Context:
Determine the number of distinct locations where a treasure could be found, given the following conditions: You are at the origin $(0,0)$, and the treasure lies within a $100$ meter radius. Person A claims to have found the treasure by walking $108$ meters from the origin, with their distance calculated as $|x| + |y|$. Person B claims to have found the treasure by walking $105$ meters from the origin, with their distance calculated as $\frac{|x|}{\sqrt{3}} + \max\left\{\frac{|x|}{\sqrt{3}}, |y|\right\}$. How many distinct locations could the treasure be located?
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Context:
What is the smallest number of $45^\circ-60^\circ-75^\circ$ triangles needed to divide a square?
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Context:
Given that the variables $a$, $b$, and $c$ are independent and identically distributed random variables with an exponential distribution with a common parameter $1$, calculate the probability that the polynomial $ax^2 + bx + c$ has real roots.
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Context:
Determine the number of alternating subsets of the set \( \{1, 2, 3, \ldots, n\} \) for any natural number \( n \), and calculate \( A(33) \).
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Context:
Within the real numbers, solve the equation$$x^5 + \lfloor x \rfloor =  20$$where $\lfloor x \rfloor$ denotes the largest whole number less than or equal to $x$.
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Context:
Find the root of the equation \( 8t^3 - 4t^2 - 4t + 1 = 0 \) that lies in the interval \( \left(0, \frac{\pi}{13}\right) \).
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Context:
Determine the conflict-free chromatic number \(\chi_{\text{cf}}(\mathbb{R}, \tau)\) for the real numbers \(\mathbb{R}\) with the Euclidean topology \(\tau\).
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Context:
Given three random variables $X$, $Y$, and $Z$ with $\mathsf{Corr}(X, Y)=0.9$ and $\mathsf{Corr}(Y, Z)=0.8$, determine the minimum possible value of $\mathsf{Corr}(X, Z)$. Provide your answer as a single numerical value.
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Context:
Compute the limit \( \lim\limits_{h\to0^+}{h\sum_{n=1}^\infty f(nh)} \) as an improper integral, where \( f(x)=\frac{\ln(x)}{1+x^2} \). Note that \( f \) is increasing on \( (0,\sqrt{e}] \) and decreasing on \( [\sqrt{e},+\infty) \).
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Context:
Find the number of permutations in the symmetric group $S_{12}$ that have the cycle structure $(2,2,0,1,2,0,0,0,0,0)$. Express your answer as a single integer.
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Context:
Determine the greatest number \( k \) such that for any sequence of \( k \) points in the plane, \( 2023-k \) points can be added so that the sequence of \( 2023 \) points is carioca.
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Context:
Evaluate the definite integral: $$\int^{\pi/4}_{0}\frac{8 \cos(2t)}{\sqrt {9-5 \sin t (2t)}}~dt.$$
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Context:
Given that \(t^3\) and \(t e^t \sin(t)\) are solutions to an nth-order linear homogeneous differential equation with constant coefficients, determine the minimum possible value of \(n\).
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Context:
Determine the quadratic character of 293 modulo 379.
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Context:
Find the whole number value of $x$ that maximizes the expression \[ \frac{100\left\lfloor\frac{10000}{\left\lfloor \frac{100000}{2x}\right\rfloor+1}\right\rfloor}{x} \] given that $100000$ is divisible by $x$. \(x\) must be a divisor of $100000$. What is the maximum value of $x$?
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Context:
Evaluate the integral \( \int_C \tan z \, dz \), where \( C \) is the positively oriented circle \( |z|=2 \).
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Context:
Provide an example of a polynomial in the vector space of polynomials with degree at most 20 that is not in the image of the second derivative operator.
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Context:
Find the number of distinct conjugacy classes of subgroups of the symmetric group $S_5$ that are isomorphic to the Klein four-group.
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Context:
Evaluate the integral \( \int_{0}^{\infty} \frac{\sin x - x \cos x}{x^2 + \sin^2 x} \, dx \).
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Context:
$n$ is a integer and $a_1, a_2, \ldots, a_n\in[-1,1]$ are real numbers with $ \sum_{i=1}^{n}a_{i}=0$ ,try to find the maximum value of
$$ \sum_{1\leq i ,  j \leq n , i\ne j}|a_{i}-a^2_j|$$
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Context:
Determine the number of elements of order 6 in the automorphism group \( \text{Aut}(\mathbb{Z}_{720}) \).
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Context:
Find the value of $y$ that satisfies the equation: \[ y = \sqrt[4]{10000 - \frac{1}{10000 - \frac{1}{10000 - \cdots}}} \]
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Context:
Determine an approximate upper bound for the maximum number of balls in a bin when dropping \( n \log n \) balls into \( n \) bins randomly, ignoring distribution dependencies and integer constraints.
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Context:
Evaluate the integral \( \int_{-1}^{1} (4t^4 - 4t^2 - 1) e^{-{(t^2 -1)}^2} \, dt \).
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Context:
Calculate the residue of the function \( f(z) = \frac{1}{z^2\sin(\pi z)} \) at the triple pole \( z=0 \) without using the Laurent series expansion.
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Context:
Find the smallest positive $m$ for which there are at least 11 even and 11 odd positive integers $n$ so that $\tfrac{n^3+m}{n+2}$ is an integer.
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Context:
Place 2005 points on the circumference of a circle. Two points \( P, Q \) are said to form a pair of neighbors if the chord \( P Q \) subtends an angle of at most \( 10^{\circ} \) at the center. Determine the smallest number of pairs of neighbors.
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Context:
Calculate the expected number of games played in a tennis match between two equally skilled players, where a player must win three consecutive games to win the match. Each game is independent, with each player having a 50% chance of winning any given game.
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Context:
Determine the number of codewords of weight 4 in a binary $[31,22,5]$ linear code, given that each weight 2 coset can contain at most one weight 4 codeword.
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Context:
Determine the field over which all 2x2 orthogonal matrices are diagonalizable.
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Context:
Calculate the volume of the region in the first octant bounded by the planes \( x + y + z = 9 \), \( 2x + 3y = 18 \), and \( x + 3y = 9 \).
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Context:
Consider a Markov Chain based random number generator with 5 points, starting at the 1st point. Each point has a 25% chance to move forward, a 25% chance to move backward, and a 50% chance to move to an adding point. The adding points add 1 to the final number and return to their connected point. The 1st point cannot move backward and instead returns to itself. If the generator moves forward from the 5th point, it reaches a final absorbing point, ending the process. Calculate the expected number of times the generator visits the adding points before reaching the final absorbing point.
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Context:
Calculate the limit: \[ \lim _{n\to \infty }\frac{2^{4n+1} \ln(2n^3+1)+n^5\cdot 10^n}{15^n+4^{2n-1} \ln (5n)} \]
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Context:
Given the parity check matrix \(H=\begin{pmatrix} 2 & 0 & 1 & 1 & 1 \\ 0 & 2 & 2 & 0 & 0 \\ 0 & 0 & 2 & 2 & 1 \\ 1 & 1 & 2 & 2 & 0 \end{pmatrix}\), determine the minimum distance \(d(C)\) of the ternary linear code \(C\).
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Context:
Find the value of \(a\) such that the equation \(x^2 = a^x\) has exactly one real root.
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Context:
Evaluate the integral \( \int_0^{2\pi} \cos(\cos(z))\cosh(\sin(z)) \, dz \) using a corollary of Cauchy's Integral Formula.
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Context:
How many 4-digit numbers exist such that deleting any one of its digits results in a 3-digit number that is a divisor of the original 4-digit number?
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Context:
Calculate the tensor product \( \mathbb{Q} / \mathbb{Z} \otimes_{\mathbb{Z}} \mathbb{Q} \).
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Context:
Using Taylor's theorem, determine the largest positive real value \( r \) for which the Maclaurin polynomial \( P_6(x) \) of \( f(x) = (\sin(x))^2 \) has an error of less than 0.2 on the interval \( 0 < x < r \).
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Context:
Find all functions \( f : \mathbb{R} \to \mathbb{R} \) that satisfy the functional equation \( f(x + y) + f(z) = f(x) + f(y + z) \) for all \( x, y, z \in \mathbb{R} \).
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Context:
令 $a_{n}$ 表示距离 $\sqrt{n}$ 最近的正整数，其中 $n \in \mathbf{N}^{*}$ ，若 $\frac{1}{a_{1}}+\frac{1}{a_{2}}+\cdots+\frac{1}{a_{n}}= 2016$，求正整数 $n$ 的值。
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Context:
In a right triangle \( ABC \) with \( AC = 16 \) and \( BC = 12 \), a circle with center at \( B \) and radius \( BC \) is drawn. A tangent to this circle is constructed parallel to the hypotenuse \( AB \) (the tangent and the triangle lie on opposite sides of the hypotenuse). The leg \( BC \) is extended to intersect this tangent. Determine by how much the leg is extended.
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Context:
The ratio of the short side of a certain rectangle to the long side is equal to the ratio of the long side to the diagonal. The original answer is in the form \(\frac{k\sqrt{m} - n}{p}\), where \(k\), \(m\), \(n\), and \(p\) are integers. What is the value of \(k + m + n + p\)?
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Context:
Let \( K(x) \) be the number of irreducible fractions \( \frac{a}{b} \) such that \( a < x \) and \( b < x \) (where \( a \) and \( b \) are natural numbers).

For example, \( K\left(\frac{5}{2}\right) = 3 \) (the fractions are \( 1, 2, \frac{1}{2} \)).

