Context:
You are playing a game. A $2 \times 1$ rectangle covers two adjacent squares (oriented either horizontally or vertically) of a $3 \times 3$ grid of squares, but you are not told which two squares are covered. Your goal is to find at least one square that is covered by the rectangle. A "turn" consists of you guessing a square, after which you are told whether that square is covered by the hidden rectangle. What is the minimum number of turns you need to ensure that at least one of your guessed squares is covered by the rectangle?
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Context:
A regular triangular prism $ABC A_1B_1C_1$ is inscribed in a sphere, with base $ABC$ and lateral edges $AA_1, BB_1, CC_1$. Segment $CD$ is a diameter of this sphere, and point $K$ is the midpoint of edge $AA_1$. Given that $CK = 2 \sqrt{6}$ and $DK = 4$, find the volume of the prism.
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Context:
Let $f$ be a function from the set $X = \{1,2, \dots, 10\}$ to itself. A partition $(S, T)$ of $X$ is called $f$-balanced if for all $s \in S$, $f(s) \in T$, and for all $t \in T$, $f(t) \in S$. (A partition $(S, T)$ is a pair of subsets $S$ and $T$ of $X$ such that $S \cap T = \emptyset$ and $S \cup T = X$. Note that $(S, T)$ and $(T, S)$ are considered the same partition.)

Let $g(f)$ be the number of $f$-balanced partitions, and let $m$ be the maximum value of $g(f)$ over all functions $f$ from $X$ to itself. If there are $k$ functions satisfying $g(f) = m$, determine $m+k$.

Find the value of $m+k$. \( \text{(2016 CCA Math Bonanza Individual #12)} \)
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Context:
The teacher gave 210 pencils to 30 students in the first classroom and 20 students in the second classroom equally. How many pencils are left for each of the two classrooms?
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Context:
Using only the paths and the directions shown, count the number of different routes from M to N. Express your answer as a single integer.
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Context:
Given that the lengths of all three sides of a triangle satisfy the equation $x^{2}-6x+8=0$, calculate all possible perimeters of this triangle. Express your answer as a list of whole numbers.
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Context:
Find the sum of the infinite series for positive real numbers $a$, $b$, and $c$:

$$ \sum_{n=0}^\infty \frac{a^n}{\Gamma(b+nc)} $$

Express your answer in terms of the Gamma function, $\Gamma$.
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Context:
On a windless day, a polar bear found itself on a small ice floe that broke off from an iceberg, floating in still water. Rescuers from a helicopter hovering above the floe noted that the animal was walking in a circle with a diameter of 9.5 meters. They were surprised when later, in a photograph, they saw the bear's trail and the diameter of this trajectory was 10 meters. Estimate the mass of the ice floe, assuming the mass of the bear is 600 kg.
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Context:
Calculate the volume of the solid $D$ defined by the inequalities: \(x^2+y^2+(z-1)^2 \le 1\) and \(z^2 \le x^2+y^2\).
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Context:
Let $M=\{0,1,2,\dots,2022\}$ and let $f:M\times M\to M$ such that for any $a,b\in M$,
\[f(a,f(b,a))=b\]
and $f(x,x)\neq x$ for each $x\in M$. How many possible functions $f$ are there $\pmod{1000}$?
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Context:
Consider a regular hexagon with vertex set $M$. Determine the number of chains of subsets $\emptyset \subset A \subset B \subset C \subset D \subset M$ such that each subset is distinct and the chain begins with the empty set and ends with the full set $M$. What is the total number of such chains?
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Context:
If $a^{2}+b^{2}=c^{2}$ $(a,b,c\in\mathbb{N}^{*})$, we call $a$, $b$, $c$ a Pythagorean triple. The following are several groups of Pythagorean triples: $3$, $4$, $5$; $5$, $12$, $13$; $7$, $24$, $25$; $9$, $40$, $41$. By analogy, the three numbers of the fifth group of Pythagorean triples are \_\_\_\_\_\_\_.
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Context:
How can we calculate the surface area of the solid enclosed by the paraboloid $z=2+x^2+y^2$ and the cone $z=4-\sqrt{x^2+y^2}$?
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Context:
Four rays spread out from a point $O$ in a 3-dimensional space such that the angle between any two rays is $a$. Determine the value of $\cos a$. \(\boxed{\text{Your answer here}}\)
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Context:
Determine the number of 3-Sylow subgroups in a simple group with 168 elements.
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Context:
Evaluate the limit \( \displaystyle \lim_{n\to \infty} \int_0^1 \frac{n^2x^{n-2}}{(x+1)^2} \, dx \).
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Context:
In the diagram, hexagon \( P Q R S T U \) has interior right angles at \( P, Q, S, T \), and \( U \) and an exterior right angle at \( R \). Also, \( P U = U T \), \( P Q = S T = 10 \), and \( Q S = 8 \). Calculate the perimeter of \( P Q R S T U \) and round your answer to the nearest whole number.
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Context:
求$y = \sin ^ { 2 } x + 2 \sin x \cdot \cos x + 3 \cos ^ { 2 } x$的最值.
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Context:
Compare the sizes of the following numbers:

\[ a = 3^{25} \]
\[ b = 4^{20} \]
\[ c = 5^{15} \]

Determine the order of \(a\), \(b\), and \(c\) using the symbols \(<\), \(=\), or \(>\).
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Context:
Denis painted some faces of his cubes gray. Vova selected 10 cubes so that all of them had different paint patterns, and then he stacked them as shown in the picture. How many white faces does the top cube have in total?
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Context:
Determine the Theta class for the recursion \( T(n) = T(3n/4) + T(n/6) + 5n \).
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Context:
Calculate the sum of the series: \( \cos \frac{2\pi}{2013} + \cos \frac{4\pi}{2013} + \cdots + \cos \frac{2010\pi}{2013} + \cos \frac{2012\pi}{2013} \).
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Context:
Find the solution set for \( x > 0 \) that satisfies the inequality \( af(x) + bx^2 f\left( \frac{1}{x} \right) \geq \sin(x) \), given that \( ab \neq 0 \) and \( a^2 \neq b^2 \).
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Context:
Given that $E[X^n] = 3n$ for $n \ge 1$, find the expected value $E[e^X]$.
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Context:
In the adjoining figure, CDE is an equilateral triangle and ABCD and DEFG are squares. Calculate the measure of angle GDA in degrees.
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Context:
Evaluate the integral \( \int_0^{\infty} x^{4n+3} e^{-x} \sin x \, dx \).
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Context:
Establish necessary and sufficient conditions on the constant  $k$  for the existence of a continuous real valued function  $f(x)$  satisfying   $$ f(f(x))=kx^9 $$   for all real  $x$ .
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Context:
Which of the following numbers is an odd integer, contains the digit 5, is divisible by 3, and lies between \(12^2\) and \(13^2\)?
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Context:
Find the smallest positive integer \( k \) such that \( 3^k \equiv 1 \pmod{3215} \), given that \( \phi(3215) = 2568 \) and \( 3^{2568} \equiv 1 \pmod{3215} \).
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Context:
In the diagram, each of the integers 1 through 9 is to be placed in one circle so that the integers in every straight row of three joined circles add to 18. The 6 and 1 have been filled in. Determine the value of the number represented by x. Express your answer as a single integer.
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Context:
If we replace the two middle digits $\overline{00}$ in 2009 with the two-digit numbers $\overline{ab}$ and $c\bar{d}$ respectively, we obtain perfect squares: $\overline{2ab9}=n^{2}$ and $\overline{2cd9}=m^{2}$, where $m > n$ and $m, n \in \mathbb{N}$. Determine the array $(m+n, \overline{ab}+\overline{cd})$.
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Context:
Given a acute triangle $PA_1B_1$ is inscribed in the circle $\Gamma$ with radius $1$. for all integers $n \ge 1$ are defined:
$C_n$ the foot of the perpendicular from $P$ to $A_nB_n$
$O_n$ is the center of $\odot (PA_nB_n)$
$A_{n+1}$ is the foot of the perpendicular from $C_n$ to $PA_n$
$B_{n+1} \equiv PB_n \cap O_nA_{n+1}$

If $PC_1 =\sqrt{2}$, find the length of $PO_{2015}$
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Context:
Find all finite sets \(A\) such that for every \(x \in A\), the expression \(x^3 - 3|x| + 4\) is also in \(A\).
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Context:
Find a constant \( M \) such that for all functions \( f \) in \( L^2(-1,1) \), the inequality \( \Vert f \Vert_{L^2} \leq M \Vert f \Vert_{H} \) holds, where \( \Vert f \Vert_H^{2} = \int_{-1}^1 \vert f(x) \vert^2 \frac{1}{1+x^2}dx \).
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Context:
[ $\left.\frac{\text { angles between bisectors }}{\text { Dihedral angle }}\right]$

The base of the pyramid $S A B C$ is a right triangle $A B C$ ( $C$ - the vertex of the right angle). All lateral faces of the pyramid are inclined to its base at the same angle, equal to $\arcsin \frac{5}{13}$. Find the area of the lateral surface of the pyramid, if $S O-$ is the height of the pyramid, $A O=1, B O=3 \sqrt{2}$.
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Context:
Determine the number of real solutions to the equation \( f(f(f(f(x)))) = \frac{1}{\sqrt{2}} \), where \( f(x)=\frac{1}{(1-x^{2011})^{\frac{1}{2011}}} \).
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Context:
In a labyrinth game, the largest room measures 10x10x8 and the smallest room measures 4x4x4. Assuming there is no empty volume between rooms and adjacency is defined by overlapping faces of at least 1 square, determine the maximum number of adjacent rooms possible.
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Context:
Find the maximum value of \( p + q \) such that the inequality \( 2px^2 + qx - p + 1 \geq 0 \) holds for all \( |x| \leq 1 \).
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Context:
In triangle \(ABC\), the bisectors \(AM\) and \(BN\) intersect at the point \(O\). Vertex \(C\) lies on the circle passing through the points \(O, M, N\). Find \(OM\) and \(ON\) if \(MN = \sqrt{3}\).
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Context:
Calculate the fractal dimension of a set using the following data for the number of boxes needed to cover the set at different shrink values: $N(1) = 7$, $N(0.3) = 201$, $N(0.22) = 478$, $N(0.12) = 2595$, and $N(0.06) = 17950$. Use these values to determine the fractal dimension.
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Context:
Determine the curves along which the function $u(x,y)$ remains constant for the partial differential equation $(x\partial_y - y\partial_x)u = 0$. Use the fact that the directional derivative along these curves is zero.
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Context:
A lattice point is a point with integer coordinates. The line y = 3x - 5 passes through square PQRS. If the coordinates of R are (2009, 2009), then find the number of lattice points on the line which are inside the square. Express your answer as a single integer.
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Context:
The total capacity of three tanks is 1620 liters. Two of them are filled with kerosene, and the third one is empty. To fill the empty tank, either all the contents of the first tank plus $1/5$ of the contents of the second tank, or all the contents of the second tank plus $1/3$ of the contents of the first tank is needed. Find the capacity of each tank.
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Context:
Using the Wright-Fisher model and the mean persistence time formula, calculate the probability of extinction of an allele in 120 generations, given that the allele starts at a frequency of 0.2.
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Context:
Given 4 equilateral triangles on a plane, each with distinct vertices, forming a set S of 12 vertices, and no three points in S are collinear, determine the maximum number of squares that can be formed using the vertices from S.
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Context:
In the diagram, points \( P, Q, R, \) and \( S \) are at intersections of gridlines in a \( 6 \times 6 \) grid. Find the perimeter of parallelogram \( PQRS \). Express your answer as a whole number.
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Context:
Find the minimum value of $p$ for which the series \( \sum_{n=1}^{\infty}(1n^{27}+8)^{27}(1+27n^{29}+8n^{30})^p \) is divergent.
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Context:
Three numbers are chosen at random between 0 and 2. What is the probability that the difference between the largest number and the smallest number is less than \( \frac{1}{4} \)?
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Context:
Consider and odd prime  $p$ . For each  $i$  at  $\{1, 2,..., p-1\}$ , let  $r_i$  be the rest of  $i^p$  when it is divided by  $p^2$ .
Find the sum: $r_1 + r_2 + ... + r_{p-1}$
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Context:
Find \( \lim_{x \to 0} g'(x) \) where the function \( g(x) \) is defined as follows:

