Context:
A perfect number is a positive integer that is equal to the sum of its proper positive divisors, that is, the sum of its positive divisors excluding the number itself. For example, $28$ is a perfect number because $1 + 2 + 4 + 7 + 14 = 28$. Let $n_i$ denote the ith smallest perfect number. Define$$f(x) =\sum_{i|n_x}\sum_{j|n_i}\frac{1}{j}$$(where $\sum_{i|n_x}$ means we sum over all positive integers $i$ that are divisors of $n_x$). Compute $f(2)$, given there are at least $50 $perfect numbers.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A regular triangular prism $ABC A_{1} B_{1} C_{1}$ with base $ABC$ and lateral edges $A A_{1}, B B_{1}, C C_{1}$ is inscribed in a sphere of radius 3. Given that segment $CD$ is a diameter of this sphere and $A D = 2 \sqrt{6}$, find the volume of the prism.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( \{z_1, z_2, z_3, z_4, z_5\} \) be the solutions of the equation \( z^5 = 2i \). Calculate the value of \( x^2 = \frac{1}{z_1} + \frac{1}{z_2} + \frac{1}{z_3} + \frac{1}{z_4} + \frac{1}{z_5} + i \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the joint moment generating function \(M_{X,Y}(t,u) = \exp[2t+3u+t^2+atu+2u^2]\), determine the value of \(a\) that makes the random variables \(X + 2Y\) and \(2X - Y\) independent.
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Context:
Given the sequence defined by \(x_{n+1} = 3x_n + \sqrt{8x^2_n + 2}\), if it is known that \(x_{2017} + x_{2023} = 990\), find the value of \(x_{2020}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the fraction of permutations of the letters A through G that, when inserted into an initially empty binary search tree, will produce the same structure as the binary search tree derived from the sequence A, E, F, G, B, D, C.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( f: \Bbb{N} \to \Bbb{N} \) be a strictly increasing function such that \( f(f(n)) = 3n \) for all \( n \in \Bbb{N} \). Find \( f(2001) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine an upper bound for the tree width of an \(n \times n\) grid graph using the elimination method, where vertices are removed and their neighbors are connected.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of distinct Jordan forms for a $6 \times 6$ matrix with the characteristic polynomial $(x+2)^4 (x-1)^2$, considering the ordering of both eigenvalues and blocks.
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Context:
Let \( x = e^{2ir\pi/7} \) where \( r \equiv 0, \pm1, \pm2, \pm3 \pmod{7} \). Evaluate the sum:
\[ \sum_{r=1}^6 \frac{x^r}{1+x^{2r}} \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Seven thieves steal diamonds and hide in the jungle. During the night, two thieves try to divide the diamonds between themselves but find one extra diamond. They wake a third thief, but still have one extra diamond. This pattern continues until they wake the seventh thief, at which point the diamonds are evenly distributed. How many diamonds did they steal in total?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate \(x + 2y + 3z\) given that the following limit is a finite non-zero number:

\[\lim_{h \to 0}\frac { \int _{ 2 }^{ h+2 }{ g(t)dt-h(xg(2)+yg(2+h/3)+zg(2+h)) } }{ { h }^{ 4 } }\]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the line integral \( \int_{C} f \, dl \) along the curve \( C \), where \( C \) is the intersection of the paraboloid \( z = x^2 + y^2 \) and the plane \( 2x + 2y - z + 2 = 0 \). The curve starts at point \((3,1,10)\) and ends at \((1,3,10)\). The function is given by \( f(x,y,z) = z - y^2 - 2x - 1 \). The projection of \( C \) in the \( xy \)-plane is the circle \((x-1)^2 + (y-1)^2 = 4\), and the parametric representation of \( C \) is:
\[
\begin{split}
x &= 1 + 2\cos t \\
y &= 1 + 2\sin t \\
z &= 6 + 4\cos t + 4\sin t
\end{split}
\]
with \( t \) ranging from the angle corresponding to point \((3,1,10)\) to the angle corresponding to point \((1,3,10)\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the distance from the vector \((1,0,0,1)\) to the subspace \(L = (\text{Sp}\{(1,0,2,1),(3,1,4,4),(1,1,0,2)\})^{\perp}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the maximum distance between any two norms on \(\mathbb{R}^n\) and a Euclidean norm.
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Context:
Find the kernel of the homomorphism \( A[Y] \to A_a \) defined by sending \( Y \) to \( 1/a \), where \( a \) is an element of \( A \). Determine the ideal \( I \) in \( A[Y] \) that forms the kernel of this homomorphism.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral: $$\int_1^{+\infty}\frac{e^{2^u}-e^{-2^u}}{2u\ln(2)}\,\d u$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the side length of the largest cube contained in the region\n\n\\[ \\{(x, y, z) : x^2+y^2+z^2 \\le 25 \\text{ and } x, y \\ge 0 \\} \\]\n\nof three-dimensional space.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $a,b,c,d,e$ be nonnegative integers such that $625a+250b+100c+40d+16e=15^3$.  What is the maximum possible value of $a+b+c+d+e$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Carol was sending out birthday invitations to her friends. Each package of invitations she bought had 5 invitations in it and she bought 3 packs. If she wants to invite 23 friends, how many extra invitations will she need to buy?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A necklace consists of 100 blue and several red beads. It is known that every segment of the necklace containing 8 blue beads contain also at least 5 red beads. What minimum number of red beads can be in the necklace?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the maximum error bound for the Taylor polynomial approximation of the function \( f(x) = x\sin(3x) \) at \( a = 0 \) with \( n = 4 \) over the interval \(-0.7 \le x \le 0.7\). The fourth-degree Taylor polynomial is given by \( T_4(x) = 3x^2 - \frac{9x^4}{2} \). Use the fifth derivative of \( f(x) \), which is \( 81(3x\cos(3x) + 5\sin(3x)) \), to find the error bound \( |R_4(x)| \le \) some error value.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Identify a nonabelian group that contains an element which is its own inverse and is not the identity element. Provide an example without using the dihedral group.
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Context:
Find the sum of all positive integers $B$ such that $(111)_B = (aabbcc)_6$, where $a, b, c$ represent distinct base $6$ digits, with $a \neq 0$.
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Context:
In an election with five candidates using a ranked-choice voting system, Candidate A secured 38% of the first-choice votes, Candidate B had 28%, and the remaining candidates (C, D, and E) each secured 11%, 15%, and 8% respectively. If after the first round, the candidate with the fewest votes (Candidate E) was eliminated and their votes were redistributed according to voters' second choices, Candidates A and B received an additional 5% each, while candidate D secured another 2% of the redistributed votes. In the end, the winning candidate had a total of 7,320 votes. Assuming no votes were spoiled, calculate the total number of valid votes cast in the election.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A group of students decided to collect as many paise from each member of the group as is the number of members. If the total collection amounts to Rs. 48.75 and it is known that 20% of the group members contributed an additional amount equivalent to half the number of members, calculate the number of members in the group.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Example 4 Given that $E$ is the midpoint of side $AB$ of quadrilateral $ABCD$, $BC=CD=CE$, $\angle B=75^{\circ}$, $\angle D=90^{\circ}$. Find the degree measure of $\angle DAB$. ${ }^{[4]}$
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Context:
Let $S_n(R)$ and $V_n(R)$ denote the surface area and volume of an $n$-sphere with radius $R$ respectively. It is well known that $$S_n(R) = \frac{n \pi^{\frac{n}{2}}R^{n-1}}{\Gamma\left(\frac{n}{2}+1\right)}\quad\text{and}\quad V_n(R) = \frac{\pi^{\frac{n}{2}}R^{n}}{\Gamma\left(\frac{n}{2}+1\right)}$$ where $\Gamma(z)$ is the gamma function: $$\Gamma(z) = \int_0^\infty e^{-x} x^{z-1}\, dx.$$ Let $v$ and $s$ be positive integers such that $S_s(1)$ and $V_v(1)$ are the maximum. What is $v-s?$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the maximum value of $p$ for natural numbers $m$, $n$, and $p$ that satisfy the equation \(2mnp=(m+2)(n+2)(p+2)\) and the inequality \(m \leq n \leq p\), given that \(m\) cannot be equal to 1 or 2.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In projective space $\mathbb{P}^n$, determine the minimum possible dimension of the intersection of a subspace of dimension $p$ and a subspace of dimension $q$, given that $p + q < n$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In an isosceles triangle $ABC$, where $A$ is the top vertex, $B$ is the right vertex, and $C$ is the left vertex, it is given that $AC = AB$. The triangle is inscribed in a circle with center $O$. The diameter $AD$ passes through $O$ and intersects $CB$ at $E$. If $AC = \sqrt{\frac{15}{2}}$ and $OE = 1$, find the radius of the circle.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the area of the portion of the conical surface defined by the equation \(x^2 + y^2 = z^2\) that is bounded between the planes \(z = 0\) and \(x + 2z = 3\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the fraction of a flux within a $20$ degree angle range ($0.111\pi$ radians) for a flux distribution given by $I(\phi) = I_0 \cos^2(\phi)$, centered at $\phi = 0$. The flux fraction is the ratio of the integrals $\int_{-\theta}^{\theta} I(\phi) \, d\phi$ and $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} I(\phi) \, d\phi$, where $\theta = 20$ degrees ($0.111\pi$ radians). Use the integral of $\cos^2(\phi)$ as $\frac{\cos(\phi)\sin(\phi)+\phi}{2}$ to compute this ratio.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Numbers $a$, $b$, and $c$ form an arithmetic sequence if $b - a = c - b$. Let $a$, $b$, and $c$ be positive integers forming an arithmetic sequence with $a < b < c$. Let $f(x) = ax^2 + bx + c$. Two distinct real numbers $r$ and $s$ satisfy $f(r) = s$ and $f(s) = r$. If $rs = 2017$, determine the smallest possible value of $a$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the maximum number of points that can be placed in \(\mathbb{R}^n\) such that no \(n+1\) points lie in the same \(\mathbb{R}^{n-1}\)-plane and the Euclidean distance between every pair of points is an integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a rectangular plot of land, a man walks in a very peculiar fashion. Labeling the corners $ABCD$, he starts at $A$ and walks to $C$. Then, he walks to the midpoint of side $AD$, say $A_1$. Then, he walks to the midpoint of side $CD$, say $C_1$, and then the midpoint of $A_1D$, which is $A_2$. He continues in this fashion indefinitely. The total length of his path, if $AB=5$ and $BC=12$, is of the form $a + b\sqrt{c}$. Find $\frac{abc}{4}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A shopping mall installed 5 colored lights, each shining in a different color: red, orange, yellow, green, or blue. The sequence in which they shine is not fixed. Let's define the ordered shining of these 5 lights once as a "flash". In each flash, only one light shines per second, and the interval between two consecutive flashes is 5 seconds. Calculate the minimum time required to achieve all different flashes. Express your answer in seconds.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A merchant bought some goods at a discount of $20\%$ of the list price. He wants to mark them at such a price that he can give a discount of $20\%$ of the marked price and still make a profit of $20\%$ of the selling price. The per cent of the list price at which he should mark them is:
Please reason step by step, and put your final answer within \boxed{}.


