Context:
Determine the isomorphism type of the quotient group $U_{26}/\langle 5 \rangle$, where $U_{26}$ is the group of units under multiplication modulo 26, the order of $U_{26}$ is 12, and the order of the subgroup generated by 5 is 4.
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Context:
A department store received 10 suitcases and 10 keys separately, with the warning that each key opens only one suitcase and that each suitcase has a corresponding key. What is the expected number of attempts required to find the right key for each suitcase?
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Context:
Find the supremum of the function \( \frac{F(x)(1-F(x))}{n} \) for \( x \in \mathbb{R} \), where \( F(x) \) is a cumulative distribution function and \( n \) is a positive integer.
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Context:
Consider the following balanced chemical reaction:

2 K2Cr2O7 + 16 HCl -> 4 KCl + 4 CrCl3 + 8 H2O + 7 Cl2

Now, calculate the mass percentage of oxygen in K2Cr2O7 using the given balanced chemical equation.
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Context:
A rectangle $ABCD$ is made up of five small congruent rectangles as shown in the given figure. Find the perimeter, in cm, of $ABCD$ if its area is $6750 cm^2$.
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Context:
Let $a \in \mathbb{R}$, and the function $f(x) = x^3 + ax^2 + (a-3)x$ has a derivative denoted by $f'(x)$. If $f'(x)$ is an even function, find the equation of the tangent line to the curve $y = f(x)$ at the origin. Express your answer in the form $y = c$, where $c$ is a constant.
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Context:
A store increased the original price of a shirt by a certain percent and then lowered the new price by the same amount. Given that the resulting price was $84\%$ of the original price, by what percent was the price increased and decreased$?$
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Context:
\(a, b, c\) are three different digits. If you add up all six two-digit numbers that can be formed using these digits, without repeating any digit in the same number, the total is 528. Find these digits.
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Context:
Find the infimum of the expression \[ \sqrt[5]{\frac{a}{b+c}}+\sqrt[5]{\frac{b}{c+d}}+\sqrt[5]{\frac{c}{d+e}}+\sqrt[5]{\frac{d}{e+a}}+\sqrt[5]{\frac{e}{a+b}} \] for positive real numbers $a, b, c, d, e$. Provide your answer as a single value.
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Context:
From a $6 \times 6$ grid square, gray triangles were cut out. What is the area of the remaining shape? The side length of each cell is 1 cm. Provide your answer in square centimeters.
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Context:
Calculate the limit:
$$\lim_{x \to 0^+}\frac{\tan x \cdot \sqrt{\tan x} - \sin x \cdot \sqrt{\sin x}}{x^3 \sqrt{x}}$$

without using L'Hôpital's Rule.
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Context:
The expression
\[2 \sqrt[3]{3 \sec^2 20^\circ \sin^2 10^\circ}\]can be expressed in the form $a + b \sec 20^\circ,$ where $a$ and $b$ are integers.  Find the ordered pair $(a,b).$
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Context:
For a positive integer $n$, let $f_n(x)=\cos (x) \cos (2 x) \cos (3 x) \cdots \cos (n x)$. Find the smallest $n$ such that $\left|f_n^{\prime \prime}(0)\right|>2023$. Please provide the value of $n$.
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Context:
In the diagram, \( Z \) lies on \( XY \) and the three circles have diameters \( XZ \), \( ZY \), and \( XY \). If \( XZ = 12 \) and \( ZY = 8 \), then calculate the ratio of the area of the shaded region to the area of the unshaded region. Express your answer as a ratio in simplest form, with the two numbers separated by a colon (:).
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Context:
Given the equation $x^4 - 4x^3 + 4x^2 + ax + b = 0$, where $a$ and $b$ are real numbers, and the equation has two distinct positive roots $\alpha$ and $\beta$ such that $\alpha + \beta = 2\alpha\beta$, determine the maximum value of $a + b$. \(\text{Express your answer as a single number.}\)
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Context:
Evaluate the integral $$\int_1^\infty\frac{e^{4x}-e^{2x}}{x(e^{4x}+1)(e^{2x}+1)}\text{d}x$$ using the Frullani integral method.
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Context:
Let the set $U = \{(x, y) | x \in \mathbb{R}, y \in \mathbb{R}\}$, $A = U \setminus \{(x, y) | x + y > m\}$, and $B = \{(x, y) | x^2 + y^2 \leq n\}$. Then, the necessary and sufficient condition for the point $(1, 2)$ to belong to $(C_U A) \cap B$ is.
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Context:
An  $m\times n(m,n\in \mathbb{N}^*)$  rectangle is divided into some smaller squares. The sides of each square are all parallel to the corresponding sides of the rectangle, and the length of each side is integer. Determine the minimum of the sum of the sides of these squares.
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Context:
Calculate the volume of the solid bounded by the surfaces $y^2 + z^2 - 2z = 0$, $x = y^2 + z^2$, and $x = 0$ using triple integrals. Determine the appropriate intervals of integration.
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Context:
Determine the positive values of $P$ for which the series \( \sum_{n=1}^{\infty}\frac{(n!)^3}{(Pn)!} \) converges.
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Context:
Consider a non-singular matrix $A$ that satisfies $A^2B = A$ with another matrix $B$. Given that $A$ is a $2 \times 2$ matrix represented as:

$$
A= \begin{bmatrix}
a & b \\
c & d \\
\end{bmatrix}
$$

where $a, b, c, d \in \{-1, 0, 1\}$. Determine the probability that the determinant of $A + B$ equals the sum of the determinants of $A$ and $B$, i.e., $\det(A + B) = \det(A) + \det(B)$. \(\boxed{?}\)
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Context:
Determine whether the limit \( \lim_{x \to 0} \frac{1}{x} \int_{0}^{x} \left| \cos \left(\frac{1}{t}\right) \right| \, dt \) exists. If it exists, find its value.
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Context:
## problem statement

Calculate the volume of the tetrahedron with vertices at points $A_{1}, A_{2}, A_{3}, A_{4_{\text {and }}}$ its height dropped from vertex $A_{4 \text { to the face }} A_{1} A_{2} A_{3}$.

$A_{1}(4 ;-1 ; 3)$

$A_{2}(-2 ; 1 ; 0)$

$A_{3}(0 ;-5 ; 1)$

$A_{4}(3 ; 2 ;-6)$
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Context:
In three of the six circles of a diagram, the numbers 4, 14, and 6 are written. In how many ways can natural numbers be placed in the remaining three circles so that the products of the triplets of numbers along each of the three sides of the triangular diagram are the same?
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Context:
What is the proof-theoretic ordinal of Primitive Recursive Arithmetic (PRA)?
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Context:
实数$x _ { 1 } , x _ { 2 } , \cdots , x _ { 2 0 0 1 }$满足$\sum _ { k = 1 } ^ { 2 0 0 0 } \vert x _ { k } - x _ { k + 1 } \vert = 2 0 0 1 ,$，令$y _ { k } = \frac { 1 } { k } ( x _ { 1 } + x _ { 2 } + \cdots +$ $x _ { k } )$ k=1,2,⋯,2001.求$\sum _ { k = 1 } ^ { 2 0 0 0 } \vert y _ { k } - y _ { k + 1 }|$ 的最大可能值.
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Context:
过半径为5的球面上的一点P作三条两两垂直的弦PA、PB、PC，且满足/PA|=2｜PB｜，则|PA|+|PB|+|PC|的最大值是
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Context:
Calculate the mass percentage of C in the biochemical compound adenosine triphosphate (ATP).
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Context:
What is the molecular weight of 8 moles of BaBr2
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Context:
Determine the value of $a$ such that the following integral equation holds:

$$\int \limits^{\infty }_{0} \frac{dx}{a^2 + \left(x - \frac{1}{x}\right)^2} = \frac{\pi}{5050}$$