\[
g(x) = \begin{cases} x^3 \left\lfloor \dfrac{1}{x^2} \right\rfloor & x \neq 0 \\
0 & x = 0 \end{cases}
\]
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Context:
Determine the minimum number of quadratic equations required to define the 10-dimensional Spinor variety in a scheme-theoretic sense.
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Context:
Given that \(a, b, c\) are unit vectors satisfying \(|a-b|^2+|b-c|^2+|c-a|^2=9\), find the value of \(|2a+5b+5c|\).
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Context:
Evaluate the limit \( \lim_{x\to0}\int_{1}^{\infty} \frac{\sin^2(xy/2)}{x+y^2} \, dy \) using the change of variable technique.
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Context:
Determine the values of \(\alpha > 1\) for which the series \(\sum_{n=1}^{\infty} |\sin n|^{cn^\alpha}\) converges.
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Context:
The lengths of the sides of a triangle are consecutive integers. Find these numbers, given that one of the medians is perpendicular to one of the angle bisectors.
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Context:
Integers with absolute value less than $4$ include ____.
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Context:
7. Given the sequence $\left\{a_{n}\right\}$ satisfies $a_{0}=0$,
$$
a_{n+1}=\frac{8}{5} a_{n}+\frac{6}{5} \sqrt{4^{n}-a_{n}^{2}}(n \geqslant 0, n \in \mathbf{N}) \text {. }
$$

Then $a_{10}=$ . $\qquad$
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Context:
What is the minimal possible value of the Shannon entropy \(H(X_1 X_2 X_3 X_4)\) for 4 pairwise independent unbiased Bernoulli random variables \(X_1, X_2, X_3, X_4\)?
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Context:
A hexagon is inscribed in a circle . Five of the sides have length $81$ and the sixth, denoted by $\overline{AB}$ , has length $31$ . Find the sum of the lengths of the three diagonals that can be drawn from $A$ .
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Context:
Evaluate the limit: \[ \lim_{n \to +\infty} \sum_{k=0}^n \left(\sqrt[n]{k} - \sqrt[n]{n}\right). \]
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Context:
In the $n$-dimensional non-negative orthant $\mathbb{R}_+^n$ with $n \geq 1$, consider an $(n-1)$-simplex $S$ defined by \[ S = \left\{x \in \mathbb{R}_+^n : \sum_{i=1}^n x_i = 1\right\} \] and its relative interior $\mathring{S}$. Let $E$ be a linear subspace of dimension $n-2$ in $\mathbb{R}^n$ such that $E \cap \mathring{S} \neq \emptyset$. The intersection $E \cap \mathring{S}$ is a convex polyhedron with a relative interior of dimension $n-3$ and has at least $n-2$ extremal points. Determine the maximum possible number of extremal points for $E \cap S$.
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Context:
What is the cardinality of the set of continuous functions from the closed interval [0, 1] to itself?
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Context:
What is the cardinality of the set of all intervals of real numbers \( \{(a,b) \mid a,b \in \mathbb{R}\} \)?
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Context:
Given a number $m$, such as $9874$, determine the number of natural numbers less than $m$ that do not contain the digit $3$. Provide your answer for $m = 9874$. Note that $1 \leq m < 10^{16}$.
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Context:
Evaluate the integral: \[ \int\limits_{0}^\infty \frac{\cos(\pi x)}{x}  \delta \left[ (x^2-1)(x-2) \right]  \mathrm{d}x \]
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Context:
Calculate the derivative of \( \nabla_{\beta} \| g(X \beta)\|_{2}^{2} \), where \( g \) is the Step function, \( X \in \mathbb{R}^{n \times d} \), and \( \beta \in \mathbb{R}^d \).
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Context:
Consider a surjective and non-decreasing function \( f: \{1, 2, 3, \ldots, 13\} \rightarrow \{1, 2, 3, \ldots, 9\} \). What is the probability that \( f(7) = 4 \)?
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Context:
Evaluate the sum \( \tan^2(1^\circ) + \tan^2(3^\circ) + \tan^2(5^\circ) + \tan^2(7^\circ) + \ldots + \tan^2(89^\circ) \).
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Context:
In $\triangle ABC$, $\angle A=30^{\circ}$, $BC=13$. Given $2$ circles $\gamma_1, \gamma_2$ ith radius $r_1,r_2$ contain $A$ and touch $BC$ at $B$ and $C$ respectively. Find $r_1r_2$.
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Context:
Determine the number of integers \(x\) in the range \(1 \leq x \leq n\) that satisfy the inequality:
\[\left\lfloor \frac{(x+1)r}{n}\right\rfloor > \left\lfloor \frac{xr}{n}\right\rfloor\]
where \(r\) and \(n\) are integers such that \(r < n\).
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Context:
Let $(X_1, X_2, \ldots)$ be an independent sequence of random variables, where each $X_n$ is uniformly distributed on the interval $[0, n]$. Determine the probability:

$$ P\left(\{w: \lim_{n \to \infty} X_n(w) = \infty\}\right) $$
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Context:
Omar made a list of all the arithmetic progressions of positive integer numbers such that the difference is equal to  $2$  and the sum of its terms is  $200$ . How many progressions does Omar's list have?
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Context:
Evaluate the line integral \( \int\limits_C (-y^3dx + x^2dy-z^3dz) \) using Stokes' Theorem, where \( C \) is the intersection of the cylinder \( x^2 + y^2 = 1 \) and the plane \( x + y + z = 1 \). Determine the value of the integral.
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Context:
Evaluate the limit: \[ \lim_{{n \to \infty}} \left( \frac{10^n+5^n}{2^n+5^n} - \frac{8^n-4^n+\sin(4^n)}{(-1)^n+4^n} \right) \] without using L'Hôpital's rule.
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Context:
给定整数 $m 、 n(2 \leqslant m<n,(m, n)=1)$.求最小的整数 $k$ ，满足对集合 $\{1,2, \cdots, n\}$ 的任意 $m$ 元子集 $I$, 若 $\sum_{i \in I} i>k$, 则存在 $n$ 个实数 $a_{1} \leqslant a_{2} \leqslant \cdots \leqslant a_{n}$, 使得
$$
\frac{1}{m} \sum_{i \in I} a_{i}>\frac{1}{n} \sum_{i=1}^{n} a_{i}
$$
若$m=3,n=5$，求$k$.
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Context:
Calculate the limit: $$\lim_{n\rightarrow\infty}\left(\frac{\prod\limits_{r=1}^n\left(1 +\frac{r}{n^2}\right)}{e^{1/2}}\right)^n.$$
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Context:
Let \( a, b, c, d, e \) be positive integers such that their sum is 2018. Define \( M = \max (a+b, b+c, c+d, d+e) \). Determine the smallest possible value of \( M \).
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Context:
Determine the number of real solutions a to the equation:
$$\left[\frac{1}{2}a\right]+\left[\frac{1}{3}a\right]+\left[\frac{1}{5}a\right]=a.$$
$\mathrm{Here,if}x\text{is a real number,then}[x]\text{denotes the greatest integer that is less than or equal to}x.$
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Context:
Through which point of triangle ABC does the trajectory of P necessarily pass, given that $O$ is a fixed point on the plane, $A$, $B$, and $C$ are three non-collinear points on the plane, and a moving point $P$ satisfies $\overrightarrow{OP} = \overrightarrow{OA} + \lambda\left(\frac{\overrightarrow{AB}}{|\overrightarrow{AB}|} + \frac{\overrightarrow{AC}}{|\overrightarrow{AC}|}\right), \lambda \in [0, +\infty)$?
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Context:
Determine the smallest integer \( n \) such that the error in the nth degree Taylor polynomial approximation of \( \ln(1-x^2) \) about \( x=0 \) is less than 0.1 for \( 0 \leq x \leq 0.5 \).
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Context:
The product  $20! \cdot 21! \cdot 22! \cdot \cdot \cdot 28!$  can be expressed in the form  $m$   $\cdot$   $n^3$ , where m and n are positive integers, and m is not divisible by the cube of any prime. Find m.
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Context:
Three spheres touch each other externally in pairs, and they also touch a certain plane at the vertices of a right triangle with one leg of length 1 and the angle opposite the leg equal to \(30^\circ\). Find the radii of the spheres.
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Context:
In Figure 1, \( AEFD \) is a unit square. The ratio of the length of the rectangle \( ABCD \) to its width is equal to the ratio of the length of the rectangle \( BCF E \) to its width. If the length of \( AB \) is \( W \) units, find the value of \( W \).
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Context:
Given that the sum of all positive integers with exactly two proper divisors, each of which is less than $30$, is $2397$, find the sum of all positive integers with exactly three proper divisors, each of which is less than $30$ (a proper divisor of $n$ is a positive integer that divides but is not equal to $n$).
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Context:
Find the mass percentage of C in Acetone
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Context:
Determine the possible values of $n$ for which an automorphism $\sigma$ of $\mathbb{Q}(\omega)$ can map $\omega$ to $\omega^n$, where $\omega = e^{\frac{2\pi i}{10}}$. Provide your answer as a set of integers.
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Context:
How many simple graphs with 10 vertices and 15 edges contain a complete subgraph with 5 vertices?
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Context:
将与105互素的所有正数，从小到大排成数列，试求出这个数列的第1000项．
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Context:
Given the reaction: CH4(g) + 2Cl2(g) ⇌ CH2Cl2(g) + 2HCl(g) with a known equilibrium constant Kc, predict the amount of HCl formed in moles at equilibrium when 3 moles of methane (CH4) and 6 moles of chlorine (Cl2) are combined in a 2L container at a certain temperature. Assume that the reaction has not yet reached equilibrium and only forward reaction is taking place initially.
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Context:
Let $S = \{c_1, \dots, c_n\}$ be a set of vectors in $\mathbb{R}^M$. Determine the maximum cardinality of a subset $S' \subset S$ such that $\dim(\text{span}(S')) < \dim(\text{span}(S))$. What is the largest size of such a subset?
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Context:
Determine the degree of the field extension \(\mathbb{Q} \left( \zeta_{14} + \zeta_{14}^9 + \zeta_{14}^{11} \right)\) over \(\mathbb{Q}\), where \(\zeta_{14}\) is a 14th primitive root of unity.
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Context:
Evaluate the limit \( \lim_{x \rightarrow 2k\pi}\frac{2(1-\cos x)+x\sin x}{2\sqrt{1-\cos x}} \) to find \( f'(2k\pi) \) for the function \( f(x)=x \sqrt{1-\cos(x)} \).
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Context:
Determine the crystallographic restriction for a discrete group of isometries whose translation group is generated by a single non-zero vector \( a \).
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Context:
Determine the highest power of 50 that divides the expression \(49^{{50}^{51}} + 51^{{50}^{49}} + 50^{{49}^{51}}\).
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Context:
Determine the winding number of the path \( \gamma:[0,2\pi] \rightarrow \mathbb{C}, \ t \mapsto a\cos(2t)+ib\sin(2t) \) around the origin, where \( a, b \in \mathbb{R} \).
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Context:
Find the number of functions $f(x)$ from $\{1,2,3,4,5\}$ to $\{1,2,3,4,5\}$ that satisfy $f(f(x)) = f(f(f(x)))$ for all $x$ in $\{1,2,3,4,5\}$.
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Context:
Calculate the limit: \[ \lim_{n\to\infty} \frac{\gamma(n,n)}{\Gamma(n)} \] where \( \gamma(s,x) = \int_0^x t^{s-1} e^{-t} \, dt \).
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Context:
Each integer in \( \{ 1,2,3,\ldots ,{2020}\} \) is coloured in such a way that,for all positive integers \( a \) and \( b \)
 such that \( a + b \leq  {2020} \) ,the numbers \( a,b \) and \( a + b \) are not coloured with three different colours
 Determine the maximum number of colours that can be used.
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Context:
已知 $x_{1}, x_{2}, x_{3}, x_{4}$ 是两两不等的实数， $x_{i 1}, x_{i 2}, x_{i 3}, x_{i 4}$ 是 $x_{1}, x_{2}, x_{3}, x_{4}$ 的任意一个排列，若记 $X=\max \left\{\min \left\{x_{i 1}, x_{i 2}\right\}, \min \left(x_{i 3}, x_{i 4}\right\}\right\}, Y=$ $\min \left\{\max \left\{x_{i 1}, x_{i 2}\right\}, \max \left\{x_{i 3}, x_{i 4}\right\}\right\}$ ，则 $X>Y$ 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:
A 5-dimensional ant starts at one vertex of a 5-dimensional hypercube of side length 1. A move is when the ant travels from one vertex to another vertex at a distance of $\sqrt{2}$ away. How many ways can the ant make 5 moves and end up on the same vertex it started at?
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Context:
On the island of Misfortune with a population of 96 people, the government decided to implement five reforms. Each reform is disliked by exactly half of the citizens. A citizen protests if they are dissatisfied with more than half of all the reforms. What is the maximum number of people the government can expect at the protest?
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Context:
Compute the turning number of the closed plane curve defined by \( \gamma(t) = \left(\left(\frac{1}{2}+3\cos(t)\right)\cos(t), \left(\frac{1}{2}+3\cos(t)\right)\sin(t)\right) \).
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Context:
Determine the asymptotic bound for the expression \( \sum_{k=0}^{\frac{n}{2^{\log(n)}}} (k+1)^{3} + \sum_{i=0}^{\frac{n}{2}} (i+1)^{2} \).
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Context:
Determine the Morley rank of the type $x = x$ in the theory of Algebraically Closed Fields of characteristic zero (ACF$_{0}$).
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Context:
Calculate the surface flux of the vector field \( \textbf{F} = x^2\textbf{i} + y\textbf{j} - xz \textbf{k} \) across the surface \( S \) defined by \( 4y=x^2 \), bounded by the planes \( x=2 \) and \( z=4 \), where \( \hat{\textbf{n}} \) points in the direction of increasing \( y \).
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Context:
Frank alternately flips two coins. The first coin has a probability of \( \frac{2}{3} \) for heads and \( \frac{1}{3} \) for tails. The second coin has a probability of \( \frac{1}{4} \) for heads and \( \frac{3}{4} \) for tails. Starting with the first coin, what is the probability that Frank sees two consecutive heads before two consecutive tails?
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Context:
Evaluate the limit: $$\lim_{n \to \infty } \left( \frac{1}{\sqrt{n^2+n+1}}+\cdots+\frac{1}{\sqrt{n^2+2n}} \right)$$
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Context:
Find the range of values for $p$ such that the interval $(3,4)$ is contained between the roots of the equation \(3^x + 3^{4-x} + p = 0.\)
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Context:
In the sum shown, different shapes represent different digits. What digit does the square represent? Express your answer as a single digit.
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Context:
在圆周上任意写上 49 个 1 及 50 个 0 , 然后进行下列运算: 在两个相同的数之间写上 0 , 在两个不同数之间写上 1 , 并擦掉原有的数字, 接着再进行同样的运算, 如此继续，问能否经过有限步这种运算使得圆周上的数字都变成 0 ?
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\boxed{不论运算多少次,都不能使所有的数字都变成 0.}


Context:
In a convex quadrilateral, three of its vertices form a triangle such that the circumcenter of this triangle is located outside the triangle. Given this information, determine the value of angle $x$ in the quadrilateral.
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Context:
Evaluate the limit \( \lim_{n \rightarrow \infty} \frac{\sum_{j=2}^{j=n+1}{k(j)}}{n} \), where for every positive integer \( n > 1 \), \( k(n) \) is the largest positive integer \( k \) such that there exists a positive integer \( m \) with \( n = m^k \).
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Context:
What is the value of  $ { \sum_{1 \le i< j \le 10}(i+j)}_{i+j=odd} $   $ - { \sum_{1 \le i< j \le 10}(i+j)}_{i+j=even} $
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Context:
Consider an event with probability \( p \). What is the probability that this event never occurs over an infinite number of trials?
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Context:
Calculate the flux of the vector field \( F = \langle z, y, zx \rangle \) through the surface of the tetrahedron enclosed by the coordinate planes and the plane \( \frac{x}{4}+\frac{y}{2}+\frac{z}{5}=1 \). Use the divergence theorem to find the correct flux value.
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Context:
Calculate the Vassiliev invariant for the knot K_1.
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Context:
Find a linear operator on \( \mathbb{C} \) (considered as a vector space over \( \mathbb{R} \)) such that the range of the operator is a proper subset of its kernel.
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Context:
Determine the number of real solutions to the equation \( f(x) = 0 \), where \( f(x) = \sum_{i=1}^{2020} \frac{i^2}{x-i} \).
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Context:
In the given figure, there are two trapezoids $ABDE$ and $ACDE$. If the area of the pentagon $ABCDE$ is 2621 and the area of the triangle $ADF$ is 300, what is the area of the triangle $EBC$?
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Context:
Problem 6. (30 points) Ten balls of the same radius are arranged in the form of a triangular pyramid such that each ball touches at least three others. Find the radius of the sphere in which the pyramid of balls is inscribed, if the radius of the sphere inscribed in the center of the pyramid of balls, touching six identical balls, is $\sqrt{6}-1$.

