Context:
A sample consisting of five observations has an arithmetic mean of 10 and a median of 12. Find the smallest possible range (largest observation minus smallest) for such a sample. Express your answer as a single integer.
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Context:
已知双曲线$C$的方程为${{x}^{2}}-\frac{{{y2}}}{8}=1$，右焦点为$F$，若点$N\left( 0,6 \right)$，$M$是双曲线$C$的左支上一点，则$\Delta FMN$周长的最小值为_____
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Context:
$ABCD$ is a convex quadrilateral such that $AB<AD$.  The diagonal $\overline{AC}$ bisects $\angle BAD$, and $m\angle ABD=130^\circ$.  Let $E$ be a point on the interior of $\overline{AD}$, and $m\angle BAD=40^\circ$.  Given that $BC=CD=DE$, determine $m\angle ACE$ in degrees.
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Context:
Determine the values of $p \in (0, \infty)$ for which the function $\frac{1}{1-z}$ belongs to the Hardy space $H^p(\mathbb{D})$. Here, $H^p(\mathbb{D})$ is defined as the space of analytic functions $f$ on the unit disk $\mathbb{D}$ such that \( \sup_{0 < r < 1} \left( \frac{1}{2 \pi} \int_{0}^{2\pi}|f(re^{i \theta})|^p d \theta \right)^{\frac{1}{p}} < \infty \).
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Context:
The internal angles of quadrilateral $ABCD$ form an arithmetic progression. Triangles $ABD$ and $DCB$ are similar with $\angle DBA = \angle DCB$ and $\angle ADB = \angle CBD$. Moreover, the angles in each of these two triangles also form an arithmetic progression. In degrees, what is the largest possible sum of the two largest angles of $ABCD$?
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Context:
Misha has accepted a job in the mines and will produce one ore each day. At the market, he can perform the following transactions:
- Buy or sell one ore for $3.
- Buy or sell bundles of three wheat for $12 each.
- Sell one wheat for one ore.

His ultimate goal is to build a city, which requires:
- Three ore
- Two wheat

How many dollars must Misha begin with in order to build a city after three days of working?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Andrew flips a fair coin  $5$  times, and counts the number of heads that appear. Beth flips a fair coin  $6$  times and also counts the number of heads that appear. Compute the probability Andrew counts at least as many heads as Beth.
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Context:
Adam, Benin, Chiang, Deshawn, Esther, and Fiona have internet accounts. Some, but not all, of them are internet friends with each other, and none of them has an internet friend outside this group. Each of them has the same number of internet friends. In how many different ways can this happen?
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Context:
A trapezoid \(ABCD\) with \(BC \parallel AD\) is inscribed in a circle with center \(O\). In the same circle, diameter \(CE\) and chord \(BE\) are drawn, intersecting \(AD\) at point \(F\). Point \(H\) is the foot of the perpendicular dropped from point \(F\) to \(CE\). Point \(S\) is the midpoint of segment \(EO\), and point \(M\) is the midpoint of \(BD\). It is known that the radius of the circle is \(R\) and \(CH = \frac{9R}{8}\). Find \(SM\).
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Context:
Let $\left\{b_{n}\right\}_{n=0}^{\infty}$ be a sequence of positive real numbers such that $b_{0}=1$, $b_{n}=2+\sqrt{b_{n-1}}-2 \sqrt{1+\sqrt{b_{n-1}}}$. Calculate the sum of $b_{n} 2^{n}$ from $n=1$ to $\infty$. The original answer includes a constant term and a logarithmic term. Please provide the integer part of the answer.
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Context:
Let \(\left\{a_{i}\right\}_{i \geq 0}\) be a sequence of real numbers defined by
$$
a_{n+1}=a_{n}^{2}-\frac{1}{2^{2020 \cdot 2^{n}-1}}
$$
for \(n \geq 0\). Determine the largest value for \(a_{0}\) such that \(\left\{a_{i}\right\}_{i \geq 0}\) is bounded.
Please reason step by step, and put your final answer within \boxed{}.


Context:
使得 n^2+2023n 为平方数的正整数 n 的最小值是__________.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that \(MN\) is a moving chord of the circumscribed circle of an equilateral triangle \( \triangle ABC \) with side length \( 2\sqrt{6} \), and \(MN = 4\), \(P\) is a moving point on the sides of \( \triangle ABC \). Find the maximum value of \( \overrightarrow{MP} \cdot \overrightarrow{PN} \).
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Context:
满足方程 \df{1}{k_1}+\df{1}{k_2}+\cdots+\df{1}{k_n}=1，k_1k_2\cdots k_n\leq 200 的所有有序正整数组 (n,k_1,\cdots,k_n) 的个数为__________.
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Context:
The number represented by a point on the number line that is a distance of $\sqrt{5}$ from the origin is ______.
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Context:
If a positive integer \( n \) makes the equation \( x^{3} + y^{3} = z^{n} \) have a positive integer solution \( (x, y, z) \), then \( n \) is called a "good number." How many good numbers are there that do not exceed 2,019?
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Context:
Using an electric stove with a power of $P=500 \mathrm{W}$, a certain amount of water is heated. When the electric stove is turned on for $t_{1}=1$ minute, the water temperature increases by $\Delta T=2^{\circ} \mathrm{C}$, and after turning off the heater, the temperature decreases to the initial value in $t_{2}=2$ minutes. Determine the mass of the heated water, assuming the thermal power losses are constant. The specific heat capacity of water is $c_{B}=4200$ J/kg$\cdot{ }^{\circ} \mathrm{C}$.
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Context:
Given the function $$f(x)=x+\int_0^1t(x+t)f(t){\rm d}t$$, find the value of the integral $$\eta=\int_0^1f(x){\rm d}x$$.
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Context:
Let $f(x)$ and $g(x)$ be odd and even functions defined on $\mathbb{R}$, respectively. When $x < 0$, $f'(x)g(x) + f(x)g'(x) > 0$, and $g(-3) = 0$, then the solution set of the inequality $f(x)g(x) < 0$ is
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Context:
If each side of a regular hexagon consists of 6 toothpicks, and there are 6 sides, calculate the total number of toothpicks used to build the hexagonal grid.
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Context:
A robot vacuum cleaner called "Orthogonal" decides to clean the coordinate plane. It starts from point $O$, located at the origin of the coordinates, and travels a distance $X_{1}$ along a straight line. At the point where it arrives, the vacuum cleaner turns and describes a circle centered at the origin. After completing the circle, it continues, without changing direction, traveling a distance $X_{2}$ along a straight line. It then turns sideways towards the origin to describe another circle centered at the origin and, after completing it, travels a distance $X_{3}$. The vacuum cleaner continues this pattern. It is known that the values $X_{1}, X_{2}, X_{3}, \ldots$ follow a certain distribution with a mean $a$ and variance $d$. Find the expected value of the area of the circle described by the $n$-th circle the vacuum cleaner travels along.
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Context:
If Leo gains 12 pounds, he will weigh 70% more than his sister Kendra. On the other hand, Leo's brother Mark weighs 20% less than the combined weight of Leo and Kendra. Currently, their combined weight (Leo, Kendra, and Mark) is 305 pounds. What is Leo's current weight?
Please reason step by step, and put your final answer within \boxed{}.


