Context:
Determine the largest two-digit number $d$ with the following property: for any six-digit number $\overline{aabbcc}$, the number $d$ is a divisor of $\overline{aabbcc}$ if and only if $d$ is a divisor of the corresponding three-digit number $\overline{abc}$.

Note: The numbers $a \ne 0$, $b$, and $c$ need not be different.
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Context:
Let  $a_1, a_2, a_3, a_4$  be integers with distinct absolute values. In the coordinate plane, let  $A_1=(a_1,a_1^2)$ ,  $A_2=(a_2,a_2^2)$ ,  $A_3=(a_3,a_3^2)$  and  $A_4=(a_4,a_4^2)$ . Assume that lines  $A_1A_2$  and  $A_3A_4$  intersect on the  $y$ -axis at an acute angle of  $\theta$ . The maximum possible value for  $\tan \theta$  can be expressed in the form  $\dfrac mn$  for relatively prime positive integers  $m$  and  $n$ . Find  $100m+n$ .

*Proposed by James Lin*
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Context:
From vertex \( B \) of parallelogram \( ABCD \), the perpendicular heights \( BK \) and \( BH \) are drawn. The segments \( KH = a \) and \( BD = b \) are known. Find the distance from point \( B \) to the orthocenter of triangle \( BKH \).
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Context:
The side edge of a regular tetrahedron \( S-ABC \) is 2, and the base is an equilateral triangle with side length 1. A section passing through \( AB \) divides the volume of the tetrahedron into two equal parts. Find the cosine of the dihedral angle between this section and the base.
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Context:
设 a_1,a_2,\cdots,a_{20}\in\{1,2,\cdots,5\}，b_1, b_2, \cdots, b_{20} \in\{1,2,\cdots,10\}. 集合 X=\{(i,j)\ |\ 1\leq i<j\leq 20,(a_i-a_j)(b_i-b_j)<0\}, 求 X 的元素个数的最大值.
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Context:
How many natural numbers from 1 to 700, inclusive, contain the digit 6 at least once?
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Context:
Among the vertices of two unequal icosahedrons, six can be chosen to form the vertices of a regular octahedron. Find the ratio of the edges of the icosahedrons.
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Context:
Given that the graph of the power function $f(x)$ passes through the point $\left( \frac {1}{2}, 8 \right),$ find the value of $f(3)$.
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Context:
Using the numbers from 1 to 22 exactly once each, Antoine writes 11 fractions. For example, he could write the fractions \(\frac{10}{2}, \frac{4}{3}, \frac{15}{5}, \frac{7}{6}, \frac{8}{9}, \frac{11}{19}, \frac{12}{14}, \frac{13}{17}, \frac{22}{21}, \frac{18}{16}, \frac{20}{1}\).

Antoine wants to have as many fractions with integer values as possible among the written fractions. In the previous example, he wrote three fractions with integer values: \(\frac{10}{2}=5\), \(\frac{15}{5}=3\), and \(\frac{20}{1}=20\). What is the maximum number of fractions that can have integer values?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $\Delta$ denote the set of all triangles in a plane. Consider the function $f: \Delta\to(0,\infty)$ defined by $f(ABC) = \min \left( \dfrac ba, \dfrac cb \right)$, for any triangle $ABC$ with $BC=a\leq CA=b\leq AB = c$. Find the set of values of $f$.
Please reason step by step, and put your final answer within \boxed{}.

\boxed{ \left[1, \frac{\sqrt{5} + 1}{2}\right) }


Context:
Given the function $f\left(x\right)=(\frac{1}{x}-a)\ln \left(1+x\right)$, if the curve $y=f(\frac{1}{x})$ is symmetric about the line $x=b$, then the value of $a+b$ is ______.
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Context:
Sandy took some money for shopping. She needed to buy items from three categories: clothing, electronics, and food. Limited-time offers were available for all items, and Sandy had to prioritize purchasing those items first.

She spent 25% of her initial money on limited-stock clothing items, 15% on limited-time electronic offers, and 10% on discounted food items. After prioritizing those purchases, she spent an additional 20% of her remaining money on non-discounted items. Finally, she paid 8% sales tax on her total purchases. With $140 remaining after shopping, what was the original amount of money Sandy took along with her?
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Context:
A circle passes through the vertices \( A \) and \( C \) of an isosceles triangle \( ABC \) (\( AB = BC \)) and intersects the sides \( AB \) and \( BC \) at points \( M \) and \( N \), respectively. The chord \( MK \) of this circle, which has a length of \( 2\sqrt{5} \), contains a point \( H \) that lies on \( AC \) and serves as the foot of the altitude of triangle \( ABC \). A line passing through point \( C \) and perpendicular to \( BC \) intersects line \( MN \) at point \( L \). Find the radius of the circle circumscribed around triangle \( MKL \) if \( \cos \angle ABK = \frac{2}{3} \).
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Context:
设A（2,0）为平面上的一个定点，$P ( \sin ( 2 t - 6 0 ^ { \circ } )$ cos(2t-$6 0 ^ { \circ } ) )$为动点，则当t由15°变到45°时，线段AP所扫过的图形的面积是 .
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Context:
The h.c.f. of two numbers is 30 and the other two factors of their l.c.m. are some values. The larger of the two numbers is 450. What is one of the other factors of their l.c.m.?
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Context:
A train of a certain length crosses an electric pole at a speed of 54 km/hr. It takes 6.666133375996587 seconds to cross the pole. What is the length of the train?
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Context:
Let there be a tiger, William, at the origin. William leaps $ 1$ unit in a random direction, then leaps $2$ units in a random direction, and so forth until he leaps $15$ units in a random direction to celebrate PUMaC’s 15th year.
There exists a circle centered at the origin such that the probability that William is contained in the circle (assume William is a point) is exactly $1/2$ after the $15$ leaps. The area of that circle can be written as $A\pi$. What is $A$?
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Context:
A bag contains an equal number of some coins, 50 paise and 25 paise coins respectively. The total value of the coins is 35 rupees. There are 20 coins of each type. What is the value of the first type of coin?
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Context:
We only know that the password of a safe consists of  $7$  different digits. The safe will open if we enter  $7$  different digits, and one of them matches the corresponding digit of the password. Can we open this safe in less than  $7$  attempts?

*(5 points for Juniors and 4 points for Seniors)*
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Context:
To a natural number \( N \), the largest divisor of \( N \) that is less than \( N \) was added, resulting in a power of ten. Find all such \( N \).
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Context:
If the inequality
$$
\left|a x^{2}+b x+a\right| \leqslant x
$$

holds for $x \in [1,2]$, then the maximum value of $3a + b$ is ___.
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Context:
7.5. Lёsha colors cells inside a $6 \times 6$ square drawn on graph paper. Then he marks the nodes (intersections of the grid lines) to which the same number of colored and uncolored squares are adjacent. What is the maximum number of nodes that can be marked?
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Context:
From the vertex of the obtuse angle \( A \) of triangle \( ABC \), a perpendicular \( AD \) is drawn. From point \( D \), a circle is drawn with radius equal to \( AD \), intersecting the sides \( AB \) and \( AC \) at points \( M \) and \( N \), respectively. Find the side \( AC \), given that \( AB = c \), \( AM = m \), and \( AN = n \).
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Context:
How many reorderings of \(2, 3, 4, 5, 6\) have the property that every pair of adjacent numbers are relatively prime?
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Context:
For distinct complex numbers $z_1,z_2,\cdots,z_{673}$, the polynomial
$$(x-z_1)^3(x-z_2)^3 \cdots (x-z_{673})^3$$
can be expressed as $x^{2019} + 20x^{2018} + 19x^{2017}+g(x)$, where $g(x)$ is a polynomial with complex coefficients and with degree at most $2016$. The sum
$$\left| \sum_{1 \le j <k \le 673} z_jz_k \right|$$
can be expressed in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
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Context:
In a race on the same distance, two cars and a motorcycle participated. The second car took 1 minute longer to cover the entire distance than the first car. The first car moved 4 times faster than the motorcycle. What portion of the distance per minute did the second car cover if it covered $\frac{1}{6}$ of the distance more per minute than the motorcycle, and the motorcycle covered the distance in less than 10 minutes?
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Context:
Given sets \( A = \{(x, y) \mid y = a x + 2\} \) and \( B = \{(x, y) \mid y = |x+1| \} \), and that \( A \cap B \) is a singleton set, determine the range of values for the real number \( a \).
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Context:
There are four entrances into Hades. Hermes brings you through one of them and drops you off at the shore of the river Acheron, where you wait in a group with five other souls. Each of these souls also came through one of the entrances, and you are all waiting to get a ride across the river.

In how many ways could the other five souls have come through the entrances such that exactly two of them came through the same entrance as you did? The order in which the souls came through the entrances does not matter, and the entrance you went through is fixed.
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Context:
Four students, named A, B, C, and D, are divided into two volunteer groups to participate in two off-campus activities. The probability that students B and C participate in the same activity is ________.
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Context:
Someone, when asked for the number of their ticket, replied: "If you add all the six two-digit numbers that can be made from the digits of the ticket number, half of the resulting sum will be exactly my ticket number." Determine the ticket number.
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Context:
Arrange the 5 natural numbers from 1 to 5 in a row from left to right, in such a way that starting from the third number, each number is either the sum or difference of the previous two numbers. How many valid arrangements satisfy these conditions? Express your answer as a single integer.
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Context:
Given the equation $\frac{{x^2}}{9-k}+\frac{{y^2}}{k-4}=1$, which represents an ellipse with foci on the y-axis, determine the range of real number k that satisfies the conditions for such an ellipse. Express your answer as an interval or range of values for k.
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Context:
If \( x - 5 \) is a factor of \( cx^3 + 23x^2 - 5cx + 55 \), find the constant \( c \).
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Context:
Let the set
\[ S = \{m \mid m \in \mathbf{Z}_{+}, \text{each prime factor of } m \text{ is less than } 10\}. \]

