Context:
Find the Frobenius number for the arithmetic sequence of integers 100, 105, 110, ..., 200.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: $$\lim_{x \to 0} \ln\left(\frac{\cos x}{1+\sin x}\right)^{\frac{1}{x}}$$.
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Context:
In a group of freshmen, any group of 3 people includes at least 2 who know each other, and any group of 4 people includes at least 2 who do not know each other. What is the maximum number of freshmen in the group?
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Context:
Alfred and Bonnie play a game in which they take turns tossing a fair coin. The winner of a game is the first person to obtain a head. Alfred and Bonnie play this game several times with the stipulation that the loser of a game goes first in the next game. Suppose that Alfred goes first in the first game, and that the probability that he wins the sixth game is $m/n$ , where $m$ and $n$ are relatively prime positive integers. What are the last three digits of $m+n$ ?
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Context:
There are  $2024$  cities in a country, every two of which are bidirectionally connected by exactly one of three modes of transportation - rail, air, or road. A tourist has arrived in this country and has the entire transportation scheme. He chooses a travel ticket for one of the modes of transportation and the city from which he starts his trip. He wants to visit as many cities as possible, but using only the ticket for the specified type of transportation. What is the largest  $k$  for which the tourist will always be able to visit at least  $k$  cities? During the route, he can return to the cities he has already visited.

*Proposed by Bogdan Rublov*
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Context:
Determine the number of connected components of the set \(T=\{(a,b,c)\in\mathbb{R}^3: a^2b^2c^2=4\}\).
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Context:
已知圆锥面 x^2+y^2=\df{z^2}{3}，记沿该圆锥面从 P(-\sqrt{3},3,6) 到 Q(\sqrt{3},0,3) 的曲线长度的最小值为 I，则 [10I]=__________.
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Context:
Evaluate the definite integral \( \int_0^\pi \sin(\sin(x))\sin(x)\,dx \).
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Context:
Given complex numbers \( z_{1}, z_{2}, \ldots, z_{n} \) such that \( |z_{1}| = |z_{2}| = \cdots = |z_{n}| = r \neq 0 \), and \( T_{s} \) is the sum of all possible products of \( s \) out of these \( n \) numbers, find \( \left|\frac{T_{s}}{T_{n-s}}\right| \) in terms of \( r \), assuming \( T_{n-s} \neq 0 \).
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Context:
Given that \( AE \) and \( BD \) are straight lines that intersect at \( C \), \( BD = 16 \), \( AB = 9 \), \( CE = 5 \), and \( DE = 3 \), calculate the length of \( AC \).
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Context:
Evaluate the integral \( \int_C \omega \) where \( \omega = \frac{-y \, dx + (x-1) \, dy}{(x-1)^2 + y^2} \) and \( C \) is the cardioid given by \( r = 1 + \cos \varphi \), positively oriented.
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Context:
When Xiaoming was doing math problems, he found that $\sqrt{1-\frac{1}{2}}=\sqrt{\frac{1}{2}}$; $\sqrt{2-\frac{2}{5}}=2×\sqrt{\frac{2}{5}}$; $\sqrt{3-\frac{3}{{10}}}=3×\sqrt{\frac{3}{{10}}}$; $\sqrt{4-\frac{4}{{17}}}=4×\sqrt{\frac{4}{{17}}}$; $\ldots$; following this pattern, if $\sqrt{a-\frac{8}{b}}=a•\sqrt{\frac{8}{b}}(a$,$b$ are positive integers), then $a+b=\_\_\_\_\_\_$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
For positive real numbers $p$ and $q$, define the remainder when dividing $p$ by $q$ as the smallest nonnegative real $r$ such that $\frac{p-r}{q}$ is an integer. For an ordered pair $(a, b)$ of positive integers, let $r_1$ and $r_2$ be the remainders when $a\sqrt{2} + b\sqrt{3}$ is divided by $\sqrt{2}$ and $\sqrt{3}$ respectively. Find the number of pairs $(a, b)$ such that $a, b \leq 20$ and $r_1 + r_2 = \sqrt{2}$.
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Context:
Find the smallest three-digit number with the following property: if you append a number that is 1 greater to its right, the resulting six-digit number will be a perfect square.
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Context:
A checker moves from the bottom-left corner of a $100 \times 100$ board to the top-right corner, with each step moving one square to the right or one square up. Let $a$ be the number of paths where exactly 70 steps are taken below the diagonal that runs from the bottom-left to the top-right, and $b$ be the number of paths where exactly 110 such steps are taken. Determine which is greater: $a$ or $b$?
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Context:
Calculate the sum of the alternating series \(\sum_{n=0}^\infty (-1)^n\binom{1/2}{n}\frac{x^{4n+1}}{4n+1}\) such that the error is less than or equal to \(\frac{0.4^9}{72}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find an example of an $S$-composite number from the set $S = \{1, 5, 9, \ldots\}$, which can be expressed as a product of $S$-primes in more than one way. Here, $S$-primes are numbers in the set with only $1$ and themselves as factors, and $S$-composites are the rest.
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Context:
Determine the values of $n$ for which the expression \( \frac{\sum_{k=0}^{n}k^{2}}{n} \) is a perfect square, excluding cases where $n$ is a prime number.
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Context:
In convex quadrilateral ABCD,diagonals AC and BD intersect at S and are perpendicular.
If diagonals of MNPQ intersect O and AD=5,BC=10,AC=10,BD=11 find value of SO
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Context:
In a row of 30 people, each person is either a knight who always tells the truth or a liar who always lies. They are numbered from 1 to 30. Each person with an odd number claims: "All people with numbers greater than mine are liars," and each person with an even number claims: "All people with numbers less than mine are liars." Determine the possible number of liars. If there are multiple solutions, list them in ascending order separated by semicolons.
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Context:
Suppose all solutions \( u(x,t) \) to the partial differential equation \( au_x + bu_t = 0 \) satisfy the condition \( u(1,2) = u(3,6) \). Determine the value of \( \frac{b}{a} \).
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Context:
Solve the equation:
\[ (x + 1)^{63} + (x + 1)^{62}(x−1) + (x + 1)^{61}(x−1)^{2} + \ldots + (x−1)^{63} = 0 \]
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Context:
Calculate the flux of the vector field \( \mathbf{F}(x, y, z) = 6x\mathbf{i} + 6y\mathbf{j} + 2\mathbf{k} \) across the surface \( G \), where \( G \) is the surface cut from the bottom of the paraboloid \( z = x^2 + y^2 \) by the plane \( z = 3 \).
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Context:
Find the non-constant polynomial $f(x)$ such that $f(g(x)) = g(f(x))$, where $g(x) = 3x^4 - 1$. Express $f(x)$ in its simplest form.
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Context:
Let $A$ and $B$ be subsets of a set $M$ and let $S_0=\{A,B\}$. For $i\geq 0$, define $S_{i+1}$ inductively to be the collection of subsets $X$ of $M$ that are of the form $C\cup D, C\cap D,$ or $M-C$ (the complement of $C$ in $M$), where $C,D \in S_i$. Let $S=\cup^\infty_{i=0} S_i$. What is the largest possible number of elements of $S$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: \[ \lim_{n\to\infty} \prod_{k=1}^\infty \left(\frac{p_n}{p_k}\right)^{\frac{p_n}{n(p_n+p_k)}} \] where \( p_n \) is the \( n \)th prime number.
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Context:
Determine the maximum number of triangles that can be formed using up to 25 nodes and up to 40 edges, allowing for overlapping triangles.
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Context:
Evaluate the integral \( \left| \oint_{|z| = R} \frac{z^2}{\log z}\,dz \right| \) for \( R > 1 \).
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Context:
Determine the values of \( k \) for which the equation \( x^3 - 3x^2 + 6x + k = 0 \) has three real roots.
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Context:
Let $X$ be an exponentially distributed random variable and $V$ be a uniformly distributed random variable on $\{-1,+1\}$, both independent of each other. Define $Y = XV$. Calculate the conditional probability $P(|Y| > b \mid Y < 0)$.
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Context:
Aubree saw 20 beavers and 40 chipmunks by a tree when going to school. While coming back from school, she realized the number of beavers had changed and the number of chipmunks had decreased by 10. She saw a total of 130 animals that day. What happened to the number of beavers when she came back from school?
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Context:
Find the number of odd numbers in the $n$-th row of Pascal's triangle.
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Context:
In triangle \(ABC\), angle \(A\) is the largest angle. Points \(M\) and \(N\) are symmetric to vertex \(A\) with respect to the angle bisectors of angles \(B\) and \(C\) respectively. Find \(\angle A\) if \(\angle MAN = 50^\circ\).
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Context:
Define a function $g$ from the positive integers to the positive integers with the following properties:
(i) $g$ is increasing.
(ii) $g(mn) = g(m)g(n)$ for all positive integers $m$ and $n$.
(iii) If $m \neq n$ and $m^n = n^m$, then $g(m) = n$ or $g(n) = m$.

