorter wait is 270 seconds shorter. How many seconds do the kids wait for the slide?
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Context:
Let  $E$  and  $F$  be the respective midpoints of  $BC,CD$  of a convex quadrilateral  $ABCD$ . Segments  $AE,AF,EF$  cut the quadrilateral into four triangles whose areas are four consecutive integers. Find the maximum possible area of  $\Delta BAD$ .
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Context:
A car and a minivan drive from Alphaville to Betatown. The car travels at a constant speed of $40 \mathrm{~km} / \mathrm{h}$ and the minivan travels at a constant speed of $50 \mathrm{~km} / \mathrm{h}$. The minivan passes the car 10 minutes before the car arrives at Betatown. How many minutes pass between the time at which the minivan arrives in Betatown and the time at which the car arrives in Betatown?
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Context:
For how many two-digit natural numbers \( n \) are exactly two of the following three statements true: (A) \( n \) is odd; (B) \( n \) is not divisible by 3; (C) \( n \) is divisible by 5?
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Context:
The line $y = a$ and the graph of the function $y = x^3 - 3x$ have three distinct intersection points. Then, the range of $a$ is \_\_\_\_\_\_\_.
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Context:
Identify a monomial that is of the same type as 2ab, meaning it contains exactly the same variables. Coefficients do not matter in this comparison. Provide your answer as a monomial.
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Context:
Of the natural numbers greater than 1000 that are composed of the digits $0, 1, 2$ (where each digit can be used any number of times or not at all), in ascending order, what is the position of 2010?
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Context:
Some mat-weavers can weave 4 mats in 4 days. At the same rate, 14 mat-weavers would weave 49 mats in 14 days. How many mat-weavers were initially weaving?
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Context:
In regular octagon $ABCDEFGH$, $M$ and $N$ are midpoints of $\overline{BC}$ and $\overline{FG}$ respectively.  Compute $[ABMO]/[EDCMO]$. ($[ABCD]$ denotes the area of polygon $ABCD$.) [asy]
pair A,B,C,D,E,F,G,H;
F=(0,0); E=(2,0); D=(2+sqrt(2),sqrt(2)); C=(2+sqrt(2),2+sqrt(2));
B=(2,2+2sqrt(2)); A=(0,2+2*sqrt(2)); H=(-sqrt(2),2+sqrt(2)); G=(-sqrt(2),sqrt(2));
draw(A--B--C--D--E--F--G--H--cycle);
draw(A--E);
pair M=(B+C)/2; pair N=(F+G)/2;
draw(M--N);

label("$A$",A,N); label("$B$",B,NE); label("$C$",C,E); label("$D$",D,E);

label("$E$",E,S); label("$F$",F,S); label("$G$",G,W); label("$H$",H,W);
label("$M$",M,NE); label("$N$",N,SW);

label("$O$",(1,2.4),E);

[/asy]The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $f(x)=\int_{0}^{x}\frac{\sin(t)}{t^{3/2}}dt$ find $\lim_{k \to \infty} \int_{0}^{\infty}k^{3/2}f(x)e^{-kx} dx$.
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Context:
A privateer discovers a merchantman $10$ miles to leeward at 11:45 a.m. and with a good breeze bears down upon her at $11$ mph, while the merchantman can only make $8$ mph in her attempt to escape. After a two-hour chase, the top sail of the privateer is carried away; she can now make only $17$ miles while the merchantman makes $15$. At what time will the privateer overtake the merchantman? Provide your answer in the format of H:MM PM, where H and MM are two-digit numbers (possibly with leading zeros). Please give the value of H + MM.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In how many ways can a barrel with a capacity of 10 liters be emptied using two containers with capacities of 1 liter and 2 liters?
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Context:
The minimum positive period of the function $y=\sin x \cdot |\cos x|$ is __________.
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Context:
In a round goblet whose section is the graph of the function $y = x^4$, a cherry — a ball of radius r — is dropped. What is the largest r for which the ball can touch the lowest point of the bottom? (In plain math words: what is the maximal radius of the disc lying in the domain $y ≥ x^4$ and containing the origin?) The original answer is in the format $\frac{m}{n}\sqrt[3]{2}$, please give the value of $m + n$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Points \( M \) and \( N \) are marked on the sides \( AB \) and \( BC \) of triangle \( ABC \) respectively such that \(\angle CMA = \angle ANC \). The segments \( MC \) and \( AN \) intersect at point \( O \), and \( ON = OM \). Find \( BC \), if \( AM = 3 \, \text{cm} \) and \( BM = 4 \, \text{cm} \).
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Context:
$\frac{3}{2}+\frac{5}{4}+\frac{9}{8}+\frac{17}{16}+\frac{33}{32}+\frac{65}{64}-7=$
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Context:
3. The continuation of the height $B H$ of triangle $A B C$ intersects the circumscribed circle around it at point $D$ (points $B$ and $D$ lie on opposite sides of line $A C$). The degree measures of arcs $A D$ and $C D$, not containing point $B$, are $120^{\circ}$ and $90^{\circ}$, respectively. Determine in what ratio the segment $B D$ is divided by side $A C$.
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Context:
A hospital's internal medicine ward has 15 nurses, who work in pairs, rotating shifts every 8 hours. After two specific nurses work the same shift together, calculate the maximum number of days required for them to work the same shift again.
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Context:
For each three-digit number, calculate the sum of its digits. The total sum of all these sums is ____.
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Context:
Let $T_k = \frac{k(k+1)}{2}$ be the $k$-th triangular number. The infinite series
$$\sum_{k=4}^{\infty}\frac{1}{(T_{k-1} - 1)(T_k - 1)(T_{k+1} - 1)}$$
has the value $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
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Context:
Find the minimum value of the expression $$A=\frac{x}{y}+2\sqrt{1+\frac{y}{z}}+3\sqrt[3]{1+\frac{z}{x}}$$ given that $x, y, z > 0$ and $x \geq \max\{y, z\}$.
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Context:
A school volleyball team played several matches. After winning the next match, the share of victories increased by $1 / 6$. In order to increase the share of victories by another $1 / 6$, the team had to win two more consecutive matches. What is the minimum number of wins the team needs to achieve to increase the share of victories by another $1 / 6$?
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Context:
Let $\omega$ and $\Gamma$ be circles such that $\omega$ is internally tangent to $\Gamma$ at a point $P$. Let $AB$ be a chord of $\Gamma$ tangent to $\omega$ at a point $Q$. Let $R\neq P$ be the second intersection of line $PQ$ with $\Gamma$. If the radius of $\Gamma$ is $17$, the radius of $\omega$ is $7$, and $\frac{AQ}{BQ}=3$, find the circumradius of triangle $AQR$.
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Context:
有 20 张卡片上分别写有数字1,2,…,20, 将它们放入一个盒子内.有4个人从中不放回地各抽取一张卡片,抽到两个较小数字的两人在同一组,抽到两个较大数字的两人在同一组.现其中有两人抽到5、14,则此两人在同一组的概率等于___(用最简分数作答)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:
A palindrome is a word that reads the same backwards as forwards, such as "eye", "race car", and "qwertyytrewq".

How many letters are in the smallest palindrome containing the letters b, o, g, t, r, and o, not necessarily in that order and not necessarily adjacent?
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Context:
Let \( m \) be a positive real number such that for any positive real numbers \( x \) and \( y \), the three line segments with lengths \( x + y \), \( \sqrt{x^2 + y^2 + xy} \), and \( m \sqrt{xy} \) can form the three sides of a triangle. Determine the range of values for \( m \).
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Context:
Find the closed form of the sum: $$g(n)=\sum_{k=0}^{n-1}\left(\left(\sum_{i=k+1}^{n}(-1)^i\binom{n}{i}\right)\cdot\left(\sum_{j=0}^{k}(-1)^j\binom{n}{j}\right)\right)$$
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Context:
Let \( h_a = \beta_b = m_c \), where \( h_a = AP \), \( \beta_b = BL \), and \( m_c = CF \) are, respectively, the altitude, angle bisector, and median drawn from three different vertices of a triangle \( ABC \). What value can the ratio of the length of the longest side of triangle \( ABC \) to the length of the shortest side take?
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Context:
Let $V$ be a 10-dimensional real vector space and $U_1, U_2$ two linear subspaces such that $U_1 \subseteq U_2$, $\dim U_1 = 3$, and $\dim U_2 = 6$. Let $\varepsilon$ be the set of all linear maps $T: V \rightarrow V$ which have $T(U_1) \subseteq U_1$ and $T(U_2) \subseteq U_2$. Calculate the dimension of $\varepsilon$.
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Context:
Let $A B C$ be a triangle with $A B=4, B C=8$, and $C A=5$. Let $M$ be the midpoint of $B C$, and let $D$ be the point on the circumcircle of $A B C$ so that segment $A D$ intersects the interior of $A B C$, and $\angle B A D=\angle C A M$. Let $A D$ intersect side $B C$ at $X$. Compute the ratio $A X / A D$.
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Context:
How many integers  $n$  with  $10 \le n \le 500$  have the property that the hundreds digit of  $17n$  and  $17n+17$  are different?