Compute the sum \( K(100) + K\left(\frac{100}{2}\right) + K\left(\frac{100}{3}\right) + \ldots + K\left(\frac{100}{99}\right) + K\left(\frac{100}{100}\right) \).
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Context:
Let $a, b, c, d, e, f$ be integers selected from the set $\{1,2, \ldots, 100\}$, uniformly and at random with replacement. Set $M=a+2 b+4 c+8 d+16 e+32 f$. What is the expected value of the remainder when $M$ is divided by 64?
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Context:
In a certain month, Mrs. Bil's earnings were a percentage of the Bil family's total income. In June, Mrs. Bil earned 10 percent more than in May. If the rest of the family's income was the same both months, then in June, Mrs. Bil's earnings were approximately 71.96 percent of the Bil family's total income. What was the percentage of Mrs. Bil's earnings in the first month?
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Context:
四个半径为 1 的球两两相切，则它们外切正四面体的棱长为多少？答案的形式为 $k(1+\sqrt{m})$，请给出 $k+m$ 的值。
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Context:
In a 200 x 200 square grid, each cell contains a real number with an absolute value not exceeding 1. The total sum of all numbers in the grid is zero. Determine the smallest value of \( S \) such that the sum of the numbers in at least one row or one column is necessarily not exceeding \( S \) in absolute value.
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Context:
In an \(m \times 10\) grid, each cell is filled with either a 0 or a 1 such that each column contains exactly three 1s. Let the sum of the numbers in the \(i^{th}\) row be denoted by \(x_i\). Furthermore, for any two columns, there is always at least one row where both columns have a 1. Let \(x = \max_{1 \leq i \leq m} \{x_i\}\). Find the minimum value of \(x\), denoted by \(x_{\text{min}} =\) ______.
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Context:
Two projective planes, each with two punctures, are joined along the two punctures. What is the Euler characteristic of the resulting surface?
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Context:
The school director, custodian, and parent committee each bought a carpet for the school assembly hall, which has dimensions $10 \times 10$. Without agreeing with each other, they decided to place all three carpets as follows: the first carpet of size $6 \times 8$ in one corner, the second carpet of size $6 \times 6$ in the opposite corner, and the third carpet of size $5 \times 7$ in one of the remaining corners (all measurements are in meters).

Find the area of the hall covered by carpets in three layers (provide the answer in square meters).
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Context:
If the function $f(x) = \ln{x} + ax^2 - 2x$ has two distinct extreme value points, find the range of the real number $a$.
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\boxed{a \in \left(0, \frac{1}{2}\right)}


Context:
Given the function $f: \mathbb{Z} \rightarrow \mathbb{Z}$ such that $f(f(x+y)) = f(x^2) + f(y^2)$, and $f(f(2020)) = 1010$, find the value of $f(2025)$.
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Context:
On a plane, a line \(\ell\) and two circles \(c_1\) and \(c_2\) of different radii are given such that \(\ell\) touches both circles at point \(P\). A point \(M \neq P\) on \(\ell\) is chosen such that the angle \(Q_1MQ_2\) is maximized, where \(Q_1\) and \(Q_2\) are the points of tangency of the tangent lines drawn from \(M\) to \(c_1\) and \(c_2\), respectively, other than \(\ell\). Find the value of \(\angle PMQ_1 + \angle PMQ_2\).
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Context:
Foma and Erema were traveling along a straight road to Moscow in a cart at a constant speed.

- At 12:00, Foma asked: "How many versts to Moscow?"
- Erema replied: "82."
- At 13:00, Foma asked: "How many versts to Moscow?"
- Erema replied: "71."
- At 15:00, Foma asked: "How many versts to Moscow?"
- Erema replied: "46."

It is known that Erema rounded the distance to the nearest whole number each time, and if there were two nearest whole numbers, he could choose either.

At 16:00, Foma asked again: "How many versts to Moscow?" This time Erema gave an exact answer, without rounding. What was Erema's answer?
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Context:
Positive integers \(a_1, a_2, \cdots, a_{2006}\) are such that the ratios \(\frac{a_1}{a_2}, \frac{a_2}{a_3}, \cdots, \frac{a_{2005}}{a_{2006}}\) are pairwise distinct. Determine the minimum number of distinct integers among \(a_1, a_2, \cdots, a_{2006}\).
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Context:
Find a possible expression for the function $f(x)$ such that the following equation holds:

$$\int f(x)\sin(2x)dx=-\frac{1}{2}f(x)\cos(2x)+\int \frac{1}{4}\cos(2x)dx$$
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Context:
A set $D$ of positive integers is called *indifferent* if there are at least two integers in the set, and for any two distinct elements $x,y \in D$, their positive difference $|x-y|$ is also in $D$. Let $M(x)$ be the smallest size of an indifferent set whose largest element is $x$. Compute the sum $M(2) + M(3) + \dots + M(100)$. \(\text{Proposed by Yannick Yao}\)
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Context:
Let the sum of the first $n$ terms of an arithmetic sequence $\{a_n\}$ be $S_n$. If $1 \leqslant a_5 \leqslant 4$ and $2 \leqslant a_6 \leqslant 3$, then the range of values for $S_6$ is ______.
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Context:
Evaluate the integral: \[ \int_0^{\frac{\pi}{2}}\left(\frac{x}{\sin x}\right)^2\text{d}x \]
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Context:
There are 3 cities, and roads must be constructed so that every pair of cities is connected. Assume that all the roads connecting any two cities have independent costs that are exponentially distributed with parameter one. Find the expected value of the minimal cost to connect all cities. That is, find \( E(c_3) \).
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Context:
Yoongi wants to subtract 57 from a four-digit number. Yoongi has mistaken 9 for 6 in the unit place of a four-digit number. When the difference Yoongi gets is 1819, find the correct result.
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Context:
Let $F_{1}(-c,0)$ and $F_{2}(c,0)$ be the two foci of the ellipse $\frac {x^{2}}{a^{2}}+ \frac {y^{2}}{b^{2}}=1(a > b > 0)$. Let $P$ be a point where the circle with diameter $F_{1}F_{2}$ intersects the ellipse. If $\angle PF_{1}F_{2}=2\angle PF_{2}F_{1}$, then the eccentricity of the ellipse equals \_\_\_\_\_\_.
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Context:
Calculate: $108^\circ 18' 25'' - 56^\circ 23' 32'' =$
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Context:
4. Determine the smallest natural number $n$ with the following property: If we choose $n$ different natural numbers less than 2006, there are two among them such that the ratio of the sum and the difference of their squares is greater than three.
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Context:
Given xyz = 1, then what is ( 5 ^ (x + y + z) ^ 2 ) / ( 5 ^ (x - y + 2z) )
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Context:
Given that $x > 0$ and $y > 0$, and $2x + y = 2$, find the minimum value of $\frac{2}{x + 1} + \frac{1}{y}$.
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Context:
Given the joint density function \(f_{X,Y}(x,y)=\left\{ \begin{array}{ll} 2 & \mbox{when } x > 0 , y > 0 \text{ and } x+y<2 \\ 0 & \mbox{otherwise } \end{array} \right.\), determine the covariance of X and Y, \(\text{Cov}(X,Y)\).
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Context:
Let \( T : \mathbb{R}^4 \to \mathbb{R}^4 \) be a linear transformation such that there exists a vector \( v \in \mathbb{R}^4 \) for which \( v, T(v), T(T(v)) \) are linearly independent and \( T(T(T(v))) = 0 \). Determine the rank of the transformation \( T \).
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Context:
Determine the values of \(\beta\) for which the improper integral \[ \iint_{\Bbb R^2}\frac{dxdy}{(1+x^2-xy+y^2)^\beta} \] converges.
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Context:
在直三棱柱 $A B C-A_{1} B_{1} C_{1}$ 中, $A B=1, B C=C C_{1}=\sqrt{3}, \angle A B C=90^{\circ}$, 点 $P$ 是平面 $A B C$上一动点, 则 $A_{1} P+\frac{1}{2} P C$ 的最小值为 $\qquad$.  原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值
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Context:
Given positive integers \( a, b, c, \) and \( d \) such that \( a > b > c > d \) and \( a + b + c + d = 2004 \), as well as \( a^2 - b^2 + c^2 - d^2 = 2004 \), what is the minimum value of \( a \)?
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Context:
The quadrilateral \(ABCD\) is a rhombus with side length \(a\), and \(\angle BAD=60^\circ\). After folding along the diagonal \(BD\) to form a dihedral angle of \(120^\circ\) \(A-BD-C\), the distance between \(AC\) and \(BD\) is \(\quad\).
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Context:
Al told Bob that he was thinking of  $2011$  distinct positive integers. He also told Bob the sum of those  $2011$  distinct positive integers. From this information, Bob was able to determine all  $2011$  integers. How many possible sums could Al have told Bob?
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Context:
Find all natural numbers that are increased ninefold if a zero is inserted between the units digit and the tens digit.
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Context:
[Ratio of areas of triangles with a common angle] Auxiliary similar triangles $\quad]$

In triangle $A B C$, angle $C$ is $30^{\circ}$, and angle $A$ is acute. A line perpendicular to side $B C$ is drawn, cutting off triangle $C N M$ from triangle $A B C$ (point $N$ lies between vertices $B$ and $C$). The areas of triangles $C N M$ and $A B C$ are in the ratio $3: 16$. Segment $M N$ is half the height $B H$ of triangle $A B C$. Find the ratio $A H: H C$.
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Context:
A Moskvich car was sent to transport mail from the post office to the airfield. The plane carrying the mail landed earlier than expected, and the delivered mail was sent to the post office by a passing truck. After driving for 30 minutes, the truck met the Moskvich on the road, which received the mail and turned back immediately without delay. The Moskvich arrived at the post office 20 minutes earlier than usual. How many minutes earlier than the scheduled time did the plane land?
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Context:
A box contains 9 good products and 3 defective products. One product is taken out at a time, and it is not put back in the box after being taken out. What is the probability that 3 defective products have been taken out before 2 good products are taken out?
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Context:
Given the following equations: observe each equation: $a+b=1$, $a^{2}+b^{2}=3$, $a^{3}+b^{3}=4$, $a^{4}+b^{4}=7$, $a^{5}+b^{5}=11$, $...$, then according to this pattern, we can get $a^{6}+b^{6}=$ $\_\_\_\_\_\_$.
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Context:
Positive integers $a$, $b$, $c$, $d$, and $e$ satisfy the equations:

1. $(a + 1)(3bc + 1) = d + 3e + 1$
2. $(b + 1)(3ca + 1) = 3d + e + 13$
3. $(c + 1)(3ab + 1) = 4(26 - d - e) - 1$