Context:
10.1. Find the sum of all such integers $a \in[0 ; 400]$, for each of which the equation $x^{4}-6 x^{2}+4=\sin \frac{\pi a}{200}-2\left[x^{2}\right]$ has exactly six roots. Here the standard notation is used: $[t]$ - the integer part of the number $t$ (the greatest integer not exceeding $t$).
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Context:
The monotonic decreasing interval of the function $f\left(x\right)={\left( \frac{1}{3}\right)}^{-{x}^{2}-4x+3}$ is \_\_\_\_\_\_.
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Context:
Junior has 34 rabbits. On Monday, he bought 8 toys for his rabbits. On Tuesday, he bought three times as many toys as he did on Monday. On Wednesday, he bought double the number of toys he bought on Tuesday. On Thursday, he bought the same amount of toys as he did on Monday. On Friday, he bought five times as many toys as he did on Monday. On Saturday, he bought half as many toys as he did on Wednesday. If he split all the toys evenly among the rabbits, how many toys would each rabbit have?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the function $f(x) = -tx^2 + 2x + 1$ (where $t < 0$, and $t$ is a constant), for any two different $x_1$, $x_2$, when $x_1$, $x_2 \in [-2, 2]$, it always holds that $|f(x_1) - f(x_2)| \leq k|x_1 - x_2|$ (where $k$ is a constant, $k \in \mathbb{R}$). Then, the range of values for the real number $k$ is.
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Context:
Find the mass percentage of Na in NaClO
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Context:
In the acute-angled triangle \(ABC\), the angle bisector \(BL\) is drawn. The circumcircle of triangle \(ABL\) intersects the side \(BC\) at point \(D\). It turns out that point \(S\), symmetric to point \(C\) with respect to line \(DL\), lies on side \(AB\) and does not coincide with its endpoints. What values can \(\angle ABC\) take?
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Context:
The graph of the function $y=\sin(\omega x+ \frac {5\pi}{6})$ where $0<\omega<\pi$ intersects with the coordinate axes at points closest to the origin, which are $(0, \frac {1}{2})$ and $( \frac {1}{2}, 0)$. Determine the axis of symmetry of this graph closest to the y-axis.
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Context:
Joe has exactly enough paint to paint the surface (excluding the bases) of a cylinder with radius 3 and height 4. It turns out this is also exactly enough paint to paint the entire surface of a cube. The volume of this cube is \( \frac{48}{\sqrt{K}} \). What is \( K \)?
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Context:
Solve the inequality:
\[ 8.57 \log_{3} \log_{4} \left(\frac{4x - 1}{x + 1}\right) - \log_{\frac{1}{3}} \log_{\frac{1}{4}} \left(\frac{x + 1}{4x - 1}\right) < 0. \]
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Context:
在正四棱雉 $P-ABCD$ 中，$M、N$ 分别为 $PA、PB$ 的中点，且侧面与底面所成二面角的正切为 $\sqrt{2}$。求异面直线 $DM$ 与 $AN$ 所成角的余弦值。答案的格式为\(\frac{k}{m}\)，请给出k+m的值。
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Context:
In each cell of a $5 \times 5$ table, a natural number is written with invisible ink. It is known that the sum of all the numbers is 200, and the sum of the three numbers inside any $1 \times 3$ rectangle is 23. What is the value of the central number in the table?
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Context:
In the geometric sequence {a_n}, a_3a_7 = 64. Find the value of a_5. Express your answer as a numerical value, including the possibility of a negative value.
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Context:
For what value of $k \in \mathbb{R}$ does the stochastic process $$Y_t = (W_t + t)e^{-W_t + kt}, \quad t \geq 0$$ have zero drift, where $W_t$ is a Wiener process?
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Context:
This morning, farmer Rita's workers collected 30,000 gallons of milk and chilled it in a storage tank. They then spent the next 4 hours pumping milk at a rate of 2,880 gallons/hour from the storage tank into a milk tanker. For the following 7 hours, the workers added more milk into the storage tank at varying rates each hour. In the first hour, they added 1,200 gallons, and every subsequent hour they increased the amount of milk produced by 200 gallons. How many gallons of milk were left in the storage tank at the end of this period?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a set  $A$  which contains  $n$  elements. For any two distinct subsets  $A_{1}$ ,  $A_{2}$  of the given set  $A$ , we fix the number of elements of  $A_1 \cap A_2$ . Find the sum of all the numbers obtained in the described way.
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Context:
The area of polygon $ABCDEF$ is 52 with $AB=8$, $BC=9$ and $FA=5$. What is the value of $DE+EF$?
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Context:
Raashan, Sylvia, and Ted play the following game. Each starts with $1$. A bell rings every $15$ seconds, at which time each of the players who currently have money simultaneously chooses one of the other two players independently and at random and gives $1$ to that player. What is the probability that after the bell has rung $2019$ times, each player will have $1$? Provide your answer in the form of \frac{k}{m}, and calculate the value of k + m.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The true discount on a bill of Rs. 74500 due 1 year hence at 15% per annum is Rs. 11175. What is the banker's discount?
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Context:
Given that the complex number $z$ and $(z+2)^{2}-8i$ are both purely imaginary, then $z=$ \_\_\_\_\_\_.
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Context:
A soccer team won 45 percent of its first 36 games in a particular season, and at least 75 percent of its remaining games. At the end of the season, the winning percentage of the team was 62 percent. The team played an even number of games, and they did not tie any of their games. What was the total number of games that the team played in that season?
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Context:
Find the most general entire functions of the form \( f(z) = u(x) + iv(y) \), where \( u \) and \( v \) depend only on one real variable.
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Context:
Determine the characteristic lines for the partial differential equation: $$yu_x + xu_y = u^2$$
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Context:
Find the number of non-archimedean extensions of the absolute value of \(\mathbb{Q}\) for the field \(\mathbb{Q}(\sqrt[n]{p})\), where \(p\) is a prime number.
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Context:
Positive real numbers \( x \) and \( y \) satisfy the following conditions: There exist \( a \in[0, x] \) and \( b \in[0, y] \) such that \( a^{2} + y^{2} = 2 \), \( b^{2} + x^{2} = 1 \), and \( ax + by = 1 \). What is the maximum value of \( x + y \)?
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Context:
Arrange all the zeros of the function $f(x)=\sin x$ ($x > 0$) in ascending order to form the sequence $\{a_n\}$. The sequence $\{b_n\}$ satisfies $b_n=3^n \cdot a_n$. Then, the sum of the first $n$ terms of the sequence $\{b_n\}$, denoted as $T_n=$ ______.
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Context:
On the side $AB$ of the convex quadrilateral $ABCD$, a point $M$ is chosen such that $\angle AMD = \angle ADB$ and $\angle ACM = \angle ABC$. The tripled square of the ratio of the distance from point $A$ to the line $CD$ to the distance from point $C$ to the line $AD$ is $2$, and $CD = 20$. Find the radius of the inscribed circle of triangle $ACD$.
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Context:
The 25th question: In country A with 2011 cities, there is a unique bidirectional flight route operated by a certain airline between any city in A and any city in country B, which also has 2011 cities. There are no flight routes between cities within the same country, and at most 19 different airlines operate these routes. Find the maximum integer $\mathrm{k}$, such that no matter how the routes are arranged, there are always $\mathrm{k}$ cities that are connected (one can travel from one city to another, not necessarily directly), and all the routes are operated by a fixed airline.
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the angle between the minute and hour hands at exactly 3:05?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the minimum value of the function
$$
f(x)=\sqrt{15-12 \cos x}+\sqrt{4-2 \sqrt{3} \sin x}+\sqrt{7-4 \sqrt{3} \sin x}+\sqrt{10-4 \sqrt{3} \sin x-6 \cos x}.
$$
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Context:
A dealer offers a certain cash discount and still makes a profit of 36% when he further allows 25 articles to be sold at the cost price of 20 articles to a particular sticky bargainer. His articles were listed 100% above the cost price. What is the cash discount percentage offered by the dealer?
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Context:
Solve for $x$ in real numbers, where $a$ and $b$ are real, in the equation:

$$\left| a^2 - 2a - b^2 - 4b - x \right| + \left| a^2 - 2a - b^2 - 4b - 3x + 2 \right| + \left| a^2 - 2a + b^2 + 4b + 2x \right| + (a^2 - 2a + b^2 + 4b + 18|x - 2|) + 11x = 20.$$
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Context:
Three partners, P, Q, and R invest in a business with the investment ratio of 3:4:5, respectively. The ratio of their profits at the end of the business period is 9:16:25. If partner P invested the money for 4 months and partner R invested for 10 months, find out for how long did partner Q invest the money in the business?
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Context:
Convex quadrilateral ABCD is given, and lines are extended from points A, B, C, and D to points B', C', D', and A' respectively, such that AB = BB' = 6, BC = CC' = 7, CD = DD' = 8, and DA = AA' = 9. If the area of quadrilateral ABCD is 10, find the area of quadrilateral A'B'C'D'. Express your answer as a whole number.
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Context:
Find the norm of the linear operator $T:C[0,1]\to C[0,1]$ defined by $Tf(t)=f(t)+\int_0^t(1-2s)f(s)ds$, where the norm is the supremum norm.
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Context:
The volume of a regular pyramid is $V$. A plane is passed through the center of a sphere inscribed in the pyramid, parallel to its base. Find the volume of the pyramid that is cut off from the given pyramid by this plane, if the dihedral angle at the base is $\alpha$.
Please reason step by step, and put your final answer within \boxed{}.

\boxed{\frac{V}{8 \cos ^{6} \frac{\alpha}{2}}}


Context:
Define the minimum real  $C$  where for any reals  $0 = a_0 < a_{1} < \dots < a_{1000}$  then   $$ \min_{0 \le k \le 1000} (a_{k}^2 + (1000-k)^2) \le C(a_1+ \dots + a_{1000}) $$   holds. Find  $\lfloor 100C \rfloor.$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the integral $$\int_{0}^{\pi/2} \ln(a^2\sin^2{x} + \cos^2{x})dx$$ for $a > 0$.
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Context:
How many nonzero coefficients can a polynomial $P(z)$ have if its coefficients are integers and $|P(z)| \leq 2$ for any complex number $z$ of unit length? Please provide the sum of all possible values.
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Context:
Points $A$ and $B$ are located on a circular alley, connected by a straight highway segment 4 km long, which is the diameter of the circular alley. A pedestrian leaves home at point $A$ and walks along the alley at 6 km/h. After 1 hour, he realizes he forgot his keys and asks a neighbor, who is a cyclist, to bring them to him. The cyclist travels at 15 km/h on the highway and 20 km/h on the alley. What is the minimum time in which the pedestrian can receive the keys, assuming he can walk towards the cyclist?
Please reason step by step, and put your final answer within \boxed{}.