*Proposed by Evan Chen*
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Context:
When every vector on the line $y = \frac{5}{2} x + 4$ is projected onto a certain vector $\mathbf{w},$ the result is always the vector $\mathbf{p}.$  Find the vector $\mathbf{p}.$
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\boxed{\begin{pmatrix} -40/29 \\ 16/29 \end{pmatrix}}


Context:
How many noncongruent triangles are there with one side of length 20, one side of length 17, and one \(60^\circ\) angle?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Distribute 7 doctors, including Doctors A and B, into three medical teams with one team having 3 members and the other two teams having 2 members each. In how many ways can this be done so that Doctors A and B are not in the same team?
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Context:
The integer sequence \(\left\{a_{n}\right\}\) is defined by \(a_{1}=1\), \(a_{2}=2\), and \(a_{n+2}=5a_{n+1}+a_{n}\). Determine the value of the expression \(\left[\frac{a_{2}}{a_{1}}\right]\left\{\left[\frac{a_{3}}{a_{2}}\right\}\left\{\frac{a_{4}}{a_{3}}\right\} \cdots \left\{\left\{\frac{a_{20225}}{a_{2024}}\right\}\left[\frac{a_{20224}}{a_{2}}\right]\right\}\right.\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the number $\overline{523abc}$ divisible by $7, 8$ and $9$.
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Context:
61. In quadrilateral $A B C D$ with angles $\angle A=60^{\circ}, \angle B=90^{\circ}, \angle C=120^{\circ}$, the point of intersection of the diagonals $M$ is marked. It turns out that $B M=1$ and $M D=2$. Find the area of $A B C D$.
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Context:
Skipper’s doghouse has a regular hexagonal base that measures one metre on each side. Skipper is tethered to a $2$-metre rope which is fixed to a vertex. What is the area of the region outside the doghouse that Skipper can reach? Calculate an approximate answer by using $\pi=3.14$ or $\pi=22/7$. The original answer is in the format $n\pi$, please give the value of n.
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Context:
On the lateral side \( CD \) of the trapezoid \( ABCD \) (\( AD \parallel BC \)), a point \( M \) is marked. A perpendicular \( AH \) is dropped from vertex \( A \) to segment \( BM \). It turns out that \( AD = HD \). Find the length of segment \( AD \), given that \( BC = 16 \), \( CM = 8 \), and \( MD = 9 \).
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Context:
已知直线$l:x+my-2-m=0(m\in R)$恒过定点$\textit{A}$，点$\textit{B}$，$\textit{C}$为圆$O:{{x}^{2}}+{{y2}}=25$上的两动点，满足$\angle BAC=90{}^\circ $，则弦$\textit{BC}$长度的最大值为___________．
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Context:
Mr. Caiden wants to do repairs to his house and requires 300 feet of metal roofing to do this. The metal roofing costs $8 per foot. The supplier offers a tiered discount system: a 10% discount is applied if purchasing 200-299 feet, and a 15% discount is applied if purchasing 300 feet or more. The shipping fee varies as well: the first 200 feet are delivered for free, but there is a $150 charge for each additional batch of 100 feet. Additionally, there is a 5% sales tax on the order if the total cost is less than or equal to $2000, and a 7% sales tax if the total cost is more than $2000.

Given that the supplier is providing the first 250 feet of metal roofing for free, how much money will Mr. Caiden be required to pay for the remaining metal roofing, taking into account the tiered discounts, varying shipping fees, and applicable sales tax?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given six cards with the digits $1, 2, 4, 5, 8$ and a comma. Using each card exactly once, various numbers are formed (the comma cannot be at the beginning or at the end of the number). What is the arithmetic mean of all such numbers?

(M. V. Karlukova)
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Context:
Two circles of the same radius 9 intersect at points \(A\) and \(B\). A point \(C\) is chosen on the first circle, and a point \(D\) is chosen on the second circle. It turns out that point \(B\) lies on the segment \(CD\), and \(\angle CAD = 90^\circ\). On the perpendicular to \(CD\) passing through point \(B\), a point \(F\) is chosen such that \(BF = BD\) (points \(A\) and \(F\) are on opposite sides of the line \(CD\)). Find the length of the segment \(CF\).
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Context:
Let $ABC$ be a triangle with $AB=4$, $AC=5$, $BC=6$, and circumcircle $\Omega$. Points $E$ and $F$ lie on $AC$ and $AB$ respectively such that $\angle ABE=\angle CBE$ and $\angle ACF=\angle BCF$. The second intersection point of the circumcircle of $\triangle AEF$ with $\Omega$ (other than $A$) is $P$. Suppose $AP^2=\frac{m}{n}$ where $m$ and $n$ are positive relatively prime integers. Find $100m+n$.
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Context:
Let  $f(x)=x^{2021}+15x^{2020}+8x+9$  have roots  $a_i$  where  $i=1,2,\cdots , 2021$ . Let  $p(x)$  be a polynomial of the sam degree such that  $p \left(a_i + \frac{1}{a_i}+1 \right)=0$  for every  $1\leq i \leq 2021$ . If  $\frac{3p(0)}{4p(1)}=\frac{m}{n}$  where  $m,n \in \mathbb{Z}$ ,  $n>0$  and  $\gcd(m,n)=1$ . Then find  $m+n$ .
Please reason step by step, and put your final answer within \boxed{}.


Context:
I made a mistake in dividing a number by 4, and instead added by 4 to get 40. Find the correctly calculated value.
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Context:
Given a set of seven positive integers with the unique mode being 6 and the median being 4, find the minimum possible sum of these seven integers.
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Context:
We have a collection of identical spheres that we stack to form a tetrahedron where all edges have $n$ spheres. Calculate, as a function of $n$, the total number of points of tangency (contacts) between the spheres in the pile.
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Context:
Ed and Ann both have lemonade with their lunch. Ed orders the regular size. Ann gets the large lemonade, which is $50\%$ more than the regular. After both consume $\frac{3}{4}$ of their drinks, Ann gives Ed a third of what she has left, and $2$ additional ounces. When they finish their lemonades they realize that they both drank the same amount. How many ounces of lemonade did they drink together?
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Context:
A basketball team won 50 percent of its first 40 games in a particular season, 60 percent of the next 30 games, and 85 percent of its remaining games. If the team won a total of 62 percent of its games that season, what was the total number of games that the team played?
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Context:
Find the percentage of people with a grade of "excellent" among the selected individuals.
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Context:
在正三棱柱 $A B C-A_{1} B_{1} C_{1}$ 中, $A B=$ $B C=C A=A A_{1}=2$, 棱 $A_{1} C_{1}$ 上的点 $D$ 满足 $B C_{1} / /$平面 $A B_{1} \dot{D}$ 。以 $D$ 为球心、 $\frac{\sqrt{7}}{2}$ 为半径的球面与侧面 $A A_{1} B_{1} B$ 的交线长的$\frac{3}{\pi}$倍是 $\qquad$.
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Context:
For an odd number $n$, we define $n!! = n \cdot (n-2) \cdot (n-4) \cdots 3 \cdot 1$. Determine how many different residues modulo $1000$ are obtained from $n!!$ when $n = 1, 3, 5, \ldots$.
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Context:
A man buys an article for $100, but he got a 10% discount on the initial purchase. After buying, he pays a 5% sales tax on the discounted price. Later, he sells the article for $130 including a 15% sales tax. Find the actual gain percent.
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Context:
Last year a bicycle cost $160 and a cycling helmet $40. This year the cost of the bicycle increased by $5\%$, and the cost of the helmet increased by $10\%$. The percent increase in the combined cost of the bicycle and the helmet is:
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Context:
In triangle $\triangle ABC$, $\angle BAC = \frac{π}{3}$, $D$ is the midpoint of $AB$, $P$ is a point on segment $CD$, and satisfies $\overrightarrow{AP} = t\overrightarrow{AC} + \frac{1}{3}\overrightarrow{AB}$. If $|\overrightarrow{BC}| = \sqrt{6}$, then the maximum value of $|\overrightarrow{AP}|$ is ______.
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Context:
11. (12 points) Five students, Jia, Yi, Bing, Ding, and Wu, ranked in the top 5 (no ties) in a math competition and stood in a row for a photo. They each said the following:

Jia said: The two students next to me are both ranked behind me;
Yi said: The two students next to me are both ranked adjacent to me;
Bing said: All the students to my right (at least 1) are ranked ahead of me;
Ding said: All the students to my left (at least 1) are ranked behind me;
Wu said: I am standing in the second position from the right.

Given that they are all honest children, Jia, Yi, Bing, Ding, and Wu ranked $A, B, C, D, E$ respectively. What is the five-digit number $\overline{\mathrm{ABCDE}}$?
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Context:
For a positive integer $n$, let $f(n)$ be the number of (not necessarily distinct) primes in the prime factorization of $k$. For example, $f(1) = 0$, $f(2) = 1$, and $f(4) = f(6) = 2$. Let $g(n)$ be the number of positive integers $k \leq n$ such that $f(k) \geq f(j)$ for all $j \leq n$. Find $g(1) + g(2) + \ldots + g(100)$.
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Context:
Given the ellipse $C$: $\dfrac{x^{2}}{4} + \dfrac{y^{2}}{3} = 1$ with its right focus $F$, and $P$ is a moving point on the ellipse $C$, and a fixed point $A(2,4)$, then the minimum value of $|PA| - |PF|$ is ______.
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Context:
Let \( A = \{x \mid -2 \leq x \leq a\} \), \( B = \{y \mid y = 2x + 3, x \in A\} \), and \( C = \{z \mid z = x^2, x \in A\} \). If \( C \subseteq B \), find the range of the real number \( a \).
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Context:
A tractor is dragging a very long pipe on sleds. Gavrila walked along the entire length of the pipe at a constant speed in the direction of the tractor's movement and counted 210 steps. When he walked in the opposite direction at the same speed, the number of steps was 100. What is the length of the pipe, if Gavrila's step is 80 cm? Round your answer to the nearest whole meter. The speed of the tractor is constant.
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Context:
The graph of the function y = sinx - $\sqrt{3}$cosx can be obtained by shifting the graph of the function y = cosx + $\sqrt{3}$sinx at least \_\_\_\_\_\_ units to the left.
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Context:
Johann originally had 120 oranges. He decided to eat 25% of them. After eating, his friend Emily borrowed 1/3 of the remaining oranges but returned 10 later. Meanwhile, two other friends, Carson and Beth, stole 1/4 and 1/5 of the remaining oranges respectively. However, feeling guilty, Carson returned half of the stolen oranges and Beth returned 8. How many oranges does Johann have now?
Please reason step by step, and put your final answer within \boxed{}.


