Context:
For a twice differentiable function \(f(x)\), it is given that \(|f(x)| \leq 3\) for all \(x \in \mathbb{R}\). Additionally, for some \(\alpha\), we have \(f(\alpha) + (f'(\alpha))^2 = 80\). Determine the number of integral values that \((f'(x))^2\) can take in the interval \((0, 77)\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine all natural numbers \( k \) such that there exists a set of \( k \) cards from a deck of 1024 cards, where each card has a unique set of digits from 0 to 9, including an empty card, with the property that no complete set of digits from 0 to 9 can be formed, but adding any additional card from the deck allows for a complete set.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the Cartesian coordinate system, consider the ellipse given by \(\frac{x^2}{2} + y^2 = 1\) with foci \(F_1\) and \(F_2\). A line \(l\) intersects the ellipse at two distinct points \(A\) and \(B\), and does not pass through \(F_1\). The distance from \(F_2\) to the line \(l\) is \(d\). If the slopes of the lines \(AF_1\), \(l\), and \(BF_1\) form an arithmetic sequence, determine the range of possible values for \(d\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
For a twice differentiable function \(f(x)\), if \(|f(x)| \leq 3\) for all \(x \in \mathbb{R}\) and there exists a point \(\alpha\) such that \(f(\alpha) + \left(f'(\alpha)\right)^2 = 80\), determine the number of integer values that \(\left(f'(x)\right)^2\) can take within the interval \((0, 77)\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine all values of $m$ such that the inequality \( \log(60x^2+120x+10m-10) > 1+3\log(x+1) \) has exactly 4 integer solutions for $x$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of distinct ellipses that define the domain of the function: \[ f(x,y)=\frac{\sqrt{x}\sqrt{9-x^{2}-y^{2}}}{\ln(\frac{x^{2}}{4}+\frac{y^{2}}{16}-1)} \] without using a graphical representation.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the area of the region in the first quadrant of the Cartesian plane defined by the inequalities $0 \leq x \leq \sqrt{y}$, $0 \leq y \leq \sqrt{x}$, and $x + y \leq \frac{3}{4}$.
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Context:
Determine the number of solutions in positive integers to the equation: \[ 37 + \frac{3(x-1)x}{2} + 3^{x+2} = y^2 \]
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Context:
A lottery has $N$ tickets numbered $1, 2, \dots, N$, where $N$ is unknown. A random sample of 5 winning tickets is drawn, and their numbers are $\{a, b, c, d, e\}$. Determine the maximum likelihood estimator for $N$ based on this sample.
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Context:
Evaluate \( x\frac{\partial z}{\partial x} + y\frac{\partial z}{\partial y} \) for the function \( z = x\tan^{-1} \frac{y}{x} + y\sin^{-1} \frac{x}{y} + 2 \) at the point (1,1).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In quadrilateral \(ABCD\), given \(\angle ABD = 70^\circ\), \(\angle CAD = 20^\circ\), \(\angle BAC = 48^\circ\), and \(\angle CBD = 40^\circ\), find the measure of \(\angle ACD\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $\ell_1$ and $\ell_2$ be two parallel lines, a distance of 15 apart. Points $A$ and $B$ lie on $\ell_1$ while points $C$  and $D$ lie on $\ell_2$ such that $\angle BAC = 30^\circ$ and $\angle ABD = 60^\circ$. The minimum value of $AD + BC$ is $a\sqrt b$, where $a$ and $b$ are integers and $b$ is squarefree. Find $a + b$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the surface integral of the vector field \( \mathbf{G}(x,y,z)= ( y^2, xz, 1 ) \) over the portion of the paraboloid \( z=x^2+y^2 \) that lies below the plane \( z=4 \), where the normal vector points upwards. Use Gauss's Divergence Theorem to find the result.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A gas station receives cars at a rate of 20 vehicles per hour and has only one gas pump. If the pump is empty, it receives one client. If a new client arrives and finds the pump occupied, they wait. Each client takes approximately 6 minutes (1/10th of an hour) to fill their vehicle. If two clients are at the pump, one filling and the other waiting, new clients leave the station directly. Given the stationary distribution of the states as \((u_1,u_2,u_3) = (1/7,2/7,4/7)\), find the average number of clients served by the gas station every hour. Express your answer as a fraction.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the flux of the vector field \( F(x,y,z) = (y, x, z) \) downwards through the surface \( z = 16 - x^2 - y^2 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
The volume of a rectangular solid each of whose side, front, and bottom faces are $12\text{ in}^{2}$, $8\text{ in}^{2}$, and $6\text{ in}^{2}$ respectively is:
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Context:
Solve the equation\( x \cdot  \left\lbrack  {x \cdot  \left\lbrack  {x \cdot  \left\lbrack  x\right\rbrack  }\right\rbrack  }\right\rbrack   = {88} \) in the set of real numbers.
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Context:
$设函数f(x)=3|x-2|+|x|.$
$求直线 y = a 与 f(x) 的图象围成的三角形的面积的最大值.$
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Context:
$ABC$ is an isosceles triangle with base $\overline{AB}$. $D$ is a point on $\overline{AC}$ and $E$ is the point on the extension of $\overline{BD}$ past $D$ such that $\angle{BAE}$ is right. If $BD = 15, DE = 2,$ and $BC = 16$, then $CD$ can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Determine $m + n$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The sequence starts at 2,187,000 and each subsequent number is created by dividing the previous number by 3. What is the last integer in this sequence?
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Context:
In triangle \( ABC \), a circle is described on side \( AC \) as its diameter, intersecting side \( AB \) at point \( M \) and side \( BC \) at point \( N \). Given that \( AC = 2 \), \( AB = 3 \), and \( AN = 1.8 \), find the cosine of angle \( BAC \).
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Context:
Given real numbers $x$, $y$, and $z$, such that \[\frac{\sin{x}+\sin{y}+\sin{z}}{\sin{(x+y+z)}}=\frac{\cos{x}+\cos{y}+\cos{z}}{\cos{(x+y+z)}}=2,\] find the value of \[\sum_{cyc}\sin x\sin y.\]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the value of $x$ such that \( \lim\limits_{n\to\infty} \sqrt{1+\sqrt{x+\sqrt{x^2+\cdots+\sqrt{x^n}}}} = 2 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the coefficient of \(x^{25}\) in the expansion of \((2x^3 + 3x^2 - 5)^{15}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the function \( f(x) \) that satisfies the functional equation \( f(x+y) = f(x) + f(y) + y\sqrt{f(x)} \) for all real numbers \( x \) and \( y \), given that \( f'(0) = 0 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the values of $k \in \mathbb{R}$ for which the fractional Sobolev space $W^{k,2}(\mathbb{R}^n)$ is continuously embedded into the space of Lebesgue-measurable functions $\mathcal{M}$ on $\mathbb{R}^n$. Specifically, for which $k$ does the identity map $C^\infty_c(\mathbb{R}^n) \hookrightarrow \mathcal{M}$ extend by continuity to a map $W^{k,2}(\mathbb{R}^n) \to \mathcal{M}$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the limit \( \lim_{x \to \infty} \left( e^{1/x}(x^3-x^2+x/2)-(x^3+x^6)^{0.5} \right) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Fiona has a deck of cards labelled $1$ to $n$, laid out in a row on the table in order from $1$ to $n$ from left to right. Her goal is to arrange them in a single pile, through a series of steps of the following form:

If at some stage the cards are in $m$ piles, she chooses $1\leq k<m$ and arranges the cards into $k$ piles by picking up pile $k+1$ and putting it on pile $1$; picking up pile $k+2$ and putting it on pile $2$; and so on, working from left to right and cycling back through as necessary.
She repeats the process until the cards are in a single pile, and then stops. So for example, if $n=7$ and she chooses $k=3$ at the first step she would have the following three piles:

$\begin{array} {ccc}7 & & \\ 4 & 5 & 6 \\ 1 & 2 & 3\\ -- & -- & --\end{array}$

If she then chooses $k=1$ at the second stop, she finishes with the cards in a single pile with cards ordered $6352741$ from top to bottom.

How many different final piles can Fiona end up with?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the values of \( \alpha \) for which the series \( \sum_{n \geqslant 1} n^{\alpha} \cdot x^{n} \cdot (1-x) \) uniformly converges on the interval \([0,1]\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
After moving his sticky toy spider one morning, Tony heads outside to play \u0022pirates\u0022 with his pal Nick, who lives a few doors down the street from the Kubiks.  Tony and Nick imagine themselves as pirates in a rough skirmish over a chest of gold.  Victorious over their foes, Tony and Nick claim the prize.  However, they must split some of the gold with their crew, which they imagine consists of eight other bloodthirsty pirates.  Each of the pirates receives at least one gold coin, but none receive the same number of coins, then Tony and Nick split the remainder equally.  If there are $2000$ gold coins in the chest, what is the greatest number of gold coins Tony could take as his share?  (Assume each gold coin is equally valuable.)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the real value \( \alpha \) such that the coordinate transformation \( \varphi(p,q) \rightarrow (P,Q) \), where \( Q = q^2 + \alpha\sqrt{q^2+p} \) and \( P = q + \sqrt{q^2+p} \), preserves the symplectic form \( dp \wedge dq \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a diagram, several circles are connected by segments. Choose a natural number \( n \) and arrange different natural numbers in the circles such that:

- If two numbers \( a \) and \( b \) are not connected by a segment, then \( |a - b| \) is coprime with \( n \).
- If two numbers \( a \) and \( b \) are connected by a segment, then \( |a - b| \) shares a common natural divisor greater than 1 with \( n \).