Compute all possible values of $g(88).$
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Context:
Point $P$ is located in the second quadrant, with a distance of $2$ to the $x$-axis and a distance of $3$ to the $y$-axis. Determine the coordinates of point $P$.
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Context:
What is the range of values for \( m \) that makes the equation \( m\left(x^{2}+y^{2}+2 y+1\right)=(x-2 y+3)^{2} \) represent an ellipse?
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Context:
The diagonals of a quadrilateral are both equal to \(a\), and the sum of its midsegments (the segments connecting the midpoints of opposite sides) is \(b\). Calculate the area of the quadrilateral.
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Context:
Question 3 Suppose the number $a$ has the property: for any four real numbers $x_{1}, x_{2}, x_{3}, x_{4}$, there always exist integers $k_{1}, k_{2}, k_{3},$ $k_{4}$, such that
$$
\sum_{1 \leqslant i<j \leqslant 4}\left(\left(x_{j}-k_{i}\right)-\left(x_{i}-k_{j}\right)\right)^{2} \leqslant a .
$$

Find the minimum value of such an $a$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
某社区组织了一次公益讲座.向社区居民普及垃圾分类知识.为了解讲座效果，随机抽取10位社区居民，让他们在讲座前和讲座后分别回答一份垃圾分类知识问卷.这10位社区居民的讲座前和讲座后答卷的正确率如下表：

| 编号/准确率 | 1号  | 2号  | 3号  | 4号  | 5号  | 6号  | 7号  | 8号  | 9号  | 10号 |
|-------------|------|------|------|------|------|------|------|------|------|------|
| 讲座前      | 65%  | 60%  | 70%  | 100% | 65%  | 75%  | 90%  | 85%  | 80%  | 60%  |
| 讲座后      | 90%  | 85%  | 80%  | 95%  | 85%  | 85%  | 95%  | 100% | 85%  | 90%  |

$从正确率不低于90\%的垃圾分类知识答卷中随机抽取3份，记随机变量x为抽中讲座前答卷的份数，求随机变量x的数学期望；$
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Context:
In an equilateral triangle $\triangle{ABC}$, points $E$, $F$, and $G$ are located inside the triangle such that they form four smaller triangles of equal area. Determine the ratio $\frac{AF}{FG}$.
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Context:
Ben flips a coin 10 times and then records the absolute difference between the total number of heads and tails he's flipped. He then flips the coin one more time and records the absolute difference between the total number of heads and tails he's flipped. If the probability that the second number he records is greater than the first can be expressed as  $\frac{a}{b}$  for positive integers  $a,b$  with  $\gcd (a,b) = 1$ , then find  $a + b$ .