Find $d^2 + e^2$.
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Context:
Let $ABC$ be a triangle with incenter $I$ such that $AB = 20$ and $AC = 19$. Point $P \neq A$ lies on line $AB$ and point $Q \neq A$ lies on line $AC$. Suppose that $IA = IP = IQ$ and that line $PQ$ passes through the midpoint of side $BC$. Suppose that $BC = \frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $100m + n$.
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Context:
Cagney can frost a cupcake every 18 seconds and Lacey can frost a cupcake every 40 seconds. Lacey starts working 1 minute after Cagney starts. Calculate the number of cupcakes that they can frost together in 6 minutes.
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Context:
A cylinder of height h is 6/7 full of water. When all of the water is poured into an empty cylinder with a certain percentage larger radius than that of the original cylinder, the new cylinder is 3/5 full. The height of the new cylinder is 70 percent of h. How much larger is the radius of the new cylinder compared to the original cylinder?
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Context:
Determine the generating function for the summation: \[ \sum_{k=0}^m\binom{m}{m-k}\binom{-n-1}{m-k} \]
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Context:
Given the hyperbola \( C: \frac{x^2}{2} - y^2 = 1 \), let \( P \) be a point on its right branch. Line \( l \) is an asymptote of the hyperbola, and \( Q \) is the projection of \( P \) onto \( l \). Let \( F_1 \) be the left focal point of the hyperbola. Determine the minimum value of \( |PF_1| + |PQ| \).
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Context:
Given that $X_t$ is a standard Brownian motion, calculate the probability $P(X_2 > 0 \mid X_1 > 0)$. Provide your answer as a single probability value.
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Context:
Compute the number of intersection points between $\mathbb{R}P^2$ and $\mathbb{C}P^1$ in $\mathbb{C}P^2$. Consider $\mathbb{R}P^2 = \{ [x_0:y_0:z_0] \mid x_i \in \mathbb{R} \}$ and $\mathbb{C}P^1 = \{ [z_0:z_1:z_2] \mid a_0 z_0 + a_1 z_1 + a_2 z_2 = 0 \}$ where $a_i \in \mathbb{C}$. Determine the number of points satisfying $a_0 x_0 + a_1 x_1 + a_2 x_2 = 0$ with $x_i \in \mathbb{R}$. Provide the number of such intersection points.
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Context:
Compute the value of the exterior 2-form \( \theta = \sum_{i=1}^{n-1} x_i \wedge x_{i+1} \) on \( \mathbb{R}^n \) at the vectors \( A = (1, 1, 1, \dots, 1) \) and \( B = (-1, 1, -1, \dots, (-1)^n) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of distinct monic quadratic polynomials with integer coefficients such that they have at least one root, all roots are powers of 3 with non-negative integer exponents, and the absolute value of their coefficients does not exceed $27^{47}$. Provide your answer as a single integer.
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Context:
Given a tetrahedron with five edges of length 2, the maximum volume of the tetrahedron is ______.
Please reason step by step, and put your final answer within \boxed{}.


Context:
$2023$ players participated in a tennis tournament, and any two players played exactly one match. There was no draw in any match, and no player won all the other players. If a player $A$ satisfies the following condition, let $A$ be "skilled player".

(Condition) For each player $B$ who won $A$, there is a player $C$ who won $B$ and lost to $A$.

It turned out there are exactly $N(\geq 0)$ skilled player. Find the minimum value of $N$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a round-robin tournament with 2010 players, each player plays against every other player exactly once. A win awards the player 1 point, while a loss gives 0 points. Determine the last three digits of the maximum possible score difference between the first and second highest scorers.
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Context:
There is a regular $17$-gon $\mathcal{P}$ and its circumcircle $\mathcal{Y}$ on the plane.
The vertices of $\mathcal{P}$ are coloured in such a way that $A,B \in \mathcal{P}$ are of different colour, if the shorter arc connecting $A$ and $B$ on $\mathcal{Y}$ has $2^k+1$ vertices, for some $k \in \mathbb{N},$ including $A$ and $B.$
What is the least number of colours which suffices?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Solve the differential equation using Laplace transform: $$xy'' + y' + 4xy = 0$$ with initial conditions $y(0) = 3$ and $y'(0) = 0$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( X_1, X_2, \) and \( X_3 \) be three independent exponential random variables with rate \( \lambda \). Find the probability that \( \min(X_1, X_2, X_3) \leq X_1 + X_2 + X_3 - \min(X_1, X_2, X_3) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of bases in the vector space \( \mathbf{Z}_5^4 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \int_0^{2\pi} \frac{\cos(3\theta)}{5-4\cos(\theta)}\mathrm{d}\theta \) using complex analysis techniques.
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Context:
Given right triangle $\triangle ABC$ has a hypotenuse $AC$ of length 10, where $A$ is at the right angle. Point $M$ is the midpoint of hypotenuse $\overline{AC}$, and $D$ is a point on leg $\overline{AB}$ such that $AD:DB = 1:2$. Determine the area of $\triangle ADM$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that the maximum value of \((x+2y+3z)^2\) is 70, find the maximum value of \(x^2 + (2y)^2 + (3z)^2\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the exact value of the norm \( \Vert g \Vert \) of the functional \( g \) defined on the normed space \( C[-1,1] \) by \[ g(x) = \int_{-1}^1 x(t)t \, \mathrm{d}t \] for all \( x \in C[-1,1] \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( I = \int_{0}^{2\pi} \frac{\cos^2(\theta) - \sin^2(\theta)}{\sin^4(\theta) + \cos^4(\theta)} \, d\theta \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
An ant is on the circumference of the base of a cone with radius $2$ and slant height $6$. It crawls to the vertex of the cone $X$ in an infinite series of steps. In each step, if the ant is at a point $P$, it crawls along the shortest path on the exterior of the cone to a point $Q$ on the opposite side of the cone such that $2QX = PX$. What is the total distance that the ant travels along the exterior of the cone?
Please reason step by step, and put your final answer within \boxed{}.


Context:
It is known that \(4 \operatorname{tg}^{2} Y + 4 \operatorname{ctg}^{2} Y - \frac{1}{\sin ^{2} \gamma} - \frac{1}{\cos ^{2} \gamma} = 17\). Find the value of the expression \(\cos ^{2} Y - \cos ^{4} \gamma\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
If $a_1, a_2, \ldots, a_{2013} \in [-2, 2]$ and $a_1 + a_2 + \cdots + a_{2013} = 0$, find the maximum of $a_1^3 + a_2^3 + \cdots + a_{2013}^3$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In figure 1, \(ABCD\) is a rectangle, and \(E\) and \(F\) are points on \(AD\) and \(DC\), respectively. Also, \(G\) is the intersection of \(AF\) and \(BE\), \(H\) is the intersection of \(AF\) and \(CE\), and \(I\) is the intersection of \(BF\) and \(CE\). If the areas of \(AGE\), \(DEHF\), and \(CIF\) are 2, 3, and 1, respectively, find the area of the grey region \(BGHI\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the interval of convergence for: $$\sum_{i=1}^\infty \frac{(x+2)^n}{(n^{1/2})(3^n)}$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
A gambler starts with an initial fortune of $2. Calculate the probability that the gambler goes broke on the 10th bet, given that their wealth reaches $6 at some point before eventually decreasing to $0.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the value of Ack(3,1) using the Ackermann function.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the double integral \( \iint 3x^2 \, dA \) over the region in the first quadrant bounded by the hyperbola \( xy = 9 \), the line \( y = x \), the y-axis (\( y = 0 \)), and the x-axis (\( x = 6 \)). Round your answer to the nearest tenth.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the maximum positive integer $r$ such that for any five 500-element subsets of the set $\{1, 2, \ldots, 1000\}$, there exist two subsets with at least $r$ common elements.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the transition matrix \( M = \begin{pmatrix} 0.25 & 0.5 & 0.25 \\ 0.5 & 0.25 & 0.25 \\ 0.5 & 0.25 & 0.25 \end{pmatrix} \) of a Markov chain with states \( S = \{0, 1, 2\} \), calculate the expected number of steps to reach state 1 starting from state 2.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In triangle $ABC$, let $I$ be the incenter, and $D, E, F$ be the points of tangency of the incircle with sides $BC, CA, AB$, respectively. Point $K$ is the intersection of line $DI$ and line $EF$. Given that $BC = 10$, $AC = 8$, and $AB = 7$, find the ratio of the areas $\frac{S_{\triangle BKC}}{S_{\triangle ABC}}$. Express your answer as a simplified fraction.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the minimum variance unbiased estimator for the total number of people in a group, given that 5 people had a birthday today, assuming birthdays are uniformly distributed over 365 days.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a convex 1950-gon, all diagonals are drawn, dividing it into smaller polygons. Determine the maximum number of sides that one of these polygons can have.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The function $f(x)=(m^{2}-m-1)x^{m^{2}+m-1}$ is a power function, and it is decreasing on $(0,+\infty)$. Find the real number $m$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Solve the equation for the unknown \( x \) where \( n \geq 2 \) is a given natural number:

$$
\sum_{i=0}^{n-2} \frac{1}{(x+i)(x+i+1)}=x(x+1)(x+2) \cdot \ldots \cdot (x+n)+\frac{n-1}{x(x+n-1)}
$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Kelvin the Frog lives in the 2-D plane. Each day, he picks a uniformly random direction (i.e., a uniformly random bearing $\theta \in [0, 2\pi]$) and jumps a mile in that direction. Let $D$ be the number of miles Kelvin is away from his starting point after ten days. Determine the expected value of $D^4$. \( \text{Express your answer as a single number.} \)
Please reason step by step, and put your final answer within \boxed{}.


Context:
In an $8 \times 8$ table, 23 cells are black, and the rest are white. In each white cell, the sum of the black cells located in the same row and the black cells located in the same column is written. Nothing is written in the black cells. What is the maximum value that the sum of the numbers in the entire table can take?
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the characteristic path length of the Petersen graph?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the sum \( S = \lim_{N\rightarrow \infty} \sum_{n=-N}^N \frac{1}{n+e} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given $f(x) = \sin \left( \frac{\pi}{3}x \right)$, and the set $A = \{1, 2, 3, 4, 5, 6, 7, 8\}$. Now, choose any two distinct elements $s$ and $t$ from set $A$. Find out the number of possible pairs $(s, t)$ such that $f(s)\cdot f(t) = 0$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Colleen has three shirts: red, green, and blue; three skirts: red, green, and grey; three scarves: red, blue, and grey; and three hats: green, blue, and grey.

How many ways are there for her to pick a shirt, a skirt, a scarf, and a hat, so that two of the four clothes are one color and the other two are one other color?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find an explicit formula for the operator $TT^*$, where $T: H \rightarrow \ell^{2}$ is the analysis operator $T x=\left\{\left\langle x, e_{n}\right\rangle\right\}_{n \in \mathbb{N}}$ and $T^{*}: \ell^{2} \rightarrow H$ is the synthesis operator $T^{*} c=\sum_{n=1}^{\infty} c_{n} e_{n}$, given an orthonormal sequence $\{e_{n}\}_{n \in \mathbb{N}}$ in a Hilbert space $H$.}
Please reason step by step, and put your final answer within \boxed{}.