Context:
An employee arrives at a bus stop at 8:10. Two buses are scheduled to depart from the stop. The first bus departs at 8:00, 8:20, and 8:40 with probabilities of 1/4, 1/2, and 1/4 respectively. The second bus departs at 9:00, 9:20, and 9:40 with probabilities of 1/4, 1/2, and 1/4 respectively. The departure times of the two buses are independent of each other. What is the probability that the employee will wait for more than 50 minutes? Express your answer as a simplified fraction.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let's determine the radius of the sphere that touches three faces of a regular tetrahedron with unit edge length, as well as the three sides of the fourth face.
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Context:
What is the smallest integer n such that any subset of {1, 2, 3,..., 20} of size n must contain two numbers that differ by 8? Express your answer as an integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Three squares, $ABCD$, $EFGH$, and $GHIJ$, each have side length $s$. Point $C$ is located at the midpoint of side $HG$, and point $D$ is located at the midpoint of side $EF$. The line segment $AJ$ intersects the line segment $GH$ at point $X$. Determine the ratio of the area of the shaded region formed by triangle $AXD$ and trapezoid $JXCB$ to the total area of the three squares.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Three trains, 650 m, 750 m, and 850 m long, are running in opposite directions on parallel tracks. The speeds of the trains are 45 km/hr, 30 km/hr, and 60 km/hr respectively. Find the time taken by the slowest train to pass a point equidistant from the drivers of the faster trains.
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Context:
Mady now has boxes each capable of holding up to 5 balls instead of 4. Under the same process as described, Mady adds balls and resets boxes. Determine the total number of balls in the boxes after her $2010$th step.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \iint_R \frac { |x+y| } { (x^2 + y^2 + 1)^2 } \, dx \, dy ,\) where \(R = \{ (x, y) \in \mathbb{R}^2 \mid x \leq 0 ,\, y \leq 0 \}.\)
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Context:
Given that \( MN \) is a chord of the circumcircle of an equilateral triangle \( \triangle ABC \) with a side length of \( 2\sqrt{6} \), and \( MN = 4 \). Let \( P \) be a point on the sides of \( \triangle ABC \). Find the maximum value of \( \overrightarrow{MP} \cdot \overrightarrow{PN} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Through the center \( O \) of the circumscribed circle of the acute-angled triangle \( ABC \), a line perpendicular to \( BO \) is drawn. This line intersects segment \( AB \) at point \( P \) and the extension of segment \( BC \) beyond point \( C \) at point \( Q \). Find \( BP \) given that \( AB=c \), \( BC=a \), and \( BQ=p \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the generator for the ideal in \( \mathbb{Z}[i] \) generated by the elements \( 3 + 4i \) and \( 4 + 7i \).
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Context:
In a match without draws, the competition ends when one person wins 2 more games than the other, and the person with more wins is the winner. It is known that in odd-numbered games, the probability that player A wins is $\frac{3}{5}$, and in even-numbered games, the probability that player B wins is $\frac{3}{5}$. Find the expected number of games played when the match ends.
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Context:
When \( x^{2} \) is added to the quadratic polynomial \( f(x) \), its maximum value increases by \( \frac{27}{2} \), and when \( 4x^{2} \) is subtracted from it, its maximum value decreases by 9. How will the maximum value of \( f(x) \) change if \( 2x^{2} \) is subtracted from it?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $A$ be the number between 0 and 1 that has 10 ones after the decimal point followed by all zeros. Determine the square root of $A$ to 20 significant figures!
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Context:
1. In a tennis tournament, 512 schoolchildren are participating. 1 point is awarded for a win, and 0 points for a loss. Before each round, pairs are formed by lottery among participants with the same number of points (those who do not find a pair are awarded a point without playing). The tournament ends as soon as a sole leader is determined. How many schoolchildren will finish the tournament with 6 points?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a qualification football round there are six teams and each two play one versus another exactly once. No two matches are played at the same time. At every moment the difference between the number of already played matches for any two teams is  $0$  or  $1$ . A win is worth  $3$  points, a draw is worth  $1$  point and a loss is worth  $0$  points. Determine the smallest positive integer  $n$  for which it is possible that after the  $n$ -th match all teams have a different number of points and each team has a non-zero number of points.
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Context:
Seven fishermen are standing in a circle. Each fisherman has a professional habit of exaggerating numbers, with a distinct measure of exaggeration (an integer) indicating by how many times the number mentioned by the fisherman exceeds the true value. For example, if a fisherman with an exaggeration measure of 3 catches two fish, he will claim to have caught six fish. When asked: "How many fish did your left neighbor catch?" the answers were (not necessarily in the seated order) $12, 12, 20, 24, 32, 42,$ and $56$. When asked: "How many fish did your right neighbor catch?" six of the fishermen answered $12, 14, 18, 32,$ $48,$ and $70$. What did the seventh fisherman answer?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the smallest integer \( n \) such that there exists a convex \( n \)-gon where the sines of all its interior angles are equal and all its side lengths are distinct.
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Context:
In the game of Galactic Dominion, players compete to amass cards, each of which is worth a certain number of points. You are playing a version of this game with only two kinds of cards: planet cards and hegemon cards. Each planet card is worth 2010 points, and each hegemon card is worth four points per planet card held. Starting with no planet cards and no hegemon cards, on each turn (starting at turn one), you take either a planet card or a hegemon card, whichever is worth more points given your current hand. Define a sequence \(\{a_{n}\}\) for all positive integers \(n\) by setting \(a_{n}\) to be 0 if on turn \(n\) you take a planet card and 1 if you take a hegemon card. What is the smallest value of \(N\) such that the sequence \(a_{N}, a_{N+1}, \ldots\) is necessarily periodic (meaning that there is a positive integer \(k\) such that \(a_{n+k}=a_{n}\) for all \(n \geq N\))?
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Context:
On a sunny day, 3000 people, including children, boarded a cruise ship. Two-fifths of the people were women, and a third were men. If 25% of the women and 15% of the men were wearing sunglasses, and there were also 180 children on board with 10% wearing sunglasses, how many people in total were wearing sunglasses?
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Context:
Seojun misunderstood the teacher's words to bring a 7/3 pizza from the next class and gave the 7/3 pizza to the next class. After giving the pizza, Seojun now has 3/2 pizza. If Seojun did the teacher's errand correctly, how many pizzas should Seojun have?
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Context:
Barney washes his kitchen towels once a week. He owns some towels and uses two at a time that he changes to clean towels daily. He missed one week of laundry. The following week, Barney will not have clean towels for 5 days. How many towels does Barney own?
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Context:
Find the mass of the region $E$ bounded by the spheres $x^2+y^2+z^2 = 4z$ and $x^2+y^2+z^2 = 2z$, given that the density of the region is $\rho(x,y,z) = (x^2+y^2+z^2)^{3/2}$.
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Context:
Edith has x number of novels stuffed on her first schoolbook shelf, which is 20% more than the number of novels she has on her second shelf. She has half as many writing books in her suitcase as the total number of novels on both shelves. If there are 56 novels on the second shelf, how many novels and writing books does Edith have altogether?
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Context:
Determine the dimension over \(\mathbb{R}\) of the space of \(n \times n\) Hermitian matrices.
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Context:
In Xiangshan County, the taxi fare is calculated based on a segmented cumulative method: a fare of 5 yuan is charged for distances within 3 kilometers (including 3 kilometers); for the part exceeding 3 kilometers but not exceeding 10 kilometers, 2 yuan per kilometer is charged; for the part exceeding 10 kilometers, 3 yuan per kilometer is charged. In addition, a fuel surcharge of 1 yuan is added for each ride. Distances less than 1 kilometer are charged as 1 kilometer. If Xiao Ming spent 38 yuan on a taxi ride from school to home, and the distance from Xiao Ming's home to the school is x kilometers, then the range of x is.
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Context:
A football invitational tournament involves 16 cities, with each city sending two teams, Team A and Team B. According to the competition rules, after several days of matches, it is observed that except for Team A from City $A$, the number of matches played by all other teams are different from each other. Find the number of matches played by Team B from City $A$.
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Context:
Let \( P=\begin{pmatrix} i & 2\\ -1 & -i \end{pmatrix} \) and \( T_P\colon M_{2\times 2}^{\mathbb{C}} \to M_{2\times 2}^{\mathbb{C}} \) be a linear map defined by \( T_P(X)=P^{-1}XP \). Determine the minimal polynomial of \( T_P \).
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Context:
7. A sphere is circumscribed around the tetrahedron $ABCD$, and another sphere with radius 1 is tangent to the plane $ABC$ and the two spheres are internally tangent at point $D$. If
$$
\begin{array}{l}
AD=3, \cos \angle BAC=\frac{4}{5}, \\
\cos \angle BAD=\cos \angle CAD=\frac{1}{\sqrt{2}},
\end{array}
$$

then the volume of the tetrahedron $ABCD$ is
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Context:
There are 4 "Outstanding Class" spots and 1 "Outstanding Group Branch" spot to be distributed among 4 classes, with each class receiving at least one spot. Calculate the total number of different distributions possible.
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Context:
Rosencrantz plays \( n \leq 2015 \) games of question, and ends up with a win rate (i.e. \(\#\) of games won / \(\#\) of games played) of \( k \). Guildenstern has also played several games, and has a win rate less than \( k \). He realizes that if, after playing some more games, his win rate becomes higher than \( k \), then there must have been some point in time when Rosencrantz and Guildenstern had the exact same win-rate. Find the product of all possible values of \( k \).
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Context:
Find the values of \(k\) such that the function \(f(x) = x^k |x|\) is three times differentiable at the origin.
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Context:
Let $X$ be a random variable following a Poisson distribution with parameter $\lambda > 0$. If $\delta(X)$ is an unbiased estimator of $g(\lambda) = e^{-\lambda}(3\lambda^2 + 2\lambda + 1)$, find the value of $\sum_{k=0}^\infty \delta(k)$. \( \text{(Hint: Use the property of unbiased estimators)} \)
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Context:
Let \( \triangle ABC \) be a triangle with sides \( AB = 13 \), \( BC = 14 \), and \( CA = 15 \). Let \( E \) and \( F \) be the feet of the altitudes from \( B \) and \( C \), respectively. Let \( \omega \) be the circumcircle of \( \triangle AEF \). We draw three lines tangent to the circumcircle \( \omega \) at \( A \), \( E \), and \( F \). Compute the area of the triangle formed by these three tangents.
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Context:
Given that the determinant of the 3x3 matrix \( A \) is \( \det(A) = -4 \), where:

\[ A = \begin{vmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{vmatrix} \]

Find the determinant of the matrix \( B \):

\[ B = \begin{vmatrix} 2a_3 & 2a_2 & 2a_1 \\ b_3-a_3 & b_2-a_2 & b_1-a_1 \\ c_3+3b_3 & c_2+3b_2 & c_1+3b_1 \end{vmatrix} \]

Use the properties of determinants to determine \( \det(B) \).
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Context:
Light of a blue laser (wavelength  $\lambda=475 \, \text{nm}$ ) goes through a narrow slit which has width  $d$ . After the light emerges from the slit, it is visible on a screen that is  $ \text {2.013 m} $  away from the slit. The distance between the center of the screen and the first minimum band is  $ \text {765 mm} $ . Find the width of the slit  $d$ , in nanometers.
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Context:
a shopkeeper sold an article offering a discount of 5 % and earned a profit of 27.3 % . what would have been the percentage of profit earned if no discount had been offered ?
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Context:
Given that $P$ is a moving point on the curve $y= \frac {1}{4}x^{2}- \frac {1}{2}\ln x$, and $Q$ is a moving point on the line $y= \frac {3}{4}x-1$, then the minimum value of $PQ$ is \_\_\_\_\_\_.
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Context:
Given that $\overrightarrow{a}=(2,3)$, $\overrightarrow{b}=(-4,7)$, and $\overrightarrow{a}+\overrightarrow{c}=\overrightarrow{0}$, find the projection of $\overrightarrow{c}$ on the direction of $\overrightarrow{b}$.
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Context:
Given a regular tetrahedron of volume 1, obtain a second regular tetrahedron by reflecting the given one through its center. What is the volume of their intersection?
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Context:
The first $9$ positive integers are placed into the squares of a $3 \times 3$ chessboard. We are taking the smallest number in a column. Let $a$ be the largest of these three smallest numbers. Similarly, we are taking the largest number in a row. Let $b$ be the smallest of these three largest numbers. How many ways can we distribute the numbers into the chessboard such that $a = b = 4$?
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Context:
Let $N$ be a three-digit integer such that the difference between any two positive integer factors of $N$ is divisible by 3 . Let $d(N)$ denote the number of positive integers which divide $N$. Find the maximum possible value of $N \cdot d(N)$.
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Context:
Determine the signature of the quadratic form \( Q(x_1, \ldots, x_n) = \sum_{i,j=1}^{n} a_i a_j x_i x_j \) in \( \mathbb{R}^n \), where \( a = (a_1, \ldots, a_n) \neq 0 \) is a vector in \( \mathbb{R}^n \).
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Context:
A new dump truck delivered sand to a construction site, forming a conical pile with a diameter of $12$ feet. The height of the cone was $50\%$ of its diameter. However, the pile was too large, causing some sand to spill, forming a cylindrical layer directly around the base of the cone. The height of this cylindrical layer was $2$ feet and the thickness was $1$ foot. Calculate the total volume of sand delivered, expressing your answer in terms of $\pi$.
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Context:
已知 \( x \) 为实数，使得 \( 2, x, x^{2} \) 互不相同，且其中有一个数恰为另一个数的 2 倍，则这样的实数 \( x \) 的个数为多少？
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Context:
A shop has a capacity of 4 customers, and any additional customers are turned away. Customers enter the shop at a random rate of 20 per hour, and the mean service time is 3 minutes. Calculate the mean length of time a customer spends in the shop.
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Context:
The park currently has 39 dogwood trees. Park workers will plant 41 more dogwood trees today, which, on average, grow by 2 trees per year, and 20 more dogwood trees tomorrow, which, on average, grow by 4 trees per year. Calculate the total number of dogwood trees expected in the park after 5 years.
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Context:
There is a set of control weights, each of them weighs a non-integer number of grams. Any
integer weight from  $1$  g to  $40$  g can be balanced by some of these weights (the control
weights are on one balance pan, and the measured weight on the other pan).What is the
least possible number of the control weights?
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Context:
Identify the group represented by the following Cayley diagram, where arrows are omitted for elements that are their own inverses.
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Context:
Given the equation $6x^2 + 2(m-13)x + 12 - m = 0$ has exactly one positive integer solution, find the value of the integer $m$.
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Context:
例1 设M={1,,2,···,2005},，A是M的子集，若对任何$a _ { i } ,$ $a _ { j } \in A ,$ $a _ { i } \neq a _ { j }$，都能以$a _ { i }$ $a _ { j }$为边长唯一地确定一个等腰三角形，求｜A｜的最大值.
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Context:
Determine all real values of $a$ such that the inequality \( \frac{1}{x}+\frac{1}{y}\geq a\left(\frac{2}{x}+\frac{3}{x+y} \right) \) holds for all positive real numbers $x$ and $y$.
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Context:
Calculate $\lg 2\lg 50 + \lg 25 - \lg 5\lg 20 = \_\_\_\_\_\_$.
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Context:
Find the solutions of the equation