Determine the smallest natural number \( n \) for which such an arrangement is possible.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the number of elements in the quotient group \(\mathbb{Z}^2/\!\ker\varphi\), where \(\varphi: \mathbb{Z}^2 \to S_{15}\) is the homomorphism defined by \(\varphi(1,0)=(2,7,3)(3,11,5)(12,13)\) and \(\varphi(0,1)=(14,15)(1,4,6,8,9,10)\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
A pendulum is suspended near a vertical wall and oscillates in a plane parallel to the wall. A thin nail is driven into the wall such that the middle of the pendulum's string hits it every time the pendulum passes through its equilibrium position from right to left. Given that the period of oscillation of this pendulum is \( T = 2.41 \) seconds and the acceleration due to gravity is \( g = 10 \, \text{m/s}^2 \), find the length of the pendulum's string.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the minimum number of variables required to construct a sentence in a first-order logic language consisting solely of the equality predicate "=" that is valid in a structure if and only if the structure's domain contains exactly n elements.
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Context:
Determine whether the infinite product \( \prod_{k=2}^\infty \left(1 - \frac{1}{k^\alpha}\right) \) converges for a given \( \alpha \in \mathbb{R} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: \[ \lim_{n\to\infty}\sum_{k=1}^n\frac{k}{n}\sin\left(\frac{2\pi k}{n}\right)\sin\left(\frac{2\pi k-\pi}{n}\right)\sin\left(\frac{\pi}{n}\right) \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the limit of the sequence \(\{a_n\}\) defined recursively by \(a_1 = 0\) and \(a_{n+1} = a_n + \sqrt{1 + a_n^2}\) as \(n\) approaches infinity. Express the limit in terms of the ratio \(\frac{a_n}{2^{n-1}}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of infinite subsets of the set of natural numbers.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The highest power of 2 that is a factor of \(15.13^{4} - 11^{4}\) needs to be determined.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the cardinality of the set $S$ of all convergent sequences of real numbers.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the treewidth of the complete bipartite graph $K_{m,n}$ using the chordal graph characterization. Recall that the treewidth $tw(G)$ is given by $\min \{\omega(H)-1 : G\subseteq H \ \wedge \ H \ \text{is chordal}\}$. Determine the treewidth of $K_{m,n}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
You want to create a three-digit number by removing one from the number cards 2, 5, 6, or 9 and using all remaining ones. Find the sum of the third largest and third smallest possible numbers.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Paul had 50 books. He went to a garage sale where they were selling mystery, science fiction, and romance books. The mystery books were on sale: buy 2, get 1 free. For the science fiction books, there was a 20% discount, and romance books cost their original price.

Here are the prices of the books:
- Mystery books: $4 each
- Science fiction books: $6 each (before discount)
- Romance books: $5 each

If Paul spent $90 at the garage sale and bought the same number of books in each category, how many books did he buy in total, considering the discounts and promotions?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of non-real roots of the equation \(z^2 = \sin(z)\) within the region \(\{z \in \mathbb{C} \setminus \mathbb{R} \mid |z| < 2\}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $X$ be a set of 9 positive integers. Define $E \subseteq X$, and let $S(E)$ represent the sum of the elements in the subset $E$. Determine the maximum value of an element $n$ in $X$ such that there exist at least two distinct subsets $A$ and $B$ of $X$ for which $S(A) = S(B)$. Find the value of $n$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The curve \( y = \sin(4x) \cdot 10e^{-x/2} \) for \( x \ge 0 \) has stationary points labeled as \( T_1, T_2, T_3, \ldots \). The x-coordinates of \( T_1 \) and \( T_2 \) are \( 0.362 \) and \( 1.147 \) respectively. Given that the x-coordinate of \( T_n \) is greater than 25, find the least possible value of \( n \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the minimum value of the integral \( \int_0^{\infty} |a+bx+cx^2+x^3|^2 e^{-x} \, dx \) with respect to \(a, b, c \in \mathbb{C}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the triple integral \( \iiint\limits_{V} xe^{-z} \, dV \) where \( V \) is the portion of the cone \( z=\sqrt{x^2+y^2} \) for \( x \geq 0 \) using cylindrical polar coordinates.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the minimum value of the integral \( \int_{-1}^{1} |x^3-a-bx-cx^2|^2 \, dx \) over all real numbers \( a, b, c \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many tori are in the connected sum obtained by encircling the wireframe of an octahedron with a small tube?
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Context:
What is the maximal number of elements in a sigma-algebra generated by a collection of N subsets of a set X?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A weight is to be suspended 10 feet below a horizontal line AB by a wire in the shape of a Y. If the points A and B are 8 feet apart, determine the shortest total length of wire that can be used.
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Context:
Determine the set of points where the partial derivative $f_x$ is continuous for the function \( f(x,y) = \frac{x^2-y^2}{\sqrt[3]{x^2+y^2}} \) for \((x,y) \neq (0,0)\) and \( f(x,y) = 0 \) for \((x,y)=(0,0)\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the minimum time complexity for checking whether a set of integers from 1 to n can be partitioned into two subsets with equal sums?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given 10 distinct points on a plane, consider the midpoints of all segments connecting all pairs of points. What is the minimum number of such midpoints that could result?
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Context:
Let $ABC$ be a triangle such that angle $C$ is obtuse. Let $D \in [AB]$ and $[DC] \perp [BC]$. If $m(\widehat{ABC}) = \alpha$, $m(\widehat{BCA}) = 3\alpha$, and $|AC| - |AD| = 10$, find the length of $|BD|$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
For a positive integer \( n \), define \( S_{n} \) as the minimum value of the sum
\[
\sum_{k=1}^{n} \sqrt{(2k-1)^{2}+a_{k}^{2}}
\]
where \( a_{1}, a_{2}, \cdots, a_{n} \) are positive real numbers whose sum is 17. There exists a unique positive integer \( n \) for which \( S_{n} \) is also an integer. Find this \( n \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
A $\textit{palindrome}$ is a number which reads the same forward as backward, for example 313 or 1001. Ignoring the colon, how many different palindromes are possible on a 12-hour digital clock displaying only the hours and minutes? (Notice a zero may not be inserted before a time with a single-digit hour value. Therefore, 01:10 may not be used.)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Two trains are moving at 80 kmph and some speed in opposite directions. Their lengths are 150 m and 100 m respectively. The time they will take to pass each other completely is 5.999520038396928 seconds. What is the speed of the second train?
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Context:
12.418 Through the vertex of the base of a regular triangular pyramid, a plane is drawn perpendicular to the opposite lateral face and parallel to the opposite side of the base. This plane forms an angle $\alpha$ with the plane of the pyramid's base. Find the plane angle at the vertex of the pyramid.
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Context:
The function $f(x)=(m^{2}-m-1)x^{m^{2}+m-3}$ is a power function, and when $x\in (0,+\infty)$, $f(x)$ is a decreasing function. Find the real number $m=$____.
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Context:
In convex quadrilateral \(ABCD\) with \(AB=11\) and \(CD=13\), there is a point \(P\) for which \(\triangle ADP\) and \(\triangle BCP\) are congruent equilateral triangles. Compute the side length of these triangles.
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Context:
在椭圆中, $A$ 为长轴的一个端点, $B$ 为短轴的一个端点, $F_{1}, F_{2}$ 为两个焦点. 若 $$ \overline{A F_{1}} \cdot \overline{A F_{2}}+\overrightarrow{B F_{1}} \cdot \overrightarrow{B F_{2}}=0, $$ 求 $\tan \angle A B F_{1} \cdot \tan \angle A B F_{2}$ 的值。原始答案为\frac{m}{n}的形式，请给出m + n的值。
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Context:
Let $m\ge 5$ be an odd integer, and let $D(m)$ denote the number of quadruples $(a_1, a_2, a_3, a_4)$ of distinct integers with $1\le a_i \le m$ for all $i$ such that $m$ divides $a_1+a_2+a_3+a_4$. There is a polynomial \[q(x) = c_3x^3+c_2x^2+c_1x+c_0\]such that $D(m) = q(m)$ for all odd integers $m\ge 5$. What is $c_1?$
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the area of the smallest circular region that can cover the triangle ABC with sides a = 2, b = 2√2, and c = √2 + √6?
Please reason step by step, and put your final answer within \boxed{}.


Context:
the h.c.f. of two numbers is 50 and the other three factors of their l.c.m. are 13, 23, and 31. The larger of the two numbers is:
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Context:
$A = 1 \cdot 4 \cdot 7 \cdots 2014$. Find the last non-zero digit of $A$ if it is known that $A \equiv 1 \pmod{3}$.
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Context:
24. [12] Find the number of subsets $S$ of $\{1,2, \ldots 6\}$ satisfying the following conditions:
- $S$ is non-empty.
- No subset of $S$ has the property that the sum of its elements is 10 .
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Context:
Adam’s wardrobe is too crowded so he decides to donate some of his clothes to a charity shop. He takes out 4 pairs of pants, 4 jumpers, 4 pajama sets (top and bottom), and some t-shirts, then asks his friends if they have anything they want to donate. 3 of his friends donate the same amount of clothing as Adam each. Then he takes another look over his clothing and decides that he actually wants to keep half of his clothes. In total, 126 articles of clothing are being donated. How many t-shirts did Adam initially take out?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a speech contest held in a class, there are a total of $5$ contestants, including $3$ females (among which is female A) and $2$ males. If the two males cannot appear consecutively, and female A cannot be the first to appear, what is the number of ways for the appearance order?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Two trains of different lengths run at the speed of 60 km/hr and 40 km/hr respectively in opposite directions on parallel tracks. One train is 450 m long, and they take 26.99784017278618 seconds to cross each other. How long is the other train?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $f(x)=x^3+7x^2+9x+10$. Find the value of $p$ such that for every integer $a,b$, the statement
\[
f(a) \equiv f(b) \ (\text{mod } p) \Rightarrow a \equiv b \ (\text{mod } p)
\]
is satisfied.
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the measure, in degrees, of the smallest positive angle \( x \) for which \( 4^{\sin ^{2} x} \cdot 2^{\cos ^{2} x} = 2 \sqrt[4]{8} \)?
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Context:
A positive integer  $N$  is given. Panda builds a tree on  $N$  vertices, and writes a real number on each vertex, so that  $1$  plus the number written on each vertex is greater or equal to the average of the numbers written on the neighboring vertices. Let the maximum number written be  $M$  and the minimal number written  $m$ . Mink then gives Panda  $M-m$  kilograms of bamboo. What is the maximum amount of bamboo Panda can get?
Please reason step by step, and put your final answer within \boxed{}.


Context:
S is a subset of {1, 2, 3, ... , 16} which does not contain three integers which are relatively prime in pairs. How many elements can S have?
Please reason step by step, and put your final answer within \boxed{}.