#
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Context:
In three out of the six circles of a diagram, the numbers 4, 14, and 6 are written. In how many ways can natural numbers be placed in the remaining three circles such that the product of the triplets along each of the three sides of the triangular diagram are the same?
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Context:
Determine the number of distinct Celtic knots of size (2, 2), considering knots that differ only by rotations and reflections as identical.
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Context:
Find the dimension of the subspace of quadric hypersurfaces in \(\mathbb{P}^4\) that vanish at eight points in general position on a rational normal curve of degree 4.
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Context:
A particle moves in the Cartesian plane according to the following rules:
From any lattice point $(a,b),$ the particle may only move to $(a+1,b), (a,b+1),$ or $(a+1,b+1).$
There are no right angle turns in the particle's path.
 How many different paths can the particle take from $(0,0)$ to $(5,5)$?
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Context:
Given triangle  $ABC$ , let  $D$ ,  $E$ ,  $F$  be the midpoints of  $BC$ ,  $AC$ ,  $AB$  respectively and let  $G$  be the centroid of the triangle. For each value of  $\angle BAC$ , how many non-similar triangles are there in which  $AEGF$  is a cyclic quadrilateral?
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Context:
Evaluate the integral \( I = \int_{C(0,2)^+} \frac{z^3}{z^3+z^2-z-1} \, dz \) using the residue theorem.
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Context:
Find the length of the latus rectum for the hyperbola given by the equation \[\frac{(3x-4y-1)^2}{100}-\frac{(4x+3y-1)^2}{225}=1.\]
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Context:
Dean, Sara, and Nick went for a swim together. The depth of the water without waves is 10 times each person's height. Dean stands at 6 feet tall, Sara at 5 feet tall, and Nick at 5.5 feet tall. They stand at distances of 5, 10, and 15 feet from the shore, respectively. The waves increase the water depth by an additional 25% at its peak. Calculate the average water depth each swimmer experienced during their swim at their respective distances from shore.
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Context:
Find the greatest common divisor (GCD) of all numbers of the form \( p^8 - 9p^4 + 8 \), where \( p \) is a prime number greater than or equal to 13.
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Context:
How many positive integers $n$ are there such that $n!(2n+1)$ and $221$ are relatively prime?
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Context:
Evaluate the limit: $$\lim_{x\rightarrow\infty}\Large{\Delta^x_{k=0}}\normalsize{\cos \frac{k\pi}{3x}}$$
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Context:
Determine the number of terms needed in the series \(\sum_{k=0}^{\infty}{\frac{(3i)^k}{k!}}\) to ensure the error is less than \(10^{-4}\).
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Context:
Find the minimum value of the function \( f(x,y) = 2x + xy + 3y \) subject to the constraints:
\[ \begin{cases} x^2 + y \geq 3 \\ x + 0.5 \geq 0 \\ y \geq 0 \end{cases} \]
Using the Kuhn-Tucker conditions, determine the point \((x, y)\) that minimizes \(f(x, y)\).
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Context:
A guidance system is designed to deliver a bomb to a target site at coordinates \((3,2)\). The bomb's landing point \((X,Y)\) follows the joint probability density function:
\[ f_{X,Y}(x,y|3,2)=\frac{1}{2\pi}\exp\left\{-\frac{(x-3)^2}{2}-\frac{(y-2)^2}{2}\right\} \]
where \(X\) and \(Y\) are measured in feet. The target will be destroyed if a bomb lands within \(r=2\) feet of it. Calculate the probability that the target will be destroyed if 10 bombs are fired at it.
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Context:
Evaluate the integral \( \int_0^4 r(\sqrt{25-r^2}-3) \, dr \).
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Context:
During a fireworks display, a body is launched upwards with an initial velocity of $c=90 \mathrm{m/s}$. We hear its explosion $t=5$ seconds later. At what height did it explode if the speed of sound is $a=340 \mathrm{m/s}$? (Air resistance is neglected.)
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Context:
Determine the largest possible constant $d$ such that the inequality \[ \frac{2^{2d+3k}\pi\sqrt{k}e\sqrt{3}}{3^{3k}} \geq 2^{dk} \] holds for all integers $k \geq 1$. Provide your answer as a single value of $d$.
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Context:
A cube with edge length 3 is divided into 27 unit cubes. The numbers 1, 2, ..., 27 are distributed arbitrarily over the unit cubes, with one number in each cube. Determine the maximum number of odd row sums that can be formed from the 27 possible row sums, where there are nine such sums of three integers for each of the three directions parallel to the edges of the cube.
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Context:
Evaluate the limit \( \lim_{x\to 0^+}\frac{\int_{x}^{1}\frac{\sin t}{t}\,\mathrm{d}t}{x} \) using L'Hopital's rule.
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Context:
Let $f(x)=30-2x-x^3$. Determine the number of positive integer values of $x$ for which $f(f(f(x))) > f(f(-x))$. Express your answer as a single integer.
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Context:
Let $n$ be a positive integer, $k \in \mathbb{C}$, and $A \in \mathcal{M}_n(\mathbb{C})$ such that $\text{Tr } A \neq 0$ and \[ \text{rank } A + \text{rank } ((\text{Tr } A) \cdot I_n - kA) = n. \] Find $\text{rank } A$. \( \text{Tr } A \) denotes the trace of matrix $A$, and $I_n$ is the $n \times n$ identity matrix.
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Context:
Two players play alternately on a  $ 5 \times 5$  board. The first player always enters a  $ 1$  into an empty square and the second player always enters a  $ 0$  into an empty square. When the board is full, the sum of the numbers in each of the nine  $ 3 \times 3$  squares is calculated and the first player's score is the largest such sum. What is the largest score the first player can make, regardless of the responses of the second player?
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Context:
Compute
$\sum\limits_{i=0}^\infty(-1)^i\sum\limits_{j=i}^\infty(-1)^j\frac{2}{j^2+4j+3}.$
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Context:
Nine lines, parallel to the base of a triangle, divide the other sides into 10 equal segments and the area into 10 distinct parts. Find the area of the original triangle, if the area of the largest of these parts is 76.
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Context:
Calculate the limit as \( n \) approaches infinity for the expression:
\[ \lim_{n\rightarrow+\infty} np^n \sum_{k=n}^{+\infty} \frac{1}{k} \binom{k-1}{n-1} (1-p)^{k-n} \]
where \( 0<p<1 \).
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Context:
Given the parametric equations for real numbers $t,$
\[(x,y) = \left( \frac{1 - t^2}{1 + t^2}, \frac{2t}{1 + t^2} \right),\]identify the type of curve on which all the plotted points lie.
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Context:
Evaluate the sum \( \sum_{k=1}^n k\cdot(-3)^k{n \choose k} \) for \( n > 0 \) and express it as a function of \( n \).
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Context:
What is the probability that you never lose in the following dice game? You start with a four-sided die numbered 1-4. In each round, you roll all your dice. If a die shows 1, you lose that die. If it shows 2, you keep the die. If it shows 3 or 4, you gain an additional identical die. You lose the game if you have no dice left. Calculate the probability that you never lose the game over infinite rounds.
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Context:
Professor Oak is feeding his 100 Pokémon. Each Pokémon has a bowl whose capacity is a positive real number of kilograms. These capacities are known to Professor Oak. The total capacity of all the bowls is 100 kilograms. Professor Oak distributes 100 kilograms of food in such a way that each Pokémon receives a non-negative integer number of kilograms of food (which may be larger than the capacity of their bowl). The dissatisfaction level of a Pokémon who received $N$ kilograms of food and whose bowl has a capacity of $C$ kilograms is equal to $|N-C|$.
Find the smallest real number $D$ such that, regardless of the capacities of the bowls, Professor Oak can distribute the food in a way that the sum of the dissatisfaction levels over all the 100 Pokémon is at most $D$.
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Context:
What is the general expression, or at least tight upper and lower bounds, for $f(n,k)$, which represents the number of ways to have at least $k$ consecutive heads in $n$ coin tosses? For instance, when $n=5$ and $k=2$, how many possibilities exist?
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Context:
Let \( k \) be a positive integer and \( w \) be an integer such that two of the roots of the cubic equation \( x^3 - 17x^2 + kx + w = 0 \) are consecutive positive integers. Find the value of \( k \) that maximizes the absolute value of \( k - w \).
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Context:
Find the value of $f(\sqrt{3})$ for a continuous function $f(x)$ defined for all real numbers $x$ such that the equation \(x^2 + x\{f(x)\} - 3 = \sqrt{3} \, f(x)\) holds, where \(\{t\}\) denotes the fractional part of \(t\).
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Context:
Given the parabola \( y^2 = 4x \), a line passing through its focus \( F \) intersects the parabola at points \( M \) and \( N \). Let \( E(m,0) \) be a point on the x-axis. The extensions of \( ME \) and \( NE \) intersect the parabola again at points \( P \) and \( Q \), respectively. If the slopes \( k_1 \) and \( k_2 \) of the lines \( MN \) and \( PQ \) satisfy \( k_1 = 3k_2 \), find the value of \( m \).
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Context:
Find the limit of \( \frac{x^x - (\sin x)^x}{x^3} \) as \( x \to 0 \).
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Context:
Let $M=\{(x,y,z) \in \mathbb{R}^3 : x+y=5, x+z=\cos^2y\}$ be a submanifold of $\mathbb{R}^3$. Consider the point $p=(5,0,-4)$ and the tangent vector $v=(-C,C,C) \in T_{(5,0,-4)}M$. Define the smooth map $F:M \rightarrow S^1$ by $F(x,y,z) = \left(\frac{x}{\sqrt{x^2+y^2}},\frac{y}{\sqrt{x^2+y^2}}\right)$. Let $\omega = -\frac{1}{x}dy$ be a 1-form on $S^1$ in a neighborhood of $(1,0)$. Compute $(F^{*}\omega)_p(v)$. Express your answer in terms of $C$.
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Context:
Find the limit: \[ \lim_{ x \to a }\left(\frac{f(x)}{g(x)}-\frac{h(x)}{k(x)}\right) \] where \( f(x), g(x), h(x), k(x) \) are polynomials, and it is given that \( \lim_{ x \to 1 }\left(\frac{1}{x-1}-\frac{3x-1}{x^2-1}\right) = -1 \).
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Context:
Given a sequence of independent and identically distributed (i.i.d.) random variables \(X_i\), each with mean \(M\) and variance \(\sigma^2\), let \(Y(z)\) denote the smallest integer such that the sum of the first \(Y(z)\) variables exceeds \(z\). As \(z\) tends to infinity, \(Y(z)\) is approximately \(z/M\). Determine the limit:
\[\lim_{{z \to \infty}} \frac{V(z) M^3}{\sigma^2 z}\]
where \(V(z)\) is the variance of \(Y(z)\).
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Context:
Determine the largest value of \( 3k \) such that the equation \( \sqrt{x^{2}-k} + 2 \sqrt{x^{3}-1} = x \) has a real root.
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Context:
Let $f_1$ and $f_2$ be holomorphic functions. Suppose $f_2$ has a zero of order $k$ at $z'_0 = f_1(z_0)$. If $f_1(z)-z_0'$ has a zero of order $m$ at $z_0$, and $f_2(z)-(z-z_0')^k$ has a zero of order $n$ at $z_0'$, determine the order of the zero of $g = f_2 \circ f_1$ at $z_0$.
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Context:
On each side of an equilateral triangle with side length  $n$  units, where  $n$  is an integer,  $1 \leq n \leq 100$ , consider  $n-1$  points that divide the side into  $n$  equal segments. Through these points, draw lines parallel to the sides of the triangle, obtaining a net of equilateral triangles of side length one unit. On each of the vertices of these small triangles, place a coin head up. Two coins are said to be adjacent if the distance between them is 1 unit. A move consists of flipping over any three mutually adjacent coins. Find the number of values of  $n$  for which it is possible to turn all coins tail up after a finite number of moves.
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Context:
An isosceles triangle has two exterior angles in the ratio of 1:4. Find the measures of its three interior angles.
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\boxed{140^	ext{\circ}, 20^	ext{\circ}, 20^	ext{\circ}}