$$
\sqrt{x+\sqrt{4 x+\sqrt{16 x+\sqrt{\ldots+\sqrt{4^{n} x+3}}}}}=1+\sqrt{x}
$$
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Context:
7.1 In the example of addition and subtraction, the student replaced the digits with letters according to the rule: identical letters are replaced by identical digits, different letters are replaced by different digits. From how many different examples could the record $0<\overline{\overline{Б A}}+\overline{\text { БА }}-\overline{\text { ЯГА }}<10$ be obtained?
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Context:
Determine the rate of convergence of the sequence \(\{a_0, a_1, \ldots\}\) where \(a_0 = n^{-\varepsilon}\) and \(a_{i+1} = \left(\frac{a_i}{1-a_i}\right)^2\).
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Context:
Evaluate the limit: $$ \lim_{x \to +\infty} \left[ (x+a)^{1+{1\over x}}-x^{1+{1\over x+a}}\right] $$
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Context:
The 40 unit squares of the $9 \times 9$ table are labeled. A horizontal or vertical row of 9 unit squares is considered "good" if it contains more labeled unit squares than unlabeled ones. How many "good" rows (both horizontal and vertical) can the table have in total?
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Context:
Determine the number of group homomorphisms from the dihedral group $D_4$ to the symmetric group $S_3$. Use the fact that the order of the image of an element must divide the order of the element.
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Context:
A shopkeeper sold an article offering a discount of 5% and earned a profit of 34.9%. What would have been the percentage of profit earned if no discount had been offered?
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Context:
Determine where the function \( f(x,y) \) is differentiable, where:
\[
f(x,y)=
\begin{cases}
\frac{x^3 y}{x^2 + y^4} & (x,y) \ne (0,0) \\
0 & (x,y)=(0,0)
\end{cases}
\]
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Context:
Originally, a certain amount of money was sufficient to buy 20 kg of rice. The price of rice first falls by 20%, then falls by an additional 10%. In the meantime, a new international trade policy affects the market, and rice prices are now given in dollars per pound. If 2.2 pounds is equivalent to 1 kilogram, and the currency exchange rate fluctuates by 5% in favor of the local currency, determine the quantity of rice (in kg) that can be bought with the initial sum of money at the new price, given the multiple price reductions, unit conversion, and inflation adjustments.
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Context:
In triangle ABC, point X is located on segment AB such that \( \frac{AX}{AB} = \frac{1}{4} \). Line CX intersects the median from A at A' and the median from B at B''. Similarly, points B', C', A'', and C'' are defined. Determine the ratio of the area of triangle A''B''C'' to the area of triangle A'B'C'.
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Context:
Evaluate the limit:
\[ \lim_{n\to\infty}\frac{\sqrt[3]{n+1}-\sqrt[3]{n+\cos\frac{3}{n}}}{\sqrt[6]{n^2+\sin\frac{2}{n}}-\sqrt[3]{n}} \]
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Context:
Evaluate the limit: \[ \lim_{x\to 0} \frac{x\sqrt{y^2-(y-x)^2}}{(\sqrt{8xy-4x^2}-\sqrt{8xy})^3} \]
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Context:
A set of  $8$  problems was prepared for an examination. Each student was given  $3$  of them. No two students received more than one common problem. What is the largest possible number of students?
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Context:
There were 3 candidates in an election. Two candidates received 2500 and 5000 votes respectively, and the winning candidate got a certain percentage of the total votes in that election. If the winning candidate received 15000 votes, what percentage of the total votes did they receive?
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Context:
Determine the fundamental group of a surface \( S_{g,m} \) with genus \( g \) and \( m \) punctures.
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Context:
找出满足下列条件的最小正整数 $n$ : 如果对平面内任意有限点集 $A$ 中任意 $n$ 个点,总有能将这 $n$ 个点覆盖的两条直线, 则存在两条直线可以覆盖 $A$ 中所有点.
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Context:
Find the volume of the region bounded between the surfaces $z = (x^2 + y^2)^2$ and $z = x$.
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Context:
Calculate the limit of the function:

$$
\lim _{x \rightarrow 2} \frac{1-2^{4-x^{2}}}{2\left(\sqrt{2 x}-\sqrt{3 x^{2}-5 x+2}\right)}
$$
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Context:
Given that the first term of an arithmetic sequence is 24 and it starts to become negative from the 10th term, find the range of the common difference d. Express your answer as an inequality using standard notation.
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Context:
Solve the equation \(a b c d e f = a + b + c + d + e + f\) in natural numbers.
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Context:
Calculate the flux of the vector field $\bar{V} = z\bar{i} + x\bar{j} + y\bar{k}$ through the surface defined by $x^2 + y^2 + z^2 = 1, x \ge 0, y \ge 0, z \ge 0$ using Gauss's theorem.
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Context:
Let $p$ be the largest prime less than $2013$ for which \[ N = 20 + p^{p^{p+1}-13} \] is also prime. Find the remainder when $N$ is divided by $10^4$.
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Context:
In a 200-meter race, Sonic is 16 meters ahead of Dash when Sonic finishes the race. The next time they race, Sonic starts 2.5 times this lead distance behind Dash, who is at the starting line. Both runners run at the same constant speed as they did in the first race. Determine the distance Sonic is ahead of Dash when Sonic finishes the second race.
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Context:
Given a set of 2023 positive numbers, some possibly zero but all being less than 1, with the sum of the elements equal to 1, determine the maximum number of ordered pairs \((j,k)\) such that \(a_k + \frac{a_j^2}{4} \ge \frac{1}{2022}\) and \(k < j\).
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Context:
Determine the number of integers \(1 \le a \le 2015\) that are primitive residues modulo 2015. An integer \(a\) is a primitive residue modulo \(n\) if it is exactly one of the following types of residues: quadratic, cubic, or quintic, modulo \(n\).
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Context:
Find the smallest positive integer $n$ that has at least $7$ positive divisors $1 = d_1 < d_2 < \ldots < d_k = n$, $k \geq 7$, and for which the following equalities hold:

$$d_7 = 2d_5 + 1 \text{ and } d_7 = 3d_4 - 1$$
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Context:
Find a vector field \( Z = a\partial_x + b\partial_y + c\partial_z \) in \( \mathbb{R}^3 \) such that \( \alpha(Z) = 1 \) and \( d\alpha(Z,.) = 0 \), where \( \alpha = dz - ydx \).
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Context:
Rachel has two indistinguishable tokens, and places them on the first and second squares of a \(1 \times 6\) grid of squares. She can move the pieces in two ways:
- If a token has a free square in front of it, then she can move this token one square to the right.
- If the square immediately to the right of a token is occupied by the other token, then she can "leapfrog" the first token; she moves the first token two squares to the right, over the other token, so that it is on the square immediately to the right of the other token.

If a token reaches the 6th square, then it cannot move forward anymore, and Rachel must move the other one until it reaches the 5th square. How many different sequences of moves for the tokens can Rachel make so that the two tokens end up on the 5th square and the 6th square?
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Context:
Given the polynomial \[ f(x)=x^{2020} + \sum_{i=0}^{2019} c_{i} x^{i} \quad (c_{i} \in\{-1,0,1\}), \] determine the maximum number of positive integer roots of \( f(x) \), including their multiplicities, given that the equation \( f(x) = 0 \) has no negative integer roots.
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Context:
Find the minimum value of $$B=\frac{x}{y}+4\sqrt{1+\frac{y}{z}}+6\sqrt[3]{1+\frac{z}{x}}$$ given that $x,y,z>0$ and $x\geq max${$y,z$}.
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Context:
Two identical cylindrical vessels are connected at the bottom by a small-diameter pipe with a valve. While the valve was closed, water was poured into the first vessel and oil into the second in such a way that the levels of the liquids were equal and reached $h = 40$ cm. At what level will the water be in the first vessel if the valve is opened? The density of water is 1000 kg/m³ and the density of oil is 700 kg/m³. The volume of the connecting pipe can be neglected. Provide the answer in centimeters.
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Context:
Evaluate the definite integral \( \int_0^a \frac{2a-x}{a+x} \sqrt{\frac{a-x}{a+x}} \, dx \).
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Context:
Jacqueline has 2 liters of soda. Liliane has 60% more soda than Jacqueline, and Alice has 40% more soda than Jacqueline. Calculate the percentage difference between the amount of soda Liliane has compared to Alice.
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Context:
A hollow box, shaped like a rectangular prism and with negligible thickness, has a volume of $108$ cubic units. The top of the box is removed, exposing the faces on the inside of the box. What is the minimum possible value for the sum of the areas of the faces on the outside and inside of the box?
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Context:
Given that $X_1, \dots, X_7$ and $Y_1, \dots, Y_8$ are independently and identically distributed (i.i.d.) according to a $N(\mu, \sigma^2)$ distribution, find the probability $P(X_1<Y_{(8)})$, where $Y_{(8)}$ denotes the maximum of $Y_1, \dots, Y_8$.
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Context:
John wants to lose weight. He eats 1800 calories a day and burns a certain amount of calories a day. If he needs to burn 4000 calories to lose 1 pound, it will take him 80 days to lose 10 pounds. How many calories does John burn a day?
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Context:
Derek is deciding between two different-sized pizzas at his favorite restaurant. The menu lists a 14-inch pizza and an 18-inch pizza. Calculate the percent increase in area if Derek chooses the 18-inch pizza over the 14-inch pizza.
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Context:
Find the remainder when\[\binom{\binom{3}{2}}{2} + \binom{\binom{4}{2}}{2} + \dots + \binom{\binom{40}{2}}{2}\]is divided by $1000$.
 ~ pi_is_3.14
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Context:
A, B, C, and D enter into a partnership. A subscribes 1/3 of the capital, B 1/4, C subscribes some fraction, and D the rest. A gets a share of Rs. 805 in a profit of Rs. 2415. What fraction of the capital did C subscribe?
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Context:
Evaluate the integral \( \int_{\{ p \}} f(x) \, \text{dvol}_{\{ p \}}(x) \), where \( f: S^2 \rightarrow \mathbb{R} \) is a continuous function on the standard sphere, \( p \in S^2 \) is a fixed point, and \( \text{dvol}_{\{ p \}}(x) \) is the Riemannian volume element of the manifold \( \{ p \} \).
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Context:
Use Horner's Rule to find the value of $v_2$ when the polynomial $f(x) = x^5 + 4x^4 + x^2 + 20x + 16$ is evaluated at $x = -2$.
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Context:
There are 7 balls of each of the three colors: red, blue, and yellow. When randomly selecting 3 balls with different numbers, determine the total number of ways to pick such that the 3 balls are of different colors and their numbers are not consecutive.
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Context:
## Task A-1.6.

