Context:
Points \( K, L, \) and \( M \) are the midpoints of sides \( AB, BC \), and \( CD \) of parallelogram \( ABCD \). It turns out that quadrilaterals \( KBLM \) and \( BCDK \) are inscribed. Find the ratio \( AC:AD \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Example 9 In $\triangle A B C$, $O$ is the intersection of $A A^{\prime}$ and $B B^{\prime}$, $A^{\prime}$ is on $B C$, $B^{\prime}$ is on $A C$, and $S_{\triangle A O R^{\prime}}=3, S_{\triangle O A B}=2, S_{\triangle O A^{\prime} B}=1$. Find $S_{\triangle C A^{\prime} B^{\prime}}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the enthalpy change (ΔH) for the reaction when 1 mole of CH3CH2OH reacts with 1.5 moles of O2 to form 1 mole of H2O and 1 mole of CH3COOH using the following bond dissociation energies:

CH3-CH2 (C-C): 347 kJ/mol
CH3-O (C-O): 358 kJ/mol
CH2-OH (O-H): 463 kJ/mol
C=O (COOH): 745 kJ/mol
O-H (COOH): 467 kJ/mol
O=O (O2): 498 kJ/mol
O-H (H2O): 467 kJ/mol
Please reason step by step, and put your final answer within \boxed{}.


Context:
A deck of playing cards without jokers has 4 suits with a total of 52 cards, each suit has 13 cards numbered from 1 to 13. Feifei draws 2 hearts, 3 spades, 4 diamonds, and 5 clubs. If the sum of the face values of these 4 drawn cards is exactly 34, how many of them are 2s?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the area of a convex octagon that is inscribed in a circle and has four consecutive sides of length 3 units and the remaining four sides of length 2 units. The answer is in the form $r + s\sqrt{t}$ with $r$, $s$, and $t$ as positive integers. Please provide the value of $r + s + t$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
设取整 $\left[\frac{10^{2022}}{10^{100}-9}\right]$ 被 $10^{n}$ 整除, 则非负整数 $n$ 的最大可能值为 $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Quadrilateral $ABCD$ satisfies the following conditions:
- $AB = 8$,
- $BC = 5$,
- $CD = 17$,
- $DA = 10$.

Let $E$ be the intersection of diagonals $AC$ and $BD$. It is given that the ratio $BE : ED = 1 : 2$.

Find the area of quadrilateral $ABCD$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A box contains $900$ cards, labeled from $100$ to $999$. Cards are removed one at a time without replacement. What is the smallest number of cards that must be removed to guarantee that the labels of at least three removed cards have equal sums of digits?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the smallest positive integer $m$ such that the coefficient of $x^2$ in the expansion of the sum $(1+x)^2 + (1+x)^3 + \ldots + (1+x)^{49} + (1+mx)^{50}$ is equal to $(3n+1)\binom{51}{3}$ for some positive integer $n$. Determine the value of $n$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The positive integer \( n \) can be divided exactly by 1990, and \( n \) has exactly 12 positive divisors (including 1 and \( n \)). Find \( n \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Jordan owns 15 pairs of sneakers. Three fifths of the pairs are red and the rest are white. Two thirds of the pairs are high-top and the rest are low-top. The red high-top sneakers make up a fraction of the collection. What is the least possible value of this fraction?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a triangle $ \triangle ABC $ with sides that are consecutive integers, and the largest angle $ \angle B $ is twice the smallest angle $ \angle A $, find the lengths of the sides of the triangle.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let the functions  $f(\alpha,x)$  and  $g(\alpha)$  be defined as \[f(\alpha,x)=\dfrac{(\frac{x}{2})^\alpha}{x-1}\qquad\qquad\qquad g(\alpha)=\,\dfrac{d^4f}{dx^4}|_{x=2}\] Then  $g(\alpha)$  is a polynomial is  $\alpha$ .  Find the leading coefficient of  $g(\alpha)$ .
Please reason step by step, and put your final answer within \boxed{}.


Context:
Perpendiculars \( AP \) and \( AK \) are dropped from vertex \( A \) of triangle \( ABC \) onto the angle bisectors of the external angles at \( B \) and \( C \) respectively. Find the length of segment \( PK \), given that the perimeter of triangle \( ABC \) is \( P \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
On Qingming Festival, students take a bus to the Martyrs' Cemetery. If the bus increases its speed by one-fifth after traveling for 1 hour at its original speed, it can arrive 20 minutes earlier than the scheduled time. If the bus first travels 72 km at its original speed and then increases its speed by one-third, it can arrive 30 minutes earlier than the scheduled time. How many kilometers is it from the school to the Martyrs' Cemetery?
Please reason step by step, and put your final answer within \boxed{}.


Context:
The coefficients of the polynomial $P(x)$ are nonnegative integers, each less than 100. Given that $P(10) = 331633$ and $P(-10) = 273373$, compute $P(1)$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $a = 4 + 3i,$ $b = 1 -2i,$ and $c = 8 - 5i.$  The complex number $d$ is such that $a,$ $b,$ $c,$ and $d$ form the vertices of a parallelogram, when plotted in the complex plane.  Enter all possible values of $d,$ separated by commas.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Figure 0 consists of a square with side length 18. For each integer \(n \geq 0\), Figure \(n+1\) consists of Figure \(n\) with the addition of two new squares constructed on each of the squares that were added in Figure \(n\). The side length of the squares added in Figure \(n+1\) is \(\frac{2}{3}\) of the side length of the smallest square(s) in Figure \(n\). Define \(A_{n}\) to be the area of Figure \(n\) for each integer \(n \geq 0\). What is the smallest positive integer \(M\) with the property that \(A_{n}<M\) for all integers \(n \geq 0\)?
Please reason step by step, and put your final answer within \boxed{}.


Context:
由数字0，1，2，3，4，5，6，7组成没有重复数字的四位数，则能被15整除且0不在个位的四位数共有________个.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given $x_1=4$, $x_2=5$, $x_3=6$, calculate the standard deviation of this sample. Express your answer as a fraction with a square root, in the form $\boxed{\frac{\sqrt{a}}{b}}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Toner Drum and Celery Hilton are both running for president. A total of  $2015$  people cast their vote, giving  $60\%$  to Toner Drum. Let  $N$  be the number of "representative'' sets of the  $2015$  voters that could have been polled to correctly predict the winner of the election (i.e. more people in the set voted for Drum than Hilton). Compute the remainder when  $N$  is divided by  $2017$ .
Please reason step by step, and put your final answer within \boxed{}.


Context:
Tom weighs 150 kg.  He manages to hold 1.5 times his weight in each hand while wearing a weight vest weighing half his weight.  How much total weight was he moving with?
Please reason step by step, and put your final answer within \boxed{}.


Context:
At a chamber music festival, six musicians have gathered. In each concert, some of the musicians perform while the rest listen from the audience. What is the minimum number of concerts required so that each of the six musicians can listen to all the others from the audience?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the geometric sequence $\{a_{n}\}$, $a_{5}a_{8}=6$, $a_{3}+a_{10}=5$, then $\frac{a_{20}}{a_{13}}=$ \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.


Context:
$5.$ Let x be a real number with  $0<x<1$  and let  $0.c_1c_2c_3...$  be the decimal expansion of x.Denote by  $B(x)$  the set of all subsequences of  $c_1c_2c_3$  that consist of 6 consecutive digits.
For instance ,  $B(\frac{1}{22})={045454,454545,545454}$ Find the minimum number of elements of  $B(x)$  as  $x$  varies among all irrational numbers with  $0<x<1$
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the diagram, line segments meet at 90 degrees as shown. If the short line segments are each 3 cm long, what is the area of the shape? Express your answer in square centimeters (cm^2) as a whole number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the diagram, $COB$ is a sector of a circle with $\angle COB=45^\circ.$ $OZ$ is drawn perpendicular to $CB$ and intersects $CB$ at $W.$ What is the length of $WZ$? Assume the radius of the circle is 10 units.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Question 233: Let the difference between any two adjacent interior angles of a convex $\mathrm{n}$-sided polygon be $20^{\circ}$, find the maximum possible value of $\mathrm{n}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the general term of the sequence $\{a_n\}$ is $a_n=n^2\left(\cos^2 \frac{n\pi}{3}-\sin^2 \frac{n\pi}{3}\right)$, calculate the value of $S_{30}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given \( a \) is a given real number, the number of subsets for the set \( M = \{ x \in \mathbf{R} \mid |x|x^2 - 3x - a^2 + 2 = 0 \} \) is what?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Two externally tangent circles $\omega_1$ and $\omega_2$ have centers $O_1$ and $O_2$, respectively. A third circle $\Omega$ passing through $O_1$ and $O_2$ intersects $\omega_1$ at $B$ and $C$ and $\omega_2$ at $A$ and $D$, as shown. Suppose that $AB = 2$, $O_1O_2 = 15$, $CD = 16$, and $ABO_1CDO_2$ is a convex hexagon. Find the area of this hexagon.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Misha drew a triangle with a perimeter of 11 and cut it into parts with three straight cuts parallel to the sides, as shown in the figure. The perimeters of the three shaded figures (trapezoids) were found to be 5, 7, and 9. Find the perimeter of the small triangle that resulted from the cutting.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Suppose that the sum of the squares of two complex numbers $x$ and $y$ is $7$ and the sum of their cubes is $10.$ List all possible values for $x+y$, separated by commas.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a three-dimensional coordinate system with origin $O,$ points $A,$ $B,$ and $C$ are on the $x$-, $y$-, and $z$-axes, respectively.   If the areas of triangles $OAB,$ $OAC,$ and $OBC$ are 4, 6, and 12, respectively, compute the area of triangle $ABC.$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the function $y= \sqrt{3}\tan \frac{x}{2}$, with $x\neq (2k+1)\pi (k\in \mathbb{Z})$, find the expression for $x$ when $y=1$. Express your answer as an equation for $x$ in terms of $k$ and $\pi$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
For any positive integer $n, S_{n}$ be the set of all permutations of \{1,2,3, \ldots, n\}. For each permutation $\pi \in S_{n}$, let $f(\pi)$ be the number of ordered pairs $(j, k)$ for which $\pi(j)>\pi(k)$ and $1 \leq j<k \leq n$. Further define $g(\pi)$ to be the number of positive integers $k \leq n$ such that $\pi(k) \equiv k \pm 1(\bmod n)$. Compute $$\sum_{\pi \in S_{999}}(-1)^{f(\pi)+g(\pi)}$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Marisa has a collection of  $2^8-1=255$  distinct nonempty subsets of  $\{1, 2, 3, 4, 5, 6, 7, 8\}$ . For each step she takes two subsets chosen uniformly at random from the collection, and replaces them with either their union or their intersection, chosen randomly with equal probability. (The collection is allowed to contain repeated sets.) She repeats this process  $2^8-2=254$  times until there is only one set left in the collection. What is the expected size of this set?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the set \( S = \{1, 2, \cdots, 100\} \), determine the smallest possible value of \( m \) such that in any subset of \( S \) with \( m \) elements, there exists at least one number that is a divisor of the product of the remaining \( m-1 \) numbers.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A certain shopping mall sells toy $A$ with a purchase price of $60$ yuan per unit. According to the requirement, the profit margin per unit at the time of sale must not exceed $40\%$. Based on market research, when the selling price is set at $80$ yuan, 200 units can be sold per day, and for every $1$ yuan decrease in the selling price, an additional 20 units can be sold per day. Find the selling price at which the shopping mall can make a profit of $2500$ yuan per day selling toy $A$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $f$ be an entire function on $\mathbb{C}$ such that $|f(z)|\le 100 \ln |z|$ for $|z|\ge 2$ and $f(i)=2i$. What is $f(1)$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the sum of the infinite series $\frac{1}{2\cdot 4} + \frac{1\cdot 3}{2\cdot 4\cdot 6} + \frac{1\cdot 3\cdot 5}{2\cdot 4\cdot 6\cdot 8} + \dots$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Using each of the digits 1-9 exactly once, form a two-digit perfect square, a three-digit perfect square, and a four-digit perfect square. What is the smallest four-digit perfect square among them?
Please reason step by step, and put your final answer within \boxed{}.