Context:
On a race track a maximum of some horses can race together at a time. There are a total of 25 horses. There is no way of timing the races. The minimum number of races needed to get the top 3 fastest horses is 7. What is the maximum number of horses that can race together at a time?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given |-x| = 2023, find the value of x. Express your answer using the ± symbol to represent both solutions.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the minimum value of  $$ x^{2014} + 2x^{2013} + 3x^{2012} + 4x^{2011} +\ldots + 2014x + 2015 $$  where  $x$  is a real number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the minimum value of the expression

\[
\sqrt{x^{2}+y^{2}-2x-2y+2} + \sqrt{x^{2}+y^{2}-2x+4y+2\sqrt{3}y+8+4\sqrt{3}} + \sqrt{x^{2}+y^{2}+8x+4\sqrt{3}x-4y+32+16\sqrt{3}}
\]

for \( (x, y) \in \mathbb{R} \)?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Call a three-digit number  $\overline{ABC}$   $\textit{spicy}$  if it satisfies  $\overline{ABC}=A^3+B^3+C^3$ . Compute the unique  $n$  for which both  $n$  and  $n+1$  are  $\textit{spicy}$ .
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Context:
In how many ways can a square with a side length of 5 units be placed on an $8 \times 8$ chessboard so that each of its vertices falls on the center of a square? (Solutions that can be transformed into each other by reflection or rotation are not considered different.)

Translating the text as requested, while preserving the line breaks and formatting.
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Context:
Positive real numbers \( x \) and \( y \) satisfy the following conditions: there exist \( a \in [0, x] \) and \( b \in [0, y] \) such that \( a^2 + y^2 = 2 \), \( b^2 + x^2 = 1 \), and \( ax + by = 1 \). What is the maximum value of \( x + y \)?
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Context:
Given $7\sqrt{3}sinθ=1+7cosθ$, find $sin(2θ+\frac{π}{6})$.
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Context:
17 Given positive numbers $a$, $b$, $c$ satisfy: $2a + 4b + 7c \leqslant 2abc$, find the minimum value of $a + b + c$.
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Context:
If the four-digit number $\overline{2 \mathrm{ABC}}$ is divisible by 13, what is the maximum value of $A + B + C$?
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Context:
There are three identical red balls, three identical yellow balls, and three identical green balls. In how many different ways can they be split into three groups of three balls each?
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Context:
A 290 meter long train running at a certain speed crosses another train running in the opposite direction at the speed of 80 kmph in 9 seconds. The length of the other train is 210.04 meters. What is the speed of the first train in kmph?
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Context:
A natural number is written on the board. If its last digit (in the units place) is erased, the remaining non-zero number is divisible by 20. If the first digit is erased, the remaining number is divisible by 21. What is the smallest number that could be on the board if its second digit is not 0?
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Context:
The area of the part of the surface $x^2+y^2+z^2=4$ which lies inside the surface $x^2+y^2=2x$ is equal to $n(\pi-2)$ for an integer n. What is the value of n?
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Context:
Determine the degree of the splitting field of the polynomial $g(x) = x^4 + 2$ over $\mathbb{Q}$.
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Context:
正 2015 边形 $A_{1} A_{2} \cdots A_{2015}$ 内接于单位圆 $O$, 任取它的两个不同的顶点 $A_{i}, A_{j}$,则 $\left|\overrightarrow{O A_{i}}+\overrightarrow{O A_{j}}\right| \geqslant 1$ 的概率是 $\qquad$.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值
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Context:
Petya marked 5 points on one face of a cube, then turned the cube and marked 6 points on the adjacent face, then turned again and marked 7 points on the next face, and so on. He marked points on each face. What is the maximum number of points that can be on two opposite faces combined?
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Context:
Let  $\mathcal{A}$  be the set of finite sequences of positive integers  $a_1,a_2,\dots,a_k$  such that  $|a_n-a_{n-1}|=a_{n-2}$  for all  $3\leqslant n\leqslant k$ . If  $a_1=a_2=1$ , and  $k=18$ , determine the number of elements of  $\mathcal{A}$ .
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Context:
In the convex quadrilateral $AEBC$, it is given that $\angle BEA = \angle CAE = 90^{\circ}$, $AB = 15$, $BC = 14$, and $CA = 13$. Let $D$ be the foot of the altitude from $C$ to the line $\overline{AB}$. If the ray $CD$ intersects $\overline{AE}$ at $F$, compute the product $AE \cdot AF$.
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Context:
Mr. Thompson is a firm believer in traditional gasoline cars and refuses to embrace electric vehicles. He drives his car 20 miles to work every day, five days a week. His car consumes 1 gallon of gasoline for every 25 miles driven, and the current price of gasoline is $4 per gallon.

Calculate how much Mr. Thompson spends on gasoline for his work commute in one week.
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Context:
Circle $\Omega$ has radius 5. Points $A$ and $B$ lie on $\Omega$ such that chord $A B$ has length 6. A unit circle $\omega$ is tangent to chord $A B$ at point $T$. Given that $\omega$ is also internally tangent to $\Omega$, find $A T \cdot B T$.
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Context:
Consider the identity \( 1+2+\cdots+n=\frac{1}{2} n(n+1) \). If we set \( P_{1}(x)=\frac{1}{2} x(x+1) \), then it is the unique polynomial such that for all positive integer \( n \), \( P_{1}(n)=1+2+\cdots+n \). In general, for each positive integer \( k \), there is a unique polynomial \( P_{k}(x) \) such that
\[ P_{k}(n)=1^{k}+2^{k}+3^{k}+\cdots+n^{k} \quad \text { for each } n=1,2, \ldots . \]

Find the value of \( P_{2010}\left(-\frac{1}{2}\right) \).
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Context:
Find the sum of the values of $x$ for which $\binom{x}{0} - \binom{x}{1} + \binom{x}{2} - \cdots + \binom{x}{2008} = 0$.
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Context:
The side lengths of a right triangle are integers, with no common divisor greater than 1. The radius of the circle tangent to the hypotenuse is 420 units. Determine the side lengths.
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Context:
Let $f: \mathbb R \to \mathbb R$ be a non-constant continuous function such that $(e^x-1)f(2x)= \left(e^{2x}-1\right)f(x)$. If $f'(0) =1 $, then $\lim_{x\to 0}\left(\dfrac {f(x)}{x}\right)^{\frac 1x}= ? $
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Context:
Sets $A$, $B$, and $C$ satisfy the following conditions:
- $|A| = 92$
- $|B| = 35$
- $|C| = 63$
- $|A \cap B| = 16$
- $|A \cap C| = 51$
- $|B \cap C| = 19$

Compute the number of possible values of $|A \cap B \cap C|$.
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Context:
Given real numbers $a$ and $b$ satisfying ${3}^{a-2}+a=\frac{1}{2}$ and $\frac{1}{3}{b}^{3}+\log_{3}b=-\frac{1}{2}$, find the value of $\frac{{3}^{a}}{{b}^{3}}$.
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Context:
There are 28 students in the class. On March 8, each boy gave each girl one flower - a tulip, a rose, or a daffodil. How many roses were given if it is known that there are 4 times more roses than daffodils, but 10 times fewer than tulips?
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Context:
A certain sunflower cell has 34 chromosomes at the late stage of the second meiotic division when forming pollen grains. Calculate the number of tetrads that can be produced by this cell during meiosis. Express your answer as a whole number.
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Context:
Given $p$: for all $x \in \mathbb{R}$, the inequality $x^2 - mx + \frac{3}{2} > 0$ always holds true, and $q$: the foci of the ellipse $\frac{x^2}{m - 1} + \frac{y^2}{3 - m} = 1$ lie on the x-axis. If the proposition $p \land q$ is true, determine the range of values for the real number $m$.
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Context:
Suppose we choose two numbers $x, y \in [0,1]$ uniformly at random. If the probability that the circle with center $(x, y)$ and radius $|x-y|$ lies entirely within the unit square $[0,1] \times [0,1]$ is written as $\frac{p}{q}$ with $p$ and $q$ relatively prime nonnegative integers, then what is $p^2+q^2$?
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Context:
Let $f(x)$ be a cubic polynomial with roots $x_{1}, x_{2}$, and $x_{3}$. Assume that $f(2 x)$ is divisible by $f^{\prime}(x)$ and compute the sum of the absolute values of the ratios $x_{1}: x_{2}: x_{3}$.
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Context:
Zyaka decided to buy a krumblik. In the store, they also sold kryambliks. Zyka bought a kryamblik and received coupons worth $50\%$ of the cost of the kryamblik. He used these coupons to pay $20\%$ of the cost of the krumblik. By paying the remaining amount, he also bought the krumblik. By what percentage did Zyka's expenses for buying both the kryamblik and the krumblik exceed his initially planned expenses for buying only the krumblik?
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Context:
Let $x,$ $y,$ and $z$ be positive real numbers such that
\[\frac{1}{x^4} + \frac{1}{y^4} + \frac{1}{z^4} = 1.\]Find the minimum value of
\[\frac{x^4 y^4 + x^4 z^4 + y^4 z^4}{x^3 y^2 z^3}.\]
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Context:
A boy needs to cover a distance of 10000 meters in 28 minutes. Find his average speed in centimeters per second.
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Context:
In a finite sequence of real numbers, the sum of any 7 consecutive terms is negative while the sum of any 11 consecutive terms is positive. What is the maximum number of terms in such a sequence?
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Context:
Let $n$ be a positive integer.
Determine the number of sequences $a_0, a_1, \ldots, a_n$ with terms in the set $\{0,1,2,3\}$ such that $$n=a_0+2a_1+2^2a_2+\ldots+2^na_n.$$
Please reason step by step, and put your final answer within \boxed{}.

\boxed{ \left\lfloor \frac{n}{2} \right\rfloor + 1 }


Context:
Circles $\mathcal{P}$ and $\mathcal{Q}$ have radii $1$ and $4$, respectively, and are externally tangent at point $A$. Point $B$ is on $\mathcal{P}$ and point $C$ is on $\mathcal{Q}$ so that line $BC$ is a common external tangent of the two circles. A line $\ell$ through $A$ intersects $\mathcal{P}$ again at $D$ and intersects $\mathcal{Q}$ again at $E$. Points $B$ and $C$ lie on the same side of $\ell$, and the areas of $\triangle DBA$ and $\triangle ACE$ are equal. This common area is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
[asy] import cse5; pathpen=black; pointpen=black; size(6cm); pair E = IP(L((-.2476,1.9689),(0.8,1.6),-3,5.5),CR((4,4),4)), D = (-.2476,1.9689); filldraw(D--(0.8,1.6)--(0,0)--cycle,gray(0.7)); filldraw(E--(0.8,1.6)--(4,0)--cycle,gray(0.7)); D(CR((0,1),1)); D(CR((4,4),4,150,390)); D(L(MP("D",D(D),N),MP("A",D((0.8,1.6)),NE),1,5.5)); D((-1.2,0)--MP("B",D((0,0)),S)--MP("C",D((4,0)),S)--(8,0)); D(MP("E",E,N)); [/asy]
Hint
$[ABC] = \frac{1}{2}ab \text{sin} C$ is your friend for a quick solve. If you know about homotheties, go ahead, but you'll still need to do quite a bit of computation. If you're completely lost and you have a lot of time left in your mocking of this AIME, go ahead and use analytic geometry.
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Context:
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. If $a=1$, $b=\sqrt{7}$, and $c=\sqrt{3}$, then $B=$ _______.
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Context:
Let $a_0 = 5/2$ and $a_k = a_{k-1}^2 - 2$ for $k \geq 1$. Compute \[ \prod_{k=0}^\infty \left(1 - \frac{1}{a_k} \right) \] in closed form.
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Context:
A positive integer is called *fancy* if it can be expressed in the form   $$ 2^{a_1}+2^{a_2}+ \cdots+ 2^{a_{100}}, $$   where  $a_1,a_2, \cdots, a_{100}$  are non-negative integers that are not necessarily distinct. Find the smallest positive integer  $n$  such that no multiple of  $n$  is a *fancy* number.