Find the smallest positive integer \( n \) such that in any \( n \)-element subset of \( S \), there exist four distinct numbers whose product is a perfect square.
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Context:
Through the edge \( BC \) of the triangular pyramid \( PABC \) and point \( M \), the midpoint of the edge \( PA \), a section \( BCM \) is drawn. The apex of a cone coincides with the apex \( P \) of the pyramid, and the base circle is inscribed in triangle \( BCM \) such that it touches the side \( BC \) at its midpoint. The points of tangency of the circle with segments \( BM \) and \( CM \) are the intersection points of the medians of faces \( APB \) and \( APC \). The height of the cone is twice the radius of the base. Find the ratio of the lateral surface area of the pyramid to the area of the base of the pyramid.
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Context:
Five students and two teachers are arranging for a group photo in a row. Calculate the probability that the two teachers are not at the ends and not adjacent to each other. Express your answer as a simplified fraction.
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Context:
设 $M=1^{4}+2^{4}+\cdots+2013^{4}$. 则 $M$ 的个位数字为 $\qquad$.
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Context:
In the diagram, \( PQR \) is a straight line segment and \( QS = QT \). Also, \( \angle PQS = x^\circ \) and \( \angle TQR = 3x^\circ \). If \( \angle QTS = 76^\circ \), the value of \( x \) is:
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Context:
Given a set of points in space, a *jump* consists of taking two points,  $P$  and  $Q,$  and replacing  $P$  with the reflection of  $P$  over  $Q$ . Find the smallest number  $n$  such that for any set of  $n$  lattice points in  $10$ -dimensional-space, it is possible to perform a finite number of jumps so that some two points coincide.
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Context:
Vika and Masha decided to decorate a room with balloons. They bought 15 packs of balloons, each containing 1, 2, 3, ..., 15 balloons respectively (the number of balloons is indicated on each pack). To make inflating balloons more fun, they came up with a game: they take turns inflating one balloon at a time from any pack, opening a new pack if needed. The player who opens the last pack loses. Which of the girls can guarantee a win regardless of the opponent's strategy, if Vika takes the first balloon?
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Context:
如果从数1,2，···，14中，按由小到大的顺序取出$a _ { 1 } , a _ { 2 } , a _ { 3 }$，使同时满足$a _ { 2 } -$ $a _ { 1 } \geq 3$与$a _ { 3 } - a _ { 2 } \geq 3$，那么所有符合上述要求的不同取法有 种.
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Context:
From a stack of 20 one-hundred yuan banknotes, which includes 5 counterfeit notes, two notes are randomly drawn. If one of the drawn notes is tested on a money detector and found to be counterfeit, calculate the probability that the other note is also counterfeit. Express your answer as a simplified fraction.
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Context:
设${{\left( {{x}^{6}}+x+1 \right)5}}={{a}_{0}}+{{a1}}x+{{a}_{2}}{{x}^{2}}+\cdots +{{a50}}{{x30}}$，其中${{a}_{0}}$、${{a1}}$、${{a}_{2}}$、$\cdots $、${{a30}}$是各项的系数，则在${{a}_{0}}$、${{a1}}$、${{a}_{2}}$、$\cdots $、${{a30}}$这$31$个系数中，值为零的个数为______．
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Context:
In a certain singles table tennis tournament, it was originally planned that each pair of players would play one match, but three players quit after each playing two matches. As a result, a total of 50 matches were played. How many matches were played among the three players? Express your answer as a whole number.
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Context:
Suppose a sequence $\{a\_n\}$ satisfies $\frac{1}{a\_{n+1}} - \frac{1}{a\_n} = d (n \in \mathbb{N}^*, d$ is a constant), then the sequence $\{a\_n\}$ is called a "harmonic sequence". It is known that the sequence $\{\frac{1}{x\_n}\}$ is a "harmonic sequence", and $x\_1 + x\_2 + ... + x\_{20} = 200$, find the maximum value of $x\_3 x\_{18}$.
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Context:
For each positive integer, define a function \[ f(n)=\begin{cases}0, &\text{if n is the square of an integer}  \left\lfloor\frac{1}{\{\sqrt{n}\}}\right\rfloor, &\text{if n is not the square of an integer}\end{cases}. \] Find the value of  $\sum_{k=1}^{200} f(k)$ .
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Context:
Let $$g(x)= \frac{1}{2}\sum_{k=0}^{\infty} \frac{(2x)^{2k+1}}{(2k+1){2k\choose k}}$$

If the value of $g(\frac{\sqrt{2}}{2})$ is $\dfrac{\pi}{a}$, where $a\in\Bbb R$, then what is the value of $a$?
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Context:
Let $f(x)=c x(x-1)$, where $c$ is a positive real number. We use $f^{n}(x)$ to denote the polynomial obtained by composing $f$ with itself $n$ times. For every positive integer $n$, all the roots of $f^{n}(x)$ are real. What is the smallest possible value of $c$?
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Context:
Suppose that \( \mathbf{G} = 10^{100} \). (\( \mathbf{G} \) is known as a googol.) How many times does the digit 9 occur in the integer equal to \( \mathbf{G} - 1009^{2} \)?
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Context:
At 12:00 hours, J starts to walk from his house at 6 kmph. At 13:30, P follows him from J's house on his bicycle at a certain speed. J will be 3 km behind P at 19.3 hours. What is P's cycling speed in kmph?
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Context:
A unit equilateral triangle is given. Divide each side into three equal parts. Remove the equilateral triangles whose bases are the middle one-third segments. Now we have a new polygon. Remove the equilateral triangles whose bases are the middle one-third segments of the sides of the polygon. After repeating these steps infinitely, find the area of the new shape. The original answer is in \(\dfrac{\sqrt{k}}{m}\) format, please give the value of k + m.
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Context:
Given the parabola $y^{2}=2px (p > 0)$ and the circle $x^{2}+y^{2}-4x-5=0$, find the value of $p$ if the parabola's axis is tangent to the circle.
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Context:
An even power of a certain number is a four-digit number with the first digit as 3 and the last digit as 5. Find this number.
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Context:
Compute the sum of the positive integers $n \le 100$ for which the polynomial $x^n + x + 1$ can be written as the product of at least 2 polynomials of positive degree with integer coefficients.
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Context:
Given that the points (2, 1) and (-1, 3) are on opposite sides of the line $3x-2y+a=0$, find the range of values for $a$. Express your answer as an interval in the format $\boxed{[lower, upper]}$.
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Context:
Given the hyperbola $x^{2}- \frac {y^{2}}{m}=1$ with its left and right foci being $F_{1}$ and $F_{2}$ respectively, a line passing through point $F_{2}$ intersects the right branch of the hyperbola at points $A$ and $B$. If $\triangle ABF_{1}$ is an isosceles triangle with $A$ as the vertex of the right angle, then the value of the real number $m$ is ______.
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Context:
Given that \(F_{1}\) and \(F_{2}\) are the foci of the ellipse \(\frac{x^{2}}{4} + y^{2} = 1\), and \(A\) and \(B\) are the left vertex and the top vertex of this ellipse, respectively. Point \(P\) is on line segment \(AB\). Determine the minimum value of \(\overrightarrow{P F_{1}} \cdot \overrightarrow{P F_{2}}\).
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Context:
Given a positive integer \( n \geq 3 \), for an \( n \)-element real array \(\left(x_{1}, x_{2}, \cdots, x_{n}\right)\), if every permutation \( y_{1}, y_{2}, \cdots, y_{n} \) of it satisfies \(\sum_{i=1}^{n-1} y_{i} y_{i+1} \geq -1\), then the real array \(\left(x_{1}, x_{2}, \cdots, x_{n}\right)\) is called "glowing". Find the largest constant \( C = C(n) \) such that for every glowing \( n \)-element real array, \(\sum_{1 \leq i < j \leq n} x_{i} x_{j} \geq C \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
A string has 150 beads of red, blue, and green colors. It is known that among any six consecutive beads, there is at least one green bead, and among any eleven consecutive beads, there is at least one blue bead. What is the maximum number of red beads that can be on the string?
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Context:
Using the digits $0,1,2,3, \cdots, 9$ exactly once, find the largest possible multiple of 12.
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Context:
Let $\mathcal{T}$ be a set of all linear maps from $\mathbf{L\,}(\mathbb{R^2})$ such that the following equality holds:

$$\mathbf{A^{4}-5A^2+4I=0}$$

Find the maximum number of elements in $\mathcal{T}$ such that no two maps are similar.
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Context:
A train 250 m long running at a certain speed crosses a platform in 15 sec. The length of the platform is 50.024 m. What is the speed of the train in kmph?
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Context:
Find a complex number $z$ such that $z^{2}+3$ is purely imaginary. Then $z=$____.
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Context:
How many of the numbers $1 \cdot 2 \cdot 3$, $2 \cdot 3 \cdot 4$, ..., $2020 \cdot 2021 \cdot 2022$ are divisible by $2020$?
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Context:
Let \omega=\cos \frac{2 \pi}{727}+i \sin \frac{2 \pi}{727}$. The imaginary part of the complex number $$\prod_{k=8}^{13}\left(1+\omega^{3^{k-1}}+\omega^{2 \cdot 3^{k-1}}\right)$$ is equal to $\sin \alpha$ for some angle $\alpha$ between $-\frac{\pi}{2}$ and $\frac{\pi}{2}$, inclusive. Find $\alpha$.
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Context:
Peter received a new jigsaw with a function for measuring the length of cuts for his birthday. To test the gift, Peter took a square piece of plywood with a side of 50 cm, and cut it into squares with sides of 10 cm and 20 cm. How many squares did he obtain in total if the jigsaw displayed a total cut length of \(2 \mathbf{m} 80 \mathrm{~cm}\)?
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Context:
Given the function $f(x)=4\sin (2x+\frac{\pi }{6})$ where $0\leqslant x\leqslant \frac{91\pi }{6}$, let ${x\_1}, {x\_2}, \cdots , {x\_n}$ denote all the zeros of the function $F(x)=f(x)-3$ in ascending order, i.e., ${x\_1} < {x\_2} < \cdots < {x\_n}$. Find the value of ${x\_1}+2{x\_2}+2{x\_3}+\cdots +2{x_{n-1}}+{x\_n}$.
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Context:
A $4 \times 4$ window is made out of 16 square windowpanes. How many ways are there to stain each of the windowpanes, red, pink, or magenta, such that each windowpane is the same color as exactly two of its neighbors?
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Context:
In a plane rectangular coordinate system, if the equation \( m (x^2 + y^2 + 2y + 1) = (x - 2y + 3)^2 \) represents an ellipse, what is the range of values for \( m \)?
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Context:
The current birth rate per certain number of people is 52, whereas the corresponding death rate is unknown per the same number of people. If the net growth rate in terms of population increase is 1.2 percent, and there are initially 3000 persons, what is the death rate per certain number of people?
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Context:
You have 2020 piles of coins in front of you. The first pile contains 1 coin, the second pile contains 2 coins, the third pile contains 3 coins, and so on. Therefore, the 2020th pile contains 2020 coins. Guess a positive integer $k$, such that the piles contain at least $k$ coins. Take away exactly $k$ coins from these piles. Find the minimum number of turns you need to take away all of these coins.
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Context:
Problem 7.2. (15 points) Given an isosceles triangle $A B C(A B=B C)$. On the sides $A B, B C, A C$ points $K, L, M$ are marked respectively such that $\angle A K M=90^{\circ}, \angle B L K=90^{\circ}$ and $K M=K L$. What is the measure of angle $C M L ?$
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Context:
Three people, including one girl, are to be selected from a group of $3$ boys and $2$ girls, determine the probability that the remaining two selected individuals are boys.
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Context:
Let $a_1, a_2, a_3, \dots$ be a sequence of positive real numbers such that $a_k a_{k+2} = a_{k+1} + 1$ for all positive integers $k$. If $a_1$ and $a_2$ are positive integers, find the maximum possible value of $a_{2014}$.
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Context:
Given a natural number $n$, consider the function $$ f_n(x, y) = x^n + x^{n−1}y + x^{n−2}y^2 + \dots + xy^{n−1} + y^n $$ of two real variables. Find the minimal number $k$ for which there exist functions $g_1, \dots , g_k, h_1, \dots , h_k$ of one real variable such that $$ f_n(x, y) = \sum_{i=1}^k g_i(x)h_i(y) $$
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Context:
A torus (donut) having inner radius $2$ and outer radius $4$ sits on a flat table.  What is the radius of the largest spherical ball that can be placed on top of the center torus so that the ball still touches the horizontal plane?  (If the $xy$-plane is the table, the torus is formed by revolving the circle in the $xz$-plane centered at $(3,0,1)$ with radius $1$ about the $z$-axis. The spherical ball has its center on the $z$-axis and rests on either the table or the donut.)The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given positive integers \(a_{1}, a_{2}, \cdots, a_{2018}\) satisfying
\[ a_{1} < a_{2} < \cdots < a_{2018}, \]