Context:
5 / 8 of the population of the country of Venezia lives in Montague province, while the rest lives in Capulet province. In the upcoming election, 80% of Montague residents support Romeo, while 70% of Capulet residents support Juliet; each resident of Venezia supports exactly one of these two candidates. What is the probability that a Juliet supporter chosen at random resides in Capulet, rounded if necessary to the nearest percent?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Inside the circle \(\omega\), there is a circle \(\omega_{1}\) that touches it at point \(K\). Circle \(\omega_{2}\) touches circle \(\omega_{1}\) at point \(L\) and intersects circle \(\omega\) at points \(M\) and \(N\). It turns out that points \(K\), \(L\), and \(M\) are collinear. Find the radius of circle \(\omega\) if the radii of circles \(\omega_{1}\) and \(\omega_{2}\) are 4 and 7, respectively.
Please reason step by step, and put your final answer within \boxed{}.


Context:
If 5 students are made to sit in a row, then 6 do not have seats. If some students are made to sit in a row, then 3 rows are empty. Each row has to have 6 students for all students to be seated. How many students were made to sit in a row in the second scenario?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $ABCDE$ be a convex pentagon such that $AB=AE=CD=1$, $\angle ABC=\angle DEA=90^\circ$ and $BC+DE=1$. Compute the area of the pentagon.

[i]Greece[/i]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Yvan and Zoé are playing the following game. Let $n \in \mathbb{N}$. The integers from 1 to $n$ are written on $n$ cards arranged in order. Yvan removes one. Zoé then removes 2 consecutive ones. Next, Yvan removes 3 consecutive ones, and Zoé finishes by removing 4 consecutive ones.

What is the smallest value of $n$ for which Zoé can ensure she can play both of her turns?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $X$ be a projective variety over $\mathbf{C}$ of dimension $n$. Consider the blow-up $\pi: Y \to X$ at a smooth point $x \in X$. Find a formula for the intersection number $E^n$, where $E$ is the exceptional divisor.
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Context:
Evaluate the integral \( \int_{-L}^L e^{i\frac{(m-n)\pi}{L}x} \, dx \), where \( 0 < L \in \mathbb{R} \) and \( m, n \in \mathbb{N} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: $$\lim_{n\to \infty}\frac{1}{\sqrt[3]{{n^3}+2n-1}}+\frac{1}{\sqrt[3]{{n^3}+2n}}+\cdots+\frac{1}{\sqrt[3]{{n^3}+4n}}$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $\oplus$ denote the xor binary operation. Define $x \star y = (x+y) - (x \oplus y).$ Compute
\[ \sum_{k=1}^{63} (k \star 45). \]

**Remark:** The xor operation works as follows: when considered in binary, the $k$th binary digit of $a \oplus b$ is $1$ exactly when the $k$th binary digits of $a$ and $b$ are different. For example, $5 \oplus 12 = 0101_2 \oplus 1100_2 = 1001_2 = 9.$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the result of the tensor product $M/IM \otimes_R R/I$, where $R$ is a ring, $I$ is an ideal in $R$, and $M$ is an $R$-module.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the Hamming distance of a linear block code with parameters (7,2), where the code is 7 bits long, has 2 data bits, and 5 error correction bits.
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Context:
X deposits $1 at time $t=0$ with a force of interest given by \( \delta_t = \frac{t^2}{k} \). Y deposits the same amount at the same time with a nominal interest rate of 8% per annum, convertible semiannually. At \( t=5 \), their accumulated values are equal. Determine the value of \( k \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the minimum number of coin flips $n$ required to ensure that the estimated probability $\hat{p} = X/n$ of a biased coin is within $5\%$ of the true probability $p$, given that $0.4 \leq p \leq 0.6$. Ensure that the probability of this confidence interval, $\mathbb{P}[0.95p \leq \hat{p} \leq 1.05p]$, is at least 0.95.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( S \) be a set of 2017 points in the plane. Define \( R \) as the radius of the smallest circle that contains all points in \( S \) either on its interior or boundary. Let \( D \) be the longest distance between any two points in \( S \). Determine the pair \((a, b)\) of real numbers such that \( a \le \frac{D}{R} \le b \) for all possible sets \( S \), where \( a \) is as large as possible and \( b \) is as small as possible.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \int_{-\infty}^{\infty} \frac{x\sin(kx)}{x^2+a^2} \, \mathrm{d}x \) using real analysis techniques, where \( k > 0 \) and \( a \) is a positive constant.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: $$\lim_{n\to\infty}\left(\frac{(n+1)^{n+1}}{n^n}-\frac{n^n}{(n-1)^{n-1}}\right)$$
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Context:
In a $2 \times 3 \times 3$ array of letters, how many five-letter strings can be formed if you are allowed to repeat letters but cannot stall and repeat? An example of an acceptable string is "DETER", while an unacceptable string is "ADDER". Disregard the requirement that the string be an actual English word.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the area of the region in the xy-plane defined by the following conditions:
\[
\left\{
\begin{array}{l}
2x + y \leq 2, \\
x \geq 0, \\
\left( x + \sqrt{x^2 + 1} \right) \left( y + \sqrt{y^2 + 1} \right) \geq 1,
\end{array}
\right.
\]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the value of \(a \in \mathbb{R}\) for which all nontrivial solutions of the differential equation \(x'' + ax' + 4x = 0\) are oscillatory about zero.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( p, q \) be two distributions over the same finite set \( X \) with a total variation distance \( TV(p,q) = \epsilon \). Given a function \( f: X \to \mathbb{R} \) such that \( 0 \leq \sum_{x \in X} f(x) p(x) \leq 1 \) and \( 0 \leq \sum_{x \in X} f(x) q(x) \leq 1 \), determine the best upper bound of \( \left| \sum_{x \in X} f(x)(p(x) - q(x)) \right| \) as a function of \( \epsilon \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: $$\lim_{x\to0}{\frac{\sqrt{\cosh{(5x^2)}}\cdot e^{2x^3}-1}{x^2\tan(3x)}}$$
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Context:
Calculate the limit: $$\lim_{n \to \infty} \sqrt[n]{|\sin n|}$$
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Context:
Determine the smallest integer \( m \) such that the inequality \( 0.\overline{2}_b < \log_{b}{2} \) holds for all bases \( b > m \), where \( 0.\overline{2}_b \) represents a repeating decimal in base \( b \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $g: \mathbb{R} \to \mathbb{R}$ be an infinitely differentiable function with 12 distinct zeros. Determine the values of $n \ge 1$ for which the $n$-th derivative $g^{(n)}$ has at least 2 zeros.
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Context:
Evaluate the limit: \[ \lim_{n \to \infty} \left( \frac{1}{\sqrt[4]{n^4+n+2}} + \frac{1}{\sqrt[4]{n^4+n+3}} + \cdots + \frac{1}{\sqrt[4]{n^4+5n-1}} \right) \]
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Context:
Evaluate the line integral \( \int_{L}{(2x-y)dx+(x-y)dy} \) without using Green's theorem, where the curve \( L \) is defined as the union of the semicircle \( \{(x,y): x^{2}+y^{2}=2y, x\geq0\} \) and the quarter-circle \( \{(x,y):x^{2}+y^{2}=4, x\leq0,y\geq0\} \), oriented counter-clockwise starting from the point \((0,0)\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Use Green's Theorem to calculate the work done by the force field \( \mathbf{F}(x,y)=x(x+y)\mathbf{i}+xy^2\mathbf{j} \) as a particle moves along the path from the origin along the \( x \)-axis to \((1,0)\), then along the line segment to \((0,1)\), and back to the origin along the \( y \)-axis.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that $w$ is a complex cube root of unity with $w \neq 1$, determine the number of non-singular matrices of the form:
\[A=\begin{bmatrix} 1 & a & b\\ w & 1 & c\\ w^2 & w & 1 \end{bmatrix}\]
where $a, b, c$ are elements of the set $S=\{w, w^2\}$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Solve for x in the equation:
\[ 2^{x^{2^{x^{\cdots}}}} = 4 \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that two congruent 30°-60°-90° triangles with hypotenuses of 12 are overlapped such that their hypotenuses exactly coincide, calculate the area of the overlapping region.
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Context:
When \((x^2 + x + 1)^{54}\) is expanded, how many coefficients of the resulting polynomial are divisible by 3?
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Context:
Find $x$ if\[\cfrac{1}{\cfrac{1}{\cfrac{1}{\cfrac{1}{x}+\cfrac12}+\cfrac{1}{\cfrac{1}{x}+\cfrac12}}+\cfrac{1}{\cfrac{1}{\cfrac{1}{x}+\cfrac12}+\cfrac{1}{\cfrac{1}{x}+\cfrac12}}}=\frac{x}{36}.\]
Please reason step by step, and put your final answer within \boxed{}.