*Senior Problems Committee of the Australian Mathematical Olympiad Committee*
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Context:
Renaldo drove 15 kilometers. Ernesto drove 7 kilometers more than one-third of Renaldo's distance. Marcos drove a negative 5 kilometers. What is the absolute difference between the total distances driven by Renaldo and Ernesto combined, and the distance driven by Marcos?
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Context:
If 2³ - 7² = 1 ; 3³ - 6² = 9 ; 5³ - 9² = 16 then 4³ - 8² = ?
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Context:
Consider the function $f(x)=e^x+2x-a$ (where $a \in \mathbb{R}$, and $e$ is the base of natural logarithms). If there exists a point $(x_0, y_0)$ on the curve $y=\sin x$ such that $f(f(y_0))=y_0$, find the range of value for $a$.
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Context:
Using only pennies, nickels, dimes, and quarters, what is the smallest number of coins Freddie would need so he could pay any amount of money less than a dollar?
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Context:
We colour all the sides and diagonals of a regular polygon $P$ with 43 vertices either red or blue in such a way that every vertex is an endpoint of 20 red segments and 22 blue segments. A triangle formed by vertices of $P$ is called monochromatic if all of its sides have the same colour. Suppose that there are 2022 blue monochromatic triangles. How many red monochromatic triangles are there?
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Context:
Twelve colored cubes are arranged in a row. There are 3 blue cubes, 2 yellow cubes, 3 red cubes, and 4 green cubes, but not in that order. A yellow cube is at one end, and a red cube is at the other end. The red cubes are all together within the row, and the green cubes are also all together within the row. The tenth cube from the left is blue. What is the color of the sixth cube from the left?
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Context:
For real numbers $x_1, x_2, x_3, \dots, x_{333} \in [-1, \infty)$, let $S_k = \sum_{i = 1}^{333} x_i^k$ for each $k$. If $S_2 = 777$, compute the least possible value of $S_3$.
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Context:
How many numbers $\overline{abcd}$ with different digits satisfy the following property: if we replace the largest digit with the digit $1$, the resulting number is a multiple of $30$?
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Context:
Evaluate: $\displaystyle \int_{0}^{\infty}\frac{\sin\left(x^{2}\right) + \cos\left(x^{2}\right) - 1} {x^{2}}\,\mathrm{d}x$.
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Context:
What is the sum of all positive integers less than 500 that are fourth powers of even perfect squares?
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Context:
Mr. and Mrs. A are opposite to each other. The distance between them is unknown. The speed of Mr. A and Mrs. A are 30 kmph and 10 kmph, respectively. One bee is running between Mr. A's nose to Mrs. A's nose and returning back to Mr. A's nose. The speed of the bee is 60 kmph. The bee traveled 180 km. What is the distance between Mr. A and Mrs. A?
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Context:
Given that the function f(x) = A sin(ωx + φ) is monotonic on $[0, \frac{\pi}{3}]$ and f(0) = f($\frac{5\pi}{6}$) = -f($\frac{\pi}{3}$), find the positive value of ω.
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Context:
For what values of $c$ does the sequence $x_n=(2c-3)(c^{n-2}+c^{n-3}+\cdots + c+1)+2$ converge as $n\to\infty$?
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Context:
Let  $ABCDEF$  be a regular hexagon, and let  $G$ ,  $H$ ,  $I$ ,  $J$ ,  $K$ , and  $L$  be the midpoints of sides  $AB$ ,  $BC$ ,  $CD$ ,  $DE$ ,  $EF$ , and  $FA$ , respectively. The intersection of lines  $\overline{AH}$ ,  $\overline{BI}$ ,  $\overline{CJ}$ ,  $\overline{DK}$ ,  $\overline{EL}$ , and  $\overline{FG}$  bound a smaller regular hexagon. Find the ratio of the area of the smaller hexagon to the area of  $ABCDEF$ .
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Context:
If (10 ^ a certain exponent * 3.456789) ^ 10 is written as a single term, there would be 20 digits to the right of the decimal place. What is the certain exponent?
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Context:
In triangle \(ABC\), the medians drawn to the sides \(AC\) and \(BC\) intersect at a right angle. Given that \(AC = b\) and \(BC = a\), find \(AB\).
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Context:
Theo's watch is 10 minutes slow, but he believes it is 5 minutes fast. Leo's watch is 5 minutes fast, but he believes it is 10 minutes slow. At the same moment, each of them looks at his own watch. Theo thinks it is 12:00. Calculate the time Leo thinks it is. Express your answer in the format HH:MM, including AM/PM designation if necessary.
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Context:
Let $a_{1}, a_{2} \ldots, a_{51}$ be non-zero elements of a field. We simultaneously replace each element with the sum of the 50 remaining ones. In this way we get a sequence $b_{1} \ldots, b_{51}$. If this new sequence is a permutation of the original one, what can be the characteristic of the field? Please provide the sum of the possible characteristics.
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Context:
Evaluate the following limit:
$$	\lim_{x \to 0}x \tan (xa+ \arctan \frac{b}{x})$$
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Context:
Let $P(z) = z^8 + \left(4\sqrt{3} + 6\right)z^4 - \left(4\sqrt{3} + 7\right)$. What is the minimum perimeter among all the $8$-sided polygons in the complex plane whose vertices are precisely the zeros of $P(z)$?
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Context:
Five students made the following statements about how many people studied math yesterday:

- Zhang: "Nobody studied math."
- Li: "One person studied math."
- Wang: "Two people studied math."
- Zhao: "Three people studied math."
- Liu: "Four people studied math."

The teacher knows that some of them studied and some didn’t. Those who studied told the truth, and those who didn’t lied. Determine the exact number of students who studied math yesterday. Express your answer as a whole number.
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Context:
If the sum of the lengths of the six edges of a trirectangular tetrahedron $PABC$ (i.e., $\angle APB=\angle BPC=\angle CPA=90^o$) is $S$, determine its maximum volume.
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Context:
Ali chooses one of the stones from a group of 2005 stones, marks this stone in a way that Betül cannot see the mark, and shuffles the stones. At each move, Betül divides stones into three non-empty groups. Ali removes the group with more stones from the two groups that do not contain the marked stone (if these two groups have equal number of stones, Ali removes one of them). Then Ali shuffles the remaining stones. The game continues until two stones remain, and Ali confesses the marked stone. What is the minimum number of moves Betül needs to guarantee finding the marked stone? Express your answer as a whole number.
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Context:
The average age of 12 men is increased by years when two of them whose ages are 21 years and 23 years are replaced by two new men. What is the average age of the two new men?
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Context:
Find the integral of $$\int \frac{1}{1+\tan x}dx$$ without using partial fractions.
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Context:
Given \( x = -2272 \), \( y = 10^3 + 10^2 c + 10 b + a \), and \( z = 1 \) which satisfy the equation \( a x + b y + c z = 1 \), where \( a, b, c \) are positive integers with \( a < b < c \), find \( y \).
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Context:
一个不均匀的骰子，掷出 $1,2,3,4,5,6$ 点的概率依次成等差数列。独立地先后郑该骰子两次，所得的点数分别记为 $a, b$ 。若事件 " $a+b=7$ " 发生的概率为 $\frac{1}{7}$ ，则事件 " $a=b$ "发生的概率为 $\qquad$。原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值
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Context:
Given the following balanced chemical equation:

2 HClO2 + 3 H2O2 → 4 H2O + 2 ClO2

Determine the weight of 6 moles of HClO2 if 4 moles of H2O2 are reacted together in this reaction under equilibrium conditions.
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Context:
Let $P$ be a moving point on the circle $(x+3)^{2}+(y-1)^{2}=2$, and $Q$ be a moving point on the line $y=x$. Determine the minimum value of $|PQ|$.
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Context:
In the diagram, \(ABCD\) is a rectangle with \(AD = 13\), \(DE = 5\), and \(EA = 12\). The area of \(ABCD\) is
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Context:
Plane A and Plane B are 500 kilometres apart and flying towards each other . If plane A is travelling at a constant speed of 240 k/m and plane B is travelling at a constant speed of 360 k/m when will the planes meet ?
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Context:
Malcolm has 240 followers on Instagram and 500 followers on Facebook. The number of followers he has on Twitter is half the number of followers he has on Instagram and Facebook combined. Meanwhile, the number of followers he has on TikTok is 3 times the number of followers he has on Twitter, and he has 510 more followers on Youtube than he has on TikTok. Additionally, he has 120 followers on Pinterest which is the double of his Snapchat followers. Over the next month, he gains 15% more followers on Instagram and 20% more followers on Facebook. In that same period, he gains 30 more followers on Twitter, 45 on TikTok, and loses 10 followers on Snapchat. How many followers does Malcolm have on all his social media accounts after this one month period?
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Context:
设集合 $A=\left\{n \left\lvert\, \frac{n}{3} \in \mathbf{N}_{+}\right.\right\}$，$B=\left\{y \mid y=x+4+\sqrt{5-x^{2}}\right\}$. 求集合 $A \cap B$ 中元素的个数。
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Context:
There is a calculator with digits from 0 to 9 and two operation symbols. Initially, the display shows the number 0. You can press any buttons. The calculator performs actions in the sequence of button presses. If an operation symbol is pressed multiple times in a row, the calculator only remembers the last press. A scattered scientist pressed a lot of buttons in a random sequence. Estimate the probability that the result of the resulting chain of operations is an odd number.
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Context:
The diagonal of a regular 2006-gon \(P\) is called good if its ends divide the boundary of \(P\) into two parts, each containing an odd number of sides. The sides of \(P\) are also called good. Let \(P\) be divided into triangles by 2003 diagonals, none of which have common points inside \(P\). What is the maximum number of isosceles triangles, each of which has two good sides, that such a division can have?
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Context:
14. If $a, b, c \in \mathbf{R}^{+}$, and satisfy $\frac{k u b c}{a+b+c} \geqslant(a+b)^{2}+(a+b+4 c)^{2}$. Find the minimum value of $k$.
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Context:
Consider a function $f: \mathbb{N} \rightarrow \mathbb{N}$ such that for any two positive integers $x, y$, the equation $f(x f(y)) = y f(x)$ holds. Find the smallest possible value of $f(2007)$.
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Context:
An airline company is planning to introduce a network of connections between the ten different airports of Sawubonia. The airports are ranked by priority from first to last (with no ties). We call such a network *feasible* if it satisfies the following conditions:

- All connections operate in both directions
- If there is a direct connection between two airports A and B, and C has higher priority than B, then there must also be a direct connection between A and C.