for \(i = 1, 2, \cdots, 2018\), let \(b_{i}\) be the number of positive integers among \(a_{1}, a_{2}, \cdots, a_{2018}\) that are less than or equal to \(i\). Then, find the value of
\[ \frac{\sum_{k=1}^{2018} a_{k} + \sum_{k=1}^{a_{2018}} b_{k}}{a_{2018} + 1}. \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that |m| = 3, |n| = 2, and mn < 0, calculate the value of m + n. Express your answer as ±a, where a is an integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given proposition $p: \forall x\geqslant 0$, $x^{2}-x\geqslant 0$, what is its negation, $\neg p$? Express your answer as a logical statement in the format of a proposition.
Please reason step by step, and put your final answer within \boxed{}.

\boxed{\exists x\geqslant 0, x^{2}-x < 0}


Context:
the bankers discount of a certain sum of money is rs . 72 and the true discount on the same sum for the same time is rs . 60 . the sum due is :
Please reason step by step, and put your final answer within \boxed{}.


Context:
A solid box is $15$ cm by $10$ cm by $8$ cm. A new solid is formed by removing a cube $3$ cm on a side from each corner of this box. What percent of the original volume is removed?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A regular $\triangle ABC$ has each of its sides divided into four equal parts. Through each division point, lines parallel to the other two sides are drawn. These lines intersect with the sides and the other parallel lines, creating 15 lattice points. If $n$ points are chosen from these 15 lattice points, determine the minimum value of $n$ such that it is guaranteed that three of the chosen points will form an isosceles triangle (including an equilateral triangle). Calculate the minimum value of $n$.
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Context:
Farmer Yang has a \(2015 \times 2015\) square grid of corn plants. One day, the plant in the very center of the grid becomes diseased. Every day, every plant adjacent to a diseased plant becomes diseased. After how many days will all of Yang's corn plants be diseased?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Example 1.12 A and B are playing table tennis, and the final score is $20: 20$. How many scenarios are there in which A has been leading B throughout the game?
Please reason step by step, and put your final answer within \boxed{}.


Context:
The product of nine consecutive natural numbers is divisible by 1111. What is the smallest possible value that the arithmetic mean of these nine numbers can take?
Please reason step by step, and put your final answer within \boxed{}.


Context:
The school is having a book-a-thon. The winning class gets a pizza party. The fifth grade has 20 students and one week to read as much as possible. The 6th grade already finished and read a total of 299 hours. How many hours does each student in 5th grade need to average per day to beat them by 1?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that $F$ is the right focus of the hyperbola $C$: ${{x}^{2}}-\dfrac{{{y}^{2}}}{8}=1$, and $P$ is a point on the left branch of $C$, $A(0,4)$. When the perimeter of $\Delta APF$ is minimized, the area of this triangle is \_\_\_.
Please reason step by step, and put your final answer within \boxed{}.


Context:
An urn is filled with coins and beads, all of which are either silver or gold. Twenty percent of the objects in the urn are beads. Forty percent of the coins in the urn are silver. What percent of objects in the urn are gold coins?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the smallest natural number that leaves a remainder of 2 when divided by 3, 4, 6, and 8.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the equation of the asymptote of the graph of $r = \cos 2 \theta \sec \theta.$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Two tribes share one shaman. One tribe lives in the river valley, and the other on a hillside. Due to the spread of the internet and rumors, the shaman had to lower the prices of his services. For the tribe living in the valley, prices decreased by $10\%$ "in furs" and $15\%$ "in fish". The ratio of fur to fish prices on the local market remains constant. For the tribe on the hillside, the price "in furs" decreased by $20\%$. By how many percent did the price "in fish" decrease for this tribe? Round the answer to one decimal place.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given vectors a=(8, 1/2x) and b=(x, 1), where x > 0, if (a-2b) || (2a+b), calculate the value of x.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In triangle ABC, the sides opposite to angles A, B, and C are a, b, and c, respectively. Let S be the area of triangle ABC. If 3a² = 2b² + c², find the maximum value of $\frac{S}{b^{2}+2c^{2}}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
How do you determine the expected number of tests needed to fix an electronic circuit with $X$ components, where the circuit functions only if all components are working, and $Y$ components are initially defective? Given that a random component is tested and replaced if defective, and the process continues until the circuit works, find the expected number of tests in the limit as both $X$ and $Y$ become large. Express your answer in terms of $X$ and Euler's constant $\gamma$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the curve
\[
(x - \arcsin \alpha)(x - \arccos \alpha) + (y - \arcsin \alpha)(y + \arccos \alpha) = 0
\]
is intersected by the line \( x = \frac{\pi}{4} \), determine the minimum value of the length of the chord intercepted as \( \alpha \) varies.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Quadrilateral $ABCD$ has right angles at $B$ and $D$, and $AC=3$. If $ABCD$ has two sides with distinct integer lengths, then what is the area of $ABCD$? Express your answer in simplest radical form.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a village in Sri Lanka, 10% of the population died due to bombardment. Of the remaining population, 20% left the village on account of fear. After some time, 5% of those who left returned to the village, and an additional 8% of the current population were injured and hospitalized. If the total population, including those hospitalized, is now 3,240, how many people were in the village in the beginning?
Please reason step by step, and put your final answer within \boxed{}.


Context:
At the cinema, guests were seated in rows. If there are 30 chairs between the guest sitting on the far left and the guest sitting on the far right, how many chairs are occupied with guests?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let the sequence $\{a_n\}$ have a sum of the first $n$ terms denoted by $S_n$, and given $a_1=1$, $a_n+a_{n+1}= \frac{1}{2^n}$ $(n=1,2,3,\ldots)$, then $S_{2n+3}=$ ______.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $P$ be the product of all non-zero digits of the positive integer $n$. For example, $P(4) = 4$, $P(50) = 5$, $P(123) = 6$, $P(2009) = 18$.
Find the value of the sum: $P(1) + P(2) + \ldots + P(2008) + P(2009)$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
17. Determine the number of ways of tiling a $4 \times 9$ rectangle by tiles of size $1 \times 2$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the smallest positive integer $b$ for which $x^2 + bx + 1764$ factors into a product of two polynomials, each having integer coefficients.
Please reason step by step, and put your final answer within \boxed{}.


Context:
We color a $98 \times 98$ checkerboard in the style of a chessboard. A move consists of selecting a rectangle made up of small squares and inverting their colors. What is the minimum number of moves required to make the entire checkerboard monochromatic?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Alex has 2 cakes - the first cake is cut into 8 slices and the second cake is cut into 12 slices. Alex gives away a fourth of the slices from the first cake and a third of the slices from the second cake to his friends. After that, he gives half of the remaining slices from both cakes to his family. Alex eats 4 slices from the first cake and 2 slices from the second cake. How many slices of the cake are left?
Please reason step by step, and put your final answer within \boxed{}.


Context:
If a, b are two positive, co-prime integers such that $$\lim _{n \rightarrow \infty}(\frac{^{3n}C_n}{^{2n}C_n})^\frac{1}{n}=\frac{a}{b}$$ then what is the value of a + b?
Please reason step by step, and put your final answer within \boxed{}.


Context:
On the extension of side $AD$ of rectangle $ABCD$ beyond point $D$, point $E$ is taken such that $DE = 0.5 AD$ and $\angle BEC = 30^\circ$.