Context:
For  $k\ge 1$ , define  $a_k=2^k$ . Let   $$ S=\sum_{k=1}^{\infty}\cos^{-1}\left(\frac{2a_k^2-6a_k+5}{\sqrt{(a_k^2-4a_k+5)(4a_k^2-8a_k+5)}}\right). $$   Compute  $\lfloor 100S\rfloor$ .
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a regular tetrahedron ABCD with an edge length of 1, points P and Q are located on segments AB and CD, respectively. Calculate the range of possible distances between points P and Q. Express your answer as a range of values, including units if necessary.
Please reason step by step, and put your final answer within \boxed{}.

\boxed{\left[ \frac{\sqrt{2}}{2}, 1 \right]}


Context:
The target below is made up of concentric circles with diameters $4$, $8$, $12$, $16$, and $20$. The area of the dark region is $n\pi$. Find $n$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a complex number $z$ satisfying the equation $|z-1|=|z+2i|$ (where $i$ is the imaginary unit), find the minimum value of $|z-1-i|$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Chim Tu has four different colored T-shirts and can wear an outfit consisting of three or four T-shirts worn in a specific order. Two outfits are distinct if the sets of T-shirts used are different or if the sets of T-shirts used are the same but the order in which they are worn is different. Given that Chim Tu changes his outfit every three days and never wears the same outfit twice, how many days of winter can Chim Tu survive without repeating an outfit?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Suppose Alan, Michael, Kevin, Igor, and Big Rahul are in a running race. It is given that exactly one pair of people tie (for example, two people both get second place), so that no other pair of people end in the same position. Each competitor has equal skill; this means that each outcome of the race, given that exactly two people tie, is equally likely. The probability that Big Rahul gets first place (either by himself or he ties for first) can be expressed in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute $m+n$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the diagram, triangles $ABC$ and $CDE$ are isosceles. The perimeter of $\triangle CDE$ is $22,$ the perimeter of $\triangle ABC$ is $24,$ and the length of $CE$ is $9.$ What is the length of $AB?$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the smallest natural number $x > 0$ such that all the following fractions are simplified:

$$\frac{3x+9}{8}, \frac{3x+10}{9}, \frac{3x+11}{10}, \ldots, \frac{3x+49}{48}$$

This means that the numerators and denominators are relatively prime.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a triangle $ABC$ with angles $\angle A = 60^{\circ}$, $\angle B = 75^{\circ}$, $\angle C = 45^{\circ}$, let $H$ be its orthocenter, and $O$ be its circumcenter. Let $F$ be the midpoint of side $AB$, and $Q$ be the foot of the perpendicular from $B$ onto $AC$. Denote by $X$ the intersection point of the lines $FH$ and $QO$. Suppose the ratio of the length of $FX$ and the circumradius of the triangle is given by $\frac{a + b \sqrt{c}}{d}$, then find the value of $1000a + 100b + 10c + d$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Joe B. first places the black king in one corner of the board. In how many of the 35 remaining squares can he place a white bishop so that it does not check the black king?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Hong and Song each have a shuffled deck of eight cards, consisting of four red cards and four black cards. On each turn, both players reveal the top two cards of their decks. The possible pairs that can be played are:

- Two black cards
- Two red cards
- One black and one red card

Determine the probability that Hong and Song play exactly the same pair of cards in each of the four turns. This probability is given as $\frac{m}{n}$, where $m$ and $n$ are positive integers such that $\gcd(m, n) = 1$. Calculate $100m + n$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let  $m$  and  $n$  be positive integers. Fuming Zeng gives James a rectangle, such that  $m-1$  lines are drawn parallel to one pair of sides and  $n-1$  lines are drawn parallel to the other pair of sides (with each line distinct and intersecting the interior of the rectangle), thus dividing the rectangle into an  $m\times n$  grid of smaller rectangles. Fuming Zeng chooses  $m+n-1$  of the  $mn$  smaller rectangles and then tells James the area of each of the smaller rectangles. Of the  $\dbinom{mn}{m+n-1}$  possible combinations of rectangles and their areas Fuming Zeng could have given, let  $C_{m,n}$  be the number of combinations which would allow James to determine the area of the whole rectangle. Given that \[A=\sum_{m=1}^\infty \sum_{n=1}^\infty \frac{C_{m,n}\binom{m+n}{m}}{(m+n)^{m+n}},\] then find the greatest integer less than  $1000A$ .

[i]Proposed by James Lin
Please reason step by step, and put your final answer within \boxed{}.


Context:
We want to create a square-based, open-top box from a circular piece of paper as shown in the figure. Determine the ratio of dimensions \( a \) and \( b \) seen in the figure such that the volume of the box is maximized.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $a,$ $b,$ $c$ be nonzero real numbers such that $a + b + c = 0,$ and $ab + ac + bc \neq 0.$ Find all possible values of
\[\frac{a^7 + b^7 + c^7}{abc (ab + ac + bc)}.\]
Please reason step by step, and put your final answer within \boxed{}.


Context:
A recipe that makes $5$ servings of hot chocolate requires $2$ squares of chocolate, $\frac{1}{4}$ cup sugar, $1$ cup water, and $4$ cups milk. Jordan has $5$ squares of chocolate, $2$ cups of sugar, lots of water, and $7$ cups of milk. If he maintains the same ratio of ingredients, what is the greatest number of servings of hot chocolate he can make? Express your answer in the form of an improper fraction \frac{k}{m}, and find the value of k + m.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given \( S = \frac{1}{9} + \frac{1}{99} + \frac{1}{999} + \cdots + \frac{1}{\text{1000 nines}} \), what is the 2016th digit after the decimal point in the value of \( S \)?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the number of ways in which the nine numbers \(1, 12, 123, 1234, \ldots, 123456789\) can be arranged in a row so that adjacent numbers are relatively prime.
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many ways are there to line up  $19$  girls (all of different heights) in a row so that no girl has a shorter girl both in front of and behind her?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the solutions to
\[\frac{x(x^2 - 56)}{4 - 7x} - \frac{21x + 22}{x^3 + 2} = 4.\]Enter all the solutions, separated by commas.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Paul laid six dominoes in a line as shown in the diagram. Dominoes are said to be arranged correctly if, for each pair of adjacent dominoes, the numbers of spots on the adjacent ends are equal. He can make a move either by swapping the position of any two dominoes (without rotating either domino) or by rotating one domino. What is the smallest number of moves he needs to make to arrange all the dominoes correctly? Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
例3 对于数集M，定义M的和为M中各数的和，记为S（M）．设M是若干个不大于15的正整数组成的集合，且M的任何两个不相交的子集有不同的和，求S（M）的最大值.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a triangle, two sides have lengths of $3$ and $8$ respectively. If the length of the third side is an even number, find the length of the third side that cannot be a valid option for the triangle. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A function $y(x)$ satisfies the differential equation $$y^{\prime}=4\sqrt{y-x^2}$$ It is known that $y(1)=2$. Find $y(3)$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $s_n$ be the number of solutions to the equation \(a_1 + a_2 + a_3 + a_4 + b_1 + b_2 = n\), where \(a_1, a_2, a_3,\) and \(a_4\) are elements of the set \(\{2, 3, 5, 7\}\) and \(b_1\) and \(b_2\) are elements of the set \(\{1, 2, 3, 4\}\). Find the number of \(n\) for which \(s_n\) is odd.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The total capacity of three buckets A, B, and C is 1440 liters. If buckets A and B are full of water and bucket C is empty, pouring all the water from bucket A and one-fifth of the water from bucket B into bucket C will completely fill bucket C. Alternatively, pouring all the water from bucket B and one-third of the water from bucket A into bucket C will also completely fill bucket C. Determine the capacities of buckets A, B, and C respectively.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Yesterday it took Robert some hours to drive from City A to City B. Today it took Robert 4.5 hours to drive back from City B to City A along the same route. If he had saved 30 minutes in both trips, the speed for the round trip would be 90 miles per hour. What is the distance between City A and City B if it took him 6 hours to drive one way?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given $a < b$, identify the incorrect step(s) in the following reasoning process. Fill in the sequence number(s) of all incorrect steps.

$\because a < b, \therefore a + a < b + a$, i.e., $2a < b + a$, ...①
$\therefore 2a - 2b < b + a - 2b$, i.e., $2(a - b) < a - b$, ...②
$\therefore 2(a - b) \cdot (a - b) < (a - b) \cdot (a - b)$, i.e., $2(a - b)^2 < (a - b)^2$, ...③
$\because (a - b)^2 > 0, \therefore$ it can be proven that $2 < 1$. ...④
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $p$ be an odd prime number, and let ${\Bbb {F}}_p$ denote the field of integers modulo $p$. Let ${\Bbb {F}}_p[x]$ be the ring of polynomials over ${\Bbb {F}}_p$, and let $q(x)\in {\Bbb {F}}_p[x]$ be given by
$$q(x)=\sum\limits_{k=1}^{p-1}a_kx^k,$$
where
$$a_k=k^{(p-1)/2}\mod p.$$
Find the greatest nonnegative integer $n$ such that $(x-1)^n$ divides $q(x)$ in ${\Bbb {F}}_p[x]$. What is the answer when p = 3?
Please reason step by step, and put your final answer within \boxed{}.