*Proposed by Vismay Sharan*
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Context:
Let $n = 2188 = 3^7+1$ and let $A_0^{(0)}, A_1^{(0)}, \ldots, A_{n-1}^{(0)}$ be the vertices of a regular $n$-gon (in that order) with center $O$. For $i = 1, 2, \dots, 7$ and $j=0,1,\dots,n-1$, let $A_j^{(i)}$ denote the centroid of the triangle \( \triangle A_j^{(i-1)} A_{j+3^{7-i}}^{(i-1)} A_{j+2 \cdot 3^{7-i}}^{(i-1)} \). Here the subscripts are taken modulo $n$. If \( \frac{|OA_{2014}^{(7)}|}{|OA_{2014}^{(0)}|} = \frac{p}{q} \) for relatively prime positive integers $p$ and $q$, find $p+q$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a matrix \(A\), determine the value of \(\theta\) such that the sum of the real parts of the eigenvalues of \(A\) equals 1.
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Context:
Given the following propositions about regular pyramids:
① A pyramid with a regular polygon as its base is a regular pyramid;
② A pyramid whose lateral edges are all equal is a regular pyramid;
③ A pyramid whose lateral edges form equal angles with the base is a regular pyramid;
④ A pyramid whose dihedral angles between the lateral faces and the base are all equal is a regular pyramid.
The definition of a regular pyramid is: the base is a regular polygon, and the projection of the apex onto the base is the center of the base polygon.
How many of these propositions are correct? Express your answer as a single integer.
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Context:
Lucky Larry's teacher asked him to substitute numbers for $a$, $b$, $c$, $d$, and $e$ in the expression $a-(b-(c-(d+e)))$ and evaluate the result. Larry ignored the parentheses but added and subtracted correctly and obtained the correct result by coincidence. The number Larry substituted for $a$, $b$, $c$, and $d$ were $1$, $2$, $3$, and $4$, respectively. What number did Larry substitute for $e$?
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Context:
The sequence \(\{\alpha_{n}\}\) is an arithmetic sequence with a common difference \(\beta\). The sequence \(\{\sin \alpha_{n}\}\) is a geometric sequence with a common ratio \(q\). Given that \(\alpha_{1}, \beta \in \mathbf{R}\), find the value of \(q\).
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Context:
In a regular triangular prism $V-ABC$, the lateral edge $VC$ has a length of 3, and the base edge $AB$ has a length of 2. A section passing through the base edge $AB$ intersects the lateral edge $VC$ at point $D$. Determine the minimum area of the triangle $	riangle ABD$. Express your answer in simplest form.
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Context:
Let n, m, p, q, and r be five integers such that the remainders when n is divided by 18, m is divided by 27, p is divided by 6, q is divided by 12, and r is divided by 3 are 10, 16, 4, 8, and 2 respectively. What is the remainder when ((3n + 2m) - (p + q) / r) is divided by 9?
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Context:
The wholesale department operates a product with a wholesale price of 500 yuan per unit and a gross profit margin of 4%. The inventory capital is 80% borrowed from the bank at a monthly interest rate of 4.2‰, and the storage and operating cost is 0.30 yuan per unit per day. Determine the maximum average storage period for the product without incurring a loss.
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Context:
Find the limit of the sequence \( b_n = \sqrt[n]{7^{3n} - 2^{n+4} - \cos(\sqrt{n})} \).
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Context:
Given a triangle $\triangle ABC$ with side lengths $a$, $b$, and $c$ opposite to angles $A$, $B$, and $C$, respectively. If $a^{2}-a- \sqrt {3}b- \sqrt {3}c=0$ and $a+ \sqrt {3}b- \sqrt {3}c+2=0$, find the cosine value of the largest angle in $\triangle ABC$.
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Context:
A man saves 20% of his monthly salary. His monthly expenses consist of rent, utilities, and groceries which are 40%, 30%, and 20% of his salary respectively. Due to an increase in the cost of living, he now has to increase his rent by 15%, utilities by 20%, and groceries by 10%. After these increases, he manages to still save, but his savings reduced to Rs. 180 per month. What is his original monthly salary?
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Context:
Determine the values of $s \in \mathbb{R}$ for which the integral \[ \iint_{(a,b)^2}\frac{1}{|x-y|^s}\mathrm{d}x\mathrm{d}y \] converges, where $(a,b)$ is a finite open interval of $\mathbb{R}$ and $s > 0$. Provide the range of $s$ for which the integral is convergent.
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Context:
Find the area of the subset \( S \) of \( \mathbb{R}^3 \) defined by the conditions: \( x^2 + y^2 + z^2 \leq 1 \) and \( \det A = 0 \), where \( A = \begin{bmatrix} 1 & x & x^2 \\ 1 & y & y^2 \\ 1 & z & z^2 \end{bmatrix} \).
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Context:
Given the function \( f(x) \) defined as \( f(x) = 0 \) for \( x \leq 0 \) and \( f(x) = e^{-x} \) for \( x > 0 \), the Fourier transform \( F(\omega) \) is calculated to be \( F(\omega) = \frac{1}{\sqrt{2\pi}(1+i\omega)} \). Use this result to evaluate the integral \( \int_{0}^{\infty} \frac{\cos(2\omega) + \omega\sin(2\omega)}{1+\omega^2} \, d\omega \).
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Context:
Gwen was organizing her book case, making room for her ever-growing collection. She had eight shelves of mystery books with six books on each shelf, five shelves of picture books with four books on each shelf, four shelves of science fiction books with seven books on each shelf, and three shelves of non-fiction books with five books on each shelf. However, she had lent out two mystery books and three science fiction books to her friends, and she had borrowed five picture books from the library. How many books did she have in her collection, excluding the borrowed books and considering the lent out books?
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Context:
How do we calculate the residue of the function $$ f(x)=\frac{x^2}{(1+x^2)^2} $$ at the pole $x=i$?
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Context:
Let $p(x)$ be a nonzero polynomial of degree less than 1992 having no nonconstant factor in common with $x^{3}-x$. Let

$$
\frac{d^{1992}}{d x^{1992}}\left(\frac{p(x)}{x^{3}-x}\right)=\frac{f(x)}{g(x)}
$$

for polynomials $f(x)$ and $g(x)$. Find the smallest possible degree of $f(x)$.
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Context:
Point \( O \) is the center of the circumscribed circle of triangle \( ABC \) with sides \( AB=5, AC=8 \), and \( BC=4 \). Find the length of the vector \( \overrightarrow{OA} - 4 \overrightarrow{OB} + 3 \overrightarrow{OC} \).
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Context:
Let $\mathcal{C}$ be the hyperbola $y^{2}-x^{2}=1$. Given a point $P_{0}$ on the $x$-axis, we construct a sequence of points $\left(P_{n}\right)$ on the $x$-axis in the following manner: let $\ell_{n}$ be the line with slope 1 passing through $P_{n}$, then $P_{n+1}$ is the orthogonal projection of the point of intersection of $\ell_{n}$ and $\mathcal{C}$ onto the $x$-axis. (If $P_{n}=0$, then the sequence simply terminates.) Let $N$ be the number of starting positions $P_{0}$ on the $x$-axis such that $P_{0}=P_{2008}$. Determine the remainder of $N$ when divided by 2008.
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Context:
In a soccer match between Team A and Team B, the final score is a 9-9 tie. During the match, Team A had the lead at one point, and later, Team B also had the lead. How many different sequences of 18 goals can represent the course of the match, considering the order in which the goals were scored?
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Context:
Calculate the sum of the infinite series \( \sum_{i=0}^{\infty} \sum_{j=0}^{\infty} \sum_{k=0}^{\infty} \frac{1}{3^i} \cdot \frac{1}{3^j} \cdot \frac{1}{3^k} \) with the condition that \( i \neq j \neq k \).
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Context:
In \( \bigtriangleup {ABC},{PQ}//{BC} \) where \( P \) and \( Q \) are points on \( {AB} \) and \( {AC} \)
 respectively. The lines \( {PC} \) and \( {QB} \) intersect at \( G \) . It is also given
 \( {EF}//{BC} \) , where \( G \in  {EF},E \in  {AB} \) and \( F \in  {AC} \) with \( {PQ} = a \) and
 \( {EF} = b \) . Find value of \( {BC} \) .
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Context:
Calculate the integral \( \int_{F(S^1)} \omega \) where \( F: S^1 \rightarrow S^1 \times S^1 \times S^1 \) is given by \( F(p) = (p,p,p) \) and \( \omega = dt_1 + dt_2 + dt_3 \) is a 1-form. Given that \( F^{*}\omega = 3\,dt \), find the value of the integral.
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Context:
Let $ABCD$ be an isosceles trapezoid with $AD \parallel BC$. Points $P$ and $Q$ are placed on segments $\overline{CD}$ and $\overline{DA}$ respectively such that $AP \perp CD$ and $BQ \perp DA$, and point $X$ is the intersection of these two altitudes. Suppose that $BX=3$ and $XQ=1$. Compute the largest possible area of $ABCD$. \(\text{Express your answer as a single number.}\)
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Context:
# Task 4.