Context:
The sides of rectangle $ABCD$ have lengths 10 and 11.  An equilateral triangle is drawn so that no point of the triangle lies outside $ABCD.$  The maximum possible area of such a triangle can be written in the form $p\sqrt{q}-r,$ where $p, q,$ and $r$ are positive integers, and $q$ is not divisible by the square of any prime number.  Find $p+q+r.$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Each time you click a toggle switch, the switch either turns from [i]off[/i] to [i]on[/i] or from [i]on[/i] to [i]off[/i]. Suppose that you start with three toggle switches with one of them [i]on[/i] and two of them [i]off[/i]. On each move you randomly select one of the three switches and click it. Let $m$ and $n$ be relatively prime positive integers so that $\frac{m}{n}$ is the probability that after four such clicks, one switch will be [i]on[/i] and two of them will be [i]off[/i]. Find $m+n$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the ancient Chinese mathematical text "Jiuzhang Suanshu" (The Nine Chapters on the Mathematical Art), there is a problem: "Given a metal rod, 5 chi in length, if one cuts 1 chi from the thicker end, it weighs 4 jin. If one cuts 1 chi from the thinner end, it weighs 2 jin. Assuming the rod varies uniformly in thickness from one end to the other, how much does each chi section of the rod weigh?" Suppose the rod varies evenly from the thick end to the thin end, we denote the weight of the rod by $M$. Now the rod is cut into 10 equal segments, and we denote the weight of the $i^{th}$ segment as $a_i$ (for $i=1,2,\ldots,10$), such that $a_1 < a_2 < \ldots < a_{10}$, and if $48a_i = 5M$, then $i = \underline{\hspace{1em}}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( ABC \) be an acute triangle and \( O \) be its circumcenter. Let \( D \) be the midpoint of \( [AB] \). The circumcircle of \( \triangle ADO \) meets \( [AC] \) at \( A \) and \( E \). If \( |AE|=7 \), \( |DE|=8 \), and \( m(\widehat{AOD}) = 45^\circ \), what is the area of \( \triangle ABC \) in the form \( x \sqrt{2} \)? Find the value of \( x \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the trapezoid \(ABCD\), the bases are given as \(AD = 4\) and \(BC = 1\), and the angles at \(A\) and \(D\) are \(\arctan 2\) and \(\arctan 3\) respectively.

Find the radius of the circle inscribed in triangle \(CBE\), where \(E\) is the intersection point of the diagonals of the trapezoid.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of solutions to the equation \(X^\top X = B^\top B\), where \(B \in \mathbb{R}^{d \times n}\) with \(d \geq n\) and \(B^\top B\) is non-singular.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Dr. Jones earns $6,000 a month. His house rental is $640 each month; his monthly food expense fluctuates between $350 and $400 depending on his needs; his electric and water bill costs 1/4 of what he makes, and his insurances cost 1/5 of what he makes. He spends 10% of his salary on taxes and 3% on transportation expenses. Additionally, Dr. Jones needs to allocate 2% of his salary for unexpected emergency costs. Dr. Jones also has a student loan payment of $300 per month and he invests 5% of his income into a retirement plan. If his food expense for a particular month is $385, how much money does he have left after paying all these expenses and making investments for that month?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the double integral: $$\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\int_0^{a\sin\theta} r^2 \, dr \, d\theta$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
 A fair six-sided die is labeled $1-6$ such that opposite faces sum to $7$. The die is rolled, but before you can look at the outcome, the die gets tipped over to an adjacent face. If the new face shows a $4$, what is the probability the original roll was a $1$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a school with 200 teachers, each teacher sends a greeting message to exactly one other teacher. Determine the minimum number of representatives, \( k \), that can be selected for a recognition ceremony such that no representative has sent a message to another representative in the group.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral: $$ \int \cos x \cdot \cos^2(2x) \cdot \cos^3(3x) \cdot \cos^4(4x) \cdot \ldots \cdot \cos^{2002}(2002x) \, dx $$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \(\int_{-1}^1 \frac{d}{dx}\ \frac{1}{1 + 2^{1/x}}\ dx\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of distinct bracelets that can be formed using 12 beads, consisting of 3 red beads, 4 green beads, and 5 blue beads.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the volume of the solid formed by rotating the region bounded by the curve \( y = e^{-x} \sin x \) and the line \( x \ge 0 \) about the x-axis.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the probability that a random undirected graph G(5, 1/2) is connected, where G(n, p) denotes a graph with n vertices and probability p of an edge between any two vertices.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the maximum achievable Hamming distance between $k$ bitvectors of length $n$, where $n$ is a power of 2.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the line integral \( \int_{K}\left(x^{2}+y z\right) d x+\left(y^{2}+x z\right) d y+\left(z^{2}-x y\right) d z \) where \( K \) is a closed curve, oriented positively, consisting of an arc defined by the parametric equation \( x=a\cos t, y=a\sin t, z=\frac{1}{2\pi}t \) and the segment \( BA, A=(a,0,0), B=(a,0,1) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: $$\lim_{x \to 0} \frac{x-\sin(\sin(\cdots\sin(x)))}{x^3},$$ where $\sin(\sin(\cdots\sin(x)))$ denotes the n-fold composite sine function.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The infinite sequence $P_0(x),P_1(x),P_2(x),\ldots,P_n(x),\ldots$ is defined as
\[P_0(x)=x,\quad P_n(x) = P_{n-1}(x-1)\cdot P_{n-1}(x+1),\quad n\ge 1.\]
Find the largest $k$ such that $P_{2014}(x)$ is divisible by $x^k$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the fundamental group of the space obtained by removing a finite number of points from \( \mathbb{R}^n \) for \( n \geq 3 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Define a sequence of polynomials $P_0\left(x\right)=x$ and $P_k\left(x\right)=P_{k-1}\left(x\right)^2-\left(-1\right)^kk$ for each $k\geq1$. Also define $Q_0\left(x\right)=x$ and $Q_k\left(x\right)=Q_{k-1}\left(x\right)^2+\left(-1\right)^kk$ for each $k\geq1$. Compute the product of the distinct real roots of\[P_1\left(x\right)Q_1\left(x\right)P_2\left(x\right)Q_2\left(x\right)\cdots P_{2018}\left(x\right)Q_{2018}\left(x\right).\]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the smallest positive integer \( n \) for which the following limit holds true for two linearly independent solutions \( \phi_1(x) \) and \( \phi_2(x) \) of the differential equation \( 3x^2y''-(2x+x^2)y'+(x^2-3)y=0 \), given that \( \phi_1(0)=0 \): \[ \lim_{x \to 0}x^n\frac{\phi_2(x)}{\phi_1(x)}=0 \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a chess tournament, a team of schoolchildren and a team of students, each consisting of 15 participants, compete against each other. During the tournament, each schoolchild must play with each student exactly once, with the condition that everyone can play at most once per day. Different numbers of games could be played on different days.

At some point in the tournament, the organizer noticed that there is exactly one way to schedule the next day with 15 games and $N$ ways to schedule the next day with just 1 game (the order of games in the schedule does not matter, only who plays with whom matters). Find the maximum possible value of $N$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Solve the equation $2^x = -2x + 11$ for $x$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the system of equations:
\[\begin{cases} x^2 + xy - y^2 = u \\ 2xy + y^2 = v \end{cases}\]
find the value of \( \left(\frac{\partial x}{\partial u}\right)_{v} \) at the point where \( x = 2 \) and \( y = -1 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the contour integral \( \int_C \frac{dz}{\sin^3 z} \), where \( C \) is the unit circle in the complex plane with positive orientation.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the net flux of the vector field \( F(x, y, z)=\frac{r}{|r|^{3}}+zi-xk \) through the upper half of the ellipsoid defined by \(\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1, \quad z\geq0,\) in the direction where the component of the outward normal along the \(y\)-axis is positive (\(N \cdot j>0\)).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a continuous-time Markov chain, Sleeping Beauty (SB) can be in one of three states while she sleeps: State 0 (flat on her back), State 1 (foetal position), and State 2 (foetal position and sucking her thumb). The transition rate matrix is given by:

\[ G = \begin{bmatrix} -\frac{1}{2} & \frac{3}{8} & \frac{1}{8} \\
\frac{1}{12} & -\frac{1}{3} & \frac{1}{4} \\
\frac{1}{4} & \frac{3}{4} & -1 \end{bmatrix} \]

What is the probability that Prince Charming finds SB in State 0 (looking truly radiant) after a long time? Assume SB has been sleeping for a very long time.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the covariance between $X$ and $X^3$ for a random variable $X$ that follows a normal distribution $N(\mu, \mu^2)$ using the moment generating function.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $V$ be the vector space of all polynomials over a field $F$. Consider a subspace $U$ of $V$ with $2 \leq \dim U \leq n$. Given distinct points $x_1, x_2 \in F$, determine the dimension of the subspace $W \cap U$, where $W$ is the subspace of polynomials that vanish at $x_1$ and $x_2$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many permutations $f \in S_8$ commute with the permutation $g = (1\,2\,3)(4\,5\,6)(7\,8)$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \int_{0}^{\pi/2}{\mathrm dx\over (1+\tan x)^6} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the chromatic number of the circulant graph $Ci_{2n}(1,n-1,n)$, where the vertex set is $V=\{a_0, \dots , a_{n-1}, b_0, \dots , b_{n-1}\}$ and the edge set is $E=\{(a_i,b_i), (a_i, a_{i+1\pmod{n}}), (b_i, b_{i+1 \pmod{n}}), (a_i, b_{i+1 \pmod{n}}), (b_i, a_{i+1 \pmod{n}} )\}$ for $0\le i <n$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the intersection point of the tangent planes to the surface $S$ in $\Bbb R^3$ given by the graph of $f(x,y) = xy^2 − 2y^2 + e^x$ at the points $(x, y) = (0, 0)$, $(0, 1)$, and $(0, 2)$. Provide the coordinates of the intersection point.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A pool table has 6 pockets. There are 10 red balls, one brown ball, and one black ball. How many ways are there to distribute the balls into the pockets if each pocket can contain no more than 9 balls and the black ball should not be in the same pocket as the brown ball?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the contour integral \( \int_C \frac{e^z}{z^2-4} \, dz \), where \( C \) is the circle defined by \( |z-1| = 1 \) with positive orientation.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \(X_1, X_2, X_3, \ldots\) be a sequence of i.i.d. uniform \((0,1)\) random variables. Calculate the value of \[ \lim_{n \to \infty} P\left(-\ln(1-X_1) - \ln(1-X_2) - \cdots - \ln(1-X_n) \geq n\right). \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Classify the following partial differential equation in cylindrical coordinates as elliptic, parabolic, or hyperbolic:

\[ \frac{\partial^2 \phi}{\partial z^2} + \frac{\partial \phi}{\partial z} + \frac{\partial^2 \phi}{\partial r^2} + \left ( \frac{1}{r} \right )\frac{\partial \phi}{\partial \theta} = 0 \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: $$ \lim_{{M \to +\infty}} \frac{1}{M} \int_1^M M^{\frac{1}{x}} \, \mathrm{d}x $$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( F(x,y,z) = (y+\sin(x),z^2+\cos(y),x^3) \) and \( \mathcal{C} \) be the curve parametrized by \( r(t)=(\sin(t),\cos(t),2\sin(2t)) \) with \( t \) ranging in \([0,2\pi)\). Use the curl theorem to evaluate \( \int_\mathcal{C} F \cdot ds \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the value of the following summation: \[ \sum_{k=0}^{13} \binom{26}{2k}\frac{(2k)!}{k!2^k} \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of distinct equivalence classes for the relation \( \sim \) on \( \mathbb{Z}_n \) defined by \( a \sim b \) if and only if there exists a unit \( k \in U_n = \mathbb{Z}_n^{\times} \) such that \( a = kb \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider the parabola given by the equation \( y^{2} = 2px \) where \( p > 0 \). The parabola has a focus \( F \) and a directrix \( l \). Points \( A \) and \( B \) lie on the parabola such that \( \angle AFB = \frac{\pi}{3} \). Let \( M \) be the midpoint of segment \( AB \), and let \( N \) be the projection of \( M \) onto the directrix \( l \). Determine the maximum value of the ratio \( \frac{|MN|}{|AB|} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find \( \lim_{n \rightarrow \infty} \mathbb{P}(X_1 + \ldots + X_n \leq 0) \), where the independent binary random variables \(X_k\) take values \(\pm 1\) with probabilities \((1 \pm k^{-1/2})/2\) for \(k = 1, 2, \ldots\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In triangle \( ABC \), \( \angle A = 45^\circ \) and \( \angle B = 60^\circ \). The feet of the altitudes from vertices \( A, B, \) and \( C \) are \( A_1, B_1, \) and \( C_1 \) respectively. The midpoints of sides \( AB, BC, \) and \( CA \) are \( C_0, A_0, \) and \( B_0 \) respectively. A circle passing through these midpoints has center \( F \). Consider the first angle trisectors of \( \angle A_0 F A_1, \angle B_0 F B_1, \angle C_0 F C_1 \), which are the rays \( F X, F Y, F Z \) such that \( \angle X F A_1 = 2 \angle A_0 F X \), \( \angle Y F B_1 = 2 \angle B_0 F Y \), and \( \angle Z F C_1 = 2 \angle C_0 F Z \). Find the angles between the rays \( F X, F Y, \) and \( F Z \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $p$ be a prime number such that $p \equiv 3 \pmod{4}$. Determine the remainder when the sum $\sum_{i=1}^{p-1}\left(\frac{i^4+1}{p}\right)$ is divided by $p$, where $\left(\frac{i^4+1}{p}\right)$ denotes the Legendre symbol.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the largest number $k = k(n)$ such that every real $n \times n$ matrix is orthogonally equivalent to a matrix with at least $k$ zero entries.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a continuous and $2\pi$-periodic function $f: \mathbb{R} \to \mathbb{R}$, if for any positive integer $p$ in the range $[0, n-1]$, both integrals $\int_{0}^{2\pi} f(t) \cos(pt)\,dt$ and $\int_{0}^{2\pi} f(t) \sin(pt)\,dt$ are zero, determine the minimum number of roots $f$ has on the interval $[0, 2\pi]$. Provide your answer as an integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Four lines are drawn on a plane such that no two are parallel and no three are concurrent. Lines are then drawn joining the points of intersection of these four lines. How many new lines are formed?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \int_C \frac{e^z}{z} \, dz \), where \( C \) consists of two circles centered at the origin with radii \( R = 2 \) and \( R = 1 \). Determine the value of the integral.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the convolution of the function \( \frac{\sin(t)}{t} \) with the constant function 1. Express your answer as a function or an expression.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider forming a subset $X$ of $\mathbb{R}$ by the following process: for each $x \in \mathbb{R}$, toss a fair coin. If the result is heads, include $x$ in $X$. Assume that the coin tosses are independent. What is the probability that $X$ is a Lebesgue measurable subset of $\mathbb{R}$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Problem 6. Point $A$ on the plane is located at the same distance from all points of intersection of two parabolas given in the Cartesian coordinate system on the plane by the equations $y=-3 x^{2}+2$ and $x=-4 y^{2}+2$. Find this distance.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that $x_1(t) = e^t$ is a solution to the differential equation $$tx''-(2t+1)x'+(t+1)x=0,$$ find a second linearly independent solution.
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Context:
Find a vector $X$ such that $U \cap W = \text{span}\{X\}$, where $U = \langle (1,1,3),(-5,-3,-15) \rangle$ and $W = \langle (4,4,9),(-44,-34,-120) \rangle$. Provide the vector $X$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the sum \( \sum_{n=1}^{101} \frac{\left(\frac{n}{101}\right)^3}{1 - 3\left(\frac{n}{101}\right) + 3\left(\frac{n}{101}\right)^2} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of solutions for the matrix equation \(X^n = \begin{pmatrix}3&6\\ 2&4\end{pmatrix}\) when \(n\) is an odd positive integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the order of the normalizer of the subgroup of elementary abelian 2-groups in the affine general linear group $AGL(3,2)$ within the symmetric group $S_8$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the range of values for $c$ such that the ellipses given by \( \frac{u^2}{(c+\frac{1}{c})^2}+\frac{v^2}{(c-\frac{1}{c})^2}=1 \) cover the entire plane, except for a specific segment on the $u$-axis.
Please reason step by step, and put your final answer within \boxed{}.


Context:
## [ Right triangle with an angle of $30^{\circ}$ ]

In a right triangle $ABC$ with acute angle $A$ equal to $30^{\circ}$, the bisector $BD$ of the other acute angle is drawn.

Find the distance between the centers of the inscribed circles of triangles $ABD$ and $CBD$, if the shorter leg is 1.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit \( \lim_{{n \to \infty}} \frac{n}{\sqrt[n]{n!}} \) without using Stirling's approximation.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \int_V \sin(\sqrt{x^2+y^2}) \, dx \, dy \) where \( V = \{(x,y) : \pi^2 \le x^2+y^2 \le 4\pi^2\} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the value of \( r > 0 \) such that with probability one, for all sufficiently large \( n \), the inequality \( M_n \le r\sqrt{\log n} \) holds, where \( M_n = \max\{|B_t - B_{n-1}| : n - 1 \le t \le n\} \) and \( B_t \) is a standard Brownian motion.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the orthogonal trajectories of the family of curves given by the equation \(x^2 + 2y^2 = k^2\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit \( \lim_{x \to \infty} x[1-F(x)] \), where \( F(x) \) is a cumulative distribution function and \( 1-F(x) \) is the tail distribution function.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the product of all non-zero elements in a finite field \( \mathbb{F}_{q} \), where \( q \) is a power of a prime.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the Galois group of the field extension $F_{p^2}$ over $\mathbb{Z}_p$, where $F_{p^2}$ is the field with $p^2$ elements.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the largest natural number \( n \ge k \) such that any selection of \( k \) numbers from a set of \( n \) natural numbers results in a composite sum.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider a topological space $(X, \tau)$ where $D$ is a closed subspace of $X$. Suppose $D$ is equipotent to $\mathbb{R}$ with a discrete induced topology, and $X$ is normal. Determine the number of continuous functions from a countable subset of $D$ to the interval $[0,1]$. Express your answer in terms of cardinality.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a standard Brownian motion \( B_t \) starting at zero, and a non-negative stochastic process \( X_t \) that satisfies the stochastic differential equation \( dX_t = \frac{1}{X_t} dt + dB_t \), calculate the expected value \( E[\sigma] \), where \( \sigma = \inf \{ t \geq 0 : X_t = 1 \} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the limit poset of the sequence \(\mathbf{0}\rightarrow\mathbf{1}\rightarrow\mathbf{2}\rightarrow\cdots\) in the category \(\mathbf{Posets}\) of posets and monotone functions.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the non-zero constant \( c \) such that the integral \( \int_{-1}^\infty \frac{e^{x/c}}{\sqrt{|x|}(x+2)}dx \) is convergent.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the values of $p$ for which the gradient of the function $A \mapsto \|A\|_p^p$ is locally Lipschitz.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of invertible $3 \times 3$ matrices over $\mathbb{Z}/3\mathbb{Z}$ that commute with the matrix $A = \begin{pmatrix} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 1 \end{pmatrix}$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider a uniform probability distribution over the interval \([0, 1]\). Determine the expected number of trials needed to select numbers from this distribution such that their product is less than \(\frac{1}{e}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
The domain of the function $f(x)=\\log_\frac{1}{2} (\\log_4(\\log_\frac{1}{4} (\\log_{16}(\log_{\frac{1}{16} }{x} ))))$ is an interval of length $\frac{m}{n} $, where $m$ and $n$ are relatively prime positive integers.  What is $m+n$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the total number of Generalized Hermitian Gell-Mann Matrices in dimension $d$, given that they consist of the $h_k^d$ matrices, where $1 \leq k \leq d$, and the $f_{k,j}^d$ matrices, where $1 \leq k, j \leq d$. How many matrices are there in total?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a graph $G$ composed of two cycles $C_x$ and $C_y$ that share an edge, where $x, y \ge 3$, find a formula for the number of spanning trees of $G$. The graph $G$ has $x + y - 1$ edges. Provide the formula in terms of $x$ and $y$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of Boolean functions $f(x_1, x_2, \dots, x_n)$ from $\{0,1\}^n$ to $\{0,1\}$ that can be constructed using only the XOR and AND operations, without using OR.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the binomial expansion of \(\left(\frac{\sqrt{x+1}}{2 \sqrt[4]{x}}\right)^{n}\), the terms are arranged in descending powers of \(x\). If the coefficients of the first two terms form an arithmetic sequence, determine the number of terms in this expansion that have integer exponents of \(x\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the domain of the self-adjoint extension of the Laplace-Beltrami operator $-\Delta$ when it is considered as an unbounded operator on $L^2(M)$, where $M$ is a complete Riemannian manifold.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the derivative of \( \text{Tr}(X^TP^TPX) \) with respect to the matrix \( P \), where \( X \) and \( P \) are real matrices.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the flux of the vector field \( F = (x^3, y^3, z^3) \) through the surface of the sphere defined by \( x^2 + y^2 + z^2 = x \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine if the random variable $T$, which follows a Bernoulli distribution with parameter $p \in (0,1)$ and is defined as:

$$T= \left\{ \begin{aligned} &1 &\text{ with probability of }1-p,\\ &1-\frac{1}{p} &\text{ with probability of }p. \end{aligned} \right.$$

is a sub-Gaussian random variable. If it is, find the variance proxy $\sigma^2$ in terms of $p$. Provide your answer for $\sigma^2$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the distribution of the random variable $X = W(s) + W(s + t)$, where $W(t)$ is a Wiener process with $W(0) = 0$. Assume that for $t \le s$, $W(s) - W(t)$ follows a normal distribution with mean $0$ and variance $(s - t)$. Provide the distribution of $X$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider a scenario with $n$ people, each having a unique ball. They place their balls into a box, and the balls are mixed. Each person then draws a ball from the box. If a person draws their own ball, they leave; otherwise, they participate in the next round. What is the expected number of rounds until everyone has retrieved their own ball?
Please reason step by step, and put your final answer within \boxed{}.


