Context:
What is the 5th largest four-digit number whose digits add to 17?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A square with a side length of 4 units has eight congruent semicircles drawn inside it. Each semicircle starts at a vertex of the square and ends at the midpoint of an edge of the square. Calculate the area of the non-shaded part of the square. Express your answer in square units.
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Context:
On the side \(AB\) of an equilateral triangle \(\mathrm{ABC}\), a right triangle \(\mathrm{AHB}\) is constructed (\(\mathrm{H}\) is the vertex of the right angle) such that \(\angle \mathrm{HBA}=60^{\circ}\). Let the point \(K\) lie on the ray \(\mathrm{BC}\) beyond the point \(\mathrm{C}\), and \(\angle \mathrm{CAK}=15^{\circ}\). Find the angle between the line \(\mathrm{HK}\) and the median of the triangle \(\mathrm{AHB}\) drawn from the vertex \(\mathrm{H}\).
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Context:
We say a point is contained in a square if it is in its interior or on its boundary. Three unit squares are given in the plane such that there is a point contained in all three. Furthermore, three points $A, B, C$, are given, each contained in at least one of the squares. Find the maximum area of triangle $A B C$.
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Context:
If the five numbers in a sample set $a$, $99$, $b$, $101$, $c$ exactly form an arithmetic sequence, then the standard deviation of this sample is $\_\_\_\_\_$.
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Context:
Ellen is baking bread. It takes 3 hours to rise 1 ball of dough, and then another 2 hours to bake it. If she makes 4 balls of dough one after another and then bakes them one after another when they're done rising, how many hours will it take?
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Context:
Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ that satisfy $|\overrightarrow{a}|=3$, $|\overrightarrow{b}|=2\sqrt{3}$, and $\overrightarrow{a} \perp (\overrightarrow{a}+\overrightarrow{b})$, find the projection of $\overrightarrow{b}$ onto $\overrightarrow{a}$.
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Context:
Given that five students from Maplewood school worked for 6 days, six students from Oakdale school worked for 4 days, and eight students from Pinecrest school worked for 7 days, and the total amount paid for the students' work was 1240 dollars, determine the total amount earned by the students from Oakdale school, ignoring additional fees.
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Context:
There are 5 people standing in a line numbered 1 through 5 in a left to right order.
Each person has the following attributes:  Nationality, Movie-Genre, Music-Genre, Beverage, Pet, Sport.
The attributes have the following possible values:
- Nationality: british, dutch, french, indonesian, polish
- Movie-Genre: action, fantasy, martial-arts, thriller, western
- Music-Genre: country, house, indie, rock, salsa
- Beverage: 7up, almond-milk, juice, lemonade, mirinda
- Pet: chinchilla, fish, horse, snake, turtle
- Sport: basketball, handball, rowing, skiing, snowboarding

Given the following premises about the line of people:
- Sport:rowing is on the far left or far right
- Music-Genre:house is on the left or right of Movie-Genre:western
- Pet:snake and Beverage:mirinda have different parity positions
- Beverage:7up and Movie-Genre:martial-arts have the same parity positions
- Movie-Genre:action != Beverage:mirinda
- Music-Genre:salsa is on the right of Beverage:lemonade
- Music-Genre:country == Movie-Genre:fantasy or Movie-Genre:fantasy == Pet:turtle or both
- Beverage:7up is not to the right of Nationality:polish
- Movie-Genre:fantasy is not to the left of Music-Genre:rock
- Sport:basketball and Beverage:mirinda have different parity positions
- Beverage:7up is somewhere to the right of Sport:skiing
- Nationality:british is not to the left of Sport:handball
- Nationality:french is not to the right of Nationality:polish
- Beverage:juice is in the middle
- Music-Genre:house is on the left of Beverage:7up
- Nationality:polish is somewhere to the left of Movie-Genre:action
- Sport:skiing and Music-Genre:indie have the same parity positions
- Sport:skiing is on the left or right of Pet:chinchilla
- Sport:snowboarding is on the left or right of Nationality:french
- Music-Genre:country is on the right of Nationality:dutch
- Sport:skiing is on the left or right of Sport:basketball
- Pet:horse is somewhere to the right of Music-Genre:country

Answer the following question:

Question: Question: What is the Beverage of the person who has/is chinchilla??
 Please provide your answer in the format: \boxed{X}, where X is the answer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
By a river lives the tribe Mumbo-Yumbo. One day, with urgent news, a young warrior Mumbo and a wise shaman Yumbo set off simultaneously to a neighboring tribe. Mumbo ran at a speed of 11 km/h to the nearest raft storage and then sailed on a raft to the neighboring tribe. Yumbo, unhurriedly, walked at a speed of 6 km/h to another raft storage and sailed to the neighboring tribe from there. As a result, Yumbo arrived earlier than Mumbo. The river is straight, and the rafts sail at the speed of the current. This speed is the same everywhere and is expressed as an integer km/h, not less than 6. What is the maximum possible value of this speed?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the number of ordered pairs of integers $(a, b) \in\{1,2, \ldots, 35\}^{2}$ (not necessarily distinct) such that $a x+b$ 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 equivalent conditions holds: - there exist polynomials $P, Q$ with integer coefficients such that $f(x)^{2}-(a x+b)=\left(x^{2}+1\right) P(x)+35 Q(x)$ - or more conceptually, the remainder when (the polynomial) $f(x)^{2}-(a x+b)$ is divided by (the polynomial) $x^{2}+1$ is a polynomial with (integer) coefficients all divisible by 35 .
Please reason step by step, and put your final answer within \boxed{}.


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

*Greece*
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Context:
设 $m, n$ 是给定的整数， $m \geq n \geq 3$ 。求具有下述性质的最小正整数 $k$ ：若将 $1,2, \cdots, k$ 中的每个数任意染为红色或者蓝色，则或者存在 $m$ 个红色的数 $x_{1}, x_{2}, \cdots, x_{m}$ （允许相同），满足 $x_{1}+x_{2}+\cdots+x_{m-1}<x_{m}$ ，或者存在 $n$ 个蓝色的数 $y_{1}, y_{2}, \cdots, y_{n}$ （允许相同），满足 $y_{1}+y_{2}+\cdots+y_{n-1}<y_{n}$ 。请给出 $m=5,n=4$时$k$ 的值。
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Context:
In the convex quadrilateral \(ABCD\), the midpoints of sides \(BC\) and \(CD\) are \(E\) and \(F\) respectively. The segments \(AE\), \(EF\), and \(AF\) divide the quadrilateral into four triangles whose areas are four consecutive integers. What is the maximum possible area of triangle \(ABD\)?
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Context:
A convex polyhedron \( P \) has 2021 edges. By cutting off a pyramid at each vertex, which uses one edge of \( P \) as a base edge, a new convex polyhedron \( Q \) is obtained. The planes of the bases of the pyramids do not intersect each other on or inside \( P \). How many edges does the convex polyhedron \( Q \) have?
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Context:
Consider a $ n\times n $ square grid which is divided into $ n^2 $ unit squares(think of a chess-board). The set of all unit squares intersecting the main diagonal of the square or lying under it is called an $n$-staircase. Find the number of ways in which an $n$-stair case can be partitioned into several rectangles, with sides along the grid lines, having mutually distinct areas.
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Context:
Let $n, k \geq 3$ be integers, and let $S$ be a circle. Let $n$ blue points and $k$ red points be chosen uniformly and independently at random on the circle $S$. Denote by $F$ the intersection of the convex hull of the red points and the convex hull of the blue points. Let $m$ be the number of vertices of the convex polygon $F$ (in particular, $m=0$ when $F$ is empty). Find the expected value of $m$. The original answer is in the form of a fraction; please provide the sum of the numerator and denominator.
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Context:
In a peculiar country called "-0 Country", they only have the digits 1 and 0. Therefore, when dealing with larger numbers, they use combinations of 1s and 0s to represent them as sums. For example, the number 12 can be represented as the sum of three numbers: $10 + 1 + 1$, or as the sum of two numbers: $11 + 1$. What is the minimum number of numbers needed to represent 20120204 in "-0 Country"?
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Context:
Two distinct numbers a and b are chosen randomly from the set $\{2, 2^2, 2^3, ..., 2^{25}\}$. What is the probability that $\mathrm{log}_a b$ is an integer?
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Context:
Say that a polynomial with real coefficients in two variable, $ x,y,$ is [i]balanced[/i] if the average value of the polynomial on each circle centered at the origin is $ 0.$ The balanced polynomials of degree at most $ 2009$ form a vector space $ V$ over $ \mathbb{R}.$ Find the dimension of $ V.$
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Context:
11.5. Each cell of a $7 \mathrm{x} 8$ table (7 rows and 8 columns) is painted in one of three colors: red, yellow, or green. In each row, the number of red cells is not less than the number of yellow cells and not less than the number of green cells, and in each column, the number of yellow cells is not less than the number of red cells and not less than the number of green cells. How many green cells can there be in such a table?
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Context:
A triangulation of a polygon is a subdivision of the polygon into triangles meeting edge to edge, with the property that the set of triangle vertices coincides with the set of vertices of the polygon. Adam randomly selects a triangulation of a regular $180$-gon. Then, Bob selects one of the $178$ triangles in this triangulation. The expected number of $1^\circ$ angles in this triangle can be expressed as $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. Compute $100a + b$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a sequence $\{a_n\}$ satisfying $a_1 = 1$ and $a_n + a_{n-1} = \left(\frac{1}{2}\right)^n$ (for $n \geq 2$), and define \( S_n = a_1 \cdot 2 + a_2 \cdot 2^2 + \ldots + a_n \cdot 2^n \). By analogy to the method presented in the textbook for deriving the formula for the sum of the first \( n \) terms of a geometric series, find the expression for $3S_n - a_n \cdot 2^{n+1}$.
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Context:
Jeffrey owns a poultry farm with various types of birds. There are 25 hens, 10 ducks, and 5 geese. The ratio of male to female birds for each type is 1:4. Each hen has 6 chicks, each duck has 8 ducklings, and each goose has 3 goslings. Calculate the total number of poultry on the farm, including all adult birds and their offspring.
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Context:
What is the smallest positive integer representable as the sum of the cubes of three positive integers in two different ways?
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Context:
Find all the real numbers $k$ such that for any non-zero real numbers $a$ and $b$, at least one of the numbers $a$, $b$, or $\frac{5}{a^2} + \frac{6}{b^3}$ is less than or equal to $k$.
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Context:
Let $P(x)$ be a polynomial with positive integer coefficients and degree 2015. Given that there exists some $\omega \in \mathbb{C}$ satisfying
$$\omega^{73} = 1\quad \text{and}$$
$$P(\omega^{2015}) + P(\omega^{2015^2}) + P(\omega^{2015^3}) + \ldots + P(\omega^{2015^{72}}) = 0,$$
what is the minimum possible value of $P(1)$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Andrew has seven pieces of wire of lengths 1 cm, 2 cm, 3 cm, 4 cm, 5 cm, 6 cm, and 7 cm. He bends some of the pieces to form a wire frame in the shape of a cube with edges of length 1 cm without any overlaps. What is the smallest number of these pieces that he can use? Express your answer as a whole number.
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Context:
For  $0\leq a\leq 2$ , find the minimum value of  $\int_0^2 \left|\frac{1}{1+e^x}-\frac{1}{1+e^a}\right|\ dx.$ *2010 Kyoto Institute of Technology entrance exam/Textile e.t.c.*
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Context:
Compute
$$
\lim _{h \rightarrow 0} \frac{\sin \left(\frac{\pi}{3}+4 h\right)-4 \sin \left(\frac{\pi}{3}+3 h\right)+6 \sin \left(\frac{\pi}{3}+2 h\right)-4 \sin \left(\frac{\pi}{3}+h\right)+\sin \left(\frac{\pi}{3}\right)}{h^{4}}
$$
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Context:
For any two sets $A$ and $B$, define $A - B = \{x \mid x \in A \text{ and } x \notin B\}$, $A \ast B = (A - B) \cup (B - A)$. Let $A = \{y \mid y \geq 0\}$ and $B = \{x \mid -3 \leq x \leq 3\}$, find $A \ast B = \_\_\_\_\_\_.$
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Context:
A train of a certain length takes 14.998800095992321 seconds to cross a bridge 80 meters long if the speed of the train is 36 kmph. How long is the train?
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Context:
Three equally spaced parallel lines intersect a circle, creating three chords of lengths $38,38,$ and $34$. What is the distance between two adjacent parallel lines?
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Context:
Ten test papers are to be prepared for the National Olympiad. Each paper has 4 problems, and no two papers have more than 1 problem in common. What is the minimum number of distinct problems needed?
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Context:
Given that the line passing through the focus \( F \) of the parabola \( y^2 = 4x \) intersects the parabola at points \( M \) and \( N \), and \( E(m,0) \) is a point on the x-axis, with the extensions of \( M E \) and \( N E \) intersecting the parabola again at points \( P \) and \( Q \) respectively. If the slopes \( k_1 \) and \( k_2 \) of the lines \( M N \) and \( P Q \) satisfy \( k_1 = 3k_2 \), find the value of the real number \( m \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the limit: $$\lim_{n \to \infty}\frac{1}{n}\sum_{k=0}^{n}{\frac{1}{1+\frac{k^6}{n^3}}}$$
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Context:
Alex starts with a rooted tree with one vertex (the root). For a vertex $v$, let the size of the subtree of $v$ be $S(v)$. Alex plays a game that lasts nine turns. At each turn, he randomly selects a vertex in the tree and adds a child vertex to that vertex. After nine turns, he has ten total vertices. Alex selects one of these vertices at random (call the vertex $v_1$). The expected value of $S(v_1)$ is of the form $\frac{m}{n}$ for relatively prime positive integers $m, n$. Find $m+n$.

