Context:
An equilateral triangle has sides of length 2 units. A second equilateral triangle is formed having sides that are $150\%$ of the length of the sides of the first triangle. A third equilateral triangle is formed having sides that are $150\%$ of the length of the sides of the second triangle. The process is continued until four equilateral triangles exist. What will be the percent increase in the perimeter from the first triangle to the fourth triangle? Express your answer to the nearest tenth.
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Context:
Given the function $f(x)=\frac{1}{9}x^3+x^2-2ax+1$, if the function $f(x)$ has an extremum on the interval $(1,2)$, then the range of real number $a$ is ______.
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Context:
Two rectangles have their edge lengths chosen uniformly from the interval [0,1]. What is the probability that the second rectangle will fit inside the first one?
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Context:
Given sets $A=\{x|x\leq-2 \text{ or } x\geq-1\}$, and $B=\{x|m-1<x\leq2m\}$, if $A\cap B=\emptyset$, and $A\cup B=A$, find the range of the real number $m$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the greatest integer $n$, such that there are $n+4$ points $A$, $B$, $C$, $D$, $X_1, \dots, X_n$ in the plane with $AB \neq CD$ that satisfy the following condition: for each $i=1,2,\dots,n$, the triangles $\triangle ABX_i$ and $\triangle CDX_i$ are equal.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the complex integral \( \int_{\gamma} \frac{-\cos(1/z)}{\sin(1/z)z^2} \, dz \), where \( \gamma \) is the positively oriented circle \( \partial B(0,2) \).
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Context:
In the two regular tetrahedra \(A-OBC\) and \(D-OBC\) with coinciding bases, \(M\) and \(N\) are the centroids of \(\triangle ADC\) and \(\triangle BDC\) respectively. Let \(\overrightarrow{OA}=\boldsymbol{a}, \overrightarrow{OB}=\boldsymbol{b}, \overrightarrow{OC}=\boldsymbol{c}\). If point \(P\) satisfies \(\overrightarrow{OP}=x\boldsymbol{a}+y\boldsymbol{b}+z\boldsymbol{c}\) and \(\overrightarrow{MP}=2\overrightarrow{PN}\), then the real number \(9x+81y+729z\) equals \(\qquad\)
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many positive integers $n$ satisfy the inequality
\[\left\lceil \frac{n}{101} \right\rceil + 1 > \frac{n}{100} \ ?\]
Recall that $\lceil a \rceil$ is the least integer that is greater than or equal to $a$.
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Context:
A circular metal wire $S$ in 3-space is described by $S=\{x^2+y^2+z^2=4\}$ and has density $\sigma (x,y,z) = \sqrt{x^2+y^2}$ at the point $(x,y,z)$. Find the mass of $S$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the value of the parameter \(a\) such that the substitution \(y(x) = z(x)^a\) transforms the differential equation \(2xy' + y = y^2\sqrt{x - x^2y^2}\) into a homogeneous equation.
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Context:
A rhombus  $\mathcal R$  has short diagonal of length  $1$  and long diagonal of length  $2023$ . Let  $\mathcal R'$  be the rotation of  $\mathcal R$  by  $90^\circ$  about its center. If  $\mathcal U$  is the set of all points contained in either  $\mathcal R$  or  $\mathcal R'$  (or both; this is known as the *union* of  $\mathcal R$  and  $\mathcal R'$ ) and  $\mathcal I$  is the set of all points contained in both  $\mathcal R$  and  $\mathcal R'$  (this is known as the *intersection* of  $\mathcal R$  and  $\mathcal R'$ , compute the ratio of the area of  $\mathcal I$  to the area of  $\mathcal U$ .
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Context:
The legs of a trapezoid are \( p \) and \( q \) (\( p < q \)), and the larger base is \( a \). The angles at the larger base are in the ratio \( 2:1 \). Find the length of the smaller base.
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Context:
A cube with edge length 6 units is painted with a blue stripe going down the center of one face, an orange stripe going horizontally across another face, and a green stripe going diagonally across the third face. It is then cut into 216 equal smaller cubes, each with side length of 1 unit. How many smaller cubes have exactly 2 faces painted with different colors?
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Context:
Compute the value of the integral \( \int_0^{\infty}{xI\left( 2x \right) \mathrm{e}^{-x^2}\mathrm{d}x} \), where the Bessel function is given by \( I\left( x \right) =\frac{1}{\pi}\int_0^{\infty}{\mathrm{e}^{x\cos \theta}\mathrm{d}\theta} \) for all real \( x \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Each week, Paul has a certain amount of homework on weeknights and 5 hours for the entire weekend. This week Paul has practice 2 nights out of the week and can't do any homework those nights. He has to average 3 hours of homework for the other nights to get his week's homework done. How many hours of homework does he have on weeknights?
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Context:
Let's call a natural number "remarkable" if it is the smallest among natural numbers with the same sum of digits. What is the sum of the digits of the two-thousand-and-first remarkable number?
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Context:
in a class , if 50 % of the boys were girls , then there would be 50 % more girls than boys . what percentage of the overall class is girls ?
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Context:
Viswam walks half a mile to get to school each day. His route consists of $10$ city blocks of equal length and he takes one minute to walk each block. Today, after walking $5$ blocks, Viswam discovers he has to take a detour, walking $3$ blocks of equal length instead of one block to reach the next corner. From the time he starts his detour what speed, in mph, must he walk in order to get to school at his usual time?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A contractor undertakes to do a job within 150 days and hires 25 people to do it. The workers have varying work efficiencies - 15 people have an efficiency of 1 unit per day, while the other 10 have an efficiency of 1.5 units per day. After 40 days, the contractor realizes that only one third of the total work is completed. Following this realization, he fires 4 less efficient workers and 3 more efficient workers. In how many more days will the work be finished?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $n$ be a positive integer.  Let $(a, b, c)$ be a random ordered triple of nonnegative integers such that $a + b + c = n$, chosen uniformly at random from among all such triples.  Let $M_n$ be the expected value (average value) of the largest of $a$, $b$, and $c$.  As $n$ approaches infinity, what value does $\frac{M_n}{n}$ approach?
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Context:
Let \(P\) be a regular polygon with 2026 sides. A diagonal of \(P\) is said to be good if its endpoints divide the contour of \(P\) into two parts, each having an odd number of sides of \(P\). The sides of \(P\) are also said to be good. Suppose that \(P\) has been subdivided into triangles by 2023 diagonals, no two of which share a common point inside \(P\). What is the maximum number of such a subdivision that can contain isosceles triangles whose two sides are good?
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Context:
Insert \( n \) real numbers between 1 and 100 such that these \( n+2 \) numbers form an increasing geometric sequence. Let the product of these \( n+2 \) numbers be denoted as \( T_n \). Furthermore, let \( a_n = \lg T_n \) for \( n \geq 1 \). Define \( b_n = \tan a_n \cdot \tan a_{n+1} \). Find the sum of the first \( n \) terms of the sequence \(\{b_n\}\) denoted as \( S_n \).
Please reason step by step, and put your final answer within \boxed{}.

\boxed{ \frac{\tan (n+3) - \tan 3}{\tan 1} - n }


Context:
Olga Ivanovna, the homeroom teacher of class 5B, is organizing a "Mathematical Ballet". She wants to arrange the boys and girls so that exactly 2 boys are at a distance of 5 meters from each girl. What is the maximum number of girls that can participate in the ballet if it is known that 5 boys are participating?
Please reason step by step, and put your final answer within \boxed{}.


Context:
设四面体ABCD中，AB=41,AC=7,AD=18,BC=36,BD=27,CD=13，设d为AB 与CD两棱中点间的距离.求$d ^ { 2 }$的值.
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Context:
In the convex quadrilateral \(ABCD\), the side lengths \(AB\) and \(BC\) are equal, \(DB\) is the angle bisector of \(\angle ADC\), and \(AD:DC = 4:3\). Find the cosine of \(\angle AKB\), if \(K\) is the intersection point of the diagonals \(AC\) and \(BD\), and \(BK:KD = 1:3\). (16 points)
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Context:
Nadya wants to cut a sheet of paper into 48 identical rectangles for a game. What is the minimum number of cuts she needs to make, given that the pieces of paper can be rearranged but not folded, and Nadya is able to cut any number of layers of paper simultaneously? (Each cut is a straight line from edge to edge of the piece.)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of spanning trees in a graph after contracting one edge in the middle of a triangle.
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Context:
Find the minimum value of \( n \) such that every subset of \( \{1, 2, \ldots, 999\} \) with \( n \) elements contains four distinct integers \( a, b, c, d \) satisfying \( a + 2b + 3c = d \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider two independent uniform random variables $X$ and $Y$ over $[a, b]$. Calculate $\mathbb{E}[X^2 - Y^2 \ | \ X < Y]$ using the correct approach.
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Context:
Determine the smallest real number \( \ell \) such that every hexagon \( H \) in the family \( \mathcal{F} \) can be covered by a strip of width \( \ell \). The family \( \mathcal{F} \) consists of hexagons with parallel opposite sides, and any three vertices of \( H \) can be covered with a strip of width 1.
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Context:
How can we find a closed-form expression for the following summation?
$$ \sum_{m\geq 0} \frac{m r^m \Gamma(m+c)}{\Gamma(m+1)} $$
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Context:
John was given $14.00 in order to buy a gift. This covered half of the cost. The store was also offering a 15% discount on the gift. Additionally, there is an 8% sales tax applied on the discounted price. Calculate the original price of the gift before the discount and tax were applied.
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Context:
Determine the smallest positive integer \(n\) such that the sum \(S = \frac{a_{1}}{1} + \frac{a_{2}}{2} + \ldots + \frac{a_{n}}{n}\), where \(a_1, a_2, \ldots, a_n\) is a permutation of \(1, 2, \ldots, n\), can take on all integer values from \(n\) to \(n+100\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Using the divergence theorem, find the flux of the vector field \( \mathbf{F} = \left(x(z - 1), y(z - 1), xy\right) \) over the region defined by the inequalities:
\[
(z-1)^2 \le 4-x^2-y^2, \quad (z-1)^2 \ge x^2+y^2, \quad z \ge 1
\]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Four consecutive digits are written one after another, and then the first two digits are swapped. The resulting four-digit number is a perfect square of a natural number. Find this number.
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Context:
Let $a,b,c$ be real numbers such that $$a\sin^2{x}+b\cos^2{x}=c,~~~\dfrac{a}{\sin^2{x}}+\dfrac{b}{\cos^2{x}}=c$$ find the value of $$\dfrac{c}{a-b}+\dfrac{a}{b-c}+\dfrac{b}{c-a}$$
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Context:
Determine the dimension of the orthogonal complement of the subspace $V_k = \mathrm{span}\{x^k, x^{k+1}, x^{k+2}, \dots\}$ in the space $L^2[-1,1]$, where $k$ is a positive integer.
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Context:
On the first day, $2^n$ students participated in a knockout table tennis tournament: initially, two students played, then the winner played against the third student, the winner of that match played against the fourth student, and this continued until the last student played. On the second day, the same students competed in a cup tournament: they were randomly paired and played in pairs, with losers being eliminated, and winners being randomly paired again until a single winner remained. It was found that the sets of pairs that played on the first and second days were identical (though possibly with different winners). Determine the largest possible value of $n$. \( \boxed{} \)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of terms required from the series \( \sum_{n=0}^\infty \frac{(-1)^{n}}{(2n+1)n!} \) to approximate \( \int_{0}^{1}e^{-x^2} \, dx \) with an error less than \( 10^{-6} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the Cartesian coordinate plane, let the vertices of triangle $ABC$ be $A(0,a)$, $B(b,0)$, $C(c,0)$, and point $P(0,p)$ be on line segment $AO$ (and not an endpoint). Given that $a$, $b$, $c$, and $p$ are non-zero real numbers, lines $BP$ and $CP$ intersect $AC$ and $AB$ at points $E$ and $F$, respectively. One student has correctly derived the equation of $OE$ as $(\frac{1}{b} - \frac{1}{c})x + (\frac{1}{p} - \frac{1}{a})y = 0$. Find the equation of $OF$: $(\_\_\_\_\_\_\_)x + (\frac{1}{p} - \frac{1}{a})y = 0$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $p(x)$ be a monic polynomial of degree 7 such that $p(0) = 0,$ $p(1) = 1,$ $p(2) = 2,$ $p(3) = 3,$ $p(4) = 4,$ $p(5) = 5,$ $p(6) = 6,$ and $p(7) = 7.$  Find $p(8).$
Please reason step by step, and put your final answer within \boxed{}.