where $a^2 \geq 0$. Provide your answer as a single value of $a$.
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Context:
Find four numbers that form a proportion, given that the sum of the first and last numbers is 14, the sum of the middle numbers is 11, and the sum of the squares of these four numbers is 221.
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Context:
4.3. Two identical cylindrical vessels are connected at the bottom by a small cross-section pipe with a valve. While the valve was closed, water was poured into the first vessel, and oil into the second, so that the level of the liquids was the same and equal to \( h = 40 \) cm. At what level will the water in the first vessel stabilize if the valve is opened? The density of water is 1000 kg/ \(\mathrm{m}^{3}\), and the density of oil is 700 kg/ \(\mathrm{m}^{3}\). Neglect the volume of the connecting pipe. Give the answer in centimeters.
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Context:
Solve the equation $\sqrt{a-\sqrt{a+ x}} = x$ for $x$.
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Context:
A square was cut into four equal rectangles, and from these, a large letter P was assembled, as shown in the diagram, with a perimeter of 56. What is the perimeter of the original square?
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Context:
A unit cube has a base and a top face, with side edges connecting them. The top face is rotated by an acute angle $\alpha$ around the line connecting the centers of the two faces. If the length of the side edges remains unchanged and the vertices' connections are intact, calculate how much closer the top face gets to the base face.
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Context:
对 $n$ 个实数 $a_{1}, a_{2}, \cdots, a_{n}$, 令
$$
b_{k}=\frac{a_{k}+a_{k+1}}{2}, c_{k}=\frac{a_{k-1}+a_{k}+a_{k+1}}{3},
$$
其中， $k=1,2, \cdots, n, a_{0}=a_{n} ; a_{n+1}=a_{1}$. 求最大的正实数 $\lambda$ ，使得对于任意正整数 $n \geqslant 3$ ，以及任意实数 $a_{1}, a_{2}, \cdots, a_{n}$ ，均有
$$
\sum_{k=1}^{n}\left(a_{k}-b_{k}\right)^{2} \geqslant \lambda \sum_{k=1}^{n}\left(a_{k}-c_{k}\right)^{2}
$$原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值
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Context:
An integer $n$ is said to be [i]good[/i] if $|n|$ is not the square of an integer. Determine all integers $m$ with the following property: $m$ can be represented, in infinitely many ways, as a sum of three distinct good integers whose product is the square of an odd integer.
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Context:
Determine the values of the parameter \( \beta \) for which the integral \( \int_0^{+\infty} \frac{\sin x}{x^\beta(1+x^3)}\, dx \) converges.
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Context:
Let $S_{n}$ be the sum of the first $n$ terms of the geometric sequence $\{a_{n}\}$, and $S_{4}=5S_{2}$. Determine all possible values of the common ratio $q$ of this sequence. List all possible values of $q$.
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Context:
Given a function $f$ for which
$f(x) = f(398 - x) = f(2158 - x) = f(3214 - x)$
holds for all real $x,$ what is the largest number of different values that can appear in the list $f(0),f(1),f(2),\cdots,f(999)?$
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Context:
Several figures can be made by attaching two equilateral triangles to the regular pentagon $ABCDE$ in two of the five positions shown. Find the number of non-congruent figures that can be constructed in this way.
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Context:
Four congruent semicircles are drawn on the surface of a sphere with radius $2$, as shown, creating a close curve that divides the surface into two congruent regions. The length of the curve is $\pi\sqrt{n}$. What is the value of $n$?
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Context:
一个等腰直角三角形的三个顶点分别在正三棱柱 的三条侧棱上, 已知正三棱柱的底面边长为 2 , 则该三角形的斜边长为 ($\quad$).
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Context:
A jailer is granting partial amnesty by the following procedure: First, he opens all the cells. Next, he locks every second cell (in this prison, all the cells are in a single row). In the third step, he turns the key in every third cell (if a cell is open, it will be locked, and if it is locked, it will be opened). Continuing in this manner, on the $n$-th step, he turns the key in every $n$-th cell. After finishing this process, he releases all prisoners who are in open cells. Indicate the numbers of the cells that contain these lucky prisoners.
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Context:
A square steel plate has had a rectangular piece of width 3 cm and another rectangular piece of width 5 cm cut off (as shown in the figure). The total area removed is 81 square centimeters. What was the area of the original square? $\qquad$ square centimeters.
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Context:
In a quadratic trinomial, the second coefficient and the constant term were switched, and the resulting trinomial was added to the original trinomial. As a result, a third quadratic trinomial was obtained, which has a single root. What can this root be? If there are multiple correct answers, list them separated by semicolons.
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Context:
A stained glass window consists of nine squares of glass arranged in a 3x3 grid. Of these nine squares, \( k \) are red, and the rest are blue. A set of windows is created such that any possible window can be formed in exactly one way by rotating and/or flipping one of the windows in the set. If the total number of red squares in the set exceeds 100, find the value of \( k \).
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Context:
A circular spinner used in a game has a radius of 15 cm. The probability of winning on one spin of this spinner is $\frac{1}{3}$ for the WIN sector and $\frac{1}{4}$ for the BONUS sector. What is the area, in square centimeters, of both the WIN sector and the BONUS sector? Express your answers in terms of $\pi$.
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Context:
What is the maximal dimension of a linear subspace  $ V$  of the vector space of real  $ n \times n$  matrices such that for all  $ A$  in  $ B$  in  $ V$ , we have  $ \text{trace}\left(AB\right) \equal{} 0$  ?
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Context:
A frame for three square photographs has a uniform width everywhere (see the figure). The perimeter of one opening is 60 cm, and the perimeter of the entire frame is 180 cm. What is the width of the frame?
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Context:
A hemisphere is placed on a sphere of radius \(100 \text{ cm}\). The second hemisphere is placed on the first one, and the third hemisphere is placed on the second one (as shown below). Find the maximum height of the tower (in cm).
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Context:
Let \( a, b, c \) be the side lengths of a right triangle, with \( a \leqslant b < c \). Determine the maximum constant \( k \) such that \( a^{2}(b+c) + b^{2}(c+a) + c^{2}(a+b) \geqslant k a b c \) holds for all right triangles, and identify when equality occurs.
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Context:
Determine the group to which the smallest group containing the functions \( f(x) = \frac{1}{x} \) and \( g(x) = \frac{x-1}{x} \), under composition, is isomorphic. The functions are defined on the set \( A = \mathbb{R} \setminus \{0,1\} \).
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Context:
A $20 \times 20 \times 20$ block is divided into 8000 non-overlapping unit cubes, each assigned a number. In each column of 20 cubes parallel to any edge of the block, the sum of their numbers is 1. One unit cube is assigned the number 10. Three $1 \times 20 \times 20$ slices parallel to the faces of the block contain this unit cube. Calculate the sum of all numbers of the cubes outside these slices.
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Context:
Consider the sequence \( B_n \) where \( n \in \mathbf{N}^* \). Let the point with coordinates \((1,1)\) be \( M \). \( P_n (x_n, y_n) \) is the circumcenter of triangle \(\triangle A_n B_n M\). Calculate the sum \( x_1 + x_2 + \cdots + x_{100} \).
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Context:
In a school with 1000 students, numbered from 1 to 1000, a group of 500 students is called a 'good group' if there exists a student in the group whose number divides the number of another student in the group. Otherwise, the group is called a 'bad group'. A 'good student' is defined as a student who does not belong to any 'bad group'. Determine the largest number of a 'good student'.
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Context:
The density function of a random variable \( x \) is given by \( f(x) = ke^{-2x^{2}+10x} \). Determine the upper 5% point of the distribution of the means for a random sample of size 25 drawn from this population.
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Context:
Let \( G = \left\{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \mid a, b, c, d \in \mathbb{Z}, ad - bc = 1 \right\} \) be the group of 2x2 integer matrices with determinant 1. Let \( H = \left\{ \begin{pmatrix} 1 & n \\ 0 & 1 \end{pmatrix} \mid n \in \mathbb{Z} \right\} \) be a subgroup of \( G \). Determine the index \([G:H]\).
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Context:
How do you find the volume of the solid bounded by the surfaces $z=x^2 + y^2$ and $x^2+y^2+z^2=20$?
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Context:
Given the equation:
\[
[x+0.1]+[x+0.2]+[x+0.3]+[x+0.4]+[x+0.5]+[x+0.6]+[x+0.7]+[x+0.8]+[x+0.9]=104
\]
where \([x]\) denotes the greatest integer less than or equal to \(x\), find the smallest value of \(x\) that satisfies this equation. Express your answer as a decimal value.
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Context:
Purple College keeps a careful count of its students as they progress each year from the freshman class to the sophomore class to the junior class and, finally, to the senior class.  Each year at the college one third of the freshman class drops out of school, $40$ students in the sophomore class drop out of school, and one tenth of the junior class drops out of school.  Given that the college only admits new freshman students, and that it wants to begin each school year with $3400$ students enrolled, how many students does it need to admit into the freshman class each year?
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Context:
Let  $a$  and  $b$  be relatively prime positive integers such that  $a/b$  is the sum of the real solutions to the equation \[\sqrt[3]{3x-4}+\sqrt[3]{5x-6}=\sqrt[3]{x-2}+\sqrt[3]{7x-8}.\] Find  $a+b$ .
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Context:
A person has 4 colors of light bulbs (enough of each color), and needs to install one light bulb at each of the 6 points $A, B, C, A_{1}, B_{1}, C_{1}$ as shown in the diagram. It is required that the light bulbs at the two ends of the same segment are of different colors. What is the number of ways to install the light bulbs, ensuring that each color is used at least once? (Answer in digits).
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Context:
For how many ordered triplets $(a, b, c)$ of positive integers less than 10 is the product $a \times b \times c$ divisible by 20?
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Context:
Consider a sphere centered at the origin $(0, 0, 0)$ in the $xyz$ coordinate system. Initially, the lower hemisphere (with negative $y$-coordinate) is painted. The sphere is then randomly rotated about any axis passing through the center. Determine the expected number of rotations required to paint the entire sphere.
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Context:
In the diagram, congruent rectangles $ABCD$ and $DEFG$ share a common vertex $D$. The sides $BC$ and $EF$ intersect at point $H$. Given that $DA = DE = 8$, $AB = EF = 12$, and $BH = 7$, find the area of the region $ABHED$.
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Context:
Find the volume of the solid obtained by rotating the curve \( y = \cosh x \) around the \( y \)-axis from \( y = 0 \) to \( y = 1 \).
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Context:
The points \( A_{1}, A_{2}, A_{3}, A_{4}, A_{5}, A_{6} \) divide a circle of radius 1 into six equal parts. A ray \( l_{1} \) is drawn from \( A_{1} \) in the direction of \( A_{2} \), a ray \( l_{2} \) is drawn from \( A_{2} \) in the direction of \( A_{3} \), and so on, with a ray \( l_{6} \) drawn from \( A_{6} \) in the direction of \( A_{1} \). From a point \( B_{1} \) on the ray \( l_{1} \), a perpendicular is dropped onto the ray \( l_{6} \), from the foot of this perpendicular another perpendicular is dropped onto \( l_{5} \), and so on. The foot of the sixth perpendicular coincides with \( B_{1} \). Find the length of the segment \( B_{1} A_{1} \).
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Context:
Find an upper bound for the operator norm \( \lvert e^A \rvert \) of a square matrix \( A \) in terms of \( \lvert A \rvert \) and the dimension \( n \) of the matrix.
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Context:
Find the variance of the stochastic process \(X_{t} = \int_{0}^{t} W^{+}_{s}dW_{s}\), where \(W_{t}\) is a Wiener process and \(a^{+} = \max(a, 0)\).
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Context:
Let $\overline{ABCD}$ be a $4$-digit number. What is the smallest possible positive value of $\overline{ABCD}- \overline{DCBA}$?
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Context:
Under the new AMC 10, 12 scoring method, $6$ points are given for each correct answer, $2.5$ points are given for each unanswered question, and no points are given for an incorrect answer. Some of the possible scores between $0$ and $150$ can be obtained in only one way, for example, the only way to obtain a score of $146.5$ is to have 24 correct answers and one unanswered question. Some scores can be obtained in exactly two ways; for example, a score of $104.5$ can be obtained with $17$ correct answers, $1$ unanswered question, and $7$ incorrect, and also with $12$ correct answers and $13$ unanswered questions. There are three scores that can be obtained in exactly three ways. What is their sum?
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Context:
A princess is choosing a groom from one hundred suitors, each better than the last, but they propose in a random order. A groom is considered outstanding if the princess likes him more than all who have proposed before him. The first groom is also outstanding by default. An outstanding groom and all those who propose after him, up to the next outstanding groom, form a sequence. Calculate the expected number of grooms in the first such sequence.
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Context:
Find the radius of convergence of the series:
\[ \sum \frac{(K/e)^{3K}}{3K!}(7x)^K. \]
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Context:
In a microcircuit with 2000 contacts, initially, any two contacts are connected by a separate wire. Hooligans Vasya and Petya take turns cutting the wires, with Vasya (who starts) cutting one wire per turn, and Petya cutting either two or three wires per turn. The hooligan who cuts the last wire connected to any contact loses. Who wins with optimal play?
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Context:
Let $X_1$ and $X_2$ be two independent uniformly distributed random variables on $[0,1]$. Calculate the expectation of $\min(X_1,X_2)$ given that $\max(X_1,X_2) > 0.5$. Provide your answer as a fraction.
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Context:
A round table has radius \(4\). Six rectangular place mats are placed on the table. Each place mat has width \(1\) and length \(x\) as shown. They are positioned so that each mat has two corners on the edge of the table, these two corners being end points of the same side of length \(x\). Further, the mats are positioned so that the inner corners each touch an inner corner of an adjacent mat. Find the value of \(x\). The original answer is in the form \(\frac{k\sqrt{n}-\sqrt{p}}{m}\). Please give the value of \(k+m+n+p\).
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Context:
Provide an example of a continuous function whose integral from 0 to infinity converges, but whose limit as x approaches infinity does not exist.
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Context:
We colour all the sides and diagonals of a regular polygon $P$ with $43$ vertices either
red or blue in such a way that every vertex is an endpoint of $20$ red segments and $22$ blue segments.
A triangle formed by vertices of $P$ is called monochromatic if all of its sides have the same colour.
Suppose that there are $2022$ blue monochromatic triangles. How many red monochromatic triangles
are there?
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Context:
Find the number of elements in the conjugacy class of the permutation $\partial = (1\ 10)(2\ 6\ 3\ 7\ 6\ 8\ 12)(4)(9\ 11)$ in the symmetric group $S_{12}$.
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Context:
Let $m,n \ge 2$ be such that $m-1$ is a divisor of $n-1$. Let $T$ be a tree with $m$ vertices. Calculate the Ramsey number $R(T,K_{1,n})$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the limit of the sequence \( \lim_{n\to\infty} \frac{(2n-1)!}{(2n)^n} \) as \( n \) approaches infinity.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the line integral \( \int_C \frac{z\arctan(z)}{\sqrt{1+z^2}}\,dz + (y-z^3)\,dx - (2x+z^3)\,dy \), where the contour \( C \) is defined by the equations \( \sqrt{1-x^2-y^2}=z \) and \( 4x^2+9y^2 = 1 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the limit:

\[ \lim_{n \to \infty} \frac{\sum\limits_{i = 0}^{n + 1} \binom{n + 1}{i}^3}{\sum\limits_{i = 0}^n \binom{n}{i}^3} \]

where \( \sum\limits_{i = 0}^n \binom{n}{i}^3 = \text{hypergeom}\left(\left[-n, -n, -n\right], [1, 1], -1\right) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the pattern in the sequence: $1, 4, 3, 9, 5, 16, 7, \quad \qquad , 36, 11, \quad \qquad$ What are the two numbers that should be filled in the blanks?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a function \( f: \mathbb{R}^2 \to \mathbb{R} \) such that \( \int_{\mathbb{R}} \left[ \int_{\mathbb{R}} f(x,y) \, dy \right] \, dx = 1 \) and \( \int_{\mathbb{R}} \left[ \int_{\mathbb{R}} f(x,y) \, dx \right] \, dy = -1 \), determine the possible range of values for \( \iint_{\mathbb{R}^2} |f| \, dm \), where \( m \) refers to the Lebesgue measure on \( \mathbb{R}^2 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $X$ be a Hilbert space and $\varphi \in X' \setminus \{0\}$. Define the set \( C = \{ x \in X : \varphi(x) = 1 \} \). Find the orthogonal complement \( C^{\perp} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit \( \lim_{N \rightarrow \infty} \frac{g(N)}{N^3} \), where \( g(N) \) is the number of quadruples \((x, y, z, w)\) of positive integers satisfying \( x + y + z + w = N \) and \( x \le y \le z \le w \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the figure, the line $t$ is parallel to the segment $EF$ and tangent to the circle. If $AE = 12$, $AF = 10$, and $FC = 14$, determine the length of $EB$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a right triangle \(ABC\) with the right angle at vertex \(B\). A median \(BD\) is drawn from \(B\). Let \(K\) be the point of tangency of side \(AD\) of triangle \(ABD\) with the incircle of this triangle. Find the acute angles of triangle \(ABC\) if \(K\) bisects \(AD\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
For which integers \( n \) from 1 to 6 does the equation

\[ a^n + b^n = c^n + n \]

have a solution in integers?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Suppose there are $m$ types of currency with denominations $p_{1}$ units, $p_{2}$ units, $\cdots, p_{m}$ units. What is the number of ways to change $r$ units of currency into these various denominations, denoted as $a_{r}$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider $9$ points in space, no four of which are coplanar. Each pair of points is joined by an edge (that is, a line segment) and each edge is either colored blue or red or left uncolored. Find the smallest value of  $\,n\,$ such that whenever exactly $\,n\,$ edges are colored, the set of colored edges necessarily contains a triangle all of whose edges have the same color.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A side of a triangle is divided into five equal parts; lines are drawn through the division points parallel to the base.

Find the lengths of the segments of these lines that are enclosed between the other two sides of the triangle, given that the base is 20 units long.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the convex quadrilateral \( ABCD \), the lines extending the opposite sides intersect at points \( K \) and \( L \), and the diagonals intersect at point \( M \). The diagonal \( AC \) intersects the line \( KL \) at point \( N \). Given that \( AM = 3 \) cm and \( MC = 2 \) cm, find the distance \( CN \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a sequence of positive integers, each sum of 17 consecutive terms is even, and each sum of 18 consecutive terms is odd. Determine the maximum number of terms such a sequence can have.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the radius of convergence for the power series \(\sum^\infty_{k=1} \frac{(2k-1)^{2k-1}}{2^{2k}(2k)!}x^k\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Amandine and Brennon play a turn-based game, with Amadine starting.
On their turn, a player must select a positive integer which cannot be represented as a sum of multiples of any of the previously selected numbers.
For example, if $3, 5$ have been selected so far, only $1, 2, 4, 7$ are available to be picked;
if only $3$ has been selected so far, all numbers not divisible by three are eligible.
A player loses immediately if they select the integer $1$.
Call a number $n$ feminist if $\gcd(n, 6) = 1$ and if Amandine wins if she starts with $n$. Compute the sum of the feminist numbers less than $40$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the integral \( \int_\gamma \frac{1}{\sin(z)} \, dz \) where \( \gamma(t) = 4e^{2 i \pi t} \) for \( t \in [0,1] \). Use the residue theorem to find the value of the integral.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Fifteen people are each randomly assigned two numbers from the set \( \{1, 2, \ldots, 25\} \). A person is considered happy if the sum of their two numbers is 25. Calculate the expected number of happy people.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( \phi : \mathbb{R}^3 \rightarrow \mathbb{R}^3 \) be a linear mapping with eigenvalues 2, 1, and -1. Determine the trace of the mapping \( \phi^{\wedge 2} : \Lambda^{2}\mathbb{R}^3 \rightarrow \Lambda^{2}\mathbb{R}^3 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the radius of convergence for the series: $$\sum_{n=1}^{\infty} \frac{z^n}{[1,2,...,n]}$$ where $[a_1,a_2,a_3,...,a_n]$ denotes the least common multiple of the numbers $a_1,a_2,...,a_n$. Provide your answer as a single numerical value.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the probability that Pollard's rho algorithm fails to find any factors of a composite number $N$ after trying all possible initial function values $x$ within the range $0 \le x \le N-1$ and polynomial constant terms $b$ within the range $1 \le b \le N-3$. Provide your answer as a probability.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the double integral \( \iint_R (x^2 - y^2) \, dA \), where \( R \) is the region defined by the inequality \( x^2 - y^2 \geq a^2 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
For the function \(f(x) = x \cdot \ln(x)\), consider its Taylor polynomial of degree 4 centered at \(a = 1\). Determine the largest value of \(x > 1\) such that the error \(|E(x)|\) is guaranteed to be less than \(10^{-4}\). The error function is given by \(\frac{-6 \cdot (x-1)^5}{z^4 \cdot 5!}\), where \(z\) lies between \(x\) and \(a\). Find the maximum \(x\) satisfying this condition.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the cardinality of the set $A$, where $A$ consists of all sequences $a = (a_n)_{n \ge 1}$ of natural numbers satisfying the following conditions for all $n \ge 1$: (a) $a_n < a_{n+1}$ (strictly increasing), and (b) the limit $\lim_{n\to \infty} \frac{\#\{j \mid a_j \le n\}}{n}$ exists.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the derivative of the function \( F(x) = \int_0^x \log(t) \, dt \) with respect to \( x \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Triangle \( A B C \) is divided by line segments as shown in the picture. Line segments \( D E \) and \( A B \) are parallel. Triangles \( C D H, C H I, C I E, \) and \( F I H \) have the same area of \( 8 \, \mathrm{dm}^2 \).

Determine the area of quadrilateral \( A F H D \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the greatest common divisor of all numbers of the form $(2^{a^2} \cdot 19^{b^2} \cdot 53^{c^2} + 8)^{16} - 1$ where $a, b, c$ are integers.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A grandfather, who is over 70 years old, and his grandson are celebrating their birthdays today. Several years ago, the grandfather's age was 10 times the age of his grandson. Several years later, he will be 8 times the age of his grandson. Find their current ages. Express your answer as two whole numbers, representing the grandfather's age and the grandson's age, respectively.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Positive integers  $x_1,...,x_m$  (not necessarily distinct) are written on a blackboard. It is known that each of the numbers  $F_1,...,F_{2018}$  can be represented as a sum of one or more of the numbers on the blackboard. What is the smallest possible value of  $m$ ?
(Here  $F_1,...,F_{2018}$  are the first  $2018$  Fibonacci numbers:  $F_1=F_2=1, F_{k+1}=F_k+F_{k-1}$  for  $k>1$ .)
Please reason step by step, and put your final answer within \boxed{}.