Context:
A soccer player takes 19 consecutive penalty shots. The probability of failing the $n$-th shot is $\frac{1}{2n+1}$. What is the probability that the player succeeds in an even number of these shots?
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Context:
Given two independent random variables $X$ and $Y$ that are both normally distributed with mean 0 and variance 1, $\mathcal N(0,1)$, find the expected value of $X^4 + Y^3$ conditioned on the sum $X + Y$. Express your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the volume of the solid obtained by revolving the area enclosed by the curve \(27ay^{2} = 4(x-2a)^{3}\), the \(x\)-axis, and the parabola \(y^{2} = 4ax\) about the \(x\)-axis.
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Context:
Using the four arithmetic operators and parentheses, find a way to combine the numbers 10, 10, 4, and 2 such that the result is 24. What is the arithmetic expression?
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Context:
Calculate the limit: \[ \lim_{n \to \infty} \frac{2\cdot6\cdot10\cdot\cdot\cdot(4n-2)}{1\cdot5\cdot9\cdot\cdot\cdot(4n-3)} \]
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Context:
Given the coin denominations 1 cent, 5 cents, 10 cents, and 50 cents, determine the smallest number of coins Lisa would need so she could pay any amount of money less than a dollar.
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Context:
Given the function \( f(x) = \cos x + \log_2 x \) for \( x > 0 \), if the positive real number \( a \) satisfies \( f(a) = f(2a) \), then find the value of \( f(2a) - f(4a) \).
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Context:
Determine the sum of the three smallest integer values of the parameter \(a\) for which the inequality \(\frac{x^{2}+(a+1) x+a}{x^{2}+5 x+4} \geq 0\) results in the union of three non-overlapping intervals.
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Context:
Let \( s_n \) be the number of solutions to the equation \( a_1 + a_2 + a_3 + a_4 + b_1 + b_2 = n \), where \( a_1, a_2, a_3, \) and \( a_4 \) are elements of the set \( \{2, 3, 5, 7\} \) and \( b_1 \) and \( b_2 \) are elements of the set \( \{1, 2, 3, 4\} \). Determine the number of values of \( n \) for which \( s_n \) is odd.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The real function $g$ has the property that, whenever $x,$ $y,$ $m$ are positive integers such that $x + y = 3^m,$ the equation
\[g(x) + g(y) = 2m^2\]holds. What is $g(2187)$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of elements in the centralizer of the permutation \((1234)\) in the alternating group \(A_4\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the circle equation x^{2}+y^{2}-4x-4y+7=0, find the number of tangent lines that have equal intercepts on the x and y axes.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Fran's school just instituted a school uniform policy. Each student needs to buy five complete uniforms, each consisting of pants, shirt, tie and socks. The pants cost $20, the shirt costs a certain ratio as the pants, the tie costs 1/5 as much as the shirt and the socks cost $3/pair. Each student needs to spend $355. What is the ratio of the cost of the shirt to the cost of the pants?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a binomial random graph $G_{n,p}$ with $n$ vertices, where each edge is present with probability $p$, consider the event that $k$ distinct sets of 3 vertices each form a triangle. Express the probability of this event, $\mathbb{P}\left(\bigcap_{j=1}^k \mathcal{E}_j\right)$, in terms of $p$. Here, $\mathcal{E}_j$ denotes the event that a set of 3 vertices forms a triangle.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the surface area of the solid generated by revolving the polar curve \( r = \sqrt{\cos 2\theta} \) around the y-axis from \( \theta = 0 \) to \( \theta = \frac{\pi}{4} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that $X$ and $Y$ are independent and identically distributed random variables, each following an exponential distribution with parameter $\lambda = 1$, compute the expected value $E(\sqrt{X+Y})$. Provide your answer to two decimal places.
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Context:
An urn contains a large number of balls with 50% black, 30% white, and 20% red. Balls are extracted randomly, one at a time, recorded, and returned to the urn. What is the average number of balls one should extract until all three colors have appeared?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: $$\lim_{x\to 0} \frac{\sqrt[5]{1+\frac{x}{5}}-\sqrt[6]{1+\frac{x}{6}}}{1-\sqrt{1-\frac{x}{3}}}$$ without using L'Hospital's rule, power expansions, or derivatives.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the value of \(1234^{1234} \pmod{5379}\) given that \(5379 = 3 \times 11 \times 163\). Do not use Euler's Theorem.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \(\Gamma\) be the set of functions \(f_n: \mathbb{C} \to \mathbb{C}\) defined by \(f_n(z) = \frac{\pi^2}{24}z^2 - \frac{1}{n^2}z + \frac{589}{20} \cdot \frac{1}{n^3}\) for \(n \in \mathbb{N}\). Let \(\Lambda\) be the set of all roots with positive imaginary part of the elements of \(\Gamma\). Find the sum of all elements of \(\Lambda\), rounding the real and imaginary parts to the nearest integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
For any two different real numbers \( x \) and \( y \), define \( D(x, y) \) as the unique integer \( d \) that satisfies
\[
2^{d} \leq |x-y| < 2^{d+1}.
\]
Given a set of real numbers \( \mathscr{F} \), for \( x \in \mathscr{F} \), the "scale" of \( x \) in \( \mathscr{F} \) is defined as the value of \( D(x, y) \) for some \( y \in \mathscr{F} \) where \( x \neq y \).

Let \( k \) be a given positive integer. Suppose each \( x \) in \( \mathscr{F} \) has at most \( k \) different scales in \( \mathscr{F} \) (these scales may vary depending on \( x \)). Find the maximum number of elements in the set \( \mathscr{F} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
设 $\omega$ 为正实数, 若存在 $a, b(\pi \leqslant a<b \leqslant 2 \pi)$, 使得 $\sin \omega a+\sin \omega b=2$, 则 $\omega$的取值范围是. 请提供满足条件的区间所有整数分子分母的和。
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Context:
Given the differential equation \( y'' + q(t)y = 0 \) where \( q(t) \) is a continuous periodic function with period \( T \), and \( y_1 \) and \( y_2 \) are linearly independent solutions, calculate the value of \( y_1(T) + y_2'(T) \) if there exists a non-trivial solution \( y_* \) that is periodic with period \( T \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find a maximal ideal \(I\) in the ring \(\mathbb{Z}[i]\) such that \(\mathbb{Z}[i]/I\) is isomorphic to \(\mathbb{Z}/521\mathbb{Z}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the contour integral \( \int_{\gamma} \frac{\log(z)}{z} \, dz \) where \( \gamma(t) = e^{it} \) for \( 0 \le t \le 2\pi \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{\ln(1+b\sin x)}{\sin x} \, dx \) where \(|b|<1\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the expression \( \frac{100^2}{100!} + \sum_{k=1}^{100} \left| (k^2 - 3k + 1) S_k \right| \), where \( S_k \) is the sum of the infinite geometric series with first term \( \frac{k-1}{k!} \) and common ratio \( \frac{1}{k} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
For a given positive integer $k$, find the values of $\alpha$ for which the series \[ \sum_{a\in \mathbb{N}^n} \left( \sum_{i=1}^n a_i^k\right)^{-\alpha} \] converges.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the range of values of $p$ for which the series \( \sum_{k = 1}^\infty \frac{\exp(-p k)}{k^{1-p}} \) converges, where \( p \geq 0 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the limit: \[ \lim_{n \to \infty} \int_0^{n^2} e^{-x^2}n\sin\frac{x}{n} \, dx. \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the following expression:
\[
1+\frac{1+\frac{1+\frac{1+\frac{1+\cdots}{2+\cdots}}{2+\frac{1+\cdots}{2+\cdots}}}{2+\frac{1+\frac{1+\cdots}{2+\cdots}}{2+\frac{1+\cdots}{2+\cdots}}}}{2+\frac{1+\frac{1+\frac{1+\cdots}{2+\cdots}}{2+\frac{1+\cdots}{2+\cdots}}}{2+\frac{1+\frac{1+\cdots}{2+\cdots}}{2+\frac{1+\cdots}{2+\cdots}}}}.
The original answer is in the form \(\frac{k+\sqrt{m}}{n}\). Please find the value of k + m + n.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the number of triples(x,a,b)where \( x \) is a real number and \( a,b \) belong to the set \( \{ 1,2,3,4,5,6,7,8,9\} \) such that
 \( {x}^{2} - a\{ x\}  + b = 0. \)
 where \( \{ x\} \) denotes the fractional part of the real number \( x \) .
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Context:
Let \( S \) be a subgroup of the group of positive rational numbers under multiplication, which is dense in the positive reals. Define the S-restricted irrationality measure by confining \( p/q \) to \( S \). If \( S \) is finitely generated, particularly if \( S \) is generated by the first \( n \) primes, determine the S-restricted irrationality measure \( \xi \) such that almost all \( x \) have measure \( \xi \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the double integral \( \int_0^2 \int_{\frac{y}{2}}^1 e^{x^2} \, dx \, dy \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that $a > 0$, $b > 0$, $c > 1$, and $a + b = 1$, find the minimum value of $( \frac{a^{2}+1}{ab} - 2) \cdot c + \frac{\sqrt{2}}{c - 1}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Bob is making partitions of 10, but he hates even numbers, so he splits 10 up in a special way. He starts with 10, and at each step, he takes every even number in the partition and replaces it with a random pair of two smaller positive integers that sum to that even integer. For example, 6 could be replaced with 1+5, 2+4, or 3+3, all with equal probability. He terminates this process when all the numbers in his list are odd. The expected number of integers in his list at the end can be expressed in the form \(\frac{m}{n}\), where \(m\) and \(n\) are relatively prime positive integers. Compute \(100m+n\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
On the diagonals $A C$ and $B D$ of trapezoid $A B C D$ with bases $B C=a$ and $A D=b$, points $K$ and $L$ are located respectively, such that

$C K: K A=B L: L D=7: 4$. Find $K L$.

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


Context:
Let $ABC$ be a scalene triangle with inradius $1$ and exradii $r_A$, $r_B$, and $r_C$ such that $20(r_B^{2r}C^2+r_C^2r_A^2+r_A^2r_B^2)=19(r_Ar_Br_C)^2.$
If
$tan\frac{A}{2}+\tan\frac{B}{2}+\tan\frac{C}{2}=2.019,$
then the area of $\triangle{ABC}$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $100m+n$.
Proposed by Tristan Shin
Please reason step by step, and put your final answer within \boxed{}.


Context:
4.1. Let \( f(x) = x^2 - x + 1 \), \( g(x) = -x^2 + x + 1 \). Find a non-constant polynomial \( h \) such that \( h(f(x)) = h(g(x)) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
令S=\ds{m=1}{\infty}\ds{n=1}{\infty}\df{m^2n}{2^m(n2^m+m2^n)},则 [100S]=__________.
Please reason step by step, and put your final answer within \boxed{}.