Context:
The heights of certain plants are normally distributed with a mean of 110 cm and a variance of 1050 cm². Two plants are chosen at random. What is the probability that the height of one plant is at least 1.5 times the height of the other plant?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the radical of the ideal \((x^3-y^6, xy-y^3)\) in \(\mathbb{C}[x,y]\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine whether there exists a positive constant \( c \in \mathbb{R} \) such that for any nonzero vectors \( a, b \in \mathbb{R}^n \), the inequality \[ \frac{1}{2}\|a-b\|^2 - \frac{1}{\pi}\sqrt{\|a\|^2\|b\|^2 - \langle a,b\rangle^2} + \langle a,b\rangle\frac{\cos^{-1}(\rho)}{\pi} \geq c\|a-b\|^2 \] holds, where \( \rho = \frac{\langle a,b\rangle}{\|a\|\,\|b\|} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Identify the elements in the quotient ring $k[x,y]/(x{y^2})$ that are nilpotent but not units.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Using Taylor's inequality, find an upper bound on the error of the approximation \( f(x) = x\ln(x) \approx T_3(x) \) on the interval \([1/2, 3/2]\), where \( T_3(x) \) is the third-degree Taylor polynomial about \( x=1 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the triple integral \( \int\limits_0^5 \int\limits_{\sqrt{25-x^2}}^{-\sqrt{25-x^2}} \int\limits_{\sqrt{25-x^2-z^2}}^{-\sqrt{25-x^2-z^2}} \frac{1}{x^2+y^2+z^2} \,\mathrm dy\,\mathrm dz\,\mathrm dx \) by converting to spherical coordinates.
Please reason step by step, and put your final answer within \boxed{}.


Context:
What minimal number of numbers from the set \( \{ 1,2,\ldots ,{1982}\} \) should be deleted to provide the property:
 none of the remained numbers equals to the product of two other remained numbers?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( f(z) \) be a function that is analytic in the unit disk \( U = \{z:|z|<1\} \) and continuous on the closed unit disk. Suppose that \( \frac{f(z)}{z^2} \) can be extended to be analytic in the open unit disk \( U \), including at the origin. Given that \( |f(z)| \leq 6 \) in the closed unit disk, determine the maximal possible value of \( f(0.4 + 0.5i) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
The first three terms of a geometric sequence are the integers \( a,{720} \) , and \( b \) , where \( a < {720} < b \) . What is the sum of the
 digits of the least possible value of \( b \) ?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $\pi:Y\rightarrow \mathbb{P}^3$ be the blow-up of two points $p,q\in\mathbb{P}^3$, and then of the strict transform of the line $L$ spanned by them. Let $E_p,E_q, E_{p,q}$ be the exceptional divisors over $p,q$, and $L$, respectively. $E_p$ and $E_q$ are isomorphic to the blow-up of a point in $\mathbb{P}^2$, and $E_{p,q}\cong \mathbb{P}^1\times\mathbb{P}^1$. Let $\sigma_{p,q}\subset E_{p,q}$ be the class of the $\mathbb{P}^1$ which is not contracted by $\pi$. Given that $E_p\cdot \sigma_{p,q} = E_q\cdot\sigma_{p,q} = 1$, find the intersection number $E_{p,q}\cdot \sigma_{p,q}$. Provide your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral: $$I=\int_{-\infty}^{\infty}\arcsin\left(\frac{1}{\cosh x}\right)\,dx.$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Simplify the expression: $$\lim_{x\to0}\left(\prod^{\frac{1}{x}-1}_{i=1}\frac{1}{\sec\frac{xi\pi}{2}}\right)^x$$
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Context:
Find the Fréchet derivative of the functional $\|Au-f\|^2$, where $A$ is a bounded linear operator on an infinite-dimensional Hilbert space. Express your answer in terms of $A$, $u$, and $f$.
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Context:
How much larger is 3↑↑↑3 compared to itself?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given two reinforced concrete buildings A and B in a seismic region, an impending earthquake might be strong (S), moderate (M), or weak (W) with probabilities P(S) = 0.02, P(M) = 0.2, and P(W) = 0.78. The probabilities of failure for each building if these earthquakes occur are 0.20, 0.05, and 0.01, respectively. If building A fails, the probability that building B will also fail is 0.50, 0.15, and 0.02 for the three types of earthquakes. Calculate the probability that the earthquake was not strong given that building A has failed and building B has survived.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral: \[ f(u) = \int_{-\infty}^{\infty} \frac{e^{-itu}}{\sqrt{1+t^2}} \, dt \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the coefficient of \(x^{30}\) in the expansion of \(\left(\frac{1-x^{11}}{1-x}\right)^3\left(\frac{1-x^{21}}{1-x}\right)\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the chemical reaction where compounds $A$ and $B$ combine to form compound $X$:

$$A + B \xrightarrow{k} X$$

If $2 \, \rm{g}$ of $A$ and $1 \, \rm{g}$ of $B$ yield $3 \, \rm{g}$ of $X$, the differential equation describing the reaction is:

$$\frac{dx}{dt} = k \left(a - \frac{2}{3}x\right)\left(b - \frac{1}{3}x\right)$$

where $a$ and $b$ are the initial amounts of $A$ and $B$ (in grams), $x(t)$ is the amount of $X$ at time $t$ in minutes, and $k$ is the reaction rate (per minute per gram). Assume that $x(0) = 0$. Express the solution in the form:

$$\ln \left(\frac{a - \frac{2}{3}x(t)}{b - \frac{1}{3}x(t)} \cdot \frac{b}{a}\right) = ct$$

where $c$ is a constant depending on $k$, $a$, and $b$. If $a = 15 \, \rm{g}$, $b = 20 \, \rm{g}$, and $15 \, \rm{g}$ of $X$ is formed after $10$ minutes, find the amount of $X$ after $20$ minutes.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the hyperbola $H: \frac{x^2}{9} - \frac{y^2}{16} = 1$, a tangent is drawn at any point $P$ on $H$. This tangent intersects another hyperbola $K: \frac{x^2}{m^2} - \frac{y^2}{n^2} = k$ at points $A$ and $B$. If $P$ is always the midpoint of $AB$, find the floor of the sum of the possible eccentricities of $K$, where $k = \pm 1$. \( \text{Provide your answer as an integer.} \)
Please reason step by step, and put your final answer within \boxed{}.


Context:
A trapezoid has side lengths $3, 5, 7,$ and $11$. The sum of all the possible areas of the trapezoid can be written in the form of $r_1 \sqrt{n_1} + r_2 \sqrt{n_2} + r_3$, where $r_1, r_2,$ and $r_3$ are rational numbers and $n_1$ and $n_2$ are positive integers not divisible by the square of a prime. What is the greatest integer less than or equal to
\[r_1 + r_2 + r_3 + n_1 + n_2?\]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given 5 observations: 4, 3, 5, 2, 6 from a \(\text{binomial}(10, p)\) random variable, find an unbiased estimate of \((1+p)^{10}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many ways can we form a group with an odd number of members (plural) from $99$ people? Express your answer in the form $a^b + c$, where $a, b$, and $c$ are integers and $a$ is prime.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $S$ be the set of all polynomials of the form $z^3+az^2+bz+c$, where $a$, $b$, and $c$ are integers. Find the number of polynomials in $S$ such that each of its roots $z$ satisfies either $\left\lvert z \right\rvert = 20$ or $\left\lvert z \right\rvert = 13$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
$\triangle ABC$ satisfies $\angle A=60^{\circ}$. Call its circumcenter and orthocenter $O, H$, respectively. Let $M$ be a point on the segment $BH$, then choose a point $N$ on the line $CH$ such that $H$ lies between $C, N$, and $\overline{BM}=\overline{CN}$. Find all possible value of
\[\frac{\overline{MH}+\overline{NH}}{\overline{OH}}\]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \int_0^{2\pi}\frac{\cos^2 3\theta\, d\theta}{5-4\cos 2\theta} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given in $\triangle ABC$, $3\sin ^{2}B+7\sin ^{2}C=2\sin A\sin B\sin C+2\sin ^{2}A$, find $\sin \left(A+\dfrac{\pi}{4} \right)=$\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.