Find the ratio of the sides of rectangle $ABCD$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Suppose that $a$, $b$, $c$ and $d$ are positive integers satisfying all of the following relations. \[abcd=2^6\cdot 3^9\cdot 5^7\] \[\text{lcm}(a,b)=2^3\cdot 3^2\cdot 5^3\] \[\text{lcm}(a,c)=2^3\cdot 3^3\cdot 5^3\] \[\text{lcm}(a,d)=2^3\cdot 3^3\cdot 5^3\] \[\text{lcm}(b,c)=2^1\cdot 3^3\cdot 5^2\] \[\text{lcm}(b,d)=2^2\cdot 3^3\cdot 5^2\] \[\text{lcm}(c,d)=2^2\cdot 3^3\cdot 5^2\] What is $\text{gcd}(a,b,c,d)$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
If the equation $a^x = 3m - 2$ ($a > 0$ and $a \neq 1$) has a negative root, find the range of the real number $m$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Anthony writes the  $(n+1)^2$  distinct positive integer divisors of  $10^n$ , each once, on a whiteboard. On a move, he may choose any two distinct numbers  $a$  and  $b$  on the board, erase them both, and write  $\gcd(a, b)$  twice. Anthony keeps making moves until all of the numbers on the board are the same. Find the minimum possible number of moves Anthony could have made.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given triangle $ABC$. Let $A_1B_1$, $A_2B_2$, $\ldots$, $A_{2008}B_{2008}$ be $2008$ lines parallel to $AB$ which divide triangle $ABC$ into $2009$ equal areas. Calculate the value of $$ \left\lfloor \frac{A_1B_1}{2A_2B_2} + \frac{A_1B_1}{2A_3B_3} + \ldots + \frac{A_1B_1}{2A_{2008}B_{2008}} \right\rfloor$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Each of two boxes contains both black and white marbles, and the total number of marbles in the two boxes is $25.$ One marble is taken out of each box randomly. The probability that both marbles are black is $27/50,$ and the probability that both marbles are white is $m/n,$ where $m$ and $n$ are relatively prime positive integers. What is $m + n$ ?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Two individuals, A and B, participate in an exam that consists of a pool of 10 questions. A can answer 6 of these questions correctly, while B can answer 8 correctly. For each test, 3 questions are randomly selected from the pool. The passing criterion is to answer at least 2 questions correctly. Calculate the probability that at least one of A or B passes the test. Express your answer as a fraction in simplest form.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Four lighthouses are located at points $A$, $B$, $C$, and $D$. The lighthouse at $A$ is $5$ kilometers from the lighthouse at $B$, the lighthouse at $B$ is $12$ kilometers from the lighthouse at $C$, and the lighthouse at $A$ is $13$ kilometers from the lighthouse at $C$. To an observer at $A$, the angle determined by the lights at $B$ and $D$ and the angle determined by the lights at $C$ and $D$ are equal. To an observer at $C$, the angle determined by the lights at $A$ and $B$ and the angle determined by the lights at $D$ and $B$ are equal. The number of kilometers from $A$ to $D$ is given by $\frac{p\sqrt{q}}{r}$, where $p$, $q$, and $r$ are relatively prime positive integers, and $r$ is not divisible by the square of any prime. Find $p + q + r$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider the function $$f(x, y, z, u) = \frac{x(1-x)y(1-y)z(1-u)}{1 - (1 - xy)z}$$ for $x, y, z, u \in (0,1)$. Find the supremum of $f(x, y, z, u)$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a sequence ${a_n}$ where the sum of the first $n$ terms is $S_n$, $a_1=1$, and it satisfies $S_n=a_{n+1}$, find the sum of the first 10 terms of the sequence ${S_n}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
An artist who needs to buy only paint and canvas finds that the cost of canvas has decreased by 40 percent and the cost of paint has decreased by 60 percent. If paint previously accounted for some times as much of the expense of painting as did canvas, the total cost for paint and canvas decreased by 55.99999999999999 percent. What is the ratio of the expense of paint to the expense of canvas before the decrease in cost?
Please reason step by step, and put your final answer within \boxed{}.


Context:
现有 11 位同学报名博物馆的志愿讲解活动，活动从上午 9 点开始到下午 5 点结束，每小时安排一场公益小讲堂，每场需要 1 位同学为参观的游客提供讲解服务. 为避免同学们劳累，馆方在排班时不会让同一人连续讲解 2 场，并且第一场与最后一场需要两位不同的同学负责. 则馆方共有__________种排班方式.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A round cake is cut into \( n \) pieces with 3 cuts. Find the product of all possible values of \( n \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
On the island of Unfortune, there live knights who always tell the truth and liars who always lie. One day, 2022 natives gathered at a round table, and each of them made the statement:

"Next to me sit a knight and a liar!"

It is known that three knights made a mistake (i.e., unintentionally lied). What is the maximum number of knights that could have been at the table?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a soap company, a soap is manufactured with 11 parts. For making one soap, you will get some parts as scrap. At the end of the day, they have 251 such scraps. From that, they can manufacture 25 more soaps. How many parts are obtained as scrap for making one soap?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A corner of a chessboard of size $(2n+1) \times (2n+1)$ is cut off. For which values of $n$ can the remaining squares be covered by $2 \times 1$ dominos in such a way that half of the dominos are horizontal?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A cylindrical can has a circumference of 24 inches and a height of 7 inches. A spiral strip is painted on the can such that it winds around the can precisely once, reaching from the bottom to the top. However, instead of reaching directly above where it started, it ends 3 inches horizontally to the right. What is the length of the spiral strip?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider a box with dimensions $10\, \text{cm} \times 16\, \text{cm} \times 1\, \text{cm}$. Determine the maximum number of balls with a diameter of $1\, \text{cm}$ that the box can contain.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $ABCD$ be a square with side length $6$. Circles $X$, $Y$, and $Z$ are congruent circles with centers inside the square such that:

- Circle $X$ is tangent to both sides $\overline{AB}$ and $\overline{AD}$.
- Circle $Y$ is tangent to both sides $\overline{AB}$ and $\overline{BC}$.
- Circle $Z$ is tangent to side $\overline{CD}$ and both circles $X$ and $Y$.

The radius of circle $X$ can be expressed as $m - \sqrt{n}$, where $m$ and $n$ are positive integers. Find $m + n$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A circle of radius \( t \) is tangent to the hypotenuse, the incircle, and one leg of an isosceles right triangle with inradius \( r = 1 + \sin \frac{\pi}{8} \). Find \( rt \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In what time will a train 250 m long cross an electric pole, if its speed is 85 km/h?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A finite increasing sequence \(a_{1}, a_{2}, \ldots, a_{n}\) of natural numbers is given, where \(n \geq 3\), and for all \(k \leq n-2\) the following equality holds: \(a_{k+2} = 3a_{k+1} - 2a_{k} - 2\). The sequence must include \(a_{k} = 2022\). Determine the maximum number of three-digit numbers, divisible by 4, that this sequence can contain.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $M$ be the least common multiple of all the integers $10$ through $30$, inclusive. Let $N$ be the least common multiple of $M$, $32$, $33$, $34$, $35$, $36$, $37$, $38$, $39$, and $40$. What is the value of $\frac{N}{M}$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Solve the equation:

$$(x+1)^{63}+(x+1)^{62}(x-1)+(x+1)^{61}(x-1)^{2}+\ldots+(x-1)^{63}=0$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
将函数 \(y=\sqrt{4+6 x-x^{2}}-2(x \in[0, 6])\) 的图象绕坐标原点逆时针方向旋转角 \(\theta(0 \leqslant \theta \leqslant \alpha)\)，得到曲线 \(C\)。若对于每一个旋转角 \(\theta\)，曲线 \(C\) 都是一个函数的图象，则 \(\alpha\) 的最大值为多少？答案的形式为\(\arctan \frac{k}{m}\)，请给出k+m的值。
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a function $f(x) = \frac {1}{2}\sin x - \frac {\sqrt {3}}{2}\cos x$ defined on the interval $[a, b]$, the range of $f(x)$ is $[-\frac {1}{2}, 1]$. Find the maximum value of $b-a$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the mathematical text "Zhang Qiujian Suanjing" from the Northern and Southern Dynasties of China, there is a problem: "There are ten ranks of people, and the palace gives gold to each rank in descending order of difference. The first three people enter and receive 4 pounds of gold, and the last three people enter and receive 3 pounds of gold. The middle four people have not yet received their share, but they will also receive gold according to their rank." Calculate the total amount of gold received by the fifth and sixth ranked people. Express your answer as a simplified fraction.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In how many ways can 8 identical rooks be placed on an $8 \times 8$ chessboard symmetrically with respect to the diagonal that passes through the lower-left corner square?
Please reason step by step, and put your final answer within \boxed{}.


Context:
For the sequence $\{c_n\}$, if there exists an arithmetic sequence $\{a_n\}$ with all terms being positive integers and a geometric sequence $\{b_n\}$ with all terms being positive integers such that $c_n = a_n + b_n$, then the sequence $\{c_n\}$ is called a "DQ sequence". It is known that the sequence $\{e_n\}$ is a "DQ sequence", and its first 5 terms are: 3, 6, 11, 20, 37. Then $e_n = \_\_\_\_\_\_$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
7. Given the equation $8 t^{3}-4 t^{2}-4 t+1=0$ has a root $x$ in $\left(0, \frac{\pi}{13}\right)$, then $x=$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Jack plays a game in which he first rolls a fair six-sided die and gets some number $n$, then, he flips a coin until he flips $n$ heads in a row and wins, or he flips $n$ tails in a row in which case he rerolls the die and tries again. What is the expected number of times Jack must flip the coin before he wins the game.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given $60\%$ of students like dancing and the rest dislike it, $80\%$ of those who like dancing say they like it and the rest say they dislike it, also $90\%$ of those who dislike dancing say they dislike it and the rest say they like it. Calculate the fraction of students who say they dislike dancing but actually like it.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the convex quadrilateral $ABCD$, points $E$ and $F$ are the midpoints of sides $BC$ and $CD$ respectively. Segments $AE$, $AF$, and $EF$ divide the quadrilateral into 4 triangles whose areas are consecutive natural numbers. What is the greatest possible area of triangle $ABD$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
2. 103 Use 94 large and small bricks, each with dimensions $4 \times 10 \times 19$, to stack into a tower 94 bricks high, where each brick can be placed to provide a height of 4, 10, or 19. If all 94 bricks are used, how many different heights can the tower be (height unit in inches)?
Please reason step by step, and put your final answer within \boxed{}.


Context:
10. For positive integer $n$ and real number $x (0 \leqslant x < n)$, define
$$
f(n, x)=(1-\{x\}) \cdot C_{n}^{[x]}+\{x\} \cdot C_{n}^{[x]+1},
$$

where $[x]$ denotes the greatest integer not exceeding the real number $x$, and $\{x\}=x-[x]$. If integers $m, n \geqslant 2$ satisfy
$$
f\left(m, \frac{1}{n}\right)+f\left(m, \frac{2}{n}\right)+\cdots+f\left(m, \frac{m n-1}{n}\right)=123,
$$

find the value of $f\left(n, \frac{1}{m}\right)+f\left(n, \frac{2}{m}\right)+\cdots+f\left(n, \frac{m n-1}{m}\right)$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a sequence with the general term formula $a_n=n^2-(6+2\lambda)n+2014$, if $a_6$ or $a_7$ is the minimum term of the sequence $\{a_n\}$, then the range of values for the real number $\lambda$ is ______.
Please reason step by step, and put your final answer within \boxed{}.