Context:
If $A$ and $B$ are vertices of a polyhedron, define the distance $d(A, B)$ to be the minimum number of edges of the polyhedron one must traverse in order to connect $A$ and $B$. For example, if $\overline{AB}$ is an edge of the polyhedron, then $d(A, B) = 1$, but if $\overline{AC}$ and $\overline{CB}$ are edges and $\overline{AB}$ is not an edge, then $d(A, B) = 2$. Let $Q$, $R$, and $S$ be randomly chosen distinct vertices of a regular icosahedron (regular polyhedron made up of 20 equilateral triangles). Find the probability that $d(Q, R) > d(R, S)$. The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the range of the function \( y = f^{[2019]}(x) \), where \( f(x) = \log_{0.5} \left(\frac{\sin x}{\sin x + 15}\right) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $\mathbf{u},$ $\mathbf{v},$ and $\mathbf{w}$ be nonzero vectors, no two of which are parallel, such that
\[(\mathbf{u} \times \mathbf{v}) \times \mathbf{w} = \frac{1}{4} \|\mathbf{v}\| \|\mathbf{w}\| \mathbf{u}.\]
Let $\phi$ be the angle between $\mathbf{v}$ and $\mathbf{w}.$  Find $\sin \phi.$
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Context:
Find all real values of the parameter  $a$  for which the system of equations
\[x^4 = yz - x^2 + a,\]
\[y^4 = zx - y^2 + a,\]
\[z^4 = xy - z^2 + a,\]
has at most one real solution.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find three prime numbers whose product is three times their sum.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Express the determinant \( f_n(x) = \left|\begin{array}{c c c c c} 1 & x & \cdots & x^{n - 1} & x^n\\ x & 1 & x & \cdots & x^{n - 1} \\ \vdots & x & \ddots & \ddots & \vdots\\ x^{n - 1} & \vdots & \ddots & 1 & x\\ x^n & x^{n - 1} & \cdots & x & 1 \end{array}\right| \) as a polynomial in terms of \( x \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the cardinality of the set of all sequences of rational numbers on the interval $[0,1]$. Consider sequences as functions from $\mathbb{N}$ to $\mathbb{Q}$. Express your answer in terms of cardinality notation.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of undirected trees with 7 labeled vertices that have exactly 2 leaves, and where vertices 3 and 5 have the same degree.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $X_1, X_2, \ldots, X_n$ be a random sample of size $n \geq 2$ from an exponential distribution with parameter $\lambda$. Given $X_{(1)} = \min(X_1, X_2, \ldots, X_n)$, find the expected value of $X_{(1)}$ conditional on $T = \sum_{i=1}^n X_i$. The cumulative distribution function (CDF) of $X_{(1)}$ is given by:
$$
F_{X_{(1)}}(x) = \begin{cases}
0, & x < 0 \\
1 - e^{-n\lambda x}, & x \geq 0
\end{cases}
$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the double integral \(\iint_{\Omega} (x+y^2) \, dx \, dy\) where \(\Omega\) is the parallelogram defined by the boundaries \(x+y=0\), \(x+y=1\), \(2x-y=0\), and \(2x-y=3\). Use a change of variables to simplify the computation of this integral.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the limit: \[ \lim_{x \to 0 } \frac{\cos{x}- (\cos{x})^{\cos{x}}}{1-\cos{x}+\log{\cos{x}}} \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
求所有的三元正整数组 $(a, b, c)$ 满足 $\frac{a}{2^{a}}=\frac{b}{2^{b}}+\frac{c}{2^{c}}$ 。请给出所有满足条件的三元组的元素乘积之和。
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Context:
Consider the $4 \times 4$ array of $16$ dots, shown below.

Counting the number of squares whose vertices are among the $16$ dots and whose sides are parallel to the sides of the grid, we find that there are nine $1 \times 1$ squares, four $2 \times 2$ squares, and one $3 \times 3$ square, for a total of $14$ squares. We delete a number of these dots. What is the minimum number of dots that must be deleted so that each of the $14$ squares is missing at least one vertex?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider the $12$-sided polygon $ABCDEFGHIJKL$, as shown. Each of its sides has length $4$, and each two consecutive sides form a right angle. Suppose that $\overline{AG}$ and $\overline{CH}$ meet at $M$. What is the area of quadrilateral $ABCM$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
After a football match, the coach lined up the team as shown in the figure and commanded: "Run to the locker room if your number is less than any of your neighbors' numbers." After a few players ran off, he repeated his command. The coach continued this until only one player remained. What is Igor's number, given that after he ran off, there were 3 players left in the line? (After each command, one or more players ran off and the line closed up, leaving no empty spaces between the remaining players.)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the mass percentage of Al in Aluminum sulfide.
Please reason step by step, and put your final answer within \boxed{}.


Context:
在1,2，···，1000中，不出现数码3，且不是3的倍数的数共有＿个．
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the smallest positive real $\alpha$, such that $$\frac{x+y} {2}\geq \alpha\sqrt{xy}+(1 - \alpha)\sqrt{\frac{x^2+y^2}{2}}$$ for all positive reals $x, y$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the standard basis $S = \{(1, 0), (0, 1)\}$ and its dual basis $S^* = \{x_1, x_2\}$ in $\Bbb R^2$, and another basis $B = \{(1, 2), (4, 7)\}$, determine the dual basis $B^*$ expressed in terms of $x_1$ and $x_2$.
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Context:
Find the total number of triples of integers $(x,y,n)$ satisfying the equation $\tfrac 1x+\tfrac 1y=\tfrac1{n^2}$, where $n$ is either $2012$ or $2013$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the sum of all the positive integers which have at most three not necessarily distinct prime factors where the primes come from the set  $\{ 2, 3, 5, 7 \}$ .
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Context:
Let $l$ some line, that is not parallel to the coordinate axes. Find minimal $d$ that always exists point $A$ with integer coordinates, and distance from $A$ to $l$ is  $\leq d$
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Context:
某参观团有2000个人，其中任意4个人中一定有某1个人认识其他3人．问：认识该参观团所有成员的人数最少有多少？
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Context:
Let  $s_1, s_2, s_3$  be the three roots of  $x^3 + x^2 +\frac92x + 9$ .  $$ \prod_{i=1}^{3}(4s^4_i + 81) $$   can be written as  $2^a3^b5^c$ . Find  $a + b + c$ .
Please reason step by step, and put your final answer within \boxed{}.


Context:
The three-dimensional figure shown in the diagram is composed of 10 unit cubes each with a side length of 1. Determine the surface area of this three-dimensional figure.
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Context:
Let $f(z)= \frac{z+a}{z+b}$ and $g(z)=f(f(z))$, where $a$ and $b$ are complex numbers. Suppose that $\left| a \right| = 1$ and $g(g(z))=z$ for all $z$ for which $g(g(z))$ is defined. What is the difference between the largest and smallest possible values of $\left| b \right|$?
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Context:
What is the molecular weight of 5 moles of C6H8O7?
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Context:
Sandy had some pennies and 31 nickels in her bank. Her dad borrowed 20 nickels from Sandy. Now, she has 11 nickels. How many pennies did Sandy have initially?
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Context:
Let  $ABCDE$  be a convex pentagon such that: $\angle ABC=90,\angle BCD=135,\angle DEA=60$  and  $AB=BC=CD=DE$ . Find angle  $\angle DAE$ .
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the smallest positive integer  $ K$  such that every  $ K$ -element subset of  $ \{1,2,...,50 \}$  contains two distinct elements  $ a,b$  such that  $ a\plus{}b$  divides  $ ab$ .
Please reason step by step, and put your final answer within \boxed{}.


Context:
In triangle \( ABC \), angle \( BAC \) is \( 60^{\circ} \), the altitude dropped from vertex \( C \) to side \( AB \) equals \( \sqrt{3} \), and the radius of the circumscribed circle around triangle \( ABC \) is 5. Find the sides of triangle \( ABC \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given circle $C$: $x^{2}+y^{2}+2x+4y+4=0$, and line $l$: $\sin \theta x + \cos \theta y - 4 = 0$, determine the relationship between line $l$ and circle $C$.
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Context:
Find all numbers of the form \(2^{n}\) (where \(n\) is a natural number) such that when the first digit of their decimal representation is removed, the result is also a power of 2.
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Context:
From identical isosceles triangles, each having an angle opposite the base equal to $45^{\circ}$ and a lateral side of length 1, a shape is formed as shown in the figure. Find the distance between points $A$ and $B$.
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Context:
给正六边形每一条边和每一条对角线染色，使任意两条有公共点的线染色不同，至少需- 种颜色.
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Context:
Find the number of positive integer solutions to the equation \( x_1 + x_2 + x_3 + x_4 + x_5 = 100 \) subject to the constraints \( x_1 \le 30 \), \( x_2 \le 40 \), \( x_3 \le 50 \), \( x_4 \le 60 \), and \( x_5 \le 70 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a circle, the number of diameters is $$\frac{1}{2}$$ of the number of radii. \_\_\_\_\_\_．(Judge right or wrong)
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Context:
The diagram shows three squares, \( PQRS \), \( TUVW \), and \( WXYZ \). Angles \( PU V \) and \( QYX \) are \( 62^{\circ} \) and \( 74^{\circ} \) respectively. Find angle \( VWX \). Express your answer in degrees as a whole number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the lengths of two sides of a triangle are 3 and 5, and the length of the third side is a root of the equation $3x^2 - 10x = 8$, then the shape of this triangle is a _______ triangle.
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Context:
Katie has a fair 2019-sided die with sides labeled $1,2, \ldots, 2019$. After each roll, she replaces her $n$-sided die with an $(n+1)$-sided die having the $n$ sides of her previous die and an additional side with the number she just rolled. What is the probability that Katie's $2019^{\text {th }}$ roll is a 2019?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the sets $M=\{x|x^{2}+x-6=0\}$ and $N=\{x|ax-1=0\}$, where $N \subseteq M$, find the value of the real number $a$.
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Context:
Find the zeros of the function f(x) = x^3 - 2x^2 - x + 2. Express your answer as a list of exact values.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the triple integral \( \iiint z \, dz \, dy \, dx \) over the region defined by \( \sqrt{x} + \sqrt{y} + \sqrt{z} \leq 1 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let
\[x^8 + 98x^4 + 1 = p(x) q(x),\]where $p(x)$ and $q(x)$ are monic, non-constant polynomials with integer coefficients.  Find $p(1) + q(1).$
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Context:
One mole of an ideal gas is expanded in such a way that during the process, the pressure of the gas is directly proportional to its volume. During this, the gas is heated by $\Delta T=100{ }^{\circ} \mathrm{C}$. Determine the work done by the gas during this process. The gas constant is $R=8.31$ J/(mol$\cdot$K).
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Context:
Consider a  $ n\times n $  square grid which is divided into  $ n^2 $  unit squares(think of a chess-board). The set of all unit squares intersecting the main diagonal of the square or lying under it is called an  $n$ -staircase. Find the number of ways in which an  $n$ -stair case can be partitioned into several rectangles, with sides along the grid lines, having mutually distinct areas.
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Context:
How many of the factorials from 1! to 100! are divisible by 9?
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Context:
14. Suppose there are 3 distinct green balls, 4 distinct red balls, and 5 distinct blue balls in an urn. The balls are to be grouped into pairs such that the balls in any pair have different colors. How many sets of six pairs can be formed?
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Context:
Is the number of ordered 10-tuples of positive integers \((a_1, a_2, \ldots, a_{10})\) such that \(\frac{1}{a_1} + \frac{1}{a_2} + \cdots + \frac{1}{a_{10}} = 1\) even or odd?
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Context:
Given the vertices of a rectangle are $A(0,0)$, $B(2,0)$, $C(2,1)$, and $D(0,1)$. A particle starts from the midpoint $P_{0}$ of $AB$ and moves in a direction forming an angle $\theta$ with $AB$, reaching a point $P_{1}$ on $BC$. The particle then sequentially reflects to points $P_{2}$ on $CD$, $P_{3}$ on $DA$, and $P_{4}$ on $AB$, with the reflection angle equal to the incidence angle. If $P_{4}$ coincides with $P_{0}$, then find $\tan \theta$. Express your answer as a single fraction or decimal.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find all integers \( n \in \mathbb{Z} \) such that \( n+1 \) divides \( n^2+1 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $ABCDE$ be a pentagon inscribed in a circle such that $AB = CD = 3$, $BC = DE = 10$, and $AE= 14$. The sum of the lengths of all diagonals of $ABCDE$ is equal to $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$ ?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Four regular pentagons with side length 1 and a square were glued together to form a box (see figure). Find the distance between points A and B.
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Context:
当1≤x≤时不等式$x ^ { 2 } - 2 a x + a < 0$恒成立，则实数a的取值范围为 .
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Context:
As shown in the figure, semicircles with diameters AB, AC, and CB are mutually tangent to each other. If CD is perpendicular to AB, calculate the ratio of the area of the shaded region to the area of the circle with radius CD. Express your answer as a simplified ratio in the form a:b.
Please reason step by step, and put your final answer within \boxed{}.