Context:
The distance on the map is 3.6 cm, and the actual distance is 1.2 mm. What is the scale of this map?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A train of a certain length running at the rate of 36 kmph takes 82.49340052795776 seconds to cross a bridge 660 meters in length. How long is the train?
Please reason step by step, and put your final answer within \boxed{}.


Context:
有长为 $2^{n}(n=0,1, \cdots, 1009)$ 的线段各三条, 则由这 $3030$ 条线段能构成不全等的三角形的个数为 $\qquad$ (用数字作答).
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many integers between 100 and 10,000 contain exactly 3 identical digits in their representation?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Ninety-eight apples who always lie and one banana who always tells the truth are randomly arranged along a line. The first fruit says "One of the first forty fruit is the banana!'' The last fruit responds "No, one of the  $\emph{last}$  forty fruit is the banana!'' The fruit in the middle yells "I'm the banana!'' In how many positions could the banana be?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that the terminal side of angle $\alpha$ lies on the ray $3x+4y=0$ ($x<0$), find the value of $$\frac {\sin(\pi-\alpha)\cos(3\pi+\alpha)\tan\alpha}{\cos(-\alpha)\sin(\pi+\alpha)}.$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given an acute triangle $ABC$, let $M$ be the orthocenter (the intersection point of its altitudes). Find angle $A$ if it is known that $AM = BC$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Abby and Ben have a little brother Carl who wants candy. Abby has $7$ different pieces of candy and Ben has $15$ different pieces of candy. Abby and Ben then decide to give Carl some candy. As Ben wants to be a better sibling than Abby, so he decides to give two more pieces of candy to Carl than Abby does. Let $N$ be the number of ways Abby and Ben can give Carl candy. Compute the number of positive divisors of $N$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral $$\displaystyle I=\int_0^{\infty} \frac{x^n-2 x+1}{\left(1+x^n\right)\left(1-x^n\right)} d x$$ where $n\geq 2$ is a real number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Three lathes \( A, B, C \) each process the same type of standard parts at a certain work efficiency. Lathe \( A \) starts 10 minutes earlier than lathe \( C \), and lathe \( C \) starts 5 minutes earlier than lathe \( B \). After lathe \( B \) has been working for 10 minutes, the number of standard parts processed by lathes \( B \) and \( C \) is the same. After lathe \( C \) has been working for 30 minutes, the number of standard parts processed by lathes \( A \) and \( C \) is the same. How many minutes after lathe \( B \) starts will it have processed the same number of standard parts as lathe \( A \)?
Please reason step by step, and put your final answer within \boxed{}.


Context:
The Elvish language consists of 4 words: "elara", "quen", "silva", and "nore". In a sentence, "elara" cannot come directly before "quen", and "silva" cannot come directly before "nore"; all other word combinations are grammatically correct (including sentences with repeated words). How many valid 3-word sentences are there in Elvish?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the real numbers $a$, $b$, and $c$ that satisfy the following equations: $$a=5^{- \frac {1}{2}}$$, $$\ln b = 1$$, and $$3c^3+c=1$$, determine the relationship among $a$, $b$, and $c$. Express your answer in the format 'X > Y > Z' or any other equivalent notation.
Please reason step by step, and put your final answer within \boxed{}.


Context:
将八个半径都为 1 的球分放两层放置在一个圆柱内，并使得每个球都和其相邻的四个球相切，且与圆柱的一个底面及侧面都相切。假设原始答案为$m+\sqrt[n]{j}$的形式，请给出m+n+j的和。
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider an arithmetic sequence $\{a\_n\}$ with a common ratio as a positive number. Let $S\_n$ denote the sum of the first $n$ terms. If $a\_1=1$ and $S\_4 - 5S\_2 = 0$, find the value of $S\_5$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \(A\) and \(B\) be two moving points on the ellipse \(x^2 + 3y^2 = 1\), and \(OA \perp OB\) (where \(O\) is the origin). Find the product of the maximum and minimum values of \( |AB| \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
A frog is at the point $(0,0)$. Every second, he can jump one unit either up or right. He can only move to points $(x, y)$ where $x$ and $y$ are not both odd. How many ways can he get to the point $(8,14)$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $u, v$ be two random vectors with $i$-th element being 1 with probability $p$. What is the probability that $u \cdot v \equiv 1 \pmod 2$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $m>1$ and $n>1$ be integers. Suppose that the product of the solutions for $x$ of the equation $$ 8(\log_n x)(\log_m x)-7\log_n x-6 \log_m x-2013 = 0 $$is the smallest possible integer. What is $m+n$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A courtyard has the shape of a parallelogram ABCD. At the corners of the courtyard there stand poles AA', BB', CC', and DD', each of which is perpendicular to the ground. The heights of these poles are AA' = 68 centimeters, BB' = 75 centimeters, CC' = 112 centimeters, and DD' = 133 centimeters. Find the distance in centimeters between the midpoints of A'C' and B'D'.
Please reason step by step, and put your final answer within \boxed{}.


Context:
If $m^2 + m - 1 = 0$, then $m^3 + 2m^2 + 2004 =$ ?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that the perimeter of a rectangle is 36, and the rectangle rotates around one of its edges to form a cylinder, the maximum lateral surface area of the cylinder formed is ______.
Please reason step by step, and put your final answer within \boxed{}.


Context:
An ant leaves the anthill for its morning exercise. It walks $4$ feet east and then makes a $160^\circ$ turn to the right and walks $4$ more feet. If the ant continues this pattern until it reaches the anthill again, what is the total distance in feet that it would have walked?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Problem 8.5.1. In the cells of a $12 \times 12$ table, natural numbers are arranged such that the following condition is met: for any number in a non-corner cell, there is an adjacent cell (by side) that contains a smaller number. What is the smallest number of different numbers that can be in the table?

(Non-corner cells are those that are not in the corner of the table. There are exactly 140 of them.)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $a$, $b$, $c$ be the three sides of a triangle, and let $\alpha$, $\beta$, $\gamma$ be the angles opposite them. If $a^2 + b^2 = 2020c^2$, determine the value of
\[\frac{\cot \gamma}{\cot \alpha + \cot \beta}.\]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $Z$ be as in problem 15. Let $X$ be the greatest integer such that $|X Z| \leq 5$. Find $X$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Petya chooses non-negative numbers \(x_{1}, x_{2}, \ldots, x_{11}\) such that their sum is 1. Vasya arranges them in a row at his discretion, calculates the products of adjacent numbers, and writes on the board the largest of the resulting ten products. Petya wants the number on the board to be as large as possible, while Vasya wants it to be as small as possible. What number will be on the board with the best play from both Petya and Vasya?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Six different digits from the set \(\{ 1,2,3,4,5,6,7,8,9\}\) are placed in the squares in the figure shown so that the sum of the entries in the vertical column is 23 and the sum of the entries in the horizontal row is 12. The sum of the six digits used is
Please reason step by step, and put your final answer within \boxed{}.


Context:
In triangle $PQR$, point $T$ is on $PR$ and point $S$ is on $PQ$ such that $TS \parallel RQ$. The measure of $\angle RPQ$ is $65^{\circ}$, and the measure of $\angle TSQ$ is $145^{\circ}$. What is the measure of $\angle PRQ$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that point \(Z\) moves on \(|z| = 3\) in the complex plane, and \(w = \frac{1}{2}\left(z + \frac{1}{z}\right)\), where the trajectory of \(w\) is the curve \(\Gamma\). A line \(l\) passes through point \(P(1,0)\) and intersects the curve \(\Gamma\) at points \(A\) and \(B\), and intersects the imaginary axis at point \(M\). If \(\overrightarrow{M A} = t \overrightarrow{A P}\) and \(\overrightarrow{M B} = s \overrightarrow{B P}\), find the value of \(t + s\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
A man saves a certain percentage of his monthly salary. If, on account of dearness of things, he is to increase his monthly expenses by 10%, he is only able to save Rs. 200 per month. His monthly salary is Rs. 20,000. What percentage of his salary does he save?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the value of $$\sum_{1 \leq a<b<c} \frac{1}{2^{a} 3^{b} 5^{c}}$$ (i.e. the sum of $\frac{1}{2^{a} 3^{b} 5^{c}}$ over all triples of positive integers $(a, b, c)$ satisfying $a<b<c$).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( m \) be the product of all positive integer divisors of 360,000. Suppose the prime factors of \( m \) are \( p_{1}, p_{2}, \ldots, p_{k} \), for some positive integer \( k \), and \( m = p_{1}^{e_{1}} p_{2}^{e_{2}} \cdot \ldots \cdot p_{k}^{e_{k}} \), for some positive integers \( e_{1}, e_{2}, \ldots, e_{k} \). Find \( e_{1} + e_{2} + \ldots + e_{k} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that $|\overrightarrow{a}|=5$, $|\overrightarrow{b}|=3$, and $\overrightarrow{a}\cdot\overrightarrow{b}=-12$, find the projection of vector $\overrightarrow{a}$ onto vector $\overrightarrow{b}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
As shown in the figure, in a rectangle with an area of 10,000 square centimeters, a large semicircle and two equal small semicircles are cut out. The remaining shaded area is $\qquad$ square centimeters. (Take $\pi$ as 3.14)
Please reason step by step, and put your final answer within \boxed{}.