Context:
I am interested in evaluating the integral $$\int_{-\infty}^{\infty} \frac{\exp( - a x^2 ) \sin( bx )}{x} \,\mathrm{d}x$$ for real $a,b>0$. I would like to find a closed-form expression for the integral, or an efficient numerical method for evaluating it.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In triangle $\triangle ABC$, if $\tan A \tan B = 1$ and $AB = \sqrt{3}$, calculate the size of angle $A$ in radians when the area of $\triangle ABC$ reaches its maximum value. Express your answer in radians, rounded to 4 decimal places if necessary.
Please reason step by step, and put your final answer within \boxed{}.


Context:
【Example 2】Using 4 different colors to color the 4 sides of a small square wooden block, with each side painted a different color, how many different coloring patterns can there be?
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Context:
What will be the cost of building a fence around an irregular shaped plot which has a square, a rectangle, and an equilateral triangle combined, if the price per foot of building the fence varies for each side? The square has an area of 289 sq ft, the rectangular side attached to the square is 45 ft long and has a height of 15 ft, and the equilateral triangle has a side length of 20 ft. The cost of fencing per foot is Rs. 55 for the square sides, Rs. 65 for the rectangular sides, and Rs. 70 for the triangular sides. Additionally, a gate will be installed on one side of the rectangle, which costs Rs. 750. Calculate the total fencing cost for the plot.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Inside triangle \( ABC \) with angles \(\angle A = 50^\circ\), \(\angle B = 60^\circ\), \(\angle C = 70^\circ\), a point \( M \) is chosen such that \(\angle AMB = 110^\circ\) and \(\angle BMC = 130^\circ\). Find \(\angle MBC\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Rectangle $EFGH$ has area $4024$. An ellipse with area $4024\pi$ passes through $E$ and $G$ and has foci at $F$ and $H$. What is the perimeter of the rectangle?
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Context:
Determine the digits $a, b, c, d, e$ such that the two five-digit numbers formed with them satisfy the equation $\overline{a b c d e} \cdot 9 = \overline{e d c b a}$.
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Context:
Given \( x \geq y > 0 \). If there exist real numbers \( a \) and \( b \) such that \( 0 \leq a \leq x \), \( 0 \leq b \leq y \), and
\[
(x - a)^{2} + (y - b)^{2} = x^{2} + b^{2} = y^{2} + a^{2},
\]
what is the maximum value of \( \frac{x}{y} \)?
Please reason step by step, and put your final answer within \boxed{}.


Context:
There are 150 'zeroes' and 151 'ones' written on a blackboard. A step involves choosing 3 digits and replacing them with a single digit. If all the 3 digits are identical, they are replaced with the same digit. Otherwise, the digit that appears twice replaces the 3 chosen digits. What is the minimum number of steps after which there will be no zero written on the blackboard?
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Context:
已知向量 $\boldsymbol{a} 、 \boldsymbol{b} 、 \boldsymbol{c}$ 满足 $|\boldsymbol{a}|=|\boldsymbol{b}|=|\boldsymbol{c}|=4$, $\boldsymbol{a} \cdot \boldsymbol{b}=8$. 则 $|3 \boldsymbol{c}-\boldsymbol{a}|+\left|\frac{1}{4} \boldsymbol{c}-\boldsymbol{b}\right|$ 的最小值的$\sqrt{133}$倍是 $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
There are 11 foreign language translators, among which there are 5 English translators, 4 Japanese translators, and the other 2 are proficient in both English and Japanese. From these, 8 people are selected to form two translation teams, with 4 people translating English and the other 4 translating Japanese, allowing these two teams to work simultaneously. How many different rosters can be made for such an arrangement?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Heinz produces tomato puree by boiling tomato juice. The tomato puree has a certain percentage of water while the tomato juice has more water. Heinz obtains 2.5 litres of tomato puree from 20 litres of tomato juice. What is the percentage of water in the tomato juice?
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Context:
Calculate the value of: $$(\frac {3}{2})^{- \frac {1}{3}} - \frac {1}{3} \times (- \frac {7}{6})^{0} + 8^{ \frac {1}{4}} \times \sqrt[4]{2} - \sqrt {(- \frac {2}{3})^{ \frac {2}{3}}}$$.
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Context:
Find the sum $\displaystyle \sum_{k = 1}^{2023}\dfrac1{1+\tan^2 \left(\dfrac{k\pi}{2\cdot 2024}\right)}$.
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Context:
At a robot racing competition, a certain number of mechanisms were presented. The robots were paired to race the same distance. The protocol recorded the differences in the finishing times between the winner and the loser in each of the races. All the differences were distinct: 1 sec, 2 sec, 3 sec, 4 sec, 5 sec, 6 sec, 7 sec, 8 sec, 9 sec, 13 sec. It is known that during the races, each robot competed against each other robot exactly once, and each robot always ran at the same speed. Determine the time of the slowest mechanism if the best time to complete the distance was 50 seconds.
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Context:
Let \( D \) be a point inside the acute triangle \( \triangle ABC \) such that
\[
\angle ADB = \angle ACB + 90^\circ
\]
and \( AC \cdot BD = AD \cdot BC \). Find the value of \(\frac{AB \cdot CD}{AC \cdot BD}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Initially, a natural number $n$ is written on the blackboard. Then, at each minute, Esmeralda chooses a divisor $d > 1$ of $n$, erases $n$, and writes $n + d$. If the initial number on the board is $2022$, what is the largest composite number that Esmeralda will never be able to write on the blackboard?
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Context:
An equilateral triangle is inscribed in a circle. A smaller equilateral triangle has one vertex coinciding with a vertex of the larger triangle and another vertex on the midpoint of a side of the larger triangle. What percent of the area of the larger triangle is the area of the smaller triangle?
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Context:
Find the $2019^{th}$ strictly positive integer $n$ such that $\binom{2n}{n}$ is not divisible by $5$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that $a$, $b$, $c$ are the sides opposite to angles $A$, $B$, $C$ in $\triangle ABC$ respectively, and $a=2$, $b=\sqrt{6}$, $A+C=2B$, find $A$.
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Context:
Create a six-digit number using the digits 1, 2, 3, 4, 5, and 6 without repetition, such that the digit 1 is not at either end. Additionally, out of the three even digits 2, 4, and 6, there must be exactly one pair of adjacent even digits. How many such six-digit numbers can be formed?
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Context:
On each OMA lottery ticket there is a  $9$ -digit number that only uses the digits  $1, 2$  and  $3$  (not necessarily all three). Each ticket has one of the three colors red, blue or green. It is known that if two banknotes do not match in any of the  $9$  figures, then they are of different colors. Bill  $122222222$  is red,  $222222222$  is green, what color is bill  $123123123$ ?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the number of real solutions $(x,y)$ to the system of equations:
$$
\begin{cases}
\sin(x^2-y) = 0 \\
|x|+|y|=2\pi
\end{cases}
$$
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Context:
Three bags of jelly beans contain 26, 28, and 30 beans.  The ratios of yellow beans to all beans in each of these bags are $50\%$, $25\%$, and $20\%$, respectively.  All three bags of candy are dumped into one bowl.  Which of the following is closest to the ratio of yellow jelly beans to all beans in the bowl?
$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a pyramid $P-ABC$ where $PA=PB=2PC=2$, and $\triangle ABC$ is an equilateral triangle with side length $\sqrt{3}$, the radius of the circumscribed sphere of the pyramid $P-ABC$ is _______.
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Context:
There exist constants $a_1,$ $a_2,$ $a_3,$ $a_4,$ $a_5$ such that
\[\cos^5 \theta = a_1 \cos \theta + a_2 \cos 2 \theta + a_3 \cos 3 \theta + a_4 \cos 4 \theta + a_5 \cos 5 \theta\]for all angles $\theta.$  Find $a_1^2 + a_2^2 + a_3^2 + a_4^2 + a_5^2.$
Please reason step by step, and put your final answer within \boxed{}.