\boxed{\left( \frac{5}{2}, \frac{9}{2} \right)}


Context:
A line \( MN \) parallel to the base \( AB \) is drawn through the center of the circle inscribed in triangle \( ABC \) (\(M\) lies on \( BC \), \(N\) on \( AC \)). Find the perimeter of the quadrilateral \( ABMN \), given that \( AB = 5 \) and \( MN = 3 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
A square $EFGH$ is inscribed in the region bound by the parabola $y = x^2 - 8x + 16$ and the $x$-axis. Find the area of square $EFGH$.

[asy]
unitsize(0.8 cm);

real parab2 (real x) {
  return(x^2 - 8*x + 16);
}

pair E, F, G, H;
real x = -2 + 2*sqrt(3);

E = (4 - x,0);
F = (4 + x,0);
G = (4 + x,-2*x);
H = (4 - x,-2*x);

draw(graph(parab2,0.5,7.5));
draw(E--H--G--F);
draw((0,0)--(8,0));

label("$E$", E, N);
label("$F$", F, N);
label("$G$", G, SE);
label("$H$", H, SW);
[/asy]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given an ellipse \(\Gamma: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 \) with \(a > b > 0\). The left and right endpoints of the major axis are \(A\) and \(B\), respectively, and the left and right foci are \(F_{1}\) and \(F_{2}\), respectively. If there exists a point \(P\) on the ellipse \(\Gamma\) such that \(P F_{1}\) and \(P F_{2}\) trisect \(\angle APB\), find the number of distinct eccentricities \(e\) of the ellipse \(\Gamma\) that satisfy this condition.
Please reason step by step, and put your final answer within \boxed{}.


Context:
If \( x \) and \( y \) are two coprime positive integers such that
\[ x + \frac{2016}{x} = 32y + \frac{63}{y}, \]
then how many pairs \((x, y)\) satisfy the condition?
Please reason step by step, and put your final answer within \boxed{}.


Context:
## Task B-2.7.

On the graph of the function $f(x)=-x^{2}+2022 x$, all integer points $(x, y)$ for which $y \geqslant-2023$ are marked in blue. On the x-axis, all integer points $(x, 0)$ are marked in red such that $a<x<b, f(a)=f(b)=0$. How many different triangles are there with vertices at two blue points and one red point?
Please reason step by step, and put your final answer within \boxed{}.


Context:
For a positive integer $n$, define a function $f_n(x)$ on the interval $[0, n+1]$ as follows:
\[ f_n(x) = \left( \sum_{i=1}^{n} |x-i| \right)^2 - \sum_{i=1}^{n} (x-i)^2. \]
Let $a_n$ be the minimum value of $f_n(x)$. Find the value of:
\[ \sum_{n=1}^{11} (-1)^{n+1} a_n. \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine how many ordered pairs of positive integers $(x, y)$, where $x < y$, have a harmonic mean of $5^{20}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given $30$ students such that each student has at most $5$ friends and for every $5$ students there is a pair of students that are not friends, determine the maximum $k$ such that for all such possible configurations, there exists $k$ students who are all not friends.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate $\int_{0}^{2\pi} \frac{\sin{3\theta}}{5+4\sin{\theta}}\,d{\theta}$ using complex integration.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The diameter of a cylindrical tin is some value and the height is 5 cm. The volume of the cylinder is 125.00000000000001 cm³. What is the diameter of the cylindrical tin?
Please reason step by step, and put your final answer within \boxed{}.


Context:
The radius of the circumcircle of the acute-angled triangle \(ABC\) is 1. It is known that on this circumcircle lies the center of another circle passing through the vertices \(A\), \(C\), and the orthocenter of triangle \(ABC\). Find \(AC\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
A fashion designer sold a pair of jeans to a wholesale distributor for 35 percent more than it cost to manufacture the pair of jeans. The wholesale distributor then sold the pair of jeans to a retail store for 25 percent more than the distributor's purchase price. Finally, a customer bought the pair of jeans for 45 percent more than the retailer paid for them. The price the customer paid was what percent greater than the cost of manufacturing the jeans?
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Context:
Sophia started saving a certain amount of money every day. In 20 days, she would have saved 2 dimes in total. How much money does she save each day?
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Context:
Each of two angles of a triangle is $60^{\circ}$ and the included side is $4$ inches. The area of the triangle, in square inches, is:
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Context:
Given 1 coin of 0.1 yuan, 1 coin of 0.2 yuan, 1 coin of 0.5 yuan, 4 coins of 1 yuan, and 2 coins of 5 yuan, how many different amounts of money can be paid using any combination of these coins?
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Context:
Adnan is trying to remember his four-digit PIN. He is sure it contains the digits 5, 3, 7, and 0 but can't recall the order in which they appear. How many different arrangements are possible for his PIN?
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Context:
A certain store sells a kind of product. Due to a decrease in the purchase price by 8% and an increase in the profit margin by 10%, what was the original profit margin of this product? (Answer in percentage) (Purchase Price × Profit Margin = Profit)
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Context:
Given a hyperbola $C_{1}$ defined by $2x^{2}-y^{2}=1$, find the area of the triangle formed by a line parallel to one of the asymptotes of $C_{1}$, the other asymptote, and the x-axis.
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Context:
4. A positive integer n is given.Find the smallest  $k$  such that we can fill a  $3*k$  gird with non-negative integers
such that: $\newline$   $i$ ) Sum of the numbers in  each column is  $n$ .
  $ii$ ) Each of the numbers  $0,1,\dots,n$  appears at least once in each row.
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Context:
The \emph{Stooge sort} is a particularly inefficient recursive sorting algorithm defined as follows: given an array $A$ of size $n$, we swap the first and last elements if they are out of order. We then (if $n \ge 3$) Stooge sort the first $\lceil \frac{2n}{3} \rceil$ elements, then the last $\lceil \frac{2n}{3} \rceil$, then the first $\lceil \frac{2n}{3} \rceil$ elements again. Given that this runs in $O(n^\alpha)$, where $\alpha$ is minimal, find the value of $(243/32)^\alpha$.
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Context:
30 students from five courses created 40 problems for the olympiad, with students from the same course creating the same number of problems, and students from different courses creating different numbers of problems. How many students created exactly one problem?
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Context:
If the point $\left(m,n\right)$ in the first quadrant is symmetric with respect to the line $x+y-2=0$ and lies on the line $2x+y+3=0$, calculate the minimum value of $\frac{1}{m}+\frac{8}{n}$.
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Context:
A regular triangular prism \(ABC A_{1}B_{1}C_{1}\) is inscribed in a sphere with the base \(ABC\) and lateral edges \(AA_{1}\), \(BB_{1}\), \(CC_{1}\). The segment \(C_{1}D\) is the diameter of this sphere, and point \(K\) is the midpoint of edge \(CC_{1}\). Find the volume of the prism if \(DK=2\) and \(DA=\sqrt{6}\).
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Context:
For positive integers $x$, let $g(x)$ be the number of blocks of consecutive 1's in the binary expansion of $x$. For example, $g(19)=2$ because $19=10011_{2}$ has a block of one 1 at the beginning and a block of two 1's at the end, and $g(7)=1$ because $7=111_{2}$ only has a single block of three 1's. Compute $g(1)+g(2)+g(3)+\cdots+g(256)$.
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Context:
We have a five-digit positive integer  $N$ . We select every pair of digits of  $N$  (and keep them in order) to obtain the  $\tbinom52 = 10$  numbers  $33$ ,  $37$ ,  $37$ ,  $37$ ,  $38$ ,  $73$ ,  $77$ ,  $78$ ,  $83$ ,  $87$ .  Find  $N$ .

*Proposed by Lewis Chen*
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Context:
Given that a high school senior year has 12 classes, with exactly 8 classes to be proctored by their own homeroom teachers, find the number of different proctoring arrangements for the math exam.
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Context:
What is the value of the sum
\[
  \sum_z \frac{1}{{\left|1 - z\right|}^2} \, ,
\]where $z$ ranges over all 7 solutions (real and nonreal) of the equation $z^7 = -1$?
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Context:
In rectangle \(ABCD\), point \(E\) is on side \(BC\) and point \(F\) is on side \(CD\), satisfying \(AE = AF\). Given that the area of \(\triangle ADF\) is 2, the area of \(\triangle ABE\) is 4, and the area of \(\triangle CEF\) is 8, find the value of \(7 \cos \angle EAF\).
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Context:
Given an arithmetic-geometric sequence $\{a_{n}\}$ where $S_{n}$ represents the sum of the first $n$ terms, if $27a_{3}-a_{6}=0$, then find the value of $\frac{S_{6}}{S_{3}}$.
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Context:
5.66 A deck of cards has a total of 54 cards. The magician divides them into several piles, and the audience writes a natural number on each card, which is equal to the number of cards in the pile where the card is located. Then the magician mixes the cards in a special way and divides them into several piles again. The audience writes a natural number on each card again, which is also equal to the number of cards in the pile where the card is now located. This process continues. How many times does this process of dividing piles and writing numbers need to be repeated at least, so that the (unordered) number arrays written on each card are all different?
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Context:
\(ABCD\) is a convex quadrilateral in which \(AC\) and \(BD\) meet at \(P\). Given \(PA = 1\), \(PB = 2\), \(PC = 6\), and \(PD = 3\). Let \(O\) be the circumcenter of \(\triangle PBC\). If \(OA\) is perpendicular to \(AD\), find the circumradius of \(\triangle PBC\).
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Context:
若复数 \(z\) 满足 \(\frac{z}{2}\) 与 \(\frac{2}{z}\) 的实部、虚部均在区间 \([-1,1]\) 上取值，则 \(z\) 在复平面上形成轨迹的面积是多少？答案的形式为 \(a-b\pi\)，请给出\(a+b\)的值。
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Context:
In triangle  $ABC$ , let  $P$  and  $R$  be the feet of the perpendiculars from  $A$  onto the external and internal bisectors of  $\angle ABC$ , respectively; and let  $Q$  and  $S$  be the feet of the perpendiculars from  $A$  onto the internal and external bisectors of  $\angle ACB$ , respectively. If  $PQ = 7, QR = 6$  and  $RS = 8$ , what is the area of triangle  $ABC$ ?
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Context:
Bobby initially had 250 pieces of candy and 175 pieces of chocolate. He ate 38 pieces of candy and then ate 36 more. After that, he shared 12 pieces of candy with his friends. Meanwhile, he ate 16 pieces of chocolate and bought 28 more. How many more pieces of candy than chocolate did Bobby have left?
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Context:
某足球邀请赛有十六个城市参加, 每市派出甲、乙两个队, 根据比赛规则, 比赛若干天后进行统计, 发现除 $A$ 市甲队外, 其它各队已比赛过的场数各不相同. 问 $A$ 市乙队已赛过多少场？
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Context:
Given $S=(x-a)^{2}+(\ln x-a)^{2}$ $(a\in\mathbb{R})$, then the minimum value of $S$ is ______________.
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Context:
A rectangular wall is covered entirely with three kinds of decorative tiles: small, regular, and jumbo. The small tiles have a length-to-width ratio of 2:1, while the regular and jumbo tiles have the same length-to-width ratio of 3:1. The wall consists of 2/5 small tiles, 3/10 regular tiles, and 1/4 jumbo tiles. Jumbo tiles have a length three times that of regular tiles. If regular tiles cover 90 square feet of the wall, and no tiles overlap, what is the area of the entire wall?
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Context:
In an acute triangle $\triangle ABC$, altitudes $\overline{AD}$ and $\overline{BE}$ intersect at point $H$. Given that $HD=6$ and $HE=3$, calculate $(BD)(DC)-(AE)(EC)$.
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Context:
A trader bought a car at a 20% discount on its original price. He then made a few modifications to the car and increased the price by 15%. Due to market conditions, he had to offer a 10% discount before finally selling the car at a 45% increase on the last discounted price. What percent of profit did he make on the original price?
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Context:
Kevin colors three distinct squares in a  $3\times 3$  grid red. Given that there exist two uncolored squares such that coloring one of them would create a horizontal or vertical red line, find the number of ways he could have colored the original three squares.
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Context:
Let $\mathcal{P}_{1}, \mathcal{P}_{2}, \mathcal{P}_{3}$ be pairwise distinct parabolas in the plane. Find the maximum possible number of intersections between two or more of the $\mathcal{P}_{i}$. In other words, find the maximum number of points that can lie on two or more of the parabolas $\mathcal{P}_{1}, \mathcal{P}_{2}, \mathcal{P}_{3}$.
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Context:
A scuba diver descends at a rate of 80 feet per minute under normal conditions. However, during this dive, the diver encounters two different currents and must make stops for decompression. From the surface to 1500 feet, the diver faces a downward current, which increases the descending speed by 30 feet per minute. Between 1500 and 3000 feet, there is an upward current slowing the diver's descent by 20 feet per minute. Below 3000 feet, the diver resumes the normal descending speed.