Some of the airports may not be served, and even the empty network (no connections at all) is allowed. How many feasible networks are there?
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Context:
Several irrational numbers are written on a blackboard. It is known that for every two numbers  $ a$  and  $ b$  on the blackboard, at least one of the numbers  $ a\over b\plus{}1$  and  $ b\over a\plus{}1$  is rational. What maximum number of irrational numbers can be on the blackboard?
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Context:
Call a positive integer "mild" if its base-3 representation never contains the digit 2. How many values of \( n \) (where \( 1 \leq n \leq 1000 \)) have the property that both \( n \) and \( n^{2} \) are mild?
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Context:
For any complex number $w=a+bi$, $|w|$ is defined to be the real number $\sqrt{a^2+b^2}$. If $w=\cos40^\circ+i\sin40^\circ$, then $|w+2w^2+3w^3+...+9w^9|^{-1}$ equals
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Context:
设 $a, b, c$ 为正数， $a<b$ 。若 $a, b$ 为一元二次方程 $a x^{2}-b x+c=0$ 的两个根，且 $a, b, c$ 是一个三角形的三边长，则 $a+b-c$ 的取值范围是$\left(\frac{m}{n}, \sqrt{p}-q\right)$，求$m+n+p+q$的值。
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Context:
Let $[x]$ denote the greatest integer less than or equal to the real number $x$, and let $\{x\}=x-[x]$. Find the sum of the squares of all natural numbers \( m \) that satisfy
$$
[(2 m+1)\{\sqrt{2 m+1}\}]=m
$$
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Context:
Seven fishermen stand in a circle. The fishermen have a professional habit of exaggerating numbers. Each fisherman has an exaggeration factor (unique integer) representing how many times the actual number is exaggerated. For example, if a fisherman with an exaggeration factor of 3 catches two fish, he will say he caught six fish. When asked, "How many fish did your left neighbor catch?", the responses (not necessarily in the order the fishermen are sitting) were $12, 12, 20, 24, 32, 42,$ and $56$. When asked, "How many fish did your right neighbor catch?", six of the fishermen responded $12, 14, 18, 32, 48,$ and $70$. What did the seventh fisherman respond?
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Context:
A lock with three buttons opens when the buttons are pressed in a specific order immediately one after the other. What is the minimum number of button presses required to ensure that the lock opens? (Any series of button presses before the correct sequence does not affect the mechanism of the lock.)
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Context:
Phil has 7 green marbles and 3 purple marbles in a bag. He removes a marble at random, records the color, puts it back, and then repeats this process until he has withdrawn 6 marbles. What is the probability that exactly three of the marbles that he removes are green? Express your answer as a decimal rounded to the nearest thousandth.
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Context:
A local school is holding a food drive. Mark brings in 4 times as many cans as Jaydon. Jaydon brings in 5 more than twice the amount of cans that Rachel brought in. Additionally, Sophie brings in cans such that the ratio of the number of cans contributed by Mark, Jaydon, and Sophie is 4:3:2. If there are 225 cans total, how many cans did Mark bring in?
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Context:
问：至多能从集合 \{1,2,\cdots,100\} 中取出多少个数，使得其中任意不同两数的和与积均不为 100 的倍数？
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Context:
Positive integers $a,b,$ and $c$ are chosen so that $a<b<c$, and the system of equations $2x + y = 2003 \quad$ and $\quad y = |x-a| + |x-b| + |x-c|$ has exactly one solution. What is the minimum value of $c$?
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Context:
Select 5 apples from a pile randomly, and measure their masses (unit: grams) as: $125, 124, 121, 123, 127$. Find the standard deviation $s=$ ______ (grams) (Answer with a number).
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Context:
There is a cube placed on stands and six different paints. How many ways can you paint all the faces of the cube (one color for each face, not necessarily using all the paints) so that adjacent faces (sharing an edge) are painted in different colors? (16 points)
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Context:
Given that $g\in C(\mathbb{R})$, $g(0) = 1$ and $g(1) = -1$, evaluate the following integral:

$$\int_{0}^1 \frac{g'(x) \cos(g(x))}{1 + g^2(x)}\mathrm{d}x.$$
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Context:
For a given real value of $x$, at what value of $n$ does the expression $x\ln x-n\ln(x!)$ change its behavior from increasing to decreasing?
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Context:
Given \( a, b > 0 \) satisfying the equation \( \sqrt{|x|} + \sqrt{|x + a|} = b \) having exactly three distinct real solutions \( x_{1}, x_{2}, x_{3} \), with \( x_{1} < x_{2} < x_{3} = b \), find the value of \( a + b \).
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Context:
Given $\lg 2=0.3010$ and $\lg 3=0.4771$, calculate the decimal place where the first non-zero digit of $\left(\frac{6}{25}\right)^{100}$ occurs. Express your answer as a whole number.
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Context:
The most common dice is a six-sided die, which is a cube. The 6 faces have 1 to 6 points on them, with the sum of points on opposite faces equal to 7. From a single point in space, you can see multiple faces of the die. The minimum visible sum of points is 1, and the maximum is 15 (15 = 4 + 5 + 6). Determine which sums in the range from 1 to 15 are not possible to see.
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Context:
At an election, 2 candidates participated. One candidate got 20% of votes and was defeated by 500 votes. There were 10 invalid votes. How many total polled votes were there?
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Context:
If the graph of the function $f(x)$ is continuous and unbroken on the interval $(0, +\infty)$, and there is a unique irrational number root $x_0$ in the interval $(2,3)$, then to find the approximate value of $x_0$ with an accuracy of 0.001 using the "bisection method", the function value at the midpoint of the interval needs to be calculated \_\_\_\_\_\_ times.
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Context:
Mrs. Hilt started by reading 2 books per day, but from the third day, she increased the number of books she read daily by 1. How many books did she read in one week?
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Context:
Person A and person B jog back and forth on a straight road that is 400 meters long, with speeds of 3 meters per second and 2.5 meters per second, respectively. They start simultaneously from opposite ends and meet a total of ____ times within 20 minutes.
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Context:
平面直角坐标系中, $\vec{e}$ 是单位向量, 向量 $\vec{a}$ 满足 $\vec{a} \cdot \vec{e}=2$, 且 $|\vec{a}|^{2} \leqslant 5|\vec{a}+t \vec{e}|$对任意实数 $t$ 成立，求 $|\vec{a}|$ 的取值范围. 请给出该区间端点之和的整数部分。
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Context:
The length of a rectangular plot is 20 meters more than its breadth. If the cost of fencing the plot is 26.50 per meter on flat ground, the price increases by an additional 10% for each meter of height the ground rises due to unevenness. If the total cost of fencing the plot is Rs. 5300, and the ground rises a total of 5 meters along the breadth of the plot, what is the length of the plot in meters?
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Context:
Find the two numbers such that their HCF is 23, the product is 98596, and the difference of these numbers is minimized. What is the greater number?
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Context:
A sequence \(a_{1}, a_{2}, a_{3}, \ldots\) of positive integers satisfies \(a_{1} > 5\) and \(a_{n+1} = 5 + 6 + \cdots + a_{n}\) for all positive integers \(n\). Determine all prime numbers \(p\) such that, regardless of the value of \(a_{1}\), this sequence must contain a multiple of \(p\).
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Context:
Given a sequence $\{a_n\}$ whose sum of the first $n$ terms is $S_n = n^2$, and in $\triangle ABC$, the ratio of the three sides is $a:b:c = a_2:a_3:a_4$, then the largest interior angle of $\triangle ABC$ equals \_\_\_\_\_\_.
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Context:
Convex quadrilateral $ABCD$ has $AB = 20$, $BC = CD = 26$, and $\angle ABC = 90^\circ$. Point $P$ is on $DA$ such that $\angle PBA = \angle ADB$. If $PB = 20$, compute the area of $ABCD$.
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Context:
Given a convex quadrilateral \( ABCD \) with an interior point \( P \) such that \( P \) divides \( ABCD \) into four triangles \( ABP, BCP, CDP, \) and \( DAP \). Let \( G_1, G_2, G_3, \) and \( G_4 \) denote the centroids of these triangles, respectively. Determine the ratio of the area of quadrilateral \( G_1G_2G_3G_4 \) to the area of \( ABCD \).
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Context:
Compare the values of $\sin 11^\circ$, $\sin 168^\circ$, and $\cos 10^\circ$. Express your answer as an inequality with the correct order of these values.
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\boxed{\sin 11^\circ < \sin 168^\circ < \cos 10^\circ}


Context:
Two high school classes took the same test. One class of $20$ students made an average grade of $80\%$; the other class of $30$ students made an average grade of $70\%$. The average grade for all students in both classes is:
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Context:
Find the least positive integer of the form $a\ b\ a\ a\ b\ a$, where $a$ and $b$ are distinct digits, such that the integer can be written as a product of six distinct primes.
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Context:
A strip of size $1 \times 10$ is divided into unit squares. The numbers $1, 2, \ldots, 10$ are written in these squares. First, the number 1 is written in one of the squares, then the number 2 is written in one of the neighboring squares, then the number 3 is written in one of the squares neighboring those already occupied, and so on (the choice of the first square and the choice of neighbor at each step are arbitrary). In how many ways can this be done?
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Context:
On the first Saturday, Jessy told eleven jokes, while Alan told seven jokes. Meanwhile, Tom told five jokes and Emily told three jokes. The following week, on each day Sunday through Wednesday, each of them doubled the number of jokes they told from the previous day. How many jokes did they tell together after eight days?
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Context:
On the board, the natural number \( N \) was written nine times (one below the other). Petya added a non-zero digit to the left or right of each of the 9 numbers; all added digits are distinct. What is the largest possible number of prime numbers that could result from these 9 new numbers?