Context:
In trapezoid \(ABCD\), diagonal \(AC\) is equal to 1 and is also its height. Perpendiculars \(AE\) and \(CF\) are drawn from points \(A\) and \(C\) to sides \(CD\) and \(AB\) respectively. Find \(AD\) if \(AD=CF\) and \(BC=CE\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
A circle intersects all three sides of a triangle, with the segments of the sides lying inside the circle being equal. One of the angles of the triangle is $60^{\circ}$. Find the remaining angles, given that the sum of the arcs of the circle lying inside the triangle is $90^{\circ}$ and one of the arcs is twice the size of another.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $G, A_{1}, A_{2}, A_{3}, A_{4}, B_{1}, B_{2}, B_{3}, B_{4}, B_{5}$ be ten points on a circle such that $G A_{1} A_{2} A_{3} A_{4}$ is a regular pentagon and $G B_{1} B_{2} B_{3} B_{4} B_{5}$ is a regular hexagon, and $B_{1}$ lies on minor arc $G A_{1}$. Let $B_{5} B_{3}$ intersect $B_{1} A_{2}$ at $G_{1}$, and let $B_{5} A_{3}$ intersect $G B_{3}$ at $G_{2}$. Determine the degree measure of $\angle G G_{2} G_{1}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the following figure, $\angle CAB = 2 \cdot \angle CBA$, $AD$ is an altitude, and $M$ is the midpoint of $AB$. If $AC = 2 \text{ cm}$, find the length of segment $DM$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Place as many points as possible on the plane such that any three points are not collinear and form an isosceles triangle.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a point $P$ in the plane, where $P \in {(x,y) | (x - 2\cos{\alpha})^2 + (y - 2\sin{\alpha})^2 = 16, \alpha \in \mathbb{R}}$, the area of the shape formed by all such points $P$ in the plane is $\_\_\_\_\_\_$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( F_{1} \) and \( F_{2} \) be the left and right foci of the hyperbola \( C: \frac{x^{2}}{4} - \frac{y^{2}}{12} = 1 \), respectively. Let point \( P \) lie on the hyperbola \( C \), and let \( G \) and \( I \) be the centroid and incenter of the triangle \( \triangle F_{1} P F_{2} \), respectively. If \( G \) and \( I \) are parallel to the x-axis, determine the circumradius \( R \) of the triangle \( \triangle F_{1} P F_{2} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral $$\int_0^{\infty}\frac{x}{x^4+1}dx$$ using complex analysis techniques. This integral is relevant in the context of the residue theorem and its applications to real integrals. Note that the given integral is not axis-symmetric, and hence standard residue theorem techniques may not directly apply.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Divide 10 mountaineers into two groups to climb the mountain in succession. Among them, 4 are familiar with the route, and each group needs 2 people who are familiar with the route. Calculate the total number of different arrangements.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In four baseball games this week, eighty people watched the second game, while 20 fewer people watched the first game than the second game. Then 15 more people watched the third game than the second game. The fourth game had 10% more viewers than the third game. If there were a total of 350 people who watched the games last week with a different pattern of viewership across the games, how many more people watched the games this week than last week with the new set of conditions?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $\{a_n\}_{n\geq 1}$ be a sequence defined by $a_n=\int_0^1 x^2(1-x)^ndx$.
Find the real value of $c$ such that $\sum_{n=1}^{\infty} (n+c)(a_n-a_{n+1})=2.$
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the maximum number of non-empty subsets that can be chosen from a set of 100 elements such that any two chosen subsets are either disjoint or one contains the other?
Please reason step by step, and put your final answer within \boxed{}.


Context:
12.66 In the interval $1 \leqslant n \leqslant 10^{6}$, how many integers $n$ are there such that the equation $n=x^{v}$ has non-negative integer solutions $x, y$, and $x \neq n$.
(Shanghai, China High School Mathematics Competition, 1990)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Use the divergence theorem to evaluate the surface integral \( \iint_S \mathbf{F} \cdot d\mathbf{S} \) for the vector field \( \mathbf{F} = (xy^2, yz^2, x^2z) \), where \( S \) is the sphere of radius 1 centered at the origin.
Please reason step by step, and put your final answer within \boxed{}.


Context:
设等比数列 $\left\{a_{n}\right\}$ 的前 $n$ 项和为 $S_{n}$, 且 $S_{n}=2^{n}+r$ ( $r$ 为常数),

记 $b_{n}=2\left(1+\log _{2} a_{n}\right) \quad\left(n \in \mathbf{N}^{*}\right)$.
若对于任意的正整数 $n$, 都有 $\frac{1+b_{1}}{b_{1}} \cdot \frac{1+b_{2}}{b_{2}} \cdot \cdots \cdot \frac{1+b_{n}}{b_{n}} \geq k \sqrt{n+1}$ 成立,求实数 $k$ 的最大值.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Replace the asterisk $(*)$ in the expression $\left(x^{3}-2\right)^{2}+\left(x^{2}+*\right)^{2}$ with a monomial such that, after squaring and combining like terms, the resulting expression has four terms.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A difficult question from the field of poultry farming. Three chicks and one duck are sold for the same amount as two geese, and one more chick, two ducks, and three geese are sold together for 25 dollars. How much does each bird cost if the prices are expressed in whole number dollars?
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the fluid that occupies the largest proportion of the total body fluid in a tiger? Express your answer as the name of the fluid.
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the smallest number of weights and their respective values needed to measure any integer weight from 1 to 40 pounds on a balance scale, given that the weights can be placed on both sides of the scale?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( A = (2, 0) \) and \( B = (8, 6) \). Let \( P \) be a point on the circle \( x^2 + y^2 = 8x \). Find the smallest possible value of \( AP + BP \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the largest positive integer $n$ less than $10,000$ such that in base $4$, $n$ and $3n$ have the same number of digits; in base $8$, $n$ and $7n$ have the same number of digits; and in base $16$, $n$ and $15n$ have the same number of digits? Express your answer in base $10$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
记集合 A=\{1,2,\cdots,2023\}，存在集合 B\subseteq A，且 B 中任意两个元素之和不是 5 的倍数；存在集合 C\subseteq B，且 C 中元素均为完全平方数. 则 |C| 的最大值为__________.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the total number of solutions to the following system of equations:

 $ \{\begin{array}{l} a^2 + bc\equiv a \pmod{37}
b(a + d)\equiv b \pmod{37}
c(a + d)\equiv c \pmod{37}
bc + d^2\equiv d \pmod{37}
ad - bc\equiv 1 \pmod{37} \end{array}$
Please reason step by step, and put your final answer within \boxed{}.


Context:
As shown in the diagram, $E$ is the midpoint of the leg $AB$ of trapezoid $ABCD$. $DF \perp EC$, $DF=10$, and $EC=24$. Find the area of trapezoid $ABCD$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The keystone arch is an ancient architectural feature. It is composed of congruent isosceles trapezoids fitted together along the non-parallel sides, as shown. The bottom sides of the two end trapezoids are horizontal. In an arch made with 9 trapezoids, let x be the angle measure in degrees of the larger interior angle of the trapezoid. Find the value of x. Express your answer in degrees as a whole number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Sort the following fractions in ascending order:

$$
t_{1}=\frac{100^{100}+1}{100^{90}+1}, \quad t_{2}=\frac{100^{99}+1}{100^{89}+1}, \quad t_{3}=\frac{100^{101}+1}{100^{91}+1}, \quad t_{4}=\frac{101^{101}+1}{101^{91}+1}, \quad t_{5}=\frac{101^{100}+1}{101^{90}+1}, \quad t_{6}=\frac{99^{99}+1}{99^{89}+1}, \quad t_{7}=\frac{99^{100}+1}{99^{90}+1}
$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
A regular dodecahedron is projected orthogonally onto a plane, and its image is an $n$-sided polygon. What is the smallest possible value of $n$ ?
Please reason step by step, and put your final answer within \boxed{}.


Context:
若正整数 $m 、 n$ 满足 $\frac{(m+n)!}{n!}=5040$,则 $m!n$ 的值为 $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Start by writing the integers $1,2,4,6$ on the blackboard. At each step, write the smallest positive integer $n$ that satisfies both of the following properties on the board. - $n$ is larger than any integer on the board currently. - $n$ cannot be written as the sum of 2 distinct integers on the board. Find the 100-th integer that you write on the board. Recall that at the beginning, there are already 4 integers on the board.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A high school has 560 freshmen, 540 sophomores, and 520 juniors. If a stratified sampling method is used to draw a sample of 81 students, how many students should be drawn from each grade? Express your answer as three numbers representing the number of students to be sampled from each grade, in the order of freshmen, sophomores, and juniors.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A box contains exactly five chips, three red and two white. Chips are randomly removed one at a time without replacement until all the red chips are drawn or all the white chips are drawn. What is the probability that the last chip drawn is white? Express your answer in the form \(\frac{k}{m}\) and find the value of \(k + m\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In $\triangle ABC$, $\angle C = 90^\circ$ and $AB = 12$. Squares $ABXY$ and $CBWZ$ are constructed outside of the triangle. The points $X$, $Y$, $Z$, and $W$ lie on a circle. What is the perimeter of the triangle?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A game works as follows: the player pays  $2$  tokens to enter the game. Then, a fair coin is flipped. If the coin lands on heads, they receive  $3$  tokens; if the coin lands on tails, they receive nothing. A player starts with  $2$  tokens and keeps playing this game until they do not have enough tokens to play again. What is the expected value of the number of tokens they have left at the end?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the given figure, a triangular frustum \( A B C-D E F \) has the ratio of the side lengths of the top base to the bottom base as \( 1:2 \). Point \( G \) is the midpoint of \( C F \). Determine the ratio of the volumes of the upper and lower parts into which the frustum is divided by the section \( A G E \). (The volume of the frustum \( V = \frac{h}{3}(t + \sqrt{fF} + F) \), where \( h \) is the height of the frustum, and \( f \) and \( F \) are the areas of the top and bottom bases, respectively.)
Please reason step by step, and put your final answer within \boxed{}.


Context:
The graph shows the forecast wind speed (in $\mathrm{km} / \mathrm{h}$) during a 7-day period. Jack can sail alone only when the forecast wind speed is less than $20 \mathrm{~km} / \mathrm{h}$. During this 7-day period, on how many days will Jack be able to sail alone?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A permutation $(a_1, a_2, a_3, \dots, a_{2012})$ of $(1, 2, 3, \dots, 2012)$ is selected at random. If $S$ is the expected value of \[ \sum_{i = 1}^{2012} | a_i - i |, \] then compute the sum of the prime factors of $S$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The following construction is used for training astronauts:
A circle $C_2$ of radius $2R$ rolls along the inside of another, fixed circle $C_1$ of radius $nR$, where $n$ is an integer greater than $2$. The astronaut is fastened to a third circle $C_3$ of radius $R$ which rolls along the inside of circle $C_2$ in such a way that the touching point of the circles $C_2$ and $C_3$ remains at maximum distance from the touching point of the circles $C_1$ and $C_2$ at all times. How many revolutions (relative to the ground) does the astronaut perform together with the circle $C_3$ while the circle $C_2$ completes one full lap around the inside of circle $C_1$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Two boards, one four inches wide and the other six inches wide, are nailed together to form an X. The angle at which they cross is 60 degrees. If this structure is painted and the boards are separated what is the area of the unpainted region on the four-inch board? (The holes caused by the nails are negligible.) Express your answer in simplest radical form.