Context:
If an income of 5000 yuan is denoted as +5000 yuan, then -400 represents ____.
Please reason step by step, and put your final answer within \boxed{}.


Context:
One of the factors of $x^4+4$ is:
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Context:
Bob and Meena play a two-person game which is won by the first person to accumulate 10 points. At each turn Bob gains a point with probability of $\frac{1}{3}$ . If he doesn't get a point, then Meena gets a point. Meena is now ahead 9 to 8. What is the probability that Meena will win? Express your answer as a common fraction.
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Context:
In a game of <i>Chomp</i>, two players alternately take bites from a 5-by-7 grid of unit squares. To take a bite, a player chooses one of the remaining squares, then removes ("eats") all squares in the quadrant defined by the left edge (extended upward) and the lower edge (extended rightward) of the chosen square. For example, the bite determined by the shaded square in the diagram would remove the shaded square and the four squares marked by $\times.$ (The squares with two or more dotted edges have been removed form the original board in previous moves.) The object of the game is to make one's opponent take the last bite. The diagram shows one of the many subsets of the set of 35 unit squares that can occur during the game of Chomp. How many different subsets are there in all? Include the full board and empty board in your count.
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Context:
About pentagon  $ABCDE$  is known that angle  $A$  and angle  $C$  are right and that the sides  $| AB | = 4$ ,  $| BC | = 5$ ,  $| CD | = 10$ ,  $| DE | = 6$ . Furthermore, the point  $C'$  that appears by mirroring  $C$  in the line  $BD$ , lies on the line segment  $AE$ . Find angle  $E$ .
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Context:
The number $201212200619$ has a factor $m$ such that $6 \cdot 10^9 < m < 6.5 \cdot 10^9$. Find $m$.
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Context:
Given a right isosceles triangle $ABC$ with hypotenuse $AB$. Point $M$ is the midpoint of side $BC$. Point $K$ is chosen on the smaller arc $AC$ of the circumcircle of triangle $ABC$. Point $H$ is the foot of the perpendicular dropped from $K$ to line $AB$. Find the angle $\angle CAK$, given that $KH = BM$ and lines $MH$ and $CK$ are parallel.
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Context:
Two players play a game on a pile of  $n$  beans. On each player's turn, they may take exactly  $1$ ,  $4$ , or  $7$  beans from the pile. One player goes first, and then the players alternate until somebody wins. A player wins when they take the last bean from the pile. For how many  $n$  between  $2014$  and  $2050$  (inclusive) does the second player win?
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Context:
Let $a,$ $b,$ $c$ be the roots of
\[x^3 - 6x^2 + 3x + 1 = 0.\]Find all possible values of $a^2 b + b^2 c + c^2 a.$  Enter all possible values, separated by commas.
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Context:
Let $\underline{xyz}$ represent the three-digit number with hundreds digit $x$, tens digit $y$, and units digit $z$, and similarly let $\underline{yz}$ represent the two-digit number with tens digit $y$ and units digit $z$. How many three-digit numbers $\underline{abc}$, none of whose digits are 0, are there such that $\underline{ab} > \underline{bc} > \underline{ca}$?
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Context:
Behind a thin lens with a focal length \( F \) and a diameter \( D \), a flat screen is placed perpendicularly to its optical axis at its focal point. A point light source is placed on the main optical axis at a distance \( d > F \) from the lens. Determine the diameter of the light spot on the screen.
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Context:
A river has a right-angle bend. Except at the bend, its banks are parallel lines of distance  $a$  apart. At the bend the river forms a square with the river flowing in across one side and out across an adjacent side. What is the longest boat of length  $c$  and negligible width which can pass through the bend?
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Context:
Evaluate the integral \( \int_{-1}^{1} \exp(x+e^{x})\,dx \), where \( \exp(x)=e^x \).
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Context:
Four balls of radius $1$ are mutually tangent, three resting on the floor and the fourth resting on the others.
A tetrahedron, each of whose edges have length $s$, is circumscribed around the balls. Then $s$ equals
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Context:
Calculate the limit: $$\lim_{n\to\infty}n\sin(5^n \pi \sqrt{2}\mathrm n!)$$
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Context:
Determine the values of \( \alpha \in \mathbb{R} \) for which the function \( g(x) = \frac{\ln(x)}{x^{\alpha}(x-1)} \) is integrable over the interval \((0,1)\).
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Context:
Two trains 121 meters and 165 meters in length respectively are running in opposite directions, one at the rate of 80 km and the other at the rate of 55 kmph. In what time will they be completely clear of each other from the moment they meet?
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Context:
Let $\alpha = \cos^{-1} \left( \frac{3}{5} \right)$ and $\beta = \sin^{-1} \left( \frac{3}{5} \right)$.
\[ \sum_{n=0}^{\infty}\sum_{m=0}^{\infty} \frac{\cos(\alpha n +\beta m)}{2^n3^m} \]
can be written as $\frac{A}{B}$ for relatively prime positive integers $A$ and $B$. Find $1000A + B$.
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Context:
Derek the Dolphin and Kevin the Frog are playing a game where they take turns taking coins from a stack of $N$ coins, except with one rule: The number of coins someone takes each turn must be a power of $6$. The person who cannot take any more coins loses. If Derek goes first, how many integers $N$ from $1$ to $6^{2021}$ inclusive will guarantee him a win? (Example: If $N = 37$, then a possible sequence of turns is: Derek takes one coin, Kevin takes $36$ coins, and Kevin wins.)
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Context:
A strip is the region between two parallel lines. Let $A$ and $B$ be two strips in a plane. The intersection of strips $A$ and $B$ forms a parallelogram $P$. If strip $A$ is rotated by $60^\circ$ to form strip $A'$, the intersection of strips $A'$ and $B$ also forms a parallelogram with the same area as $P$. Determine the greatest possible value of one interior angle of parallelogram $P$ in degrees.
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Context:
$ABC$  is triangle.  $l_1$ - line passes through  $A$  and parallel to  $BC$ ,  $l_2$  -  line passes through  $C$  and parallel to  $AB$ . Bisector of  $\angle B$  intersect  $l_1$  and  $l_2$  at  $X,Y$ .  $XY=AC$ . What  value can take  $\angle A- \angle C$  ?
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Context:
Find the sum of the infinite series: $$\sum_{n=1}^{\infty}\left\lfloor\frac{3^{20}+2^{n-1}}{2^n}\right\rfloor$$ Given that you've identified a finite support for the sequence, how can you determine the sum?
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Context:
Determine the smallest real value \( C \) such that for all functions \( f: [0,1] \to \mathbb{R} \) in the set \( M \), the inequality \( f(x) \le Cx \) holds for all \( x \in [0,1] \). The set \( M \) consists of functions satisfying the following properties:

- \( f(x) \ge 0 \) for all \( x \in [0,1] \),
- \( f(1) = 1 \),
- \( f(x+y) \ge f(x) + f(y) \) for all \( x, y \) such that \( x+y \in [0,1] \).
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Context:
Compute the limit \( \lim_{r\to 0}\frac{f(z+re^{i\alpha})-f(z)}{re^{i\alpha}} \) where \( f(z)=\bar{z} \).
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Context:
Find the derivative of the function \( g: M(2, \mathbb{R}) \to M(2, \mathbb{R}) \) defined by \( g(A) = A^2 \), where \( M(2, \mathbb{R}) \) is the vector space of \( 2 \times 2 \) real matrices.
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Context:
Determine the number of rotational symmetries of the 120-cell.
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Context:
Find the cardinality of the set \( B = \{ g \in C[0, 1] : g(0) = 0, g(1) = 1, |g'(t)| \geq 1 \text{ for all } t \in [0, 1] \} \).
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Context:
Let \( f: (0,\infty )\rightarrow \mathbb{R} \) and \( F(x) = \int_{0}^{x}tf(t)dt \). If \( F(x^2)= x^{4} + x^{5} \), find the value of \( \sum_{r=1}^{12}f(r^{2}) \).
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Context:
2019 numbers are arranged in a circle. For any two neighboring numbers \( x \) and \( y \), the inequalities \( |x - y| \geq 2 \) and \( x + y \geq 6 \) hold. Determine the smallest possible sum of the numbers arranged in the circle.
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Context:
Evaluate the integral of the function \( f(x,y) = x^2 + y^2 \) over the region \( R \) bounded by the curves \( y = x^2 \), \( y = x^2 + 4 \), \( x = y \), and \( x = y + 2 \).
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Context:
Ana, Bob, and Cao bike at constant rates of $8.6$ meters per second, $6.2$ meters per second, and $5$ meters per second, respectively. They all begin biking at the same time from the northeast corner of a rectangular field whose longer side runs due west. Ana starts biking along the edge of the field, initially heading west, Bob starts biking along the edge of the field, initially heading south, and Cao bikes in a straight line across the field to a point $D$ on the south edge of the field. Cao arrives at point $D$ at the same time that Ana and Bob arrive at $D$ for the first time. The ratio of the field's length to the field's width to the distance from point $D$ to the southeast corner of the field can be represented as $p : q : r$, where $p$, $q$, and $r$ are positive integers with $p$ and $q$ relatively prime. Find $p+q+r$.
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Context:
Evaluate the expression \(\frac{I}{J}\), where:
\[I=\int_{0}^{1}\frac{x^{\frac{5}{2}}(1-x)^{\frac{7}{2}}}{12}\,dx\]
and
\[J=\int_{0}^{1}\frac{x^{\frac{5}{2}}(1-x)^{\frac{7}{2}}}{(x+3)^8}\,dx\]
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Context:
Find the smallest positive integer \( n \) such that the symmetric group \( S_n \) has a subgroup of order \( p^2 \), where \( p \) is a prime number.
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Context:
Each face of a cube is given a single narrow stripe painted from the center of one edge to the center of the opposite edge.  The choice of the edge pairing is made at random and independently for each face.  What is the probability that there is a continuous stripe encircling the cube?
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Context:
Determine the number of independent parameters required to specify all possible conditional probabilities for a set of $n$ events $X_1, X_2, ..., X_n$. Consider all probabilities such as $P(X_i)$, $P(X_i, X_j)$ for $i \neq j$, $P(X_i | X_j)$ for $i \neq j$, $P(X_i, X_j, X_k)$ for $i \neq j \neq k$, $P(X_i | X_j, X_k)$ for $i \neq j \neq k$, and $P(X_i, X_j | X_k)$ for $i \neq j \neq k$. How many of these probabilities must be specified to determine all others using the standard laws of probability?
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Context:
If a matrix $X$ has $m$ rows and $n$ columns and is row orthogonal with rank $m$, what is the minimum number of columns that a new matrix $Y$ must have to ensure that it also has rank $m$, assuming that the columns of $Y$ are sampled from $X$?
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Context:
设 $\alpha, \beta$ 为实数, 若对任意实数 $x, y, z$. 有 $a(x y+y z+z x) \leq M \leq \beta\left(x^{2}+y^{2}+z^{2}\right)$ 恒成立, 其中 $M=\sqrt{x^{2}+x y+y^{2}} \cdot \sqrt{y^{2}+y z+z^{2}}+\sqrt{y^{2}+y z+z^{2}} \cdot \sqrt{z^{2}+z x+x^{2}}+\sqrt{z^{2}+z x+x^{2}} \cdot \sqrt{x^{2}+x y+y^{2}}$.求 $\alpha$ 的最大值和 $\beta$ 的最小值.
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Context:
Josh takes a walk on a rectangular grid of \( n \) rows and 3 columns, starting from the bottom left corner. At each step, he can either move one square to the right or simultaneously move one square to the left and one square up. In how many ways can he reach the center square of the topmost row?
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Context:
The polynomial  $f(x)=x^{2007}+17x^{2006}+1$  has distinct zeroes  $r_1,\ldots,r_{2007}$ .  A polynomial  $P$  of degree  $2007$  has the property that  $P\left(r_j+\dfrac{1}{r_j}\right)=0$  for  $j=1,\ldots,2007$ .  Determine the value of  $P(1)/P(-1)$ .
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Context:
Evaluate the limit: $$\lim_{n\to\infty} \frac{\int_0^{\pi/4}(\tan x)^{2n}dx}{\int_0^{\pi/4}(\tan x)^{2n+1}dx}$$
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Context:
How many of the letters in MATHCOUNTS have a horizontal line of symmetry?
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Context:
Let  $k$  be a fixed positive integer. The  $n$ th derivative of  $\tfrac{1}{x^k-1}$  has the form  $\tfrac{P_n(x)}{(x^k-1)^{n+1}}$ , where  $P_n(x)$  is a polynomial. Find  $P_n(1)$ .
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Context:
For a positive integer n, let  $w(n)$  denote the number of distinct prime
divisors of n. Determine the least positive integer k such that
 $2^{w(n)} \leq k \sqrt[4]{n}$
for all positive integers n.
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Context:
Consider the set  $S$  of permutations of  $1, 2, \dots, 2022$  such that for all numbers  $k$  in the
permutation, the number of numbers less than  $k$  that follow  $k$  is even.