Context:
函数 $f(x) = \left|\log_{\frac{1}{2}} x\right|$ 的单调增区间为何?
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Context:
All three-digit numbers from 100 to 999 are written consecutively without spaces. Kostya underlined $k$ consecutive digits in this sequence, and Andrey underlined another $k$ consecutive digits in this sequence. The $k$-digit numbers underlined by the boys are equal. What is the largest possible value of $k$ for which this could happen?
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Context:
Given that $x$, $y$ are positive integers such that $x(x+1)$ divides $y(y+1)$, but neither $x$ nor $x+1$ divides either $y$ or $y+1$. Find the smallest possible value of $x^2 + y^2$.
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Context:
Three (not necessarily distinct) points in the plane which have integer coordinates between $ 1$ and $2020$, inclusive, are chosen uniformly at random. The probability that the area of the triangle with these three vertices is an integer is $a/b$ in lowest terms. If the three points are collinear, the area of the degenerate triangle is $0$. Find $a + b$.
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Context:
Evaluate the $\int \int_SF* dS$ for the given vector field F and the oriented surface S. for closed surfaces, use the positive (outward) orientation. F(x,y,z) = xi +yj +5k. S is the boundary of the region enclosed by the cylinder $x^2 + z^2$ = 1 and the planes y = 0 and x + y = 2.
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Context:
In a granary, there are 3318 stones of grain. A sample of grain is taken, which contains 168 grains, and 22 of them are imperfect. Using this information, calculate the approximate amount of imperfect grains in the granary. Express your answer in stones, rounded to the nearest whole number.
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Context:
Call a set of integers spacy if it contains no more than one out of any three consecutive integers. How many subsets of $\{1,2,3,\cdots,12\},$ including the empty set, are spacy?
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Context:
The area of the region bounded by the graph of\[x^2+y^2 = 3|x-y| + 3|x+y|\]is $m+n\pi$, where $m$ and $n$ are integers. What is $m + n$?
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Context:
The sequence $\{a\_n\}$ satisfies: $a\_1= \frac {1}{2}$, $a\_{n+1}= \frac {1+a\_n}{1-a\_n} (n\geqslant 1)$, find $a\_{2016}=$ \_\_\_\_\_\_.
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Context:
From 49 balls numbered 1 to 49, select some balls and arrange them in a circle so that the product of the numbers on any two adjacent balls is less than 100. What is the maximum number of balls that can be selected?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the function $f(x)=\cos(\omega x+\varphi)$, where $\omega \in N$, $\varphi \in [0,\pi]$, there are exactly two extreme points in the interval $x \in \left(-\frac{2}{3}\pi,\frac{1}{3}\pi\right)$, and $f\left(-\frac{2}{3}\pi\right)+f\left(\frac{1}{3}\pi\right)=0$. Find the set of all possible values of $\varphi$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In measuring the sides of a rectangle, one side is taken 12% in excess and the other 5% in deficit. What is the error percentage in the area calculated from these measurements?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In triangle \(ABC\), points \(M\), \(N\), and \(P\) are chosen such that \(M\) and \(N\) lie on sides \(AC\) and \(BC\) respectively, and \(P\) lies on segment \(MN\). The following conditions are satisfied:

\[
\frac{|AM|}{|MC|} = \frac{|CN|}{|NB|} = \frac{|MP|}{|PN|}
\]

Find the area of triangle \(ABC\) given that the areas of triangles \(AMP\) and \(BNP\) are \(T\) and \(Q\) respectively.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Write down an irrational number $x$ such that $1 \lt x \lt 4$. $x$ can be ____ (just write down one $x$ that satisfies the condition).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of integers $n$ between 1500 and 1515 such that $5$ divides $\left\lfloor\frac{\sqrt{3}}{2}(1+\sqrt{3})^n\right\rfloor$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the right trapezoid \(ABCD\) (with \(BC\) parallel to \(AD\) and \(AB\) perpendicular to \(AD\)), the shorter base \(AD\) is equal to 3, and the leg \(CD\) is equal to 6. Point \(E\), the midpoint of side \(CD\), is connected by a line segment to point \(B\).

It is known that the angle \(CBE\) is \(\alpha\). Find the area of trapezoid \(ABCD\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given Liam has written one integer three times and another integer four times. The sum of these seven numbers is 131, and one of the numbers is 17, determine the value of the other number.
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Context:
We make colored cubes according to the following specifications:

1. Each face of the cube is divided into two triangles by a diagonal, such that every drawn diagonal connects with two other such diagonals at each end point.
2. Each face's two triangles must be of different colors.
3. Triangles that are adjacent along an edge must be of the same color.

How many different cubes can be made using 6 colors? (Two cubes are not considered different if they can be positioned so that their painted faces look the same from any viewpoint.)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the area bounded by the graph of \( y = \arcsin(\cos x) \) and the \( x \)-axis on the interval \( \left[0, 2\pi\right] \).
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Context:
The distance between every two telephone poles by the roadside is 50 meters. Xiao Wang is traveling at a constant speed by car. He counted 41 telephone poles within 2 minutes after seeing the first pole. How many meters per hour is the car traveling?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Solve the fractional equation application problem. A maglev train is a train that is propelled by magnetic levitation force. The construction of maglev trains is a brilliant leap in the history of China's transportation development. The distance between station $A$ and station $B$ is $30$ km. The average speed of the maglev train running between them is $6.25$ times the average speed of the subway. Moreover, the time taken to ride the maglev train is $\frac{7}{10}$ hours less than the time taken to ride the subway. Find the average speed of the maglev train.
Please reason step by step, and put your final answer within \boxed{}.


Context:
There are three piles of peaches with the following counts: $93, 70, 63$. A monkey moves peaches among the 3 piles. It is known that the monkey can move at most 5 peaches at a time, and will eat 1 peach while carrying it from one pile to another. When the number of peaches in the 3 piles is equal, how many peaches will the monkey have eaten at least?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the trapezoid \(ABCD\), the bases \(AD\) and \(BC\) are 8 and 18 respectively. It is known that the circumcircle of triangle \(ABD\) is tangent to the lines \(BC\) and \(CD\). Find the perimeter of the trapezoid.
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Context:
Haman's father has an egg business supplying the local market. On a Wednesday morning, his father sends him to go and collect 10 trays of eggs for sale from their store. While loading the eggs into the car, he accidentally drops some trays. He calls his father telling him this, and is told to add 7 more trays for sale. They sold 540 eggs that day. How many trays did Haman accidentally drop?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that the ellipse $\frac{x^{2}}{a} + \frac{y^{2}}{b} = 1 (a > b > 0)$ and the hyperbola $\frac{x^{2}}{m} - \frac{y^{2}}{n} = 1 (m, n > 0)$ share the same foci $F\_1$, $F\_2$, and $P$ is the point of intersection of the two curves, find the value of $|PF\_1| \cdot |PF\_2|$. Express your answer as a single algebraic expression in terms of $a$ and $m$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the polar coordinates of the center of the circle given by the equation $\rho = \cos\left(\theta + \frac{\pi}{3}\right)$. Express your answer in polar coordinates as $(\rho, \theta)$.
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Context:
Michael picks a random subset of the complex numbers $\left\{1, \omega, \omega^{2}, \ldots, \omega^{2017}\right\}$ where $\omega$ is a primitive $2018^{\text {th }}$ root of unity and all subsets are equally likely to be chosen. If the sum of the elements in his subset is $S$, what is the expected value of $|S|^{2}$? (The sum of the elements of the empty set is 0.)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let the function $f(n)=k$ where $n\in\mathbb{N}^*$ and $k$ is the $n$-th digit after the decimal point of $\pi$, $\pi=3.141\ 592\ 653\ 5\ldots$, then $\{f\ldots f[f(10)]\}=$ ______.
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Context:
Let $\{a_{n}\}$ be an integer sequence such that for any $n \in \mathbf{N}^{*}$, the condition \((n-1) a_{n+1} = (n+1) a_{n} - 2 (n-1)\) holds. Additionally, \(2008 \mid a_{2007}\). Find the smallest positive integer \(n \geqslant 2\) such that \(2008 \mid a_{n}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the integer \( n \), \( -150 < n < 150 \), such that \( \tan n^\circ = \tan 1340^\circ \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Tommy's mother lets him ride his bike 2 blocks north of his block, 3 blocks east, 2 blocks west, and 2 blocks south. His friend's father lets his friend ride his bike in a certain area. The area his friend can ride in is 4 times greater than Tommy's. How many square blocks can his friend ride his bike in?
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Context:
Define a sequence recursively by $t_1 = 20$, $t_2 = 21$, and\[t_n = \frac{5t_{n-1}+1}{25t_{n-2}}\]for all $n \ge 3$. Then $t_{2020}$ can be expressed as $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.
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Context:
At 8:00 AM, Xiao Cheng and Xiao Chen set off from locations A and B respectively, heading towards each other. They meet on the way at 9:40 AM. Xiao Cheng says: "If I had walked 10 km more per hour, we would have met 10 minutes earlier." Xiao Chen says: "If I had set off half an hour earlier, we would have met 20 minutes earlier." If both of their statements are correct, how far apart are locations A and B? (Answer in kilometers).
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Context:
How many non-singular $3 \times 3$ matrices exist with elements either $1$ or $-1$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given \( x, y, z \in \mathbb{Z}_{+} \) with \( x \leq y \leq z \), how many sets of solutions satisfy the equation \(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=\frac{1}{2}\) ?
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Context:
在平面直角坐标系中, 双曲线 $\Gamma: x^{2}-y^{2}=1$ 的右顶点为 $A$.将圆心在 $y$ 轴上，且与 $\Gamma$ 的两支各恰有一个公共点的圆称为 "好圆".若两个好圆外切于点 $P$ ，圆心距为 $d$ ，求 $\frac{d}{|P A|}$ 的所有可能的值.原始的答案是k\sqrt{m}+n的形式，请给出k + m + n的值。
Please reason step by step, and put your final answer within \boxed{}.


Context:
6. How many pairs of natural numbers $(a ; b)$ exist such that the number $5 a-3$ is divisible by $b$, and the number $5 b-1$ is divisible by $a$? Enter the number of such pairs of numbers in the provided field.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a parabola $x^2=4y$ with focus $F$ and the point $A(-1, 8)$, if $P$ is a point on the parabola, then the minimum value of $|PA| + |PF|$ is \_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.


Context:
There are a certain number of identical plastic bags that can be nested within each other. If all the other bags are inside one of the bags, we call this situation a "bag of bags." Calculate the number of ways to make a "bag of bags" from 10 bags.

Explanation: Use parentheses to denote a bag.

If we had one bag, the way to make a "bag of bags" would be: (). Two bags can be nested in only one way: (()).

Three bags can be nested in two different ways: $(()())$ and $((()))$, and so on.