[asy]
draw(6dir(150)--15dir(-30),linewidth(1));
draw((6dir(150)+12/sqrt(3)*dir(30))--(15dir(-30)+12/sqrt(3)*dir(30)),linewidth(1));

draw(6dir(210)--(0,0),linewidth(1));
draw((9dir(210)+8/sqrt(3)*dir(-30))--8/sqrt(3)*dir(-30),linewidth(1));

draw(12/sqrt(3)*dir(30)--(12/sqrt(3)+6)*dir(30),linewidth(1));
draw(12/sqrt(3)*dir(30)+8/sqrt(3)*dir(-30)--(12/sqrt(3)+9)*dir(30)+8/sqrt(3)*dir(-30),linewidth(1));

draw(2dir(150)--2dir(150)+6dir(60),dashed);
draw(2dir(210)--2dir(210)+4dir(-60),dashed);

dot((2,0));
dot((4,-1));
dot((8,1));
dot((6,2));

label("$60^{\circ}$", (11,1), E);
label(rotate(30)*"$4^{\prime\prime}$", .5*(2dir(210)+2dir(210)+4dir(-60))+(0,-.5),W);
label(rotate(-30)*"$6^{\prime\prime}$", .5*(2dir(150)+2dir(150)+6dir(60))+(1,1),W);
[/asy]The answer is in the form k\sqrt{m}+n,. Please provide the value of k + m + n.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Right triangle $ACD$ with right angle at $C$ is constructed outwards on the hypotenuse $\overline{AC}$ of isosceles right triangle $ABC$ with leg length $1$, as shown, so that the two triangles have equal perimeters. What is $\sin(2\angle BAD)$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the derivative.

$$
y=\sin \sqrt[3]{\operatorname{tg} 2}-\frac{\cos ^{2} 28 x}{56 \sin 56 x}
$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that $X_1$, $X_2$, $X_3$, and $X_4$ are independent standard normally distributed random variables, find the constant $c$ such that the random variable $Y = \frac{c(X_1 + X_2)}{\sqrt{(X_3)^2 + (X_4)^2}}$ follows a t-distribution.
Please reason step by step, and put your final answer within \boxed{}.


Context:
João calculated the product of the non zero digits of each integer from  $1$  to  $10^{2009}$  and then he summed these  $10^{2009}$  products. Which number did he obtain?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the least number which when divided by 5, 67, and 8 leaves a remainder 3, but when divided by another number leaves no remainder. The least number is 1683. What is the number that leaves no remainder when the least number is divided by it?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Ggrmanova T.A.

Tanya sequentially wrote down numbers of the form $\{n \wedge 7-1\}$ for natural numbers $n=2,3, \dots$ and noticed that for $n=8$ the resulting number is divisible by 337. For what smallest $n>1$ will she get a number divisible by $2022?$
Please reason step by step, and put your final answer within \boxed{}.


Context:
In triangle \(ABC\), the median \(BK\), the angle bisector \(BE\), and the altitude \(AD\) are given. If the lines \(BK\) and \(BE\) divide the segment \(AD\) into three equal parts, and \(AB=4\), find the length of side \(AC\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
The coordinates of the foci of the hyperbola $5x^{2}-4y^{2}+60=0$ are ______.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider the sequence of numbers defined recursively by $t_1=1$ and for $n>1$ by $t_n=1+t_{n/2}$ when $n$ is even and by $t_n=\frac{1}{t_{n-1}}$ when $n$ is odd. Given that $t_n=\frac{19}{87}$, find $n.$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that $a = \log_2{6}$, $b = \log_5{15}$, and $a > \log_3{m} > b$, where $m \in \mathbb{N}^*$, find the possible value of $m$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Tarek wants to shade two further squares on the diagram shown so that the resulting pattern has a single axis of symmetry. How many different ways can he complete his pattern symmetrically? Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
How can we find the limit of the following expression?

$$\lim_{n \to \infty} \sqrt[n^2]{\frac{(n+1)^{n+1}(n+2)^{n+2} \cdot (...) \cdot (2n)^{2n}}{n^{n+1}n^{n+2} \cdot (...) \cdot n^{2n}}}$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $A, B, C, \ldots, Z$ be 26 nonzero real numbers. Suppose that $T = TNYWR$. Compute the smallest possible value of \[ \left\lceil A^2 + B^2 + \cdots + Z^2 \right\rceil. \] (The notation $\left\lceil x \right\rceil$ denotes the least integer $n$ such that $n \geq x$.)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Real numbers \(a, b, c\) and a positive number \(\lambda\) such that \(f(x)=x^3 + ax^2 + bx + c\) has three real roots \(x_1, x_2, x_3\), satisfying
1. \(x_2 - x_1 = \lambda\);
2. \(x_3 > \frac{1}{2}(x_1 + x_2)\).

Find the maximum value of \(\frac{2a^3 + 27c - 9ab}{\lambda^3}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many solutions does the equation $\tan(2x)=\cos(\frac{x}{2})$ have on the interval $[0,2\pi]?$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the coordinates of the four points on the curve $C$ defined by the equation $2x^2 + y^2 = 18$ where the normal to the curve passes through the point $(1, 0)$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many ways are there to permute the first  $n$  positive integers such that in the permutation, for each value of  $k \le n$ , the first  $k$  elements of the permutation have distinct remainder mod  $k$ ?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the area of a right-angled triangle if the altitude drawn from the right angle divides it into two triangles with the radii of the inscribed circles equal to 3 and 4.
Please reason step by step, and put your final answer within \boxed{}.


Context:
There are seven cards in a hat, and on the card  $k$  there is a number  $2^{k-1}$ ,  $k=1,2,...,7$ . Solarin picks the cards up at random from the hat, one card at a time, until the sum of the numbers on cards in his hand exceeds  $124$ . What is the most probable sum he can get?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the dimension of the complex vector space of meromorphic 1-forms on a compact Riemann surface $X$ with a pole of order at most 3 at a point $p$, assuming the genus of $X$ is $g$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Rectangles $R_1$ and $R_2,$ and squares $S_1, S_2,$ and $S_3,$ shown below, combine to form a rectangle that is 3322 units wide and 2020 units high. What is the side length of $S_2$ in units?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the value of x by calculating (47% of some number - 36% of 1412) + 65, given that the value of x is 5. What is the number?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the least positive integer   $n$  satisfying the following statement: for eash pair of positive integers  $a$  and  $b$  such that  $36$  divides   $a+b$  and  $n$  divides  $ab$  it follows that  $36$  divides both   $a$  and  $b$ .
Please reason step by step, and put your final answer within \boxed{}.


Context:
$$
\frac{\cos \left(\frac{5}{2} \pi-6 \alpha\right)+\sin (\pi+4 \alpha)+\sin (3 \pi-\alpha)}{\sin \left(\frac{5}{2} \pi+6 \alpha\right)+\cos (4 \alpha-2 \pi)+\cos (\alpha+2 \pi)}=\operatorname{tg} \alpha
$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
"The four interior angles of a quadrilateral are equal" is a \_\_\_\_\_\_\_\_\_ condition for "the quadrilateral is a square".
Please reason step by step, and put your final answer within \boxed{}.

\boxed{\text{necessary but not sufficient}}


Context:
Find the number of pairs of integers  $x, y$  with different parities such that  $\frac{1}{x}+\frac{1}{y} = \frac{1}{2520}$ .
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $O$ be the center and let $F$ be one of the foci of the ellipse $25x^2 +16 y^2 = 400$. A second ellipse, lying inside and tangent to the first ellipse, has its foci at $O$ and $F$. What is the length of the minor axis of this second ellipse?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given an arithmetic sequence $\{a_n\}$ where $a_1=1$ and $a_n=70$ (for $n\geq3$), find all possible values of $n$ if the common difference is a natural number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $P$ be a regular $2006$-gon. A diagonal is called *good* if its endpoints divide the boundary of $P$ into two parts, each composed of an odd number of sides of $P$. The sides of $P$ are also called *good*.

Suppose $P$ has been dissected into triangles by $2003$ diagonals, no two of which have a common point in the interior of $P$. Find the maximum number of isosceles triangles having two good sides that could appear in such a configuration.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the number \(2016*****02*\), each of the 5 stars must be replaced by any of the digits \(0, 2, 4, 5, 7, 9\) (digits can repeat) so that the resulting 11-digit number is divisible by 15. How many ways can this be done?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Eliška placed cakes into boxes and then built a pyramid from these boxes, as shown in the picture. Each box in a higher row contained as many cakes as the sum of cakes in the two adjacent boxes in the row below. In the three boxes marked with stars, there were three, five, and six cakes respectively.

Eliška noticed that if the marked boxes were interchanged in any way (and the cake counts in other boxes adjusted according to the previous rule), the total number of cakes would not be smaller.

How many cakes were in the marked box in the second row from the bottom?

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


Context:
A circle \( k_{1} \) with a radius of 8 cm lies inside a circle \( k \). Both circles intersect with a circle \( k_{2} \) with a radius of 15 cm, as shown in the figure. What is the radius of \( k \) if the shaded area inside \( k \) but outside \( k_{1} \) is equal to the total shaded areas inside \( k_{2} \)?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Two identical cylindrical sheets are cut open along the dotted lines and glued together to form one bigger cylindrical sheet. The smaller sheets each enclose a volume of 100. What volume is enclosed by the larger sheet?
Please reason step by step, and put your final answer within \boxed{}.


Context:
8. A thin beam of light falls normally on a plane-parallel glass plate. Behind the plate, at some distance from it, stands an ideal mirror (its reflection coefficient is equal to one). The plane of the mirror is parallel to the plate. It is known that the intensity of the beam that has passed through this system is 256 times less than the intensity of the incident beam. The reflection coefficient at the glass-air boundary is assumed to be constant regardless of the direction of the beam. Neglect absorption and scattering of light in air and glass. Find the reflection coefficient at the glass-air boundary under these conditions. (10 points)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that point $P(\frac{1}{2}, 8)$ lies on the graph of the power function $f(x)$, determine the value of $f(2)$.
Please reason step by step, and put your final answer within \boxed{}.