For example, for  $n=4; S = \{[3,4,1,2]; [3,1,2,4]; [1,2,3,4]; [1,4,2,3]\}$ If  $|S| = (a!)^b$  where  $a, b \in \mathbb{N}$ , then find the product  $ab$ .
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Context:
Determine the Krull dimension of the injective hull $E = E_R(R/\mathfrak{m})$, where $(R, \mathfrak{m})$ is a Noetherian local ring.
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Context:
How many subsets of {1,2,3,4,5,6,7,8,9,10,11,12} have the property that no two of its elements diﬀer by more than 5? For example, count the sets {3}, {2,5,7}, and {5,6,7,8,9} but not the set {1,3,5,7}.
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Context:
Evaluate the limit: $$\lim_{x\to0}\frac{x\sin{(\sin{x})}-\sin^2{x}}{x^6}$$
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Context:
Find the area of the part of the cylinder \(x^2+y^2=2ay\) that lies inside the sphere \(x^2+y^2+z^2=4a^2\).
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Context:
In the triangle \(ABC\), it is known that \(AB=BC\) and \(\angle BAC=45^\circ\). The line \(MN\) intersects side \(AC\) at point \(M\), and side \(BC\) at point \(N\). Given that \(AM=2 \cdot MC\) and \(\angle NMC=60^\circ\), find the ratio of the area of triangle \(MNC\) to the area of quadrilateral \(ABNM\).
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Context:
Determine the Galois group of the polynomial \( f(X) = X^4 + 2X^2 + 4 \) over \( \mathbb{Q} \).
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Context:
Two rectangles, each measuring 7 cm in length and 3 cm in width, overlap to form the shape shown on the right. What is the perimeter of this shape in centimeters?
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Context:
Find the coefficient of \(x^8\) in the expansion of \((1-2x+3x^2-4x^3+5x^4-6x^5+7x^6)^6\).
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Context:
Find the mass percentage of Cl in Chlorous acid
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Context:
Given a positive integer $n$, it can be shown that every complex number of the form $r+si$, where $r$ and $s$ are integers, can be uniquely expressed in the base $-n+i$ using the integers $0,1,2,\ldots,n^2$ as digits. That is, the equation
$r+si=a_m(-n+i)^m+a_{m-1}(-n+i)^{m-1}+\cdots +a_1(-n+i)+a_0$
 is true for a unique choice of non-negative integer $m$ and digits $a_0,a_1,\ldots,a_m$ chosen from the set $\{0,1,2,\ldots,n^2\}$, with $a_m\ne 0$. We write
$r+si=(a_ma_{m-1}\ldots a_1a_0)_{-n+i}$
 to denote the base $-n+i$ expansion of $r+si$. There are only finitely many integers $k+0i$ that have four-digit expansions
$k=(a_3a_2a_1a_0)_{-3+i}~~$
$~~a_3\ne 0.$
 Find the sum of all such $k$,
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Context:
Denis painted some of the faces of his cubes gray. Vova selected 10 cubes so that all of them had different patterns of painting and then stacked them as shown in the figure. How many white faces does the top cube have?
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Context:
Find the value of $n$ that satisfies the equation: $$\frac{_{365}P_n}{365^n} \approx \frac{1}{5}.$$
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Context:
A table  $110\times 110$  is given, we define the distance between two cells  $A$  and  $B$  as the least quantity of moves to move a chess king from the cell  $A$  to cell  $B$ . We marked  $n$  cells on the table  $110\times 110$  such that the distance between any two cells is not equal to  $15$ . Determine the greatest value of  $n$ .
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Context:
Let $\triangle A_0B_0C_0$ be a triangle whose angle measures are exactly $59.999^\circ$, $60^\circ$, and $60.001^\circ$. For each positive integer $n$, define $A_n$ to be the foot of the altitude from $A_{n-1}$ to line $B_{n-1}C_{n-1}$. Likewise, define $B_n$ to be the foot of the altitude from $B_{n-1}$ to line $A_{n-1}C_{n-1}$, and $C_n$ to be the foot of the altitude from $C_{n-1}$ to line $A_{n-1}B_{n-1}$. Find the least positive integer $n$ for which $\triangle A_nB_nC_n$ is obtuse.
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Context:
Let be given a triangle  $ABC$  and points  $D,M,N$  belong to  $BC,AB,AC$ , respectively. Suppose that  $MD$  is parallel to  $AC$  and  $ND$  is parallel to  $AB$ . If  $S_{\vartriangle BMD} = 9$  cm  $^2, S_{\vartriangle DNC} = 25$  cm $^2$ , compute  $S_{\vartriangle AMN}$ ?
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Context:
In $\triangle ABC$, $BD$ is a median. $CF$ intersects $BD$ at $E$ so that $\overline{BE}=\overline{ED}$. Point $F$ is on $AB$. Then, if $\overline{BF}=5$, $\overline{BA}$ equals:
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Context:
Let  $u, v$  be real numbers. The minimum value of  $\sqrt{u^2+v^2} +\sqrt{(u-1)^2+v^2}+\sqrt {u^2+ (v-1)^2}+ \sqrt{(u-1)^2+(v-1)^2}$  can be written as  $\sqrt{n}$ . Find the value of  $10n$ .
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Context:
In the diagram, each scale shows the total mass (weight) of the shapes on that scale. Using the information provided by the scales, calculate the mass (weight) of a triangle. Express your answer as a single number.
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Context:
Determine the number of loops in the curve defined by \(y^4 = b^2(y^2 - x^2)\).
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Context:
Find the smallest positive number $\lambda $ , such that for any complex numbers ${z_1},{z_2},{z_3}\in\{z\in C\big| |z|<1\}$ ,if  $z_1+z_2+z_3=0$, then $$\left|z_1z_2 +z_2z_3+z_3z_1\right|^2+\left|z_1z_2z_3\right|^2 <\lambda .$$
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Context:
Given a linearly independent set of functionals in the dual space of a vector space \( V \), determine the nullity of a linear transformation from \( V \) to \( \mathbb{R}^m \).
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Context:
An isosceles right triangle has a leg length of 36 units. Starting from the right angle vertex, an infinite series of equilateral triangles is drawn consecutively on one of the legs. Each equilateral triangle is inscribed such that their third vertices always lie on the hypotenuse, and the opposite sides of these vertices fill the leg. Determine the sum of the areas of these equilateral triangles.
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Context:
In the diagram, \( AE \) and \( BD \) are straight lines that intersect at \( C \). Given that \( BD = 16 \), \( AB = 9 \), \( CE = 5 \), and \( DE = 3 \), calculate the length of \( AC \). Express your answer as a whole number.
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Context:
Determine the number of non-trivial ring homomorphisms from the ring of integers modulo 20, $\mathbb{Z}_{20}$, to the ring of integers modulo 44, $\mathbb{Z}_{44}$. Provide your answer as a single integer.
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Context:
Solve the recurrence relation: \[ T(n) = 2T(n - 1) + (\log n)T([\log n]) + 1 \]
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Context:
Evaluate whether the rising factorial $0^{\bar{0}}$ is equal to 1 using the relationship $a^{\bar{n}}=\frac{\Gamma(a+n)}{\Gamma(a)}$. Consider the case when $a=n=0$.
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Context:
Evaluate the limit by expressing it as a definite integral: $$\lim_{n\to\infty}\frac{\pi}{2n}\left[\cos\left(\frac{\pi}{2n}\right)+\cos\left(\frac{\pi}{n}\right)+\cos\left(\frac{3\pi}{2n}\right)+\cdots+\cos\left(\frac{(n-1)\pi}{2n}\right)\right]$$
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Context:
Evaluate the limit: $$\lim_{{x \to 0^+}} \frac{\log(x+\sqrt{1+x^2})-\sin x}{(2^{x^2}-1)(\sin(e^{x^2}-1))}.$$
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Context:
There are $n$ mathematicians seated around a circular table with $n$ seats numbered $1,$ $2,$ $3,$ $...,$ $n$ in clockwise order. After a break they again sit around the table. The mathematicians note that there is a positive integer $a$ such that
 ($1$) for each $k,$ the mathematician who was seated in seat $k$ before the break is seated in seat $ka$ after the break (where seat $i + n$ is seat $i$);
 ($2$) for every pair of mathematicians, the number of mathematicians sitting between them after the break, counting in both the clockwise and the counterclockwise directions, is different from either of the number of mathematicians sitting between them before the break.
 Find the number of possible values of $n$ with $1 < n < 1000.$
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Context:
Find the annihilator operator of the function \( f(x) = x(\sin x + 3x - x^{-1}) - \cos x \), where the operator is \( \frac{d}{dt} \equiv D \).
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Context:
Determine the complexity of the function \( f(n) = f(7n/15) + f(8n/15) + n^2 \) for all \( n \geq 0 \) using the Akra-Bazzi theorem.
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Context:
The new PUMaC tournament hosts  $2020$  students, numbered by the following set of labels  $1, 2, . . . , 2020$ . The students are initially divided up into  $20$  groups of  $101$ , with each division into groups equally likely. In each of the groups, the contestant with the lowest label wins, and the winners advance to the second round. Out of these  $20$  students, we chose the champion uniformly at random. If the expected value of champion’s number can be written as  $\frac{a}{b}$ , where  $a, b$  are relatively prime integers, determine  $a + b$ .
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Context:
Cynthia loves Pokemon and she wants to catch them all. In Victory Road, there are a total of  $80$  Pokemon. Cynthia wants to catch as many of them as possible. However, she cannot catch any two Pokemon that are enemies with each other. After exploring around for a while, she makes the following two observations:
1. Every Pokemon in Victory Road is enemies with exactly two other Pokemon.