Context:
For any natural number \(n \ge 3\), find the maximum number of odd entries in a matrix \(B\in\mathcal{M}_{n}\left(\mathbb{N}\right)\) such that \(\det(B)=1\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the lower bound for the expression \(-\frac{1}{4}\sqrt{(1-2a+x)^2-8(-3-4a-3x-2ax)}-\frac{1}{4}(1-2a+x)\) given that \(x \leq 0\) and \(a > 0\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the number of ordered tuples $\left(C,A,M,B\right)$ of non-negative integers such that\[C!+C!+A!+M!=B!\]
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Context:
Consider the family of holomorphic functions $\mathcal F = \{f \text{ holomorphic in } \Bbb D \text{ with } f(\Bbb D)\subset\Bbb D, f\left(\frac{1}{2}\right)=f'\left(\frac{1}{2}\right)=0\}$, where $\Bbb D$ is the unit disc. Compute $\sup_{f\in\mathcal F}|f(0)|$. \( \boxed{?} \)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit \( \lim_{x \to 1^-} \sqrt{1-x} \sum_{n=0}^{\infty} x^{n^3} \).
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Context:
In a pyramid \(S-ABC\) with a height of \(SO = 3\) and a square base with side length 6, a perpendicular is drawn from point A to the opposite face \(SBC\), with the foot of the perpendicular being \(O'\). On the line segment \(AO'\), a point \(P\) is chosen such that \(\frac{AP}{PO'} = 8\). Determine the area of the cross-section through point \(P\) that is parallel to the base.
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Context:
已知集合 A=\{1,2,\cdots,10\}. 定义函数 f:A\rightarrow A，满足：(1) 对任意 x,y\in A，若 x\leq y，则 f(x)\leq f(y)；(2) f(3)=4. 这样的函数 f 的个数为__________.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral: \[ \int_0^2 \left(\sum_{n=0}^\infty \frac{(x-1)^{2n+1}}{2n+1}\right) dx \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $g(x),g:\mathbb{R}\to \mathbb{R}$ be a non-constant continuous function satisfying the equation $$\left(e^{2x}-1\right)g(x)=\left(e^x-1\right)g(2x).$$ If $g'(0)=2$, find the value of $$\lim_{x\to \infty}\left(\frac{g(x)}{x}\right)^{2/x}.$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
An ant starts at one vertex of a regular dodecagon and wants to reach the opposite vertex by moving to adjacent vertices. Let \( p_n \) represent the number of such paths with length \( n \). Compute the sum \( p_1 + p_2 + p_3 + \ldots + p_{12} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider a sample $X_1, X_2, \ldots, X_{100}$ from a Poisson distribution with parameter $\lambda = 3.5$. Calculate the approximate probability that the total sum $T = \sum_{i=1}^{100} X_i$ exceeds 360.
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Context:
Evaluate the integral \( I = \oint_{|z|=1} \frac{\cos(z)}{z(e^{z}-1)} \, dz \) along the unit circle.
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Context:
Determine the structure of the quotient group $G/G'$ for the group $G$ defined by the presentation $\langle a, b \mid a^6 = 1, b^2 = a^3, bab^{-1} = a^{-1} \rangle$, where the derived subgroup is $G' = \{e, a^2, a^4\}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the residue of the function \( \frac{1}{e^{\frac{1}{z}}-1} \) at the point \( z=0 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of homomorphisms $\psi: D_4 \longrightarrow \mathbb{Z}_4$, where $D_4$ is the dihedral group of order 8.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a connected network with 2008 computers, no two cycles share a common vertex. A hacker and an administrator play a game on this network. On the 1st move, the hacker selects and hacks one computer. On the 2nd move, the administrator selects and protects another computer. For every 2k+1th move (k > 0), the hacker hacks a computer that is directly connected to a previously hacked computer and is not protected. For every 2k+2th move, the administrator protects a computer that is directly connected to a previously protected computer and is not hacked. The game ends when neither can make a move. What is the maximum number of computers the hacker can guarantee to hack by the end of the game?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the values of \( \alpha \) for which the limit \( \lim_{n\rightarrow \infty}\int_{1}^{n}\frac{\sin{x^2}}{x^{\alpha}}dx \) is finite.
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Context:
Call an ordered triple of sets $(A, B, C)$ nice if $|A \cap B| = |B \cap C| = |C \cap A| = 2$ and $|A \cap B \cap C| = 0$. How many ordered triples of subsets of $\{1, 2, · · · , 9\}$ are nice?
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Context:
Determine the bases for which there exists an integer N such that there are no N-digit prime numbers.
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Context:
Find the smallest constant \( M \) such that the inequality \[ \frac{x_{1}}{x_{1}+x_{2}}+\frac{x_{2}}{x_{2}+x_{1}}+\frac{x_{3}}{x_{3}+x_{4}}+\cdots+\frac{x_{2009}}{x_{2009}+x_{1}} < M \] holds for all positive real numbers \( x_{1}, x_{2}, \ldots, x_{2009} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the fundamental group of the space \((S^1 \times S^1)/((S^1 \times \{x\})\), where \(x\) is a point in \(S^1\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the work done by the vector field \( \vec{G}=(x^2y+z^2)\vec{i}+(xy^2-z)\vec{j}+(yz+x)\vec{k} \) in moving a particle along the curve formed by the intersection of the plane \( x+y+z=1 \) and the cylinder \( x^2+y^2=4 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the series: \[ \sum_{k=1}^{\infty} (-1)^{k+1}\frac{\pi^{6k}}{(6k)!} \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $F$ be a field of order $32$. Determine the number of non-zero solutions $(a,b) \in F \times F$ to the equation $x^2 + xy + y^2 = 0$. Provide your answer as a single integer.
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Context:
In an alphabet with $n > 1$ letters, a word is defined as any finite sequence of letters where any two consecutive letters are different. A word is called "good" if it is not possible to delete all but four letters from it to obtain a sequence of the form $a a b b$, where $a$ and $b$ are different letters. Determine the maximum possible number of letters in a "good" word.
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Context:
Find the sum of the infinite series \(\sum_{n=1}^{\infty} \frac{f(n)}{n(n+1)}\), where \(f(n)\) is the number of ones in the binary representation of the integer \(n\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the spectrum of a bounded self-adjoint operator $S: H \rightarrow H$ on a Hilbert space $H$ that satisfies the equation $g(S) = 0$, where $g(z) = z^4 + z^3 + z^2 + z + 1$. Provide your answer as a set of values.
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Context:
A unit square is divided into rectangles, each colored either yellow or blue. For a blue rectangle, the number written inside it is the rectangle's width divided by its height. For a yellow rectangle, the number is the rectangle's height divided by its width. Let $x$ be the sum of all these numbers. If the total area of the blue rectangles equals the total area of the yellow rectangles, find the smallest possible value of $x$. \( \text{(Express your answer as a single number.)} \)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \int_\gamma \frac{dz}{(z-\frac{1}{2}-i)(z-1-\frac{3i}{2})(z-1-\frac{i}{2})(z-\frac{3}{2}-i)} \), where \( \gamma \) is the polygon with vertices \( 0, 2, 2+2i, 2i, 0 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many positive three-digit integers $\\underline{a}\\underline{b}\\underline{c}$ can represent a valid date in $2013$, where either $a$ corresponds to a month and $\\underline{b}\\underline{c}$  corresponds to the day in that month, or $\\underline{a}\\underline{b}$ corresponds to a month and $c$ corresponds to the day?  For example, 202 is a valid representation for February 2nd, and 121 could represent either January 21st or December 1st.  \n\n(Note:  During the actual test they had to write the number of days in each month so don feel bad if you have to google that .
Please reason step by step, and put your final answer within \boxed{}.


Context:
Solve for \( t \) in the equation:
\[ 5000 = 21t - 111.37e^{-\frac{12t}{6.5625}} + 1207.12 \]
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Context:
Let \( N \) be the set \( \{ 1,2,\ldots ,{2018}\} \) . For each subset \( A \) of \( N \) with exactly 1009 elements,define
 \( f\left( A\right)  = \mathop{\sum }\limits_{{i \in  A}}i\mathop{\sum }\limits_{{j \in  N,j \notin  A}}j. \)
 If \( \mathbb{E}\left\lbrack  {f\left( A\right) }\right\rbrack \) is the expected value of \( f\left( A\right) \) as \( A \) ranges over all the possible subsets of \( N \) with exactly 1009 elements, find
 remainder when the sum of the distinct prime factors of \( \mathbb{E}\left\lbrack  {f\left( A\right) }\right\rbrack \) is divided by 1000 .
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the limit \(\lim_{n\to \infty} \frac{\ln^2n}{n}\sum_{k=2}^{n-2}\frac{1}{\ln k\ln(n-k)}\).
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Context:
Find the range of values for $a$ such that the system of inequalities \( \left\{\begin{array}{l}x^{2}-x+a-a^{2}<0 \\ x+2a>1\end{array}\right. \) has exactly two integer solutions for $x$. \( \boxed{} \)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of non-decreasing tuples of integers \((a_1, a_2, \dots, a_{16})\) such that \(0 \leq a_i \leq 16\) for all \(i\), and the sum of all \(a_i\) is even.
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Context:
Find the volume of the region in \( \mathbb{R}^3 \) described by the inequalities \( x^2 + y^2 + z^2 \le 1 \) and \( |y| \le x^2 + y^2 \).
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Context:
Determine the set of values of $a > 0$ for which the function \( f(x) = \frac{\sin([x])}{a} + \frac{\cos([x])}{a} \), where \([ \cdot ]\) denotes the fractional part of \(x\), can attain its maximum value.
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Context:
The measures of the interior angles of a convex polygon of $n$ sides are in arithmetic progression. If the common difference is $5^{\circ}$ and the largest angle is $160^{\circ}$, then $n$ equals:
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Context:
If the eccentricity of the hyperbola $x^{2}+ky^{2}=1$ is $2$, then the value of the real number $k$ is ______.
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Context:
Find the derivative of the expression $a^{T}Xb$ with respect to the vector $b$, where $a$ and $b$ are $d$-dimensional vectors and $X$ is a $d \times d$ matrix.
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Context:
Two differentiable functions $f(x)$ and $g(x)$ are defined on the interval $[0, 2]$. Given that $f''(x) - g''(x) = 0$, $f'(A) = 2$, $g'(A) = 4$, $f(B) = 3$, and $g(B) = 9$, find the value of $f\left(\frac{3}{2}\right) - g\left(\frac{3}{2}\right)$. \( \boxed{?} \)
Please reason step by step, and put your final answer within \boxed{}.


Context:
If the inequality $mx^2 + 4mx - 4 < 0$ holds for all real numbers $x$, then the range of the real number $m$ is __________.
Please reason step by step, and put your final answer within \boxed{}.