Four brigades were developing an open coal deposit for three years, working with a constant productivity for each brigade. In the second year, due to weather conditions, work was not carried out for four months, and for the rest of the time, the brigades worked in rotation (one at a time). The ratio of the working times of the first, second, third, and fourth brigades and the amount of coal mined respectively are: in the first year 4:1:2:5 and 10 million tons; in the second year $2: 3: 2: 1$ and 7 million tons; in the third year 5:2:1:4 and 14 million tons. How much coal would these four brigades have mined in 4 months if they worked together?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the operator norm \( \|T\| \) for the operator \( T: \ell^p \to \ell^p \) defined by \( Tx = (c_1 x_1, c_2 x_2, \ldots) \), where \( c_k \in \mathbb{C} \) and \( \lim_{k \to \infty} c_k = 0 \), for \( 1 \le p \le \infty \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the value of n such that in a lottery where 5 balls are drawn without replacement from a set of balls numbered 1 through n, the probability of matching exactly 0 balls is as close as possible to the probability of matching exactly 1 ball. The goal is to create a scenario where matching at least one ball is slightly favored.
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Context:
In a Cartesian coordinate system, if the equation \(m\left(x^{2}+y^{2}+2 y+1\right)=(x-2 y+3)^{2}\) represents an ellipse, then what is the range of values for \(m\)?
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the number of degrees in $x$? [asy]
import olympiad; import geometry; defaultpen(linewidth(0.8)); size(150);
pair A = origin, B = (Cos(50),Sin(50)), C = (Cos(50) + Sin(50)*sqrt(3),0);
draw(A--B--C--cycle);
label("$30^\circ$",C-(0.3,-.03),NW);
label("$2x$",B + (-.10,-0.1),SE);
label("$x$",A+(0.15,.03),NE);
[/asy]
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Context:
Given the lines $l_{1}$: $\left(3+a\right)x+4y=5-3a$ and $l_{2}$: $2x+\left(5+a\right)y=8$, if $l_{1}$ is parallel to $l_{2}$, determine the value of $a$.
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Context:
The complex numbers \( z_{1}, z_{2}, z_{3}, z_{4} \) satisfy \( \left|z_{1}\right|=\left|z_{2}\right|=\left|z_{3}\right|=\left|z_{4}\right|=1 \), and \( z_{1}+z_{2}+z_{3}+z_{4}=0 \). What is the shape of the quadrilateral whose vertices correspond to these 4 complex numbers?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the smallest natural number that divides $2^n + 15$ for some natural number $n$ and can be expressed in the form $3x^2 - 4xy + 3y^2$ for some integers $x$ and $y$.
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Context:
Find the smallest positive integer $n$, such that for any sequence of $n$ positive integers $a_{1}, a_{2}, \cdots, a_{n}$ satisfying $\sum_{i=1}^{n} a_{i}=2007$, there must be a sum of some consecutive terms equal to 30. (Provided by Tao Pingsheng)
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Context:
Given a complex number z that satisfies the equation (1-i)/(1+i) - z = 1, calculate the magnitude of 2 + i times the conjugate of z. Express your answer as a numerical value, including the square root symbol if necessary.
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Context:
Let $ABC$ be an equilateral triangle with each side of length 1. Let $X$ be a point chosen uniformly at random on side $\overline{AB}$. Let $Y$ be a point chosen uniformly at random on side $\overline{AC}$. (Points $X$ and $Y$ are chosen independently.) Let $p$ be the probability that the distance $XY$ is at most $\frac{1}{\sqrt[4]{3}}$. What is the value of $900p$, rounded to the nearest integer?
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Context:
Let \( P \) be a regular 2006-sided polygon. A diagonal of \( P \) is called a "good edge" if both parts of \( P \)'s boundary divided by this diagonal contain an odd number of edges. Each side of \( P \) is also defined as a "good edge."

Given that 2003 non-intersecting diagonals inside \( P \) divide it into several triangles, determine the maximum number of isosceles triangles with two "good edges" in this division.
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Context:
已知 S_n 为数列 \{a_n\} 的前 n 项和，且 S_n=\df{9}{8}a_n-\df{3}{8}\cdot 3^n+\df{3}{8}. 则使 \ds{i=1}{n}\df{S_i}{a_i}<2024 成立的最大正整数 n 的值为__________.
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Context:
A circle passes through the point (1, 2) and has the equation $x^2+y^2=1$. The area of the quadrilateral formed by the two tangents to the circle from this point and the x-axis and y-axis is \_\_\_\_\_.
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Context:
Thomas buys a weight vest that weighed 60 pounds and worked well for him in the beginning. After a bit of training, he decides he wants to increase the weight by 60%. The weights come in 2-pound steel ingots. Each ingot cost $5. If you buy a certain number of ingots, you get a discount. It costs $72 to get the weight he needs. What is the discount percentage if you buy more than 10 ingots?
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Context:
Let  $S = {1, 2, \cdots, 100}.$   $X$  is a subset of  $S$  such that no two distinct elements in  $X$  multiply to an element in  $X.$  Find the maximum number of elements of  $X$ .

*2022 CCA Math Bonanza Individual Round #3*
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Context:
The sequence \(a_{n} = b[\sqrt{n+c}] + d\) is given, where the successive terms are
\[
1, 3, 3, 3, 5, 5, 5, 5, 5, \cdots
\]
In this sequence, each positive odd number \(m\) appears exactly \(m\) times consecutively. The integers \(b\), \(c\), and \(d\) are to be determined. Find the value of \(b+c+d\).
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Context:
Simplify first, then evaluate: $\frac{{a}^{2}-{b}^{2}}{a}÷(a+\frac{2ab+{b}^{2}}{a})$, where $a=\sqrt{2}+1$ and $b=\sqrt{2}-1$.
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Context:
Schoolboy Alexey told his parents that he is already an adult and can manage his finances independently. His mother suggested using a duplicate bank card from her account. For participation in a charitable Christmas program, Alexey wants to buy 40 "Joy" chocolate bars and donate them to an orphanage. However, the bank, where Alexey's parents are clients, has implemented a new system to protect against unauthorized card payments. The protection system analyzes the root mean square (RMS) value of expenses for the last 3 purchases (S) using the formula \(S=\sqrt{\frac{x_{1}^{2}+x_{2}^{2}+x_{3}^{2}}{3}}\), where \(x_{1}, x_{2}\), and \(x_{3}\) are the costs of the previous purchases, and compares the value of \(S\) with the cost of the current purchase. If the cost of the current payment exceeds the value \(S\) by 3 times, the bank blocks the payment and requires additional verification (e.g., a call from mom to the call center). In the last month, payments made on the card were only for cellphone bills in the amount of 300 rubles each. Into how many minimum number of receipts should Alexey split the purchase so that he can buy all 40 "Joy" chocolate bars at a cost of 50 rubles each?
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Context:
How many different $T_0$ topologies can be defined on an infinite set, such as the set of real numbers?
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Context:
Suppose a parabola has vertex $\left(\frac{3}{2},-\frac{25}{4}\right)$ and follows the equation $y = ax^2 + bx + c$, where $a < 0$ and the product $abc$ is an integer. Find the largest possible value of $a$.
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Context:
Find $x$ such that $$\sum_{k=1}^{2014} k^k \equiv x \pmod {10}$$
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Context:
Paula the painter and her two helpers each paint at constant, but different, rates. They always start at 8:00 AM, and all three always take the same amount of time to eat lunch. On Monday the three of them painted 50% of a house, quitting at 4:00 PM. On Tuesday, when Paula wasn't there, the two helpers painted only 24% of the house and quit at 2:12 PM. On Wednesday Paula worked by herself and finished the house by working until 7:12 P.M. How long, in minutes, was each day's lunch break?
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Context:
The sum of the coefficients in the expansion of the binomial $$(3x - \frac{1}{2 \cdot 3x})^n$$ is $\frac{1}{64}$. Find the constant term in the expansion.
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Context:
For an integer $30 \le k \le 70$, let $M$ be the maximum possible value of:
\[ \frac{A}{\gcd(A,B)} \]
where $A = \binom{100}{k}$ and $B = \binom{100}{k+3}$. Find the remainder when $M$ is divided by $1000$.
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Context:
记 $I=\sum\limits_{m=1}^{+\infty} \sum\limits_{n=1}^{+\infty} \frac{1}{n!n!+m!n!}$, 则 $[10 I]=$ $\qquad$.
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Context:
There is a set of points \( M \) on a plane and seven different circles \( C_{1}, C_{2}, \dots, C_{7} \). Circle \( C_{7} \) passes through exactly 7 points in \( M \); circle \( C_{6} \) passes through exactly 6 points in \( M \); ..., circle \( C_{1} \) passes through exactly 1 point in \( M \). Determine the minimum number of points in set \( M \).
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Context:
[Help me] Determine the smallest value of the sum M =xy-yz-zx where x; y; z are real numbers satisfying the following condition  $x^2+2y^2+5z^2 = 22$ .
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Context:
2. The chord $AB$ divides a circle with radius $r$ into two arcs in the ratio $1:2$. In the larger segment, a square is inscribed such that one of its sides lies on this chord. Express the length of the side of the square in terms of the radius $r$.
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Context:
在 \(\triangle A B C\) 中, 已知 \(\cos C=\frac{\sin A+\cos A}{2}=\frac{\sin B+\cos B}{2}\), 求 \(\cos C\) 的值。原始答案为 \(\frac{\sqrt{m}}{n}\) 的格式，请给出 m + n 的值。
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Context:
Fill the five numbers $2015, 2016, 2017, 2018, 2019$ into the five boxes labeled " $D, O, G, C, W$ " such that $D+O+G=C+O+W$. How many different ways can this be done?
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Context:
The base of a pyramid is an equilateral triangle. One of the lateral faces of the pyramid is perpendicular to the plane of the base. Find the cosine of the angle between the two other lateral faces, given that they form an angle $\alpha$ with the plane of the base.
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Context:
Let $x_1, x_2, \ldots, x_n$ be distinct real numbers. Define the function $g(t)$ as the number of functions $f_k(t)$ that are equal to 1, where each $f_k(t)$ is defined based on the original question's context. Determine the maximum value of $g(t)$. \( \text{Provide your answer as a single integer.} \)
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Context:
A fast train with a weight of $P=150$ tons travels at a maximum speed of $v=72 \frac{\text{km}}{\text{hour}}$ on a horizontal track with a friction coefficient of $\rho=0,005$. What speed can it reach on a track with the same friction conditions but with an incline having $e=0.030$?