Context:
For an entire month of June, Owen purchased 2 burgers each day. If each burger costs 12 dollars and there was a special deal where Owen could buy 2 burgers and get 1 free, how much money did Owen save in June?
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Context:
Suppose \( x_0, x_1, x_2, \cdots \) is a sequence of numbers such that \( x_0 = 1000 \), and
\[ x_n = -\frac{1000}{n}\left(x_0 + x_1 + x_2 + \cdots + x_{n-1}\right) \]
for all \( n \geq 1 \). Find the value of
\[ \frac{1}{2^{2}} x_0 + \frac{1}{2} x_1 + x_2 + 2 x_3 + 2^2 x_4 + \cdots + 2^{997} x_{999} + 2^{998} x_{1000} \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( a, b \), and \( c \) be complex numbers such that \( |a|=|b|=|c|=1 \). If
\[ \frac{a^{2}}{b c}+\frac{b^{2}}{c a}+\frac{c^{2}}{a b}=1 \]
as well, determine the product of all possible values of \( |a+b+c| \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the work done by the vector field \( F = (x^2 + y^2)\mathbf{i} + (x^2 - y^2)\mathbf{j} \) along the closed curve formed by the line segments from \((0, 0)\) to \((2, 0)\), from \((2, 0)\) to \((2, 2)\), and from \((2, 2)\) to \((0, 0)\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Julian has 80 Facebook friends and 150 Instagram followers. If 60% of his Facebook friends are boys and 40% are girls, and 70% of his Instagram followers are girls while the rest are boys. Meanwhile, Boyd has twice as many friends who are girls on Facebook than Julian and three times the boys among his Instagram followers compared to Julian. Boyd's total number of friends and followers across these two social media platforms are 200. What percentage of Boyd’s friends on both platforms are boys?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the line integral \( \int_C ze^{\sqrt{x^2+y^2}} \, \mathrm{ds} \) where \( C \) is the curve of intersection of the sphere \( x^2+y^2+z^2=a^2 \) and the plane \( x+y=0 \), with \( a > 0 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the partial derivative \( \frac{\partial c^{*}}{\partial c} \) where \( c \in \mathbb{C} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \((X_n)_{n>1}\) be a sequence of independent random variables such that \(X_n \sim \text{Bern}(1-\frac{1}{n^a})\) where \(a>0\) is a constant. Define \(Y_n=\prod_{j=1}^n X_j.\) Determine the values of \(a\) for which \(E[Y_n^2]\) converges to zero.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a natural number \( n \), determine the largest number \( k \) such that for any \( n \) rays in space, no two of which have common points, it is possible to mark \( k \) points on these rays that lie on the same sphere.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $(X_{n})_{n\geq1}$ be a sequence of i.i.d. random variables such that $\mathbb{P}(X_{n} = -1) = \frac{1}{3}$ and $\mathbb{P}(X_{n} = 1) = \frac{2}{3}$. Let $S_{0} = 0$ and for $n\geq 1$ let $S_{n} = \sum_{k=1}^{n} X_{k}$. Determine the value of $\mathbb{P}(\liminf S_{n} = -\infty)$.
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Context:
Determine the values of \( n \) for which an \( n \times n \) board can be completely covered by T-shaped tetrominoes without overlapping or extending off the board.
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Context:
There is an oak tree growing by the road from Yolkino to Palkino, and it is twice as close to Yolkino as it is to Palkino. Fedya, traveling at a constant (and positive) speed from Yolkino to Palkino, was twice as close to the oak tree as he was to Yolkino at 12:00. At 12:40, it turned out again that Fedya was twice as close to the oak tree as he was to Yolkino. When will Fedya arrive in Palkino?
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Context:
Tall Tuna has twice as many tuna fish as Jerk Tuna but also has 30% mackerel in addition. Swell Tuna has fifty percent more tuna fish than Tall Tuna and has 25% more mackerel than Jerk Tuna's mackerel count. If Jerk Tuna has one hundred forty-four tuna fish and eighty mackerel, how many fish (tuna and mackerel) do they have together?
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Context:
The binomial expansion of $\left(\frac{\sqrt{x+1}}{2 \sqrt[4]{x}}\right)^{n}$ is arranged in descending powers of $x$. If the coefficients of the first two terms form an arithmetic sequence, how many terms in this expansion have integer exponents of $x$?
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Context:
John carves a giant marble rectangular prism 8 meters tall with a 2-meter square base.  It has a density of 2700 kg per cubic meter.  How much does it weigh?
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Context:
The 110th anniversary celebration of Guiyang No.1 High School was held on September 30, 2016. During the celebration, 2 volunteers from the 10th grade and 4 volunteers from the 11th grade were randomly selected to serve at Gate 1 to welcome the alumni. The probability that at least one of the selected volunteers is from the 10th grade is ______.
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Context:
从全体正整数1,2,3，···中划去3和4的倍数，但其中凡是5的倍数都保留（例如15,20,60，···等都保留），划完后，将剩下的数从小到大排成一个数列：$a _ { 1 } = 1$,$a _ { 2 } = 2 ,$ $a _ { 3 } = 5$, $a _ { 4 } = 7 ,$.$a _ { 5 } = 1 0 ,$,⋯,求a2011之值．
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Context:
Find \(\sum_{i=1}^{\infty} \sum_{j=1}^{\infty} \frac{1}{i^2 j + 2ij + ij^2}\).
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Context:
Compute the argument of the sum:
\[ e^{2\pi i/40} + e^{6\pi i/40} + e^{10\pi i/40} + e^{14\pi i/40} + e^{18\pi i/40} + e^{22\pi i/40} + e^{26\pi i/40} + e^{30\pi i/40} + e^{34\pi i/40} + e^{38\pi i/40} \]
and express it in the form \( r e^{i \theta} \), where \( 0 \le \theta < 2\pi \).
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Context:
There are 3 boys and 3 girls arranged in a row. If it is required that any two boys and any two girls cannot be adjacent, and boy A and girl B must be adjacent, then find the total number of such arrangements. Express your answer as a numerical value.
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Context:
Given a square grid of size $2023 \times 2023$, with each cell colored in one of $n$ colors. It is known that for any six cells of the same color located in one row, there are no cells of the same color above the leftmost of these six cells and below the rightmost of these six cells. What is the smallest $n$ for which this configuration is possible?
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Context:
Fibonacci, an Italian mathematician, is known for the Fibonacci sequence, where the sequence $\{a_{n}\}$ satisfies $a_{1}=a_{2}=1$, and $a_{n+2}=a_{n+1}+a_{n}$, then the sequence $\{a_{n}\}$ is called the Fibonacci sequence. Given that the sequence $\{a_{n}\}$ is a Fibonacci sequence, and the sequence $\{b_{n}\}$ satisfies ${b_{n+3}}+{(-1)^{{a_n}}}{b_n}=n$, if the sum of the first $12$ terms of the sequence $\{b_{n}\}$ is $86$, then $b_{1}+b_{2}=$____.
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Context:
Initially, five variables are defined: $a_1 = 1, a_2 = 0, a_3 = 0, a_4 = 0, a_5 = 0$. On a turn, Evan can choose an integer $2 \le i \le 5$. Then, the integer $a_{i-1}$ will be added to $a_i$. For example, if Evan initially chooses $i = 2$, then now $a_1 = 1, a_2 = 0 + 1 = 1, a_3 = 0, a_4 = 0, a_5 = 0$. Find the minimum number of turns Evan needs to make $a_5$ exceed 1,000,000.
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Context:
Find the limit of the expression as \(x \to -\infty\): \[ \frac{5}{\sqrt{x}\left( (1+\frac{9}{x^2})^{1/2}-(1+\frac{2}{x^2})^{1/2} \right)} \]
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Context:
Consider a Brownian motion $W_t$ and the Bachelier model $S_t = 1 + \mu t + \sigma W_t$ for the stock price $S_t$. Find the value of an option that pays $1(S_1 > 1)$. Use the risk-neutral valuation argument to determine the option's value.
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Context:
Evaluate the definite integral \( \int_{-\infty}^{\infty} \frac{\sin^2(\pi Rx)}{R(\pi x)^2}dx \) for \( R > 0 \).
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Context:
For a function $h(t)$, it is given that $\int_{-\infty}^{\infty} h(t)e^{-j\omega t}dt=\frac{1}{\omega ^2+1}$. If $x(t)=\int_{-\infty}^{t}h(\tau)d\tau$, find the value of $\int_{-\infty}^{\infty} x(t)e^{-j\omega t}dt$ at $\omega=1$. \(\boxed{?}\)
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Context:
Determine one of the symmetry axes of the function $y = \cos 2x - \sin 2x$.
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Context:
Let \( a \in \mathbf{R} \) be a constant and consider the function \( f(x) = (a - x)|x| \). Suppose \( f(x) \) has an inverse function \( f^{-1}(x) \). Determine the range of values for the real number \( m \) such that the inequality \( f^{-1}(x^{2} + m) < f(x) \) holds for all \( x \in [-2,2] \).
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Context:
Given the prime numbers \( p \) and \( q \) satisfying \( p q + p^{2} + q^{2} = 199 \) where \( p < q \), find the value of the algebraic expression \(\frac{2}{p(p+1)}+\frac{2}{(p+1)(p+2)}+\cdots+\frac{2}{(q-1) q} \).
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Context:
Determine the number of onto relations from the set \( A = \{1, 2, 3, \ldots, m\} \) to the set \( B = \{1, 2, 3, \ldots, n\} \), where the range of the relation is exactly \( B \).
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Context:
In triangle \( ABC \), the sides \( AC = 14 \) and \( AB = 6 \) are known. A circle with center \( O \), constructed on side \( AC \) as its diameter, intersects side \( BC \) at point \( K \). It is given that \(\angle BAK = \angle ACB\). Find the area of triangle \( BOC \).
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Context:
已知多项式 f(z)=z^3+az^2+bz+c(a,b,c\in\bZ) 的所有根的模均为 20 或 15，则这样的多项式的个数为__________.
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Context:
Consider a graph $G$ where each vertex represents a 3-element subset of the set $\{1, 2, 3, 4, 5, 6\}$. Two vertices, corresponding to subsets $A$ and $B$, are connected by an edge if they share exactly one element, and the cost of the edge is equal to the shared element. Determine the minimum cost of a spanning tree for this graph.
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Context:
Calculate the quadratic variation of the process $Y_t = B_{2t}$, where $B_t$ is a Brownian motion with $\\langle B \\rangle_t = t$. Express the quadratic variation $\\langle Y \\rangle_t$ in terms of $t$.
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Context:
Determine the intersection of the fields $\mathbb{F}_{2^{4}}$ and $\mathbb{F}_{2^{6}}$ within the algebraic closure $\overline{\mathbb{F}}_2$. Use the fact that the lattice of finite extensions of a finite field is isomorphic to the lattice of positive integers.
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Context:
Find the range of values for $p$ such that the interval $(1,2)$ is between the roots of the equation $2^x + 2^{2-x} + p = 0$.
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Context:
On Monday a school library was attended by 5 students, on Tuesday, by 6, on Wednesday, by 4, on Thursday, by 8, and on Friday, by 7. None of them have attended the library two days running. What is the least possible number of students who visited the library during a week?
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Context:
After a storm deposits 120 billion gallons of water into the city reservoir, the reservoir is 85% full. If the original contents of the reservoir totaled a certain amount, and the reservoir was approximately 55.00000000000001% full before the storm, how many gallons of water can the reservoir hold when it is full?
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Context:
Last year Jessica paid $1000 for rent, $200 for food, and $100 for car insurance each month. This year her rent goes up by some percentage, food costs increase by 50%, and the cost of her car insurance triples because she was at fault in an accident. Jessica pays $7200 more for her expenses over the whole year compared to last year. By what percentage did Jessica's rent increase this year?
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Context:
An archipelago consists of an infinite number of islands along the southern shore of an endless sea. Each island is connected to the next by a bridge, and each island is also connected to the shore by a bridge. During a strong earthquake, each bridge independently has a probability of 0.5 of being destroyed. What is the probability that it will still be possible to travel from the first island to the shore using the remaining bridges?
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Context:
Define a  $\textit{tasty residue}$  of  $n$  to be an integer  $1<a<n$  such that there exists an integer  $m>1$  satisfying \[a^m\equiv a\pmod n.\] Find the number of tasty residues of  $2016$ .
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Context:
Call a number orz if it is a positive integer less than $2024$. Call a number admitting if it can be expressed as $a^2-1$ where $a$ is a positive integer. Finally call a number muztaba if it has exactly $4$ positive integer factors. Find the number of muztaba admitting orz numbers.
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Context:
Suppose $x$ and $y$ are real numbers such that\[x^2+xy+y^2=2\qquad\text{and}\qquad x^2-y^2=\sqrt5.\]The sum of all possible distinct values of $|x|$ can be written in the form $\textstyle\sum_{i=1}^n\sqrt{a_i}$, where each of the $a_i$ is a rational number. If $\textstyle\sum_{i=1}^na_i=\frac mn$ where $m$ and $n$ are positive realtively prime integers, what is $100m+n$?