Determine the smallest natural number $n$ such that half of $n$ is the square of some natural number, a third of $n$ is the cube of some natural number, and a fifth of $n$ is the fifth power of some natural number.
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Context:
Calculate the variance of the difference between the maximum and minimum values of a set of n independent and identically distributed (i.i.d.) uniform variables, $X_i$, where each $X_i$ is uniformly distributed on the interval (0,1). Specifically, find $variance[\max(\{X_i\}) - \min(\{X_i\})]$.
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Context:
How can you measure 15 minutes using a 7-minute hourglass and an 11-minute hourglass?
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Context:
Given that F1 and F2 are the left and right foci of the hyperbola $$\frac {x^{2}}{a^{2}}- \frac {y^{2}}{b^{2}}=1(a＞0，b＞0)$$ respectively, if there exists a point P on the left branch of the hyperbola such that |PF1|=|F1F2|, and the distance from F1 to the line PF2 is $$\sqrt {7}a$$, then the eccentricity of the hyperbola e is ___.
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Context:
Determine the interval for the parameter \( c \) such that the expression \( \frac{x^2-x+c}{x^2+x+2c} \) can take all real values for any real number \( x \).
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Context:
Four objects of different masses are available. What is the minimum number of weighings on a balance scale without weights needed to determine the heaviest and the lightest of these objects?
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Context:
Find the volume of a tetrahedron whose vertices are located at points with coordinates \(\left(F_{n}, F_{n+1}, F_{n+2}\right), \quad\left(F_{n+3}, F_{n+4}, F_{n+5}\right), \quad\left(F_{n+6}, F_{n+7}, F_{n+8}\right),\) and \(\left(F_{n+9}, F_{n+10}, F_{n+11}\right)\), where \(F_{i}\) is the \(i\)-th term of the Fibonacci sequence: \(1, 1, 2, 3, 5, 8 \ldots\).
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Context:
Suppose that  $a_1 = 1$ , and that for all  $n \ge 2$ ,  $a_n = a_{n-1} + 2a_{n-2} + 3a_{n-3} + \ldots + (n-1)a_1.$  Suppose furthermore that  $b_n = a_1 + a_2 + \ldots + a_n$  for all  $n$ . If  $b_1 + b_2 + b_3 + \ldots + b_{2021} = a_k$  for some  $k$ , find  $k$ .

*Proposed by Andrew Wu*
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Context:
Determine the convergence of the series:

$$ \sum_{k=1}^{\infty} (\sqrt k - 2\sqrt {k + 1} + \sqrt {k + 2}) $$

Assume that methods like the Root Test or comparing the series directly are not applicable.
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Context:
Let $f(x) = g(x) \cdot |x-1| \cdot |x-2| \cdot |x-3|^2 \cdot |x-4|^3$, where $g(x) = x^3 + ax^2 + bx + c$. If $f(x)$ is differentiable for all $x$ and $f'(3) + f'''(4) = 0$, find the third factor of $g(x)$ given that $x-1$ and $x-2$ are factors and $f'(3) = 0$.
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Context:
Find the length of a curve of constant width \( h \), composed of an odd number of arcs of circles with radius \( h \).
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Context:
Five cards labeled 1, 3, 5, 7, 9 are laid in a row in that order, forming the five-digit number 13579 when read from left to right. A swap consists of picking two distinct cards and then swapping them. After three swaps, the cards form a new five-digit number \( n \) when read from left to right. Compute the expected value of \( n \).
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Context:
由曲线 $x^{2}=4 y, x^{2}=-4 y, x=4, x=-4$ 围成的图形绕 $y$ 轴旋转一周所得的旋转体的体积为 $V_{1}$; 满足 $x^{2}+y^{2} \leq 16, x^{2}+(y-2)^{2} \geqslant 4, x^{2}+(y+2)^{2} \geqslant 4$ 的点绕 $y$ 轴旋转一周所得的旋转体的体积为 $V_{2}$。设$V_{1}=aV_{2}+b$，请给出a+b的值。
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Context:
What is the number of elements of order $p^n$ in the group ${\bf Z}_{p^n}\oplus{\bf Z}_p$?
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Context:
Determine if the limit \( L = \lim_{n \rightarrow \infty} \int_{0}^{n} \frac{f(x/n)}{(1+x)^{4}} \, dx \) exists for a continuous function \( f : [0, \infty) \rightarrow \mathbb{R} \) such that \( |f(x)| \leq 1 + x^{2} \) for all \( x \geq 0 \), and find its value.
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Context:
Determine the Galois group of the polynomial $x^5 - 5$ over the field of rational numbers $\mathbb{Q}$.
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Context:
Virginia had 13 Sweettarts. She gave equal numbers of Sweettarts to some friends and then she ate what was left. Each person ate 3 Sweettarts. How many friends did Virginia give Sweettarts to?
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Context:
Find the function $y(x)$ that makes the integral \( \int_{0}^{1} (x^{2} + y^{'2}) \, dx \) stationary, subject to the constraints \( \int_{0}^{1} y^2 \, dx = 2 \), \( y(0) = 0 \), and \( y(1) = 0 \).
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Context:
in a fraction , if numerator is added by 2 and denominator by 1 it results 1 , and if numerator added again by 4 and denominator by 2 it results 1 / 2 . find the fraction ?
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Context:
Calculate the probability that a Brownian motion $W$ with $T_1 = 1$ year and $T_2 = 2$ years satisfies $W_{T_1} > 0$ and $W_{T_2} < 0$.
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Context:
Akash's birthday cake is a $4 \times 4 \times 4$ cube. The top and four sides of the cake are frosted, but the bottom is not. If the cake is cut into 64 smaller $1 \times 1 \times 1$ cubes, how many of the smaller cubes have frosting on exactly two faces?
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Context:
$O$ 是凸四边形 $A B C D$ 对角线 $A C$ 和 $B D$ 的交点。已知三角形 $A O B, B O C, C O D, D O A$ 的周长相同，且三角形 $A O B, B O C, C O D$ 的内切圆半径分别为 $3, 4, 6$。求三角形 $D O A$ 的内切圆半径。答案的形式为\(\frac{k}{m}\)，请给出k+m的值。
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Context:
The faces of a cube are inscribed with six natural numbers (possibly repeating) such that the numbers on adjacent faces differ by more than 1. What is the smallest possible sum of these six numbers?
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Context:
Determine the possible degree of the polynomial \(P(x)\) with real coefficients, given that \(P(a_1) = 0\) and \(P(a_{i+1}) = a_i\) for \(i = 1, 2, \ldots\), where \(\{a_i\}_{i=1,2,\ldots}\) is an infinite sequence of distinct natural numbers.
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Context:
Solve the exponential equation \(2^x + 3^x = 10000\) for \(x\).
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Context:
The triangular pyramid \(ABCD\) intersects with the plane \(P\) along the quadrilateral \(EFGH\) such that the vertices \(E\) and \(F\) lie on the edges \(AB\) and \(AC\), respectively. It is known that the plane \(P\) is parallel to the edges \(AD\) and \(BC\), the ratio of segment \(EA\) to segment \(EB\) is 2, and the edges \(AD\) and \(BC\) are equal. Find the ratio \(EF:EH\).
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Context:
In $\triangle ABC$, point $D$ is on side $AB$, and $\overrightarrow{BD}=\dfrac{1}{2}\overrightarrow{DA}$. Let $\overrightarrow{CB}=a$ and $\overrightarrow{CA}=b$. Calculate $\overrightarrow{CD}$ in terms of $a$ and $b$, expressing your answer as a linear combination of $a$ and $b$.
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Context:
$A 、 B$ 为 $y=1-x^{2}$ 上在 $y$ 轴两侧的点, 求过 $A 、 B$ 的切线与 $x$ 轴围成面积的最小值。原始答案为 \frac{m \sqrt{n}}{p} 的形式，请给出 m + n +p的值。
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Context:
A person forgot the last digit of a phone number and dialed randomly. Calculate the probability of connecting to the call in no more than 3 attempts.
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Context:
Determine the radius of convergence of the product of two power series, each with radii of convergence \( R \) and \( S \), respectively, where the functions are analytic in the complex plane.
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Context:
Determine the number of solutions to the equation \(x^3 = (1\ 2\ 3\ 4)\) in the symmetric group \(S_7\).
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Context:
Given a cube \(ABCD-A_1B_1C_1D_1\), construct a line \(l\) passing through vertex \(A_1\) such that the angles between \(l\) and the lines \(AC\) and \(BC_1\) are both \(60^\circ\). How many such lines \(l\) are there? Express your answer as a single integer.
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Context:
If \(n\) is a multiple of \(4\), find the sum \(s=1+2i+3i^2+ \ldots +(n+1)i^{n}\), where \(i=\sqrt{-1}\). The original answer is in the form \(\frac{k}{m}(n+a-b\cdot i)\). Please calculate the value of \(k + m + a + b\).
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Context:
A child spends their time drawing pictures of Native Americans (referred to as "Indians") and Eskimos. Each drawing depicts either a Native American with a teepee or an Eskimo with an igloo. However, the child sometimes makes mistakes and draws a Native American with an igloo.