2. Due to her inability to catch Pokemon that are enemies with one another, the maximum number of the Pokemon she can catch is equal to  $n$ .
What is the sum of all possible values of  $n$ ?
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Context:
Evaluate the infinite series \( \sum_{r=1}^{\infty} \frac{r^3 + (r^2 + 1)^2}{(r^4 + r^2 + 1)(r^2 + r)} \).
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Context:
Given an element \( v = (v_1, \dots, v_d) \in \mathbb{Z}^d \) with \( \gcd(v_1, \dots, v_d) = 1 \), there exists a regular integer matrix \( A \in \mathop{GL}_d(\mathbb{Z}) \) such that \( A(v) = (1, 0, \dots, 0) \). Assume \( |v_1| + \dots + |v_d| = n \). Determine the smallest \( \ell \in \mathbb{N} \) such that the image \( A\left([-1, 1]^d\right) \) is contained in the cube \( [-\ell, \ell]^d \). Find the asymptotic behavior of \( \ell(n) \) up to a multiplicative constant.
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Context:
Let $A B C$ be a triangle where $A B=9, B C=10, C A=17$. Let $\Omega$ be its circumcircle, and let $A_{1}, B_{1}, C_{1}$ be the diametrically opposite points from $A, B, C$, respectively, on $\Omega$. Find the area of the convex hexagon with the vertices $A, B, C, A_{1}, B_{1}, C_{1}$.
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Context:
Evaluate the triple integral $\iiint_S {z{\sqrt{x^2+y^2}}}\,dx\,dy\,dz$ over the region $S$ bounded by $x^2+y^2=1$, $z=0$, and $z=x$ with $z \ge 0$.
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Context:
If \( a < b \), then which inequality sign ( \( > \) or \( < \) ) should be placed between \( \frac{1}{a} \) and \( \frac{1}{b} \) ?
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Context:
有 20 个队参加全国足球冠军决赛，为了使任何三个队中都有两个队相互比赛过，问最少要进行多少场比赛?
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Context:
Which of the following numbers is an odd integer, contains the digit 5, is divisible by 3, and lies between \(12^{2}\) and \(13^{2}\)?
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Context:
How can we solve the integral $$\int \limits_0^{2 \pi}e^{\cos t} \cos(nt - \sin t) \mbox{d}t$$?
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Context:
Let $ F_0=\ln x.$ For $ n\ge 0$ and $ x>0,$ let $ \displaystyle F_{n+1}(x)=\int_0^xF_n(t)\,dt.$ Evaluate $ \displaystyle\lim_{n\to\infty}\frac{n!F_n(1)}{\ln n}.$
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Context:
Given the change of basis matrix \( S = \begin{pmatrix} p & q \\ 1 & 1 \end{pmatrix} \), the vector \( x = (8, 3) \) transforms to \( x' = (1, 2) \) and the vector \( y = (5, 2) \) transforms to \( y' = (1, 1) \). Determine the new coordinates of the vector \( z = (-1, 0) \) after the change of basis.
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Context:
Calculate the derivative of the conjugate of a complex number $c$ with respect to $c$ itself. Specifically, find $\frac{\partial (c^*)}{\partial c}$, where $c \in \mathbb{C}$.
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Context:
Given that the moment generating function of a random variable $X$ is $M_X(u) = e^{5u + 2u^2}$, determine the probability that the stock price $S = e^X$ is between $70$ and $90$, given that $S > 50$. Assume $X$ follows a normal distribution with mean $\mu = 5$ and variance $\sigma^2 = 4$. Provide your answer as a probability.
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Context:
Five computers are connected in a network. For each pair of computers, a coin is flipped. If it is heads, a direct link is established between the two computers; if it is tails, no link is established. A message received by a computer is automatically forwarded to all computers directly linked to it. Calculate the probability that every computer can receive messages from every other computer.
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Context:
Steve wrote the digits $1$, $2$, $3$, $4$, and $5$ in order repeatedly from left to right, forming a list of $10,000$ digits, beginning $123451234512\ldots.$ He then erased every third digit from his list (that is, the $3$rd, $6$th, $9$th, $\ldots$ digits from the left), then erased every fourth digit from the resulting list (that is, the $4$th, $8$th, $12$th, $\ldots$ digits from the left in what remained), and then erased every fifth digit from what remained at that point. Find the sum of the three digits that were then in the positions $2019, 2020, 2021$.
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Context:
Evaluate the line integral \( \int_{C} (x^{4}+5y^{2})\,dx + (xy-y^{5})\,dy \), where \( C \) is the polygonal path joining the vertices \( (-1, 0), (4, 0), (6, 4), (4, 8), (-1, 8), \) and \( (-1, 0) \), oriented counterclockwise.
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Context:
Each of the points $A,B,C,D,E,$ and $F$ in the figure below represents a different digit from $1$ to $6.$ Each of the five lines shown passes through some of these points. The digits along each line are added to produce five sums, one for each line. The total of the five sums is $47.$ What is the digit represented by $B?$
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Context:
How many games must be played between two opponents to have 90% confidence that the overall win percentage approaches a single number?
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Context:
Given the expression \( y = \sin^{-1}(\sin 8) - \tan^{-1}(\tan 10) + \cos^{-1}(\cos 12) - \sec^{-1}(\sec 9) + \cot^{-1}(\cot 6) - \csc^{-1}(\csc 7) \), find the value of \( a - b \) if \( y \) simplifies to \( a\pi + b \).
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Context:
Circle  $\omega$  has radius  $5$  and is centered at  $O$ .  Point  $A$  lies outside  $\omega$  such that  $OA=13$ .  The two tangents to  $\omega$  passing through  $A$  are drawn, and points  $B$  and  $C$  are chosen on them (one on each tangent), such that line  $BC$  is tangent to  $\omega$  and  $\omega$  lies outside triangle  $ABC$ .  Compute  $AB+AC$  given that  $BC=7$ .
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Context:
Determine the number of functions \( g: \{1, 2, \cdots, 10\} \rightarrow \{1, 2, \cdots, 10\} \) such that if \((x, y, z)\) satisfies \(xy = z\), then \((g(x), g(y), g(z))\) also satisfies this equation.
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Context:
Nine numbers  $a, b, c, \dots$  are arranged around a circle. All numbers of the form  $a+b^c, \dots$  are prime. What is the largest possible number of different numbers among  $a, b, c, \dots$ ?
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Context:
Arrange $n+1$ squares with a side length of 1 as shown in figure 4. Points $A, A_1, A_2, \cdots, A_{n+1}, M_0, M_1, M_2, \cdots, M_n$ are vertices of the squares. Connect $A M_i \ (i=1, 2, \cdots, n)$ and intersect with side $A_i M_{i-1}$ at point $N_i$. Let the area of quadrilateral $M_i N_i A_i A_{i+1}$ be $S_i \ (i=1,2, \cdots, n)$. What is $S_{2017}$?
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Context:
A square is drawn in the Cartesian coordinate plane with vertices at \((2, 2)\), \((-2, 2)\), \((-2, -2)\), \((2, -2)\). A particle starts at \((0,0)\). Every second it moves with equal probability to one of the eight lattice points (points with integer coordinates) closest to its current position, independently of its previous moves. In other words, the probability is \(1/8\) that the particle will move from \((x, y)\) to each of \((x, y + 1)\), \((x + 1, y + 1)\), \((x + 1, y)\), \((x + 1, y - 1)\), \((x, y - 1)\), \((x - 1, y - 1)\), \((x - 1, y)\), or \((x - 1, y + 1)\). The particle will eventually hit the square for the first time, either at one of the 4 corners of the square or at one of the 12 lattice points in the interior of one of the sides of the square. The probability that it will hit at a corner rather than at an interior point of a side is \(m/n\), where \(m\) and \(n\) are relatively prime positive integers. What is \(m + n\)?
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Context:
Let $k$ be a positive integer with the following property: For every subset $A$ of $\{1,2,\ldots, 25\}$ with $|A|=k$, we can find distinct elements $x$ and $y$ of $A$ such that $\tfrac23\leq\tfrac xy\leq\tfrac 32$.  Find the smallest possible value of $k$.
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Context:
Find all twice differentiable functions \( f\left( x\right) \) such that \( {f}^{\prime \prime }\left( x\right)  = 0,f\left( 0\right)  = {19} \) ,and \( f\left( 1\right)  = {99} \) .
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Context:
The denominator of the fraction $15 \cdot 18$ in simplest form is 30. Find the sum of all such positive rational numbers less than 10.
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Context:
On a circle, 40 red points and one blue point are marked. All possible polygons with vertices at the marked points are considered. Which polygons are more numerous, those with a blue vertex, or those without it, and by how many?
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Context:
The product of four consecutive integers is a perfect square. What are these integers?
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Context:
How many functions \( f : \mathbb{N} \rightarrow  \mathbb{N} \) are there such that \( f\left( 0\right)  = {2011},f\left( 1\right)  = {111} \) ,and \( f\left( {\max \{ x + y + 2,{xy}\} }\right)  = \min \{ f\left( {x + y}\right) ,f\left( {{xy} + 2}\right) \} \)for all non-negative integers \( x,y \) ?
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Context:
Ali defines a pronunciation of any sequence of English letters to be a partition of those letters into substrings such that each
 substring contains at least one vowel. For example, \( \Lambda \left| {\mathrm{{THEN}} \mid  \Lambda ,\Lambda \mathrm{{TH}} \mid  \mathrm{E} \mid  \mathrm{{NA}},\text{ ATHENA,and AT }}\right| \mathrm{{HEN}} \mid  \Lambda \) are all
 pronunciations of the sequence ATHENA. How many distinct pronunciations does YALEMATHCOMP have? (Y is not a
 vowel.)