(Note: In this problem, $\rho$ is the friction coefficient and $e$ is the sine of the inclination angle.)
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Context:
5. The set
$$
A=\left\{\left.(x, y)\left|(x-t)^{2}+y^{2} \leqslant\left(1-\frac{t}{2}\right)^{2},\right| t \right\rvert\, \leqslant 2\right\}
$$

represents a plane region with an area of $\qquad$ .
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Context:
For real numbers $a, b$, and $c$ the polynomial $p(x) = 3x^7 - 291x^6 + ax^5 + bx^4 + cx^2 + 134x - 2$ has $7$ real roots whose sum is $97$. Find the sum of the reciprocals of those $7$ roots.
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Context:
2 cows and 10 goats are brought for Rs. 2000. If the individual prices of goats are Rs. 65, Rs. 70, Rs. 75, Rs. 80, Rs. 75, Rs. 71, Rs. 82, Rs. 64, Rs. 78, and Rs. 79, respectively, find the individual prices of each cow.
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Context:
8*. Among the numbers from 1 to 1000, how many are divisible by 4 but do not have the digit 4 in their representation?
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Context:
The diameter $[AC]$ of a circle is divided into four equal segments by points $P$, $M$, and $Q$. Consider a segment $[BD]$ that passes through $P$ and intersects the circle at points $B$ and $D$, such that $PD = \frac{3}{2} AP$. Given that the area of triangle $[ABP]$ is $1$ cm$^2$, calculate the area of quadrilateral $[ABCD]$.
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Context:
Suppose that f is a function $f : R_{\ge 0} \to R$ so that for all $x, y \in R_{\ge 0}$ (nonnegative reals) we have that $$f(x)+f(y) = f(x+y+xy)+f(x)f(y).$$ Given that $f\left(\frac{3}{5} \right) = \frac12$ and$ f(1) = 3$, determine
$$\lfloor \log_2 (-f(10^{2021} - 1)) \rfloor.$$
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Context:
$5$ 级方阵 $A$ 满足: 第一行元素均为 $1$, 第二行元素从左至右依次为 $1,-1,1,-1,1$,且 $\operatorname{trace}\left(A A^{\prime}\right)=28$. 记 $\operatorname{det} A$ 的最大值为 $M$, 则 $[M]=$ $\qquad$.
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Context:
Compute the minimal number of generators, \( \mu(mA_m) \), of the module \( mA_m \), where \( m \) is the maximal ideal \( (\bar{x} - 1, \bar{y} - 1) \) of \( A = \mathbb{C}[x,y]/(x^3-y^2) \).
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Context:
Given integers \(a, b\) and odd prime numbers \(p, q\), find the total number of distinct integer solutions \((a, b)\) to the equation \(a^2 + 2b^2 = p^{11}q^{13}\).
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Context:
已知 \(y^{2}=4x\)，过 \(A(-2,3)\) 做抛物线两条切线，交 y 轴于 \(B, C\) 两点，求 \(\triangle ABC\) 的外接圆方程。答案的形式为\((x + \frac{k}{2})^2 + (y - \frac{m}{2})^2 = \frac{n}{p}\)，请给出k + m + n + p的值，其中\(\frac{n}{p}\)为最简分数。
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Context:
During a certain play's run, the performances were spread across three weeks, and tickets were sold at varying ratios of full price to reduced price. During the first week, 5400 tickets were sold, all at a reduced price. In the second week, twice as many tickets were sold at full price as were sold at the reduced price, and a total of 7200 tickets were sold during the week. In the third week, seven times as many tickets were sold at the full price as were sold at reduced price, and a total of 13400 tickets were sold during the week.

Additionally, there were three levels of ticket pricing: reduced, regular full price, and premium full price. During the first week, tickets were sold only at the reduced price. During the second week, 1 out of every 10 full-priced tickets was sold at the premium price, and the rest were sold at the regular full price. In the third week, 2 out of every 15 full-priced tickets were sold at the premium price, and the rest were sold at the regular full price.