Context:
On a backpacking trip with 10 people, in how many ways can I choose 2 cooks and 1 medical helper if any of the 10 people may fulfill these roles?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the covariance of the random variables X and Y given the joint density function:
\[ f_{X,Y}(x,y) = \frac{1}{2} \int_{0}^{\infty} x^3 e^{-x(y+2)} \text{ d}x \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Problem 4. Simplify the expression

$$
M=\frac{2}{\sqrt{4-3 \sqrt[4]{5}+2 \sqrt[4]{25}-\sqrt[4]{125}}}
$$
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Context:
Find the norm of the operator $N_x: l_2 \to l_2$ defined by $N_x(y) = (x_n y_n)$, where $x = (x_n) \in l_\infty$. Express your answer as a single value.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a city with 10 parallel streets and 10 streets intersecting them at right angles, what is the minimum number of turns a closed bus route passing through all intersections can have?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a machine with two components, it will fail if either component fails. Let $X$ and $Y$ be the time-to-failure random variables for the first and second components, respectively, measured in hours. The joint moment generating function of $X$ and $Y$ is given by: $$ M_{X,Y}(s,t)=\frac{1}{(1-3s-2t+6ts)}. $$ Determine the probability that the machine will fail within the first hour.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Three faces of a right rectangular prism have areas of 54, 56, and 60 square units. Calculate the volume of the prism in cubic units and round it to the nearest whole number.
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Context:
Each of the symbols $\diamond$ and $\circ$ represents an operation in the set $\{+,-,\times,\div\}$, and $\frac{15 \diamond 3}{8 \circ 2} = 3$. What is the value of $\frac{9 \diamond 4}{14 \circ 7}$? Express your answer as a common fraction.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find a $n\in\mathbb{N}$ such that for all primes $p$, $n$ is divisible by $p$ if and only if $n$ is divisible by $p-1$.
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Context:
Given that θ ∈ [0, 2π), if the circle (x - cosθ)² + (y - sinθ)² = r² (r > 0) intersects with the line 2x - y - 10 = 0, find the minimum value of r.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral
$$ \int_0^1 \frac{\ln\left(\left(x+\sqrt{1-x^2}\right)^2 / \left(x-\sqrt{1-x^2}\right)^2\right)}{1-x^2} x \, dx $$
Please reason step by step, and put your final answer within \boxed{}.


Context:
The sum of the digits of [(10 ^ x) ^ a] - 64 = 279, where a is a positive integer. The value of xy is 32. What is the value of a?
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Context:
A book contains 30 stories. Each story has a different number of pages under 31. The first story starts on page 1 and each story starts on a new page. What is the largest possible number of stories that can begin on odd page numbers?
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Context:
Calculate the value of the expression: \[ \arctan \frac{21 \pi}{\pi^2-54} + \arctan \frac{\pi}{18} + \arctan \frac{\pi}{3} \] without using a calculator or Maclaurin series.
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Context:
In the Cartesian coordinate plane, the circle \( C_1: x^2 + y^2 - a = 0 \) is symmetric about a line \( l \). The resulting circle is \( C_2: x^2 + y^2 + 2x - 2ay + 3 = 0 \). Determine the equation of the line \( l \).
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Context:
For a set $P$ of five points in the plane, with no three points being collinear, determine the maximum number of acute triangles that can be formed using the points in $P$ as vertices.
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Context:
An urn contains 5 balls numbered 0, 1, 2, 3, and 4. A ball is drawn from the urn, its number is recorded, and then it is returned to the urn. This process is repeated until the balls numbered 1, 2, and 3 have each been drawn at least once. What is the probability that this process is repeated exactly 5 times?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Assume that the target population is divided into three demographic groups with different response rates: Group A with a 70% response rate, Group B with a 55% response rate, and Group C with a 40% response rate. If the percentage of each group in the target population is 40% for Group A, 35% for Group B, and 25% for Group C, and a total of 300 responses are needed, what is the minimum number of questionnaires that should be mailed accounting for the weighted average of response rates?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find a generator of the maximal ideal of the local ring of rational functions on the algebraic curve $C: Y^2 = X + X^3$ at the point $P = (0,0)$. This local ring is a discrete valuation ring and thus a principal ideal domain.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Through the right focus \( F \) of the ellipse \( C: \frac{x^2}{2} + y^2 = 1 \), draw a line \( l \) that intersects the ellipse \( C \) at two different points \( A \) and \( B \). Suppose \( \overrightarrow{FA} = \lambda \overrightarrow{FB} \) and \( T(2,0) \). If \( \lambda \in [-2, -1] \), find the range of values of \( |\overrightarrow{TA} + \overrightarrow{TB}| \).
Please reason step by step, and put your final answer within \boxed{}.

\boxed{\left[ 2, \frac{13 \sqrt{2}}{8} \right]}


Context:
Determine the number of real solutions to the equation \(\sin(4x) = \frac{x}{100}\) for \(x \in (0, 2\pi)\) without using graphical methods.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the range of values of \(\alpha\) for which the Lyapunov condition is not satisfied for a collection of independent random variables \(\{X_{i}\}_{i\geq1}\).
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Context:
Evaluate the integral \( \int_{a}^{d} \delta(x-b)\delta(x-c)f(x)\,dx \), where \( a \le b \le d \) and \( a \le c \le d \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Identify the original function \( f(x) \) for which the Newton's method iteration is given by:
\[ x_{n+1} = 2x_n - x_n^2y \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the contour integral \( \oint_{|z|=4} \frac{e^{1 /(z-1)}}{z-2} \, dz \) using the residue theorem.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Two circles with centers \( P \) and \( Q \) intersect at points \( M \) and \( N \). The lines \( PM \) and \( QN \) intersect the first circle at points \( A \) and \( B \), and the second circle at points \( D \) and \( C \) respectively. The point \( M \) lies between \( A \) and \( D \), and the point \( N \) lies between \( B \) and \( C \). It is known that \( AB = CD \), and the area of the quadrilateral \( ABCD \) is \( 24 \sqrt{3} \). If the distance between the intersection points \( M \) and \( N \) is 4, find the distance between the centers \( P \) and \( Q \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the order of convergence of the modified Newton's method given by \(x_{n+1}=x_n-m\frac{f(x_n)}{f'(x_n)}\), where \(m\) is the multiplicity of the root.
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Context:
Evaluate the recursively defined sequence $T_n$ where $T_n = n + \frac{T_{n+2}}{n+1}$, starting with $T_1$. Express the result as a single number.
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Context:
Evaluate the integral \( \frac{1}{2\pi i}\int_C \frac{\sum_{n=0}^{15}z^n}{(z-i)^3}dz \), where \( C \) is the contour \(|z-i|=2\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Alice and Bob are each secretly given a real number between 0 and 1 uniformly at random. Alice states, \u201cMy number is probably greater than yours.\u201d Bob repudiates, saying, \u201cNo, my number is probably greater than yours!\u201d Alice concedes, muttering, \u201cFine, your number is probably greater than mine.\u201d If Bob and Alice are perfectly reasonable and logical, what is the probability that Bob\u2019s number is actually greater than Alice\u2019s?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \int_{0}^{2\pi} e^{-\sin^{2}{x}} \cos\left(6x - \frac{\sin{2x}}{2}\right) \, dx \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
A triangle is formed by selecting three lattice points (points with integer coordinates) at random with replacement within the square \([-99, 100] \times [-99, 100]\). Determine the probability that the area of this triangle (which may be degenerate) is an integer. Express this probability as a fraction \(\frac{m}{n}\), where \(m\) and \(n\) are coprime positive integers, and calculate \(m + n\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the expected time until there are two students in a help center, given that student arrivals follow a Poisson process with rate $r$, and the first student leaves after an exponential time with rate $2r$. Provide your answer in terms of $r$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find \(\alpha \in \mathbb{R}\) such that there exists a Lie algebra representation of \(\mathfrak{sl}_2 (\mathbb{R})\) on \(C^\infty(\mathbb{R})\) with image equal to \(\text{span}_\mathbb{C} \{ X , Y , Z_\alpha \}\), where the differential operators are defined as follows:

\[X f(t) = f''(t)\]
\[Y f(t) = t^2 f(t)\]
\[Z_\alpha f(t) = t f'(t) + \alpha f(t)\]

for \(f \in C^\infty(\mathbb{R})\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the precise decay rate of the sequence \(a_n = \min_{1 \le N \le n} |\langle N \alpha \rangle|\) for an irrational number \(\alpha\).
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Context:
For discrete probability distributions \( P \) and \( Q \) defined on the same probability space, \( \mathcal{X} \), the relative entropy from \( Q \) to \( P \) is defined as \[ D_{\mathrm{KL}}(P \| Q)=\sum_{x \in \mathcal{X}} P(x) \log \left(\frac{P(x)}{Q(x)}\right) \] Given the distributions \( P=[p, 0, \ldots,0, 1-p] \) and \( Q=\text{binomial}(n,q)=\left(\begin{array}{l} n \\ k \end{array}\right) q^{k}(1-q)^{n-k}, \forall k \in \{0,1,2,...n\} \), find the parameter \( q \) that minimizes \( D_{\mathrm{KL}}(P \| Q) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the expression:
\[ \left(a_{0}-4 a_{2}+16 a_{4}+\cdots-2^{14} a_{14}\right)^{2}+\left(2 a_{1}-8 a_{3}+32 a_{5} \cdots-2^{15} a_{15}\right)^{2} \]
where for a complex number \(z\), we have the binomial expansion \((1+z)^{15}=a_{0}+a_{1} z+\cdots+a_{15} z^{15}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that $i$ is an annual effective interest rate, $1$ accumulates to $3$ in $r$ years, $2$ accumulates to $7$ in $s$ years, and $3$ accumulates to $16$ in $t$ years. Determine the value of $10(1+i)^{(2r+s-t)}$. Provide your answer as a numerical value.
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Context:
Professor Liyung wants to form a math club with his 40 students. Each student is enemies with two other students, and no student will join the club if any of their enemies is already a member. Let \( M \) be the maximum number of members the club can have. Find the sum of all possible values of \( M \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the cardinality of a Banach space $X$ that has a countable dense subset?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: $$\lim_{n\to\infty} \int_0^\infty \frac{\sin(x/n)}{x+x^2}dx$$
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Context:
Given a countable collection of pairwise disjoint sets $\{ X_n \mid n \in \omega \}$, each with cardinality $c$, where $c = 2^{\aleph_0}$ (the cardinality of the continuum), determine the cardinality of their union $\bigcup_n X_n$.
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Context:
Evaluate the integral \[ I = \frac{1}{\pi^2} \int_{0}^{\infty} \frac{(\ln{x})^2}{\sqrt{x}(1-x)^2} \, \mathrm{d}x \] using suitable substitutions and techniques.
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Context:
Suppose the power series \( \sum^\infty_{n=0}{ a_n x^n } \) converges with a radius of convergence \( R_a \) and the power series \( \sum^\infty_{n=0}{ b_n x^n } \) converges with a radius of convergence \( R_b \). Determine the radius of convergence of the power series \( \sum^\infty_{n=0}{ (a_n+b_n) x^n } \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( M \) be a \( 3 \times 3 \) matrix such that \( M \begin{bmatrix} -2 \\ 1 \\ 0 \end{bmatrix} = \begin{bmatrix} 6 \\ -3 \\ 0 \end{bmatrix} \). Suppose that \( M^3 \begin{bmatrix} 1 \\ -\frac{1}{2} \\ 0 \end{bmatrix} = \begin{bmatrix} \alpha \\ \beta \\ \gamma \end{bmatrix} \) for some \( \alpha, \beta, \gamma \in \mathbb{R} \). Find the value of \( |\alpha| \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of zeros of the polynomial \( f(z) = z^{4}+z^{3}+4z^{2}+2z+3 \) that lie in the first quadrant of the complex plane.
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Context:
Bryan Ai has the following 8 numbers written from left to right on a sheet of paper: \( \textbf{1 4 1 2 0 7 0 8} \). In each of the 7 gaps between adjacent numbers, Bryan Ai places one of \( + \), \( - \), or \( \times \). If he picks a random placement out of the \( 3^7 \) possible placements, what is the expected value of the expression, considering the order of operations?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the smallest integer \( k \) such that if the Fourier coefficients \( a_n \) and \( b_n \) of a function \( f \) satisfy \( |a_n| \leq Cn^{-k} \) and \( |b_n| \leq Cn^{-k} \) for some constant \( C \), then \( f \) is in \( C^1(T) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the triple integral \( \int\int\int_D z(x^2+y^2+z^2)^{-\frac{3}{2}} \, dV \) over the solid \( D \) that lies above the plane \( z=3 \) and below the sphere \( x^2+y^2+z^2=36 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: \( \lim_{x \to \infty} x^2\ln\left(x\cot^{-1}(x)\right) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Two painters are painting a fence that surrounds garden plots. They come every other day and paint one plot (there are 100 plots) in either red or green. The first painter is colorblind and mixes up the colors; he remembers which plots he painted, but cannot distinguish the color painted by the second painter. The first painter aims to maximize the number of places where a green plot borders a red plot. What is the maximum number of such transitions he can achieve (regardless of how the second painter acts)?