Proposed by David Altizio
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Context:
Let $(a_n)$ and $(b_n)$ be the sequences of real numbers such that
\[ (2 + i)^n = a_n + b_ni \]for all integers $n\geq 0$, where $i = \sqrt{-1}$. What is
\[\sum_{n=0}^\infty\frac{a_nb_n}{7^n}\,?\]
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Context:
For a sequence $s = (s_1, s_2, . . . , s_n)$, define
$$F(s) =\sum^{n-1}_{i=1} (-1)^{i+1}(s_i - s_{i+1})^2.$$
Consider the sequence $S =\left(2^1, 2^2, . . . , 2^{1000}\right)$. Let $R$ be the sum of all $F(m)$ for all non-empty subsequences $m$ of $S$. Find the remainder when $R$ is divided by $1000$.

Note: A subsequence is a sequence that can be obtained from another sequence by deleting some non-negative number of values without changing the order.
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Context:
Determine the total surface area of a cube if the distance between the non-intersecting diagonals of two adjacent faces of this cube is 8. If the answer is not an integer, round it to the nearest whole number.
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Context:
Find the area of an isosceles trapezoid if its height is $h$ and the lateral side is seen from the center of the circumscribed circle at an angle of $60^{\circ}$.
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Context:
Let \(\Gamma\) be a circle in the plane and \(S\) be a point on \(\Gamma\). Mario and Luigi drive around the circle \(\Gamma\) with their go-karts. They both start at \(S\) at the same time. They both drive for exactly 6 minutes at constant speed counterclockwise around the track. During these 6 minutes, Luigi makes exactly one lap around \(\Gamma\) while Mario, who is three times as fast, makes three laps.

While Mario and Luigi drive their go-karts, Princess Daisy positions herself such that she is always exactly in the middle of the chord between them. When she reaches a point she has already visited, she marks it with a banana.

How many points in the plane, apart from \(S\), are marked with a banana by the end of the race?
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Context:
If the shortest distance from a point on the ellipse $\frac{y^2}{16} + \frac{x^2}{9} = 1$ to the line $y = x + m$ is $\sqrt{2}$, find the minimum value of $m$.
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Context:
A circle touches the extensions of two sides \( AB \) and \( AD \) of the square \( ABCD \) with a side length of \( 2 \sqrt{3} \) cm. From point \( C \), two tangents are drawn to this circle. Given that the angle between the tangents is \( 30^{\circ} \), find the radius of the circle. Use the fact that \( \sin 15^{\circ} = \frac{\sqrt{3}-1}{2 \sqrt{2}} \).
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Context:
In an isosceles trapezoid \(ABCD\), \(AB\) is parallel to \(CD\) and \(AB > CD\). Define a hyperbola with foci at \(A\) and \(B\) that passes through point \(D\), and an ellipse with foci at \(C\) and \(D\) that passes through point \(A\). Let the eccentricities of the hyperbola and the ellipse be \(e_1\) and \(e_2\) respectively. Find the value of \(e_1 e_2\).
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Context:
In regular hexagon $ABCDEF$, points $W$, $X$, $Y$, and $Z$ are chosen on sides $\overline{BC}$, $\overline{CD}$, $\overline{EF}$, and $\overline{FA}$ respectively, so lines $AB$, $ZW$, $YX$, and $ED$ are parallel and equally spaced. What is the ratio of the area of hexagon $WCXYFZ$ to the area of hexagon $ABCDEF$?
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Context:
Consider the set $X = \mathbb{R}^2 - \{(0,0),(1,0),(2,0),(3,0)\}$. Let $V$ be the vector space of irrotational vector fields over $X$, and let $W$ be the vector space of conservative vector fields over $X$. Determine the dimension of the quotient vector space $V/W$. Provide your answer as a single integer.
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Context:
Find the minimum value of the sum \( \sum_{i=1}^{n} a_{i} \). Express your answer in terms of \( n \).
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Context:
Let $a_n$ ($n=2,3,4,\ldots$) be the coefficient of the term containing $x$ in the expansion of $(3- \sqrt{x})^n$. Calculate the value of $$\frac{1}{a_2} + \frac{3}{a_3} + \frac{9}{a_4} + \ldots + \frac{3^{2011}}{a_{2013}}$$ and express your answer as a single fraction.
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Context:
Let $\tau = \frac{1 + \sqrt{5}}{2}.$  Find
\[\sum_{n = 0}^\infty \frac{\lfloor \tau^n \rceil}{2^n}.\]Note: For a real number $x,$ $\lfloor x \rceil$ denotes the integer closest to $x.$
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Context:
Dexter's Laboratory has $2024$ robots, each with a program set up by Dexter. One day, his naughty sister Dee Dee intrudes and writes an integer in the set $\{1, 2, \dots, 113\}$ on each robot's forehead. Each robot detects the numbers on all other robots' foreheads and guesses its own number based on its program, individually and simultaneously.

Find the largest positive integer $k$ such that Dexter can set up the programs so that, no matter how the numbers are distributed, there are always at least $k$ robots who guess their numbers correctly.
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Context:
What is the smallest integer $n$ , greater than one, for which the root-mean-square of the first $n$ positive integers is an integer?
$\mathbf{Note.}$ The root-mean-square of $n$ numbers $a_1, a_2, \cdots, a_n$ is defined to be \[\left[\frac{a_1^2 + a_2^2 + \cdots + a_n^2}n\right]^{1/2}\]
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Context:
The numbers $1, 2,..., 2023$ are written on the board in this order. We can repeatedly perform the following operation with them: We select any odd number of consecutively written numbers and write these numbers in reverse order. How many different orders of these $2023$ numbers can we get?

Example: If we start with only the numbers $1, 2, 3, 4, 5, 6$, we can perform the following steps
$$1, 2, 3, 4, 5, 6 \to 3, 2, 1,4, 5, 6 \to 3, 6, 5, 4, 1, 2 \to 3, 6, 1, 4, 5, 2 \to ...$$
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Context:
In isosceles triangle \( \triangle ABC \), where \( \overline{AD} \) is a median and \( \overline{EC} \) is a cevian, it is given that \( \overline{AD} = 2\overline{EC} \). Find the value of \( \cos{B} \).
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Context:
What is the perimeter of the triangle formed by the points of tangency of the incircle of a 5-7-8 triangle with its sides?
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Context:
How many pairs of positive integer solutions \((x, y)\) satisfy \(\frac{1}{x+1} + \frac{1}{y} + \frac{1}{(x+1) y} = \frac{1}{1991}\)?
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Context:
Let $SP_1P_2P_3EP_4P_5$ be a heptagon. A frog starts jumping at vertex $S$. From any vertex of the heptagon except $E$, the frog may jump to either of the two adjacent vertices. When it reaches vertex $E$, the frog stops and stays there. Find the number of distinct sequences of jumps of no more than $12$ jumps that end at $E$.
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Context:
Seryozha arranged the numbers from 1 to 8 in circles so that each number, except one, was used exactly once. It turned out that the sums of the numbers on each of the five lines are equal. Which number did Seryozha not use?
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Context:
Find the maximum possible value of  $k$  for which there exist distinct reals  $x_1,x_2,\ldots ,x_k $  greater than  $1$  such that for all  $1 \leq i, j \leq k$ ,  $$ x_i^{\lfloor x_j \rfloor }= x_j^{\lfloor x_i\rfloor}. $$  *Proposed by Morteza Saghafian*
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Context:
My frisbee group often calls "best of five" to finish our games when it's getting dark, since we don't keep score. The game ends after one of the two teams scores three points (total, not necessarily consecutive). If every possible sequence of scores is equally likely, find the expected score of the losing team. The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.
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Context:
Evaluate the limit \(\lim_{x\to \infty} \frac{\Gamma(x+1,x)}{\Gamma(x+1)}\), where \(x\) is a positive integer.
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Context:
Find the expected value $E(X_t)$ for the stochastic differential equation given by $dX_t = (X_t + 2t) \,dW_t + (t^2 + 1) \,dW_t$ with the initial condition $X_0=0$.
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Context:
Consider a collection of 20 marbles drawn from an infinite pool of pink and blue marbles, with replacement. The probability of drawing a pink marble is 0.4, and the probability of drawing a blue marble is 0.6. You draw 20 marbles and check if there are at least 12 pink marbles in the collection. If true, you note this result. Then, you randomly discard one marble from the collection and draw a new marble to replace it. You repeat this process of inspecting, noting the result, discarding, and replenishing a total of 100 times. What is the probability that you will encounter at least one instance where the collection has at least 12 pink marbles during these 100 cycles?
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Context:
Determine the value of $n$ for which the space $\mathbb{R}^{2n} - \mathbb{R}^{n}$ is path connected, where $\mathbb{R}^{n}$ is embedded in the first $n$ coordinates of $\mathbb{R}^{2n}$.
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Context:
A shop owner professes to sell his articles at a certain cost price, but he uses false weights with which he cheats by 12% while buying and by 30% while selling. What is his percentage profit?
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Context:
Square $AXYZ$ is inscribed in equiangular hexagon $ABCDEF$ with $X$ on $\overline{BC}$, $Y$ on $\overline{DE}$, and $Z$ on $\overline{EF}$. Suppose that $AB=40$, and $EF=41(\sqrt{3}-1)$. What is the side-length of the square?
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Context:
Alice and the Cheshire Cat play a game. At each step, Alice either (1) gives the cat a penny, which causes the cat to change the number of (magic) beans that Alice has from $n$ to $5n$ or (2) gives the cat a nickel, which causes the cat to give Alice another bean. Alice wins (and the cat disappears) as soon as the number of beans Alice has is greater than 2008 and has last two digits 42. What is the minimum number of cents Alice can spend to win the game, assuming she starts with 0 beans?
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Context:
11 个黑球， 9 个红球，依次取出，剩下全是一种颜色就结束，求最后只剩下红球的概率？原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值
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Context:
Let $A,B,C$ be angles of an acute triangle with
$\begin{aligned}
\cos^2 A + \cos^2 B + 2 \sin A \sin B \cos C &= \frac{15}{8} \text{ and} \\
\cos^2 B + \cos^2 C + 2 \sin B \sin C \cos A &= \frac{14}{9}.
\end{aligned}$
There are positive integers $p$, $q$, $r$, and $s$ for which\[ \cos^2 C + \cos^2 A + 2 \sin C \sin A \cos B = \frac{p-q\sqrt{r}}{s}, \]where $p+q$ and $s$ are relatively prime and $r$ is not divisible by the square of any prime. Find $p+q+r+s$.