**Note:** In a rooted tree, the subtree of $v$ consists of its indirect or direct descendants (including $v$ itself).
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Context:
Let $g(x) = ax^7 + bx^3 + dx^2 + cx - 8$. If $g(-7) = 3$, then find $g(7)$.
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Context:
A parallelogram $ABCD$ has a base $\overline{BC}$ of 6 units and a height from $A$ to line $BC$ of 3 units. Extend diagonal $AC$ beyond $C$ to point $E$ such that $\overline{CE}$ is 2 units, making $ACE$ a right triangle at $C$. Find the area of the combined shape formed by parallelogram $ABCD$ and triangle $ACE$.
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Context:
Let the complex number \( z \) satisfy the conditions that both the real and imaginary parts of \( \frac{z}{10} \) and \( \frac{10}{\bar{z}} \) are positive numbers less than 1. What is the area of the trajectory of the point corresponding to \( z \) in the complex plane?
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Context:
For a given positive integer \( k \), let \( f_{1}(k) \) represent the square of the sum of the digits of \( k \), and define \( f_{n+1}(k) = f_{1}\left(f_{n}(k)\right) \) for \( n \geq 1 \). Find the value of \( f_{2005}\left(2^{2006}\right) \).
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Context:
Triangles $OAB,OBC,OCD$ are isoceles triangles with $\angle OAB=\angle OBC=\angle OCD=\angle 90^o$. Find the area of the triangle $OAB$ if the area of the triangle $OCD$ is 12.
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Context:
7.8. The distances from the midpoint of the height of a regular triangular pyramid to the lateral face and to the lateral edge are 2 and $\sqrt{14}$, respectively. Find the volume of the pyramid. If necessary, round your answer to two decimal places.
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Context:
Tiffany attends the carnival and her mother gives her $3 to play on a ring toss game. For every red bucket she tosses a ring into she gets 2 points. For every green bucket she gets three points, and for every rare blue bucket, she gets 5 points. She gets zero points for a miss. Blue buckets are harder to hit, with only a 10% success rate. Every play costs her $1 and she gets 5 rings per play. She's played two games and already gotten 4 red buckets, 5 green buckets, and 1 blue bucket. If she plays until her money runs out, what is the most total points she can get for all three games?
Please reason step by step, and put your final answer within \boxed{}.


Context:
The owner of an apartment insured it for 3,750,000 rubles (the actual cost of the apartment is 7,500,000 rubles). The actual damage amounted to 2,750,000 rubles. The insurance compensation under the proportional liability system was 1,350,000 rubles. Determine the amount of the deductible provided for in the contract.
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Context:
Given $p$: $x^{2}+2x-3 > 0$, $q$: $x > a$, and the sufficient but not necessary condition for $\lnot q$ is $\lnot p$, determine the range of values for $a$.
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Context:
设$S = \{ 1 , 2 , 3 , 4 \} , n$项数列$a _ { 1 } , a _ { 2 } , \cdots , a _ { n }$有以下性质：对于S的任何一个非空子集B，在该数列中有相邻的｜B｜项恰好组成集合B.求n的最小值.
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Context:
正整数$[ \frac { 1 0 ^ { 9 3 } } { 1 0 ^ { 3 1 } + 3 } ]$的末两位数字是
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Context:
Find the number of real solutions of the equation
\[\frac{x}{50} = \cos x.\]
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Context:
12. (10 points) In a math competition, each team can only score 0 points, 3 points, or 5 points per question. At the end of the competition, the total score of three teams is 32 points. If any team's total score can reach 32 points, how many different combinations of total scores are there for these three teams?
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Context:
There are 12 numbers from a certain starting number to 47 that are exactly divisible by 3. What is the starting number?
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Context:
The positive integers $a$ and $b$ are such that the numbers $15a + 16b$ and $16a - 15b$ are both squares of positive integers. What is the least possible value that can be taken on by the smaller of these two squares?
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Context:
Let $ABCD$ be a rectangle with $AB=10$ and $BC=26$. Let $\omega_1$ be the circle with diameter $\overline{AB}$ and $\omega_2$ be the circle with diameter $\overline{CD}$. Suppose $\ell$ is a common internal tangent to $\omega_1$ and $\omega_2$ and that $\ell$ intersects $AD$ and $BC$ at $E$ and $F$ respectively. What is $EF$?
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Context:
It is known that a cone with an acute triangular cross-section has a slant height of $4$. If the range of the areas of all cross-sections passing through the vertex of the cone is $(0,4\sqrt{3}]$, then the central angle of the sector in the lateral surface development of the cone is ______.
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Context:
One end of the diameter of a semicircle coincides with the vertex of the angle at the base of an isosceles triangle, and the other end belongs to this base. Find the radius of the semicircle, given that it touches one of the legs of the triangle and divides the other leg into segments of 5 cm and 4 cm, measured from the base.
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Context:
Let $V$ be a 10 -dimensional real vector space and $U_{1}$ and $U_{2}$ two linear subspaces such that $U_{1} \subseteq U_{2}, \operatorname{dim}_{\mathbb{R}} U_{1}=3$ and $\operatorname{dim}_{\mathbb{R}} U_{2}=6$. Let $\mathcal{E}$ be the set of all linear maps $T: V \longrightarrow V$ which have $U_{1}$ and $U_{2}$ as invariant subspaces (i.e., $T\left(U_{1}\right) \subseteq U_{1}$ and $T\left(U_{2}\right) \subseteq U_{2}$ ). Calculate the dimension of $\mathcal{E}$ as a real vector space.
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Context:
Given circle $O$, point $C$ is on one side of diameter $\overline{AB}$, and point $D$ is on the opposite side. If $\angle AOC = 40^{\circ}$, and $\angle DOB = 30^{\circ}$, calculate the ratio of the area of the smaller sector $COD$ to the area of the circle.
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Context:
For positive integers $n$ and $k$, let $\mho(n, k)$ be the number of distinct prime divisors of $n$ that are at least $k$. Find the closest integer to $$\sum_{n=1}^{\infty} \sum_{k=1}^{\infty} \frac{\mho(n, k)}{3^{n+k-7}}$$
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Context:
Given the sets \( M = \{x, y, \lg(xy)\} \) and \( N = \{0, |x|, y\} \) with \( M = N \), find the value of \(\left(x + \frac{1}{y}\right) + \left(x^{2} + \frac{1}{y^{2}}\right) + \left(x^{3} + \frac{1}{y^{3}}\right) + \cdots + \left(x^{2001} + \frac{1}{y^{2001}}\right) \).
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Context:
While playing a trivia game Adam scored 283 points total. He scored a certain amount of points each round and played about 4 rounds. How many points did Adam score each round?
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Context:
In triangle \(ABC\), \(AB = 6\), \(BC = 7\), and \(CA = 8\). Let \(D\), \(E\), and \(F\) be the midpoints of sides \(BC\), \(AC\), and \(AB\), respectively. Also, let \(O_A\), \(O_B\), and \(O_C\) be the circumcenters of triangles \(AFD\), \(BDE\), and \(CEF\), respectively. Find the area of triangle \(O_A O_B O_C\).
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Context:
Teacher Xiaoyu stands in a 9-row by 9-column square formation, and she notices that there are 2 people directly in front of her. After the entire formation turns right, Teacher Xiaoyu notices that there are now 4 people directly in front of her. If the entire formation turns right again, how many people will Teacher Xiaoyu notice directly in front of her?
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Context:
Let $n = (10^{2020} + 2020)^2$. Find the sum of all the digits of $n$.
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Context:
On a side of a triangle, four points \( K, P, H, \) and \( M \) are taken, which are respectively the midpoint of this side, the foot of the angle bisector of the opposite angle of the triangle, the point of tangency with this side of the inscribed circle in the triangle, and the foot of the corresponding altitude. Find \( KH \) if \( KP = a \) and \( KM = b \).
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Context:
Seryozha chose two different natural numbers \(a\) and \(b\). He wrote down four numbers in his notebook: \(a, a+2, b\), and \(b+2\). Then he listed on the board all six pairwise products of the numbers from the notebook. What is the maximum number of perfect squares that can be among the numbers on the board?
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Context:
The sum of the absolute values of the terms of a finite arithmetic progression is equal to 100. If all its terms are increased by 1 or all its terms are increased by 2, in both cases the sum of the absolute values of the terms of the resulting progression will also be equal to 100. What values can the quantity \( n^{2} d \) take under these conditions, where \( d \) is the common difference of the progression, and \( n \) is the number of its terms?
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Context:
The square quilt block shown is made from 16 unit squares, four of which have been divided in half to form triangles. Additionally, two squares are completely filled while others are empty. What fraction of the square quilt is shaded? Express your answer as a common fraction.
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Context:
The Houson Association of Mathematics Educators decides to hold a grand forum on mathematics education and invites a number of politicians from around the United States to participate. Around lunch time, the politicians decide to play a game. In this game, players can score $19$ points for pegging the coordinator of the gathering with a spitball, $9$ points for downing an entire cup of the forum’s interpretation of coffee, or $8$ points for quoting more than three consecutive words from the speech Senator Bobbo delivered before lunch.