Additionally, the diver must make two decompression stops, one at 1800 feet for 5 minutes and another at 3600 feet for 8 minutes. The lost ship is located at a depth of 4000 feet below sea level.

Considering these factors, how long will it take the diver to reach the lost ship?
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Context:
Jane started babysitting when she was 20 years old. Whenever she babysat for a child, that child was no more than half her age at the time. Jane stopped babysitting 10 years ago when she was 22 years old. She babysat for kids with age gaps between them not more than 1 year. If Jane is currently 32 years old, what is the current age of the oldest person for whom Jane could have babysat, given that this person's age is a prime number?
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Context:
Given the line $L: x+y-9=0$ and the circle $M: 2x^2+2y^2-8x-8y-1=0$, point $A$ is on line $L$, and points $B$ and $C$ are on circle $M$. In $\triangle ABC$, $\angle BAC=45^\circ$, and $AB$ passes through the center $M$ of the circle. The range of the $x$-coordinate of point $A$ is \_\_\_\_\_\_.
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Context:
Find the number which when multiplied by 15 is increased by 196. What is the result?
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Context:
A sample of 5 individuals has values a, 0, 1, 2, 3. Given that the average value of the sample is 1, calculate the variance of the sample using the formula for variance. Express your answer as a single numerical value.
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Context:
Consider nine points in space, no four of which lie in a plane. We connect each pair of points with an edge (i.e., a straight line segment), and color each such edge red, blue, or leave it uncolored. Determine the smallest value of $n$ such that whenever the number of colored edges is exactly $n$, the set of colored edges necessarily contains a triangle whose edges are all the same color.
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Context:
Find the domain of the function $f(x)= \frac{ \sqrt{x}}{\ln (2-x)}$. Express your answer as an interval or union of intervals.
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Context:
Split a face of a regular tetrahedron into four congruent equilateral triangles. How many different ways can the seven triangles of the tetrahedron be colored using only the colors orange and black? (Two tetrahedra are considered to be colored the same way if you can rotate one so it looks like the other.)

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Context:
Split a face of a regular tetrahedron into four congruent equilateral triangles. How many different ways can the seven triangles of the tetrahedron be colored using only the colors orange and black? (Two tetrahedra are considered to be colored the same way if you can rotate one so it looks like the other.)
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Context:
Evaluate the contour integral $$\oint\limits_{|z-3|=4}^{}\frac{z}{\cos z-1}dz$$ using the residue theorem.
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Context:
Let $O$ be the set of odd numbers between 0 and 100. Let $T$ be the set of subsets of $O$ of size 25. For any finite subset of integers $S$, let $P(S)$ be the product of the elements of $S$. Define $n = \sum_{S \in T} P(S)$. If you divide $n$ by 17, what is the remainder?
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Context:
A positive integer is called oddly even if the sum of its digits is even. Find the sum of the first 2013 oddly even numbers.
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Context:
Evaluate the following integral:
$$I=\int_0^1 \frac{\log^3(1-x)}{x}dx$$
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Context:
A robot walks 1m in a certain direction, then turns left 45°, walks another 1m, then turns left 45° again, and repeats the instruction "walk 1m, then turn left 45°". When the robot returns to the starting point for the first time, it has walked     m.
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Context:
The decimal representation of
$$\dfrac{1}{20^{20}}$$
consists of a string of zeros after the decimal point, followed by a $9$ and then several more digits. How many zeros are in that initial string of zeros after the decimal point?
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Context:
Sebastian plays drums for percussion in an orchestra seated in various sections. In the percussion section, there are three more players besides Sebastian: one playing the timpani, another playing the marimba, and a third who switches between the xylophone and the triangle.

He sits next to the brass section where five people play trombone, four play trumpet, two play the tuba, two play a French horn and one person alternates between the French horn and trumpet. In front of them are the strings section with seven violinists, five violists, four cellists, and two contrabassists. Among the violinists, one also plays the viola, and a cellist sometimes switches to double bass.

Adjacent to the strings are the woodwinds section with three clarinet players, a quartet of flute players, two bassoonists, and an oboist. One of the clarinet players also doubles as a bass clarinet player. A pianist, a harpist, and a musician who plays both the celesta and the organ join the orchestra as well in the keyboards and harp section.

The maestro conducts them all. How many people are in the orchestra considering the musicians who play multiple instruments?
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Context:
If the largest angle in a triangle is twice the size of the smallest angle, and the lengths of its sides are three consecutive natural numbers, find the side lengths of the triangle. Express your answer as three consecutive natural numbers.
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Context:
What is the maximum number of \(4 \times 1\) rectangles that can be placed in a \(6 \times 6\) square (without crossing the cell boundaries)?
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Context:
A ball of radius R and mass m is magically put inside a thin shell of the same mass and radius 2R. The system is at rest on a horizontal frictionless surface initially. When the ball is, again magically, released inside the shell, it sloshes around in the shell and eventually stops at the bottom of the shell. Calculate the distance the shell moves from its initial contact point with the surface. Express your answer in terms of R.
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Context:
How many natural numbers \(n\) are there such that

\[ 100 < \sqrt{n} < 101 ? \]
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Context:
3. (1990 Vietnam Olympiad) Let $a, b, c \in \mathbf{R}^{+}$, and $a b c + a + c = b$, try to determine the maximum value of $p = \frac{2}{a^{2}+1} - \frac{2}{b^{2}+1} + \frac{3}{c^{2}+1}$.
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Context:
Tuition at the music school is $45 per person for regular enrollments. There is a $15 discount for early bird enrollments, a $15 discount for the first sibling, and a $10 discount for each additional sibling. Additionally, there is a $20 extra cost per person for weekend classes and a $10 discount per person if enrolled in more than one musical instrument.

Ali, an early bird registrant, and her brother, Matt, signed up for weekday classes for one instrument each. Their two sisters, Jane and Sarah, enlisted for weekend classes to learn two instruments each. How much will it cost for all four children?
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Context:
5. [5 points] On a plane with a given rectangular Cartesian coordinate system, a square is drawn with vertices at points $(0 ; 0),(0 ; 59),(59 ; 59)$, and $(59 ; 0)$. Find the number of ways to choose two grid nodes inside this square (not including its boundary) such that at least one of these nodes lies on one of the lines $y=x$ or $y=59-x$, but neither of the chosen nodes lies on any line parallel to any of the coordinate axes.
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Context:
Let $A=\{x\in \mathbb{N} | -2 \leq x \leq 2\}$ and $B=\{2,4\}$. Find the union of sets A and B, denoted as A∪B. Express your answer as a set of numbers, including all unique elements from both sets.
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Context:
In the polar coordinate system, the curve $\rho=4\sin \left( \theta- \frac{\pi}{3} \right)$ is symmetric about what axis?
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Context:
An arrangement is defined as any way to seat $n$ couples at a round table with $2n$ places. A couple is said to be together if the two members are seated next to each other. What is the average number of couples that are together per arrangement?
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Context:
Each segment with endpoints at the vertices of a regular 100-gon is colored red if there is an even number of vertices between its endpoints, and blue otherwise (in particular, all sides of the 100-gon are red). Numbers were placed at the vertices such that the sum of their squares equals 1, and at the segments, the products of the numbers at the endpoints were placed. Then, the sum of the numbers on the red segments was subtracted by the sum of the numbers on the blue segments. What is the largest possible value that could be obtained?