Note: due to an oversight by the exam-setters, there is no acute triangle satisfying these conditions. You should instead assume $ABC$ is obtuse with $\angle B > 90^{\circ}$.
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Context:
Find the limit of the following sequence: $$\lim_{n\to \infty} \frac{\prod_{i = 0}^{n}\pars{i + 2/3}}{\prod_{j = 0}^{n}\pars{j + 1/3}}$$
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Context:
Find the limit of the series as \( n \to \infty \):
\[ u_n(x) = \frac{1}{x+1} + \frac{2}{(x+1)(x+2)} + \frac{3}{(x+1)(x+2)(x+3)} + \cdots + \frac{n x^{n-1}}{(x+1)(x+2)\cdots(x+n)} \]
Evaluate \( \lim_{n \to \infty} u_n(x) \).
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Context:
A ship travels downstream from point $A$ to point $B$ and takes 1 hour. On the return trip, the ship doubles its speed and also takes 1 hour. How many minutes will it take to travel from point $A$ to point $B$ if the ship initially uses double its speed?
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Context:
Find the smallest positive real number $c$ such that for all nonnegative real numbers $x, y,$ and $z$, the following inequality holds:
\[\sqrt[3]{xyz} + c |x - y + z| \ge \frac{x + y + z}{3}.\]
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Context:
Circle $\omega$ has radius $5$ and is centered at $O$.  Point $A$ lies outside $\omega$ such that $OA=13$.  The two tangents to $\omega$ passing through $A$ are drawn, and points $B$ and $C$ are chosen on them (one on each tangent), such that line $BC$ is tangent to $\omega$ and $\omega$ lies outside triangle $ABC$.  Compute $AB+AC$ given that $BC=7$.
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Context:
Let $g$ be a non-negative function defined on the interval $[0, \pi]$. If $\int_0^x \cos g'(t) dt = \int_0^x g(t) dt$ for $0 \leq x \leq \pi$ and $g(0) = 0$, find the possible values of $g\left(\frac{\pi}{2}\right)$.
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Context:
Xiao Hua, Xiao Li, and Xiao Xia all need to travel from place A to place B. Xiao Hua and Xiao Li set off from place A together at 6 am. Xiao Hua walks at a speed of 5 kilometers per hour, and Xiao Li walks at a speed of 4 kilometers per hour. Xiao Xia does not leave place A until 8 am. By 6 pm, Xiao Hua and Xiao Xia arrive at place B simultaneously. The question is: At what time does Xiao Xia catch up with Xiao Li?
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Context:
Positive real numbers $x \neq 1$ and $y \neq 1$ satisfy $\log_2{x} = \log_y{16}$ and $xy = 64$. What is $(\log_2{\tfrac{x}{y}})^2$?
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Context:
Evaluate the limit \( \lim_{n\to\infty} \sqrt[n]{\sum_{i=1}^m |x_i|^n} \) for any vector \(x\in\mathbb{R}^m\), where \(x_p=\min_{1\le i\le m} |x_i|\) and \(x_q=\max_{1\le i\le m} |x_i|\).
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Context:
Let \( F = \langle a, b \rangle \) be a free group. If \( N \) is the normal subgroup of \( F \) generated by \( a^2 \) and \( b^2 \), find the index of \( N \) in \( F \).
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Context:
what is the remainder when 121212 . . . . . . . ( 300 ) digits is divided by 99 .
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Context:
Determine the quadratic variation $[X]_t$ of the process $X_t = \int_0^t B_s \, ds$, where $B_t$ is a standard Brownian motion.
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Context:
Evaluate the limit \( \lim_{n\to\infty} \sum_{r=0}^{p-k}\frac{1}{k(n+r)} \) using integration techniques.
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Context:
Calculate the variance of the present value of a whole life annuity, $Y$, using the equation $$VarY=\frac{^2A_x-(A_x)^2}{d^2},$$ where $$A_x=\sum_{j=0}^\infty v^{j+1} {}_j| q_{x};$$ $$^2 A_x=\sum_{j=0}^\infty (v^2)^{j+1} {}_j| q_{x}$$ and $\ddot{a}_x=\frac{1-A_x}{d}$ with $d=\frac{i}{1+i}$. Given that $\ddot{a}_x=10$ for $i'=1/24=e^\delta-1$ and $\ddot{a}_x=6$ for $i=e^{2\delta}-1$, find the variance of $Y$.}
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Context:
The average age of 50 students in a class is 20 years. Out of these, the average age of 15 students is 18 years, that of another 15 students is 22 years, the average age of 10 students is 25 years, and the average age of 9 students is 24 years. Find the age of the 50th student.
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Context:
Let the expression x # y # z denote the product of the consecutive multiples of 3 between x and y, and between y and z, inclusive. What is the sum of the exponents in the prime factorization of 21 # 30 # 36 with the constraint that the sum of exponents of all prime factors other than 3 must be even?
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Context:
Points $A$, $B$, $C$, and $D$ lie on a circle such that chords $\overline{AC}$ and $\overline{BD}$ intersect at a point $E$ inside the circle. Suppose that $\angle ADE = \angle CBE = 75^\circ$, $BE=4$, and $DE=8$. The value of $AB^2$ can be written in the form $a+b\sqrt{c}$ for positive integers $a$, $b$, and $c$ such that $c$ is not divisible by the square of any prime. Find $a+b+c$.
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Context:
Find the smallest natural number for which there exist that many natural numbers such that the sum of the squares of their squares is equal to  $ 1998. $ *Gheorghe Iurea*
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Context:
Given a sequence of sums involving six integers, determine the minimum number of even integers among them. The sequence is as follows: the sum of the first two integers is 26, adding two more integers results in a sum of 41, and adding the final two integers to the previous sum gives 57. Express your answer as a single integer representing the minimum number of even integers required.
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Context:
设 $A=\{x \mid-2 \leqslant x \leqslant a\}, B=\{y \mid y=2 x+3, x \in A\}, C=\left\{z \mid z=x^2, x \in A\right\}$,且 $C \subseteq B$, 则实数 $a$ 的取值集合是 $\qquad$ .
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Context:
Let's determine

$$
\left(\frac{7 \sqrt[3]{x}}{2 \sqrt{y}}-\frac{\sqrt[3]{y}}{7 \sqrt[5]{x^{2}}}\right)^{8}
$$

the sixth term of the expansion.
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Context:
Determine the values of parameter \( a \) for which the inequality

$$
\sqrt{1 + x^4} > 1 - 2ax + x^2
$$

holds for all positive \( x \).
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Context:
Let $V=\{(x,y,z)\in \mathbb{R}:\frac{1}{4}\le x^2+y^2+z^2\le1\}$ and $\vec{F}=\frac{x\hat{i}+y\hat{j}+z\hat{k}}{(x^2+y^2+z^2)^2}$ for $(x,y,z)\in V$. Find the flux of $\vec{F}$ through the entire boundary of $V$.
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Context:
Given sets $A=\{2, a-1\}$, $B=\{a^2-7, -1\}$, and $A\cap B=\{2\}$, find the real number $a$.
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Context:
Let  $s_n$  be the number of solutions to  $a_1 + a_2 + a_3 +a _4 + b_1 + b_2 = n$ , 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  $\{ 1, 2, 3, 4\}$ . Find the number of  $n$  for which  $s_n$  is odd.

*Author: Alex Zhu*

Clarification$s_n$  is the number of *ordered* solutions  $(a_1, a_2, a_3, a_4, b_1, b_2)$  to the equation, where each  $a_i$  lies in  $\{2, 3, 5, 7\}$  and each  $b_i$  lies in  $\{1, 2, 3, 4\}$ .
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Context:
Let \( ABCD \) be a quadrilateral with an inscribed circle \( \omega \) that has center \( I \). If \( IA = 5 \), \( IB = 7 \), \( IC = 4 \), and \( ID = 9 \), find the value of \( \frac{AB}{CD} \).
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Context:
There are four positive integers that are divisors of each number in the list $$20, 40, 100, 80, 180.$$ Find the sum of these four positive integers.
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Context:
Let  $ABCD$  be a square with the point of intersection  $O$  of the diagonals and let  $P,\ Q,\ R,\ S$  be the points which are on the segments  $OA,\ OB,\ OC,\ OD$ , respectively such that  $OP=3,\ OQ=5,\ OR=4$ . If the points of intersection of the lines  $AB$  and  $PQ$ , the lines  $BC$  and  $QR$ , the lines  $CD$  and  $RS$  are collinear, then find the length of the segment  $OS$ .
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Context:
Let \( n \) be the integral part of \( \frac{1}{\frac{1}{1980}+\frac{1}{1981}+\cdots+\frac{1}{2009}} \). Find the value of \( n \).
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Context:
Find the sum of the infinite series \(\sum_{k=1}^\infty \frac{(5k+7)}{k^3+4k^2+3k}\).
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Context:
Given that $\log_3M=a_1+b_1$ and $\log_5 M=a_2+b_2$, where $a_1,a_2$ are natural numbers and $b_1,b_2 \in [0,1)$, and $a_1a_2=6$, determine the number of integral values of $M$.
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Context:
设 $A, B, C$ 是 $\triangle A B C$ 的三个内角, 则 $3 \cos A+2 \cos 2 B+\cos 3 C$ 的取值范围为 $\qquad$. 答案的标准形式为[-\frac{a}{b},c]，请给出该范围的两端点a+b+c之和。
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Context:
Determine how many roots the polynomial \( z^4 + 3z^2 + z + 1 \) has in the right-half complex plane, where \( \text{Re}(z) > 0 \).
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Context:
Express the infinite series \( \sum_{n=1}^{\infty} \frac{i^n(1+i)}{2} \) as a specific value.
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Context:
XI OM - I - Task 3