A psychologist noticed the following:
1. The number of Native Americans drawn is twice the number of Eskimos.
2. The number of Eskimos with teepees is equal to the number of Native Americans with igloos.
3. Each teepee drawn with an Eskimo is matched with three igloos.

Based on this information, determine the proportion of Native Americans among the inhabitants of teepees.
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Context:
Mr. Curry is playing basketball. Suppose that, on average, he makes two successful three-point shots out of every five attempts. What is the probability that he will be able to make at least two successful 3-point shots in a row out of four attempts?
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Context:
5. $\underbrace{11 \cdots 1}_{100 \uparrow 1}+\underbrace{22 \cdots 2}_{50 \uparrow 2}+\underbrace{44 \cdots 4}_{25 \uparrow 4}+2010$ The sum of the digits of the calculation result is $\qquad$
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Context:
There was only enough room for 21 people on a raft if no one wore life jackets. There were 7 fewer people who could fit on the raft if everyone wore life jackets. If a certain number of people on the raft needed life jackets, 17 people could fit on the raft. How many people needed life jackets?
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Context:
给定正整数 $m, n(2 \leq m \leq n)$, 设 $a_{1}, a_{2}, \cdots, a_{m}$ 是 $1,2, \cdots, n$ 中任取 $m$ 个互不相同的数构成的一个排列，如果存在 $k \in\{1,2, \cdots, m\}$ 使得 $a_{k}+k$ 为奇数，或者存在整数 $k, l(1 \leq k<l \leq m)$ ，使得 $a_{k}>a_{l}$ ，则称 $a_{1}, a_{2}, \cdots, a_{m}$ 是一个 "好排列"，试确定所有好排列的个数。当m=4, n=6时，请给出答案。
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Context:
Let \( f(n) \) be the number of 0's in the decimal representation of the positive integer \( n \). For example, \( f(10001123) = 3 \) and \( f(1234567) = 0 \). Find the value of
\[ f(1) + f(2) + f(3) + \ldots + f(99999) \]
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Context:
Evaluate the integral $ \iiint_{R} \frac{dxdydz}{\sqrt{(x-a)^{2}+y^{2}+z^{2}}} $ over the solid unit sphere $ 0 \leq x^{2}+y^{2}+z^{2} \leq 1 $ for $ a > 1 $.
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Context:
What is the cardinality of a single equivalence class of Cauchy sequences in the rational numbers, corresponding to a single real number?
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Context:
Micah bought envelopes to send, and depending on the weight of the envelope he will need more stamps. If an envelope weighs more than a certain weight, he will need 5 stamps. If it weighs less than that, it will only need 2 stamps. He bought 52 stamps with 6 envelopes that weigh less than that weight. Micah needed to buy 14 envelopes in total. What is the weight limit for an envelope to need only 2 stamps?
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Context:
What is the value of $ { \sum_{1 \le i< j \le 10}(i+j)}_{i+j=odd} $ $ - { \sum_{1 \le i< j \le 10}(i+j)}_{i+j=even} $
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Context:
In a school of 1,500 students, 35% wear blue shirts, 20% wear red shirts, 10% wear yellow shirts, 5% wear green shirts, and the remaining students wear other colors. Among those wearing blue shirts, 40% have stripes, 15% have polka dots, 10% have floral patterns, and the rest have no patterns. Among those wearing red shirts, 30% have stripes, 25% have polka dots, 5% have floral patterns, and the rest have no patterns. Among those wearing yellow shirts, 20% have stripes, 10% have polka dots, and the remaining 70% have no patterns. Among those wearing green shirts, 10% have stripes, 30% have polka dots, 50% have floral patterns, and the remaining 10% have no patterns.

In addition to their shirts, 18% of the students wear glasses, 12% wear a hat, and 10% wear a scarf regardless of their other features.

How many students wear a blue shirt with stripes and glasses, a red or yellow shirt with polka dots and a hat, or a green shirt with floral patterns and a scarf?
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Context:
9. Tim has a working analog 12 -hour clock with two hands that run continuously (instead of, say, jumping on the minute). He also has a clock that runs really slow-at half the correct rate, to be exact. At noon one day, both clocks happen to show the exact time. At any given instant, the hands on each clock form an angle between $0^{\circ}$ and $180^{\circ}$ inclusive. At how many times during that day are the angles on the two clocks equal?
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Context:
Compute $\int_{\gamma}{\log(z)\over z}dz$ for $\gamma(t)=e^{it}$, $0\le t\le 2\pi$.
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Context:
Let $Z$ denote the set of points in $\mathbb{R}^n$ whose coordinates are 0 or 1. (Thus $Z$ has $2^n$ elements, which are the vertices of a unit hypercube in $\mathbb{R}^n$.) Given a vector subspace $V$ of $\mathbb{R}^n$, let $Z(V)$ denote the number of members of $Z$ that lie in $V$. Let $k$ be given, $0 \leq k \leq n$. Find the maximum, over all vector subspaces $V \subseteq \mathbb{R}^n$ of dimension $k$, of the number of points in $V \cap Z$.
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Context:
Given that the general term of the sequence $\{a_n\}$ is $a_n=2^{n-1}$, and the general term of the sequence $\{b_n\}$ is $b_n=3n$, let set $A=\{a_1,a_2,\ldots,a_n,\ldots\}$, $B=\{b_1,b_2,\ldots,b_n,\ldots\}$, $n\in\mathbb{N}^*$. The sequence $\{c_n\}$ is formed by arranging the elements of set $A\cup B$ in ascending order. Find the sum of the first 28 terms of the sequence $\{c_n\}$, denoted as $S_{28}$.
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Context:
15.1. $[9.3$ (15 points)] In trapezoid $ABCD$, diagonal $AC$ is equal to 1 and is also its height. Perpendiculars $AE$ and $CF$ are drawn from points $A$ and $C$ to sides $CD$ and $AB$ respectively. Find $AD$, if $AD=CF$ and $BC=CE$.
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Context:
In a factory, a total of $100$ parts were produced. Among them, A produced $0$ parts, with $35$ being qualified. B produced $60$ parts, with $50$ being qualified. Let event $A$ be "Selecting a part from the $100$ parts at random, and the part is qualified", and event $B$ be "Selecting a part from the $100$ parts at random, and the part is produced by A". Then, the probability of $A$ given $B$ is \_\_\_\_\_\_.
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Context:
A point $Q$ is chosen in the interior of $\triangle DEF$ such that when lines are drawn through $Q$ parallel to the sides of $\triangle DEF$, the resulting smaller triangles $u_{1}$, $u_{2}$, and $u_{3}$ have areas $16$, $25$, and $36$, respectively. Furthermore, a circle centered at $Q$ inside $\triangle DEF$ cuts off a segment from $u_3$ with area $9$. Find the area of $\triangle DEF$.
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Context:
Determine the number of increasing sequences of positive integers $a_1 \le a_2 \le a_3 \le \cdots \le a_8 \le 1023$ such that $a_i-i$ is even for $1\le i \le 8$. The answer can be expressed as $\binom{m}{n}$ for some $m > n$. Compute the remainder when $m$ is divided by 1000.
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Context:
Find the mass percentage of O in Aluminum oxide.
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Context:
4. To celebrate Barbara's birthday, Alberto proposes the following game: given the set of numbers $0,1, \ldots, 1024$, Barbara removes $2^{9}$ numbers from this set. In the next step, Alberto removes $2^{8}$ numbers from the remaining ones. It's Barbara's turn again, and she removes $2^{7}$ numbers from the remaining ones, and so on, until only 2 numbers $a$ and $b$ are left. At this point, Alberto must pay Barbara $|a-b|$ euros.

Determine the maximum amount of euros that Barbara is sure to collect, regardless of the strategy adopted by Alberto.
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Context:
Find the expression for the optimal value of \(\alpha\) in the capacitor charging equation to maximize the slope of the voltage curve at time \(T\).
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Context:
Let square $WXYZ$ have sides of length $8$. An equilateral triangle is drawn such that no point of the triangle lies outside $WXYZ$. Determine the maximum possible area of such a triangle.
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Context:
Given a sector that is one-fourth of a circle with radius $R$. Find the length of the tangent drawn from the midpoint of the arc to its intersection with the extensions of the sector's radial boundaries.
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Context:
Find the last three digits of
\[2008^{2007^{\cdot^{\cdot^{\cdot ^{2^1}}}}}.\]
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Context:
A digit was crossed out from a six-digit number, resulting in a five-digit number. When this five-digit number was subtracted from the original six-digit number, the result was 654321. Find the original six-digit number.
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Context:
A survey is conducted on the website of the football club "Rostov" to determine which of the $m$ football players visitors consider the best based on the season's results. Each visitor votes once for one football player. The website displays the rating of each football player as the percentage of votes they have received, rounded to the nearest whole number. After several visitors have voted, the total rating of the nominees added up to $95 \%$. What is the smallest possible value of $m$ for which this is possible?
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Context:
Create a cube $C_{1}$ with edge length 1. Take the centers of the faces and connect them to form an octahedron $O_{1}$. Take the centers of the octahedron's faces and connect them to form a new cube $C_{2}$. Continue this process infinitely. Find the sum of all the surface areas of the cubes and octahedrons.
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Context:
Compute
$\sum_{n=3}^{\infty}{\frac{n^{2}-2}{\left(n^{2}-1\right)\left(n^{2}-4\right)}}.$
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Context:
Let $ABC$ be a triangle with $AB=20$, $AC=34$, and $BC=42$. Let $\omega_1$ and $\omega_2$ be the semicircles with diameters $\overline{AB}$ and $\overline{AC}$ erected outwards of $\triangle ABC$. Denote by $\ell$ the common external tangent to $\omega_1$ and $\omega_2$. The line through $A$ perpendicular to $\overline{BC}$ intersects $\ell$ at $X$ and $BC$ at $Y$. The length of $\overline{XY}$ can be written in the form $m+\sqrt{n}$ where $m$ and $n$ are positive integers. Find $100m+n$.
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Context:
Calculate the cardinality of the set \(\left(\{1,2,3\}^{\mathbb{N}} - \{1,2\}^{\mathbb{N}}\right)\cap\mathcal{P}(\mathbb{N}\times\mathbb{N})\), where \(A^B\) denotes the set of all functions from \(B\) to \(A\).
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Context:
Calculate the total number of different possible staffing arrangements for three shot put match courts, where each court is staffed by two judges from different grades (10, 11, and 12), and there are two judges from each grade available. Provide your answer as a single whole number.
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Context:
Calculate the line integral of $\frac{1}{z}$ over a contour that consists of a square and a circle, both centered at the origin. Assume the contour is oriented counterclockwise.
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Context:
In the expansion of \((-xy + 2x + 3y - 6)^6\), what is the coefficient of \(x^4 y^3\)? (Answer with a specific number)
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Context:
Several irrational numbers are written on a blackboard. It is known that for every two numbers $ a$ and $ b$ on the blackboard, at least one of the numbers $ a\over b\plus{}1$ and $ b\over a\plus{}1$ is rational. What maximum number of irrational numbers can be on the blackboard?