Context:
Determine the automorphism group of the graph \( \Gamma \), where \( \Gamma \) is defined as follows: Let \( S_6 \) be the symmetric group on six letters, and let \( X \) denote the conjugacy class containing \((12)(34)\). The vertex set of \( \Gamma \) is \( X \), and the edges are precisely the 2-element subsets of \( X \) that commute as elements of \( S_6 \).
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Context:
If 5a + some b = 70, where a and b are positive integers, what is the largest possible value of g for which exactly one pair of integers (a, b) makes the equation true?
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Context:
The smallest three positive proper divisors of an integer n are $d_1 < d_2 < d_3$ and they satisfy $d_1 + d_2 + d_3 = 57$. Find the sum of the possible values of $d_2$.
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Context:
Given a cube $A B C D A 1 B 1 C 1 D 1$ with edge length $a$. On the rays $C 1 C$, $C 1 B 1$, and $C 1 D 1$, segments $C 1 M$, $C 1 N$, and $C 1 K$ are respectively laid out, each equal to $\frac{5}{2} a$. Construct the section of this cube by the plane passing through points $M$, $N$, and $K$, and find the area of the resulting section.
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Context:
The diagram shows eight small squares. Six of these squares are to be shaded so that the shaded squares form the net of a cube. In how many different ways can this be done? Express your answer as a whole number.
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Context:
Boys and girls are standing in a circle (there are both), a total of 20 children. It is known that each boy's neighbor in the clockwise direction is a child in a blue T-shirt, and each girl's neighbor in the counterclockwise direction is a child in a red T-shirt. Can you uniquely determine how many boys are in the circle?
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Context:
Let $O$ be a fixed point on a plane, and let $A$, $B$, and $C$ be three non-collinear points on the same plane. A moving point $P$ satisfies the equation $\overrightarrow{OP} = \overrightarrow{OA} + λ(\frac{\overrightarrow{AB}}{|\overrightarrow{AB}|} + \frac{\overrightarrow{AC}}{|\overrightarrow{AC}|}), λ ∈ [0, +∞)$. Determine which center of $\triangle ABC$ the trajectory of $P$ will always pass through.
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Context:
Vera has several identical matches, from which she makes a triangle. Vera wants any two sides of this triangle to differ in length by at least  $10$  matches, but it turned out that it is impossible to add such a triangle from the available matches (it is impossible to leave extra matches). What is the maximum number of matches Vera can have?
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Context:
Let $ABCD$ be a rhombus with angle $\angle A = 60^\circ$. Let $E$ be a point, different from $D$, on the line $AD$. The lines $CE$ and $AB$ intersect at $F$. The lines $DF$ and $BE$ intersect at $M$. Determine the angle $\angle BMD$ as a function of the position of $E$ on $AD.$
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Context:
Points  $A, B$  and  $C$  lie on the same line so that  $CA = AB$ . Square  $ABDE$  and the equilateral triangle  $CFA$ , are constructed  on the same side of line  $CB$ . Find the acute angle between straight lines  $CE$  and  $BF$ .
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Context:
$ AA_{3}$  and  $ BB_{3}$  are altitudes of acute-angled  $ \triangle ABC$ . Points  $ A_{1}$  and  $ B_{1}$  are second points of intersection lines  $ AA_{3}$  and  $ BB_{3}$  with circumcircle of  $ \triangle ABC$  respectively.  $ A_{2}$  and  $ B_{2}$  are points on  $ BC$  and  $ AC$  respectively.  $ A_{1}A_{2}\parallel AC$ ,  $ B_{1}B_{2}\parallel BC$ . Point  $ M$  is midpoint of  $ A_{2}B_{2}$ .  $ \angle BCA \equal{} x$ . Find  $ \angle A_{3}MB_{3}$ .
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Context:
The faces of each of $7$ standard dice are labeled with the integers from $1$ to $6$. Let $p$ be the probabilities that when all $7$ dice are rolled, the sum of the numbers on the top faces is $10$. What other sum occurs with the same probability as $p$?
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Context:
Evaluate the sum
$\sum_{k=0}^{\infty}\left(3\cdot\frac{\ln(4k+2)}{4k+2}-\frac{\ln(4k+3)}{4k+3}-\frac{\ln(4k+4)}{4k+4}-\frac{\ln(4k+5)}{4k+5}\right)$
$=3\cdot\frac{\ln2}{2}-\frac{\ln3}{3}-\frac{\ln4}{4}-\frac{\ln5}{5}+3\cdot\frac{\ln6}{6}-\frac{\ln7}{7}-\frac{\ln8}{8}-\frac{\ln9}{9}+3\cdot\frac{\ln10}{10}-\cdots.$
(As usual, $lnx$ denotes the natural logarithm of x.)
 The original answer is in $(\log m)^{n}$format, please give the value of m + n.
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Context:
Find four positive integers such that the sums of the numbers taken three at a time are 6, 7, 8, and 9.
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Context:
Triangle $ABC$ has $AC = 450$ and $BC = 300$. Points $K$ and $L$ are located on $\overline{AC}$ and $\overline{AB}$ respectively so that $AK = CK$, and $\overline{CL}$ is the angle bisector of angle $C$. Let $P$ be the point of intersection of $\overline{BK}$ and $\overline{CL}$, and let $M$ be the point on line $BK$ for which $K$ is the midpoint of $\overline{PM}$. If $AM = 180$, find $LP$.
[asy] import markers; defaultpen(fontsize(8)); size(300); pair A=(0,0), B=(30*sqrt(331),0), C, K, L, M, P; C = intersectionpoints(Circle(A,450), Circle(B,300))[0]; K =  midpoint(A--C); L = (3*B+2*A)/5; P = extension(B,K,C,L); M = 2*K-P; draw(A--B--C--cycle); draw(C--L);draw(B--M--A); markangle(n=1,radius=15,A,C,L,marker(markinterval(stickframe(n=1),true))); markangle(n=1,radius=15,L,C,B,marker(markinterval(stickframe(n=1),true))); dot(A^^B^^C^^K^^L^^M^^P); label("$A$",A,(-1,-1));label("$B$",B,(1,-1));label("$C$",C,(1,1)); label("$K$",K,(0,2));label("$L$",L,(0,-2));label("$M$",M,(-1,1)); label("$P$",P,(1,1)); label("$180$",(A+M)/2,(-1,0));label("$180$",(P+C)/2,(-1,0));label("$225$",(A+K)/2,(0,2));label("$225$",(K+C)/2,(0,2)); label("$300$",(B+C)/2,(1,1)); [/asy]
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Context:
Let $a$, $b$, and $c$ be positive integers with $a\ge$ $b\ge$ $c$ such that $a^2-b^2-c^2+ab=2011$ and $a^2+3b^2+3c^2-3ab-2ac-2bc=-1997$. What is $a$?
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Context:
Identical gases are in two thermally insulated vessels with volumes $V_{1}=1$ L and $V_{2}=2$ L. The gas pressures are $p_{1}=2$ atm and $p_{2}=3$ atm, and their temperatures are $T_{1}=300$ K and $T_{2}=400$ K, respectively. The gases are mixed. Determine the temperature that will be established in the vessels after mixing.
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Context:
On the number line, consider the point $x$ that corresponds to the value $10$. Consider $24$ distinct integer points $y_1, y_2, \ldots, y_{24}$ on the number line such that for all $k$ where $1 \leq k \leq 12$, $y_{2k-1}$ is the reflection of $y_{2k}$ across $x$. Find the minimum possible value of
\[\sum_{n=1}^{24} (|y_n-1| + |y_n+1|).\]
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Context:
Using eight $2 \times 1$ small rectangles as shown in the diagram, a $4 \times 4$ square can be formed. If a formed square pattern is considered the same as another if it can be rotated to match, the question is: How many different square patterns can be formed such that it has only one diagonal as its axis of symmetry?
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Context:
Given that point $P$ moves on the circle $x^{2}+(y-2)^{2}=1$, and point $Q$ moves on the ellipse $\frac{x^{2}}{9}+y^{2}=1$, find the maximum value of the distance $PQ$.
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Context:
Given a circle of radius 1, starting from a fixed point, points are placed along the circumference at curved distances of 0, 1, 2, ..., respectively. Determine the minimum number of these points needed to ensure that at least two of them are less than a distance of \( \frac{1}{5} \) apart.
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Context:
Find the principal value of \( \left(\left(\frac{1+\sqrt{3}i}{2}\right)^{-3}\right)^{1-i} \).
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Context:
Let \( f(x)=e^{x+1}-1 \) and \( g(x)=100|f(x)|-\sum_{k=1}^n|f(x^k)| \), where \( n \in \mathbb{N} \). It is given that \( g(x) \) is a differentiable function over the complete real numbers. Find the sum of all possible values of \( n \).
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Context:
Determine the number of distinct $k$-dimensional subspaces in an $n$-dimensional vector space $V$ over a finite field with $q$ elements.
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Context:
Let  $a_i=min\{ k+\dfrac{i}{k}|k \in N^*\}$ , determine the value of  $S_{n^2}=[a_1]+[a_2]+\cdots +[a_{n^2}]$ , where  $n\ge 2$  . ( $[x]$  denotes the greatest integer not exceeding x)
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Context:
Let $x,$ $y,$ $z$ be positive real numbers.  Find the minimum value of
\[\frac{(1 + 5z)(4z + 3x)(5x + 6y)(y + 18)}{xyz}.\]
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Context:
In  $\triangle ABC$ , point  $D$  lies on side  $AC$  such that  $\angle ABD=\angle C$ . Point  $E$  lies on side  $AB$  such that  $BE=DE$ .  $M$  is the midpoint of segment  $CD$ . Point  $H$  is the foot of the perpendicular from  $A$  to  $DE$ . Given  $AH=2-\sqrt{3}$  and  $AB=1$ , find the size of  $\angle AME$ .
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Context:
For how many natural numbers \( n \) not exceeding 600 are the triples of numbers

\[ \left\lfloor \frac{n}{2} \right\rfloor, \left\lfloor \frac{n}{3} \right\rfloor, \left\lfloor \frac{n}{5} \right\rfloor \quad\text{and}\quad \left\lfloor \frac{n+1}{2} \right\rfloor, \left\lfloor \frac{n+1}{3} \right\rfloor, \left\lfloor \frac{n+1}{5} \right\rfloor \]

distinct? As always, \(\lfloor x \rfloor\) denotes the greatest integer less than or equal to \( x \).
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Context:
Knot must travel from point \( K \) to point \( T \) on a figure where \( OK = 2 \) and \( OT = 4 \). Before reaching \( T \), Knot must first reach both solid lines depicted in the figure in order to collect the two pieces of the Lineforce. What is the minimal distance Knot must travel to accomplish this?
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Context:
From the positive integers, $m, m+1, \dots, m+n$, only the sum of digits of $m$ and the sum of digits of $m+n$ are divisible by $8$. Find the maximum value of $n$.
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Context:
An empty fuel tank with a capacity of 300 gallons was filled partially with three different types of fuel: fuel A, fuel B, and fuel C. Fuel A contains 10% ethanol by volume, fuel B contains 15% ethanol by volume, and fuel C contains 20% ethanol by volume. If the total volume of ethanol in the full fuel tank is 'x' gallons, where 30 <= x <= 60, find an equation that represents the relationship between the volumes of fuel A, B, and C added to the tank.
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Context:
How many ways are there to choose distinct positive integers  $a, b, c, d$  dividing  $15^6$  such that none of  $a, b, c,$  or  $d$  divide each other? (Order does not matter.)
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Context:
Given the expansion of $(a+b)^n$, if the fourth term has the maximum binomial coefficient, find all possible values of $n$. Express your answer as a list of values separated by commas.
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Context:
Let $\mathcal{P}$ be a set of monic polynomials with integer coefficients of the least degree, with root $k \cdot \cos\left(\frac{4\pi}{7}\right)$, as $k$ spans over the positive integers. Let $P(x) \in \mathcal{P}$ be the polynomial so that $|P(1)|$ is minimized. Find the remainder when $P(2017)$ is divided by $1000$.
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Context:
已知正整数n不超过2000，并且能表示成不少于60个连续正整数之和，那么，这样的n的个数是 .
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Context:
There are 20 points, each pair of adjacent points are equally spaced. By connecting four points with straight lines, you can form a square. Using this method, you can form _ squares.
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Context:
Let $P$ and $Q$ be points on line $l$ with $P Q=12$. Two circles, $\omega$ and $\Omega$, are both tangent to $l$ at $P$ and are externally tangent to each other. A line through $Q$ intersects $\omega$ at $A$ and $B$, with $A$ closer to $Q$ than $B$, such that $A B=10$. Similarly, another line through $Q$ intersects $\Omega$ at $C$ and $D$, with $C$ closer to $Q$ than $D$, such that $C D=7$. Find the ratio $A D / B C$.
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Context:
A square was cut into four equal rectangles and used to form a large letter П, as shown in the figure, with a perimeter of 56. What is the perimeter of the original square?
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Context:
What is the mass percentage of Ca in a compound if the mass percentage of Ca in Calcium carbonate is the same as in that compound?
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Context:
Compute the largest possible number of distinct real solutions for  $x$  to the equation \[x^6+ax^5+60x^4-159x^3+240x^2+bx+c=0,\] where  $a$ ,  $b$ , and  $c$  are real numbers.