(I. Efremov)
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Context:
What is the minimum number of points in set M, given that circle C7 passes through exactly 7 points in M, circle C6 passes through exactly 6 points in M, and so on, with circle C1 passing through exactly 1 point in M?
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Context:
Find the third side of an acute triangle if its two sides are \( a \) and \( b \) and it is known that the medians to these sides intersect at a right angle.
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Context:
Let  $ABC$  be a right triangle with  $m(\widehat{A})=90^\circ$ . Let  $APQR$  be a square with area  $9$  such that  $P\in [AC]$ ,  $Q\in [BC]$ ,  $R\in [AB]$ . Let  $KLMN$  be a square with area  $8$  such that  $N,K\in [BC]$ ,  $M\in [AB]$ , and  $L\in [AC]$ . What is  $|AB|+|AC|$ ?
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Context:
While waiting for customers, a watermelon seller weighed 20 watermelons (each weighing 1 kg, 2 kg, 3 kg, ..., 20 kg) by balancing a watermelon on one side of the scale with one or two weights (possibly identical) on the other side of the scale. The seller recorded the weights he used on a piece of paper. What is the minimum number of different numbers that could appear in his notes if the weight of each weight is an integer number of kilograms?
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Context:
2. Given are two circles $k_{1}$ and $k_{2}$ with centers $P$ and $R$ respectively, which touch each other externally at point $A$. Let $p$ be their common tangent, which does not pass through $A$, and touches $k_{1}$ and $k_{2}$ at points $B$ and $C$, respectively. The line $P R$ intersects the line $B C$ at point $E$, and the circle $k_{2}$ at points $A$ and $D$. If $\overline{A C}=\frac{\overline{A B}}{2}$, find the ratio $\frac{\overline{B C}}{\overline{D E}}$.
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Context:
Initially, there are 41 crayons and 26 pencils in the drawer. Rachel removes 8 crayons from the drawer, and then Sam adds 12 crayons back into the drawer. After that, the number of crayons in the drawer increases by 10%. How many crayons are in the drawer now?
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Context:
Given the sequence ${a_n}$ with the sum of its first $n$ terms denoted as $S_n$, and $a_n= \frac {n\cdot 2^{n}-2^{n+1}}{(n+1)(n^{2}+2n)}\ (n\in\mathbb{N_+})$, find the expression for $S_n$.
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Context:
During Teacher's Day at a certain school, there are 200 teachers who each send a greeting message to exactly one other teacher. Determine the minimum value of \( k \) such that it is possible to select at least \( k \) representatives to attend a recognition ceremony, where no one in this group has sent a message to anyone else in the group.
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Context:
A train of a certain length crosses an electric pole in 4.499640028797696 seconds, with its speed being 72 km/hr. How long is the train?
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Context:
Two differentiable functions $f(x)$ and $g(x)$ are defined on the interval $[0, 2]$. Given that $f''(x) - g''(x) = 0$, $f'(A) = 2$, $g'(A) = 4$, $f(B) = 3$, and $g(B) = 9$, find the value of $f(\frac{3}{2}) - g(\frac{3}{2})$.
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Context:
Eliana walked 200 steps for her morning exercise, did some press-ups, then added some more steps to her count for the first day. The next day, she walked twice the number of steps she walked on the first day. And on the third day, Eliana walked an additional 100 steps. The total number of steps Eliana walked during these three days is 1600. How many additional steps did Eliana add to her count on the first day after doing press-ups?
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Context:
Given the proposition p: ∀x ∈ ℝ, 2^x > 0, what is the proposition ¬p? Express your answer as a logical proposition using correct logical notation and quantifiers.
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Context:
Evaluate the integral
$$I = \int_0^{\infty} \frac{\log(x)}{x^2 - 1} \, dx$$
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Context:
In triangle \(ABC\) with a \(120^\circ\) angle at vertex \(A\), the bisectors \(AA_1\), \(BB_1\), and \(CC_1\) are drawn. Find the angle \(C_1 A_1 B_1\).
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Context:
Let \( S = \{1, 2, \ldots, 98\} \). Find the smallest natural number \( n \) such that from any subset of \( S \) containing \( n \) elements, we can always select 10 numbers. No matter how we divide these 10 numbers into two groups, there will always be one group containing a number that is coprime with the other 4 numbers in its group, and another group containing a number that is not coprime with the other 4 numbers in its group.
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Context:
Let $l$ be a tangent line at the point $T\left(t,\ \frac{t^2}{2}\right)\ (t>0)$ on the curve $C: y=\frac{x^2}{2}.$ A circle $S$ touches the curve $C$ and the $x$ axis.
Denote by $(x(t),\ y(t))$ the center of the circle $S$. Find $\lim_{r\rightarrow +0}\int_r^1 x(t)y(t)\ dt.$
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Context:
4 and a certain number are factors of 60 n, and the minimum value of n is 8. What is the other factor?
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Context:
25. How many even four-digit numbers can be formed from the digits $0,1,2,3,4,5$?
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Context:
A show debut and 200 people buy tickets. For the second showing, a certain number of people show up. If each ticket cost $25, the show made $20,000. What is the ratio of the number of people who attended the second showing to the number of people who attended the debut show?
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Context:
The segment \( AB \) is the diameter of a circle. A second circle with center at point \( B \) has a radius of 2 and intersects the first circle at points \( C \) and \( D \). The chord \( CE \) of the second circle is part of a tangent to the first circle and is equal to 3. Find the radius of the first circle.
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Context:
Let $z_1,$ $z_2,$ $z_3,$ and $z_4$ be the four distinct complex solutions of the equation
\[
  z^4 - 6z^2 + 8z + 1 = -4(z^3 - z + 2)i.
\]Find the sum of the six pairwise distances between $z_1,$ $z_2,$ $z_3,$ and $z_4$ in the complex plane.The answer is in the form k\sqrt{m}+n,. Please provide the value of k + m + n.
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Context:
$\triangle ABC$ is inscribed in a circle of radius $r$ such that its base $AB$ coincides with the diameter $AB$. Point $C$ lies somewhere on the circle different from points $A$ and $B$. Let $s = AC + BC$. Determine the maximum possible value of $s^2$ given all permissible positions of $C$ on the circle.
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Context:
A traveler visited a village where each person either always tells the truth or always lies. The villagers stood in a circle, and each told the traveler whether the neighbor to their right was truthful. Based on these statements, the traveler was able to unequivocally determine the fraction of liars among the villagers. Determine what this fraction is.
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Context:
Let $a_1 = 11^{11}$, $a_2 = 12^{12}$, $a_3 = 13^{13}$, and
\[a_n = |a_{n-1} - a_{n-2}| + |a_{n-2} - a_{n-3}|, \quad n \ge 4.\]
Determine $a_{14^{14}}$.
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Context:
In what base number systems can a three-digit number be equal to twice the number formed by reversing the order of its digits?
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Context:
What is the negation of the proposition ∀x ∈ R, e^x - x - 1 ≥ 0? Express your answer as a logical statement using proper mathematical notation.
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Context:
10.3. 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:
On an \(8 \times 8\) chessboard, 6 black rooks and \(k\) white rooks are placed on different cells so that each rook only attacks rooks of the opposite color. Compute the maximum possible value of \(k\). (Two rooks attack each other if they are in the same row or column and no rooks are between them.)
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Context:
Let $M = \{y | y = x^2, x \in \mathbb{Z}\}$ and $N = \{x | x^2 - 6x - 27 \geq 0, x \in \mathbb{R}\}$, with the universal set $U = \mathbb{R}$. Calculate the number of proper subsets of $M \cap (\complement_U N)$. Express your answer as a single integer.
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Context:
Seven little children sit in a circle. The teacher distributes pieces of candy to the children in such a way that the following conditions hold. - Every little child gets at least one piece of candy. - No two little children have the same number of pieces of candy. - The numbers of candy pieces given to any two adjacent little children have a common factor other than 1. - There is no prime dividing every little child's number of candy pieces. What is the smallest number of pieces of candy that the teacher must have ready for the little children?
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Context:
The sequence $x_1, x_2, x_3, \ldots$ is defined by $x_1 = 2022$ and $x_{n+1} = 7x_n + 5$ for all positive integers $n$. Determine the maximum positive integer $m$ such that \[ \frac{x_n(x_n - 1)(x_n - 2) \cdots (x_n - m + 1)}{m!} \] is never a multiple of $7$ for any positive integer $n$.
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Context:
Evaluate the line integral $\int_\mathcal{E} F*dr$ where $F(x,y,z) = x\,i+xy\,j+x^2yz\,k$ and $\mathcal{E}$ is the ellipse $x^2 + 4y^2 - 8y + 3 = 0$ in the xy-plane, without using parameterization.
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Context:
Determine the range of values for the real number $a$ such that the inequality $(a-2)x^{2}+2(a-2)x-4 < 0$ holds true for all real numbers $x$.
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Context:
Given \(3^{s} + 13^{t} = 17^{s}\) and \(5^{s} + 7^{t} = 11^{t}\), determine the relationship between \(s\) and \(t\) (fill in \(>, <\) or \(=\)).
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Context:
We draw a radius of a circle. We draw a second radius $23^\circ$ clockwise from the first radius. We draw a third radius $23^\circ$ clockwise from the second. This continues until we have drawn $40$ radii, each $23^\circ$ clockwise from the one before it. What is the measure in degrees of the smallest angle between any two of these $40$ radii?
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Context:
"Pre-assembled" unit cube has a base and a top, as well as its sides. The top is rotated by an acute angle $\alpha$ around the line connecting the centers of the two faces, relative to the base. By how much does the top get closer to the base if the side edges remain the same length and the connection between the vertices also remains unchanged?
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Context:
Given a triangle with sides of lengths 3, 4, and 5. Three circles are constructed with radii of 1, centered at each vertex of the triangle. Find the total area of the portions of the circles that lie within the triangle.
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Context:
Find the value of $\left(1+\cos \frac{\pi}{5}\right)\left(1+\cos \frac{3 \pi}{5}\right)$. The original answer is in the form of \frac{k}{m}, please provide the value of k + m.
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Context:
For a cone and a cylinder with the same base diameter and height, what is the ratio of their lateral surface areas?
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Context:
Find the number of different ways to arrange all natural numbers from 1 to 9 inclusively, one in each cell of a 3x3 grid, such that the sums of the numbers in each row and each column are equal. The table cannot be rotated or reflected.
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Context:
Marco loves trading cards. He has a total of 750 cards on hand, and a third of them are duplicates. Marco decides to trade one-fifth of these duplicates for completely new cards to his friend Josh. After trading with Josh, Marco trades an additional one-third of these new cards with his friend Alex for more distinct cards. Finally, Marco exchanges half of the remaining new cards from the trades with Josh and Alex with his friend Mia. How many distinct new cards is Marco left with after all three trades?
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Context:
Given that the universal set U = ℝ, let set P = {x | x ≥ 3} and M = {x | x < 4}. Calculate the intersection of set P with the complement of set M with respect to the universal set U, denoted as P ∩ (C_U M). Express your answer as a set definition using proper set notation.
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Context:
4. Find the smallest positive number $c$ such that for any integer $n \geqslant 4$, and any set $A \subseteq\{1,2, \cdots, n\}$, if $|A|>c n$, then there exists a function $f: A \rightarrow\{1,-1\}$, satisfying $\left|\sum_{a \in A} f(a) \cdot a\right| \leqslant 1$.
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Context:
The domain of $y= \frac{\sqrt{\sin x} + \lg(\cos x)}{\tan x}$ is __________.
Please reason step by step, and put your final answer within \boxed{}.