Each side of a triangle with a given area $ S $ is divided into three equal parts, and the points of division are connected by segments, skipping one point to form two triangles. Calculate the area of the hexagon that is the common part of these triangles.
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Context:
Evaluate the integral \( \int_0^\infty \frac{\sqrt{x}}{x^3+1}\ \mathrm{d}x \) using residue theory with a keyhole contour to exclude the branch cut.
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Context:
For which values of \( b > 0 \) is the set \( \{e_n(t) = e^{int} : n \in \mathbb{Z}\} \) a basis for \( L^2([0,b]) \), meaning that the orthogonal complement \( V^\perp = 0 \) where \( V = \text{span}\{e_n : n \in \mathbb{Z}\} \)?
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Context:
Thomas is now 6 years younger than Roy but 4 years older than Kelly. Roy is now 8 years older than Julia and half of that amount older than Kelly. If in 2 years, Roy will be 3 times as old as Julia, and Thomas will be twice as old as Kelly, what would be the product of Roy's, Kelly's, and Thomas' ages in 2 years?
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Context:
Use the Divergence Theorem to evaluate the surface integral \( \iint_S F \cdot dS \), where \( F = (x^3y + z)i + xze^yj + (xy − 3z^4)k \) and \( S \) consists of the five lower faces of the cube \([0, 1]\times[0, 1]\times[0, 1]\), excluding the face with \( z = 1 \). Hint: Close the surface to apply the theorem.
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Context:
During the weekends, Eli delivers milk in the complex plane. On Saturday, he begins at $z$ and delivers milk to houses located at $z^{3}, z^{5}, z^{7}, \ldots, z^{2013}$, in that order; on Sunday, he begins at 1 and delivers milk to houses located at $z^{2}, z^{4}, z^{6}, \ldots, z^{2012}$, in that order. Eli always walks directly (in a straight line) between two houses. If the distance he must travel from his starting point to the last house is $\sqrt{2012}$ on both days, find the real part of $z^{2}$.
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Context:
Evaluate the integral $\int_{-2}^{2}(x^2\sin x + \sqrt{4-x^2})dx$.
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Context:
Determine the smallest value of $k$ such that the inequality $$2A+3B+k(B+2C) \le \frac{k+3}{6} (2A+5B+12C)$$ holds for all positive real numbers $A, B, C$ where $A=x^3+y^3+z^3$, $B=x^2y+xy^2+z^2x+zx^2+y^2z+yz^2$, and $C=xyz$.
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Context:
Find the length of the arc of the semicubical parabola defined by the equation \( y^{2}=x^{3} \) from the origin to the point where \( x=5 \).
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Context:
Determine the number of distinct intersection points formed when perpendicular lines are drawn from three non-collinear points $Q_1, Q_2, Q_3$, located on the same side of a straight line $l$, to the lines formed by joining any two of the four collinear points $P_1, P_2, P_3, P_4$, excluding the point from which the perpendicular is drawn.
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Context:
How many lines in a three dimensional rectangular coordiante system pass through four distinct points of the form $(i,j,k)$ where $i,j,$ and $k$ are positive integers not exceeding four?
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Context:
Determine the number of times a biased coin must be flipped to estimate the probability of it landing heads with an accuracy of 0.01 and a confidence level of 99%.
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Context:
Determine the number of real numbers required to describe all possible arrangements of a quantum register consisting of 3 qubits.
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Context:
An urn is filled with coins and beads, all of which are either silver or gold. Twenty percent of the objects in the urn are beads. Forty percent of the coins in the urn are silver. What percent of the objects in the urn are gold coins?
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Context:
Let $X_1, X_2, X_3$ be independent random variables each uniformly distributed on the interval $(0, 1)$. Find the expected value of the expression $E\left[\frac{X_1 + X_2}{X_1 + X_2 + X_3}\right]$. Provide your answer as a single value.
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Context:
In a programming contest with 2008 participants, the participants are divided into two groups in each round. Determine the minimum number of rounds required so that every pair of participants has been in the same group at least once.
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Context:
Determine the norm of the operator \( A: L^{1}[1,2] \to L^{1}[0,1] \) defined by \( (Af)(x) = \int_{1}^{2} \frac{f(t)}{x+t} \, dt \).
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Context:
In triangle $\triangle ABC$, the in-centers are $I(0,0)$, $I_1(2,3)$, and $I_2(5,7)$, and the ex-center is $I_3$. Calculate the distance between the orthocenters of triangles $\triangle II_1I_3$ and $\triangle I_1I_2I_3$. Provide your answer as a single numerical value.
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Context:
Find the second derivative $F''(\pi)$ for the function $$F(x)=\int_{x}^{\sin(x)} \left(\int_{0}^{\sin(t)}\sqrt{1+u^4}\text{d}u\right)\text{d}t.$$ Use the Fundamental Theorem of Calculus and the chain rule to evaluate this double integral.
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Context:
Determine the number of endomorphisms on the direct sum of the groups \( \mathbb{Z}_2 \) and \( \mathbb{Z}_4 \).
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Context:
Determine the minimum width required for an infinite strip of paper to be able to cut out any triangle with an area of 1.
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Context:
Given that the equation \( ax^3 + bx^2 + cx + d = 0 \) has three distinct solutions, determine the number of distinct solutions for the equation:
\[
4(ax^3 + bx^2 + cx + d)(3ax + b) = (3ax^2 + 2bx + c)^2.
\]
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Context:
Given that the service time follows an exponential distribution with a rate of 12 customers per hour, calculate the probability that a customer waits for less than 3 minutes.
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Context:
Consider a square matrix \( M \) with dimensions \( n \times n \), containing only 0s and 1s. Each column of \( M \) sums to a constant \( k \). Additionally, the following equation holds: \( M^T M = kI + J \), where \( I \) is the identity matrix and \( J \) is a matrix of all 1s. Determine the relationship between \( n \) and \( k \).
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Context:
Solve the initial value problem: \( y' + y = \delta(t) \) with \( y(0) = 0 \).
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Context:
Let $L^2[0,10]=\{f:[0,10]\rightarrow \mathbb{R} \mid f \text{ is Lebesgue measurable and } \int_0^{10} f^2(x)\,dx < \infty \}$ be equipped with the norm $\| f \|=\left(\int_0^{10} f^2(x)\,dx \right)^{1/2}$. Let $T$ be the linear functional on $L^2[0,10]$ given by $T(f)=\int_0^2 f(x)\,dx - \int_3^{10} f(x)\,dx$. Find the norm $\| T \|$.
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Context:
An explorer is trapped in a cave with three tunnels. The first tunnel leads to the surface after 1 day, the second leads back to the cave in 4 days, and the third leads back to the cave in 2 days. The explorer chooses the first, second, and third tunnels with probabilities 0.2, 0.3, and 0.5, respectively. Given that the expectation of the number of days until the explorer is free has been calculated, compute the variance of the number of days until the explorer is free.
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Context:
Determine the values of $p$ for which the series \( \sum_{n=N}^\infty \frac{1}{n(\log(n))(\log\log(n))\dotsm(\log_{j}(n))(\log_{j+1}(n))^p} \) converges, where $N$ is a positive integer and $\log_j(n)$ denotes the $j$th iteration of the logarithm function.
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Context:
Given that each particle maintains a direction towards the particle at the next corner, the time when particles will meet each other is given by \( \frac{2a}{\sqrt{n} v} \). Find the value of \( n \).
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Context:
Given that the solution set of $f(\cos x) \geq 0$ is $\left[0, \frac{\pi}{2}\right]$, find the solution set of the inequality $f(\sin x) \geq 0$. Provide your answer in interval notation.
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Context:
For each integer $  n \geq 2  $, let $  S_n  $ be the sum of all products $  j k  $, where $  j  $ and $  k  $ are integers and $  1 \leq j < k \leq n  $. What is the sum of the 10 least values of $  n  $ such that $  S_n  $ is divisible by 3?
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Context:
Determine the center of gravity of a hemispherical shell with radius $a$, assuming a uniform density of 1. The shell is positioned with its base on the $xy$-plane.
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Context:
Solve the equation for $x$ without using inverse functions or a calculator: $$2 - 3 \cos 2x - 5 \sin 2x + \cos 4x - \sin 4x = 0$$
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Context:
Determine all possible values of \(\frac{a}{a+b+d} + \frac{b}{a+b+c} + \frac{c}{b+c+d} + \frac{d}{a+c+d}\) for positive reals \(a, b, c, d\).
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Context:
Reese joined a Pinterest group where members contributed an average of 10 pins per day. The group owner deleted older pins at the rate of 5 pins per week per person. The group has 20 people. After Reese had been a member for a month, there were 6,600 pins. What was the initial total number of pins in the group?
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Context:
Find \( g(2022) \) if for any real numbers \( x, y \) the following equality holds:

\[ g(x-y) = g(x) + g(y) - 2021(x+y) \]
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Context:
$M$ is the midpoint of the height $B D$ in an isosceles triangle $A B C$. Point $M$ serves as the center of a circle with radius $M D$. Find the angular measure of the arc of the circle enclosed between the sides $B A$ and $B C$, given that $\angle B A C = 65^{\circ}$.
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Context:
设 $a, b$ 为实数, 若一元二次不等式 $x^{2}+b<a x$ 的解集为 $(1,5)$, 请计算一元二次不等式 $x^{2}-a>b x$ 的解集的区间端点和。
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Context:
A tournament in Anchurian checkers is conducted over multiple rounds. If the number of participants in a round is even, they are split into random pairs. If the number is odd, random pairs are created, and one player is randomly left out and does not participate in the round. No ties are possible, and the loser of each pair is eliminated from the tournament, while the winners and the player without a pair, if any, move to the next round under the same rules. This continues until only two players remain, who then play a final match to determine the tournament winner.