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. If Nathaniel goes first, determine the probability that he ends up winning.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A certain art school randomly arranges one Chinese, one Mathematics, one Foreign Language, and three other art classes among its six classes in a day. What is the probability that there are at most one art class between any two adjacent cultural classes? (Answer with a number).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Fonzie, Aunt Bee, Lapis, Skylar, and Orion paid $7,000, $8,500, $9,000, $7,500, and $8,000 respectively to buy a map to Capt. Plaidstache's treasure. When they found it, the treasure turned out to be worth a staggering $1,200,000. How much should each person receive, given their individual investments in acquiring the map?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a regular polygon with $2n+1$ sides. How many triangles can be formed whose vertices are the vertices of the polygon and contain the center of the polygon?
Please reason step by step, and put your final answer within \boxed{}.


Context:
As shown in the diagram, in parallelogram \(ABCD\), \(EF \parallel AD\), \(GH \parallel AB\), the area of \(\triangle EBH\) is 6, the area of \(\triangle DGF\) is 8, and the area of \(\triangle ECG\) is 18. Find the area of parallelogram \(ABCD\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In an abandoned chemistry lab, Gerome found a two-pan balance scale and three 1-gram weights, three 5-gram weights, and three 50-gram weights. By placing one pile of chemicals and as many weights as necessary on the pans of the scale, Gerome can measure out various amounts of the chemicals in the pile. Find the number of different positive weights of chemicals that Gerome could measure.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Two men at points $R$ and $S$, $76$ miles apart, set out at the same time to walk towards each other. The man at $R$ walks uniformly at the rate of $4\tfrac{1}{2}$ miles per hour; the man at $S$ walks at the constant rate of $3\tfrac{1}{4}$ miles per hour for the first hour, at $3\tfrac{3}{4}$ miles per hour for the second hour, and so on, in arithmetic progression. If the men meet $x$ miles nearer $R$ than $S$ in an integral number of hours, then $x$ is:
Please reason step by step, and put your final answer within \boxed{}.


Context:
On the edges \(AB\), \(BC\), and \(AD\) of the tetrahedron \(ABCD\), points \(K\), \(N\), and \(M\) are chosen, respectively, such that \(AK:KB = BN:NC = 2:1\) and \(AM:MD = 3:1\). Construct the section of the tetrahedron by the plane passing through points \(K\), \(M\), and \(N\). In what ratio does this plane divide edge \(CD\)?
Please reason step by step, and put your final answer within \boxed{}.


Context:
双曲线$\frac{{{x}^{2}}}{{{a2}}}-\frac{{{y}^{2}}}{{{b2}}}=1(a>0,b>0)$的左焦点为$F(-2,0)$，点$A(0,\sqrt{5})$，点$P$为双曲线右支上的动点，且$\vartriangle APF$周长的最小值为8，则双曲线的离心率为______.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The base of a rectangular prism is an equilateral triangle with sides of unit length. A plane intersects the lateral surface of the prism in an isosceles right triangle. What are the side lengths of this intersecting triangle?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $N=\overline{5 A B 37 C 2}$, where $A, B, C$ are digits between 0 and 9, inclusive, and $N$ is a 7-digit positive integer. If $N$ is divisible by 792, determine all possible ordered triples $(A, B, C)$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In an institute, there are truth-tellers, who always tell the truth, and liars, who always lie. One day, each of the employees made two statements:

1) There are fewer than ten people in the institute who work more than I do.
2) In the institute, at least one hundred people have a salary greater than mine.

It is known that the workload of all employees is different, and their salaries are also different. How many people work in the institute?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that $x > 2$, $y > 0$ and they satisfy $2^{x \cdot 2^{y}} = 16$, calculate the minimum value of $\frac {2}{x-2}$ + $\frac {2}{y}$. Express your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the areas of the regions bounded by the curves given in polar coordinates.

$$
r=\cos 2 \phi
$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
A box contains $2$ pennies, $4$ nickels, and $6$ dimes. Six coins are drawn without replacement, with each coin having an equal probability of being chosen. What is the probability that the value of coins drawn is at least $50$ cents?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a chess tournament, 12 participants played. After the tournament, each participant compiled 12 lists. The first list includes only the participant himself, the second list includes himself and those he won against, the third list includes everyone from the second list and those they won against, and so on. The twelfth list includes everyone from the eleventh list and those they won against. It is known that for any participant, there is a person in their twelfth list who was not in their eleventh list. How many drawn games were played in the tournament?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the coefficient of the $x^{3}$ term in the expansion of $(x^{2}+1)^{2}(x-1)^{6}$. Express your answer as an integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The graph of the linear function $y=kx+2$ intersects the $x$-axis at a point that is $2$ units away from the origin. The value of $k$ is \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In triangle $ABC$, $AB=AC$ and $\measuredangle A=80^\circ$. If points $D, E$, and $F$ lie on sides $BC, AC$ and $AB$, respectively, and $CE=CD$ and $BF=BD$, then $\measuredangle EDF$ equals
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given two lines $l_{1}$: $mx+2y-2=0$ and $l_{2}$: $5x+(m+3)y-5=0$, if $l_{1}$ is parallel to $l_{2}$, determine the value of $m$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the equation of the ellipse $x^{2}+2y^{2}=4$, calculate the coordinates of the foci. Express your answer as a pair of coordinates $(x, y)$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the figure, there are graphs of two linear functions \( y = f(x) \) and \( y = g(x) \), with the coordinates of their intersections with the axes indicated. Find the maximum point of the function \( y = f(x) \cdot g(x) \) (i.e., the value of \( x \) at which the function \( f(x) \cdot g(x) \) is maximized). If the answer is a fractional number, it must be recorded using a decimal point, e.g., \( 0.15 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given 2023 identical-looking coins, where two of them are counterfeit coins with the same weight and the remaining 2021 coins are genuine coins with the same weight, the counterfeit coins have a different weight than the genuine coins. Using a balance scale, what is the minimum number of weighings needed to determine whether a counterfeit coin or a genuine coin is heavier?
Please reason step by step, and put your final answer within \boxed{}.


Context:
M4. The squares on each side of a right-angled scalene triangle are constructed and three further line segments drawn from the corners of the squares to create a hexagon, as shown. The squares on these three further line segments are then constructed (outside the hexagon). The combined area of the two equal-sized squares is \(2018 \mathrm{~cm}^{2}\). What is the total area of the six squares?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the area of the region bounded by a function  $y=-x^4+16x^3-78x^2+50x-2$  and the tangent line which is tangent to the curve at exactly two distinct points.

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


Context:
It is known that point $M$ is a point on the parabola $y^2 = 4x$, and $F$ is the focus of the parabola. Point $A$ is on the circle $C$: $(x-4)^2 + (y-1)^2 = 1$. Find the minimum value of $|MA| + |MF|$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $AD$ be a median of $\triangle ABC$ such that $m(\widehat{ADB})=45^{\circ}$ and $m(\widehat{ACB})=30^{\circ}$. If the measure of $\widehat{ABC}$ is $x^\circ$, what is the value of $x$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
The equation of a circle is given as $x^2+y^2-2x=0$. Find the coordinates of the center and the radius of the circle. Express your answer in the format: Center: (x, y), Radius: r.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Using the digits 2, 3, 4, 5, and 6, form two three-digit numbers that are squares. What is the digit that is placed in the center of the grid? Express your answer as a single digit.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In triangle \(ABC\), the height \(AM\) is not less than \(BC\), and the height \(BH\) is not less than \(AC\). Find the angles of triangle \(ABC\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
4 设X＝｛00,01，··，98,99｝是100个二位数码的集合，A是X的子集，满足：对任何一个由0到9中的数字构成的无穷序列中，都有两个相邻的数字组成的二位数码属于A，求｜A｜的最小值．（第52届莫斯科数学奥林匹克试题）
Please reason step by step, and put your final answer within \boxed{}.


Context:
Carl and Bob can demolish a building in 6 days, Anne and Bob can do it in  $3$ , Anne and Carl in  $5$ .  How many days does it take all of them working together if Carl gets injured at the end of the first day and can't come back?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Five boys and six girls are to be seated in a row of eleven chairs so that they sit one at a time from one end to the other. The probability that there are no more boys than girls seated at any point during the process is  $\frac{m}{n}$  , where  $m$  and  $n$  are relatively prime positive integers. Evaluate  $m + n$ .
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let  $p$  be a prime number. A rational number  $x$ , with  $0 < x < 1$ , is written in lowest terms. The rational number obtained from  $x$  by adding  $p$  to both the numerator and the denominator differs from  $x$  by  $1/p^2$ . Determine all rational numbers  $x$  with this property.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Two cars of length 120 m and some length are running towards each other on parallel lines at 42 kmph and 30 kmph respectively. They will be clear of each other in 19.99840012798976 seconds from the moment they meet. What is the length of the second car?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the largest real number $k$, such that for any positive real numbers $a,b$,
$$(a+b)(ab+1)(b+1)\geq kab^2$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that \(a, b, c\) are positive integers satisfying
$$
a+b+c=\operatorname{gcd}(a, b)+\operatorname{gcd}(b, c)+\operatorname{gcd}(c, a)+120 \text {, }
$$
determine the maximum possible value of \(a\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Two of the vertices of a cube are to be chosen at random. What is the probability that they will be the endpoints of a space diagonal of the cube?
Please reason step by step, and put your final answer within \boxed{}.


Context:
At a physical education lesson, 29 seventh graders attended, some of whom brought one ball each. During the lesson, sometimes one seventh grader would give their ball to another seventh grader who did not have a ball.

At the end of the lesson, $N$ seventh graders said, "I received balls less often than I gave them away!" Find the largest possible value of $N$, given that no one lied.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A square is divided into  $25$  unit squares by drawing lines parallel to the sides of the square. Some diagonals of unit squares are drawn from such that two diagonals do not share points. What is the maximum number diagonals that can be drawn with this property?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the equation $\left\lfloor \left\lfloor \frac{91}{n} \right\rfloor \div n \right\rfloor = 1$, where $n$ is a positive integer, find all possible values of $n$. Express your answer as a list of integers.
Please reason step by step, and put your final answer within \boxed{}.