I. Bogdanov
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Context:
Find the minimum value of $k$ such that the inequality $$\sin{a}\leq\frac{a}{\sqrt{1+\frac{a^2}{k}}}$$ holds for all $a>0$.
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Context:
There are 3 garages at Liam's house. In the first garage, there are 5 bicycles, 6 tricycles, 9 unicycles, and 3 quadracycles, but 2 of the bicycles are missing a wheel. Each bicycle typically has 2 wheels, each tricycle has 3 wheels, each unicycle has 1 wheel, and each quadracycle has 4 wheels. In the second garage, there are 2 bicycles, 1 tricycle, 3 unicycles, 4 quadracycles, and 2 pentacycles, with 1 of the pentacycles missing 2 wheels. Each pentacycle typically has 5 wheels. In the third garage, there are 3 bicycles, 4 tricycles, 2 unicycles, 2 quadracycles, 1 pentacycle, and a hexacycle with a sidecar, but 1 of the unicycles is missing a wheel, and the hexacycle with a sidecar is missing 1 wheel. Each hexacycle typically has 6 wheels, and a sidecar adds 1 extra wheel. How many wheels are there in total across all three garages?
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Context:
A 470 m long train is running at a speed of 55 km/hr. It crossed a platform of a certain length in 64.79481641468682 seconds. What is the length of the platform?
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Context:
Ian started saving a certain amount of money every day. After 40 days, he would have saved 4 dimes in total. How much money does Ian save each day?
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Context:
Five points lie on a straight line. Alex finds the distances between every pair of points. He obtains, in increasing order, 2, 5, 6, 8, 9, \( k \), 15, 17, 20, and 22. What is the value of \( k \)?
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Context:
Let $G = (V,E)$ be a connected graph and $\alpha \in \mathbb{R}$. A graph $H =(V_H, E_H)$ is called an $\alpha$-Spanning Tree of $G$ if $V_H = V$, $E_H\subseteq E$, and for every pair of vertices $u,v$, the inequality $d_H(u,v) \leq \alpha \cdot d_G(u,v)$ applies. What is the smallest possible value of $\alpha$ such that a grid graph $M_{n,n}$ has an $\alpha$-spanning tree?
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Context:
Let $f(x)$ be an odd function with a period of 3 defined over $\mathbb{R}$. Given that $f(2) = 0$, the number of solutions to the equation $f(x) = 0$ in the interval $[0, 6]$ is \_\_\_\_\_.
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Context:
已知 \(z, w \in \mathbf{C},|z+w|=1,\left|z^{2}+w^{2}\right|=4\)，求 \(|z w|\) 的最大值。原始答案的形式为 \(\frac{k}{m}\)，请给出\(k+m\)的值。
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Context:
两个半径都是 1 的球 $O_{1}$ 和球 $O_{2}$ 相切，且均与直二面角 $\alpha-l-\beta$ 的两个半平面都相切，另有一个半径为 $r(r<1)$ 的小球 $O$与这个二面角的两个半平面也都相切, 同时,与球 $O_{1}$ 和球 $O_{2}$ 都外切。则 $r$ 的值为原始答案为 $n-\sqrt{k}$ 的形式，请给出n + k的值。
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Context:
Four points are marked on a straight line and one point is marked outside the line. There are six triangles that can be formed with vertices at these points.

What is the maximum number of these triangles that can be isosceles?
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Context:
In a convex 13-gon, all the diagonals are drawn. They divide it into polygons. Consider a polygon among them with the largest number of sides. What is the maximum number of sides it can have?
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Context:
Esmeralda writes $2n$ real numbers $x_1, x_2, \dots , x_{2n}$, all belonging to the interval $[0, 1]$, around a circle and multiplies all the pairs of numbers neighboring to each other, obtaining, in the counterclockwise direction, the products $p_1 = x_1x_2$, $p_2 = x_2x_3$, $\dots$ , $p_{2n} = x_{2n}x_1$. She adds the products with even indices and subtracts the products with odd indices. What is the maximum possible number Esmeralda can get?
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\boxed{ \left\lfloor \frac{n}{2} \right\rfloor }


Context:
While there do not exist pairwise distinct real numbers  $a,b,c$  satisfying  $a^2+b^2+c^2 = ab+bc+ca$ , there do exist complex numbers with that property. Let  $a,b,c$  be complex numbers such that  $a^2+b^2+c^2 = ab+bc+ca$  and  $|a+b+c| = 21$ . Given that  $|a-b| = 2\sqrt{3}$ ,  $|a| = 3\sqrt{3}$ , compute  $|b|^2+|c|^2$ .
<details><summary>Clarifications</summary>

- The problem should read  $|a+b+c| = 21$ .  An earlier version of the test read  $|a+b+c| = 7$ ; that value is incorrect.
- $|b|^2+|c|^2$  should be a positive integer, not a fraction; an earlier version of the test read ``... for relatively prime positive integers  $m$  and  $n$ .  Find  $m+n$ .''
</details>

*Ray Li*
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Context:
Let $ABCD$ be a square with vertices $A=(0,0)$ and $B=(1,1)$. A point $P\left(\frac{2}{7},\frac{1}{4}\right)$ is located inside the square. An ant begins its journey from point $P$, touches three sides of the square, and returns to point $P$. The shortest possible distance the ant can travel is expressed as $\frac{\sqrt{a}}{b}$, where $a$ and $b$ are integers, and $a$ is not divisible by any square number other than $1$. Determine the value of $(a+b)$.
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Context:
Find the sum of the two smallest odd primes  $p$  such that for some integers  $a$  and  $b$ ,  $p$  does not divide  $b$ ,  $b$  is even, and  $p^2=a^3+b^2$ .

*2021 CCA Math Bonanza Individual Round #13*
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Context:
A fair six-sided die has faces numbered 1, 2, 3, 4, 5, 6. The die is rolled four times, resulting in numbers a, b, c, d. Calculate the probability that one of the numbers in the set {a, a+b, a+b+c, a+b+c+d} equals 4. Express your answer as a simplified fraction.
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Context:
The new individual income tax law has been implemented since January 1, 2019. According to the "Individual Income Tax Law of the People's Republic of China," it is known that the part of the actual wages and salaries (after deducting special, additional special, and other legally determined items) obtained by taxpayers does not exceed $5000$ yuan (commonly known as the "threshold") is not taxable, and the part exceeding $5000$ yuan is the taxable income for the whole month. The new tax rate table is as follows:

2019年1月1日后个人所得税税率表

| 全月应纳税所得额 | 税率$(\%)$ |
|------------------|------------|
| 不超过$3000$元的部分 | $3$ |
| 超过$3000$元至$12000$元的部分 | $10$ |
| 超过$12000$元至$25000$元的部分 | $20$ |
| 超过$25000$元至$35000$元的部分 | $25$ |

Individual income tax special additional deductions refer to the six special additional deductions specified in the individual income tax law, including child education, continuing education, serious illness medical treatment, housing loan interest, housing rent, and supporting the elderly. Among them, supporting the elderly refers to the support expenses for parents and other legally supported persons aged $60$ and above paid by taxpayers. It can be deducted at the following standards: for taxpayers who are only children, a standard deduction of $2000$ yuan per month is allowed; for taxpayers with siblings, the deduction amount of $2000$ yuan per month is shared among them, and the amount shared by each person cannot exceed $1000$ yuan per month.