\boxed{(2k\pi, \frac{\pi}{2} + 2k\pi), k \in \mathbb{Z}}


Context:
Let $A B C$ be a triangle and $\omega$ be its circumcircle. The point $M$ is the midpoint of arc $B C$ not containing $A$ on $\omega$ and $D$ is chosen so that $D M$ is tangent to $\omega$ and is on the same side of $A M$ as $C$. It is given that $A M=A C$ and $\angle D M C=38^{\circ}$. Find the measure of angle $\angle A C B$.
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Context:
Consider the function \( f(x) = \frac{1}{(1+2x)^4} \), which has a linear approximation \( L(x) = 1 - 8x \) around \( x = 0 \). Determine the interval of \( x \) values for which the linear approximation is accurate within an error of \( 0.1 \). Express this as an inequality: \( -0.1 \leq f(x) - L(x) \leq 0.1 \).
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Context:
Given the quadratic equation $x(kx+1)-x^{2}+3=0$, find the largest integer value of $k$ for which the equation has real roots.
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Context:
A circle with diameter $AB$ is drawn, and the point $ P$ is chosen on segment $AB$ so that $\frac{AP}{AB} =\frac{1}{42}$ . Two new circles $a$ and $b$ are drawn with diameters $AP$ and $PB$ respectively. The perpendicular line to $AB$ passing through $ P$ intersects the circle twice at points $S$ and $T$ . Two more circles $s$ and $t$ are drawn with diameters $SP$ and $ST$ respectively. For any circle $\omega$ let $A(\omega)$ denote the area of the circle. What is $\frac{A(s)+A(t)}{A(a)+A(b)}$?
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Context:
In the trapezoid \(ABCD\), the side \(CD\) is equal to the diagonal \(AC\). On the smaller arc \(BC\) of the circumcircle of triangle \(BCD\), a point \(E\) is chosen such that \(CD = CE\). Find the angle \(\angle AEB\).
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Context:
In a random arrangement of 3 red, 2 white, and 2 blue balls, what is the probability of having 2 adjacent white balls or 2 adjacent blue balls? Express your answer as a simplified fraction in the form \(\boxed{\frac{a}{b}}\).
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Context:
The tower function of twos is defined recursively as follows: $T(1) = 2$ and $T(n + 1) = 2^{T(n)}$ for $n\ge1$. Let $A = (T(2009))^{T(2009)}$ and $B = (T(2009))^A$. What is the largest integer $k$ for which $\underbrace{\log_2\log_2\log_2\ldots\log_2B}_{k\text{ times}}$ is defined?
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Context:
How many positive integers $n$ satisfy \[\dfrac{n+1000}{70} = \lfloor \sqrt{n} \rfloor?\](Recall that $\lfloor x\rfloor$ is the greatest integer not exceeding $x$.)
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Context:
Zang is at the point $(3,3)$ in the coordinate plane. Every second, he can move one unit up or one unit right, but he may never visit points where the $x$ and $y$ coordinates are both composite. In how many ways can he reach the point $(20, 13)$?
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Context:
Solve the equation or inequality using your calculator: $\ln (x)=e^{-x}$
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Context:
How many times longer is the staircase to the fourth floor of a building compared to the staircase to the second floor of the same building?
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Context:
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. If $a^{2}=3b^{2}+3c^{2}-2\sqrt{3}bc\sin A$, find the value of angle $C$.
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Context:
15. Xiaohua wrote a computer program. One minute after the program runs, some bubbles first appear on the computer screen, and at each subsequent integer minute, the same number of new bubbles appear. Half a minute after the 11th appearance of bubbles, one bubble bursts. Thereafter, every minute, more bubbles burst, with the number increasing by 1 each minute (i.e., half a minute after the 12th appearance of bubbles, 2 bubbles burst...). At a certain moment, the total number of burst bubbles is exactly equal to the total number of bubbles that have appeared on the screen, meaning all bubbles have disappeared at this moment. How many bubbles were there at most on the screen at any given time during the entire process of the program running?
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Context:
A number, when divided by the sum of a certain number and 445, gives 2 times their difference as quotient and 40 as remainder. The number is 220040. What is the first number?
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Context:
Given the function $f(x)=f'( \frac {\pi}{6})\sin x+\cos x$, then the value of $f( \frac {\pi}{6})$ is \_\_\_\_\_\_.
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Context:
Charlie noticed his golden ticket was golden in two ways! In addition to being gold, it was a rectangle whose side lengths had ratio the golden ratio $\varphi = \tfrac{1+\sqrt{5}}{2}$. He then folds the ticket so that two opposite corners (vertices connected by a diagonal) coincide and makes a sharp crease (the ticket folds just as any regular piece of paper would). The area of the resulting shape can be expressed as $a + b \varphi$. What is $\tfrac{b}{a}$?
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Context:
The numbers 1 to 2031 are written on a blackboard.
1. Select any two numbers on the blackboard, find the absolute value of their difference, and erase these two numbers.
2. Then select another number on the blackboard, find the absolute value of its difference from the previous absolute value obtained, and erase this number.
3. Repeat step (2) until all numbers on the blackboard are erased.
What is the maximum final result?
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Context:
A solid $\Omega$ is bounded by the planes $x=2$, $y=0$, $y=1$, $z=0$ and the cylinder $z^2 = 6x$. The volume density of the material at each point is proportional to the distance from the $XOY$ plane. Find the moment of inertia of the solid $\Omega$ relative to the $z$-axis.
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Context:
If $A$ and $B$ are two different points on the parabola $y^{2}=4x$, and the perpendicular bisector of segment $AB$ intersects the $x$-axis at point $D(4,0)$, then the maximum value of $|AB|$ is ______.
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Context:
Define a sequence of functions: $f_{1}(x)=f(x)= \dfrac {x}{1-x}$, $f_{2}(x)=f(f_{1}(x))$, $f_{3}(x)=f(f_{2}(x))$,..., $f_{n}(x)=f(f_{n-1}(x))$. Find the coordinates of the intersection point between the graph of the function $y=f_{2017}(x)$ and the curve $y= \dfrac {1}{x-2017}$. Express your answer as an ordered pair (x, y) with exact values.
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Context:
Evaluate the limit $\lim_{x \to 0} \frac {\sin(xe^x)-\sin(xe^{-x})}{x^3}$ using only asymptotic simplifications and Taylor series.
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Context:
A line segment AB has a length of 2, with its endpoints A and B sliding along the x-axis and y-axis, respectively. What geometric figure is formed by the trajectory of the midpoint of line segment AB?
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Context:
Given two lines $l_{1}: 2x-ay-1=0$ and $l_{2}: ax-y=0$. If $l_{2} \nparallel l_{1}$, then the real number $a=$ ______.
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Context:
1. (YUG) Find all real numbers $x \in[0,2 \pi]$ such that  $$ 2 \cos x \leq|\sqrt{1+\sin 2 x}-\sqrt{1-\sin 2 x}| \leq \sqrt{2} $$
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Context:
On the segment \( AB \), point \( C \) is taken, and segments \( AB \) and \( CB \) serve as diameters of circles. Chord \( AM \) touches the smaller circle at point \( D \). Line \( BD \) intersects the larger circle at point \( N \). Given that \( \angle DAB = \boldsymbol{\alpha} \) and \( AB = 2R \), find the area of quadrilateral \( ABMN \).
Please reason step by step, and put your final answer within \boxed{}.

\boxed{R^2 \cos \alpha (1 + \sin \alpha)}


Context:
Let \( n \geqslant 4, \alpha_{1}, \alpha_{2}, \cdots, \alpha_{n} ; \beta_{1}, \beta_{2}, \cdots, \beta_{n} \) be two sets of real numbers, satisfying \( \sum_{j=1}^{n} \alpha_{j}^{2}<1 \) and \( \sum_{j=1}^{n} \beta_{j}^{2}<1 \). Define
\[ A^{2}=1-\sum_{j=1}^{n} \alpha_{j}^{2}, \, B^{2}=1-\sum_{j=1}^{n} \beta_{j}^{2}, \, W=\frac{1}{2}\left(1-\sum_{j=1}^{n} \alpha_{j} \beta_{j}\right)^{2} . \]

Find all real numbers \(\lambda\) such that the equation \( x^{n}+\lambda\left(x^{n-1}+\cdots+x^{3}+W x^{2}+A B x+1\right)=0 \) has only real roots.
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Context:
Define a function $g : \mathbb{N} \rightarrow \mathbb{R}$ such that:
\[
g(x) = \sqrt{4^x + \sqrt{4^{x+1} + \sqrt{4^{x+2} + \cdots}}}
\]
Find the last two digits in the decimal representation of $g(2021)$.
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Context:
The distance from $A$ to $B$ is 999 km. Along the road, there are kilometer markers that indicate the distance to $A$ and to $B$ as follows:

$0|999, 1|998, \ldots, 999|0$.