Note: The garden plots are arranged in a single line.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the norm \( ||T|| \) of the operator \( T \) defined on \( H = L^2(0,1) \) by:
\[ Tf(s) = \int_0^1 (5s^2t^2 + 2)f(t) \, dt \]
where \( f \in L^2(0,1) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
There are 7 volunteers to be arranged to participate in the promotional activities for the Shanghai World Expo on Saturday and Sunday, with 6 people participating each day and 3 people arranged for each day. There are __________ different arrangements possible (answer with a number).
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Context:
Evaluate the limit: \[ \lim_{n\to\infty}\left[\dfrac{1}{2^{n/2}\Gamma(n/2)} \int_{n-\sqrt{2n}}^{\infty} t^{\frac{n}{2}-1}e^{-t/2}\,dt\right] \]
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Context:
Calculate the number of ways to obtain exactly one set of three of a kind when rolling 9 dice.
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Context:
Evaluate the contour integral \( \int_\gamma \frac{1+z}{1-\cos z} \, dz \), where \( \gamma \) is a circle centered at the origin with a radius of 7.
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Context:
An insurance company pays out claims following an exponential distribution. When a deductible \( d \) is applied, the expected claim payment decreases by 10\%. Calculate the percentage reduction in the variance of the claim payment.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a uniform random variable $U$ over all $n$-bit binary strings, and another random variable $X$ that depends on $U$ and also takes values in $n$-bit binary strings, if the mutual information between $X$ and $U$ is bounded as $I(X;U) \le \epsilon$, determine the tightest lower bound on the entropy of the XOR of $X$ and $U$, denoted by $H(X \oplus U)$. Specifically, can we establish a bound like $H(X \oplus U) \ge n - \epsilon$?
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Context:
Let $  a_1 = \sqrt{7}  $ and $  b_i = \lfloor a_i \rfloor  $. $  a_{i+1} = \frac{1}{b_i - [b_i]}  $ for each $  i \geq i  $.

What is the smallest integer $  n  $ greater than 2004 such that $  b_n  $ is divisible by 4?

($  [x]  $ denotes the largest integer less than or equal to $  x  $)
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Context:
Determine the number of Hamiltonian paths in a tournament that contains a cycle and has at least 4 vertices.
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Context:
Given the function $f(x)=\sin (2x+\varphi)$, if the graph is shifted to the left by $\dfrac {\pi}{6}$ units and the resulting graph is symmetric about the $y$-axis, determine the possible value of $\varphi$.
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Context:
Initially, the men and women in a room were in the ratio of 7 : 9. Then, 5 men entered the room and 6 women left the room. Afterwards, 3 more men joined the room and 4 women returned. The number of women then doubled and now there are 28 men in the room. How many women are currently in the room?
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Context:
Determine the number of real solutions in the interval \((0, \pi)\) for the equation:
\[ \sin(14u) - \sin(12u) + 8\sin(u) - \cos(13u) = 4 \]
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Context:
Evaluate the limit: $$\lim \limits_{m\to \infty }\cos^2 \left(\pi \sqrt{m^2-m}\right)$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let y = f(x) be a function defined on R with a period of 1. If g(x) = f(x) + 2x and the range of g(x) on the interval [1,2] is [-1,5], find the range of the function g(x) on the interval [-2020,2020]. Express your answer as an interval [min, max].
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Context:
Given \( 0 < x < \frac{\pi}{4} \), let \( A = \cos (x^{\sin (x^{\sin x)})} \), \( B = \sin (x^{\cos (x^{\sin x)})} \), and \( C = \cos (x^{\sin (x x^{\cos x})}) \). Determine the relationship of magnitudes between \( A \), \( B \), and \( C \).
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Context:
Quadrilateral \(ABCD\) is inscribed in a circle. Point \(X\) lies on its side \(AD\), with \(BX \parallel CD\) and \(CX \parallel BA\). Find \(BC\) if \(AX = \frac{3}{2}\) and \(DX = 6\).
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Context:
There are 6 spots for top students, planned to be distributed among three classes. The probability that exactly one class does not receive a spot for a top student is ____.
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Context:
5.1. At the vertices of a cube, numbers $\pm 1$ are placed, and on its faces - numbers equal to the product of the numbers at the vertices of that face. Find all possible values that the sum of these 14 numbers can take. In your answer, specify their product.
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Context:
Let $a$ and $b$ be positive real numbers with $a\ge b$. Let $\rho$ be the maximum possible value of $\frac {a}{b}$ for which the system of equations \[a^2 + y^2 = b^2 + x^2 = (a - x)^2 + (b - y)^2\] has a solution in $(x,y)$ satisfying $0\le x < a$ and $0\le y < b$. Then $\rho^2$ can be expressed as a fraction $\frac {m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
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Context:
If $x > y > 0$, then $\frac{x^{y}y^{x}}{y^{y}x^{x}} = $
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Context:
Jerry went to a shop and bought several items worth a total of Rs. 120. Out of these, 40% of the items were taxed at a 6% tax rate, 30% of the items were taxed at an 8% tax rate, and the remaining items were tax-free. Additionally, Jerry received a discount of 5% on items in the second tax bracket. What was the cost of the tax-free items?
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Context:
Determine the number of elements in the quotient group $\mathbb{Z}^2/\ker\varphi$ given the homomorphism $\varphi: \mathbb{Z}^2 \to S_{15}$ where \[ \varphi(1,0) = (2,7,3)(3,11,5)(12,13) \quad \text{and} \quad \varphi(0,1) = (14,15)(1,4,6,8,9,10). \]
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Context:
Evaluate the limit \( \lim_{n\to\infty}{ \int_{1}^{\infty}{\dfrac{x^n}{2+x^n}}dx } \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the line integral: $$ \int_{L} \frac{-y \, dx}{x^{2}+y^{2}}+\frac{x \, dy}{x^{2}+y^{2}} $$
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Context:
Determine the cardinality of a basis for the vector space of the real numbers over the field of rational numbers.
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Context:
The positive sequence $\{a_{n}\}$ satisfies the equation \(\frac{1}{a_{n} a_{n+1}}+\frac{1}{a_{n} a_{n+2}}+\frac{1}{a_{n+1} a_{n+2}}=1\) for all natural numbers \(n\). Given that \(a_{1} + a_{3} = 6\), and \(a_{1}\), \(a_{2}\), and \(a_{3}\) form a strictly increasing geometric progression, let \(S_{n}\) be the sum of the first \(n\) terms of the sequence \(\{a_{n}\}\). Find the value of \(\left\lfloor S_{2008} \right\rfloor\), where \(\left\lfloor x \right\rfloor\) denotes the greatest integer less than or equal to \(x\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that \( \int_0^\infty (g(x) - f(x)) \, dx = 0 \), where both \( g(x) \) and \( f(x) \) are continuous, differentiable, positive, and monotone decreasing functions with \( \lim_{x \to \infty} g(x) = 0 \) and \( \lim_{x \to \infty} f(x) = 0 \), and assuming \( f(x) \) and \( g(x) \) are distinct such that the set \( S = \{x: f(x) \neq g(x) \} \) has a non-zero measure, determine the cardinality of the set \( X = \{x \in \mathbb{R}: f(x) = g(x), \text{ and for some } \epsilon > 0, \ [f(x-\epsilon) < g(x) < f(x+\epsilon)] \lor [f(x-\epsilon) > g(x) > f(x+\epsilon)] \} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the volume of the $d$-dimensional $\ell_1$-surface $\mathbb{S}_d$ using the parameterization and integral approach.
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Context:
Consider the sequence defined by \(a_1 = 7\) and \(a_{n+1} = \frac{2^n + n^{10^{17}}}{(1 + \frac{1}{n})^{(n^2)} - \frac{1}{n}} \cdot a_n\). Given that \(\lim_{n \to \infty}{(1+\frac{1}{n})^{(n^2)} - \frac{1}{n}} = e^2\), determine the limit of this sequence as \(n\) approaches infinity.
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Context:
Two chess players, X and Y, play a series of games with the following rules: The probability of X winning a particular game against Y is \( \frac{1}{3} \), and the probability of Y winning the game is \( \frac{2}{3} \). X wins the series if X wins two consecutive games, while Y wins the series if Y wins four consecutive games. They continue playing until one of them wins the series. What is the probability that Y wins the series?
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Context:
How many positive integers $n\leq100$ satisfy $\left\lfloor n\pi\right\rfloor=\left\lfloor\left(n-1\right)\pi\right\rfloor+3$? Here $\left\lfloor x\right\rfloor$ is the greatest integer less than or equal to $x$; for example, $\left\lfloor\pi\right\rfloor=3$.
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Context:
Let the incircle of  $\triangle ABC$  be tangent to  $AB,BC,AC$  at points  $M,N,P$ , respectively. If  $\measuredangle BAC=30^\circ$ , find  $\tfrac{[BPC]}{[ABC]}\cdot\tfrac{[BMC]}{[ABC]}$ , where  $[ABC]$  denotes the area of  $\triangle ABC$ .
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Context:
Given the sequence $\{a_n\}_{n\in\mathbb{N}}$ defined by $a_0 = -1$, $a_1 = 1$, and $a_{n+2} = 6a_{n+1} - a_n$ for all $n\in\mathbb{N}$, find all indices $n$ such that $a_n = 8k^2 - 1$ for some $k\in\mathbb{N}$. Provide your answer as a set of all such $n$.
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Context:
Determine the precise value of the uniform density function $f(y_1, y_2)$ for the continuous bivariate random variable $(Y_1, Y_2)$ with support $S = \{(y_1, y_2) : y_2 \leq 1-y_1^2, y_1 \leq 0, y_2 \leq 0\}$.
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Context:
Determine the number of semi-direct products \(H \rtimes Q\) that can be constructed for \(H \cong C_{42}\) and \(Q \cong C_{3}\).
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Context:
Evaluate the surface integral \( \iint_Y \mathbf{F} \cdot \mathbf{N} \, \mathrm{d}S \) where \( \mathbf{F}= (x^2 yz + x \sin z , x^2 + y(1 - \sin z ), x + y - xy z^2) \) and \( Y \) is the part of the conical surface \( 4x^2 + (y-1)^2 = z^2 \) which lies between \( z = 1 \) and \( z = 2 \). The normal vector points away from the \( z \)-axis. Use Gauss's theorem to assist in the calculation.
Please reason step by step, and put your final answer within \boxed{}.