[i]Proposed by Tristan Shin
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Context:
In triangle $ABC$, if it is known that $\angle A : \angle B : \angle C = 2 : 3 : 5$, then triangle $ABC$ is a _______ triangle.
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Context:
For all positive integers $n$, denote by $\sigma(n)$ the sum of the positive divisors of $n$ and $\nu_p(n)$ the largest power of $p$ which divides $n$. Compute the largest positive integer $k$ such that $5^k$ divides \[ \sum_{d|N}\nu_3(d!)(-1)^{\sigma(d)}, \] where $N=6^{1999}$.
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Context:
As illustrated, to set up a "gesture password" on a mobile phone in a $2 \times 2$ grid containing 9 points, start at one point and consecutively connect multiple points with line segments. Each segment cannot pass through any unused points, and the endpoints of each segment cannot both be previously used points. If a person's gesture password starts at the center point and uses only three points, how many connection methods are there?
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Context:
In the figure below, parallelograms $ABCD$ and $BFEC$ have areas $1234\, \text{cm}^2$ and $2804\, \text{cm}^2$, respectively. Points $M$ and $N$ are chosen on sides $AD$ and $FE$, respectively, so that segment $MN$ passes through $B$. Find the area of $\triangle MNC$.
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Context:
Let  $ABC$  be a triangle with  $\angle BAC = 90^\circ$ . Construct the square  $BDEC$  such as  $A$  and the square are at opposite sides of  $BC$ . Let the angle bisector of  $\angle BAC$  cut the sides  $[BC]$  and  $[DE]$  at  $F$  and  $G$ , respectively. If  $|AB|=24$  and  $|AC|=10$ , calculate the area of quadrilateral  $BDGF$ .
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\boxed{The area of quadrilateral BDGF is 338.}


Context:
试求出具有如下性质的最小的正整数：它可以表示为2002个各位数字和相等的正整数之和，又可以表示为2003个各位数字之和相等的正整数之和．
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Context:
Let  $ n\geq 3 $  be an integer and let  $ x_1,x_2,\ldots,x_{n-1} $  be nonnegative integers such that

\begin{eqnarray*} \ x_1 + x_2 + \cdots + x_{n-1} &=& n  x_1 + 2x_2 + \cdots + (n-1)x_{n-1} &=& 2n-2. \end{eqnarray*}

Find the minimal value of  $ F(x_1,x_2,\ldots,x_n) = \sum_{k=1}^{n-1} k(2n-k)x_k $ .
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Context:
Determine the maximum number of bishops that we can place in a  $8 \times 8$  chessboard such that there are not two bishops in the same cell, and each bishop is threatened by at most one bishop.

Note: A bishop threatens another one, if both are placed in different cells, in the same diagonal. A board has as diagonals the  $2$  main diagonals and the ones parallel to those ones.
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Context:
$A$ rectangle has sides of length $a$ and $36$. $A$ hinge is installed at each vertex of the rectangle, and at the midpoint of each side of length $36$. The sides of length $a$ can be pressed toward each other keeping those two sides parallel so the rectangle becomes a convex hexagon as shown. When the figure is a hexagon with the sides of length $a$ parallel and separated by a distance of $24$, the hexagon has the same area as the original rectangle. Find $a^2$.
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Context:
In right triangle  $ ABC$  with right angle at  $ C$ ,  $ \angle BAC < 45$  degrees and  $ AB \equal{} 4$ .  Point  $ P$  on  $ AB$  is chosen such that  $ \angle APC \equal{} 2\angle ACP$  and  $ CP \equal{} 1$ .  The ratio  $ \frac{AP}{BP}$  can be represented in the form  $ p \plus{} q\sqrt{r}$ , where  $ p,q,r$  are positive integers and  $ r$  is not divisible by the square of any prime.  Find  $ p\plus{}q\plus{}r$ .
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Context:
Rory uses four identical standard dice to build a solid. Whenever two dice touch, the numbers on the touching faces are the same. The numbers on some faces of the solid are shown. Given that the numbers on opposite faces of a standard die add up to 7, what number is written on the face marked with a question mark? Express your answer as a single number.
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Context:
Let  $ n > 1, n \in \mathbb{Z}$  and  $ B \equal{}\{1,2,\ldots, 2^n\}.$  A subset  $ A$  of  $ B$  is called weird if it contains exactly one of the distinct elements  $ x,y \in B$  such that the sum of  $ x$  and  $ y$  is a power of two. How many weird subsets does  $ B$  have?
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Context:
The incircle  $\Gamma$  of a scalene triangle  $ABC$  touches  $BC$  at  $D, CA$  at  $E$  and  $AB$  at  $F$ . Let  $r_A$  be the radius of the circle inside  $ABC$  which is tangent to  $\Gamma$  and the sides  $AB$  and  $AC$ . Define  $r_B$  and  $r_C$  similarly. If  $r_A = 16, r_B = 25$  and  $r_C = 36$ , determine the radius of  $\Gamma$ .
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Context:
计算 $\frac{3^3\div3(3)+3}{\frac{3}{3}}+3!.$
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Context:
Evaluate the improper integral by converting to polar coordinates: $$\int_{0}^\infty \int_{-\infty}^{\infty} \exp[-((y^2+z^2)/2)]\,dy\,dz$$
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Context:
The complex number \( \left(\sqrt{\frac{1}{2} + \frac{\sqrt{3}}{2}i}\right)^{1/3} \) can be expressed in polar form as \( r(\cos x + i\sin x) \) with \( r < 0 \). Find the smallest positive value of \( x \) in degrees.
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Context:
Cassandra sets her watch to the correct time at noon. At the actual time of $ \text{1: \!\,00 PM}$, she notices that her watch reads $ \text{12: \!\,57}$ and $ 36$ seconds. Assuming that her watch loses time at a constant rate, what will be the actual time when her watch ﬁrst reads $ \text{10: \!\,00 PM}$?
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Context:
On a planet,far,far away,the Yaliens have defined: \( x \) "equals" \( y \) if and only if
 \( \left| {x - y}\right|  \leq  3 \) . Let \( S \) be a set of positive integers. What is the smallest possible number
 of elements in \( S \) such that,for any positive integer \( r \) ,where \( 1 \leq  r \leq  {2024},r \) "equals"
 some element in \( S \) ?
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Context:
Determine the number of partitions of the natural numbers into countably many countable sets.
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Context:
Determine the radius of convergence for the differential equation \((t^3-3t^2+3t)y''+4(t-1)y'+y = 0\) at \(t = 1\).
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Context:
What is the expected number of points in the interval \([0,1]\) for the sine point process, which is a determinantal point process with kernel \(K(x,y)=\frac{\sin \pi(x-y)}{\pi(x-y)}\) when \(x\neq y\) and \(1\) when \(x=y\)?
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Context:
Let $A(z_1)$, $B(z_2)$, and $C(z_3)$ be the vertices of a triangle $ABC$ such that $|z_1|=|z_2|=|z_3|=1$. Given that $z_1+z_2\cos\alpha+z_3\sin\alpha=0$ for some $\alpha\in(0,\frac\pi2)$, and $z_2\overline{z_3}=\pm i$, find the length of the longest side of triangle $ABC$. Express your answer in simplest form.
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Context:
Consider a permutation of the numbers $1, 2, 3, \ldots, 2020$ denoted by $a_1, a_2, \ldots, a_{2020}$. Determine the number of permutations such that the absolute difference between each term and its position is constant, i.e., $|a_1 - 1| = |a_2 - 2| = \ldots = |a_{2020} - 2020|$. Provide your answer as a single integer.
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Context:
Find the curvature of the curve \( c(t) = \int^t_0 r(u) \, du \), where \( r(t) \) is a unit speed curve on a sphere of radius \( R \), in terms of \( R \). Do not use the formula for curvature of a non-unit speed curve.
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Context:
$ A$ is a set of points on the plane, $ L$ is a line on the same plane. If $ L$ passes through one of the points in $ A$, then we call that $ L$ passes through $ A$.
Find the biggest integer $ r$, so that if we divide all the rational points on the plane into $ 100$ pairwisely non-intersecting point sets with infinity elements with any method, then there is at least one line that passes through $ r$ sets of the $ 100$ point sets.
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Context:
Let \( n \) denote the product of the first 2013 primes. Find the sum of all primes
 \( p \) with \( {20} \leq  p \leq  {150} \) such that
 (i) \( \frac{p + 1}{2} \) is even but is not a power of 2,and
 (ii) there exist pairwise distinct positive integers \( a,b,c \) for which
 \( {a}^{n}\left( {a - b}\right) \left( {a - c}\right)  + {b}^{n}\left( {b - c}\right) \left( {b - a}\right)  + {c}^{n}\left( {c - a}\right) \left( {c - b}\right) \)
 is divisible by \( p \) but not \( {p}^{2} \) .
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Context:
Evaluate the line integral \( \oint_C \mathbf{F} \cdot d\mathbf{r} \) where \( \mathbf{F}(x,y,z) = -x^{2}\mathbf{i} + xz\mathbf{j} + yx\mathbf{k} \) and \( C \) is the boundary of the surface \( S \) given by the graph of \( f(x,y) = 4x - 8y + 30 \) over the rectangle \( R = \{ (x,y) \mid -2 < x < 3, 0 < y < 2 \} \), oriented in the counterclockwise direction when viewed from above.
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Context:
Determine the minimal possible order of a group $G$ that contains a subset $A \subset G$ with $|A| = 2n$, such that for every $a \in A$, there exists a unique $b \in A$ with $[a, b] \neq e$. Provide your answer as a single integer.
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Context:
John folds a rope in half five times then cuts the folded rope with four knife cuts, leaving five stacks of rope segments. How many pieces of rope does he now have?
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Context:
Determine the angle between the integral curves of the differential equations \( x' = x - t^2 \) and \( x' = 2x + t^2 \) at the point \( M = (1, -2) \).
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Context:
Let \( a,b,c,p,q,r > 0 \) such that(a,b,c)is a geometric progression and(p,q,r)is an arithmetic progression. If
 \( {a}^{p}{b}^{q}{c}^{r} = 6\; \) and \( \;{a}^{q}{b}^{r}{c}^{p} = {29} \)
 then compute \( \left\lfloor  {{a}^{r}{b}^{p}{c}^{q}}\right\rfloor \) .
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Context:
Let $n > 1$ be an odd number. Let $A$ be an $n \times n$ matrix defined as follows:
\[
a_{i, j} = \begin{cases} 1, & \text{for}\ i - j  \equiv \pm 2 \pmod n\\ 2, & \text{for}\ i = j\\ 0, & \text{otherwise}.\end{cases}
\]
Calculate the determinant of matrix $A$. \( \boxed{?} \)
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Context:
Find the equation of the circle that is transformed into a straight line by the function \( f(z) = \frac{z-1}{iz+i} \) and passes through the point \( z = 1 \).
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Context:
In a city, 100 graduates applied to 5 different universities. Each university was unable to reach exactly half of its applicants during the first and second waves. Additionally, representatives from at least three universities were unable to reach the graduates. What is the maximum number of graduates who could have been of interest to the military recruitment office?
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Context:
Compute the Laplacian of the scalar field \( \dfrac{(\mathbf{\mu} \cdot \mathbf{r})}{r^3} \), where \( r = \Vert \mathbf{r} \Vert \).
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Context:
Determine the value of the Ramsey number $R(K_3, C_5)$. Use the hint that $R(3,4) = 9$ to assist in your calculation.
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Context:
Evaluate \( \lim_{n\to\infty} I_n \), where \( I_n = \int\int_{r\leq 1} \frac{n}{r\cos \pi r+n^2r^3} \ dxdy \) and \( r = \sqrt{x^2+y^2} \).
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Context:
Evaluate the limit of the integral \(\lim_{R\to\infty}\int_C \frac{2z^2-5}{(z^2+1)(z^2+4)}\,dz\) where \(C\) is the upper half-circle \(|z|=R\), \(\operatorname{Im} z>0\).
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Context:
Determine the number of integers  $2 \le n \le 2016$  such that  $n^n-1$  is divisible by  $2$ ,  $3$ ,  $5$ ,  $7$ .
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Context:
Let $S$ be the vector space of real $n \times n$ skew-symmetric matrices. For a nonsingular matrix $A$, compute the determinant of the linear map $T_A : S \rightarrow S$ where $T_A(X) = AXA^{T}$. What is the determinant of this map?
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Context:
Evaluate the limit: $$\lim_{r \to \infty} \frac{\int_0^{\pi/2}\left(\frac{x}{\pi/2}\right)^{r-1}\cos x\,dx}{\int_0^{\pi/2}\left(\frac{x}{\pi/2}\right)^r\cos{x}\,dx}$$
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Context:
Find the remainder when \(3^{3^{3^{\cdots}}}\) (with 2020 copies of 3) is divided by 46.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the supremum of the ratio between the area of a centrally-symmetric convex curve $B$ and the area of an arbitrary planar convex set $A$, where $B$ is the smallest such curve that circumscribes $A$. Provide your answer as a single value.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $ABC$ be an equilateral triangle, and let $D$ be a point inside $ABC$. Let $E$ be a point such that $ADE$ is an equilateral triangle and suppose that segments $DE$ and $AB$ intersect at point $F$. Given that $\angle CAD = 15^o$, compute $\angle DFB$ in degrees.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The focus of a vertically oriented, rotational paraboloid-shaped tall vessel is at a distance of 0.05 meters above the vertex. If a small amount of water is poured into the vessel, what angular velocity $\omega$ is needed to rotate the vessel around its axis so that the water overflows from the top of the vessel?
Please reason step by step, and put your final answer within \boxed{}.