The order of bags inside the bag matters. For example, the arrangement $(())())$ is different from $(()(()))$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A brand of orange juice is available in shop $A$ and shop $B$ at an original price of $\$2.00$ per bottle. Shop $A$ provides a "buy 4 get 1 free" promotion and shop $B$ provides a $15\%$ discount if one buys 4 bottles or more. Find the minimum cost (in cents) if one wants to buy 13 bottles of the orange juice.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The intersection of two squares with a perimeter of $8$ is a rectangle with a diagonal length of $1$. Given that the distance between the centers of the two squares is $2$, the perimeter of the rectangle can be expressed as $P$. Find $10P$.
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Context:
Find the limit of the following integral as $n$ approaches infinity:

$$ \lim_{n\to\infty} \int_0^1 \dfrac{n x^p+x^q}{x^p+n x^q} dx$$

where $q<p+1$.
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Context:
Consider the grid of points $X = \{(m,n) \mid 0 \leq m,n \leq 4 \}$. We say a pair of points $\{(a,b),(c,d)\}$ in $X$ is a knight-move pair if $(c = a \pm 2$ and $d = b \pm 1)$ or $(c = a \pm 1$ and $d = b \pm 2)$. The number of knight-move pairs in $X$ is:
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the gradient of the function $$f(A)=Trace\left[\left(AB\right)^{T}Q\left(AB\right)\right]$$ where $Q = Q^T>0$ with respect to $A$.
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Context:
Javier is excited to visit Disneyland during spring break. He plans on visiting five different attractions, but he is particularly excited about the Space Mountain ride and wants to visit it twice during his tour before lunch. How many different sequences can he arrange his visits to these attractions, considering his double visit to Space Mountain?
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Context:
Let $Q$ be a quadratic polynomial. If the sum of the roots of $Q^{100}(x)$ (where $Q^i(x)$ is defined by $Q^1(x)=Q(x)$, $Q^i(x)=Q(Q^{i-1}(x))$ for integers $i \geq 2$) is $8$ and the sum of the roots of $Q$ is $S$, compute $|\log_2(S)|$.
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Context:
Alice and Mary were searching attic and found scale and box with weights. When they sorted weights by mass, they found out there exist  $5$  different groups of weights. Playing with the scale and weights, they discovered that if they put any two weights on the left side of scale, they can find other two weights and put on to the right side of scale so scale is in balance. Find the minimal number of weights in the box
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Context:
Given proposition $p$: The equation $x^{2}-ax+4=0$ has real roots; proposition $q$: The function $y=2x^{2}+ax+4$ is increasing on the interval $(3,+\infty)$. If "proposition $p$ or proposition $q$" is true, and "proposition $p$ and proposition $q$" is false, find the range of the real number $a$.
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Context:
Consider a modified octahedron with an additional ring of vertices. There are 4 vertices on the top ring, 8 on the middle ring, and 4 on the bottom ring. An ant starts at the highest top vertex and walks down to one of four vertices on the next level down (the middle ring). From there, without returning to the previous vertex, the ant selects one of the 3 adjacent vertices (excluding the one it came from) and continues to the next level. What is the probability that the ant reaches the bottom central vertex?
Please reason step by step, and put your final answer within \boxed{}.


Context:
It is known that from these five segments it is possible to form four different right triangles. Find the ratio of the largest segment to the smallest.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the greatest value of the positive integer $x$, such that the number \( A = 2^{182} + 4^x + 8^{700} \) is a perfect square.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Two beads, each of mass  $m$ , are free to slide on a rigid, vertical hoop of mass  $m_h$ . The beads are threaded on the hoop so that they cannot fall off of the hoop. They are released with negligible velocity at the top of the hoop and slide down to the bottom in opposite directions. The hoop remains vertical at all times. What is the maximum value of the ratio  $m/m_h$  such that the hoop always remains in contact with the ground? Neglect friction.
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filldraw(circle((1.6,1.99499),0.1), gray(.3));
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Please reason step by step, and put your final answer within \boxed{}.


Context:
Find every real values that  $a$  can assume such that  $$ \begin{cases}
x^3 + y^2 + z^2 = a
x^2 + y^3 + z^2 = a
x^2 + y^2 + z^3 = a
\end{cases} $$  has a solution with  $x, y, z$  distinct real numbers.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a biased coin with probabilities of $\frac{3}{4}$ for heads and $\frac{1}{4}$ for tails, determine the difference between the probability of winning Game A, which involves 4 coin tosses and at least three heads, and the probability of winning Game B, which involves 5 coin tosses with the first two tosses and the last two tosses the same.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The function $y=2\cdot a^{|x-1|}-1 (a > 0, a \neq 1)$ passes through the fixed point $\boxed{\text{answer}}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the function $$ F(x) = \int_x^{\sin x} \left( \int_0^{\sin t} \sqrt{1+u^4} \, du \right) dt, $$ find the second derivative $F''(\pi)$ using the fundamental theorem of calculus.
Please reason step by step, and put your final answer within \boxed{}.


Context:
From the numbers 1, 2, 3,..., 7, select several numbers such that the sum of the even numbers is equal to the sum of the odd numbers. How many combinations satisfy this condition? Express your answer as a single integer.
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Context:
The number $2013$ is expressed in the form   $2013 = \frac {a_1!a_2!...a_m!}{b_1!b_2!...b_n!}$,where $a_1 \ge a_2 \ge \cdots \ge a_m$ and $b_1 \ge b_2 \ge \cdots \ge b_n$ are positive integers and $a_1 + b_1$ is as small as possible. What is $|a_1 - b_1|$?
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Context:
For each integer $n$ greater than 1, let $F(n)$ be the number of solutions of the equation $\sin x = \sin nx$ on the interval $[0, \pi]$.  What is $\sum_{n=2}^{2007} F(n)$?
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Context:
If the fractional equation in terms of $x$, $\frac{{x+5}}{{x-3}}=2-\frac{m}{{3-x}}$, has a positive root, then the value of the constant $m$ is ______.
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Context:
Consider 9 points in the plane with no alignment. What is the minimum value of $n$ such that if we color $n$ edges connecting two of the points in red or blue, we are sure to have a monochromatic triangle regardless of the coloring?

(IMO 1992)
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Context:
What is the largest possible number of subsets of the set \(\{1, 2, \ldots, 2n + 1\}\) such that the intersection of any two subsets consists of one or several consecutive integers?
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Context:
Compute the number of positive integers that divide at least two of the integers in the set $\{1^1, 2^2, 3^3, 4^4, 5^5, 6^6, 7^7, 8^8, 9^9, 10^{10}\}$.
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Context:
Call a \(2n\)-digit base-10 number special if we can split its digits into two sets of size \(n\) such that the sum of the numbers in the two sets is the same. Let \(p_{n}\) be the probability that a randomly-chosen \(2n\)-digit number is special. (We allow leading zeros in \(2n\)-digit numbers). The sequence \(p_{n}\) converges to a constant \(c\). Find \(c\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the definite integral:

$$
\int_{\pi / 2}^{\pi} 2^{4} \cdot \sin ^{6} x \cos ^{2} x \, dx
$$
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Context:
Find the minimum number of terms required in the Maclaurin series expansion of $f(x) = \log \left(\frac{1+x}{1-x}\right)$ to approximate the value of $\log(2)$ with six exact decimal places when $x = \frac{1}{3}$. Note that $\log$ represents the natural logarithm ($\ln$).
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Context:
Zhendarov R.G.

Cells of a $9 \times 9$ board are painted in a checkerboard pattern with black and white colors (corner cells are white). What is the minimum number of rooks needed to be placed on this board so that all white cells are under attack by these rooks? (A rook attacks all cells in the row and column where it is placed.)
Please reason step by step, and put your final answer within \boxed{}.


Context:
3. [30] Let $M$ be a $2014 \times 2014$ invertible matrix, and let $\mathcal{F}(M)$ denote the set of matrices whose rows are a permutation of the rows of $M$. Find the number of matrices $F \in \mathcal{F}(M)$ such that $\operatorname{det}(M+F) \neq 0$.
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Context:
Let \(ABCD\) be a convex quadrilateral such that \(AB + BC = 2021\) and \(AD = CD\). We are also given that \(\angle ABC = \angle CDA = 90^\circ\). Determine the length of the diagonal \(BD\).
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Context:
Given that \( \boldsymbol{m} \) is a non-zero vector, \( \boldsymbol{n} \) is a unit vector, \( \boldsymbol{m} \neq \boldsymbol{n} \), and the angle between \( \boldsymbol{m} \) and \( \boldsymbol{m} - \boldsymbol{n} \) is \( 60^{\circ} \), with \( |\boldsymbol{m}| \in (0, a] \), find the minimum value of \( a \).
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Context:
在正方体的每个面上随机填入 1,2,\cdots,6 中的一个数字（各个面上的数字互不相同）. 再对八个顶点进行编号，使得每个顶点的编号数为其相邻三个面上的数字之积. 则八个顶点的编号数之和的最大值为__________.
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Context:
In the isosceles right triangle \(AOB\), points \(P, Q, R,\) and \(S\) are chosen on the sides \(OB, OA,\) and \(AB\) respectively, such that \(PQRS\) forms a square. If the lengths of \(OP\) and \(OQ\) are \(a\) and \(b\) respectively, and the area of the square \(PQRS\) is \(\frac{2}{5}\) of the area of the triangle \(AOB\), determine the value of \(\frac{a}{b}\).
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Context:
People were sitting in a circle. The 7th one is directly opposite to the 18th one. How many people were there in that group?
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Context:
Given the planar vectors $\overrightarrow {e_{1}}$ and $\overrightarrow {e_{2}}$ that satisfy $|\overrightarrow {e_{1}}| = |3\overrightarrow {e_{1}} + \overrightarrow {e_{2}}| = 2$, determine the maximum value of the projection of $\overrightarrow {e_{1}}$ onto $\overrightarrow {e_{2}}$.
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Context:
The power function $f(x) = (2m^2 + m)x^m$ is monotonically increasing on the interval $[0, +\infty)$. Determine the value of $m$.
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Context:
Define the minimum real $C$ such that for any real numbers $0 = a_0 < a_1 < \dots < a_{1000}$, the inequality $$\min_{0 \le k \le 1000} (a_k^2 + (1000-k)^2) \le C(a_1 + \dots + a_{1000})$$ holds. Find $\lfloor 100C \rfloor$.
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Context:
Given $F$ as the focus of the parabola $y^2 = 3x$, $P$ as any point on the parabola, and $A(3,2)$ as a fixed point on the plane, the minimum value of $|PF|+|PA|$ is __________.
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Context:
已知$x _ { i } \in R ^ { + } ( i = , 2 , \cdots , n ) , \sum _ { i = } ^ { n } x _ { i } = m , p , q \in R ^ { + } , 2 \leq k \in N ^ { + }$.求函数$f ( x ) = \sum _ { i = 1 } ^ { n } \sqrt [ k ] { p x _ { i } + q }$的最大值.
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Context:
A circle with a radius of 3 is centered at the midpoint of one side of an equilateral triangle each side of which has a length of 9. Determine the difference between the area inside the circle but outside the triangle and the area inside the triangle but outside the circle.
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Context:
In the arithmetic sequence {a_n}, it is known that a_2 - a_3 - a_7 - a_11 - a_13 + a_16 = 8. Find the value of a_9. Express your answer as a single integer.
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Context:
Kobar and Borah are playing a game on a whiteboard with the following rules: They start with two distinct positive integers on the board. On each step, beginning with Kobar, each player takes turns changing the numbers on the board as follows:

1. From $P$ and $Q$ to $2P-Q$ and $2Q-P$, or
2. From $P$ and $Q$ to $5P-4Q$ and $5Q-4P$.

The game ends if a player writes an integer that is not positive. That player is declared to lose, and the opponent is declared the winner.