A taxpayer has only one older sister, and both of them meet the conditions for supporting the elderly as specified. If the taxpayer's personal income tax payable in May 2020 is $180$ yuan, then the taxpayer's monthly salary after tax in that month is ____ yuan.
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Context:
Points $F$, $G$, $H$, $I$, and $J$ are positioned in 3-dimensional space with $FG = GH = HI = IJ = JF = 3$ and $\angle FGH = \angle HIJ = \angle JIF = 90^\circ$. The plane of triangle $FGH$ is parallel to $\overline{IJ}$. Determine the area of triangle $GIJ$.
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Context:
Find the maximum value of the following expression:
$$
|\cdots|\left|x_{1}-x_{2}\right|-x_{3}\left|-\cdots-x_{1990}\right|,
$$
where \( x_{1}, x_{2}, \cdots, x_{1990} \) are distinct natural numbers from 1 to 1990.
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Context:
In trapezoid \(ABCD\), \(AB\) is parallel to \(DC\) and \(\angle DAF = 90^\circ\). Point \(E\) is on \(DC\) such that \(EB = BC = CE\). Point \(F\) is on \(AB\) such that \(DF\) is parallel to \(EB\). In degrees, what is the measure of \(\angle FDA\)?
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Context:
如图， $A B$ 是单位圆的直径，在 $A B$ 上任取一点 $D$ ，作 $D C \perp A B$, 交圆周于 $C$ 。若点 $D$ 的坐标为 $(x, 0)$,则当 $x \in$ $(a, b)$时，线段 $A D 、 B D 、 C D$ 可以构成锐角三角形。请给出a + b的值。
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Context:
The parabola $P$ has its focus at $(0,0)$ and passes through the points $(4,3)$ and $(-4,-3)$. Determine the number of points $(x,y) \in P$ with integer coefficients such that $|4x + 3y| \leq 1000$.
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Context:
某班有 47 个学生，所用教室有 6 排，每排有 8个座位，用 $(i, j)$ 表示位于第 $i$ 排第 $j$ 列的座位。新学期准备调整座位，设某学生原来的座位为 $(i, j)$ ，如果调整后的座位为 $(m, n)$ ，则称该生作了移动 $[a$, $b]=[i-m, j-n]$ ，并称 $a+b$ 为该生的位置数。所有学生的位置数之和记为 $S$ 。求 $S$ 的最大可能值与最小可能值之差。
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Context:
Teams A, B, and C are to complete two projects, Project A and Project B. The workload of Project B is 25% more than that of Project A. The time it takes for teams A, B, and C to individually complete Project A are 20 days, 24 days, and 30 days, respectively. To complete these two projects together, team A was first assigned to Project A, while teams B and C worked on Project B. After a few days, team C was reassigned to work with team A to finish Project A. How many days did teams B and C work together?
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Context:
Let \( n \in \mathbf{Z}_{+} \). A volleyball team has \( n \) male players and \( n \) female players. Initially, each player is assigned to one of the positions numbered \( 1, 2, \cdots, 2n \), with only positions 1 and \( n+1 \) being outside the field. During the game, position swaps occur, with each player at position \( i \) (for \( i = 1, 2, \cdots, 2n \)) moving to \( i+1 \). Position \( 2n+1 \) is considered position 1. How many different initial arrangements of the players ensure that no matter how many swaps are made, at least \( n-1 \) female players are always on the field?
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Context:
Hugo can fold a small box in 3 seconds and a medium one in twice that time. Tom can fold both the small and medium boxes in 4 seconds. If Hugo and Tom want to leave as early as possible, how long (in seconds) will it take them to fold 2400 small boxes and 1800 medium boxes?
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Context:
Given point $P(-1,1)$, let $C$ be the family of circles described by the equation $(x-t)^{2}+(y-t)^{2}=1$, where $t \in R$. Let $A$ and $B$ be the points of tangency of the tangent lines to circle $C$ that pass through point $P$. Find the minimum value of $\overrightarrow{PA} \cdot \overrightarrow{PB}$.
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Context:
Alice and Bob are playing a game where Alice declares, "My number is 36." Bob has to choose a number such that all the prime factors of Alice's number are also prime factors of his, but with the condition that the exponent of at least one prime factor in Bob's number is strictly greater than in Alice's. What is the smallest possible number Bob can choose?
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Context:
Find the smallest value x such that, given any point inside an equilateral triangle of side 1, we can always choose two points on the sides of the triangle, collinear with the given point and a distance x apart.
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Context:
Let $a_{0} = 2$, $a_{1} = 5$, and $a_{2} = 8$, and for $n > 2$ define $a_{n}$ recursively to be the remainder when $4$($a_{n-1}$ $+$ $a_{n-2}$ $+$ $a_{n-3}$) is divided by $11$. Find $a_{2018} \cdot a_{2020} \cdot a_{2022}$.
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Context:
In the diagram shown, \( M \) is the midpoint of \( PQ \). The line \( PS \) bisects \( \angle RPQ \) and intersects \( RQ \) at \( S \). The line \( ST \) is parallel to \( PR \) and intersects \( PQ \) at \( T \). The length of \( PQ \) is 12 and the length of \( MT \) is 1. The angle \( SQT \) is \( 120^{\circ} \). Calculate the length of \( SQ \). Express your answer as a single number.
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Context:
Two trains of length 90 m and 100 m are 200 m apart. They start moving towards each other on parallel tracks, at speeds 71 kmph and 89 kmph. After how much time will the trains meet?
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Context:
Bookman purchased 55 copies of a new book released recently, 10 of which are hardback and sold for $20 each, and the rest are paperback and sold for $10 each. If 14 copies were sold, what was the total value of the remaining books?
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Context:
Which type of quadrilateral is not necessarily a symmetrical figure? Provide your answer as the name of the quadrilateral.
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Context:
An apartment and an office are sold for $15,000 each. The apartment was sold at a loss of 25% and the office at a gain of 25%. Determine the net effect of the transactions.
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Context:
Let $f(t) = \frac{t}{1-t}$, where $t \neq 1$. If $y = f(x)$, express $x$ in terms of $y$. The original answer is in the form \frac{m}{n}, please give the value of m + n.
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Context:
A frog starts at point $(2, 3)$ on a grid and makes jumps of length $1$ parallel to the coordinate axes. Each jump direction (up, down, right, or left) is chosen randomly. The jumping sequence stops when the frog reaches any point on the boundary of the rectangle defined by vertices $(0,0), (0,5), (5,5),$ and $(5,0)$. Calculate the probability that the sequence of jumps ends at either the top or bottom horizontal side of the rectangle.
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Context:
In $\triangle ABC$, $b=2$, $B=45^\circ$. If there are two possible values for $a$, find the range of $a$.
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Context:
In the sequence  $00$ ,  $01$ ,  $02$ ,  $03$ ,  $\cdots$ ,  $99$  the terms are rearranged so that each term is obtained from the previous one by increasing or decreasing one of its digits by  $1$  (for example,  $29$  can be followed by  $19$ ,  $39$ , or  $28$ , but not by  $30$  or  $20$ ). What is the maximal number of terms that could remain on their places?
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Context:
Tristan is eating his favorite cereal, Tiger Crunch, which has marshmallows of two colors, black and orange. He eats the marshmallows by randomly choosing from those remaining one at a time, and he starts out with $17$ orange and $5$ black marshmallows. If $\frac{p}{q}$ is the expected number of marshmallows remaining the instant that there is only one color left, and $p$ and $q$ are relatively prime positive integers, find $p + q$.
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Context:
Given vectors a=(8, 1/2x) and b=(x, 1), where x > 0, find the value of x such that (a-2b) || (2a+b). Express your answer as a single number.
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Context:
Consider the polynomial $f(x) = x^5 - 4x + 2$. Let $L$ be the complex splitting field of $f(x)$ over $\mathbb{Q}$. How many subfields $K$ of $L$ are there such that $(K : \mathbb{Q}) = 2$?
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Context:
Find the value of \( \sin^{4} \frac{\pi}{8} + \sin^{4} \frac{3\pi}{8} + \sin^{4} \frac{5\pi}{8} + \sin^{4} \frac{7\pi}{8} \).
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Context:
A sequence of consecutive positive integers starting with 1 is written on the blackboard. If one of the numbers is erased, the arithmetic mean of the remaining numbers is 35 7/17. Determine the erased number. Express your answer as a single integer.
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Context:
Evaluate the following limit:

$$\lim_{h \rightarrow 0} \frac{\tan(a+3h)-3\tan(a+2h)+3\tan(a+h)-\tan a}{h}$$
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Context:
If 5 times a number is 2, then 100 times the reciprocal of the number is
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Context:
A 180 m long train is running at 72 kmph. It crossed a platform of length 220.03199999999998 m in a certain amount of time. How many seconds did it take for the train to cross the platform?
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Context:
For how many integers $n$ between $1$ and $50$, inclusive, is $\frac{(n^2-1)!}{(n!)^n}$ an integer?
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Context:
In a multinational office in Singapore, there are 60% female employees and 40% male employees, working in various job roles such as engineers, managers, and support staff. The engineers constitute 35% of the total workforce, the managers account for 25%, and the support staff make up the remaining 40%. Out of the total 1400 employees, 62% are computer literate. Furthermore, the computer literacy rates among the employees are different depending on their gender and job roles as follows:

- Male engineers: 80%
- Female engineers: 75%
- Male managers: 55%
- Female managers: 60%
- Male support staff: 40%
- Female support staff: 50%

Calculate the joint probability of a randomly selected employee being both female and computer literate, considering the different job roles and computer literacy rates.
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Context:
Determine the number of integers  $ n$  with  $ 1 \le n \le N\equal{}1990^{1990}$  such that  $ n^2\minus{}1$  and  $ N$  are coprime.
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Context:
Given two concentric circles, a tangent to the smaller circle divides the length of the arc of the larger circle in the ratio $1: 5$. Find the ratio of the areas of the circles bounded by these circles.
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Context:
Given that the function \( f(x) \) satisfies the equation \( 2 f(x) + x^{2} f\left(\frac{1}{x}\right) = \frac{3 x^{3} - x^{2} + 4 x + 3}{x + 1} \) and \( g(x) = \frac{5}{x + 1} \), determine the minimum value of \( f(x) + g(x) \).
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Context:
Among the permutations of integers $1, 2, \cdots, n$, how many permutations are there such that each number is either greater than all the numbers preceding it or less than all the numbers preceding it?
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Context:
Given $1994$ points on a plane, it is known that any $100$ of them can be selected such that $98$ of these can be enclosed by a circle with a diameter of $1$. Determine the minimum number of circles with a radius of $1$ required to cover all $1994$ points.
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Context:
On side \( CB \) of triangle \( ABC \), a point \( D \) is taken such that \( |CD| = \alpha |AC| \). The radius of the circumcircle of \( \triangle ABC \) is \( R \). Find the distance between the center of the circumcircle of \( \triangle ABC \) and the center of the circumcircle of \( \triangle ADB \).
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Context:
For each positive integer $n$, define $S(n)$ to be the smallest positive integer divisible by each of the positive integers $1, 2, 3, \ldots, n$. How many positive integers $n$ with $1 \leq n \leq 100$ have $S(n) = S(n+4)$?
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Context:
Two spheres, each with a radius of 1, are externally tangent to each other and internally tangent to a cylinder with a radius of 1. Another smaller sphere is externally tangent to both of these spheres and internally tangent to the same cylinder. If the plane passing through the center of the small sphere and the center of one of the larger spheres intersects the cylinder to form an ellipse, find the maximum possible value of the eccentricity \( e \) of this ellipse.
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Context:
There are 3 females and 3 males to be arranged in a sequence of 6 contestants, with the restriction that no two males can perform consecutively and the first contestant cannot be female contestant A. Calculate the number of different sequences of contestants.
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Context:
Evaluate the integral: $$\int_0^{\frac{\pi}{2}}\ln^2\left(\cos^2x\right)\text{d}x$$
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Context:
Five apples have weights of 125g, $a$, 121g, $b$, and 127g respectively. If the median and the mean of the sample weights are both 124g, find the standard deviation $s$ of the sample.
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Context:
\(ABCD\) is a convex quadrilateral where \(AB = 7\), \(BC = 4\), and \(AD = DC\). Also, \(\angle ABD = \angle DBC\). Point \(E\) is on segment \(AB\) such that \(\angle DEB = 90^\circ\). Find the length of segment \(AE\).
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Context:
A cryptographer devises the following method for encoding positive integers. First, the integer is expressed in base $5$. Second, a 1-to-1 correspondence is established between the digits that appear in the expressions in base $5$ and the elements of the set $\{V, W, X, Y, Z\}$. Using this correspondence, the cryptographer finds that three consecutive integers in increasing order are coded as $VYZ$, $VYX$, $VVW$, respectively. What is the base-10 expression for the integer coded as $XYZ$?
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Context:
Let $a$, $b \in \mathbb{R}$, $c \in [0,2\pi)$. If for any real number $x$, the equation $2\sin (3x- \frac {\pi}{3})=a\sin (bx+c)$ holds, then how many sets of $(a, b, c)$ satisfy the condition? Express your answer as a single integer.
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Context:
When 100 is divided by positive integer x, the remainder is 3. What is the remainder when 197 is divided by x?
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Context:
Find the minimum value of the expression \(\frac{13 x^{2}+24 x y+13 y^{2}+16 x+14 y+68}{\left(9-x^{2}-8 x y-16 y^{2}\right)^{5 / 2}}\). Round the answer to the nearest hundredth if needed.
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Context:
What is the sum of all $k \leq 25$ such that one can completely cover a $k \times k$ square with $T$ tetrominos without any overlap?
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Context:
Cassandra sets her watch to the correct time at noon. At the actual time of $1:00$ PM, she notices that her watch reads $12:57$ and $36$ seconds. Assuming that her watch loses time at a constant rate, calculate the total time discrepancy in seconds between her watch and the actual time when her watch first reads $10:00$ PM. Please provide your answer as an integer representing the number of seconds.
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Context:
What is the sum of the six positive integer factors of 30?
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