If a total of 25200 tickets were sold during the entire play's run, what is the total number of tickets sold at full price (both regular and premium combined) during these three weeks?
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Context:
Positive integers $a$, $b$, $c$, and $d$ are such that $a<b<c<d$, and the system of equations
\[ 2x + y = 2007 \quad\text{and}\quad y = |x-a| + |x-b| + |x-c| + |x-d| \]
has exactly one solution. What is the minimum value of $d$?
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Context:
Out of 1500 people surveyed, $25\%$ do not like television, and out of those who do not like television, $15\%$ also do not like books. How many people surveyed do not like both television and books?
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Context:
How many $6$-digit positive integers whose digits are different from $0$ are there such that each number generated by rearranging the digits of the original number is always divisible by $7$? Find the total number of such integers.
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Context:
An integer, whose decimal representation reads the same left to right and right to left, is called symmetrical. For example, the number 513151315 is symmetrical, while 513152315 is not. How many nine-digit symmetrical numbers exist such that adding the number 11000 to them leaves them symmetrical?
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Context:
Two given quadratic trinomials differ by exchanging the free term and the second coefficient. The sum of these trinomials has a unique root. What value does this sum take at one?
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Context:
4. The unit square in the first quadrant of the coordinate plane ( $0 \leq x, y \leq 1$ ) is divided into squares with side length $2 \cdot 10^{-4}$. How many nodes of this partition (inside the unit square) lie on the parabola $y=x^{2} ?$
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Context:
Sheela deposits Rs. 4500 in her bank savings account. If this is a certain percentage of her monthly income, and her monthly income is Rs. 16071.42857142857, what is the percentage of her income that she deposits in her bank savings account?
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Context:
A shopkeeper applies an initial 10% discount on an article's cost price, followed by a 5% tax on the discounted price. Then, the shopkeeper sells the article with a 20% profit for $1800. Find the original cost price.
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Context:
There are 3,250 more red apples than green apples in a grocery store. A truck arrives and delivers another 3,600 green apples and 1,300 red apples. Simultaneously, 750 green apples and 810 red apples are sold to customers. If there were originally 1,200 green apples, how many more green apples than red apples are there in the store now?
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Context:
Given an arithmetic sequence $\{a_n\}$ with a common difference $d \neq 0$ and the first term $a_1 = d$, the sum of the first $n$ terms of the sequence $\{a_n^2\}$ is $S_n$. A geometric sequence $\{b_n\}$ has a common ratio $q$ less than 1 and consists of rational sine values, with the first term $b_1 = d^2$, and the sum of its first $n$ terms is $T_n$. If $\frac{S_3}{T_3}$ is a positive integer, then find the possible value of $q$.
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Context:
Let $(a_1,a_2,a_3,\ldots,a_{15})$ be a permutation of $(1,2,3,\ldots,15)$ for which
$a_1>a_2>a_3>a_4>a_5>a_6>a_7 \mathrm{\ and \ } a_7<a_8<a_9<a_{10}<a_{11}<a_{12}<a_{13}<a_{14}<a_{15}.$
An example of such a permutation is $(7,6,5,4,3,2,1,8,9,10,11,12,13,14,15).$ Find the number of such permutations.
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Context:
Let point \( O \) be located inside a regular tetrahedron \( ABCD \), and let \( G \) be the centroid of the tetrahedron. The line \( OG \) intersects each face of the tetrahedron at points \( A_1, B_1, C_1, \) and \( D_1 \). What is the value of \( \frac{A_1 O}{A_1 G} + \frac{B_1 O}{B_1 G} + \frac{C_1 O}{C_1 G} + \frac{D_1 O}{D_1 G} \)?
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Context:
Let \(\mu = e^{2\pi i/9}\). Define \(u_j = (\mu^j, \mu^{2j}, \mu^{3j}, \ldots, \mu^{9j})^T \in \mathbb{C}^9\) for \(j = 1, \ldots, 9\). Let \(V\) be the vector subspace of \(\mathbb{C}^9\) spanned by \(\{u_2, u_3, u_4, u_5, u_6, u_7\}\). It is known that there exists a matrix \(A \in M_{3 \times 6}(\mathbb{C})\) such that \(V = \{(x_1, \ldots, x_9) \in \mathbb{C}^9 : (x_3, x_6, x_9)^T = A(x_1, x_2, x_4, x_5, x_7, x_8)^T\}\). If the statement "On a special kind of 6-dimensional vector subspace of \(\mathbb{C}^9\)" is true, then \(A\) is unique and has the form \(A = \begin{pmatrix} a_1 & a_2 & a_3 & a_4 & a_5 & a_6 \\ a_5 & a_6 & a_1 & a_2 & a_3 & a_4 \\ a_3 & a_4 & a_5 & a_6 & a_1 & a_2 \end{pmatrix}\). Let \(L\) be the smallest subfield of \(\mathbb{C}\) generated by the entries of \(A\). It is known that \(L \subseteq \mathbb{Q}(\mu)\), and hence \(L/\mathbb{Q}\) is a Galois extension. Determine the extension degree \([L : \mathbb{Q}]\). Given that \([\mathbb{Q}(\mu) : \mathbb{Q}] = \phi(9) = 6\), the possible values for \([L : \mathbb{Q}]\) are 1, 2, 3, or 6. Find the exact value.
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Context:
Regular decagon `ABCDEFGHIJ` has its center at `K`. Each of the vertices and the center are to be associated with one of the digits `1` through `10`, with each digit used once, in such a way that the sums of the numbers on the lines `AKE`, `BKF`, `CKG`, `DLH` and `EJI` are all equal. In how many ways can this be done?
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Context:
Two identical rectangles $PQRS$ and $P_{1}Q_{1}R_{1}S_{1}$ are inscribed in the triangle $ABC$ such that points $P$ and $P_{1}$ lie on side $AB$, points $Q$ and $Q_{1}$ lie on side $BC$, and points $R, S, R_{1}$, and $S_{1}$ lie on side $AC$. It is known that $PS = 12$ and $P_{1}S_{1} = 3$. Find the area of triangle $ABC$.
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Context:
Given an arithmetic sequence $\{a_n\}$, let the sum of the first $n$ terms be $S_n$. If $S_{m-1} = -2$, $S_m = 0$, and $S_{m+1} = 3$, find the value of $m$. Express your answer as a single integer.
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Context:
Given the three-view diagrams of a solid, determine the volume of the solid. The side view is an isosceles right triangle with leg length 1. The top view is an equilateral triangle with a base length of 1 and a height of 1. The front view is an isosceles triangle with a base length of 1 and a height of 1. Calculate the volume of the solid. Express your answer as a fraction or decimal.
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Context:
An $n$-sided regular polygon with side length $1$ is rotated by $\frac{180^\circ}{n}$ about its center. The intersection points of the original polygon and the rotated polygon are the vertices of a $2n$-sided regular polygon with side length $\frac{1-\tan^2 10^\circ}{2}$. What is the value of $n$?
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Context:
In an acute-angled triangle $ABC$, let $H$ be the orthocenter. From points $A$ and $C$, tangents $AK$ and $CT$ are drawn to the circle with diameter $BH$. Given that the lengths of these tangents are $15$ and $19$ respectively, find the square of the smallest possible length of side $AC$. Express your answer as a single integer.
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Context:
Consider a $4$ state Markov Chain with the row-stochastic transition matrix $$P = \left(\begin{array}{cccc} 1/4&1/4&1/4&1/4 \\ 1/2&0&0&1/2 \\ 1/3&1/3&1/3&0 \\ 0&0&0&1 \end{array} \right).$$ If the chain starts in state $1$, find the expected length of time intervals until absorption in state $4$.
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Context:
$n$ 是任意自然数，$1^{3}+2^{3}+3^{3}+\cdots+n^{3}$ 的个位数不可能是______。
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Context:
Evaluate the integral $\int_{-\pi}^{\pi} \left(\frac{1+r^2}{1 - 2r\cos \theta + r^2}-\frac{(1-r^2)^2}{(1 - 2r\cos \theta + r^2)^2}\right) d\theta$ for $r \in \mathbb{R}$.
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Context:
Find the value of $n$ for which the following integral is equal to $\frac{1}{\sqrt{n}}$:

$$\int_{-\infty}^{\infty}(-1)^xe^{\frac{-\pi x}{2}(nx+1)}dx=\frac{1}{\sqrt{n}}$$

Assume that $n > 0$.
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Context:
Determine the values of $i$ for which the homology groups $H_i(K^{(p)})$ and $H_i(K)$ are isomorphic.
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Context:
Let $z$ and $z+1$ be complex numbers such that both are $n^{\text{th}}$ roots of unity. Given that $n$ is a multiple of 5, find the minimum value of $n + z^3$. \( \text{(Express your answer as a single integer.)} \)
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Context:
集合 $A=\{1,2, \cdots, n\}$.若存在非空集合 $B C$, 使得 $B \cap C=\varnothing, B \cup C=A$, 且 $B$ 中元素的平方和为 $M, C$ 中元素的平方和为 $N$ ，满足 $M-N=2016$ ，求 $n$ 的最小值.
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Context:
Let Sn represent the sum of the first n terms of the geometric sequence {an}. If a1 = 1, S3 = $$\frac{3}{4}$$, then S4 = \_\_\_\_\_\_.
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Context:
A retailer buys 120 pens at the market price of 100 pens from a wholesaler. If the wholesaler offers an additional 5% discount on bulk purchases and the retailer sells these pens giving a discount of 4%, what is the profit%?
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Context:
Let $\mathbb{N}_0$ be the set of non-negative integers. Consider a triple $(f,a,b)$, where $f$ is a function from $\mathbb{N}_0$ to $\mathbb{N}_0$ and $a,b \in \mathbb{N}_0$ satisfying the following conditions:

1. $f(1) = 2$
2. $f(a) + f(b) \leq 2\sqrt{f(a)}$
3. For all $n > 0$, $f(n) = f(n-1)f(b) + 2n - f(b)$

Find the sum of all possible values of $f(b+100)$.
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Context:
Three segments are drawn from a point inside an equilateral triangle, each parallel to one of the triangle's sides, with lengths 2 m, 3 m, and 6 m. Find the perimeter of the equilateral triangle. Express your answer in meters as a whole number.
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Context:
Find the number of matrices in \(GL_{3}(\mathbb{Z}_{3})\) such that \([1 \ 1 \ 1]^{T}\) is an eigenvector.
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Context:
Provide an example of a function \( f:[a, b] \to \mathbb{R} \) such that \( f \in R[a, c] \) for all \( c \in [a, b) \), but \( f \notin R[a, b] \).
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Context:
For any positive integer $x$, let $f(x) = x^x$. Suppose that $n$ is a positive integer such that there exists a positive integer $m$ with $m \neq 1$ such that $f(f(f(m))) = m^{m^{n+2020}}$. Compute the smallest possible value of $n$.
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Context:
Given the matrix \( A = \begin{bmatrix}\lambda & 1 & 0 & \ldots & 0 \\
0 & \lambda & 1 & \ldots & 0 \\
\vdots & \vdots & \lambda & \ddots & 1 \\
0 & \ldots & \ldots & \ldots & \lambda \end{bmatrix} \), find the minimal polynomial of \( A \). The characteristic polynomial is given by \((\lambda-x)^n=0\).
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Context:
The value of an item is $C$ dollars. After a loss of $x \%$ of the selling price, it is sold for $100. Then, it is sold again with a profit of $x \%$ of the new selling price $S$. If the difference between $S$ and $C$ is $1 \frac{1}{9}$ dollars, what is $x$?
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Context:
Consider the discrete-time state-space system:
\[x(t+1) = Ax(t)+Bu(t), \quad y(t) = Cx(t)\]
with
\[
A =\begin{bmatrix}
0 & 1 & 0 \\
1 & 0 & 1 \\
1 & 1 & 0 \\
\end{bmatrix},
\quad B =\begin{bmatrix}
0 \\
0 \\
1 \\
\end{bmatrix},
\quad C =\begin{bmatrix}
1 & 0 & 0\\
\end{bmatrix}
\]
Determine the minimal time \(T \in \mathbb{N}\) such that there exists an input signal \(u : \mathbb{N} \rightarrow \mathbb{R}\) for which the state \(x(T,0,u) = x_{f}\) with \(x_{f} = \begin{bmatrix} 0 & 1 & 1\end{bmatrix} ^{T}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
At a certain university, 70% of the professors are women, and 70% of the professors are tenured. Among the tenured professors, 15% are above the age of 60. In addition, 50% of the professors are associate or full professors. If 90% of the professors are women, tenured, or both, and 80% of the associate or full professors are men, then what percent of the male associate or full professors are tenured?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A basketball player scored 18, 22, 15, and 20 points respectively in her first four games of a season. Her points-per-game average was higher after eight games than it was after these four games. If her average after nine games was greater than 19, determine the least number of points she could have scored in the ninth game.
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Context:
在正三棱锥 $P-A B C$ 中, 已知侧棱与底面所成的角为 $45^{\circ}$ 。则相邻两侧面所成角的余弦值为 $\qquad$.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值
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Context:
Let \(y=f(x)\) be an infinitely differentiable function on real numbers such that \(f(0) \neq 0\) and \(\frac{d^n(y)}{dx^n} \neq 0\) at \(x=0\) for \(n=1,2,3,4\). If \[\lim_{x\to0} \frac{f(4x) + af(3x) + bf(2x) + cf(x) + df(0)}{x^4}\] exists, find the value of \(25a + 50b + 100c + 500d\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
From a point given on the circumference of a circle, a diameter and a chord equal to the radius are drawn. Find the angle between them.
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Context:
John is flipping his favorite bottle, which currently contains $10$ ounces of water. However, his bottle is broken from excessive flipping, so after he performs a flip, one ounce of water leaks out of his bottle. When his bottle contains $k$ ounces of water, he has a $\frac{1}{k+1}$ probability of landing it on its bottom. What is the expected number of flips it takes for John’s bottle to land on its bottom?
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Context:
Positive integers \( d, e, \) and \( f \) are chosen such that \( d < e < f \), and the system of equations
\[ 2x + y = 2010 \quad \text{and} \quad y = |x-d| + |x-e| + |x-f| \]
has exactly one solution. What is the minimum value of \( f \)?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Peter is randomly filling boxes with candy. If he has 10 pieces of candy and 5 boxes in a row labeled $A$, $B$, $C$, $D$, and $E$, how many ways can he distribute the candy so that no two adjacent boxes are empty?
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Context:
Given $p$: $-4 < x-a < 4$, and $q$: $(x-1)(2-x) > 0$. If $\neg p$ is a sufficient condition for $\neg q$, then the range of the real number $a$ is __________.
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Context:
Professor Oscar has to evaluate his students on the final exams, but this semester, he designed 5 types of exams with varying question types and components for different class sizes. Here's the breakdown of each class, the number of students, and their exams:

Class A, with 30 students:
- 12 multiple-choice questions
- 3 essay questions
- 1 individual presentation

Class B, with 25 students:
- 15 multiple-choice questions
- 5 short-answer questions
- 2 essay questions
- 1 group project with presentations, involving 5 student groups

Class C, with 35 students:
- 10 multiple-choice questions
- 3 essay questions
- 1 interactive group activity per group of 7 students