[i]Author: Alexander Golovanov[/i]
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Context:
Evaluate the contour integral \( \oint_{C} \left( z + \cfrac{z}{z+\cfrac{z}{\vdots}} \right) dz \), where \( C \) is a contour oriented counterclockwise around (but not including) the interval \([-4, 0]\) in the complex plane.
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Context:
Determine the number of distinct values of the expression

$$
\frac{n^{2}-2}{n^{2}-n+2}
$$

where \( n \in \{1, 2, \ldots, 100\} \).
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Context:
For positive integers $n$, let $c_n$ be the smallest positive integer for which $n^{c_n} - 1$ is divisible by $210$, if such a positive integer exists, and $c_n = 0$ otherwise. What is $c_1 + c_2 + \dots + c_{210}$?
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Context:
Let $C$ be the class of all real valued continuously differentiable functions $f$ on the interval $0<x<1$ with $f(0)=0$ and $f(1)=1$. Determine the largest real number $u$ such that
$$
u \leqslant \int_{0}^{1}\left|f^{\prime}(x)-f(x)\right| d x$$
for all $f$ in $C$. The original answer is in \(\frac{m}{ne}\) format, please provide the value of m+ n.
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Context:
在凸四边形 ABCD 中，\overrightarrow{BC}=2\overrightarrow{AD}，点 P 是四边形 ABCD 所在平面上一点，满足 \overrightarrow{PA}+2020\overrightarrow{PB}+\overrightarrow{PC}+2020\overrightarrow{PD}=\overrightarrow{0}. 设 s,t 分别为四边形 ABCD 与 \triangle PAB 的面积，则 \df{t}{s}=__________.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:
Find the interval $(n, n+1)$ (where $n$ is a natural number) within which the root of the function $f(x) = 3x - 7 + \ln(x)$ lies.
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Context:
Find the minimum value of the expression $x^3 + y^3 + z^3$ subject to the constraints $x^2 + y^2 + z^2 = 1$ and $xy + xz + yz = 0$.
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Context:
a cricket bat is sold for $ 850 , making a profit of $ 215 . the profit percentage would be
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Context:
Given that $a > 0$, suppose proposition $p$: the function $y=a^{x}$ is monotonically increasing on $\mathbb{R}$; proposition $q$: the inequality $x^{2}-ax+1 > 0$ holds for all $x \in \mathbb{R}$. If $p$ and $q$ are false, and $\neg p$ is false, find the range of values for the real number $a$.
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Context:
Find minimum number \( n \) that:
1)\( {80} \mid  n \)
2) we can permute 2 different numbers in \( n \) to get \( m \) and \( {80} \mid  m \)
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Context:
An ancient Greek was born on January 7, 40 B.C., and died on January 7, 40 A.D. How many years did he live?
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Context:
Let \( z_{1}, z_{2}, \cdots, z_{12} \) be the roots of the equation \( z^{12}=2^{36} \). Let \( w_{j} (j=1,2, \cdots, 12) \) be one of \( z_{j} \) or \( iz_{j} \). Find the maximum value of \( \operatorname{Re} \sum_{j=1}^{12} w_{j} \).
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Context:
Example 4-24 Use two colors to color the 6 faces and 8 vertices of a cube. How many different schemes are there?
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Context:
Given $f(x)=\sin {x}^{2} - \sqrt{3} \cos x\cos \left(x+ \frac{π}{2}\right)$, find the interval(s) on which $f(x)$ is monotonically increasing within $\left[0, \frac{π}{2}\right]$.
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Context:
Two people take turns breaking a chocolate bar that is 6×8. During a move, they are allowed to make a straight-line break along any of the grooves. The player who cannot make a move loses. Who wins this game?
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Context:
Given a regular triangular pyramid $P-ABC$ (the base triangle is an equilateral triangle, and the vertex $P$ is the center of the base) with all vertices lying on the same sphere, and $PA$, $PB$, $PC$ are mutually perpendicular, and the side length of the base equilateral triangle is $\sqrt{2}$, calculate the volume of this sphere.
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Context:
In the sequence $\{a_n\}$, $a_n = n^2 - kn$. If for any positive integer $n$, $a_n \geq a_3$ holds, then the range of values for $k$ is.
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Context:
In a regular tetrahedron \( P-ABCD \) with lateral and base edge lengths both equal to 4, find the total length of all curve segments formed by a moving point on the surface at a distance of 3 from vertex \( P \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
On a line \( l \) in space, points \( A \), \( B \), and \( C \) are sequentially located such that \( AB = 18 \) and \( BC = 14 \). Find the distance between lines \( l \) and \( m \) if the distances from points \( A \), \( B \), and \( C \) to line \( m \) are 12, 15, and 20, respectively.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $S=\left\{ \frac{p}{q} \mid q \leq 2009, \frac{p}{q} < \frac{1257}{2009}, p,q \in \mathbb{N} \right\}$. If the maximum element of $S$ is $\frac{p_0}{q_0}$ in reduced form, find $p_0+q_0$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
If $m$ and $n$ are the roots of $x^2+mx+n=0$, $m \ne 0$, $n \ne 0$, then the sum of the roots is:
Please reason step by step, and put your final answer within \boxed{}.


Context:
Barney's grocery store sold out all of its items at the beginning of the pandemic, so they ordered extra items to restock the shelves. They sold another 1561 items that day, and have 575 items in the storeroom. They have 3,472 items left in the whole store. How many items did they order to restock the shelves?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Jack bought an ice cream cone before jogging to the beach. The ice cream cone will normally melt in 10 minutes. However, the weather conditions are affecting the melting rate of the ice cream: there's a 15mph wind blowing in the opposite direction of Jack's running path, and the temperature is 85°F, causing the ice cream to melt 25% faster than in normal conditions. The beach is 16 blocks away, and each block is 1/8th of a mile. How fast does Jack need to jog (in miles per hour) to get to the beach before the ice cream melts, considering the wind and temperature factors?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a school of wizardry, there are 13 students. Before a clairvoyance exam, the teacher seated them around a round table and asked each to guess who would receive the diploma of clairvoyance. Each student humbly did not mention themselves or their two neighbors and wrote: "None of these ten will receive it!" Of course, all those who passed the exam guessed correctly, and all the other students guessed incorrectly. How many wizards received the diploma?
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Context:
The sequence of positive integers $a_1, a_2, \dots$ has the property that $\gcd(a_m, a_n) > 1$ if and only if $|m - n| = 1$. Find the sum of the four smallest possible values of $a_2$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Two circles with radii \(\sqrt{19}\) and \(\sqrt{76}\), which touch each other externally, are inscribed in a semicircle (i.e., each of the circles touches this semicircle and its diameter). Find the radius of the semicircle.
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Context:
A person walks on a 400-meter circular track at a speed of 6 kilometers per hour. They start from point A and walk clockwise for 1 minute, then counter-clockwise for 3 minutes, and again clockwise for 5 minutes. To return to the starting point A, the minimum time required is     minutes.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the isosceles trapezoid \( KLMN \), the base \( KN \) is equal to 9, and the base \( LM \) is equal to 5. Points \( P \) and \( Q \) lie on the diagonal \( LN \), with point \( P \) located between points \( L \) and \( Q \), and segments \( KP \) and \( MQ \) perpendicular to the diagonal \( LN \). Find the area of trapezoid \( KLMN \) if \( \frac{QN}{LP} = 5 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In hexagon $ABCDEF$, $AC$ and $CE$ are two diagonals. Points $M$ and $N$ divide $AC$ and $CE$ internally such that $\frac{AM}{AC}=\frac{CN}{CE}=r$. Given that points $B$, $M$, and $N$ are collinear, find $r$.
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Context:
Let $\mathbb{Z}_{\ge 0}$ denote the set of nonnegative integers.

Define a function $f:\mathbb{Z}_{\ge 0} \to \mathbb{Z}$ with $f\left(0\right)=1$ and
\[ f\left(n\right)=512^{\left\lfloor n/10 \right\rfloor}f\left(\left\lfloor n/10 \right\rfloor\right) \]
for all $n \ge 1$. Determine the number of nonnegative integers $n$ such that the hexadecimal (base $16$) representation of $f\left(n\right)$ contains no more than $2500$ digits.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the common rational root \( k \) of the following polynomial equations which is not integral:
\[45x^4 + ax^3 + bx^2 + cx + 8 = 0\]
\[8x^5 + dx^4 + ex^3 + fx^2 + gx + 45 = 0\]
This root \( k \) is assumed to be a negative non-integer.
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Context:
In the isosceles trapezoid ABCD, where AB is parallel to DC, AB = 998, DC = 1001, and AD = 1999, find the number of points P on segment AD such that ∠BPC = 90°. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
9. The real quartic $P x^{4}+U x^{3}+M x^{2}+A x+C$ has four different positive real roots. Find the square of the smallest real number $z$ for which the expression $M^{2}-2 U A+z P C$ is always positive, regardless of what the roots of the quartic are.
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Context:
How many numbers are in the list $-50, -44, -38, \ldots, 68, 74$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the smallest number $x$ such that $10^x \equiv 1 \pmod{1997}$, using Euler's totient function, which indicates $\phi(1997) = 1996 = 4 \times 499$.
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Context:
Two congruent right circular cones each with base radius $5$ and height $12$ have the axes of symmetry that intersect at right angles at a point in the interior of the cones a distance $4$ from the base of each cone. Determine the maximum possible value of the radius $r$ of a sphere that lies within both cones.
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Context:
Find the mass percentage of Cl in Barium chloride
Please reason step by step, and put your final answer within \boxed{}.