At the beginning of the game, the two numbers on the board are $2024$ and $A$. It is known that Kobar does not lose on his first move. Determine the largest possible value of $A$ so that Borah can win this game.
Please reason step by step, and put your final answer within \boxed{}.


Context:
There are 456 natives on an island, each of whom is either a knight who always tells the truth or a liar who always lies. All residents have different heights. Once, each native said, "All other residents are shorter than me!" What is the maximum number of natives who could have then said one minute later, "All other residents are taller than me?"
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Context:
In a $100 \times 100$ grid, the numbers $1, 2, \ldots, 10000$ are placed, each appearing exactly once, such that numbers differing by 1 are located in adjacent cells. After this, the distances between the centers of all pairs of cells whose numbers differ by exactly 5000 are calculated. Let $S$ be the minimum of these distances. What is the maximum possible value of $S$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
5. In a row, all natural numbers from 1 to 100 inclusive are written in ascending order. Under each number in this row, the product of its digits is written. The same procedure is applied to the resulting row, and so on. How many odd numbers will be in the fifth row?
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Context:
$A$ 是 10 阶实矩阵, 且 $A^{5}=O$, 则 $\operatorname{rank}(A)$ 的所有可能值之和为 $\qquad$.
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Context:
A regular football match is being played. A draw is possible. The waiting time for the next goal is independent of previous events in the match. It is known that the expected total number of goals in football matches of these teams is 2.8. Find the probability that an even number of goals will be scored during the match.
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Context:
On a table, there are five clocks with hands. It is allowed to move any number of them forward. For each clock, the time by which it is moved will be referred to as the translation time. It is required to set all clocks such that they show the same time. What is the minimum total translation time needed to guarantee this?
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Context:
$2020$ positive integers are written in one line. Each of them starting with the third is divisible by previous and by the sum of two previous numbers. What is the smallest value the last number can take?

A. Gribalko
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Context:
Circle \(O\) has chord \(AB\). A smaller circle is tangent to circle \(O\) at \(T\) and tangent to \(AB\) at \(X\) such that \(AX = 2XB\). What is \(\frac{AT}{BT}\)?
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Context:
The eight vertices of a cube are randomly labelled with the integers from 1 to 8 inclusive. Judith looks at the labels of the four vertices of one of the faces of the cube. She lists these four labels in increasing order. After doing this for all six faces, she gets the following six lists: (1,2,5,8),(3,4,6,7),(2,4,5,7),(1,3,6,8),(2,3,7,8), and (1,4,5,6). What is the label of the vertex of the cube that is farthest away from the vertex labelled 2? Provide your answer as a single integer.
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Context:
Given that the real numbers \( a \) and \( b \) satisfy that both quadratic trinomials \( x^2 + ax + b \) and \( x^2 + bx + a \) have two distinct real roots each, and the product of their roots results in exactly three distinct real roots. Find all possible values of the sum of these three distinct roots.
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Context:
For a positive integer $n$ not divisible by $211$, let $f(n)$ denote the smallest positive integer $k$ such that $n^k - 1$ is divisible by $211$. Find the remainder when $\sum_{n=1}^{210} nf(n)$ is divided by $211$.
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Context:
Given the parabola \( y^2 = 2px \) (where \( p > 0 \)), two mutually perpendicular chords \( FA \) and \( FB \) pass through the focus \( F \). Find the minimum area of \( \triangle FAB \).
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Context:
There are 6 positive integers \( a, b, c, d, e, f \) arranged in order, forming a sequence, where \( a=1 \). If a positive integer is greater than 1, then the number that is one less than it must appear to its left. For example, if \( d > 1 \), then one of \( a, b, \) or \( c \) must have the value \( d-1 \). For instance, the sequence \( 1,1,2,1,3,2 \) satisfies the requirement; \( 1,2,3,1,4,1 \) satisfies the requirement; \( 1,2,2,4,3,2 \) does not satisfy the requirement. How many different permutations satisfy the requirement?
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Context:
Three pirates divided the diamonds they collected during the day: Bill and Sam got twelve each, and the rest went to John, who did not know how to count. At night, Bill stole one diamond from Sam, Sam stole one diamond from John, and John stole one diamond from Bill. As a result, the average mass of Bill's diamonds decreased by one carat, the average mass of Sam's diamonds decreased by two carats, but the average mass of John's diamonds increased by four carats. How many diamonds did John receive?
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Context:
Sandy walked some distance towards south. Then Sandy turned to her left and walked the same distance. She then turned to her left and walked the same distance. She then turned to her right and walked the same distance. Sandy is 40 meters away from the starting point in the north direction. How far did she walk in each segment?
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Context:
正整数 $9+95+995+\cdots+\underbrace{99 \cdots 9}_{2016} 5$ 的十进制表示中数值 $1$ 的个数为多少？
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Context:
Given that the line $y=ax+1$ intersects the hyperbola $3x^2-y^2=1$ at points A and B. If the circle with diameter AB passes through the origin, find the value of $a$.
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Context:
Let $f(x) = \cos(\omega x + \varphi)$ (where $\omega > 0$, $|\varphi| < \frac{π}{2}$). If the distance between the axis of symmetry $x = \frac{π}{6}$ of the graph of the function $y = f(x)$ and its center of symmetry is $\frac{π}{8}$, then $f(x) = \_\_\_\_\_\_$.
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Context:
Five years ago, the price of a movie ticket was $100. In the first year, the price increased by 12%. In the second year, the price decreased by 5%. In the third year, the price increased by 8%. In the fourth year, the price decreased by 4%. In the fifth and final year, the price increased by 6%. After all these changes, the local government introduced a 7% entertainment tax on the movie ticket price, and a particular movie theater offered a 10% special discount on the ticket's price. What is the final price of the movie ticket after incorporating taxes and discounts?
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Context:
Given a real number $a$ satisfying $a^2 - a - 1 = 0$. Find the value of $a^8 + 7a^{-4}$.
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Context:
Marisa has two identical cubical dice labeled with the numbers  $\{1, 2, 3, 4, 5, 6\}$ . However, the two dice are not fair, meaning that they can land on each face with different probability. Marisa rolls the two dice and calculates their sum. Given that the sum is  $2$  with probability  $0.04$ , and  $12$  with probability  $0.01$ , the maximum possible probability of the sum being  $7$  is  $p$ . Compute  $\lfloor 100p \rfloor$ .
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Context:
A positive integer has exactly 8 divisors. The sum of its smallest 3 divisors is 15. This four-digit number has a prime factor such that the prime factor minus 5 times another prime factor equals twice the third prime factor. What is this number?
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Context:
Given the power function f(x) = (m^2 - 2m - 2)x^m is monotonically decreasing on (0, +∞), find the value of m.
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Context:
Determine the smallest natural number written in the decimal system such that the product of its digits equals $10! = 1 \times 2 \times 3 \times \ldots \times 9 \times 10$.
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Context:
Find the largest natural number in which all digits are different, and the sum of any two of its digits is a prime number.
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Context:
Calculate: $|-5|-202{3}^{0}+tan45°+\sqrt{9}$.
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Context:
A collection of seven positive integers has a mean of 6, a unique mode of 4, and a median of 6. If a 12 is added to this collection, what is the new median?
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Context:
On a circle, there are 2018 points. Each of these points is labeled with an integer. Each number is greater than the sum of the two numbers immediately preceding it in a clockwise direction.

Determine the maximum possible number of positive numbers among these 2018 numbers.

(Walther Janous)
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Context:
The U.S. produces about 8 million tons of apples each year. Initially, $30\%$ of the apples are mixed with other products. If the production increases by 1 million tons, the percentage mixed with other products increases by $5\%$ for each additional million tons. Of the remaining apples, $60\%$ is used to make apple juice and $40\%$ is sold fresh. Calculate how many million tons of apples are sold fresh.
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Context:
Compute the following limit:
$$ \lim _{n \to \infty} \frac{1}{n}\sum_{k=0}^{n-1}\left[\frac{k+1}{n}-\frac{1}{2}\right]\log(2n-2k-1) .$$
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Context:
Consider a scenario where a random variable $X$ represents the number of failures in a sequence of Bernoulli trials with success probability $p$ until the first success occurs. $X$ follows a geometric distribution with parameter $p$: $X \sim Geo(p)$. Now, in this process, after each failure, we toss two fair coins and record the number of heads and tails. When the first success occurs, we check if the number of heads equals the number of tails. Let $B$ be a binomial random variable representing the count of heads out of $2X$ coin tosses with probability $\frac{1}{2}$: $B \sim Bin(2X, \frac{1}{2})$. Determine the probability $P(B = X)$.
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Context:
已知四面体 $ABCD$ 内接于球 $O$，且 $AD$ 为球 $O$ 的直径。若$\triangle A B C$与$\triangle B C D$均是边长为 1 的等边三角形，求四面体 $ABCD$ 的体积。答案的形式为$\frac{\sqrt{k}}{m}$，请给出k+m的值。
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Context:
已知点$M = \{ ( x , y ) \vert \sqrt { 1 - x ^ { 2 } } \cdot \sqrt { 1 - y ^ { 2 } }$