The tournament started with 26 participants, all of whom play equally well, meaning each player has an equal chance of winning against any opponent. Among these participants are Denis and Oleg. Find the probability that Denis and Oleg will play against each other at some point during the tournament.
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Context:
What is the minimum number of sides that an odd-sided polygon (not necessarily convex) can have, which can be divided into parallelograms?
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Context:
In the phrases "虚有其表", "表里如一", "一见如故", and "故弄玄虚", each Chinese character represents one of 11 consecutive non-zero natural numbers. The same character represents the same number, and different characters represent different numbers. Additionally, "表" > "一" > "故" > "如" > "虚", and the sum of the numbers represented by the four characters in each phrase is 21. What is the maximum value that "弄" can represent?
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Context:
How many distinct 3x3 matrices with entries in \(\{0,1\}\) are there such that each row is non-zero, distinct, and each matrix is of real rank 2 or 3, considering that if a matrix \(M\) is included in the count, \(PM\) where \(P\) is a permutation matrix, is not included?
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Context:
When a positive integer $N$ is fed into a machine, the output is a number calculated according to the rule shown below.

For example, starting with an input of $N=7,$ the machine will output $3 \cdot 7 +1 = 22.$ Then if the output is repeatedly inserted into the machine five more times, the final output is $26.$  $7 \to 22 \to 11 \to 34 \to 17 \to 52 \to 26$ When the same $6$-step process is applied to a different starting value of $N,$ the final output is $1.$ What is the sum of all such integers $N?$ $N \to \rule{0.5cm}{0.15mm} \to \rule{0.5cm}{0.15mm} \to \rule{0.5cm}{0.15mm} \to \rule{0.5cm}{0.15mm} \to \rule{0.5cm}{0.15mm} \to 1$
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Context:
The integers \(a, b,\) and \(c\) satisfy the equations \(a + 5 = b\), \(5 + b = c\), and \(b + c = a\). What is the value of \(b\)?
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Context:
Given the numbers \(\log _{\left(\frac{x}{2}-1\right)^{2}}\left(\frac{x}{2}-\frac{1}{4}\right)\), \(\log _{\sqrt{x-\frac{11}{4}}}\left(\frac{x}{2}-1\right)\), \(\log _{\frac{x}{2}-\frac{1}{4}}\left(x-\frac{11}{4}\right)^{2}\). For which \(x\) are two of these numbers equal, and the third one greater than them by 1?
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Context:
Compute: $\left(\frac{1}{2} \right)^{3} \cdot \left(\frac{1}{7} \right)$.
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Context:
Given that $\tan \alpha = \frac{1}{3}$ and $\tan (\alpha + \beta) = \frac{1}{2}$, find the value of $\tan \beta$.
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Context:
Quadrilateral $ABCD$ has right angles at $B$ and $D$. The length of the diagonal $AC$ is $5$. If two sides of $ABCD$ have integer lengths and one of these lengths is an odd integer, determine the area of $ABCD$.
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Context:
A collection of coins includes only pennies (1 cent), nickels (5 cents), dimes (10 cents), and quarters (25 cents). What is the smallest possible number of coins in the collection that can be used to create any amount of money less than one dollar (100 cents)? Express your answer as a single integer.
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Context:
A submarine has three navigational devices but can remain at sea if at least two are working. Suppose that the failure times are exponential with means 1, 1.5, and 3 years. What is the average length of time the boat can remain at sea?
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Context:
Find the binormal vector for a curve given by the parametric equations: $x = 5\sin t$, $y = 2$, $z = 5\cos t$.
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Context:
A circle is inscribed in an equilateral triangle. Three nested sequences of circles are then constructed as follows: each circle touches the previous circle and has two edges of the triangle as tangents.

Find the ratio of the area of the largest circle to the combined area of all the other circles. The original answer is in the format \(\frac{k}{m}\), where \(k\) and \(m\) are integers. Please calculate the value of \(k + m\).
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Context:
The walls of a room are in the shape of a triangle  $ABC$  with  $\angle ABC = 90^\circ$ ,  $\angle BAC = 60^\circ$ , and  $AB=6$ .  Chong stands at the midpoint of  $BC$  and rolls a ball toward  $AB$ .  Suppose that the ball bounces off  $AB$ , then  $AC$ , then returns exactly to Chong.  Find the length of the path of the ball.
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Context:
Let the set \( I = \{1, 2, \cdots, n\} (n \geqslant 3) \). If two non-empty proper subsets \( A \) and \( B \) of \( I \) satisfy \( A \cap B = \varnothing \) and \( A \cup B = I \), then \( A \) and \( B \) are called a partition of \( I \). If for any partition \( A \) and \( B \) of the set \( I \), there exist two numbers in \( A \) or \( B \) such that their sum is a perfect square, then \( n \) must be at least \(\qquad\).
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Context:
A flea jumps on the number line. It starts at 0, each jump has a length of 1, and the next jump has a probability $p$ of being in the same direction as the previous one, and a probability $1-p$ of being in the opposite direction. What is the probability that it returns to 0?
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Context:
Convex quadrilateral $ABCD$ satisfies $\angle{CAB} = \angle{ADB} = 30^{\circ}, \angle{ABD} = 77^{\circ}, BC = CD$ and $\angle{BCD} =n^{\circ}$ for some positive integer $n$. Compute $n$.
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Context:
A modified octahedron consists of two pyramids, each with a pentagonal base, glued together along their pentagonal bases, forming a polyhedron with twelve faces. An ant starts at the top vertex and walks randomly to one of the five adjacent vertices in the middle ring. From this vertex, the ant walks again to another randomly chosen adjacent vertex among five possibilities. What is the probability that the second vertex the ant reaches is the bottom vertex of the polyhedron? Express your answer as a common fraction.
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Context:
If any two adjacent digits of a three-digit number have a difference of at most 1, it is called a "steady number". How many steady numbers are there?
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Context:
In every acyclic graph with 2022 vertices we can choose  $k$  of the vertices such that every chosen vertex has at most 2 edges to chosen vertices. Find the maximum possible value of  $k$ .
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Context:
The base of a pyramid is an isosceles triangle with a side length of \(a\) and a vertex angle of \(\alpha\). All the lateral edges are inclined to the plane of the base at an angle \(\beta\). Find the volume of the pyramid.
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\boxed{ \frac{a^3 \sin \frac{\alpha}{2} \tan \beta}{6} }


Context:
数列 $\left\{a_{n}\right\}$ 满足: $a_{1}=\frac{5}{2}, a_{n+1}=a_{n}^{2}-2$, 求 $\left[a_{2023}\right]$ 除以 7 的余数。
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Context:
A train 120 m long is running at 60 kmph. It takes 21.598272138228943 seconds to pass a platform of a certain length. How long is the platform?
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Context:
Bret sets off on his journey to Boston, which includes a total of five train transfers. His entire trip takes 20 hours. On the first train, he reads a book for 2 hours and spends another 1 hour eating dinner. On the second train, he watches movies on his computer for 3 hours and spends 1 hour chatting with a fellow passenger. During the transfer between the second and third trains, he spends 45 minutes browsing a souvenir shop and 30 minutes waiting for the next train.

On the third train, he spends 2 hours working on his laptop and takes a 1-hour break to read a magazine. When transferring to the fourth train, he encounters a 1-hour delay and spends the time listening to music. On the fourth train, he spends an additional 1 hour browsing the internet on his phone. During the final transfer, he has a 30-minute phone call with a friend while waiting for the last train.