Context:
For a positive constant \( a \), consider the curve \( C: y=\frac{a}{2}(e^{\frac{x}{a}}+e^{-\frac{x}{a}}) \). Let \( l(t) \) be the length of the segment of \( C \) where \( a \leq y \leq t \), and let \( S(t) \) be the area bounded by the line \( y=t \) (where \( a<t \)) and the curve \( C \). Determine \( \lim_{t\to\infty} \frac{S(t)}{l(t)\ln t} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the 1000th decimal place of \( \sqrt{\underbrace{111\ldots111}_{1998 \text{ times}}} \).
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Context:
In how many ways can the letters in BEEKEEPER be rearranged so that two or more Es do not appear together?
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Context:
Find the number of permutations \(a_1, a_2, \ldots, a_{100}\) of the numbers \(1, 2, \ldots, 100\) such that:
\[|a_1 - 1| = |a_2 - 2| = \ldots = |a_{100} - 100|\]
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Context:
In triangle \(ABC\), let \(\omega\) be the incircle tangent to \(BC, CA,\) and \(AB\) at points \(D, E,\) and \(F\) respectively. A point \(P\) on \(EF\) is such that \(PD\) is perpendicular to \(BC\). The line \(AP\) intersects \(BC\) at \(Q\). Circles \(\omega_1\) and \(\omega_2\) pass through \(B\) and \(C\) respectively, both tangent to \(AQ\) at \(Q\). Circle \(\omega_1\) meets \(AB\) again at \(X\), and circle \(\omega_2\) meets \(AC\) again at \(Y\). The line \(XY\) intersects \(BC\) at \(Z\). Given \(AB=15\), \(BC=14\), and \(CA=13\), find \(\lfloor XZ \cdot YZ \rfloor\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a list of non-intersecting rectangles $W$, where each rectangle $w$ is represented as $(x, y, \text{width}, \text{height})$ and the list is sorted by $y$-coordinates, determine the maximum factor by which an input rectangle $R$ can grow while maintaining its aspect ratio without intersecting any rectangles in $W$. For example, if $W=\{(10, 2, 2, 8), (5, 5, 3, 3), (7, 7, 2, 2)\}$ and $R=(1, 1, 1, 1)$, find the maximum growth factor for $R$. The aspect ratio of $R$ is $\frac{w}{h}=1$. Determine the maximum factor by which $R$ can grow before it intersects with any rectangle in $W$. Output the maximum growth factor.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the condition number of the matrix product $X \cdot Y$ with respect to the Frobenius norm, where $X \in \mathbb{R}^{M \times N}$ and $Y \in \mathbb{R}^{N \times P}$. Provide your answer as a single numerical value.
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Context:
Given a random vector \(X = (Y, X, Z)\) that follows a Gaussian distribution with mean \(\mu = (1, 2, 4)^T\) and covariance matrix \(\Sigma = \begin{pmatrix} 2 & 3 & 1\\ 3 & 5 & 2\\ 1 & 2 & 6 \end{pmatrix}\), calculate the conditional variance \(\operatorname{Var}(Y | X, Z)\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of homomorphisms from the dihedral group \(D_5\) to the symmetric group \(S_3\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the minimum number of unit squares that must be removed from a \(202 \times 202\) grid so that it can be tiled with T-tetrominoes.
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Context:
Determine the big-O bound for the expression: $$\sum^{ N}_{k=0}{2^k \log_2\left(\frac{2^N}{2^k}\right)}$$ where $N = \log M$. Provide your answer in big-O notation.
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Context:
Find the intersection multiplicity of the curves given by the equations \( f(x,y) = x^5 + x^4 + y^2 \) and \( g(x,y) = x^6 - x^5 + y^2 \) at the point \( P = (0,0) \).
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Context:
You have a standard six-sided die and you roll it repeatedly, writing down the numbers that come up. You win when two of your rolled numbers add up to 7. How many turns do you expect to play before you win?
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Context:
Determine the form of a matrix $A$ in $\mathbb{R}^3$ that commutes with every other matrix $X$ in the same space, i.e., $AX = XA$ for all matrices $X$. What is the form of matrix $A$?
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Context:
What is the value of $\aleph_1^{\aleph_0}$?
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Context:
Calculate the surface area of the solid formed by rotating the astroid given by the equation \(x^{2/3} + y^{2/3} = 1\) around the \(y\)-axis, for \(y\) in the interval \([0, 1]\).
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Context:
$a_1,a_2, \cdots , a_{2003}$ are integers such that $|a_1| = 1$ and $|a_{i+1}|=|a_i+1|$ $(1\leq i \leq 2002)$. What is the minimal value of $|a_1+a_2+\cdots + a_{2003}|$?
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Context:
Determine the minimum number of black points, $m$, required such that every unit circle centered at a black point contains exactly 10 white points. Assume $m \ge n$ and that the maximum intersection between any two subsets of size 10 is 2.
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Context:
Let \(P_1, P_2, P_3, P_4, P_5\) be five distinct planes in \(\mathbb{R}^3\) such that the intersection of any two distinct planes is a line, the intersection of any three distinct planes is a point, and the intersection of any four distinct planes is empty. Let \(X = \bigcup_{i=1}^5 P_i\) with the topology as a subspace of \(\mathbb{R}^3\). Determine the Euler characteristic of \(X\).
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Context:
In a right triangle \(ABC\) with the right angle at \(C\), the inscribed circle touches the sides \(AB\), \(BC\), and \(CA\) at points \(C_1\), \(A_1\), and \(B_1\) respectively. The altitudes of triangle \(A_1B_1C_1\) intersect at point \(D\). Given that the legs of triangle \(ABC\) are 3 and 4, find the distance between points \(C\) and \(D\).
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Context:
Triangle $AB_0C_0$ has side lengths $AB_0 = 12$, $B_0C_0 = 17$, and $C_0A = 25$. For each positive integer $n$, points $B_n$ and $C_n$ are located on $\overline{AB_{n-1}}$ and $\overline{AC_{n-1}}$, respectively, creating three similar triangles $\triangle AB_nC_n \sim \triangle B_{n-1}C_nC_{n-1} \sim \triangle AB_{n-1}C_{n-1}$. The area of the union of all triangles $B_{n-1}C_nB_n$ for $n\geq1$ can be expressed as $\tfrac pq$, where $p$ and $q$ are relatively prime positive integers. Find $q$.
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Context:
In a game with 2022 nuts on the table, two players, Alice and Bob, take turns eating nuts. On each turn, a player rolls a regular six-sided die and eats the number of nuts shown on the top face. The player who cannot make a move loses the game. Determine the starting number of nuts, \(a\), for which Bob can force a win.
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Context:
Calculate the line integral of the scalar field \( x+y \) over the curve \( L \), where \( L \) is the right loop of the polar equation \( r^2 = 2\cos(2\theta) \).
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Context:
Calculate the fundamental group \(\pi_1(Y)\) of the space \(Y\), where \(Y\) is the quotient space obtained from \(X = S^2 \times [0,1]\) by identifying each point \(x \in S^2 \times \{1\}\) with its antipodal in \(S^2 \times \{0\}\).
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Context:
Let $Z_1$ and $Z_2$ be independent identically distributed standard Gaussian random variables with mean 0 and variance 1. Determine the distribution of the random variable \( X = \frac{(Z_1 + Z_2)^2}{(Z_1 - Z_2)^2} \).
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Context:
Evaluate the principal value of the improper integral: $$\int_{-1}^{1} \frac{1}{x^{2}}dx$$.
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Context:
Calculate the limit: \[ \lim_{n \rightarrow \infty} \left( \frac{n \pi}{4} - \left( \frac{n^2}{n^2+1^2} + \frac{n^2}{n^2+2^2} + \cdots + \frac{n^2}{n^2+n^2} \right) \right). \]
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Context:
Determine the constant $k$ such that the probability density function \( p(t) = k(1+\sin(20t))e^{-\frac{t}{1.5}} \) is properly normalized and satisfies \( p(t) \leq 1 \) for all \( t > 0 \).
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Context:
Determine the order of the group $GL_3(\mathbb{Z_4})$, which consists of all invertible $3 \times 3$ matrices with entries from $\mathbb{Z}_4$. Provide your answer as a single integer.
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Context:
Find the explicit Cartesian equation for the surface obtained by revolving the curve \( y = \frac{1}{x} \) about the line \( y = x \).
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Context:
Consider a sequence of 10 independent tosses of a biased coin, where the probability of Heads is \( p = \frac{3}{4} \) and the probability of Tails is \( 1 - p \). A reward of one unit is given at time \( k \), for \( k \in \{1, 2, \ldots, 10\} \), if the toss at time \( k \) resulted in Tails and the toss at time \( k-1 \) resulted in Heads. Otherwise, no reward is given at time \( k \). Let \( R \) be the sum of the rewards collected at times \( 1, 2, \ldots, 10 \). Calculate \( E(R^2) \).
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Context:
The amount of a claim that a car insurance company pays out follows an exponential distribution. By imposing a deductible of $d$, the insurance company reduces the expected claim payment by 10%. What is the percentage reduction in the variance of the claim payment?
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Context:
Calculate the probability \( P(N(4) \ge 6 \mid S(4) = 3) \), where \( N(t) \) is the number of events that have occurred between 0 and \( t \), and \( S(n) \) is the arrival time of the \( n \)th event in a Poisson process.
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Context:
How many times must a fair coin be tossed to be at least 99% confident that the percentage of heads will fall between 49.5% and 50.5%?
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Context:
A casino uses two kinds of dice: a fair die with a probability of \(\frac{1}{6}\) for each number, and a loaded die with a probability of \(\frac{1}{2}\) for rolling a 6 and \(\frac{1}{10}\) for each of the other numbers. The probability of switching from the fair to the loaded die is 0.1, and from the loaded to the fair die is 0.05. Given the observed sequence of numbers \(4, 6, 6, 6\), determine the most likely sequence of dice used, assuming the casino starts with the fair die.
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Context:
In a tournament with $8$ players, each pair of players played exactly one match. Each match ended either with a win for one player, who received $2$ points while the opponent received $0$ points, or in a draw, where both players received $1$ point each. After all matches were completed, it was found that every player had the same number of points. Determine the minimal total number of draws.
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Context:
Compute the number of sequences of integers $(a_1,\ldots,a_{200})$ such that the following conditions hold.
· $0\leq a_1<a_2<\cdots<a_{200}\leq 202.$
· There exists a positive integer $N$ with the following property: for every index $i\in\{1,\ldots,200\}$ there exists an index $j\in\{1,\ldots,200\}$ such that $a_i+a_j-N$ is divisible by $203$.
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Context:
Determine the minimum value of the integral
\[ \mathcal{I}[y] = \int_{-\infty}^{\infty} \left( \frac{1}{2} y'^2 + 1 - \cos y \right) \, dx \]
subject to the boundary conditions \(y(-\infty) = 0\) and \(y(\infty) = 2\pi\).
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Context:
Find all solutions for $x$ that satisfy the equation:
\[ \frac{1}{2(x-1)} = \int_0^\infty \frac{\sin (x\arctan y)}{(1+y^2)^{x/2}(e^{y\pi}+1)}\,dy \]
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Context:
Given a trapezoid \(ABCD\) with \(AD \parallel BC\). It turns out that \(\angle ABD = \angle BCD\). Find the length of segment \(BD\) if \(BC = 36\) and \(AD = 64\).
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Context:
Calculate the probability of generating a dense graph with $n$ vertices from a graphon with a specific set $S$ of positive measure.
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Context:
已知数列 a_n=n\cos\frac{n\pi}{2},n=1,2,\cdots，函数 f(x)=\df{x^2}{x^2-2x+2}. 则数列 \{f(a_n)\} 的前 2024 项的和 T_{2024} 为__________.
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Context:
In the figure, $ABC$ is an obtuse triangle. $BC = 6$ cm, $AB = 5$ cm, and the height $AD$ from $A$ to side $BC$ is 4 cm. If this triangle moves upward along the direction of $DA$ at a speed of 3 cm per second, what is the area swept by the triangle after 2 seconds?
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Context:
Let $Z \sim N(0,1)$ be a standard normal random variable. Calculate the expected value $E[\max\{Z-1,0\}]$. Express your answer as a single numerical value.
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Context:
Find the least possible value of \( N \) such that the error in approximating the integral \( \int_0^1 3e^{x^2} \, dx \) using Simpson's Rule is less than or equal to \( 1 \times 10^{-9} \). Use the error bound formula for Simpson's Rule: \( \text{Error}(S_N) \le \frac{K(b-a)^5}{180N^4} \), where \( K \) is the maximum value of the fourth derivative of the integrand on the interval \([a, b]\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In triangle $ABC$, the angles $A$, $B$, and $C$ have tangents that form a geometric sequence: $\tan A$, $\tan B$, $\tan C$. Given that $1 \le \tan A + \tan B + \tan C \le 2015$, determine the number of possible integer values for $\tan B$. (Note: $\tan A$ and $\tan C$ do not need to be integers.)
Please reason step by step, and put your final answer within \boxed{}.