What is the product of the two greatest scores that a player cannot score in this game?
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Context:
Compute
\[
\left( 1 - \sin \frac {\pi}{8} \right) \left( 1 - \sin \frac {3\pi}{8} \right) \left( 1 - \sin \frac {5\pi}{8} \right) \left( 1 - \sin \frac {7\pi}{8} \right).
\]
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Context:
A polynomial $p(x) = \sum_{j=1}^{2n-1} a_j x^j$ with real coefficients is called [i]mountainous[/i] if $n \ge 2$ and there exists a real number such that the polynomial's coefficients satisfy $a_1=1, a_{j+1}-a_j=k$ for $1 \le j \le n-1,$ and $a_{j+1}-a_j=-k$ for $n \le j \le 2n-2;$ we call $k$ the [i]step size[/i] of $p(x).$ A real number $k$ is called [i]good[/i] if there exists a mountainous polynomial $p(x)$ with step size $k$ such that $p(-3)=0.$ Let $S$ be the sum of all good numbers $k$ satisfying $k \ge 5$ or $k \le 3.$ If $S=\tfrac{b}{c}$ for relatively prime positive integers $b,c,$ find $b+c.$
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Context:
Two boys started running simultaneously around a circular track from the same point at speeds of 60 km/hr and 100 km/hr. They met for the first time anywhere on the track after 107.99136069114472 minutes, while moving in opposite directions. What is the length of the track?
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Context:
Given $\sin\alpha + \cos\alpha = \frac{\sqrt{2}}{3}$, where $\alpha \in (0, \pi)$, calculate the value of $\sin\left(\alpha + \frac{\pi}{12}\right)$.
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Context:
Starting with an empty string, we create a string by repeatedly appending one of the letters $H$, $M$, $T$ with probabilities $\frac{1}{4}$, $\frac{1}{2}$, $\frac{1}{4}$, respectively, until the letter $M$ appears twice consecutively. What is the expected value of the length of the resulting string?
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Context:
For positive integers $m$ and $n$, the decimal representation for the fraction $\tfrac{m}{n}$ begins $0.711$ followed by other digits. Find the least possible value for $n$.
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Context:
From a deck of 32 cards which includes three colors (red, yellow, and blue) with each color having 10 cards numbered from $1$ to $10$, plus an additional two cards (a small joker and a big joker) both numbered $0$, a subset of cards is selected. The score for each card is calculated as $2^{k}$, where $k$ is the number on the card. If the sum of these scores equals $2004$, the subset is called a "good" hand. How many "good" hands are there?

(2004 National Girls' Olympiad problem)
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Context:
Find the extremum of the function $f(x) = (1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \dots + \frac{x^n}{n!})e^{-x}$ where $n \geq 4$ is a positive integer.
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Context:
There are 5 telephone poles, 2 of which are electrified and will cause birds to be electrocuted and fall if touched. Given that 5 birds randomly choose one of the telephone poles to rest on, what is the probability that only 2 of the telephone poles have birds on them?
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Context:
There is a greatest number which on dividing a certain number and 2037 leaves remainders 7 and 5 respectively. The greatest number is 2. What is the first number?
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Context:
There are three types of people, A, B, and C, with a total of 25 people. Type A people always tell the truth, Type B people always lie, and Type C people alternate between telling the truth and lying (for example, if a Type C person tells the truth this time, their next statement will certainly be a lie, and the statement after that will be true again).

The priest asks each person, "Are you a Type A person?" 17 people answer "yes."
The priest then asks each person, "Are you a Type C person?" 12 people answer "yes."
The priest finally asks each person, "Are you a Type B person?" 8 people answer "yes."

How many of the 25 people are Type C people?
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Context:
A train of a certain length is running at 60 kmph. It takes 23.998080153587715 seconds to pass a platform 260 m long. How long is the train?
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Context:
计算：125(7/8)/(11(3/4)-4(3/20)+2.25-(7/20))．
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Context:
Given the numbers: one and nine zeros. It is allowed to select two numbers and replace each with their arithmetic mean. What is the smallest number that can appear in place of the one after a series of such operations?
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Context:
Given the median AD and BE of $\triangle ABC$ intersect at K, and $AB = \sqrt{3}$, with points K, D, C, E being concyclic, find the length of CK.
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Context:
Given the Euler function $\phi(q)=\prod_{n = 1}^{\infty}(1-q^{n})$ which is a modular form where $q=\exp(2\pi i \tau)$, $|q|\lt1$. What is the limit $\lim_{q\rightarrow 1}\frac{\phi^5(q)_{\infty}}{\phi(q^5)_{\infty}}$ ?
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Context:
Given that the magnitude of the star Altair is $0.75$ and the magnitude of the star Vega is $0$, determine the ratio of the luminosity of Altair to Vega.
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Context:
Let $T$ be the triangle in the coordinate plane with vertices $(0,0)$, $(4,0)$, and $(0,3)$. Consider the following five isometries (rigid transformations) of the plane: rotations of $90^{\circ}$, $180^{\circ}$, and $270^{\circ}$ counterclockwise around the origin, reflection across the $x$-axis, and reflection across the $y$-axis. Find how many of the $125$ sequences of three of these transformations (not necessarily distinct) will return $T$ to its original position.
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Context:
The degree-$6$ polynomial $f$ satisfies the following conditions:
- $f(7) - f(1) = 1$,
- $f(8) - f(2) = 16$,
- $f(9) - f(3) = 81$,
- $f(10) - f(4) = 256$,
- $f(11) - f(5) = 625$.

Compute $f(15) - f(-3)$.
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Context:
Given the universal set U = {x | x is a positive integer less than 9}, set M = {1, 2, 3}, and set N = {3, 4, 5, 6}, find the intersection of the complement of M in U and N. Express your answer as a set of integers.
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Context:
Given that the "optimized sum" of the sequence $a_1, a_2,..., a_{2018}$ is 2019, calculate the "optimized sum" of the sequence $2, a_1, a_2,..., a_{2018}$. Express your answer as a single number.
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Context:
The diagonals $AC$ and $BD$ of a convex cyclic quadrilateral $ABCD$ intersect at point $E$. Given that $AB = 39, AE = 45, AD = 60$ and $BC = 56$, determine the length of $CD.$
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Context:
In January, Luke had 48 dollars. He spent 30% of his money during February. In March, he spent another 11 dollars and received 21 dollars from his mom. If he saved 10% of his remaining money each month from April to June, how much money would he have at the end of June?
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Context:
A triangular pyramid stack of apples is formed with a base consisting of 6 apples on each side. Each higher layer is formed by 3 apples per row, with the top layer consisting of a single apple. Calculate the total number of apples used in the entire stack.
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Context:
Considering all numbers of the form $n = \left\lfloor \frac{k^3}{2012} \right\rfloor$, where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$, and $k$ ranges from $1$ to $2012$, how many of these $n$’s are distinct?
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Context:
Sides \(\overline{AB}\) and \(\overline{AC}\) of equilateral triangle \(ABC\) are tangent to a circle at points \(B\) and \(C\) respectively. Find the fraction of the area of \(\triangle ABC\) that lies outside the circle. The original answer is in the form \(\frac{k}{m}-\frac{n\sqrt{3}\pi}{p}\). Please give the value of \(k + m + n + p\).
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Context:
A house worth $9000 is sold by Mr. A to Mr. B at a 10% loss. Mr. B then sells the house back to Mr. A at a 10% gain. Calculate Mr. A's net loss or gain after both transactions. Express your answer as a dollar value.
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Context:
Alice buys three burgers, two sodas, and one fry for \$4.20, and Bill buys two burgers, one soda, and two fries for \$3.40. Determine the cost of one fry if a burger costs 80 cents and a soda costs 60 cents.
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Context:
Let \( R \) be the region in the first quadrant bounded by the x-axis, the line \( 2y = x \), and the ellipse \( \frac{x^2}{9} + y^2 = 1 \). Let \( R' \) be the region in the first quadrant bounded by the y-axis, the line \( y = mx \), and the ellipse. Find \( m \) such that \( R \) and \( R' \) have the same area.
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Context:
Let \( \triangle ABC \) be a triangle such that \( AB = 7 \), and let the angle bisector of \( \angle BAC \) intersect line \( BC \) at \( D \). If there exist points \( E \) and \( F \) on sides \( AC \) and \( BC \), respectively, such that lines \( AD \) and \( EF \) are parallel and divide triangle \( ABC \) into three parts of equal area, determine the number of possible integral values for \( BC \).
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Context:
The numbers \( x, y, \) and \( z \) are such that \( \frac{x}{y+z}+\frac{y}{z+x}+\frac{z}{x+y}=1 \). What values can the expression \( \frac{x^{2}}{y+z}+\frac{y^{2}}{z+x}+\frac{z^{2}}{x+y} \) take?
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Context:
Let $x, y, z$ be real numbers such that $x+y+z=-2$ and
\[
(x^2+xy+y^2)(y^2+yz+z^2) + (y^2+yz+z^2)(z^2+zx+x^2) + (z^2+zx+x^2)(x^2+xy+y^2) = 625+ \frac{3}{4}(xy+yz+zx)^2.
\]
Compute $|xy+yz+zx|$.
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Context:
For a Poisson distribution with parameter $\lambda$, calculate the following limit as $\lambda \rightarrow \infty$ and $k \rightarrow \infty$ such that $k/\lambda \rightarrow a$, where $a < 1$:

$$\lim \frac{\mathbb{P}(N < k)}{\mathbb{P}(N = k)}$$
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Context:
设 $a_{n}=\left(\frac{1+q}{1-q}\right) \cdot\left(\frac{1+q^{2}}{1-q^{2}}\right)^{\frac{1}{2}} \cdot\left(\frac{1+q^{4}}{1-q^{4}}\right)^{\frac{1}{4}} \cdots\left(\frac{1+q^{2^{n}}}{1-q^{2^{n}}}\right)^{\frac{1}{2^{n}}}$, 其中 $q=\frac{11}{13}$ 。记 $a=\lim _{n \rightarrow+\infty} a_{n}$ ，则 $[a]=$ $\qquad$.
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Context:
The initial terms of the sequence $\{a_n\}$ are 1, 3, 5, 7, 9, 11, 13,... In the sequence $\{b_n\}$, $b_1=a_1$, $b_2=a_2$, $b_8=a_8$,... What is the value of $b_{20}$?
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Context:
The sequence $\left\{a_{n}\right\}$ is: $1,1,2,1,1,2,3,1,1,2,1,1,2,3,4,\cdots$. The sequence is formed as follows: start with $a_{1}=1$ and copy this term to obtain $a_{2}=1$. Then add the number 2 as $a_{3}=2$. Next, copy all terms $1,1,2$ and place them as $a_{4}, a_{5}, a_{6}$, then add the number 3 as $a_{7}=3$, and so on. What is $a_{2021}$?
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Context:
An elevator starts from rest at the top of a building and travels to the bottom. The elevator accelerates from rest between 2 s and 4 s, then travels at a constant speed, and finally decelerates to a stop between 22 s and 24 s. At what time interval does the elevator have its maximum downward velocity? Express your answer as a time interval in seconds (s).
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Context:
The dividend is 12, the divisor is 17, the quotient is some number, and the remainder is 8. What is the quotient?
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Context:
Given three different points \( L, M \), and \( N \) on a straight line (with \( M \) between \( L \) and \( N \), and \( LN \neq MN \)), semicircles are constructed with diameters \( LM \), \( MN \), and \( LN \) such that the centers of these semicircles are points \( A, B \), and \( C \) respectively. Point \( C \) lies on one side of the line \( LN \), while points \( A \) and \( B \) lie on the other side of the line \( LN \). Find the ratio of the area of the figure enclosed by these three semicircles to the area of the triangle \( ABC \).
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Context:
The vertices of an equilateral triangle lie on the hyperbola \( xy = 3 \). The centroid of this triangle is at the origin, \( (0,0) \). What is the square of the area of the triangle?
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Context:
Place the integers 1 through 9 on three separate cards, with three numbers on each card. The condition is that the difference between any two numbers on the same card cannot also be on the same card. Currently, the numbers 1 and 5 are on the first card, number 2 is on the second card, and number 3 is on the third card. Find the remaining number to be written on the first card.
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Context:
圆内接四边形 $A B C D$ 中, $A B=136, B C=80, C D=150, D A=102$, 求它的外接圆直径的值。
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Context:
设数列 \{a_n\} 满足 a_1=8，且对任意正整数 n，均有a_{n+1}=\left(1+\df{3}{n}\right)a_n+\left(1-\df{3}{n}\right)\cdot 2^n,则 a_{2023} 除以 5 的余数是__________.（李纪琛供题）
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Context:
Emily and John each solved three-quarters of the homework problems individually and the remaining one-quarter together. Emily correctly answered 70% of the problems she solved alone, achieving an overall accuracy of 76% on her homework. John had an 85% success rate with the problems he solved alone. Calculate John's overall percentage of correct answers.
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Context:
In a number game between two very smart students A and B, a referee first writes the integers \(2, 3, \cdots, 2020\) on a blackboard, then deletes one of these numbers. Then A and B take turns deleting one number at a time (B deletes the first number, and then A deletes one). This continues until two numbers remain that are coprime (mutually prime). If the final two numbers are coprime, student A wins; otherwise, student B wins. According to these game rules, what is the probability that student A wins?
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Context:
Let \( a_{1}, a_{2}, \ldots, a_{9} \) be nine real numbers, not necessarily distinct, with average \( m \). Let \( A \) denote the number of triples \( 1 \leq i<j<k \leq 9 \) for which \( a_{i}+a_{j}+a_{k} \geq 3m \). What is the minimum possible value of \( A \)?
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Context:
Tom's algebra notebook consists of 50 pages, 25 sheets of paper. Specifically, page 1 and page 2 are the front and back of the first sheet of paper, page 3 and page 4 are the front and back of the second sheet of paper, and so on. One day, Tom left the notebook on the table while he went out, and his roommate took away several consecutive pages. The average of the remaining page numbers is 19. How many pages did the roommate take away?
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Context:
A circle inscribed in a right triangle \(A B C\) \(\left(\angle A B C = 90^{\circ}\right)\) is tangent to sides \(A B\), \(B C\), and \(A C\) at points \(C_{1}\), \(A_{1}\), and \(B_{1}\) respectively. An excircle is tangent to side \(B C\) at point \(A_{2}\). \(A_{0}\) is the center of the circle circumscribed around triangle \(A_{1} A_{2} B_{1}\); similarly, point \(C_{0}\) is defined. Find the angle \(A_{0} B C_{0}\).
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Context:
The angle between the generatrix and the axis of a cone is $30^{\circ}$. Then the central angle $\alpha \left(\alpha\ \ \gt 0\right)$ of the unfolded diagram of the cone's lateral surface is ______.
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Context:
8.3. In the tetrahedron $K L M N$, it is known that $K L = M N = 4$, $K M = L N = 5$, $K N = M L = 6$. Points $P, Q, R, S$ are the centers of the inscribed circles of triangles $K L M, K L N$, $K M N$ and $L M N$. Find the volume of the tetrahedron $P Q R S$. Round the answer to the nearest hundredth if necessary.
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Context:
A shopkeeper sells his goods at cost price but uses three faulty meters that weigh 900 grams, 960 grams, and 980 grams for every kilogram, respectively. The shopkeeper uses these meters in the ratio of 2:3:5. Calculate the overall profit percentage using a weighted average approach.
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Context:
A large $5 \times 5 \times 5$ cube is formed using 125 small $1 \times 1 \times 1$ cubes. There are three central columns, each passing through the small cube at the very centre of the large cube: one from top to bottom, one from front to back, and one from left to right. All of the small cubes that make up these three columns are removed. What is the surface area of the resulting solid?
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Context:
How many ways can we write digits (1) in place of the letters so that we get a correct subtraction? (Of course, different letters are replaced by different digits, and zero is not written as the leading digit of a number.)

$$
\overline{FTC}-\overline{ETO}=\overline{KK}
$$
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Context:
Let $P$ be a point on the plane of $\triangle ABC$ that satisfies $3PA + 4\vec{u}P = m\vec{B}$ with $m > 0$. If the area of $\triangle ABP$ is 8, find the area of $\triangle ABC$.
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Context:
The non-intersecting diagonals of two adjacent faces of a rectangular parallelepiped are inclined at angles $\alpha$ and $\beta$ to the plane of the base. Find the angle between these diagonals.
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Context:
Let $n$ be the 200th smallest positive real solution to the equation $x - \frac{\pi}{2} = \tan x$. Find the greatest integer that does not exceed \(\frac{n}{2}\).
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Context:
For \(\theta < \pi\), find the largest real number \(\theta\) such that
$$
\prod_{k=0}^{10} \cos 2^{k} \theta \neq 0, \prod_{k=0}^{10}\left(1+\frac{1}{\cos 2^{k} \theta}\right)=1.
$$
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Context:
Given that two children, A and B, and three adults, 甲, 乙, and 丙, are standing in a line, A is not at either end, and exactly two of the three adults are standing next to each other. The number of different arrangements is $\boxed{\text{answer}}$.
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Context:
How many positive integers $a$ with $a \le 154$ are there such that the coefficient of $x^a$ in the expansion of
\[(1+x^{7}+x^{14}+ \cdots +x^{77})(1+x^{11}+x^{22}+\cdots +x^{77})\]
is zero?
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Context:
A cube has six faces. Each face has some dots on it. The numbers of dots on the six faces are 2, 3, 4, 5, 6, and 7. Harry removes one of the dots at random, with each dot equally likely to be removed. When the cube is rolled, each face is equally likely to be the top face. What is the probability that the top face has an odd number of dots on it?
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Context:
The corners of a $2$-dimensional room in the shape of an isosceles right triangle are labeled $A$, $B$, $C$ where $AB = BC$. Walls $BC$ and $CA$ are mirrors. A laser is shot from $A$, hits off each of the mirrors once, and lands at a point $X$ on $AB$. Let $Y$ be the point where the laser hits off $AC$. If $\frac{AB}{AX} = 64$, and $\frac{CA}{AY} = \frac{p}{q}$ for coprime positive integers $p$ and $q$, find $p + q$.
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Context:
In the expression \((x+y+z)^{2024} + (x-y-z)^{2024}\), the parentheses are expanded and like terms are combined. How many monomials \(x^{a} y^{b} z^{c}\) have a non-zero coefficient?
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Context:
For any natural number \( k \):
- If \( k \) is even, divide it by 2.
- If \( k \) is odd, add 1 to it.