Context:
8 distinct nonzero natural numbers are arranged in increasing order. The average of the first 3 numbers is 9, the average of all 8 numbers is 19, and the average of the last 3 numbers is 29. What is the maximum possible difference between the second largest number and the second smallest number?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A nonempty word is called pronounceable if it alternates in vowels (A, E, I, O, U) and consonants (all other letters) and it has at least one vowel. How many pronounceable words can be formed using the letters P, U, M, A, C at most once each? Words of length shorter than $5$ are allowed.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the expression \( \frac{\sum_{k=0}^{6}\csc^2\left(x+\frac{k\pi}{7}\right)}{7\csc^2(7x)} \) when \( x=\frac{\pi}{8} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
We have a 99x99 checkerboard. There is a fly on each square. In one step, each fly must move exactly one square diagonally (multiple flies can end up on the same square). After one step, what is the minimum number of empty squares?
Please reason step by step, and put your final answer within \boxed{}.


Context:
4. The continuation of the height $B H$ of triangle $A B C$ intersects the circumscribed circle around it at point $D$ (points $B$ and $D$ lie on opposite sides of line $A C$). The degree measures of arcs $A D$ and $C D$, not containing point $B$, are $120^{\circ}$ and $90^{\circ}$, respectively. Determine in what ratio the segment $B D$ is divided by side $A C$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In an isosceles triangle, the base is twice the height. Find the angles of the triangle.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a convex quadrilateral \(ABCD\) with side lengths \(AB = 7\), \(BC = 4\), \(CD = 5\), and \(DA = 6\), determine the range \((a, b]\) of its area \(S\). Then, find the value of \(a + b\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
$A$ is located 60 kilometers west of $B$. Individuals A and B depart from location $A$ while individuals C and D depart from location $B$ at the same time. A, B, and D all travel east, while C travels west. It is known that the speeds of A, B, C, and D form an arithmetic sequence, with A having the highest speed. After $n$ hours, B and C meet, and $n$ hours later, A catches up to D at location $C$. What is the distance between locations $B$ and $C$ in kilometers?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let  $2000 < N < 2100$  be an integer. Suppose the last day of year  $N$  is a Tuesday while the first day of year  $N+2$  is a Friday. The fourth Sunday of year  $N+3$  is the  $m$ th day of January. What is  $m$ ?

*Based on a proposal by Neelabh Deka*
Please reason step by step, and put your final answer within \boxed{}.


Context:
A triathlete competes in a triathlon in which the swimming, biking, and running segments are all of the same length. The triathlete swims at a rate of 3 kilometers per hour, bikes at a rate of 20 kilometers per hour, and runs at a rate of 10 kilometers per hour. Which of the following is closest to the triathlete's average speed, in kilometers per hour, for the entire race?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the volume of the region given by the inequality
\[
|x + y + z| + |x + y - z| + |x - y + z| + |-x + y + z| \le 6.
\]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Janet rolls a standard $6$-sided die $4$ times and keeps a running total of the numbers she rolls. What is the probability that at some point, her running total will equal $3$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Solve the system of equations: \[ \begin{aligned} x + y + z &= 1 \\ x^3 + y^3 + z^3 &= 1 \\ xyz &= -16. \end{aligned} \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the minimum number of times you need to spin a bottle to have at least a 50% chance of pointing to each of the 10 people in a circle at least once.
Please reason step by step, and put your final answer within \boxed{}.


Context:
On the Island of Misfortune, there are knights who always tell the truth and liars who always lie. One day, $n$ islanders gathered in a room.

The first one said: "Exactly every second person in this room is a liar."

The second one said: "Exactly every third person in this room is a liar."

and so on.

The person with number $n$ said: "Exactly every $(n - 1)$-th person in this room is a liar."

How many people could be in the room, knowing that not all of them are liars?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the expected value of the absolute value of the second pivot in a 4x4 matrix after performing Gauss-Jordan elimination, given that the matrix elements are drawn from a uniform distribution over [0, 1] and the first pivot is fixed at 1.
Please reason step by step, and put your final answer within \boxed{}.


Context:
各位数字之和等于 12 的四位数共有多少个?
Please reason step by step, and put your final answer within \boxed{}.


Context:
The integer $m$ is the largest positive multiple of $18$ such that every digit of $m$ is either $9$ or $0$. Compute $\frac{m}{18}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A cafe has 3 tables and 5 individual counter seats. People enter in groups of size between 1 and 4, inclusive, and groups never share a table. A group of more than 1 will always try to sit at a table, but will sit in counter seats if no tables are available. Conversely, a group of 1 will always try to sit at the counter first. One morning, $M$ groups consisting of a total of $N$ people enter and sit down. Then, a single person walks in, and realizes that all the tables and counter seats are occupied by some person or group. What is the minimum possible value of $M+N$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given \( x_{0} > 0 \), \( x_{0} \neq \sqrt{3} \), a point \( Q\left( x_{0}, 0 \right) \), and a point \( P(0, 4) \), the line \( PQ \) intersects the hyperbola \( x^{2} - \frac{y^{2}}{3} = 1 \) at points \( A \) and \( B \). If \( \overrightarrow{PQ} = t \overrightarrow{QA} = (2-t) \overrightarrow{QB} \), then \( x_{0} = \) _______.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the maximum dimension of the annihilator \( W^\circ \) of the subspace \( W = \{ A^m ; m \geq 0 \} \) within the vector space \( V = M_n(\mathbb{R}) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the longest finite arithmetic progression with a common difference of 6 consisting of prime numbers.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a doctor's clinic, patients arrive at an average rate of 10 per hour, following a Poisson distribution. The time taken for each patient's consultancy, which is exponentially distributed, averages 6 minutes (or 0.1 hour). The doctor doesn't attend to any patient if there are already 10 patients waiting. The system can be modeled as an M/M/1/11 queue (1 doctor, 10 waiting). Calculate the expected number of patients waiting in the queue at steady state.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the sets $S=\{x\mid (x-2)^{2} > 9\}$ and $T=\{x\mid a < x < a+8\}$, with $S\cup T=\mathbb{R}$, find the range of values for the real number $a$. Express your answer as an interval.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Together, Kenneth and Ellen pick a real number $a$. Kenneth subtracts $a$ from every thousandth root of unity (that is, the thousand complex numbers $\omega$ for which $\omega^{1000}=1$) then inverts each, then sums the results. Ellen inverts every thousandth root of unity, then subtracts $a$ from each, and then sums the results. They are surprised to find that they actually got the same answer! How many possible values of $a$ are there?
Please reason step by step, and put your final answer within \boxed{}.


Context:
p3. A multi-set of 11 positive integers has a median of 10, a unique mode of 11, and a mean of 12. What is the largest possible
 number that can be in this multi-set? (A multi-set is a set that allows repeated elements.)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the computational complexity, in terms of steps, for solving a linear system $Ax = b$ where $A \in \mathbb{R}^{n \times n}$ is a rank-deficient matrix with rank $r < n$. Express the complexity in terms of $r$ and $n$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
6.33 8 people participate in a chess round-robin tournament, where the winner gets 1 point, a draw gets 0.5 points, and the loser gets 0 points. It is known that the 8 people's scores are all different, and the second-place player's score is the sum of the scores of the last 4 players. What was the result of the match between the third-place player and the seventh-place player?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let  $1 = d_1 < d_2 < ...< d_k = n$  be all natural divisors of the natural number  $n$ . Find all possible values of the number  $k$  if   $n=d_2d_3 + d_2d_5+d_3d_5$ .
Please reason step by step, and put your final answer within \boxed{}.


Context:
5. (COL) A cube is assembled with 27 white cubes. The larger cube is then painted black on the outside and disassembled. A blind man reassembles it. What is the probability that the cube is now completely black on the outside? Give an approximation of the size of your answer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Three circles with radius one pass through a common point. Their second points of intersection are \( A \), \( B \), and \( C \). What is the radius of the circle \( ABC \)?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A plane cuts a sphere of radius  $ 1$  into two pieces, one of which has three times the surface area of the other. What is the area of the disk that the sphere cuts out of the plane?
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the largest 4-digit integer congruent to $7 \pmod{19}$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the correlation coefficient between $X_{(1)}$ and $X_{(3)}$, where $X_{(1)}$ and $X_{(3)}$ are the order statistics of three independent random variables $X_1, X_2, X_3$, each uniformly distributed over $[0,1]$. Given that $X_{(k)}$ follows a Beta distribution with parameters $k$ and $4-k$, and the variance of $X_{(k)}$ is $\frac{k(4-k)}{80}$ with mean $\frac{k}{3}$, use the theorem for covariance to find the correlation coefficient.
Please reason step by step, and put your final answer within \boxed{}.


Context:
James decides to replace his car and buy a new car for his wife. He sold his $20,000 car for 80% of its value and then was able to haggle to buy a $30,000 sticker price car for 90% of its value. Additionally, he sold his wife's $15,000 car for 70% of its value and bought another car with a $25,000 sticker price at an 85% discounted price.

During these transactions, a 7% sales tax was applied to the purchase of each new car, and James had to pay a 2% processing fee for each car sale. Considering all these transactions, how much total amount was James out of pocket?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the volume of the solid enclosed by the surface \((x^2+y^2+z^2)^2 = x\) using spherical coordinates \((r, \theta, \phi)\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
The length of a circular race track is $1 \mathrm{~km}$. Two motorcyclists start simultaneously from a given point $A$ on the track, traveling in opposite directions. One of them travels at a constant speed, while the other accelerates uniformly. They first meet at point $B$ on the track, and then for the second time at point $A$. How far did the first motorcyclist travel from the starting point to point $B$?
Please reason step by step, and put your final answer within \boxed{}.

\boxed{\frac{-1 + \sqrt{5}}{2} \text{ km}}


Context:
James visits 20 houses to try and make a sale. He manages to sell something in every house. The next day he visits twice as many houses. He sold two things at each house in each of the two days and sold 104 items in total. What percentage of houses did he sell to on the second day?
Please reason step by step, and put your final answer within \boxed{}.


Context:
An  $8\times 8$  chessboard is made of unit squares. We put a rectangular piece of paper with sides of length 1 and 2. We say that the paper and a single square overlap if they share an inner point. Determine the maximum number of black squares that can overlap the paper.
Please reason step by step, and put your final answer within \boxed{}.