Context:
For a Brownian motion \( B_t \), let \( T_a = \inf\{t \geq 0 \mid B_t = a\} \) be the hitting time of level \( a \). Compute the probability \( P(T_2 < T_{-1} < T_1) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the annihilator operator of the function \( f(x) = x(\sin x + 3x - x^{-1}) - \cos x \), where the operator is \( \frac{d}{dx} \equiv D \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In how many ways may thirteen beads be placed on a circular necklace if each bead is either blue or yellow and no two yellow beads may be placed in adjacent positions? (Beads of the same color are considered to be identical, and two arrangements are considered to be the same if and only if each can be obtained from the other by rotation).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the residue at $z = 0$ of the function $(1-e^{-z})^{-n}$, where $n$ is a positive integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Two joggers each run at their own constant speed and in opposite directions from one another around an oval track. They meet every 36 seconds. The first jogger completes one lap of the track in a time that, when measured in seconds, is a number (not necessarily an integer) between 80 and 100. The second jogger completes one lap of the track in a time, \(t\) seconds, where \(t\) is a positive integer. The product of the smallest and largest possible integer values of \(t\) is:
Please reason step by step, and put your final answer within \boxed{}.


Context:
Jessica's family is some distance away from New York. They are traveling at the rate of 50 km/h and stop to rest for 30 minutes every 2 hours. It takes them 7 hours to reach New York. How far away is Jessica's family from New York?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Polycarp, the top student, wrote a huge number by writing out all natural numbers from 1 to 500 in sequence: 123...10111213...499500. The underachieving Kolya erased the first 500 digits of this number. What do you think the first digit of the remaining number is?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $B = (20, 14)$ and $C = (18, 0)$ be two points in the plane. For every line $\ell$ passing through $B$, we color red the foot of the perpendicular from $C$ to $\ell$. The set of red points enclose a bounded region of area $\mathcal{A}$. Find $\lfloor \mathcal{A} \rfloor$, which is the greatest integer not exceeding $\mathcal{A}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $f(x) = \frac{x}{a(x+2)}$, and the equation $x = f(x)$ has a unique solution. Given that $f(x_0) = \frac{1}{1008}$ and $f(x_{n-1}) = x_n$ for $n=1, 2, 3, \ldots$, calculate $x_{2015}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the minimum possible value of the sum \(\sum_{i=1}^{11}x_i\), where \(x_i\) are positive integers for \(i=1,2,...,11\). The conditions \(x_i+x_{i+1}\geq 100\) and \(|x_i-x_{i+1}|\geq 20\) hold for \(i=1,2,...,10\), and additionally, \(x_{11}+x_{1}\geq 100\) and \(|x_{11}-x_1|\geq 20\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Each frog needs to eat 30 flies per day to live, but they also cooperate with herons, collectively catching 60 flies per heron present. Each fish needs to eat 8 frogs per day to live. However, the fish and frogs also compete for a common prey: tadpoles. Each fish needs to eat either 8 frogs or 24 tadpoles, and each frog needs to eat either 30 flies or 2 tadpoles per day.

Each gharial needs to eat 15 fish a day to live. Additionally, each heron needs to eat 5 frogs and 3 fish per day to live, and each caiman needs to eat 2 gharials and 2 herons per day to live.

Considering the above interactions and consumption requirements, how many flies get eaten every day in this swamp if it has 9 gharials, 12 herons, 7 caimans, 20 fish, and 50 frogs? Assume that a given ratio of tadpoles and frogs can be determined and use it to calculate the daily total of consumed flies.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of pairs \((x, y)\) of real numbers that satisfy the inequalities \(x^6+y^6 < 1\) and \(x^3+y^3 > 1\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Among the eight vertices of the cube ABCD-A1B1C1D1, a plane passes through four of them, and the remaining four vertices are all at equal distances from the plane. Find the number of such planes.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that an agent's task completion times follow an exponential distribution, Agent 1 has Task 1 and Task 2 with mean completion times of 6 and 4 units, respectively. Agent 2's completion times for Task 1 and Task 2 have means of 3 and 1 unit, respectively. Task 2 can only begin after Task 1 is completed by both agents. Assuming all times are independent, calculate the probability that Agent 1 finishes Task 2 before Agent 2 does.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the sum of \( \frac{\sigma(n) \cdot d(n)}{ \phi (n)} \) over all positive \( n \) that divide \( 60 \).

Note: The function \( d(i) \) outputs the number of divisors of \( i \), \( \sigma (i) \) outputs the sum of the factors of \( i \), and \( \phi (i) \) outputs the number of positive integers less than or equal to \( i \) that are relatively prime to \( i \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
A round robin tournament is held with $2016$ participants. Each player plays each other player once, and no games result in ties. We define a pair of players $A$ and $B$ as a \textit{dominant pair} if all other players either defeat both $A$ and $B$ or are defeated by both $A$ and $B$. Find the maximum number of dominant pairs.
Please reason step by step, and put your final answer within \boxed{}.


Context:
For the nodal cubic defined by \(y^2 = x^2(x+1)\), consider the parametrization given by \(x(t) = t^3 - 3t\) and \(y(t) = t^4 - 4t^2\). Find the values of \(t\) where this parametrization fails to be regular.
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Context:
Ricsi, Dénes, and Attila often play ping-pong against each other, where two of them team up on one side. Dénes and Attila win against Ricsi three times as often as they lose; Dénes wins against Ricsi and Attila as often as he loses; finally, Attila wins against Ricsi and Dénes twice as often as he loses. Recently, they spent an entire afternoon playing six matches, playing each combination twice. What is the probability that Ricsi won at least once, either alone or with a partner?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the limiting value of the ratio of integrals \( \frac{\int_{0}^{
fty}x^{m+1}e^{x}dx}{\int_{0}^{
fty}x^{m}e^{x}dx} \) as the upper limit approaches infinity, for \( m \in \mathbb{N} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Martha needs to paint all four walls in her kitchen, which has a certain length, 16 feet width, and 10 foot high ceilings. Each wall needs one coat of primer and two coats of paint. Martha can paint 40 square feet per hour, and it takes her 42 hours to paint the kitchen. What is the length of the kitchen?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the integer values of \( n \) for which the inequality \( n^{\sqrt{n+2}} > (n+1)^{\sqrt{n+1}} \) holds.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( P(x) \) be a polynomial of degree 4, having extrema at \( x=1 \) and \( x=2 \). Given that \( \lim_{x\to 0}\frac{x^2+P(x)}{x^2}=2 \), find the value of \( P(2) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the number of integers $n$ such that $1 \leq n \leq 2017$ and $(n-2)(n-0)(n-1)(n-7)$ is a multiple of $1001$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Integers $1 \le n \le 200$ are written on a blackboard just one by one. We surrounded just $100$ integers with circle. We call a square of the sum of surrounded integers minus the sum of not surrounded integers $score$ of this situation. Calculate the average score in all ways.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the value of \(c\) in the joint probability density function \(f_{X,Y}(x,y) = c\) for \((x,y) \in \left[\frac{1}{2}, 1\right] \times \left[-x+\frac{3}{2}, 1\right] \cup \left[0, 1\right] \times \left[-x+\frac{1}{2}, -x+1\right]\) and \(0\) otherwise, given that the total probability is 1.
Please reason step by step, and put your final answer within \boxed{}.