This is considered one operation. Let \( a_n \) be the number of numbers that become 1 exactly after \( n \) operations. Find \( a_{15} \).
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Context:
James takes 5 oranges and breaks each orange into a certain number of pieces. He splits the pieces between 4 people. If an orange has 80 calories and each person gets 100 calories, into how many pieces did James break each orange?
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Context:
For some positive real $\alpha$, the set $S$ of positive real numbers $x$ with $\{x\}>\alpha x$ consists of the union of several intervals, with total length 20.2. The value of $\alpha$ can be expressed as $\frac{a}{b}$, where $a, b$ are relatively prime positive integers. Compute $100a+b$. (Here, $\{x\}=x-\lfloor x\rfloor$ is the fractional part of $x$.)
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Context:
How many positive integers $k$ are there such that $\frac{k}{2013}(a+b) = \text{lcm}(a,b)$ has a solution in positive integers $(a,b)$?
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Context:
There are $N$ natural numbers written on a board, where $N \geq 5$. It is known that the sum of all the numbers is 80, and the sum of any five of them is no more than 19. What is the smallest possible value of $N$?
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Context:
Let \(A B C D E F\) be a regular hexagon. A frog starts at vertex \(A\). Each time, it can jump to one of the two adjacent vertices. If the frog reaches vertex \(D\) within 5 jumps, it stops jumping. If it does not reach vertex \(D\) within 5 jumps, it stops after completing the 5 jumps. How many different sequences of jumps are possible from the start until the frog stops?
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Context:
Erica took a quiz with 20 questions. She scored 7 points for each correct answer, lost 4 points for each incorrect answer, and received 0 points for each question left blank. Given that Erica scored a total of 100 points, calculate the number of questions she left blank. Express your answer as a whole number.
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Context:
Evaluate the line integral $\int_C xy^2\mathrm{d}x + 2x^2y\mathrm{d}y , $ where $C$ is the triangle with vertices $(0,0), (2,2),(2,4)$.
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Context:
Let f(x) be a monotonically decreasing odd function defined on R. If x1+x2>0, x2+x3>0, and x3+x1>0, determine the sign of f(x1)+f(x2)+f(x3). Express your answer as a single inequality in the format f(x1)+f(x2)+f(x3) [inequality] 0.
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Context:
In a sign pyramid, a cell gets a "+" if the two cells below it have the same sign, and it gets a "-" if the two cells below it have different signs. The diagram below illustrates a sign pyramid with four levels. Find the number of possible ways to fill the four cells in the bottom row to produce a "+" at the top of the pyramid.
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Context:
For all integers $n \geq 2$, let $f(n)$ denote the largest positive integer $m$ such that $\sqrt[m]{n}$ is an integer. Evaluate
\[f(2) + f(3) + \cdots + f(100).\]
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Context:
Let $n$ be a positive integer. A sequence of $n$ positive integers (not necessarily distinct) is called [b]full[/b] if it satisfies the following condition: for each positive integer $k\geq2$, if the number $k$ appears in the sequence then so does the number $k-1$, and moreover the first occurrence of $k-1$ comes before the last occurrence of $k$. For each $n$, how many full sequences are there ?
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Context:
In a single-round-robin tournament, 10 chess players are participating. After what minimum number of rounds can it happen that a sole winner has already been determined prematurely? (In each round, the participants are divided into pairs. Win - 1 point, draw - 0.5 points, loss - 0).
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Context:
Determine the number of 8-tuples of nonnegative integers \(\left(a_{1}, a_{2}, a_{3}, a_{4}, b_{1}, b_{2}, b_{3}, b_{4}\right)\) satisfying \(0 \leq a_{k} \leq k\), for each \(k=1,2,3,4\), and \(a_{1}+a_{2}+a_{3}+a_{4}+2 b_{1}+3 b_{2}+4 b_{3}+5 b_{4}=19\).
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Context:
Find the angle between the generatrix and the base of a truncated cone whose total surface area is twice the surface area of an inscribed sphere.
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Context:
Xiao Zhao, Xiao Qian, Xiao Sun, and Xiao Li were asked who among them had visited the Great Wall, and their responses were as follows:
Xiao Zhao said: I haven't been there;
Xiao Qian said: Xiao Li has been there;
Xiao Sun said: Xiao Qian has been there;
Xiao Li said: I haven't been there.
Given that only one of the four students is lying, determine the name of the student who definitely visited the Great Wall.
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Context:
What is the greatest common divisor of $121^2 + 233^2 + 345^2$ and $120^2 + 232^2 + 346^2$?
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Context:
Find the smallest real number \( c \) such that for any sequence of positive numbers \(\{x_n\}\), if
\[
x_1 + x_2 + \cdots + x_n \leq x_{n+1}, \quad n = 1, 2, 3, \ldots
\]
then
\[
\sqrt{x_1} + \sqrt{x_2} + \cdots + \sqrt{x_n} \leq c \sqrt{x_1 + x_2 + \cdots + x_n}, \quad n = 1, 2, 3, \ldots
\]
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Context:
Find $\limsup_{n\to\infty}(\frac{2\cdot5\cdot8\cdot\cdots\cdot(3n-4)}{3^nn!})^{1/n}$
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Context:
Let  $a_1$ ,  $a_2$ ,  $a_3$ ,  $a_4$ ,  $a_5$  be real numbers satisfying
\begin{align*}
  2a_1+a_2+a_3+a_4+a_5 &= 1 + \tfrac{1}{8}a_4
      2a_2+a_3+a_4+a_5 &= 2 + \tfrac{1}{4}a_3
          2a_3+a_4+a_5 &= 4 + \tfrac{1}{2}a_2
              2a_4+a_5 &= 6 + a_1
\end{align*}
Compute  $a_1+a_2+a_3+a_4+a_5$ .

*Proposed by Evan Chen*
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Context:
If \( \tan (\pi \cos \theta) = \cot (\pi \sin \theta) \), then \( \cos \left(\theta - \frac{\pi}{4}\right) = \) ______
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Context:
33rd Putnam 1972 Problem B2 A particle moves in a straight line with monotonically decreasing acceleration. It starts from rest and has velocity v a distance d from the start. What is the maximum time it could have taken to travel the distance d? Solution
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Context:
If the point corresponding to the complex number (2-i)z is in the second quadrant of the complex plane, calculate the value of z.
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Context:
Inside the square \(ABCD\), points \(K\) and \(M\) are marked (point \(M\) is inside triangle \(ABD\), point \(K\) is inside \(BMC\)) such that triangles \(BAM\) and \(DKM\) are congruent \((AM = KM, BM = MD, AB = KD)\). Find \(\angle KCM\) if \(\angle AMB = 100^\circ\).
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Context:
Given an isosceles triangle \( \triangle ABC \) with base angles \( \angle ABC = \angle ACB = 50^\circ \), points \( D \) and \( E \) lie on \( BC \) and \( AC \) respectively. Lines \( AD \) and \( BE \) intersect at point \( P \). Given \( \angle ABE = 30^\circ \) and \( \angle BAD = 50^\circ \), find \( \angle BED \).
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Context:
7.7. The distances from the midpoint of the height of a regular triangular pyramid to the lateral face and to the lateral edge are 2 and $\sqrt{13}$, respectively. Find the volume of the pyramid. If necessary, round your answer to two decimal places.
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Context:
Each vertex of the parallelogram $ABCD$ lies on the same side of the plane $S$ such that the distances of the vertices $A, B$, and $C$ from the plane $S$ are 4 cm, 6 cm, and 8 cm, respectively. The area of the projection of the parallelogram onto the plane $S$, which forms the quadrilateral $A'B'C'D'$, is $10 \text{ cm}^2$. What is the volume of the solid $ABCD A'B'C'D'$?
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Context:
Mac is trying to fill 2012 barrels with apple cider. He starts with 0 energy. Every minute, he may rest, gaining 1 energy, or if he has \( n \) energy, he may expend \( k \) energy \((0 \leq k \leq n)\) to fill up to \( n(k+1) \) barrels with cider. What is the minimal number of minutes he needs to fill all the barrels?
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Context:
An electrician was called to repair a garland of four light bulbs connected in series, one of which has burned out. It takes 10 seconds to unscrew any bulb from the garland and 10 seconds to screw it back in. The time spent on other actions is negligible. What is the minimum time in which the electrician can definitely find the burned-out bulb, given that he has one spare bulb?
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Context:
In any permutation of the numbers \(1, 2, 3, \ldots, 18\), we can always find a set of 6 consecutive numbers whose sum is at least \(m\). Find the maximum value of the real number \(m\).
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Context:
Express
$$\sum_{n=1}^{\infty} \sum_{m=1}^{\infty} \frac{1}{m^{2} n+m n^{2}+2 m n}$$
as a rational number. The original answer is in \(\frac{m}{n}\) format, please give the value of m + n.
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Context:
Let \( f(x) = c x (x - 1) \), where \( c \) is a positive real number. We use \( f^{n}(x) \) to denote the polynomial obtained by composing \( f \) with itself \( n \) times. For every positive integer \( n \), all the roots of \( f^{n}(x) \) are real. What is the smallest possible value of \( c \)?
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Context:
Four points are chosen at random on the surface of a sphere. What is the probability that the center of the sphere lies inside the tetrahedron whose vertices are at the four points? (It is understood that each point is independently chosen relative to a uniform distribution on the sphere.) The original answer is in \frac{m}{n} format, please give the value of m + n.
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Context:
9.5. In the parliament of the island state of Promenade-and-Tornado, 2019 indigenous inhabitants were elected, who are divided into knights and liars: knights always tell the truth, liars always lie. At the first meeting, 2016 of them sat in the parliamentary seats arranged in the hall in a rectangle of $42 \times 48$, three - in the seats of the chairman and his deputies in the presidium. During the meeting, each one stated that among their neighbors in the seat there are both knights and liars (neighbors are those who sit to the left, right, in front, behind, and diagonally: there can be from 3 to 8 in the hall and 1 or 2 in the presidium). Determine the minimum number of liars at the meeting.
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Context:
Given the graphs of $y=\sin (\frac{1}{2}x-\frac{\pi }{6})$, determine the horizontal shift required to obtain the graph of $y=\sin \frac{1}{2}x$.
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Context:
In a circle with radius $R$, draw the chord that, when rotating the circle around its diameter parallel to the chord, describes a cylindrical surface with the maximum possible area.
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Context:
Let \( S = \{1, 2, 3, 4\} \). A sequence of \( n \) terms \( a_1, a_2, \cdots, a_n \) has the following property: for any non-empty subset \( B \) of \( S \) (denote the number of elements in \( B \) as \( |B| \)), there exist \( |B| \) adjacent terms in the sequence that exactly form the subset \( B \). Find the minimum value of \( n \).
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Context:
Find the sum of all $x$ from $2$ to $1000$ inclusive such that \( \prod_{n=2}^x \log_{n^n}(n+1)^{n+2} \) is an integer.
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Context:
The task is given a finite increasing sequence \( a_{1}, a_{2}, \ldots, a_{n} \) (\(n \geq 3\)) of natural numbers, and for all \( k \leq n-2 \), the equality \( a_{k+2}=3 a_{k+1}-2 a_{k}-1 \) holds. The sequence must necessarily contain the term \( a_{k}=2021 \). Determine the maximum number of three-digit numbers divisible by 25 that this sequence can contain.
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Context:
If $\log_3 (x+5)^2 + \log_{1/3} (x - 1) = 4,$ compute $x.$
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Context:
Find all integers $n\ge 3$ with the following property: for all real numbers $a_1,a_2,\dots,a_n$ and $b_1,b_2,\dots, b_n $ satisfying $|a_k|+|b_k| = 1$ for $1\le k\le n$, there exist $x_1,x_2,\dots,x_n$ each of which is either $-1$ or $1$, such that $$ |\sum _{k=1}^nx_ka_k | + |\sum _{k=1}^nx_kb_k| \le 1 $$
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Context:
已知不等式 $\left|a x^{2}+b x+a\right| \leqslant x$ 对 $x \in$ $[1,2]$ 恒成立. 则 $3 a+b$ 的最大值为 _____.
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Context:
Jeannie hikes the 12 miles to Mount Overlook at a pace of 4 miles per hour on level ground. The trail up the mountain has an elevation gain of 3,000 feet. Her pace decreases by 1 mile per hour for every 1,000 feet of elevation gain. After reaching Mount Overlook, she continues to hike another 9 miles to Lake Astonishing at a pace of 4 miles per hour but with 2,000 feet of elevation loss. Her pace increases by 1 mile per hour for every 1,000 feet of elevation loss. She then returns to her starting point on a different trail that is 15 miles long with 5,000 feet of elevation change (3,000 feet up and 2,000 feet down). Her uphill pace decreases by 1 mile per hour for every 1,000 feet of elevation gain, and her downhill pace increases by 1 mile per hour for every 1,000 feet of elevation loss.