How many of these markers have signs that contain only two different digits?
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Context:
Find the integer $n,$ $-180 \le n \le 180,$ such that $\cos n^\circ = \cos 430^\circ.$
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Context:
Triangle \(\triangle A B C\) has \(A B=21, B C=55\), and \(C A=56\). There are two points \(P\) in the plane of \(\triangle A B C\) for which \(\angle B A P=\angle C A P\) and \(\angle B P C=90^{\circ}\). Find the distance between them.
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Context:
Convert the number 380,180,000,000 to scientific notation. Express your answer in the form a × 10^n, where 1 ≤ a < 10.
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Context:
To understand the scores of 1252 students in a knowledge competition at a certain school, it was decided to use systematic sampling to draw a sample of 50. The number of individuals randomly eliminated from the population is __________.
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Context:
Let $a,$ $b,$ and $c$ be the roots of the equation $x^3 - 15x^2 + 25x - 10 = 0$. Calculate the value of $(1+a)(1+b)(1+c)$.
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Context:
设多项式 \(f(x)\) 的各项系数都是非负实数，且 \(f(1)=f^{\prime}(1)=f^{\prime \prime}(1)=f^{\prime \prime \prime}(1)=1\)。求 \(f(x)\) 的常数项的最小值。答案的形式为\(\frac{k}{m}\)，请给出k+m的值。
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Context:
We are given five watches which can be wound forward. What is the smallest sum of winding intervals that allows us to set them to the same time, regardless of their initial settings?
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Context:
Three circles with radius $2$ are mutually tangent. What is the total area of the circles and the region bounded by them, as shown in the figure? Express your answer in the form $a\pi + b\sqrt{3}$, where $a$ and $b$ are constants. Please find the value of $a + b$.
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Context:
A circle centered at point \( p \) is tangent to the circle given by the equation \((x+1)^{2} + y^{2} = 1\) externally and is tangent to the circle given by the equation \(x^{2} + y^{2} - 2x - 8 = 0\) internally. Determine the coordinates of the point \( p \) that is the farthest distance from the origin.
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Context:
Out of ten disks, three have both sides red, two have both sides blue, and the remaining five have one side blue and the other side red. The ten disks are placed in a hat, one is drawn, and one side of it is observed. How should we guess the color of the other side of the drawn disk?
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Context:
A box contains 11 balls, numbered 1, 2, 3, ..., 11. If 6 balls are drawn simultaneously at random, find the probability that the sum of the numbers on the balls drawn is odd. Express your answer as a fraction in simplest form, \(\frac{k}{m}\), and provide the value of k + m.
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Context:
Given $F_{1}$ and $F_{2}$ are the foci of a pair of related curves, and $P$ is the intersection point of the ellipse and the hyperbola in the first quadrant, when $\angle F_{1}PF_{2}=60^{\circ}$, determine the eccentricity of the ellipse in this pair of related curves.
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Context:
长度分别为 $1, a, a, a, a, a$ 的线段能成为同一个四面体的 6 条棱的充要条件是 $a$ 满足的条件是 \frac{\sqrt{k}}{m}，请给出k+m的值。
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Context:
In triangle $ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively. Given that $\sin C + \sin (B-A) = \sin 2A$, $\sin C= \frac{\sqrt{3}}{3}$, and $a-b=3- \sqrt{6}$, find the area of $\triangle ABC$.
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Context:
Paco had some salty cookies and 34 sweet cookies. He ate 15 sweet cookies and 56 salty cookies. After that, Paco had 19 sweet cookies left. How many salty cookies did Paco have initially?
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Context:
在底面半径为 6 的圆柱内, 有两个半径也为 6 的球面，其球心距为 13 。若作一平面与这二球面相切，且与圆柱面相交成一栯圆，则这个粨圆的长轴长与短轴长之和是 $\qquad$.
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Context:
The product of two positive integers plus their sum equals 119. The integers are relatively prime, and each is less than 25. What is the sum of the two integers?
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Context:
The "Hua Luogeng" Golden Cup Junior Math Invitational Contest was first held in 1986, the second in 1988, and the third in 1991, and has subsequently been held every 2 years. The sum of the digits of the year of the first "Hua Cup" is: \( A_1 = 1 + 9 + 8 + 6 = 24 \). The sum of the digits of the years of the first two "Hua Cup" contests is: \( A_2 = 1 + 9 + 8 + 6 + 1 + 9 + 8 + 8 = 50 \). Find the sum of the digits of the years of the first 50 "Hua Cup" contests, \( A_{50} \).
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Context:
In a convex quadrilateral \(ABCD\), side \(AB\) is equal to diagonal \(BD\), \(\angle A=65^\circ\), \(\angle B=80^\circ\), and \(\angle C=75^\circ\). What is \(\angle CAD\) (in degrees)?
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Context:
If 0.5% of a equals 95 paise, b is three times the amount of a minus 50, and c is the difference between a and b squared, what is the combined value of a, b, and c in rupees, considering that a and c must be positive integers?
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Context:
设正六边形 $A B C D E F$ 的边长为 1. 则 $(\overrightarrow{A B}+\overrightarrow{D C}) \cdot(\overrightarrow{A D}+\overrightarrow{B E})=$
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Context:
In triangle \(ABC\), side \(BC\) is equal to \(a\), and the radius of the inscribed circle is \(r\).

Find the radii of two equal circles that touch each other, if one of them touches sides \(BC\) and \(BA\), and the other touches sides \(BC\) and \(CA\).
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Context:
What is the negation of the proposition '$\forall x > 1$, $x^{2}+1\geqslant 0$'? Express your answer as a logical statement.
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Context:
设$n=1(mod4)$且$n>1$, $P = ( a _ { 1 } , a _ { 2 } , \cdots , a _ { n } )$是$(1，2,⋯,n)$的任意排列，$k_p$表示使不等式$a _ { 1 } + a _ { 2 } + \cdots + a _ { k } < a _ { k + 1 } + a _ { k + 2 } + \cdots + a _ { n }$成立的最大下标$k$.试对一切可能的不同排列$P$，求对应的$k_p$的和.
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Context:
A pentagon is drawn by placing an isosceles right triangle on top of a square as pictured. What percent of the area of the pentagon is the area of the right triangle?

[asy]
size(50);
draw((0,0)--(0,-1)--(1,-1)--(1,0)--(0,0)--(.5,.5)--(1,0));
[/asy]
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Context:
Two cars, A and B, start from points A and B respectively and travel towards each other at the same time. They meet at point C after 6 hours. If car A maintains its speed and car B increases its speed by 5 km/h, they will meet 12 km away from point C. If car B maintains its speed and car A increases its speed by 5 km/h, they will meet 16 km away from point C. What was the original speed of car A?
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Context:
Find a recursive formula for the integral $I_n = \int_0^2 x^n(4 - x^2)^{1/2} dx$, where $n \geq 2$, using a suitable method.
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Context:
The  $42$  points  $P_1,P_2,\ldots,P_{42}$  lie on a straight line, in that order, so that the distance between  $P_n$  and  $P_{n+1}$  is  $\frac{1}{n}$  for all  $1\leq n\leq41$ . What is the sum of the distances between every pair of these points? (Each pair of points is counted only once.)
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Context:
For an ordered $10$-tuple of nonnegative integers $a_1,a_2,\ldots, a_{10}$, we denote:
$$f(a_1,a_2,\ldots,a_{10})=\left(\prod_{i=1}^{10} {\binom{20-(a_1+a_2+\cdots+a_{i-1})}{a_i}}\right) \cdot \left(\sum_{i=1}^{10} {\binom{18+i}{19}}a_i\right).$$
When $i=1$, we take $a_1+a_2+\cdots+a_{i-1}$ to be $0$. Let $N$ be the average of $f(a_1,a_2,\ldots,a_{10})$ over all $10$-tuples of nonnegative integers $a_1,a_2,\ldots, a_{10}$ satisfying:
$$a_1+a_2+\cdots+a_{10}=20.$$
Compute the number of positive integer divisors of $N$.
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Context:
Given an ellipse with a chord passing through the focus and perpendicular to the major axis of length $\sqrt{2}$, and a distance from the focus to the corresponding directrix of $1$, determine the eccentricity of the ellipse.
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Context:
一珍稀物种出现在地球，对每个珍稀生物，每天有如下事件发生：有 p (0\leq p\leq 1) 的概率消失，有 \df{1-p}{3} 的概率保持不变，有 \df{1-p}{3} 的概率分裂成两个，有 \df{1-p}{3} 的概率分裂成三个. 对所有新产生的生物每天也会发生上述事件. 假设开始只有一个这样的珍稀生物，若希望最终这种生物灭绝的概率不超过 \df{1}{2}，则 p 至多为__________.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值
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Context:
Two workers started paving paths in a park simultaneously from point $A$. They pave with constant speeds: the first one on the section $A-B-C$, and the second one on the section $A-D-E-F-C$. They finished their work at the same time, spending 9 hours on it. It is known that the second worker works 1.2 times faster than the first. How many minutes did the second worker spend paving the section $DE$?
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Context:
Evaluate the integral \( \int \frac{\cos 2x}{\cos^2 x \sin^2 x} \, dx \).
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Context:
If a number is selected at random from the set of all five-digit numbers in which the sum of the digits is equal to $43$, find the probability that this number is divisible by $11$. The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.
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Context:
Humanity has discovered 15 habitable planets, where 7 are "Earth-like" and 8 are "Mars-like". Colonizing an Earth-like planet requires 3 units of colonization, while a Mars-like planet requires 1 unit. If humanity has 21 units available for colonization, determine how many different combinations of planets can be occupied given that all planets are distinct.
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Context:
Given an arithmetic sequence $\{a_n\}$ with a common difference $d \neq 0$, and $a_3^2 \neq a_1a_9$. Find the value of $\frac {a_{3}}{a_{6}}$.
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Context:
The number of years of service of the 8 employees in a production department are some values. The range in the number of years of service of the 8 employees is 14. If the second lowest number of years of service is 10, what is the lowest number of years of service?
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Context:
Given positive numbers $a$ and $b$ that satisfy $5 - 3a \leq b \leq 4 - a$ and $\ln{b} \geq a$, find the range of values for $\frac{b}{a}$.
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Context:
For each positive integer n, let $f(n) = \sum_{k = 1}^{100} \lfloor \log_{10} (kn) \rfloor$. Find the largest value of $n$ for which $f(n) \le 300$.
Note: $\lfloor x \rfloor$ is the greatest integer less than or equal to $x$.
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