≥xy}，则平面直角坐标系中点集M覆盖区域的面积为 ·
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Context:
If $9:y^3 = y:81$, what is the value of $y$?
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Context:
For any set \( S \) of five points in the plane, no three of which are collinear, let \( M(S) \) and \( m(S) \) denote the greatest and smallest areas, respectively, of triangles determined by three points from \( S \). What is the minimum possible value of \( \frac{M(S)}{m(S)} \)?
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Context:
In each cell of a chessboard, there is a knight. What is the minimum number of knights that should be removed from the board so that there is not a single knight left on the board that can attack exactly four other knights? (A knight attacks cells that are one square away horizontally and two squares vertically, or vice versa.)
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Context:
An island has $10$ cities, where some of the possible pairs of cities are connected by roads. A [i]tour route[/i] is a route starting from a city, passing exactly eight out of the other nine cities exactly once each, and returning to the starting city. (In other words, it is a loop that passes only nine cities instead of all ten cities.) For each city, there exists a tour route that doesn't pass the given city. Find the minimum number of roads on the island.
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Context:
Four students from a class are assigned to serve in three different pavilions, A, B, and C, with at least one person assigned to each pavilion. If student A requests not to be assigned to pavilion A, calculate the number of different assignment plans. Express your answer as a single integer.
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Context:
Assuming a Poisson process with a rate of 2.5 mistakes per year, calculate the probability that an individual dies before reaching the age of 67.2 years, given that death occurs after 196 mistakes. Use the gamma distribution with parameters 196 and 2.5 to find this probability.
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Context:
Evaluate the following limit using advanced techniques or alternative approaches, without necessarily finding the value of the integral directly:
$$\lim_{p\to0} \left[p \int_{-\infty}^\infty \frac{\ln^2(1+x)}{(p^2 +x^2)^2} \, \mathrm dx\right]$$
Attempts to simplify the integral by manipulating the small parameter $p$ and applying suitable substitutions have been unsuccessful.
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Context:
10. A rectangle $24 \times 60$ is divided by lines parallel to its sides into unit squares. Into how many parts will this rectangle be divided if we also draw its diagonal?
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Context:
In the sequence \(\left\{a_{n}\right\}\),
\[
a_{0}=a \quad (a \in \mathbf{Z}_{+}), \quad a_{n+1}=\frac{a_{n}^{2}}{a_{n}+1}.
\]

Find the largest integer not exceeding \(a_{n}\) when \(0 \leqslant n \leqslant \frac{a}{2}+1\).
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Context:
In a triangle, if two sides have lengths of $3$ and $5$, then the length of the third side can be _______. (Fill in only one number)
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Context:
Using a set of student triangles with internal angles (45°, 45°, 90° and 30°, 60°, 90°), find the number of different angles that can be drawn greater than 0° and less than 176°. Express your answer as a whole number.
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Context:
Determine the largest positive integer \( n \) such that for any positive real numbers \( a_1, a_2, \ldots, a_n \) with \( \prod_{i=1}^{n} a_i = 1 \), the inequality \[ \sum_{i=1}^n \frac{a_i}{a_{i+1}} + \sum_{i=1}^n \frac{a_{i+1}}{a_i} \geq 2 \sum_{i=1}^n a_i \] holds, where \( a_{n+1} = a_1 \).
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Context:
Minimize the function \( f(x,y,z,w) = 2x^2 + 3y^2 + 3z^2 + w^2 - 2xy + xz \) subject to the constraints:

\[ \begin{cases} x + 2y - z = 7 \\ x - z + w = 6 \end{cases} \]
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Context:
Find the coefficient of degree \(-2\) in the differential form \(\frac{df}{f}\) for the formal Laurent power series \(f=\sum_{i\ge m} a^i t^i\), where \(m\in \mathbb{Z}\).
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Context:
Determine the total number of matrices that are similar to the matrix \(\begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}\) over the finite field \(\mathbb{Z}_7\).
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Context:
A and B are taking turns shooting with a six-shot revolver that has only one bullet. They randomly spin the cylinder before each shot. A starts the game. Find the probability that the gun will fire while A is holding it.
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Context:
If $x$ and $y$ are real numbers, find the minimum value of $\sqrt{4+y^{2}}+\sqrt{x^{2}+y^{2}-4 x-4 y+8}+\sqrt{x^{2}-8 x+17}$.
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Context:
函数 $f(x)=a^{2 x}+3 a^{x}-2(a>0, a \neq 1)$ 在区间 $x \in[-1,1]$ 上的最大值为 8, 则它在这个区间上的最小值是。原始答案为 $-\frac{m}{n}$ 的形式，请给出 m + n 的值。
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Context:
Consider all the positive integers $N$ with the property that all of the divisors of $N$ can be written as $p-2$ for some prime number $p$. Then, there exists an integer $m$ such that $m$ is the maximum possible number of divisors of all numbers $N$ with such property. Find the sum of all possible values of $N$ such that $N$ has $m$ divisors.
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Context:
Let $n$ be a positive integer, and let $V_n$ be the set of integer $(2n+1)$-tuples $\mathbf{v}=(s_0,s_1,\cdots,s_{2n-1},s_{2n})$ for which $s_{0}=s_{2n}=0$ and $|s_j-s_{j-1}|=1$ for $j=1,2,\cdots,2n.$ Define $$q(\mathbf{v})=1+\sum_{j=1}^{2n-1}3^{s_{j}},$$ and let $M(n)$ be the average of $\frac{1}{q(\mathbf{v})}$ over all $\mathbf{v}\in V_n.$ Evaluate $M(2020). The original answer is in \frac{m}{n} format, please give the value of m + n.
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Context:
已知幂函数$f(x) = (m - 1 )^2  x^{m^2 - 4m + 2}$在（0，+∞）上单调递减，则$\textit{m}$=______．
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Context:
Let $P(x)=x^{2}-1$. How many distinct real solutions does the following equation have:
$$
\underbrace{P(P(\ldots(P}_{2004}(x)) \ldots))=0? \quad[20 \text { points }]
$$
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Context:
Let $\mathcal{S} = \sum_{i=1}^{\infty}\left(\prod_{j=1}^i \dfrac{3j - 2}{12j}\right).$ Then $(\mathcal{S} + 1)^3 = \frac{m}{n}$ with $m$ and $n$ coprime positive integers. Find $10m + n$.
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Context:
Margot wants to prepare a jogging timetable. She wants to jog exactly twice a week, and on the same days every week. She does not want to jog on two consecutive days. How many different timetables could Margot prepare?
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Context:
40 kids are running a race. 10% of them pass the finish line in less than 6 minutes. A certain multiple of that number finish in less than 8 minutes. 1/6 of the remaining kids take more than 14 minutes, and 4 kids take more than 14 minutes. What is the multiple of the number of kids who finish in less than 6 minutes that finish in less than 8 minutes?
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Context:
Task 5. In class 5A, a survey was conducted on what fruits the students like. It turned out that 13 students like apples, 11 like plums, 15 like peaches, and 6 like melons. A student can like more than one fruit. Every student who likes plums also likes either apples or peaches (but not both at the same time). And every student who likes peaches also likes either plums or melons (but not both at the same time). What is the minimum number of people that can be in $5 A$? (20 points)
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Context:
21.2.10 ** Divide a circle with a circumference of 24 into 24 equal segments. Choose 8 points from the 24 points such that the arc length between any two points is not equal to 3 or 8. How many different ways are there to select such a group of 8 points?
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Context:
设正三棱雉 $P-A B C$ 的底面边长为 1, 高为 $\sqrt{2}$, 过底边 $B C$ 作此三棱雉的截面，则截面面积的最小值为$\frac{m \sqrt{n}}{k}$的形式，请给出$m + n + k$的值。
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Context:
The Razorback shop makes $192 dollars off each t-shirt and a certain amount off each jersey. During the Arkansas and Texas tech game, they sold 157 t-shirts and 19 jerseys. A t-shirt costs $158 more than a jersey. How much does the shop make off each jersey?
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Context:
在平面直角坐标系中，设点集
$\{P_{1},P_{2},\cdots,P_{4n+1}\}$
$=\left\{(x,y)|x,y\text{ 为整数},|x|\leq n,|y|\leq n,xy=0\right\}$,
其中$n\in\mathbf{N}_+.$求
$$(P_{1}P_{2})^{2}+(P_{2}P_{3})^{2}+\cdots+(P_{n}P_{4n+1})^{2}+(P_{4n+1}P_{1})^{2}$$
的最小值。请提供当 $n=1$ 时的最小值。
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Context:
Let $z_1=18+83i,~z_2=18+39i,$ and $z_3=78+99i,$ where $i=\sqrt{-1}.$ Let $z$ be the unique complex number with the properties that $\frac{z_3-z_1}{z_2-z_1}~\cdot~\frac{z-z_2}{z-z_3}$ is a real number and the imaginary part of $z$ is the greatest possible. Find the real part of $z$.
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Context:
The country Omega grows and consumes only vegetables and fruits. It is known that in 2014, 1200 tons of vegetables and 750 tons of fruits were grown in Omega. In 2015, 900 tons of vegetables and 900 tons of fruits were grown. During the year, the price of one ton of vegetables increased from 90,000 to 100,000 rubles, and the price of one ton of fruits decreased from 75,000 to 70,000 rubles. By what percentage (%) did the real GDP of this country change in 2015, if the base year in Omega is 2014? Round your answer to two decimal places. If the real GDP of the country decreased, put a minus sign in the answer, and if it increased, put a plus sign.
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Context:
Given that $(1-2x)^{9}=a_{9}x^{9}+a_{8}x^{8}+...+a_{2}x^{2}+a_{1}x+a_{0}$, find the value of $a_{1}+a_{2}+...+a_{8}+a_{9}$.
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Context:
6. 135 Let $N$ be the set of all positive integers, and $f$ a function from $N$ to $N$ itself, such that for any $s$ and $t$ in $N$, the following holds:
$$f\left(t^{2} f(s)\right)=s(f(t))^{2}$$