How long did her entire hike take, in hours?
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Context:
On the sides \(AB\) and \(AD\) of square \(ABCD\), points \(E\) and \(F\) are marked such that \(BE : EA = AF : FD = 2022 : 2023\). Segments \(EC\) and \(FC\) intersect the diagonal \(BD\) of the square at points \(G\) and \(H\), respectively. Find the ratio \(GH : BD\).
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Context:
What is the maximum number of rooks that can be placed in an \(8 \times 8 \times 8\) cube so that they do not attack each other?
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Context:
The distance from a point $A$ on the parabola $y^{2}=4x$ to its focus $F$ is $4$. What is the slope of line $AF$?
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Context:
If n is an integer, f(n) = f(n-1) - n. The value of f(6) is 1. What is the value of f(n-1) when f(n) = 4?
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Context:
The axis cross-section $SAB$ of a cone with an equal base triangle side length of 2, $O$ as the center of the base, and $M$ as the midpoint of $SO$. A moving point $P$ is on the base of the cone (including the circumference). If $AM \perp MP$, then the length of the trajectory formed by point $P$ is ( ).
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Context:
In $\vartriangle ABC, \angle A = 52^o$ and $\angle B = 57^o$. One circle passes through the points $B, C$, and the incenter of $\vartriangle ABC$, and a second circle passes through the points $A, C$, and the circumcenter of $\vartriangle ABC$. Find the degree measure of the acute angle at which the two circles intersect.
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Context:
Dana creates a "rectangular spiral" by drawing line segments of lengths, in cm, of \(1,1,2,2,3,3,4,4, \ldots\). Dana's pen runs out of ink after the total of all the lengths he has drawn is \(3000 \, \text{cm}\). Find the length of the longest line segment that Dana draws. Express your answer as a single integer (length in cm).
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Context:
In $\triangle ABC$, $\angle ABC = 90^{\circ}$, and $BC = 8$. Points $D$ and $E$ lie on $AB$ and $AC$ respectively, such that $CD = DE$ and $\angle DCB = \angle EDA$. If the area of $\triangle EDC$ is 50, what is the length of $AB$?
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Context:
Find the largest value for the constant $\beta$ in the inequality: $$xy^2 \ge 3x^3+\beta y^3$$ assuming $x,y$ are positive numbers.
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Context:
Find the value of \(m + n\) where \(m\) and \(n\) are integers defined as follows: The positive difference between the roots of the quadratic equation \(5x^2 - 9x - 14 = 0\) can be expressed as \(\frac{\sqrt{m}}{n}\), with \(m\) not divisible by the square of any prime number.
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Context:
设 $x, y, z \in[0,1]$, 则 $M=\sqrt{|x-y|}+\sqrt{|y-z|}+\sqrt{|z-x|}$ 的最大值是 $\qquad$.  原始的答案是k\sqrt{m}+n的形式，请给出k + m + n的值。
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Context:
Let  $ABC$  be a triangle such that  $AB = 7$ , and let the angle bisector of  $\angle BAC$  intersect line  $BC$  at  $D$ . If there exist points  $E$  and  $F$  on sides  $AC$  and  $BC$ , respectively, such that lines  $AD$  and  $EF$  are parallel and divide triangle  $ABC$  into three parts of equal area, determine the number of possible integral values for  $BC$ .
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Context:
Anička received a rectangular cake for her birthday. She cut the cake with two straight cuts. The first cut was made such that it intersected both longer sides of the rectangle at one-third of their length. The second cut was made such that it intersected both shorter sides of the rectangle at one-fifth of their length. Neither cut was parallel to the sides of the rectangle, and at each corner of the rectangle, there were either two shorter segments or two longer segments of the divided sides joined.

Anička ate the piece of cake marked in grey. Determine what portion of the cake this was.
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Context:
Solve the equation:
$$(1 + x + x^ 2 )(1 + x + . . . + x^{10}) = (1 + x + . . . + x^6 ) ^2.$$
Please write the sum of all solutions for $x$.
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Context:
在正方体的8个顶点中任取3个顶点构成一个三角形，求该三角形为等腰三角形的概率。答案的格式为\frac{k}{m}，请给出k+m的值。
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Context:
If a = 105 and a ^ 5 = 3^3 × 5^2 × 7^2 × 11^2 × 13 × c, what is the value of c?
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Context:
The area of the figure formed by the parabola $y=x^2$ and its tangent line at $x=1$ and the x-axis is \_\_\_\_\_\_.
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Context:
A polyhedron with vertices at the midpoints of the edges of a cube is called a cuboctahedron. The cross-section of a cuboctahedron with a plane results in a regular polygon. What is the maximum number of sides this polygon can have?
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Context:
Nathaniel and Obediah play a game in which they take turns rolling a fair six-sided die and keep a
running tally of the sum of the results of all rolls made. A player wins if, after he rolls, the number on the running tally is a multiple of 7. Play continues until either player wins, or else indenitely. If Nathaniel goes first, determine the probability that he ends up winning.
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Context:
Use the division algorithm to find the gcd of $f(x)= 3x^3 + 2x^2 -x +1$ and $g(x)= x^2 + 2x -3$ in $\mathbb Z[x]$.
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Context:
For nonnegative integer $n$, the following are true:
- $f(0) = 0$
- $f(1) = 1$
- $f(n) = f\left(n - \frac{m(m-1)}{2}\right) - f\left(\frac{m(m+1)}{2} - n\right)$ for integer $m$ satisfying $m \geq 2$ and $\frac{m(m-1)}{2} < n \leq \frac{m(m+1)}{2}$.