Context:
$\begin{aligned} & \text{ Let }ABC\text{ be an acute triangle, and let }D\mathrm{~and~}E\text{ be the feet of the altitudes to }BC\mathrm{~and~}CA,\text{respectively. Suppose that} \\ & \mathrm{segments~}AD\mathrm{~and~}BE\text{ intersect at point }H\mathrm{~with~}AH=20\mathrm{~and~}HD=13.\mathrm{Compute~}BD\cdot CD. \end{aligned}$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the value of the sum: \[ \sum_{i=1}^{2000}\gcd(i,2000)\cos\left(\frac{2\pi i}{2000}\right) \] where \( \gcd \) denotes the greatest common divisor.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $x_0,x_1,x_2,\dotsc$ be a sequence of numbers, where each $x_k$ is either $0$ or $1$. For each positive integer $n$, define \[S_n = \sum_{k=0}^{n-1} x_k 2^k\] Suppose $7S_n \equiv 1 \pmod{2^n}$ for all $n \geq 1$. What is the value of the sum \[x_{2019} + 2x_{2020} + 4x_{2021} + 8x_{2022}?\]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the infimum of \( \mu(P) = \sup_{|z| = 1} |P(z)| \) over all complex polynomials \( P \) of degree \( n \) in the form \( P = X^n + \sum_{k=0}^{n-1} a_kX^k \), denoted as \( \mu_n \), for \( n \geq 1 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Draw the lines parallel to two sides of a triangle that bisect the area of the triangle. In what ratio does the line parallel to the third side of the triangle, passing through their intersection point, divide the area of the triangle?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the pattern of positive odd numbers shown below, find the 6th number from the left in the 21st row.
Please reason step by step, and put your final answer within \boxed{}.


Context:
During one hour, Tom can read 12 pages of a book. How many pages would he be able to read during 2 hours if he could increase his reading speed by a certain factor, given that he reads 72 pages in that time?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $n$ be a positive integer. Compute the number of words $w$ that satisfy the following three properties.

1. $w$ consists of $n$ letters from the alphabet $\{a,b,c,d\}.$

2. $w$ contains an even number of $a$'s

3. $w$ contains an even number of $b$'s.

For example, for $n=2$ there are $6$ such words: $aa, bb, cc, dd, cd, dc.$
Please reason step by step, and put your final answer within \boxed{}.


Context:
给定整数 $n \geqslant 3$, 设 $a_{1}, a_{2}, \cdots, a_{2 n}, b_{1}, b_{2}, \cdots, b_{2 n}$ 是 $4 n$ 个非负实数, 满足$$a_{1}+a_{2}+\cdots+a_{2 n}=b_{1}+b_{2}+\cdots+b_{2 n}>0$$且对任意 $i=1,2, \cdots, 2 n$, 有 $a_{i} a_{i+2} \geqslant b_{i}+b_{i+1}$ （这里 $a_{2 n+1}=a_{1}, a_{2 n+2}=a_{2}$, $b_{2 n+1}=b_{1}$ ), 求 $a_{1}+a_{2}+\cdots+a_{2 n}$ 的最小值。请提供当 $n=3$ 和 $n\geqslant 4$ 时最小值的和。
Please reason step by step, and put your final answer within \boxed{}.


Context:
N是由5个不同的非零数码组成的5位数，且N等于这5个数码中取3个不同的数码构成的所有三位数的和，求所有的这种5位数N．
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the Galois group \( Gal \left(\frac{\mathbb{Q}(\sqrt[3]{3},\sqrt{-3})}{\mathbb{Q}}\right) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the cartesian coordinate system xOy, point P is located at (1, $\sqrt {3}$). If $|$$\overrightarrow {OA}$$|=|$$\overrightarrow {OB}$$|=|$$\overrightarrow {OC}$$|=1, and $$\overrightarrow {OA}$$+ $$\overrightarrow {OB}$$+ $$\overrightarrow {OC}$$\= $$\overrightarrow {0}$$, determine the range of values for $$\overrightarrow {PB}$$• $$\overrightarrow {OA}$$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
There are 25 individuals numbered $1, 2, \ldots, 25$. Each individual randomly and independently chooses 12 others from this group to call. Let $X$ be the number of calls received by Individual 1, and $Y$ be the number of calls between Individual 1 and Individual 3. Calculate $E(X|Y=1)$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a certain game, two "electronic dogs," one black and one yellow, start from vertex A of a cube with edge length 1, labeled ABCD-A1B1C1D1, and crawl along the edges. Completing one edge is referred to as "completing one segment." The black "electronic dog" follows the path AA1→A1D1→…, while the yellow "electronic dog" follows the path AB→BB1→…, both adhering to the rule that the (i+2)th segment must be on a skew line with the ith segment (where i is a positive integer). Assuming the black "electronic dog" completes 2006 segments and the yellow "electronic dog" completes 2007 segments, and they both stop at some vertex of the cube, calculate the distance between the black and yellow "electronic dogs" at that time. Express your answer as a single numerical value.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $a_0=5/2$ and $a_k=a^2_{k-1}-2$ for $k\geq1.$ Compute $$\prod\limits_{k=0}^{\infty}\left(1-\frac{1}{a_k}\right)$$ in closed form. The original answer is in \frac{m}{n} format, please give the value of m + n.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider a Brownian motion $X_t$, defined for $t \geqslant 0$. Let $T_a$ denote the stopping time at which $X_t$ reaches level $a$, i.e., $T_a = \inf \{t \mid X_t = a\}$. Compute the probability $\mathbb{P}\{T_{2} < T_{-1} < T_{3}\}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Helen the hippo and her friends are preparing for Thanksgiving at Helen's house. Helen baked 31 cookies three days ago, 270 cookies two days ago, and 419 cookies the day before yesterday. Unfortunately, some cookies were lost due to various reasons. On the day she baked 31 cookies, her mischievous bird friend, Beaky, ate 5 of them. On the second day, 15% of the cookies crumbled during a cookie stacking competition. And on the third day, Helen generously gave 30 cookies away to her neighbors.

In addition to these losses, Helen received a surprise gift of 45 cookies from her friend Lucy. Considering all these events, how many cookies are available at Helen's house for Thanksgiving as of last night?
Please reason step by step, and put your final answer within \boxed{}.


Context:
某单位有A，B两个餐厅为员工提供午餐与晚餐服务，甲、乙两位员工每个工作日午餐和晚餐都在单位就餐，近100个工作日选择餐厅就餐情况统计如下：

|选择餐厅情况(午餐，晚餐)|(A，A)|(A，B)|(B，A)|(B，B)|
|---|---|---|---|---|
|甲员工|30天|20天|40天|10天|
|乙员工|20天|25天|15天|40天|

假设甲、乙员工选择餐厅相互独立，用频率估计概率.
$记X为甲、乙两员工在一天中就餐餐厅的个数，求X的数学期望E(X)；$
Please reason step by step, and put your final answer within \boxed{}.


Context:
The base of the quadrilateral pyramid \( S A B C D \) is a quadrilateral \( A B C D \) with \( A B = A D = 6 \), \( C B = C D = 2 \), and the sides \( A B \) and \( B C \) are perpendicular to each other. All the lateral faces are inclined to the plane of the base at an angle of \( 60^{\circ} \). Find the volume \( V \) of the pyramid, given that \( V > 12 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find all continuous functions \( f: \mathbb{R} \to \mathbb{R} \) that satisfy the equation \( f(x) = f(x^2 + C) \) for all \( x \in \mathbb{R} \), where \( C > 0 \) is a constant.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider the set $L$ of binary strings of length less than or equal to $9$. For a string $w$, define $w^{+}$ to be the set $\{w, w^2, w^3, \ldots\}$ where $w^k$ represents $w$ concatenated to itself $k$ times. How many ways are there to pick an ordered pair of (not necessarily distinct) elements $x, y \in L$ such that $x^{+} \cap y^{+} \neq \varnothing$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the integers \( 1, 2, 3, \ldots, 40 \), find the greatest possible sum of the positive differences between the integers in twenty pairs, where the positive difference is either 1 or 3.
Please reason step by step, and put your final answer within \boxed{}.


Context:
$
\begin{aligned}
 & \text{Given a positive integer }n,\text{find the proportion of the subsets of }\{1,2,\ldots,2n\}\mathrm{such~that} \\
 & \text{their smallest element is odd.}
\end{aligned}
$
Please reason step by step, and put your final answer within \boxed{}.


Context:
A rectangle is cut at one corner to form a pentagon with side lengths of $13, 19, 20, 25,$ and $31$ (the order of lengths is not necessarily this). What is the area of the pentagon?
Please reason step by step, and put your final answer within \boxed{}.


Context:
For a positive integer $n$, denote by $\tau(n)$ the number of positive integer divisors of $n$, and denote by $\phi(n)$ the number of positive integers that are less than or equal to $n$ and relatively prime to $n$. Call a positive integer $n$ $\emph{good}$ if $\varphi (n) + 4\tau (n) = n$. For example, the number $44$ is good because $\varphi (44) + 4\tau (44)= 44$.

Find the sum of all good positive integers $n$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Ann, Bohan, Che, Devi, and Eden each ordered a different meal at a restaurant and asked for their own bill. Although the correct five bills arrived at the table, none of them were given to the correct person. If the following four statements are true, who was given the bill for Bohan's meal?

- Che and Eden ordered meals with the same cost.
- Devi's bill was given to the same person whose bill was given to Devi.
- Ann, Bohan, and Eden were each given a bill for less money than the meal they ordered.
- Ann's bill was given to Che.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit:

\[ \lim_{x \to 0} \frac{1}{x} \left[ \sqrt[3]{\frac{1 - \sqrt{1 - x}}{\sqrt{1 + x} - 1}} - 1 \right] \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider the equation \( {x}^{4} = a{x}^{3} + b{x}^{2} + {cx} + {2007} \) , where \( a,b,c \) are real numbers. Determine the largest value of \( b \) for
 which this equation has exactly three distinct solutions, all of which are integers.
Please reason step by step, and put your final answer within \boxed{}.


Context:
For a differential operator $P$ of order $p$ and a multi-index $\beta$, determine the maximum possible order of the commutator $[P, \partial_x^\beta]$. Provide your answer as an expression in terms of $p$ and $|\beta|$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the probability that the minimum of a Brownian motion \((W_t)_{t \in [0, T]}\) is zero, given that \(W_0 = 0\) almost surely. Express this probability as \(\mathbb{P} \left(\min_{t \in [0, T]} W_t = 0\right)\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a set of numbers \([1, 2, 3, 4, 5, 6, 7, 8, 9, 10]\), calculate the expected value of the largest number when selecting 3 numbers at random from the set. Assume each number is selected with equal probability and selections are independent. What is the expected value of the largest number in such a selection?
Please reason step by step, and put your final answer within \boxed{}.