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Context:
Find the smallest integer \( n \) such that there exists a cyclic binary code of length \( n \) with the generator polynomial \( g(x) = 1 + x^4 + x^5 \).
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Context:
Solve for $X$ and $Y$ such that
\[\frac{Yx + 8}{x^2 - 11x + 30} = \frac{X}{x - 5} + \frac{7}{x - 6}\].
Find $X+Y$.
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Context:
Evaluate the surface integral \( \iint_S \mathbf{F} \cdot \mathbf{n} \, dS \) for the surface \( S \) defined by the boundary of the triangle with vertices \((1,0,0)\), \((0,1,0)\), and \((0,0,1)\). The vector field is \( \mathbf{F}(x,y,z) = x\mathbf{i} + y\mathbf{j} + z\mathbf{k} \), and \( \mathbf{n} \) is the unit normal to \( S \) with a non-negative \( z \)-component. Use the parameterization \( \mathbf{r}(u,v) = (u+v)\mathbf{i} + (u-v)\mathbf{j} + (1-2u)\mathbf{k} \), where \( u \) and \( v \) are parameters.
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Context:
Determine the sub-interval of \([0,1]\) on which the sequence of functions \(f_n(x) = x^{1/n}\) converges uniformly to the constant function 1.
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Context:
Points \( A \), \( B \), \( C \), \( D \), and \( E \) are located in 3-dimensional space with \( AB = BC = CD = DE = 3 \) and \( \angle ABC = \angle CDE = 60^\circ \). Additionally, the plane of triangle \( ABC \) is perpendicular to line \( \overline{DE} \). Determine the area of triangle \( BDE \).
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Context:
In an old estate, the house is surrounded by tall trees arranged in a circle, including spruces, pines, and birches. There are 96 trees in total. These trees have a peculiar property: for any coniferous tree, among the two trees that are two trees away from it, one is coniferous and the other is deciduous; also, among the two trees that are three trees away from any coniferous tree, one is coniferous and the other is deciduous. How many birches have been planted around the house?
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Context:
Let $n(S)$ denote the number of subsets of a set $S$. Given that $n(X) + n(Y) + n(Z) = n(X \cup Y \cup Z)$ and $|X| = |Y| = 200$, find the minimum possible value of $|X \cap Y \cap Z|$. Provide your answer as a single integer.
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Context:
设函数 $f(x)$ 满足: 对任意非零实数 $x$, 均有 $f(x)=f(1) \cdot x+\frac{f(2)}{x}-1$, 则 $f(x)$ 在 $(0,+\infty)$ 上的最小值为原始答案为\sqrt{3}-1的形式，请给出m + n的值。
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Context:
Find the area of one petal of the rose curve given by \( r = |\cos(2\theta)|^{1/2} \).
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Context:
The CMU Kiltie Band is attempting to crash a helicopter using a grappling hook. The helicopter starts parallel to the ground at an angle of $0$ degrees. Each time the band members pull the hook, they tilt the helicopter forward by either $x$ or $x+1$ degrees, with equal probability, if the helicopter is currently at an angle $x$ degrees with the ground. Determine the expected number of pulls required to tilt the helicopter to $90$ degrees or beyond, causing it to crash.
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Context:
Consider the Banach space $c_0$ with the supremum norm. Let $F$ be a subset of $c_0$ defined as:
$$F=\{x=(x_n)\in c_0:\sum_{n=1}^\infty 3^{-n}x_n=1\}.$$ Find the distance $d(F,r)$, where $r=(2,0,0,0,...)$. Provide your answer as a single number.
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Context:
Let \( \{ a_n \} \) be a sequence of non-negative real numbers such that the series \( \sum_{n=1}^{\infty} a_n \) is convergent. Determine the range of the real number \( q \) for which the series \( \sum \frac{a_n}{n^q} \) also converges.
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Context:
Find the smallest constant M, so that for any real numbers \( {a}_{1},{a}_{2},\ldots {a}_{2023} \in  \left\lbrack  {4,6}\right\rbrack \) and
 \( {b}_{1},{b}_{2},\ldots {b}_{2023} \in  \left\lbrack  {9,{12}}\right\rbrack \) following inequality holds:
 \( \sqrt{{a}_{1}^{2} + {a}_{2}^{2} + \cdots  + {a}_{2023}^{2}} \cdot  \sqrt{{b}_{1}^{2} + {b}_{2}^{2} + \cdots  + {b}_{2023}^{2}} \leq  M \cdot  \left( {{a}_{1}{b}_{1} + {a}_{2}{b}_{2} + \cdots  + {a}_{2023}{b}_{2023}}\right) \)
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Context:
Determine the number of functions \( f: A \to A \), where \( A = \{1, 2, 3, 4, 5, 6, 7\} \), such that \( f(f(f(x))) = x \) for all \( x \in A \).
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Context:
In the center of a circular field, there is a house of geologists. Eight straight roads emanate from the house, dividing the field into 8 equal sectors. Two geologists set off on a journey from their house at a speed of 4 km/h choosing any road randomly. Determine the probability that the distance between them after an hour will be more than 6 km.
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Context:
Consider the sequence \(a_{1}, a_{2}, a_{3}, \cdots, a_{21}\) such that \(\left|a_{n+1}-a_{n}\right| = 1\) for \(n = 1, 2, 3, \cdots, 20\). Additionally, \(a_{1}, a_{7}, a_{21}\) form a geometric sequence. Given that \(a_{1}=1\) and \(a_{21}=9\), determine the number of distinct sequences that satisfy these conditions.
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Context:
Evaluate the limit of the fractional derivative of the softplus activation function at $x=0$ and $a=0.9$. What is the value of this limit?
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Context:
In a rectangular parallelepiped \( ABCD A_{1} B_{1} C_{1} D_{1} \) with edges \( AB=4 \), \( AD=2 \), \( AA_{1}=3\sqrt{2} \), a plane passes through the diagonal \( BD_{1} \) and intersects the edge \( AA_{1} \) such that the cross-section of the parallelepiped with this plane has the smallest perimeter. Find the area of this cross-section.
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Context:
Let $X_{1}, X_{2}, \ldots, X_{n}, \ldots$ be independent and identically distributed random variables with distribution $P(X_{n}=0)=P(X_{n}=1)=P(X_{n}=2)=P(X_{n}=3)=\frac{1}{4}$. Assume that the sequence $\{Y_n\}$ is defined as: $Y_{0}=0$ and for all $n\in\mathbb{N}$ we have $Y_n=\begin{cases} 3 &\text{if } X_n=3,\\\min{\{Y_{n-1},X_n\}} &\text{if } X_n<3. \end{cases} $ Compute $\displaystyle \lim_{n\to +\infty}E[Y_{n}Y_{n-1}]$. \(\text{Express your answer as a single number.}\)
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Context:
Determine all values of \( \alpha \) for which some non-trivial solutions of the differential equation \( y'' + (2\alpha - 3)y' + \alpha(\alpha - 3)y = 0 \) remain bounded and some become unbounded as \( t \to \infty \).
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Context:
In a free group with two generators $a$ and $b$, you start with a word of length 1 at time $0$. Every second, you multiply the current word by one of the elements $a$, $b$, $a^{-1}$, or $b^{-1}$, each chosen with probability $\frac{1}{4}$. What is the probability that the reduced word $a$ ever appears in this process?
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Context:
Determine the number of integer-coordinate points inside the \(\ell_1\) ball with radius \(m\), excluding the boundary, where \(m\) is a natural number.
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Context:
Find the function \( \phi(x) \) that minimizes the integral \[ I = \int_a^b \left\{ \frac{1}{4} m \left( \frac{d\phi(x)}{dx} \right)^2 + \frac{1}{4} n^2 \cdot \omega \cdot \sin^2 \phi(x) \right\} dx, \] where \( m, n, \omega, a, \) and \( b \) are constants.
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Context:
We color the numbers $1, 2, 3, \ldots, 20$ with two colors: white and black, ensuring that both colors are used. Determine the number of ways to perform this coloring such that the greatest common divisor (GCD) of the product of the white numbers and the product of the black numbers is equal to $1$.
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Context:
Let \(\alpha \in \mathbb{R}\). If \((3,0,0,\beta)\) is an optimal solution of the linear programming problem, minimize \(x+y+z-\alpha t\) subject to the constraints \(2x-y+z=6\), \(-x+y+t=3\), and \(x,y,z,t \ge 0\). What is the maximum value of \(\beta - \alpha\)?
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Context:
What is the greatest number of identical bouquets that can be made out of 21 white tulips, 91 red tulips, 37 blue tulips, and 67 yellow tulips if no flowers are to be left out and each bouquet has a ratio of 3 white tulips: 7 red tulips: 5 blue tulips: 9 yellow tulips? Additionally, each bouquet must have at least 24 flowers and no more than 50 flowers.
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Context:
Solve for $x$ in the equation:

$$26280 = 2 \sqrt{\frac{149598000000 - x}{1.4}} + \frac{x}{10217588910387300000}$$
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Context:
From point $M$ on the circle, three chords are drawn: $M N=1, M P=6, M Q=2$. The angles $N M P$ and $P M Q$ are equal. Find the radius of the circle.
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Context:
Cole wants to fence his backyard on three sides. His backyard is 9 feet along the sides and 18 feet along the back. The neighbor behind him agreed to pay for a portion of their shared side of his fence, and the neighbor on his left agreed to pay for a third of their shared side. Fencing costs $3 per foot. Cole has to pay $72 for the fence. What is the ratio of the amount the neighbor behind him pays for their shared side to the total cost of that side?
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Context:
Calculate \( \frac{d}{dx}f(x,f(x,x^2)) \) at \( x=1 \), given that \( f(x,y) \) is differentiable at \((1,1)\), and \( f(1,1)=\frac{\partial f}{\partial x}(1,1)=\frac{\partial f}{\partial y}(1,1) \).
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