Find the smallest integer $n$ such that $f(n) = 4$.
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Context:
Let $g_{1}(x)=\frac{1}{3}\left(1+x+x^{2}+\cdots\right)$ for all values of $x$ for which the right hand side converges. Let $g_{n}(x)=g_{1}\left(g_{n-1}(x)\right)$ for all integers $n \geq 2$. What is the largest integer $r$ such that $g_{r}(x)$ is defined for some real number $x$ ?
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Context:
The new paint can was weighed 24 kilograms (kg). After using half the paint, I reweighed the can and it weighed 14 kilograms (kg). What is the weight in kilograms (kg) of the paint in the new can, excluding the weight of the can?\
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Context:
If a, b, c, d, e, f, g, h, i, and j are integers and (ab + cdef + ghij) < 0, then what is the maximum number w of integers that can be negative?
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Context:
There are 15 members in a family. It takes 180 days for a woman to complete the work, however, a man completes it in 120 days. They take 17 days to complete the work if men work on alternate days and women work every third day. If all of them started together on the 1st day, how many women are there in the family?
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Context:
Given the following numbers: 20172017, 20172018, 20172019, 20172020, and 20172021. Is there a number among them that is relatively prime to all the others? If so, which one?
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Context:
已知 $a_{n}=\mathrm{C}_{200}^{n} \cdot(\sqrt[3]{6})^{200-n} \cdot\left(\frac{1}{\sqrt{2}}\right)^{n}(n=1,2, \cdots, 95)$, 则数列 $\left\{a_{n}\right\}$ 中整数项的个数为 $\qquad$.
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Context:
Given that ABCD is a unit square and P is the midpoint of side AB, the square is folded along PC and PD to form a tetrahedron PACD (where A coincides with B). Calculate the volume of tetrahedron PACD. Express your answer as a fraction with a square root.
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Context:
Given that the function f(x) has two zero points (x1, x2) where x1 < 0 < x2 < 1, and the function g(x) = x - ln(x^2), find the maximum number of real roots of the equation f[g(x)] = 0. Express your answer as a single integer.
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Context:
In the geometric sequence $\{a_n\}$, if $a_2a_5 = -\frac{3}{4}$ and $a_2 + a_3 + a_4 + a_5 = \frac{5}{4}$, calculate the value of $\frac{1}{a_2} + \frac{1}{a_3} + \frac{1}{a_4} + \frac{1}{a_5}$.
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Context:
From the set $\{1, 2, 3, 4, \ldots, 20\}$, select four different numbers $a, b, c, d$ such that $a+c=b+d$. If the order of $a, b, c, d$ does not matter, calculate the total number of ways to select these numbers.
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Context:
At exactly noon, a truck left a village for the city, and at the same time, a car left the city for the village. If the truck had left 45 minutes earlier, they would have met 18 kilometers closer to the city. If the car had left 20 minutes earlier, they would have met $k$ kilometers closer to the village. Find $k$.
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Context:
Baker made some cakes and 148 pastries. He sold 15 cakes and 103 pastries. Baker still has 45 pastries. How many cakes did Baker make initially?
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Context:
Determine the limiting value of the infinite product $$P = \prod_{k=2}^{\infty} \frac{k^3 - 1}{k^3 + 1}$$
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Context:
Positive integers \(a\), \(b\), and \(c\) are randomly and independently selected with replacement from the set \(\{1, 2, 3, \cdots, 2010\}\). Find the probability that \(abc + ab + a\) is divisible by \(3\). The original answer is in \(\frac{k}{m}\) format, please give the value of \(k + m\).
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Context:
There are two fair dice and their sides are positive integers \( a_{1}, \ldots, a_{6} \) and \( b_{1}, \ldots, b_{6} \), respectively. After throwing them, the probability of getting a sum of \( 2, 3, 4, \ldots, 12 \) respectively is the same as that of throwing two normal fair dice. Suppose that \( a_{1}+\cdots+a_{6} < b_{1}+\cdots+b_{6} \). What is \( a_{1}+\cdots+a_{6} \)?
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Context:
On the Cartesian plane $\mathbb{R}^{2}$, a circle is said to be nice if its center is at the origin $(0,0)$ and it passes through at least one lattice point (i.e. a point with integer coordinates). Define the points $A=(20,15)$ and $B=(20,16)$. How many nice circles intersect the open segment $A B$ ?
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Context:
Peter knows that the distance an airplane travels between Spain and Russia is 7019 km, while the distance between Spain and Germany is 1615 km. Peter took a flight from Spain, which made a stopover in Germany, while being there, he receives a call from his brother asking him how many kilometers he has to fly to get to Russia and then return to Spain.  What will Peter answer?
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Context:
The number $1$ is special. The number $2$ is special because it is relatively prime to $1$. The number $3$ is not special because it is not relatively prime to the sum of the special numbers less than it, $1 + 2$. The number $4$ is special because it is relatively prime to the sum of the special numbers less than it. So, a number bigger than $1$ is special only if it is relatively prime to the sum of the special numbers less than it. Find the twentieth special number.
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Context:
We take $100$ consecutive natural numbers $a_1, a_2, \ldots, a_{100}$. Determine the last two digits of the number $a_1^8 + a_2^8 + \ldots + a_{100}^8$.
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Context:
A school's sports department has 2 identical basketballs and 3 identical volleyballs to distribute among 4 different classes, with each class receiving 1 ball. Calculate the total number of different methods to distribute the balls.
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Context:
Complex numbers \(a\), \(b\), \(c\) form an equilateral triangle with side length 24 in the complex plane. If \(|a + b + c| = 48\), find \(|ab + ac + bc|\).
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Context:
Armen paid \$190 to buy movie tickets for a group of \( t \) people, consisting of some adults and some children. Movie tickets cost \$5 for children and \$9 for adults. How many children's tickets did he buy?
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Context:
A group of adventurers is showing their loot. It is known that exactly 4 adventurers have rubies; exactly 10 have emeralds; exactly 6 have sapphires; and exactly 14 have diamonds. Additionally, it is known that:

- If an adventurer has rubies, they also have either emeralds or diamonds (but not both simultaneously).
- If an adventurer has emeralds, they also have either rubies or sapphires (but not both simultaneously).

What is the minimum number of adventurers that can be in such a group?
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Context:
Let $f(x)=x^{4}+14 x^{3}+52 x^{2}+56 x+16$. Let $z_{1}, z_{2}, z_{3}, z_{4}$ be the four roots of $f$. Find the smallest possible value of $|z_{a} z_{b}+z_{c} z_{d}|$ where $\{a, b, c, d\}=\{1,2,3,4\}$.
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Context:
Three people are sitting in a row of eight seats. If there must be an empty seat on each side of each person, the number of different seating arrangements is ____.
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Context:
Define the annoyingness of a permutation of the first \(n\) integers to be the minimum number of copies of the permutation that are needed to be placed next to each other so that the subsequence \(1,2, \ldots, n\) appears. For instance, the annoyingness of \(3,2,1\) is 3, and the annoyingness of \(1,3,4,2\) is 2. A random permutation of \(1,2, \ldots, 2022\) is selected. Compute the expected value of the annoyingness of this permutation.
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Context:
Find the natural number "a" for which:

$$\sum_{k=0}^n f(a+k)=16(2^n-1)$$

where the function $f$ satisfies the relation

$$f(x+y)=f(x)*f(y)$$

for all natural numbers $x,y$ and $f(1)=2$.
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Context:
In the Cartesian coordinate system, the point $P\left(a-3,a+3\right)$ cannot be in the ________ quadrant.
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Context:
In a 6-story building, three people, A, B, and C, enter the elevator on the first floor. Find the total number of possible situations in which exactly one person wants to get off at the 6th floor, and A does not get off on the 2nd floor. Express your answer as a single integer.
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Context:
If \((1.0025)^{10}\) is calculated accurate to five decimal places, what is the digit in the fifth decimal place?
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Context:
Anna works everyday and gets her salary on a weekly basis. If Anna didn't go to work on 2 days this week, her deducted salary was 985 dollars. What is her usual weekly salary?
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Context:
Example 10 Find the smallest positive integer $n$, such that in any two-coloring of the complete graph $K_{n}$ with $n$ vertices, there always exist $m$ monochromatic edges, no two of which share a common endpoint.
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Context:
There are $22$ chairs around a round table. Find the minimum number $n$ such that for any group of $n$ people sitting at the table, there will always be two people with exactly $2$ or $8$ chairs between them.
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Context:
Evaluate $\displaystyle\int_{-\infty}^\infty\frac{\sin(t)}{t(1+t^2)}\,\mathrm{d}t.$
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Context:
Evaluate the limit: \[ \lim_{k\rightarrow \infty}\binom{2k+1}{3}\frac{1}{2^{2(2k-1)}}\left[\binom{2k-2}{k-2}-\binom{2k-2}{k-1}\right]^2. \]
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Context:
For which positive integers \(n\) do there always exist positive integers \(a, b, c, d\) such that \(x < \frac{a}{b} < \frac{c}{d} < y\) and \(a^n + b^n = c^n + d^n\)?
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Context:
Determine the order of the quotient module $M/K$ for the ring $D = \mathbb{Z}[\sqrt{-2}]$ and the free $D$-module $M = D^3$, where $K$ is the submodule generated by $(1, 1 + \sqrt{-2}, 2), (0, 1, 3), (2, 0, 1)$. Provide your answer as a number or expression.
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Context:
Evaluate the limit: $$\lim_{n \to \infty} \left( \lim_{x \to 0} \left(1+\sin^2(x)+\sin^2(2x)+\cdots+\sin^2(nx)\right)^\frac{1}{n^3\cdot x^2}\right)$$
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Context:
4. A rectangle of 19 cm $\times$ 65 cm is divided by lines parallel to its sides into squares with a side of 1 cm. Into how many parts will this rectangle be divided if we also draw its diagonal?
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Context:
Determine the minimum number of charts required to cover a torus, given that each chart must consist of simply connected open sets.
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Context:
Determine the maximum number of integer solutions \( k \) to the equation \( p(k) = k^3 \), where \( p(x) \) is a polynomial with integer coefficients and \( p(100) = 100 \).
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Context:
Determine the cardinality of the quotient ring of Gaussian integers when factored by a non-zero element with relatively prime components.
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Context:
Determine the maximum depth of subnormality for subgroups of a nilpotent group with nilpotency class $n$.
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Context:
Evaluate the limit: $$\lim_{x \to \frac{\pi}{2}} \biggl|\frac{1}{(1+\sin x)\log(\sin x)}+\frac{1}{\cos^2x} \biggl| $$
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Context:
Let $A$ be a square matrix consisting of $1$s and $0$s. If $A^2=0$, determine the minimum number of $1$s that $A$ can have.
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Context:
Given the equation of the conic $5x^2 + 6xy + 5y^2 = 8$, find the equation of its auxiliary circle.
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Context:
Calculate the flux of the curl of the vector field \( F(x, y, z) = (x - yz, xz, y) \) across the surface \( S \), which is obtained by rotating the graph of \( x = 2^z + 3^z \) for \( z \in [0, 1] \) around the \( z \)-axis, with normal vectors oriented outward.
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Context:
For any positive integers $a$ and $b$ with $b>1$, let $s_b(a)$ be the sum of the digits of $a$ when it is written in base $b$. Suppose $n$ is a positive integer such that
$$
\sum_{i=1}^{\lfloor\log_{23}n\rfloor}s_{20}\left(\left\lfloor\frac{n}{23^i}\right\rfloor\right)=103\mathrm{\quad and}\sum_{i=1}^{\lfloor\log_{20}n\rfloor}s_{23}\left(\left\lfloor\frac{n}{20^i}\right\rfloor\right)=115
$$
Compute $s_{20}(n) - s_{23}(n)$.
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Context:
Consider a random walk on a finite state space $0, 1, \\ldots, N$ with absorbing barriers at states $0$ and $N$. At each state $1, \\ldots, N-1$, the probability of moving to the adjacent states is $\\frac{1}{2}$ each. At states $0$ and $N$, the process is absorbed, meaning it stays in the same state with probability $1$. How many stationary measures does this Markov chain have?
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Context:
Determine the minimal dimension of an affine subspace in \(\mathbb{R}^n\) that can divide an open, contractible set \(U \subseteq \mathbb{R}^n\) into disconnected components, where \(n \ge 3\).
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Context:
How many distinguishable large equilateral triangles can be formed using four congruent equilateral triangles, each with a different color on each side, considering that the colors on both sides of the triangles can be used?
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Context:
Given that \(a < b\) and \(\frac{f(b)/b}{f(a)/a}<2\), find a tighter upper bound for the expression \(\frac{a}{b}+\frac{b}{a}\).
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Context:
Evaluate the limit \( \lim_{n\to\infty}\int_{1}^{1+\delta}\frac{n}{x^2(1+x^n)}dx \) using Lebesgue's Dominated Convergence Theorem or the Sandwich Theorem, where \( \delta > 0 \) is a fixed number.
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Context:
Find the values of \( \alpha \) for which the matrix \( A_\alpha \) has a repeated eigenvalue.
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Context:
Determine the number of distinct solutions to the equation \( xyz + yvv = xxzy \), where each letter represents a different digit from 0 to 9. Each digit must be unique to a letter.
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Context:
Given the joint probability mass function $p(x,y)$ for $x \in \{0, 1\}$ and $y \in \{0,1,2\}$, where $3p(1,1) = p(1,2)$ and $p(1,1)$ maximizes the variance of $XY$, calculate the probability that $X$ or $Y$ is zero.
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Context:
In a Gaussian random walk, where $x_n$ is the sum of $n$ normal random variables, calculate the probability $P(x_1 > 0, x_2 > 0)$. Assume that the events $x_1 > 0$ and $x_2 > 0$ are not independent.
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Context:
Calculate the variance of $12(X_1 + X_2 + X_3)$, where $X_1$ is uniformly distributed on $(0,1)$, $X_2$ is uniformly distributed on $(0, X_1)$, and $X_3$ is uniformly distributed on $(0, X_2)$. You do not need to determine the distribution of $X_1 + X_2 + X_3$. Provide your answer as a single numerical value.
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Context:
There are 4 different types of coupons, divided into two groups: the first group consists of types 1 and 2, and the second group consists of types 3 and 4. Each new coupon obtained is type $i$ with probability $p_i$, where $p_1 = p_2 = 1/8$ and $p_3 = p_4 = 3/8$. Find the expected number of coupons that one must obtain to have at least all the types of either group.
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Context:
Calculate the surface area of the portion of the surface defined by \( z^2 = 2xy \) within the region \( 0 \le x \le 2 \) and \( 0 \le y \le 1 \).
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Context:
Given a figure composed of $3n^{2}$ rhombuses, determine the minimum number of rearrangement operations required to transform it into a specified target figure.
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Context:
In a game involving a positive integer \( n \geqslant 2 \), Player A selects \( 2n \) non-negative numbers \( x_1, x_2, \ldots, x_{2n} \) such that their sum is 1. Player B arranges these numbers in a circle and calculates the product of each pair of adjacent numbers, writing the largest product on the board. Determine the number that will be written on the board if both players play optimally.
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Context:
Find all functions $f: \Bbb{R} \to \Bbb{R}$ that satisfy the functional equation $f(a^2 + b^2) = f(a^2 - b^2) + f(2ab)$ for all real numbers $a$ and $b$.
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Context:
How many even 11-digit numbers, with no leading zeros, contain at least three occurrences of the digit 3?
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Context:
Determine the number of subgroups of $\mathbb{Z}_{p^2} \oplus \mathbb{Z}_{p^2}$ that are isomorphic to $\mathbb{Z}_{p^2}$, where $p$ is a prime.
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Context:
Determine the number of unique ways to distribute 5 blue, 4 green, 2 yellow, 1 red, and 1 black ball into 3 buckets such that each bucket has a distinct combination of balls and all balls must be placed in a bucket. An empty bucket is considered a distinct combination.
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Context:
Given a sequence of independent identically distributed (i.i.d.) random variables $X_1, X_2, \cdots$, each following a normal distribution with mean 1 and variance 1, and a Poisson random variable $N$ with mean 2, independent of the sequence $X_i$'s, find the variance of the sum $X_1 + X_2 + \cdots + X_{N+1}$. \( \text{Express your answer as a single number.} \)
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Context:
Determine the minimum distance of the ternary linear code with the following parity check matrix:
\[
H=\begin{pmatrix}
2 & 0 & 1 & 1 & 1 \\
0 & 2 & 2 & 0 & 0 \\
0 & 0 & 2 & 2 & 1 \\
1 & 1 & 2 & 2 & 0
\end{pmatrix}.
\]
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Context:
Determine the constant $c$ such that the solution to the differential equation $$(1-2x)y' = y$$ with boundary conditions $y(0) = -1$ and $y(1) = 1$ is real and finite for all $0 \leq x \leq 1$. The solution is given by $\ln y = \ln\left(\frac{2}{1-2x}\right) + c$. Find the value of $c$.
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Context:
In a $11\times11$ grid, calculate the number of non-decreasing grid walks from $(0,0)$ to $(10,10)$ that do not pass through the points $(3,3)$, $(7,2)$, $(3,7)$, or $(7,7)$. The total number of unrestricted paths is $\binom{20}{10}$. Determine the number of valid paths by subtracting the number of paths that pass through any of the forbidden points.
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Context:
Calculate the probability of obtaining a sum of at least 100 when rolling 30 dice, each with 10 sides numbered from 1 to 10.
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Context:
An urn initially contains 6 black balls and 0 white balls. In each turn, one ball is drawn at random, removed, and replaced with a ball of the opposite color. Calculate the expected number of turns until all 6 balls in the urn are white.
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Context:
Determine the sign of the second-order term in the Taylor expansion of the functional \( I[1+v] \), where \( I(u)=\int_{0}^{1}u^2(2-u)^2dx \) and \( u \in Y \), with \( Y=\{u\in C([0,1]):u(0)=1,u(1)=1\} \).
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Context:
How many times changes the sign of the function \[ f(x)=\cos x\cos\frac{x}{2}\cos\frac{x}{3}\cdots\cos\frac{x}{2009}\]at the interval $ \left[0, \frac{2009\pi}{2}\right]$?
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Context:
Let \(n \in \mathbb{Q}\) be a number such that \(\sqrt{n} + \sqrt[3]{n} \in \mathbb{N}\). Determine the form of \(n\).
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Context:
Find a closed-form expression for the integral \( \int_0^\pi \log(\sin x) \, dx \).
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Context:
Consider a system where two repairmen serve three machines, meaning at most two machines can be under repair at the same time. The time until a machine needs repair is exponentially distributed with a mean of \(1/3\), and each repair time is exponentially distributed with a mean of \(1/2\). These times are independent of each other. Let \(X_t\) represent the number of machines under repair or needing repair at time \(t\). Given the transition rate matrix:
\[
Q = \begin{bmatrix}
 -9 &  9  &  0  &  0  \\
  2 & -8  &  6  &  0  \\
  0 &  4  & -7  &  3  \\
  0 &  0  &  2  & -2
\end{bmatrix}
\]
Determine \(\lim_{t \to \infty} \mathbb{P}_i(X_t = 0)\) for \(i = 0, 1, 2, 3\).
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Context:
Given a right circular cone with an isosceles right triangle as its axial cross-section, where \( P \) is the vertex, \( PA \) is a generatrix, and \( B \) is a point on the base, find the maximum value of \( \frac{PA + AB}{PB} \).
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Context:
Let $w$ be a positive integer. Determine the range of values for $w$ such that there exist $a, b, c$ with $\pi \le a < b < c \le 2\pi$ satisfying the equation:
\[ \sin(aw) + \sin(bw) + \sin(cw) = 3. \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the radius of the sphere that touches the faces of the unit cube passing through vertex $A$ and the edges passing through vertex $B$.
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Context:
Let $ABC$ be a triangle with $AB=17$ and $AC=23$. Let $G$ be the centroid of $ABC$, and let $B_1$ and $C_1$ be on the circumcircle of $ABC$ with $BB_1\parallel AC$ and $CC_1\parallel AB$. Given that $G$ lies on $B_1C_1$, the value of $BC^2$ can be expressed in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Determine $100m+n$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the expected value of the random variable \(\tau\), which represents the number of draws needed for the sum of values drawn from a uniform distribution on \((0, 1)\) to exceed 2.
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Context:
Determine the values of $p$ for which the $p$-norm on $\mathbb{R}^n$ is differentiable everywhere except at the origin.
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Context:
Find the solution for $u_1(x, t)$ given the system of partial differential equations with $a = 0$ and the initial conditions $u_1(x, 0) = f(x)$ and $u_2(x, 0) = 0$.
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Context:
An employer is hiring one new employee from a group of N candidates, each rated on a scale from 1 to N. Candidates are interviewed in random order, and a decision to hire is made immediately after each interview. The employer can hire the mth candidate only if they are better than all previous m-1 candidates. If a candidate is hired on the ith trial, what is the probability that this candidate is the best among all N candidates?
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Context:
Suppose 55% of a large population of voters favor candidate A. Determine the minimum size of a random sample needed to ensure a 99% probability that the majority of voters in the sample will favor candidate A.
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Context:
Evaluate the line integral \( \oint_C \vec{F} \cdot d\vec{R} \) using Stokes' theorem, where the vector field is \( \vec{F} = (ye^x, x^2+e^x, z^2e^z) \) and the curve \( C \) is given by \( \vec{r}(t) = (1+\cos t, 1+\sin t, 1-\cos t-\sin t) \) for \( 0 \leq t \leq 2\pi \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the surface parameterized by:
\[ x = u(3v^2 - u^2 - 1/3), \quad y = v(3u^2 - v^2 - 1/3), \quad z = 2uv \ . \]
Find the angle between the coordinate lines at a point on the surface.
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Context:
A cube is sectioned by 33 planes parallel to its faces, forming multiple parallelepipeds. If the number of parallelepipeds with faces coinciding with the faces of the cube is half of the total number of parallelepipeds, find the total number of parallelepipeds formed.
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Context:
Determine the number of solutions to the equation $e^z - z = 0$.
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Context:
Determine the eigenvalues of the following $4 \times 4$ matrix:
$$J(\epsilon)=\begin{bmatrix} -\mu & 0 & 0 & -\frac{\beta b}{\mu}\\ 0 & -\sigma & 0 & \frac{\beta b}{\mu}\\ 0 & \sigma & -(\phi + d) & 0\\ 0 & 0 & \alpha & -(\phi + \epsilon) \end{bmatrix}$$
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Context:
Find the equation of the directrix of a parabola with a focus at (-1, -1) and a tangent at the point (7, 13) given by the equation 3x - y - 8 = 0.
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Context:
Find all values of \( p > 1 \) such that the series \( \sum_{n=1}^{\infty} f_n \) converges in \( L^p(\mathbb{R}) \), where:
\[
f_n(x) = \frac{1}{1+n^2\sqrt{x}}\chi_{[\exp(2n),2\exp(2n+2)]}(x)
\]
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Context:
Determine the number of inequivalent two-dimensional complex representations of the group $Z_4$. Provide your answer as a single integer.
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Context:
Find the coefficient of \(x^{12}y^{24}\) in the expansion of \((x^3 + 2xy^2 + y + 3)^{18}\) using the multinomial theorem.
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Context:
A polynomial \( p(x) \) is called self-centered if it has integer coefficients and satisfies \( p(100) = 100 \). If \( p(x) \) is a self-centered polynomial, determine the maximum number of integer solutions \( k \) to the equation \( p(k) = k^3 \).
Please reason step by step, and put your final answer within \boxed{}.