Context:
When I saw Eleonora, I found her very pretty. After a brief trivial conversation, I told her my age and asked how old she was. She answered:

- When you were as old as I am now, you were three times older than me. And when I will be three times older than I am now, together our ages will sum up to exactly a century.

How old is this capricious lady?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the line integral \( \oint_{C_1} \left(\frac{y\hat{i} - x\hat{j}}{x^2 + y^2}\right) \cdot d\vec{r} \), where \( C_1 \) is a circle of radius 3 centered at the origin in \( \mathbb{R}^2 \). Determine the result without relying solely on parametrization.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A quadrilateral is given, where the lengths of two sides are $1$ and $4$. Additionally, a diagonal of length $2$ divides the quadrilateral into two isosceles triangles. Determine the perimeter of this quadrilateral.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the line integral \( \int_C \mathbf{F} \cdot d\mathbf{r} \) where \( C \) is the curve joining the origin \((0, 0, 0)\) to the point \((1, 2, 3)\) defined by the parametric curve \( \mathbf{r}(t) = \frac{\ln t}{\ln 2} \mathbf{i} + t^{\frac{3}{2}} \mathbf{j} + t \cos(\pi t) \mathbf{k} \), \( 1 \leq t \leq 4 \), and \( \mathbf{F}(x, y, z) = 3x^2 z \mathbf{i} + z^2 \mathbf{j} + (x^3 + 2yz) \mathbf{k} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the integral \( \int_{0}^{\pi} e^{a\cos(\theta)} \cos(a\sin(\theta)) \, d\theta \) using complex analysis.
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Context:
Find the numerator of

$$
\frac{\overbrace{1010 \, 11 \ldots 11 \, 0101}^{2011 \text{ ones}}}{\underbrace{1100 \, 11 \ldots 11 \, 0011}_{2011 \text{ ones}}}
$$

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


Context:
I was walking with my mother-in-law. We were walking slowly at a speed of 3 km/h along Saint Catherine Street in Bordeaux, which, as everyone (or maybe not everyone) knows, is straight. Suddenly, I remembered that I needed to mail a letter as soon as possible in a mailbox located a bit further down the street. Leaving my mother-in-law, who continued walking peacefully, I hurried forward, mailed the letter, and then walked back towards my mother-in-law at a speed of 5 km/h. I clearly remember that I left my mother-in-law alone for only 3 minutes.

At what distance from the mailbox were we when we parted ways?
Please reason step by step, and put your final answer within \boxed{}.


Context:
The sequence 12, 15, 18, 21, 51, 81, $\ldots$ consists of all positive multiples of 3 that contain at least one digit that is a 1. What is the $50^{\mathrm{th}}$ term of the sequence?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the total number of times the digit ‘ $2$ ’ appears in the set of integers  $\{1,2,..,1000\}$ . For example, the digit ’ $2$ ’ appears twice in the integer  $229$ .
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the largest integer $K$ such that for each $k∈$ {$2,3,4,...,K$}, the number $2^{30}-k$ has a prime factor less than $30$.
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Context:
What fraction of the entire wall is painted red if Matilda paints half of her section red and Ellie paints one third of her section red?
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Context:
Randomly select seven numbers from the set $\{1, 2, \cdots, 63\}$. Let the smallest of these numbers be denoted as $x$. What is the expected value $\mathrm{E}(x)$ of $x$?
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Context:
On their way driving to Los Angeles, Benjamin and James see lots of billboard ads. Curious, Benjamin starts to count how many they see within specific periods each hour. In the first hour, Benjamin notes the billboards during a 25-minute period and counts 15 billboards. In the second hour, he only counts during a 45-minute period and sees 31 billboards. In the third hour, he counts during a 20-minute period and sees 12 billboards. James asks Benjamin the average number of billboards they've seen per hour considering the varying periods he counted. What should Benjamin tell him?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Sean originally has 55 blocks. He decided to share them with his 5 friends, giving each friend an equal number of blocks. Before they get a chance to play with the blocks, a hippopotamus eats 29 of the remaining blocks. How many blocks does Sean have left now?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $PQR$ be an equilateral triangle with sides of length three units. $U$, $V$, $W$, $X$, $Y$, and $Z$ divide the sides into lengths of one unit. Find the ratio of the area of the shaded quadrilateral $UWXY$ to the area of the triangle $PQR$.  The answer is in the form \frac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the smallest possible value of n such that there are n distinct lines in the plane, one of which intersects exactly 5 of the n lines, another intersects exactly 9 of the n lines, and yet another intersects exactly 11 of the n lines.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A five-digit number $\overline{ABCDE}$ is a multiple of 2014, and $\overline{CDE}$ has exactly 16 factors. What is the smallest value of $\overline{ABCDE}$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
The apothem of a regular hexagonal pyramid is $h$, and the dihedral angle at the base is $60^{\circ}$. Find the total surface area of the pyramid.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the polynomial equation
$$
x^{5}+3 x^{4}-4044118 x^{3}-12132362 x^{2}-12132363 x-2011^{2}=0
$$
where all roots are integers, find the number of distinct roots of the equation. Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of ways to choose a set of $n$ distinct non-negative integers $i_1, i_2, \ldots, i_n$ from the set $\{0, 1, \ldots, n-1\}$ such that their sum is a multiple of $n$. Each $i_j$ must be unique.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the following limit: $$\lim_{n\to\infty}\left(1-\frac{1}{2}\right)\left(1-\frac{1}{2^2}\right)\left(1-\frac{1}{2^4}\right)\cdots\left(1-\frac{1}{2^{2^n}}\right)$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the cardinality of the set \( \prod_{n\in\mathbb{N}} (n+1) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Two sisters were picking currants: the older one was using a 4-liter bucket, and the younger one was using a 3.5-liter bucket. The older sister was always working faster than the younger one. When the older sister had collected three-quarters of her bucket, and the younger sister had collected more than half of hers, the girls exchanged buckets. Continuing to work at the same speeds as before exchanging buckets, they finished at the same time. How many times faster was the older sister picking berries compared to the younger sister?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find a bijective function that maps the real numbers to the interval $(-1, 1)$, different from the standard arctangent function.
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Context:
The function $y=(m^{2}-m-1){x}^{m^{2}-2m-1}$ is a power function, and it is a decreasing function for $x\in (0,+\infty)$. Find the real number $m$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of ways to distribute 15 distinct balls into 5 indistinguishable boxes.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $\Gamma_1, \Gamma_2, \Gamma_3$ be three pairwise externally tangent circles with radii $1,2,3,$ respectively. A circle passes through the centers of $\Gamma_2$ and $\Gamma_3$ and is externally tangent to $\Gamma_1$ at a point $P.$ Suppose $A$ and $B$ are the centers of $\Gamma_2$ and $\Gamma_3,$ respectively. What is the value of $\frac{{PA}^2}{{PB}^2}?$
Please reason step by step, and put your final answer within \boxed{}.


Context:
In how many ways can a random number of identical men be arranged in a line of 15 chairs such that no two men sit next to each other, and each empty chair has at least one man sitting next to it?
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Context:
Given a point $P(\frac{1}{2},0)$, determine the number of lines passing through point $P$ that intersect the hyperbola $4x^{2}-y^{2}=1$ at only one point.
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Context:
Given the sum of $n=14$ iid random variables $X_1, \dots, X_{14}$ with PDF \( f_{X_i}(x) = \frac{1}{\theta} e^{-\frac{x-\delta}{\theta}}, \quad x > \delta \) and 0 otherwise, where \( \theta = 2 \) and \( \delta = 0 \), calculate the probability \( P(Y < 2.467) \) for \( Y = \frac{\sqrt{28}Z}{\sqrt{S_{14}}} \) and \( Z \sim \mathcal{N}(0,1) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In order to assess the number of students in various classes at a school participating in an extracurricular group, 5 classes are randomly selected. The number of students participating in the group from each class is taken as sample data. It is known that the sample mean is 7, the sample variance is 4, and the sample data are all distinct. Find the maximum value in the sample data.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $N$ be a positive integer. Brothers Michael and Kylo each select a positive integer less than or equal to $N$, independently and uniformly at random. Let $p_{N}$ denote the probability that the product of these two integers has a units digit of 0. The maximum possible value of $p_{N}$ over all possible choices of $N$ can be written as $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. Compute $100 a+b$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the right trapezoid \( ABCD \), it is known that \(\angle A = \angle D = 90^\circ\), \(DE \perp AC\) at point \(E\), \(\angle ACD = \angle EBC = 30^\circ\), and \(AD = \sqrt{3}\). Find \(BC\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( A = \left(a_1, a_2, \ldots, a_{2001}\right) \) be a sequence of positive integers. Let \( m \) be the number of 3-element subsequences \( (a_i, a_j, a_k) \) with \( 1 \leq i < j < k \leq 2001 \) such that \( a_j = a_i + 1 \) and \( a_k = a_j + 1 \). Considering all such sequences \( A \), find the greatest value of \( m \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
A watch loses $2\frac{1}{2}$ minutes per day. It is set right at $1$ P.M. on March 15. Let $n$ be the positive correction, in minutes, to be added to the time shown by the watch at a given time. When the watch shows $9$ A.M. on March 21, what is the value of $n$ in the form of an improper fraction \frac{k}{m}? Please provide the sum of k and m.
Please reason step by step, and put your final answer within \boxed{}.


Context:
For natural number $n$, define the function $f(n)$ to be the number you get by $f(n)$ adding the digits of the number $n$. For example, $f(16)=7$, $f(f(78))=6$, and $f(f(f(5978)))=2$.
Find the least natural number $n$ such that $f(f(f(n)))$ is not a one-digit number.
Please reason step by step, and put your final answer within \boxed{}.