Determine the smallest possible value of $f(1998)$ among all such functions $f$.
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Context:
A box contains red, green, blue, and white balls, totaling 111 balls. If you take out 100 balls without looking, there will always be 4 balls of different colors among them. What is the smallest number of balls you must take out without looking to ensure that among them there will always be balls of at least 3 different colors?
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Context:
In triangle $ABC$, the median $AM$ is perpendicular to the angle bisector $BD$. Find the perimeter of the triangle given that $AB = 1$ and the lengths of all sides are integers.
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Context:
Given the function $f(x)=2\sin (2x+ \frac {\pi}{4})$, let $f_1(x)$ denote the function after translating and transforming $f(x)$ to the right by $φ$ units and compressing every point's abscissa to half its original length, then determine the minimum value of $φ$ for which $f_1(x)$ is symmetric about the line $x= \frac {\pi}{4}$.
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Context:
If vector $\overrightarrow{a}=(1,-x)$ and vector $\overrightarrow{b}=(x,-6)$ are in opposite directions, then $x=$ ______.
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Context:
Four two-inch squares are placed with their bases on a line. The second square from the left is lifted out, rotated 45 degrees, then centered and lowered back until it touches its adjacent squares on both sides. Determine the distance, in inches, of point P, the top vertex of the rotated square, from the line on which the bases of the original squares were placed.
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Context:
数列$\left\{ {{a}_{n}} \right\}$满足${{an+1}}={{a}_{n}}\cdot \cos (n\pi )+2n-1,n\in {{N}^{*}}$，若${{a6}}+{{a}_{20}}=2m$，则实数$m=$___.
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Context:
The numbers 2, 3, 4, 5, 6, 7, 8 are to be placed, one per square, in a 2x2 grid such that the four numbers in the horizontal row add up to 21 and the four numbers in the vertical column add up to 21. Calculate the value of x that satisfies these conditions. Express your answer as a single integer.
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Context:
9. Let set $A$ consist entirely of positive integers, and for any $x, y \in A (x \neq y)$, we have $|x-y| \geqslant \frac{1}{25} x y$. How many numbers can $A$ contain at most?
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Context:
If the equation $(x-2)(x^2-4x+m)=0$ has three roots, and these three roots can exactly serve as the lengths of the sides of a triangle, then the range of $m$ is.
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Context:
The number in each box below is the product of the numbers in the two boxes that touch it in the row above. For example, $30 = 6\times5$. What is the missing number in the top row?

[asy] unitsize(0.8cm); draw((-1,0)--(1,0)--(1,-2)--(-1,-2)--cycle); draw((-2,0)--(0,0)--(0,2)--(-2,2)--cycle); draw((0,0)--(2,0)--(2,2)--(0,2)--cycle); draw((-3,2)--(-1,2)--(-1,4)--(-3,4)--cycle); draw((-1,2)--(1,2)--(1,4)--(-1,4)--cycle); draw((1,2)--(1,4)--(3,4)--(3,2)--cycle); label("600",(0,-1)); label("30",(-1,1)); label("6",(-2,3)); label("5",(0,3)); [/asy]

Find the missing number in the top row.
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Context:
On a cubic planet, cubic mice live only on the faces of the cube and not on the edges or vertices. The number of mice on different faces is different and on any two neighboring faces, this number differs by at least 2. What is the minimum number of cubic mice that can live on this planet if there is at least one mouse on each face?
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Context:
Three pipes, a, b, & c are attached to a tank. Pipe a can fill it in 20 minutes, b can fill it in 30 minutes, and c can empty it in some time. If a, b & c are kept open successively for 4 minute each, the tank will be filled in 12 minutes. How soon can pipe c empty the tank?
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Context:
What is the value of $o-w$, if $gun^2 = wowgun$ where $g,n,o,u,w \in \{0,1,2,\dots, 9\}$?
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Context:
A dot is marked at each vertex of a triangle $A B C$. Then, 2,3 , and 7 more dots are marked on the sides $A B, B C$, and $C A$, respectively. How many triangles have their vertices at these dots?
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Context:
Find the third even number in a sequence of 6 consecutive even numbers with a sum of 180.
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Context:
There are three piles with 40 stones each. Petya and Vasya take turns, starting with Petya. Each turn, a player must combine two piles, then divide the combined stones into four piles. The player who cannot make a move loses. Which player (Petya or Vasya) can guarantee a win, regardless of the opponent's moves?
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Context:
In three-dimensional space, find the number of lattice points that have a distance of 5 from the origin.

Note: A point is a lattice point if all its coordinates are integers.
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Context:
Randomly and without replacement, select three numbers \(a_1, a_2, a_3\) from the set \(\{1, 2, \cdots, 2014\}\). Then, from the remaining 2011 numbers, again randomly and without replacement, select three numbers \(b_1, b_2, b_3\). What is the probability that a brick with dimensions \(a_1 \times a_2 \times a_3\) can fit inside a box with dimensions \(b_1 \times b_2 \times b_3\)?
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Context:
John saves 10% of his monthly salary and invests an additional 5% into a retirement fund. At the end of each month, he is able to save $400 in a general savings account, after accounting for his expenses and a fluctuating increase of 5-7% in expenses due to dearness of things. Furthermore, his retirement fund has a cap of $1500 per month. During some months, John receives an extra allowance with a conversion rate of Rs. 70 per dollar. He pays income tax at a rate of 20% on his base salary and 10% on his extra allowance. The increase in expenses depends on specific monthly conditions related to his consumption patterns. What is John's monthly base salary in dollars, accounting for the various savings, expenses, and additional parameters?
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Context:
Determine the number of zeros of the function $f(z)=z^4+2z^3+3z^2+z+2$ in the annulus $1<|z|<2$ using Rouche's theorem.
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Context:
Nora's mother tells her to go and pick oranges from their farm. She picks 80 oranges from the first tree, 60 from the second tree, 120 from the third tree, 45 from the fourth tree, 25 from the fifth tree, and 97 from the sixth tree. After picking oranges, she gives half of the oranges from the second and fourth trees to her neighbor. Additionally, she finds out that 20% of the oranges from the third and fifth trees are not ripe and decides not to take these unripe oranges. What's the total number of oranges she picked after considering all the changes?
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Context:
Given a tetrahedron \( A B C D \) with side lengths \( A B = 41 \), \( A C = 7 \), \( A D = 18 \), \( B C = 36 \), \( B D = 27 \), and \( C D = 13 \), let \( d \) be the distance between the midpoints of edges \( A B \) and \( C D \). Find the value of \( d^{2} \).
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Context:
Draw 63 south lines on the coordinate plane: $y=b, y=\sqrt{3} x+2 b, y=-\sqrt{3} x+2 b$, where $b=-10$, $-9,-8, \cdots, 8,9,10$. These lines divide the plane into several equilateral triangles. The number of equilateral triangles with side length $\frac{2}{\sqrt{3}}$ is $\qquad$
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Context:
Last year, Manfred received 26 paychecks. Each of his first 6 paychecks was $750, and each of his remaining paychecks was $20 more than each of his first 6 paychecks. Additionally, Manfred had a bonus of $250 added to his 10th paycheck and a medical deductible of $120 taken from his 20th paycheck. To the nearest dollar, what was the average (arithmetic mean) amount of his paychecks for the year, taking into account his bonus and the medical deductible?
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Context:
Richard starts with the string HHMMMMTT. A move consists of replacing an instance of HM with MH , replacing an instance of MT with TM, or replacing an instance of TH with HT. Compute the number of possible strings he can end up with after performing zero or more moves.
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Context:
Problem 9.6. A white checkered $8 \times 8$ table is given. In it, 20 cells are painted black. What is the minimum number of pairs of adjacent white cells that could remain?
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Context:
Black and white balls are arranged in a circle, with black balls being twice as many as white balls. It is known that among the pairs of neighboring balls, there are three times as many pairs of the same color as there are pairs of different colors. What is the minimum number of balls that could have been arranged?
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Context:
Find the value of $\cot{t}$ if $$\sum_{i=1}^{\infty} \cot^{-1}\biggl(\frac{1}{2i^{2}}\biggr)= t.$$
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Context:
At a chamber music festival, six musicians have gathered. At each concert, some of the musicians perform while the others listen from the audience. What is the minimum number of concerts needed for each of the six musicians to listen (from the audience) to all the others?
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Context:
Find the angle of the minor arc corresponding to the chord intercepted by the line $6x+8y-10=0$ on the circle $x^{2}+y^{2}=4$.
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Context:
Three concentric circles form two rings. The inner and middle circumference of the first ring are 352 / 7 m and 528 / 7 m, respectively. The outer circumference of the second ring is 704 / 7 m. Find the combined width of the two rings.
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Context:
In the $xy$-coordinate plane, the $x$-axis and the line $y=x$ are mirrors. If you shoot a laser beam from the point $(126, 21)$ toward a point on the positive $x$-axis, there are $3$ places you can aim at where the beam will bounce off the mirrors and eventually return to $(126, 21)$. They are $(126, 0)$, $(105, 0)$, and a third point $(d, 0)$. What is $d$?