Context:
A store sells three items: a radio, a television, and a refrigerator. The cost prices for these items are Rs. 1500, Rs. 8000, and Rs. 25000, respectively. The items were sold at the following prices: Rs. 1110, Rs. 7500, and Rs. 23000, respectively. Calculate the overall loss percentage for the store.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the constant term in the expansion of \((1+x)(e^{-2x}-e^{x})^{9}\). Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the regular tetrahedron \( P-ABC \), \( AB = 1 \), \( AP = 2 \). The plane \( \alpha \) passing through \( AB \) bisects its volume. Find the cosine of the angle between edge \( PC \) and plane \( \alpha \).
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Context:
$y''' + y'' -y' -y=0$ $y(0)=7,y'(0)=-3,y''(0)=\alpha$ Find all values of $\alpha$ for which the solution obeys $y(x) \rightarrow 0 $ as $x \rightarrow \infty$
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Context:
Let $T:l_2 \to l_2$ be defined as $T((x_1,x_2,\ldots,x_n,\ldots))=(x_2-x_1,x_3-x_2,\ldots,x_{n+1}-x_n,\ldots)$. Determine the range of possible values for $\|T\|$.
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Context:
Let $a>0$ be a real number, and $\{x_n\}$ be a sequence such that $x_1=1$ and $ax_n=x_1+x_2+\cdots+x_{n+1}$ for all $n\ge 1$. Find the minimum value of $a$ such that $x_n>0$ for all $n\ge 1$.
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Context:
Timmy's parents have a 2 car garage, which has both cars inside it. Also inside is a riding lawnmower, a bicycle for Timmy as well as each of his parents, a tricycle for Timmy's little brother Joey, a unicycle that Timmy's dad practices riding on, a skateboard, a wheelbarrow, and a four-wheeled wagon for Timmy and Joey to play with. How many wheels in total are in this garage?
Please reason step by step, and put your final answer within \boxed{}.


Context:
已知 $[x]$ 表示不超过 $x$ 的整数, 如 $[1.2]=1,[-1.2]=-2$. 已知 $\alpha=\frac{1+\sqrt{5}}{2}$, 则 $\left[\alpha^{12}\right]=$ $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A monkey climbs up and down a ladder with $n$ steps, ascending 18 steps at a time or descending 10 steps. If it can climb from the ground to the top step and then return to the ground, what is the minimum value of $n$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a rectangle of size $3 \times 4$, 4 points are chosen. Find the smallest number $C$ such that the distance between some two of these points does not exceed $C$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A circle with radius $R$ is drawn through the vertices $A$, $B$, and $C$ of a right trapezoid $ABCD$ (with $\angle A = \angle B = 90^\circ$). The circle intersects segments $AD$ and $CD$ at points $M$ and $N$ respectively, such that $AM : AD = CN : CD = 1 : 3$. Find the area of the trapezoid.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A circle is inscribed in a square, and within this circle, a smaller square is inscribed such that one of its sides coincides with a side of the larger square and two vertices lie on the circle. Calculate the percentage of the area of the larger square that is covered by the smaller square.
Please reason step by step, and put your final answer within \boxed{}.


Context:
If two lines \( l \) and \( m \) have equations \( y = -2x + 8 \), and \( y = -3x + 9 \), what is the probability that a point randomly selected in the 1st quadrant and below \( l \) will fall between \( l \) and \( m \)? Express your answer as a decimal to the nearest hundredth.
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Context:
Five people are sitting at a round table. Let $f\geq 0$ be the number of people sitting next to at least 1 female and $m\geq0$ be the number of people sitting next to at least one male. Find the number of possible ordered pairs $(f,m)$. Express your answer as a single integer.
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Context:
Given an $m$ by $n$ matrix $A$ whose elements are all 0 or 1, and each row/column of $A$ is weakly monotone, how many such matrices are there?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Allen, Shiela's brother, likes to play with blocks. Shiela repainted Allen's old blocks in different colors and shapes. If Allen has 100 identical blocks, and there are 4 shapes (cubes, cylinders, spheres, and pyramids) with different sizes and 5 blocks for each color of paint used, determine the number of color and shape combinations Shiela created.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider the set of 30 parabolas defined as follows: all parabolas have as focus the point $(0,0)$ and the directrix lines have the form $y=ax+b$ with $a$ and $b$ integers such that $a\in \{-2,-1,0,1,2\}$ and $b\in \{-3,-2,-1,1,2,3\}$. No three of these parabolas have a common point. How many points in the plane are on two of these parabolas?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the smallest positive integer $M$ such that there exists a positive integer $n$ for which:

- $M$ is the sum of the squares of some $n$ consecutive positive integers.
- $2M$ is the sum of the squares of some $2n$ consecutive positive integers.
Please reason step by step, and put your final answer within \boxed{}.


Context:
4. Consider the game "Battleship" on a $5 \times 5$ square grid. What is the minimum number of shots needed to guarantee hitting a ship of size $1 \times 4$ cells?
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Context:
Given that $\alpha$ is an angle in the third quadrant, simplify $\sqrt{ \frac{1+\cos\left(\frac{9\pi}{2}-\alpha\right)}{1+\sin(\alpha-5\pi)}} - \sqrt{ \frac{1-\cos\left(-\frac{3\pi}{2}-\alpha\right)}{1-\sin(\alpha-9\pi)}}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
after getting 2 successive discounts , a shirt with a list price of rs 150 is available at rs 105 . if the second discount is 12.55 , find the first discount .
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the set of 10 integers {1, 2, 3, ..., 9, 10}, choose any 3 distinct numbers to be the coefficients of the quadratic function f(x) = ax^2 + bx + c. Determine the number of ways to choose the coefficients such that f(1)/3 is an integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Suppose $f_n(x)=x^n\ln{x}$, $ n \in \mathbb{N}$. Find $$ \lim_{n\to\infty} \frac{f_n^{(n)}(\frac{1}{n})}{n!} $$
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Context:
某银行一年期定期储蓄年利率为2.25%，如果存款到期不取出继续留存于银行，银行自
动将本金及80%的利息（利息须交纳20%利息税，由银行代交）自动转存一年期定期储蓄，
某人以一年期定期储蓄存入银行20万元，则5年后，这笔钱款交纳利息税后的本利和为
________元.（精确到1元）
Please reason step by step, and put your final answer within \boxed{}.


Context:
16. On the plane there are two triangles, each with side lengths 18,24 and 30 . If the two triangles do not completely overlap, but share the same circumcircle as well as the same inscribed circle, find the area of the region common to both triangles.
(2 marks)
平面上有兩個三角形，邊長都是 18、24、30。若兩個三角形並非完全重疊，
但它們的外接圓相同, 內切圓也相同, 求兩個三角形重疊部分的面積。
Please reason step by step, and put your final answer within \boxed{}.


Context:
Along the path between Neznayka's and Sineglazka's houses, there were flowers planted in a row: 15 peonies and 15 tulips mixed together. While visiting Neznayka, Sineglazka watered all the flowers she encountered. After watering the 10th tulip, she ran out of water, leaving 10 flowers unwatered. The next day, while visiting Sineglazka, Neznayka picked all the flowers he encountered. After picking the 6th tulip, he decided he had enough for a bouquet. How many flowers were left growing along the path?
Please reason step by step, and put your final answer within \boxed{}.


Context:
At the baseball stadium, the snack stand offers popcorn and soda. The price before tax for popcorn is $14.70 for 5 bags and the price for soda is $2 per can. If you wanted to buy 4 bags of popcorn and 3 cans of soda, a 10% discount is given if you buy more than 3 bags of popcorn and a 5% discount applies if you buy more than 2 cans of soda. The sales tax rate is 7% for soda and 6% for popcorn. Calculate the total cost for your purchases, including the respective discounts and sales tax for each item.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the function $f(x) = 2 + \log_{3}x$, where $x \in [1,9]$, find the range of the function $y = [f(x)]^{2} + f(x^{2})$.
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Context:
Among 30 people with different ages, select two groups: the first group with 12 people and the second group with 15 people, such that the oldest person in the first group is younger than the youngest person in the second group. Determine the number of ways to make this selection.
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Context:
The set of real numbers $x$ for which

$$\dfrac{1}{x-2009}+\dfrac{1}{x-2010}+\dfrac{1}{x-2011}\ge1$$

is the union of intervals of the form $a<x\le b$. What is the sum of the lengths of these intervals?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given complex numbers \( z \) and \( \omega \) satisfying the following two conditions:
1. \( z + \omega + 3 = 0 \);
2. \( |z|, 2, |\omega| \) form an arithmetic sequence.

Is there a maximum value for \( \cos(\arg z - \arg \omega) \)? If so, find it.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of $8$-tuples of nonnegative integers $(a_1,a_2,a_3,a_4,b_1,b_2,b_3,b_4)$ satisfying $0 \le a_k \le k$, for each $k = 1,2,3,4$, and $a_1 + a_2 + a_3 + a_4 + 2b_1 + 3b_2 + 4b_3 + 5b_4 = 19$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the constant term in the expansion of (1+x)(e^(-2x)-e^x)^9. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider a trapezoidal field where it's planted uniformly with wheat. The trapezoid has the following measurements: side $AB$ is 150 m, base $AD$ (the longest side) is 300 m, and the other base $BC$ is 150 m. The angle at $A$ is $75^\circ$, and the angle at $B$ is $105^\circ$. At harvest time, all the wheat is collected at the point nearest to the trapezoid's perimeter. What fraction of the crop is brought to the longest side $AD$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In triangle \( ABC \), the angle bisector \( AL \) is drawn. Points \( E \) and \( D \) are marked on segments \( AB \) and \( BL \), respectively, such that \( DL = LC \) and \( ED \parallel AC \). Find the length of segment \( ED \), given \( AE = 15 \) and \( AC = 12 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
A square flag has a cross of equal width. The cross is symmetric about the diagonals of the square, with a small blue square in the center and the rest being white. If the entire cross (including the red and blue portions) occupies 36% of the flag's area, then what percentage of the flag's area is occupied by the blue square? Express your answer as a whole number percentage.
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Context:
Three friends met at a café: sculptor Belov, violinist Chernov, and artist Ryzhov. "It's remarkable that each of us has white, black, and red hair, but none of us has hair color matching our last name," said the black-haired person. "You're right," said Belov. What is the artist's hair color?
Please reason step by step, and put your final answer within \boxed{}.