Context:
$设F为椭圆C：x^2/2+y^2=1的右焦点，过点F且与x轴不重合的直线l交椭圆C于A，B两点。$
$在x轴上是否存在异于F的定点Q，使 \frac{k_{QA}}{k_{QB}}为定值(其中k_{QA},k_{QB}分别为直线QA,QB的斜率)？若存在，求出Q的坐标；若不存在，请说明理由.$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the derivative of \( \text{tr}(AXB) \) with respect to the matrix \( X \), where \( A, B \in \mathbb{R}^{2 \times 2} \) and \( X \in \mathbb{R}^{2 \times 2} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find a closed formula for the infinite series \(\sum_{n=1}^{\infty}\frac{n^4}{4^n}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a $2021 \times 2021$ square table, each cell is colored in one of $n$ colors. It is known that for any four cells of the same color located in a single column, there are no cells of the same color to the right of the top cell and to the left of the bottom cell. Determine the minimum value of $n$ for which this coloring is possible.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the area of the cone defined by the equation \(z=\sqrt{2x^2+2y^2}\) that is inscribed in the sphere \(x^2+y^2+z^2=12^2\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the character of the standard representation of the symmetric group $S_3$ on $\mathbb{R}^3$. Express your answer in terms of the conjugacy classes of $S_3$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the maximum number of points $P$ in a plane such that all triangles formed by any three points in $P$ are not obtuse. Consider degenerate triangles as well.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the valuation \( v_{\pi_L}(1-\zeta^{a-1}) \) in the cyclotomic field \( L=\mathbb{Q}[\zeta] \), where \( \pi_L=1-\zeta \) is the ideal above \( p \) and \( (a,p)=1 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the remainder when the integer $a$, which satisfies the equation \( \frac{1}{1}+\frac{1}{2}+\ldots+\frac{1}{23}=\frac{a}{23!} \), is divided by 13.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the Lie derivative of the volume form \(\Omega = dx\wedge dy\wedge dz\) on a 3-dimensional orientable manifold along the vector field \( X = (x+y)\partial_x + (x-y)\partial_y+ (x^2+y^2)\partial_{z} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the greatest integer  $A$  for which in any permutation of the numbers  $1, 2, \ldots , 100$  there exist ten consecutive numbers whose sum is at least  $A$ .
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the surface area of the portion of the cylinder defined by \(x^2 + y^2 = 2ay\) that is contained within the sphere \(x^2 + y^2 + z^2 = 4a^2\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the average squared distance between two points chosen randomly on the disk \( D = \{(x,y) \in \mathbb{R}^2 \mid x^2 + y^2 \leq 1\} \). Use the integral formula: \[ \frac{1}{m(D)^2} \int \int_D \int \int_D ((x_1 - x_2)^2 + (y_1 - y_2)^2) \, dx_1 \, dy_1 \, dx_2 \, dy_2 \] where \( m(D) \) is the Lebesgue measure of \( D \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the line integral \( \oint_C (4x \, dx + 9y \, dy + 3(x^2 + y^2) \, dz) \) where \( C \) is the boundary of the surface \( z = 4 - x^2 - y^2 \) in the first octant (\( x, y, z \ge 0 \)), oriented counterclockwise as viewed from above, using Stokes' Theorem.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A sequence $\{a_n\}$ is defined by the recurrence relation $a_n = a_{n-1} + 2a_{n-2} - a_{n-3}$ with initial conditions $a_1 = a_2 = \frac{a_3}{3} = 1$. Find the value of the infinite series $$a_1 + \frac{a_2}{2} + \frac{a_3}{2^2} + \cdots$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the smallest integer \( n \) such that the Lagrange interpolation error \(|f(x) - Q_n(x)|\) for the function \( f(x) = e^x \) on the interval \([-4, 0]\) using equally spaced nodes is less than 0.1. Assume \( M_{n+1} = 1 \) and use the error bound formula \(|f(x) - P_n(x)| \leq \frac{M_{n+1}}{(n+1)!}|(x - x_0)\cdot...\cdot(x - x_n)|\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $T$ be a nonzero bounded linear operator in $B(H)$, where $H$ is an infinite dimensional Hilbert space, and let $\|T\|$ be the norm of $T$. Compute the norm $\|I-T\|$, where $I$ is the identity operator.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the sum \( \sum_{k=2}^{\infty}\log{(1-1/k^2)} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
A pyramid with a square base, and all its edges of length $2$, is joined to a regular tetrahedron, whose edges are also of length $2$, by gluing together two of the triangular faces. Find the sum of the lengths of the edges of the resulting solid.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let  $r$  and  $s$  be positive real numbers, and let  $A=(0,0)$ ,  $B=(1,0)$ ,  $C=(r,s)$ , and  $D=(r+1,s)$  be points on the coordinate plane. Find a point  $P$ , whose coordinates are expressed in terms of  $r$  and  $s$ , with the following property: if  $E$  is any point on the interior of line segment  $AB$ , and  $F$  is the unique point other than  $E$  that lies on the circumcircles of triangles  $BCE$  and  $ADE$ , then  $P$  lies on line  $\overleftrightarrow{EF}$ .
Please reason step by step, and put your final answer within \boxed{}.

\boxed{\left( \frac{3r+1}{2}, \frac{s}{2} \right)}


Context:
Find the smallest positive integer $n$ such that for every integer $m$,
with $0<m<1993$, there exists an integer $k$ for which
$$\frac{m}{1993}<\frac{k}{n}<\frac{m+1}{1994}.$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine all real numbers $x$ such that the series \( S = \sum_{n=1}^{\infty} \frac{1}{n^2+n} \cdot \tan^2\left(\frac{1}{\sqrt{n}}\right) \cdot n^x \) converges.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the function $f(x)=x^{3}+x$, for any $m\in[-2,2]$, the inequality $f(mx-2)+f(x) < 0$ always holds true. Determine the range of values for $x$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The diagram shows a square PQRS with side length 1. W is the center of the square, and U is the midpoint of RS. Line segments TW, UW, and VW split the square into three regions of equal area. Find the length of SV. Express your answer as a fraction.
Please reason step by step, and put your final answer within \boxed{}.