Note: When light bounces off a mirror, the angle of incidence has the same measure as the angle of reflection.
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Context:
Nick has a $3 \times 3$ grid and wants to color each square in the grid one of three colors such that no two squares that are adjacent horizontally or vertically are the same color. Compute the number of distinct grids that Nick can create.
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Context:
Find the smallest positive integer \( n \) such that a cube with side length \( n \) can be divided into 1996 smaller cubes, each with side length a positive integer.
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Context:
Arrange the numbers 1 to 9 in a row from left to right such that every three consecutive numbers form a three-digit number that is a multiple of 3. How many arrangements are there?
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Context:
For the power function $f(x) = (m^2 - m - 1)x^{m^2 + m - 3}$ to be a decreasing function on the interval $(0, +\infty)$, then $m = \boxed{\text{answer}}$.
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Context:
$ABCDE$ is a regular pentagon inscribed in a circle of radius 1. What is the area of the set of points inside the circle that are farther from $A$ than they are from any other vertex?
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Context:
Find the number of solutions in positive integers \(\left(k; a_{1}, a_{2}, \ldots, a_{k}; b_{1}, b_{2}, \ldots, b_{k}\right)\) to the equation
\[
a_{1}\left(b_{1}\right) + a_{2}\left(b_{1}+b_{2}\right) + \cdots + a_{k}\left(b_{1}+b_{2}+\cdots+b_{k}\right)= 7.
\]
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Context:
Find the surface area of the part of the paraboloid \( z = x^2 - y^2 \) that is enclosed between the paraboloids \( z = 3x^2 + y^2 - 2 \) and \( z = 3x^2 + y^2 - 4 \).
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Context:
On the banks of an island, which has the shape of a circle (viewed from above), there are the cities $A, B, C,$ and $D$. A straight asphalt road $AC$ divides the island into two equal halves. A straight asphalt road $BD$ is shorter than road $AC$ and intersects it. The speed of a cyclist on any asphalt road is 15 km/h. The island also has straight dirt roads $AB, BC, CD,$ and $AD$, on which the cyclist's speed is the same. The cyclist travels from point $B$ to each of points $A, C,$ and $D$ along a straight road in 2 hours. Find the area enclosed by the quadrilateral $ABCD$.
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Context:
Evaluate the expression:
\[4(1+4(1+4(1+4(1+4(1+4(1+4(1+4(1))))))))\]
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Context:
Segments \(AA', BB'\), and \(CC'\), each of length 2, all intersect at a point \(O\). If \(\angle AOC'=\angle BOA'=\angle COB'=60^{\circ}\), find the maximum possible value of the sum of the areas of triangles \(AOC', BOA'\), and \(COB'\).
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Context:
There is a set of points M on a plane and seven different circles C1, C2,..., C7. Circle C7 passes through exactly 7 points in M, circle C6 passes through exactly 6 points in M, and so on, with circle C1 passing through exactly 1 point in M. Find the minimum number of points in M. Express your answer as a whole number.
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Context:
How many ways can you color the squares of a $2 \times 2008$ grid in 3 colors such that no two squares of the same color share an edge?
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Context:
Heinz produces a mixed vegetable puree by combining and boiling tomato juice, carrot juice, and spinach juice. The final mixed vegetable puree has 25% water content. The tomato juice has 90% water content, the carrot juice has 88% water content, and the spinach juice has 91% water content. They use 20 liters of tomato juice, 12 liters of carrot juice, and 8 liters of spinach juice. How many liters of mixed vegetable puree will be obtained from this combination of vegetable juices?
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Context:
In how many ways can two disjoint subsets be selected from a set with $n$ elements?
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Context:
Let $ABCD$ be a rhombus with $\angle ADC = 46^\circ$. Let $E$ be the midpoint of $\overline{CD}$, and let $F$ be the point on $\overline{BE}$ such that $\overline{AF}$ is perpendicular to $\overline{BE}$. What is the degree measure of $\angle BFC$?
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Context:
For any positive integer \( n \), let \( f(n) \) denote the number of 1's in the base-2 representation of \( n \). For how many values of \( n \) with \( 1 \leq n \leq 2002 \) do we have \( f(n) = f(n+1) \)?
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Context:
设 $f(x)$ 是定义在 $\mathbf{R}$ 上的函数, 若 $f(x)+x^{2}$ 是奇函数, $f(x)+2^{x}$ 是偶函数,则 $f(1)$ 的值为 \qquad. 原始答案为 $-\frac{m}{n}$ 的格式，请给出 $|m + n|$ 的值。
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Context:
If n divided by a certain number has a remainder of 2, and the remainder when 2 times n is divided by the same number is 4, what is that number?
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Context:
Calculate the limit:
$$\lim_{{x \to 0}} \frac{\cos(\sin x) - \cos x}{x^4}$$
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Context:
Given the floor function \( \left\lfloor x \right\rfloor \) represents the largest integer not greater than \( x \), find all positive integer values of \( n \) that satisfy the equation:
\[
\left\lfloor \frac{\left\lfloor \frac{91}{n} \right\rfloor}{n} \right\rfloor = 1
\]
Express your answer as a list of integers.
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Context:
Circles $\omega$ and $\gamma$, both centered at $O$, have radii $20$ and $17$, respectively. Equilateral triangle $ABC$, whose interior lies in the interior of $\omega$ but in the exterior of $\gamma$, has vertex $A$ on $\omega$, and the line containing side $\overline{BC}$ is tangent to $\gamma$. Segments $\overline{AO}$ and $\overline{BC}$ intersect at $P$, and $\dfrac{BP}{CP} = 3$. Then $AB$ can be written in the form $\dfrac{m}{\sqrt{n}} - \dfrac{p}{\sqrt{q}}$ for positive integers $m$, $n$, $p$, $q$ with ${gcd}(m,n) = {gcd}(p,q) = 1$. What is the value of $m+n+p+q$?
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Context:
7. [6] Let $a_{1}, a_{2}$, and $a_{3}$ be nonzero complex numbers with non-negative real and imaginary parts. Find the minimum possible value of
$$
\frac{\left|a_{1}+a_{2}+a_{3}\right|}{\sqrt[3]{\left|a_{1} a_{2} a_{3}\right|}} .
$$
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Context:
Given the function $y=\cos \left(x+ \frac {\pi}{3}\right)$, derive the horizontal shift of the graph of the function $y=\sin x$.
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Context:
Jackson begins at $1$ on the number line. At each step, he remains in place with a probability of $85\%$ and increases his position on the number line by $1$ with a probability of $15\%$. Let $d_n$ be his position on the number line after $n$ steps, and let $E_n$ be the expected value of $\frac{1}{d_n}$. Find the least $n$ such that $\frac{1}{E_n} > 2017$.
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Context:
Points \(A\) and \(C\) lie on a circle centered at \(O\), each of \(\overline{BA}\) and \(\overline{BC}\) are tangent to the circle, and \(\triangle ABC\) is equilateral. The circle intersects \(\overline{BO}\) at \(D\). The original answer is in the form \(\frac{k}{m}\). Please find the value of \(k + m\).
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Context:
How many solutions does the equation $\tan x = \tan (\tan x)$ have in the interval $0 \le x \le \tan^{-1} 1000$? Assume $\tan \theta > \theta$ for $0 < \theta < \frac{\pi}{2}$.
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Context:
Let the function $f(x)$ have a derivative $f'(x)$ for all $x \in \mathbb{R}$, and for all $x \in \mathbb{R}$, it holds that $f(-x) + f(x) = x^2$. If $f'(x) < x$ for $x \in (0, +\infty)$ and $f(2-m) - f(m) > 2 - 2m$, then the range of values for the real number $m$ is \_\_\_\_\_\_.
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Context:
What is the minimum number of bits required to represent the numbers 25, 18.5, and 9.25 precisely, using a system similar to the IEEE floating point standard?
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Context:
In the diagram, \( S \) lies on \( R T \), \( \angle Q T S = 40^{\circ} \), \( Q S = Q T \), and \( \triangle P R S \) is equilateral. Find the value of \( x \) in degrees.
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Context:
Nikita schematically drew the graph of the quadratic polynomial \( y = ax^{2} + bx + c \). It turned out that \( AB = CD = 1 \). Consider the four numbers \(-a, b, c\), and the discriminant of the polynomial. It is known that three of these numbers are equal to \( \frac{1}{4}, -1, -\frac{3}{2} \) in some order. Find the value of the fourth number.
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Context:
Meredith drives in a series of directions:

1. 5 miles to the northeast.
2. 15 miles to the southeast.
3. 25 miles to the southwest.
4. 35 miles to the northwest.
5. 20 miles to the northeast.

Calculate how many miles Meredith is from her starting point after completing her journey.
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Context:
A circle can be inscribed in and circumscribed around quadrilateral \(ABCD\). The diagonals of this quadrilateral are mutually perpendicular. Find its area if the radius of the circumscribed circle is \(R\) and \(AB = 2BC\).
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Context:
# Task 14. (1 point)

Calculate the amount of personal income tax (NDFL) paid by Sergey for the past year, if he is a Russian resident and during this period had a stable income of 30000 rub./month and a one-time vacation bonus of 20000 rub. In the past year, Sergey sold a car he inherited two years ago for 250000 rub. and bought a land plot for building a residential house for 300000 rub. Sergey claimed all the tax deductions he was entitled to. (Provide the answer without spaces and units of measurement.)
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Context:
For the function $y=\left(m-2\right)x^{|m|}-4$ with respect to $x$, $m=$____.
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Context:
There is a set of points \( M \) on a plane and seven different circles \( C_{1}, C_{2}, \cdots, C_{7} \). Circle \( C_{7} \) passes through exactly 7 points in \( M \), circle \( C_{6} \) passes through exactly 6 points in \( M \), and so on, with circle \( C_{1} \) passing through exactly 1 point in \( M \). Determine the minimum number of points in set \( M \).
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Context:
设 3 阶实对称矩阵 A 的三个特征值分别为 $-1,-1,2$, 且 $(1,1,-1)^{T}$ 是特征值 2所对应的特征向量。记 A中所有元素的平方之和为 I, 则 $[I]=$ $\qquad$.
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Context:
Five students named Fred, Gail, Henry, Iggy, and Joan are seated around a circular table in that order. They play a game of "countdown" where Henry starts by saying '34', and then each student says the next number in the countdown in their circular order. Who will eventually say '1'? Provide the name of the student.
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Context:
There are  $27$  unit cubes. We are marking one point on each of the two opposing faces, two points on each of the other two opposing faces, and three points on each of the remaining two opposing faces of each cube. We are constructing a  $3\times 3 \times 3$  cube with these  $27$  cubes. What is the least number of marked points on the faces of the new cube?
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Context:
Raman's salary was initially decreased by 40%, then increased by 20%, and finally decreased by 10%. What is the overall percentage change in his salary?
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Context:
已知$p:{{\left( x-a \right)}^{2}}<9$，$q:{{\log }_{3}}\left( x+2 \right)<1$．若$\neg p$是$\neg q$的充分不必要条件，则$a$的取值范围是________．
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Context:
4. Given $n$ positive integers $x_{1}, x_{2}, \cdots, x_{n}$ whose sum is 2016. If these $n$ numbers can be divided into 32 groups with equal sums and also into 63 groups with equal sums, find the minimum value of $n$.
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Context:
We placed 6 different dominoes in a closed chain on the table. The total number of points on the dominoes is $D$. What is the smallest possible value of $D$? (The number of points on each side of the dominoes ranges from 0 to 6, and the number of points must be the same on touching sides of the dominoes.)
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Context:
Alice refuses to sit next to either Bob or Carla. Derek refuses to sit next to Eric. Find the number of ways for the five of them to sit in a row of $5$ chairs under these conditions.
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Context:
Two spheres are inscribed in a dihedral angle such that they touch each other. The radius of one sphere is 4 times that of the other, and the line connecting the centers of the spheres forms an angle of \(60^\circ\) with the edge of the dihedral angle. Find the measure of the dihedral angle. Provide the cosine of this angle, rounded to two decimal places if necessary.
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Context:
The sum of \(\frac{1}{2 \times 3 \times 4}+\frac{1}{3 \times 4 \times 5}+\frac{1}{4 \times 5 \times 6}+\ldots+\frac{1}{13 \times 14 \times 15}+\frac{1}{14 \times 15 \times 16}\) is \(\frac{m}{n}\) in its lowest terms. Find the value of \(m+n\).
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