On the fifth train, he plays a game on his phone for an hour and 30 minutes before deciding to take a nap. How many hours does he have left to take a nap?
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Context:
Let \( n \) be a positive integer. There are \( 2018n + 1 \) cities in the Kingdom of Sellke Arabia. King Mark wants to build two-way roads that connect certain pairs of cities such that for each city \( C \) and for each integer \( 1 \leq i \leq 2018 \), there are exactly \( n \) cities that are a distance \( i \) away from \( C \). (The distance between two cities is the least number of roads on any path between the two cities.) For which \( n \) is it possible for Mark to achieve this?
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Context:
A circle given by the equation \(x^{2}+(y+1)^{2}=3\) is rotated about the line \(kx-y-1=0\) for one full revolution. Determine the surface area of the resulting geometric shape.
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Context:
Tina is a professional boxer. She wins her first 10 fights of her career. She then goes on to win some more before losing her first fight, and then doubles her number of wins before losing again. She then retires. At the end of her career, she has 28 more wins than losses. How many fights did she win before losing her first fight?
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Context:
3. An electronic clock displays time from 00.00.00 to 23.59.59. How much time during a day does the clock display exactly four digit 3s?
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Context:
One hundred and one of the squares of an $n \times n$ table are colored blue. It is known that there exists a unique way to cut the table into rectangles along the boundaries of its squares with the following property: every rectangle contains exactly one blue square. Find the smallest possible $n$.
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Context:
For each positive integer $n$, find the number of $n$-digit positive integers that satisfy both of the following conditions:
[list]
[*] no two consecutive digits are equal, and
[*] the last digit is a prime.
[/list]
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Context:
已知数列 $\left\{a_{n}\right\}$ 满足 $a_{n}=\sqrt{\frac{2 n-1}{4 n^{2}+1}}$, 前 n 项和为 $S_{n}$, 求与 $S_{128}-S_{32}$ 最接近的整数.
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Context:
A square is initially divided into sixteen equal smaller squares. The center four squares are then each divided into sixteen smaller squares of equal area, and this pattern continues indefinitely. In each set of sixteen smaller squares, the four corner squares are shaded. What fractional part of the entire square is shaded?
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Context:
Let \( \{a_n\} \) be a sequence of positive numbers with the sum of its first \( n \) terms being \( b_n \). Let \( \{b_n\} \) be such that the product of its first \( n \) terms is \( c_n \), and \( b_n + c_n = 1 \). Find the number in the sequence \( \{1/a_n\} \) that is closest to 2002.
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Context:
Determine the number of inequivalent spin structures on the manifold \( M = \#_{k}(S^2 \times S^1) \).
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Context:
Your math teacher asks you to rationalize the denominator of the expression $\frac{a}{b + \sqrt{c}}$, where $a$, $b$, and $c$ are integers and $c$ is not divisible by the square of any prime. You find that $\frac{a}{b + \sqrt{c}}$ is equal to $\frac{30 - 5\sqrt{14}}{11}$. Compute the triple $(a,b,c)$.
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Context:
Determine the signature of the quadratic form given by \( f(x, y, z) = xy + yz + xz \). The quadratic form is represented as a matrix product: \( f(x,y,z) = \begin{bmatrix}x&y&z\end{bmatrix} \cdot \begin{bmatrix}0&1/2&1/2\\1/2&0&1/2\\1/2&1/2&0\end{bmatrix} \cdot \begin{bmatrix}x\\y\\z\end{bmatrix} \).
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Context:
Find the largest positive integer \( n \) such that for any real number \( c \), there exists at most one base-\( n \) polynomial \( P(x) \) for which \( P(\sqrt{2} + \sqrt{3}) = c \).
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Context:
Determine the group that \( \langle (135)(246), (12)(34)(56) \rangle \subset S_6 \) is isomorphic to, using only the groups \( D_n, S_n, \mathbb{Z}_n \), and their direct sums, where \( S_n \) is the permutation group of \( n \) elements, and \( D_n \) is the dihedral group of order \( 2n \).
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Context:
A fisherman continues fishing until the weight of the last fish caught is less than the one before. Calculate the probability that he catches exactly $k$ fish.
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Context:
Let $X=(X,d)$ be a metric space and $r \ge 0$. Determine the smallest possible upper bound for $\inf_T \text{Lip}(T)$, where $\text{Lip}(T) := \underset{x,x' \in X,\;x \ne x'}{\sup}\; \frac{d(T(x),T(x'))}{d(x,x')}$, and the infimum is taken over all functions $T:X \rightarrow X$ such that $\underset{x \in X}{\sup}\;d(T(x),x) \le r$. Provide your answer as a single expression or number.
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Context:
Let $X_t$ be a stochastic process. Using the rules of Ito's calculus, find the value of $\langle dX_t, dX_t \rangle$ when $dX_t = Y_t \, dt$, where $Y_t$ is a progressive stochastic process.
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Context:
Given the sequence defined by the initial conditions $a_0=a_1=a_2=a_3=a_4=0$ and $a_5=1$, and the recurrence relation $$a_{n+6} = \frac{a_{n+5} + a_{n+4} + a_{n+3} + a_{n+2} + a_{n+1} + a_{n}}{6},$$ find the limit of $a_n$ as $n$ approaches infinity.
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Context:
Find the value of \( f(f(100)) \) given the function \( f(x) = ([\{x\}] + 3 - x^7)^{1/7} \).
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Context:
Find the Ramsey number for the graphs $(m+1)K_4$ and $K_4$, given that $R(mK_4,K_4) + 4 \leq R((m+1)K_4,K_4)$ for $m \geq 2$. Provide your answer as a single number.
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Context:
Determine the number of subfields of the splitting field of the polynomial \(x^{42} - 1\) over \(\mathbb{Q}\).
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Context:
Misha rolls a standard, fair six-sided die until she rolls $1$-$2$-$3$ in that order on three consecutive rolls. The probability that she will roll the die an odd number of times is $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
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Context:
Consider a Markov chain with a transition probability matrix $P$ where each transition probability is 0.5, except for the transitions from the first and last states, which have a probability of 1. Calculate the expected number of steps required to reach state $j$ starting from state 1.
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Context:
Consider a 100-dimensional hypersphere with a radius of 4000 miles. What fraction of its volume lies within 40 miles of its surface?
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Context:
Find the number of ordered pairs of integers $(a,b)\in\{1,2,\ldots,35\}^2$ (not necessarily distinct) such that $ax+b$ is a "quadratic residue modulo $x^2+1$ and $35$", i.e. there exists a polynomial $f(x)$ with integer coefficients such that either of the following $\textit{equivalent}$ conditions holds:
there exist polynomials $P$, $Q$ with integer coefficients such that $f(x)^2-(ax+b)=(x^2+1)P(x)+35Q(x)$;
or more conceptually, the remainder when (the polynomial) $f(x)^2-(ax+b)$ is divided by (the polynomial) $x^2+1$ is a polynomial with integer coefficients all divisible by $35$.
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Context:
Determine the smallest non-abelian group $G$ such that the power map $f(x) = x^n$ is a homomorphism for some integer $n > 2$. What is this group?
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Context:
Calculate the supremum of \( \int_{-1}^1 f \, dg \) for \( f \in \mathcal{F} \), where \( g(x) = x^2 \) on \([-1, 1]\) and \( \mathcal{F} = \{ f \in \mathcal{C}[-1,1] : |f(x)| \leq 5 \text{ for all } x \in [-1,1] \} \).
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Context:
Calculate the flux of the vector field \( F(x,y,z) = (x,y,z) \) across the surface defined by \( x^2 - y^2 + z^2 = 0 \) with \( y \in [0, 1] \).
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Context:
Find the volume enclosed between the surfaces \(x+y+2z=2\) and \(2x+y+z=4\) in the first octant.
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Context:
Given a polynomial \( P(x) = a_{2n} x^{2n} + a_{2n-1} x^{2n-1} + \ldots + a_1 x + a_0 \) where each coefficient \( a_i \) belongs to the interval \([100,101]\), determine the smallest natural number \( n \) such that this polynomial can have a real root.
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Context:
Find the value of \(\alpha\) such that the roots of the polynomial \(x^5 + 5\lambda x^4 - x^3 + (\lambda \alpha - 4)x^2 - (8\lambda + 3)x + \lambda\alpha - 2 = 0\) are independent of \(\lambda\).
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Context:
Find the value of \( b \) such that the tangent line to the function \( f_a(x) = (x-a)e^{a+2-x} \) at \( x = b \) intersects the y-axis at the point \( A(0, 2012) \).
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Context:
For given integers $x$ and $d$, determine the largest number of $\mathbb{Z}^d$ lattice points that are mutually $x$-visible. Denote this number as $g(x,d)$. For instance, it is known that $g(0,2) = 4$ and $g(1,2) \ge 9$. What is $g(x,d)$?
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Context:
Find the interval of convergence for the power series with coefficients \(a_n= \frac{(\ln n)^2}{n}.\)
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Context:
King Arthur has two advisors, Merlin and Percival, each providing correct answers with probability \( p \). If both advisors agree, the king follows their advice; if they disagree, he flips a coin. Arthur considers firing one advisor if it doesn't decrease the probability of making correct decisions. Merlin, aiming to avoid being fired, will give an answer that maximizes his chance of staying. What is the probability that Merlin will be fired?
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Context:
Given that the function $f(x,y)$ is differentiable at $(1,1)$ and $f(1,1)=\frac{\partial f}{\partial x}(1,1)=\frac{\partial f}{\partial y}(1,1)$, find the value of $\frac{d}{dx}f(x,f(x,x))$ when $x=1$. Provide your answer as a single number.
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Context:
Determine if there is a particular solution to the equation $x = \log(x) + c$ without using numerical methods.
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Context:
Find the real number \( x \) represented by the continued fraction \([12;2,2,12,2,2,12,2,2,12\dots]\).
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Context:
Given a matroid $M$ with a ground set $E$ of size $2n$, where there exists a subset $A \subseteq E$ of size $n$ such that both $A$ and $E \setminus A$ are independent, determine the minimum number of subsets $B \subseteq E$ such that both $B$ and $E \setminus B$ are independent.
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Context:
Let $A$ be the matrix defined as follows:

$$A := \begin{pmatrix} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 1 \end{pmatrix}$$

Determine the number of matrices similar to $A$ over the field $\mathbb{Z}/3\mathbb{Z}$.
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Context:
Find an element of order 3 in the multiplicative group \( \mathbb{Z}_{67}^* \).
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Context:
Given positive real numbers $x$, $y$, and $z$ such that $\frac{1}{3} \leq xy + yz + zx \leq 3$, find the range of possible values for $x + y + z$. Express your answer as an interval.
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Context:
Question 150: Four spheres with a radius of 1 are pairwise externally tangent. Then the volume of the circumscribed regular tetrahedron is $\qquad$ .
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Context:
Find the center of gravity of a hemispherical shell with radius $a$ and uniform density of 1, positioned with its base on the $x-y$ plane. Use the method of slicing parallel to the $x-y$ plane to determine the center of gravity.
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Context:
The monotonically decreasing interval of the function $f(x)= \sqrt {3-2x-x^{2}}$ is $\boxed{\text{answer}}$ .
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Context:
The science club has 25 members: 10 boys and 15 girls.  A 5-person committee is chosen at random.  What is the probability that the committee has at least 1 boy and at least 1 girl?The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.
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Context:
Triangle $PQR$ has vertices $P(0, 10)$, $Q(3, 0)$, $R(9, 0)$. A line through $Q$ bisects the area of $\triangle PQR$. Find the sum of the slope and $y$-intercept of this line.
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Context:
Determine the values of \( n \) for which it is possible to assemble a cube with an edge length of \( 2n \) cm from smaller cubes obtained by dividing a solid cube with an edge length of \( n \) cm into 1 cm edge length cubes. The assembled cube may have a hollow interior.
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Context:
在正四面体 P-ABC 中，点 E,F 分别在棱 PB, PC 上. 若 PE\neq PF，且 AE=AF=2, EF=\sqrt{3}，则四面体 P-AEF 的体积为__________.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值
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Context:
Given six students – three boys and three girls – standing in a row, where girls A, B, and C are not allowed to stand next to each other and girls are not allowed to stand at the ends of the row, calculate the number of different arrangements.
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Context:
Equilateral triangles \(ABC\) and \(A_1B_1C_1\) with side length 10 are inscribed in the same circle such that point \(A_1\) lies on arc \(BC\) and point \(B_1\) lies on arc \(AC\). Find \(AA_1^2 + BC_1^2 + CB_1^2\).
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Context:
There are 8 distinct points on a plane, where no three are collinear. An ant starts at one of the points, then walks in a straight line to each one of the other points, visiting each point exactly once and stopping at the final point. This creates a trail of 7 line segments. What is the maximum number of times the ant can cross its own path as it walks?
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Context:
A circle with radius $r$ is contained within the region bounded by a circle with radius $R$. The area bounded by the larger circle is $\frac{a}{b}$ times the area of the region outside the smaller circle and inside the larger circle. Then $R:r$ equals:
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Context:
If the coordinates of point $N$ are $\left(a,2a-1\right)$, then point $N$ must not be in the ________ quadrant.
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Context:
What is the name of the operation $U^TRU$, where $U$ is an isometric embedding from $\mathbb{R}^d$ to $\mathbb{R}^n$ with $d<n$?
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Context:
There are 40 students in Jimin's class. Jimin and Jungkook's math scores are 98 and 100, respectively. Excluding these two students, the average score in math is 79. What is the average math score of Jimin's class?
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Context:
In the orchard, 170 apples, 268 tangerines, and 120 mangoes were loaded equally into each of trucks. There are 8 apples left, 2 tangerines short, and 12 mangoes left. Find the maximum number of trucks in the orchard.
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