Context:
We choose 100 points in the coordinate plane. Let  $N$  be the number of triples  $(A,B,C)$  of distinct chosen points such that  $A$  and  $B$  have the same  $y$ -coordinate, and  $B$  and  $C$  have the same  $x$ -coordinate. Find the greatest value that  $N$  can attain considering all possible ways to choose the points.
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Context:
Crisp All, a basketball player, is *dropping dimes* and nickels on a number line. Crisp drops a dime on every positive multiple of  $10$ , and a nickel on every multiple of  $5$  that is not a multiple of  $10$ . Crisp then starts at  $0$ . Every second, he has a  $\frac{2}{3}$  chance of jumping from his current location  $x$  to  $x+3$ , and a  $\frac{1}{3}$  chance of jumping from his current location  $x$  to  $x+7$ . When Crisp jumps on either a dime or a nickel, he stops jumping. What is the probability that Crisp *stops on a dime*?
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Context:
For how many different values of integer $n$, one can find $n$ different lines in the plane such that each line intersects with exacly $2004$ of other lines?
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Context:
Evaluate the limit: \[ \lim_{n \to \infty} n^2 \int_0^1 \frac{1}{(1+x^2)^n} \, dx. \]
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Context:
In measuring the sides of a rectangle, one side is taken 6% in excess, and the other in deficit. The error percent in the area calculated from these measurements is 0.7%. What is the percentage in deficit for the other side?
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Context:
Given that \(X_1\) and \(X_2\) are independent and identically distributed discrete random variables, the moment generating function of \(X_1 + X_2\) is given by:
\[ M(t) = 0.01e^{-2t} + 0.15e^{-t} + 0.5925 + 0.225e^{t} + 0.0225e^{2t} \]
Find the probability \(P[X < 0]\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a polynomial \( P(x) = a_{2n} x^{2n} + a_{2n-1} x^{2n-1} + \ldots + a_1 x + a_0 \) where each coefficient \( a_i \) belongs to the interval \([100,101]\), what is the smallest natural number \( n \) such that this polynomial can have a real root?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In triangle \(ABC\), where \(AB = 9\), \(BC = 10\), and \(CA = 17\), let \(\Omega\) be its circumcircle. Points \(A_1\), \(B_1\), and \(C_1\) are diametrically opposite to \(A\), \(B\), and \(C\) on \(\Omega\), respectively. Calculate the area of the convex hexagon with vertices \(A, B, C, A_1, B_1, C_1\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( r_{1}, \ldots, r_{n} \) be the distinct real zeros of the equation
\[ x^{8} - 14x^{4} - 8x^{3} - x^{2} + 1 = 0. \]
Evaluate \( r_{1}^{2} + \cdots + r_{n}^{2} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Express the total installed capacity of 16,780,000 kilowatts in scientific notation in the format a × 10^n, where 1 ≤ a < 10 and n is an integer.
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Context:
Calculate the fundamental group of the space $S^3 - S^1$. Provide your answer as a well-defined mathematical group.
Please reason step by step, and put your final answer within \boxed{}.


Context:
考虑所有满足以下两个条件的实数序
列 $x_{0}, x_{1}, \cdots, x_{100}$ :
(1) $x_{0}=0$;
（2）对任意整数 $i(1 \leqslant i \leqslant 100)$ 有
$1 \leqslant x_{i}-x_{i-1} \leqslant 2$.
求最大的正整数 $k \leqslant 100$ ，使得对任意这样的序列 $x_{0}, x_{1}, \cdots, x_{100}$, 均有
$$
x_{k}+x_{k+1}+\cdots+x_{100} \geqslant x_{0}+x_{1}+\cdots+x_{k-1}
$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit \(\lim_{n\to\infty}\frac{n!}{x^n}\) using the Stolz-Cesaro theorem, where \(x\) is a real number such that \(|x|>1\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the cardinality of the set $F$ of all entire functions $f$ such that $f(x) = |x^k|$ for all $x \in (-1, 1)$, where $K$ is an odd integer $\geq 1$. What is the cardinality of $F$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $f(n)=\displaystyle\sum_{k=2}^\infty \dfrac{1}{k^n\cdot k!}.$ Calculate $\displaystyle\sum_{n=2}^\infty f(n)$.
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Context:
Determine the maximum length of a roof that can be obtained by stacking an unlimited number of identical rectangular bricks with some offset so that they do not fall.
Please reason step by step, and put your final answer within \boxed{}.


Context:
There is a graph with 30 vertices. If any of 26 of its vertices with their outgoiing edges are deleted, then the remained graph is a connected graph with 4 vertices.\nWhat is the smallest number of the edges in the initial graph with 30 vertices?
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Context:
Determine the number of possible values for the tuple \((a, b, c, d)\) of real numbers that satisfy the following equations: \(abc = d\), \(bcd = a\), \(cda = b\), and \(dab = c\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
The necessary and sufficient condition for the lines $ax+2y+1=0$ and $3x+(a-1)y+1=0$ to be parallel is "$a=$ ______".
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the number of elements in the intersection of $A_7$ and the normalizer $N_G(H)$, where $G = S_7$ and $H = \langle (1\ 2\ 3\ 4\ 5\ 6\ 7) \rangle$. Given that $|N_G(H)| = 42$ and $|A_7| = 2520$, use the internal product formula to determine $|A_7 \cap N_G(H)|$. Provide your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: $$\lim_{n\to\infty}\frac{\int_1^{n^{1/n}} y^{n^2}(\arctan y) ny^{n-1}\,dy}{n^n}$$
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Context:
Determine the set of complex values of $z$ for which the series \( \sum_{n=0}^{\infty}\sum_{k=0}^{\infty}\dfrac{(-z)^{n+2nk}}{1+z^{2n}} \) is convergent.
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Context:
For the sequence defined by \(x_1=\frac{1}{2}\) and \(x_{n+1}=x_n-x_n^2\), determine if there exists a value of \(\alpha\) such that \(\lim\limits_{n\to\infty}n^\alpha \cdot \frac{1}{x_n} \neq 0\).
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Context:
Along the diagonals $AC$ and $B_{1}D_{1}$ of the base of a cube $ABCDEFGH$ with edge length $a$, two ants, Gosha and Lesha, are crawling. They started moving simultaneously from points $A$ and $B_{1}$ respectively at constant speeds, with Lesha's speed being three times the speed of Gosha, and finished when Lesha reached point $D_{1}$. What was the minimum distance between Gosha and Lesha during their movement?
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Context:
Consider a matrix of size $n\\times n$ whose entries are real numbers of absolute value not exceeding $1$. The sum of all entries of the matrix is $0$. Let $n$ be an even positive integer. Determine the least number $C$ such that every such matrix necessarily has a row or a column with the sum of its entries not exceeding $C$ in absolute value.\n\n[i]Proposed by Marcin Kuczma, Poland[\/i]
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Context:
Evaluate the double integral \( \iint_D \left|x^2 + y^2 - 2y\right| \, d\sigma \), where \( D = \{(x, y) \mid x^2 + y^2 \le 4\} \).
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Context:
 Square $ABCD$ has side length $2$. Square $ACEF$ is drawn such that $B$ lies inside square $ACEF$. Compute the area of pentagon $AFECD$.
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Context:
Given a Markov chain $(X_n)_{n\geq1}$ with state space $S=\{A,B,C,D,E \}$ and the transition matrix
$$ P = \left( \begin{matrix} 0.6 & 0.4 & 0 & 0 & 0 \\ 0.3 & 0.7 & 0 & 0 & 0 \\ 0.2 & 0 & 0.4 & 0 & 0.4 \\ 0.2 & 0.2 & 0.2 & 0.2 & 0.2 \\ 0 & 0 & 0 & 0 & 1 \end{matrix} \right) $$
compute $\lim_{n \to \infty} P(X_n=A|X_0=C)$.
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Context:
Given the following transition matrix for a Markov chain with state space $S=\{1,2,3,4,5\}$:
\[
\begin{pmatrix}
0 & 0 & 1 & 0 & 0 \\
0 & 0 & \frac{4}{5} & \frac{1}{5} & 0 \\
0 & \frac{1}{6} & \frac{2}{3} & 0 & \frac{1}{6} \\
0 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & 1
\end{pmatrix}
\]
Calculate the probability that the Markov chain is absorbed at state 4.
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Context:
Find \(\lim_{x \to \infty} x^2\big(\ln(x\cot^{-1}(x))\).
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Context:
Find a polynomial \( g(t) \) such that the linear map \( g(T) \) has the characteristic polynomial \( p(t) = (11-t)^r(17-t)^{n-r} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the permutation $x$ such that for $y = (1274)(356)$, the equation $x^{-1}yx = (254)(1736)$ holds.
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Context:
Find the harmonic conjugate of the function \( v(r,\theta) = -\ln r \) in polar coordinates, where \( r \) and \( \theta \) are the radial and angular coordinates, respectively.
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Context:
On the sides $AB$ and $BC$ of triangle $ABC$, points $K$ and $M$ are chosen such that the quadrilaterals $AKMC$ and $KBMN$ are cyclic , where $N = AM \cap CK$ . If these quads have the same circumradii, find$\angle ABC$
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Context:
Calculate the sum \( \sum_{k \in S(m,n)} \varphi(k) \) where \( S(a,b) = \{ k \in \mathbb{Z} \mid a \equiv b \pmod{k} \} \) and \( \varphi \) is Euler's totient function.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the line integral \( \int_{C} \frac{x \, dy - y \, dx}{x^{2}+y^{2}} \) along the oriented broken line \( C \) with vertices \((2,-2)\), \((4,4)\), \((-5,5)\) oriented counterclockwise.
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Context:
Cheburashka has a set of 36 stones with masses of 1 g, 2 g, ..., 36 g, and Shapoklyak has super glue, with which one drop can glue two stones into one (it is possible to glue three stones with two drops and so on). Shapoklyak wants to glue the stones in such a way that Cheburashka cannot choose one or more stones with a total mass of 37 g from the resulting set. What is the minimum number of glue drops she needs to carry out her intention?
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Context:
Given circle \( O: x^{2} + y^{2} = 4 \) and curve \( C: y = 3|x - t| \). Points \( A(m, n) \) and \( B(s, p) \) \((m, n, s, p \in \mathbb{N}^{*})\) are on curve \( C \). It is required that the ratio of the distance from any point on circle \( O \) to point \( A \) to the distance to point \( B \) is a constant \( k \) (\( k > 1 \)). Find the value of \( t \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the cardinality of the set \( A = \{ f: \mathbb{R} \to \mathbb{R} \mid f \text{ is continuous and } f(\mathbb{Q}) \subset \mathbb{Q} \} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the coefficient of \(x^{30}\) in the expansion of the generating function \(\frac{(1-x^{10})^4}{(1-x)^4}\) using the Negative Binomial Series.
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Context:
Let \( a \) be a root of the equation \( x^{11} + x^{7} + x^{3} = 1 \). Determine all natural numbers \( n \) such that \( a^{4} + a^{3} = a^{n} + 1 \).
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Context:
If three, standard, 6-faced dice are rolled, what is the probability that the sum of the face up integers is 16?The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.
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Context:
Evaluate the limit: \[ L = \lim_{n \rightarrow \infty} \prod_{r=3}^{n} \frac{r^3 - 8}{r^3 + 8} \]
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Context:
For how many ordered pairs of positive integers $(x, y)$, with $x < y$, is the harmonic mean of $x$ and $y$ equal to $9^{15}$?
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Context:
The perimeter of the pentagonal square ABCDE is 2000 meters. Person A and person B start from points A and C, respectively, at the same time and walk around the square in the direction A→B→C→D→E→A→… Person A walks at a speed of 50 meters per minute, and person B walks at a speed of 46 meters per minute. After how many minutes will person A and person B be walking on the same side for the first time?
Please reason step by step, and put your final answer within \boxed{}.