Context:
6. [4 points] Two parallel lines $\ell_{1}$ and $\ell_{2}$ touch the circle $\omega_{1}$ with center $O_{1}$ at points $A$ and $B$ respectively. The circle $\omega_{2}$ with center $O_{2}$ touches the line $\ell_{1}$ at point $D$, intersects the line $\ell_{2}$ at points $B$ and $E$, and intersects the circle $\omega_{1}$ again at point $C$ (with point $O_{2}$ lying between the lines $\ell_{1}$ and $\ell_{2}$). It is known that the ratio of the area of quadrilateral $B O_{1} C O_{2}$ to the area of triangle $O_{2} B E$ is $\frac{6}{5}$. Find the ratio of the radii of the circles $\omega_{2}$ and $\omega_{1}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of $p$-element subsets $A$ of the set $\{1, 2, 3, \ldots, p\}$ such that the sum of the elements in $A$ is divisible by $p$, where $p$ is an odd prime number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
在7×7的正方形方格纸中，选择k个小方格的中心，使得它们中任意四个都不构成为一个矩形的顶点（矩形的边与原来正方形的边平行），求k的最大值.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the smallest real number $a$ such that $e^x > x^5$ for all $x > a$. Provide your answer as a single value of $a$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \int_{|z|=2} \frac{\operatorname{Im} z}{z-\alpha} \, dz \) where \( \alpha \) is a complex number satisfying \( 0<|\alpha|<1 \), and \( \operatorname{Im} z \) denotes the imaginary part of \( z \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many permutations of the word "arrangement" have exactly two of the following consecutive letter pairs: aa, ee, rr, nn?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \int_{0}^{\infty} e^{-x} f(x) \, dx \), where the function \( f(x) \) is defined by the series \( f(x) = \sum_{n=0}^{\infty} (-1)^n \frac{1}{(2n!)^2} (2015x)^{2n} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider the fractional order algebraic equation \( x^{\pi} - 1 = 0 \). Let \( U \) be the set of unit magnitude complex numbers defined as \( U = \{ x \mid x \in \mathbb{C}, |x|=1, x^{\pi} \neq 1 \} \). What percentage of the unit circle does the set \( U \) constitute?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Two rectangles are drawn on a sheet of graph paper. The first rectangle has a vertical side shorter than its horizontal side, while the second rectangle has a vertical side longer than its horizontal side. Each rectangle covers more than 2010 but less than 2020 cells. Determine the maximum possible area of their intersection.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: $$\lim_{x \to 0}\frac{1-\frac{1}{3} x^3 - \cos\left(\frac{x}{1-x^3}\right)}{x^6}$$ without using L'Hopital's rule.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the line integral \(\int_C (y^2z^3\,dx + 2xyz^3\,dy + 3xy^2z^2\,dz)\) where \(C\) is the straight line path from \(a = (1,1,1)\) to \(b = (2,2,2)\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \int_1^\infty \frac{1-(-1)^{\lfloor u \rfloor}}{2u^2} \, \mathrm{d}u \) using Abel's Summation formula.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given two ElGamal signatures over the finite field $\mathbb{F}_{2027}$ with the cyclic group $\mathbb{G}$ of order 1013, the signatures are for hash values $h(m_1) = 345$ and $h(m_2) = 567$, given by $(r_1, s_1) = (365, 448)$ and $(r_2, s_2) = (365, 969)$. Compute Alice's secret key $x$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of independent components of the Riemann curvature tensor $R_{iklm}$ in a 4-dimensional spacetime.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider a projection valued measure $E$ from the complex plane into the projections on a Hilbert space $\mathcal{H}$, with compact support $K \subseteq \mathbb{C}$. Determine the spectrum of the normal operator $\int_K z \, dE(z)$ in $\mathcal{B}(\mathcal{H})$. \( \mathcal{B}(\mathcal{H}) \) denotes the bounded linear operators on $\mathcal{H}$. \( K \) is the compact support of the measure.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the transfer function \( G(s) = \frac{1}{(s+7)(s+12)} \) and the input \( x(t) = \left(1 + \frac{1}{t}\right)^t \), calculate the limit of the output \( y(t) \) as \( t \to \infty \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the unique polynomial \( r(x) \) such that for all \( p(x) \in \mathbb{P}^2 \), the functional \( f(p(x)) = p'(-15) + 8p(-1) \) can be expressed as \( \langle p, r \rangle = \int_0^1 p(x)r(x)dx \). Here, \( \mathbb{P}^2 \) is the space of polynomials of degree less than two.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $(B_t)_{t \ge 0}$ be a standard one-dimensional Brownian motion, independent of $W_1$. Calculate the probability that $W_1 + B_t > 0$ for all $t \in (0, 1)$. Provide your answer as a probability value.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \int_{-\infty}^\infty \frac{\sin(xt)}{1+t^2} \, dt \) using the residue theorem.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A piece of paper in the shape of an isosceles triangle is folded such that the vertex $A$ of its base is brought to the midpoint of side $BC$. As a result, one endpoint of the crease (folding line) divides the side $AC$ in a ratio of $p: q$. Determine the ratio in which the other endpoint of the crease divides the base $AB$. Provide your answer in the form of a ratio.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $f:[0,1]\rightarrow [0,1]$ be a continuous function such that $f(0)=f(1)$. Define the set \( A = \{ (t,s) \in [0,1] \times [0,1] \mid t \neq s \text{ and } f(t)=f(s) \} \). Determine the number of elements in the set $A$. \( \text{(Express your answer as a number or state if it is infinite.)} \)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine all real numbers \( \alpha \) such that the set \( \{(x,y)\in\mathbb{R}^2: \sin(x+y)=\alpha y\} \) represents the graph of a differentiable function \( g:\mathbb{R}\rightarrow\mathbb{R} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the product: \[ \prod_{k=0}^{2001} \left( 1 + \left( \frac{1+i}{2} \right)^{2^k} \right). \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
During the softball season, Judy had 35 hits. Among her hits were 1 home
 run, 1 triple and 5 doubles. The rest of her hits were single. What percent of
 her hits were single?

Please reason step by step, and put your final answer within \boxed{}.


Context:
In preparation for the World Cup, the coach of country $F$ evaluates players $A_{1}, A_{2}, \ldots, A_{7}$ in 3 training matches, each lasting 90 minutes. Only one player is on the field at any time. Players $A_{1}, A_{2}, A_{3},$ and $A_{4}$ each spend a total time on the field that is divisible by 7 minutes. Players $A_{5}, A_{6},$ and $A_{7}$ each spend a total time on the field that is divisible by 13 minutes. With unlimited substitutions allowed, how many different scenarios are possible for the total time each player spends on the field?
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the best known bound on the index of an abelian subgroup in a discrete subgroup of the Euclidean isometry group \( \isom(\mathbb{R}^n) \)?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that $a$ and $b$ are positive integers where $b$ is a multiple of $a$, and \[ \log\left(\left(\frac{b}{a}\right)^{\frac{b}{3}}\right) + \log\left(\left(\sqrt[3]{\frac{a}{b}}\right)^{9a}\right) = 1, \] find the value of $a^2 + b^2$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find \( \lim_{x \to 0} \frac{e^{\arcsin x} - e^{\sin x}}{e^{\arctan x} - e^{\tan x}} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that the composition of linear transformations $T_1: \mathbb{R}^n \rightarrow \mathbb{R}^m$ and $T_2: \mathbb{R}^m \rightarrow \mathbb{R}^m$ is bijective, find the ranks of $T_1$ and $T_2$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A stock's price follows a model where, if the current price is \( s \), after one period, it will be either \( us \) with probability \( p \) or \( ds \) with probability \( 1 - p \). Assuming successive movements are independent, approximate the probability that the stock's price will be up at least 30% after the next 1000 periods, given \( u = 1.012 \), \( d = 0.990 \), and \( p = 0.52 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the maximum number of real zeros that a polyquaral with a degree of 5 and 3 nested square roots can have?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( f \) be a function analytic on the annulus \( \{ z \in \mathbb{C} : 1/2 < |z| < 2 \} \) except for a simple pole at \( z = 1 \). The residue of \( f \) at \( z = 1 \) is 1. Let \( \sum_n a_n z^n \) and \( \sum_n b_n z^n \) be the Laurent series expansions of \( f \) on the annuli \( \{ z : 1/2 < |z| < 1 \} \) and \( \{ z : 1 < |z| < 2 \} \), respectively. Compute \( b_n - a_n \) for every integer \( n \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Carol stands at a flag pole that is $21$ feet tall. She begins to walk in the direction of the flag's shadow to say hi to her friends. When she has walked $10$ feet, her shadow passes the flag's shadow. Given that Carol is exactly $5$ feet tall, how long in feet is her shadow?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the distribution of the random variable \( W = \sqrt{\sum_{i=1}^{n} X_i^2} \), where \( X_1, X_2, \ldots, X_n \) are independent and identically distributed as \( \mathcal{N}(0,1) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate \( E(B_{1}^2B_{2}B_{3}) \), where \( B_{i} \) represents Brownian motion.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the value of the following expression: \[ \frac{\int_0^\pi x^3\ln(\sin x)\,dx}{\int_0^\pi x^2\ln(\sqrt{2}(\sin x))\,dx} \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
On a circular board, five numbers $2, 0, 1, 9, 0$ are written in the order specified clockwise (the last zero is written next to the first two). At each step, the sum of each pair of neighboring numbers is inserted between them. After 5 steps, calculate the difference between the sum of all numbers from the first zero (initially between 2 and 1) to the second zero (initially between 9 and 2) when traversing the circle clockwise, and the sum of all the other numbers.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the determinant of the weighted orthogonal projection matrix $J = A \left( A^\intercal W A \right)^{-1} A^\intercal W$, where $A$ is a non-square matrix, and $W$, $J$, and $A^\intercal W A$ are square matrices.
Please reason step by step, and put your final answer within \boxed{}.


Context:
For the series \( e^{\tan x} \), estimate the series by finding the best integer value \( k \) such that:
\[ e^{\tan x} = 1 + x + \frac{x^2}{2!} + o(x^k) \]
as \( x \to 0 \). Determine the largest integer \( k \) for which this statement holds true.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the characteristic polynomial of a matrix \( A \) as \( p(\lambda) = (\lambda+1)(\lambda-2)^2(\lambda+3)^2 \), find \( \det(A^{-1}) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Suppose \( x,y,z \) , and \( w \) are positive reals such that
 \( {x}^{2} + {y}^{2} - \frac{xy}{2} = {w}^{2} + {z}^{2} + \frac{wz}{2} = {36} \)
 \( {xz} + {yw} = {30}. \)
 Find the largest possible value of \( {\left( xy + wz\right) }^{2} \) .
Please reason step by step, and put your final answer within \boxed{}.


Context:
Ana and Banana play a game on a $1 \times 2024$ board, with squares initially painted white. Ana starts the game. Each move by Ana consists of choosing any $k \leq m$ white squares on the board and painting them all green. Each move by Banana consists of choosing any sequence of consecutive green squares and painting them all white. Determine the smallest positive integer $m \leq 2024$ for which Ana can guarantee that, after one of her moves, the entire board will be painted green.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A flagpole is originally5meters tall. A hurricane snaps the flagpole at a point \( x \) meters above the ground so that the upper part,
 still attached to the stump,touches the ground 1 meter away from the base. What is \( x \) ?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate \( \alpha^2(\alpha^2+\alpha+1) \) in the Galois field \( GF(27) \), where \( \alpha \) is a root of the irreducible polynomial \( x^3+2x+1 \) over \( \mathbb{Z}_3 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the fundamental period of the continuous time signal given by:
\[ x(t) = \sum_{n=-\infty}^{\infty}e^{-(2t-n)}u(2t-n) \] where \( u(t) \) is the unit step function.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the limit \( \lim_{x\rightarrow\infty}\frac{x}{\overline{F}_{X}(x)}\int_{x}^{\infty}\overline{F}_{X}(s)ds \), where \( X\sim\text{Norm}(0,1) \) is a standard normally distributed random variable and \( \overline{F}_{X}(x) = 1 - F_{X}(x) \) is the survival function of \( X \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $(B_t)_{t \ge 0}$ be a standard Brownian motion. Define the stopping time \( T := \inf\{t \ge 0 : B_t = 5t - 2\} \). Calculate \( E[T] \).
Please reason step by step, and put your final answer within \boxed{}.