Class D, with 40 students:
- 11 multiple-choice questions
- 4 short-answer questions
- 3 essay questions
- 1 individual research paper

Class E, with 20 students:
- 14 multiple-choice questions
- 5 short-answer questions
- 2 essay questions
- 1 in-class debate per group of 4 students

Given the varying number of questions, students, and exam components for each of the 5 classes, how many evaluations must Professor Oscar review in total? Consider every component (multiple-choice, short-answer, essay, individual presentation, group project presentation, interactive group activity, individual research paper, and debate) as a separate evaluation.
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Context:
The set $\{(x, y) \in R^2| \lfloor x + y\rfloor \cdot \lceil x + y\rceil = (\lfloor x\rfloor + \lceil y \rceil ) (\lceil x \rceil  + \lfloor y\rfloor), 0 \le x, y \le 100\}$ can be thought of as a collection of line segments in the plane. If the total length of those line segments is $a + b\sqrt{c}$ for $c$ squarefree, find $a + b + c$.
($\lfloor z\rfloor$ is the greatest integer less than or equal to $z$, and $\lceil z \rceil$ is the least integer greater than or equal to $z$, for $z \in R$.)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate \(\sec \frac{2 \pi}{9} + \sec \frac{4 \pi}{9} + \sec \frac{6 \pi}{9} + \sec \frac{8 \pi}{9}\).
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Context:
Evaluate the integral \( \int_C \frac{x\mathrm{d}y-y\mathrm{d}x}{x^2+y^2} \), where \( C \) is the curve defined by the equation \( x^{48} + y^{76} = 987654321^{123456789} \).
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Context:
A car radiator contains 10 liters of a 30 % antifreeze solution. 2.85714285714 liters of the solution had to be replaced with pure antifreeze to achieve a certain percentage of antifreeze. What is the percentage of antifreeze in the resulting solution?
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Context:
The area of the largest regular hexagon that can fit inside of a rectangle with side lengths 20 and 22 can be expressed as $a \sqrt{b}-c$, for positive integers $a, b$, and $c$, where $b$ is squarefree. Compute $100 a+10 b+c$.
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Context:
8.1. Parallelogram $A B C D$ was folded along the diagonal $B D$ so that vertex $C$ remained in place, and vertex $A$ took the position $A^{\prime}$. Segments $B C$ and $A^{\prime} D$ intersected at point $K$, and $B K: K C=3: 2$. Find the area of triangle $A^{\prime} K C$, if the area of parallelogram $A B C D$ is 27.
Please reason step by step, and put your final answer within \boxed{}.


Context:
For the numerical sequence \( \{x_n\} \), where all terms starting from \( n \geq 2 \) are distinct, the relation \( x_{n} = \frac{x_{n-1} + 198 x_{n} + x_{n+1}}{200} \) holds. Find \( \sqrt{\frac{x_{2023} - x_{1}}{2022} \cdot \frac{2021}{x_{2023} - x_{2}}} + 2022 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
$M$ is the midpoint of side $BC$ in $\triangle ABC$, and $\angle MAC = 15^\circ$. Find the maximum value of $\angle B$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \(a_n\) denote the angle opposite to the side of length \(4n^2\) units in an integer right-angled triangle with side lengths \(4n^2\), \(4n^4+1\), and \(4n^4-1\), where \(n \in \mathbb{N}\). Find the value of:
\[
\lim_{p \to \infty} \sum_{n=1}^p a_n
\]
The original answer is in the format \(\frac{k\pi}{m}\). Please give the value of \(k + m\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
$48n$ is the smallest positive integer that satisfies the following conditions:
1. $n$ is a multiple of 75;
2. $n$ has exactly 75 positive divisors (including 1 and itself).

Find $\frac{n}{75}$.
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Context:
In a football tournament, 15 teams participated, each playing exactly once against every other team. A win awarded 3 points, a draw 1 point, and a loss 0 points.

After the tournament ended, it was found that some 6 teams each scored at least $N$ points. What is the maximum possible integer value of $N$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
If the sum of 5 consecutive even positive integers is some value, then the sum of the next 5 even consecutive integers in terms of that value is 50. What is the sum of the first 5 consecutive even positive integers?
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Context:
For any positive integer $x$, define $\operatorname{Accident}(x)$ to be the set of ordered pairs $(s, t)$ with $s \in \{0,2,4,5,7,9,11\}$ and $t \in\{1,3,6,8,10\}$ such that $x+s-t$ is divisible by 12. For any nonnegative integer $i$, let $a_{i}$ denote the number of $x \in\{0,1, \ldots, 11\}$ for which $|\operatorname{Accident}(x)|=i$. Find $$a_{0}^{2}+a_{1}^{2}+a_{2}^{2}+a_{3}^{2}+a_{4}^{2}+a_{5}^{2}$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the following limit: $$\lim_{n\to\infty} \sum_{k=1}^n \int_0^{\pi} e^{-nx}\cos (kx) dx$$
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Context:
Given the cross-section of cone PQ is an equilateral triangle and plane α is parallel to PQ, the curve obtained by intersecting the cone's lateral surface with plane α, denoted as C, lies on a hyperbola. Calculate the eccentricity of this hyperbola. Express your answer as a fraction or decimal value.
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Context:
The class mean score on a test was 60, and the standard deviation was 10. Jack's score is within the top 5% of his class, while also being within 2 standard deviations of the mean. What is the lowest score he could have received?
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Context:
设ΔABC的内角A、B、C的对边分别为a、b、c，且$A - C = \frac { \pi } { 2 }$ 0b、c成等差数列，则cosB的值是
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Context:
Let $A B C$ be a triangle with $A B=13, B C=14, C A=15$. Let $I_{A}, I_{B}, I_{C}$ be the $A, B, C$ excenters of this triangle, and let $O$ be the circumcenter of the triangle. Let $\gamma_{A}, \gamma_{B}, \gamma_{C}$ be the corresponding excircles and $\omega$ be the circumcircle. $X$ is one of the intersections between $\gamma_{A}$ and $\omega$. Likewise, $Y$ is an intersection of $\gamma_{B}$ and $\omega$, and $Z$ is an intersection of $\gamma_{C}$ and $\omega$. Compute $$\cos \angle O X I_{A}+\cos \angle O Y I_{B}+\cos \angle O Z I_{C}$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
The value of the definite integral $\int_{0}^{-1} (\sqrt{1-x^{2}}+x)dx$ is ______.
Please reason step by step, and put your final answer within \boxed{}.


Context:
For each integer  $1\le j\le 2017$ , let  $S_j$  denote the set of integers  $0\le i\le 2^{2017} - 1$  such that  $\left\lfloor \frac{i}{2^{j-1}} \right\rfloor$  is an odd integer. Let  $P$  be a polynomial such that
\[P\left(x_0, x_1, \ldots, x_{2^{2017} - 1}\right) = \prod_{1\le j\le 2017} \left(1 - \prod_{i\in S_j} x_i\right).\]
Compute the remainder when
\[ \sum_{\left(x_0, \ldots, x_{2^{2017} - 1}\right)\in\{0, 1\}^{2^{2017}}} P\left(x_0, \ldots, x_{2^{2017} - 1}\right)\]
is divided by  $2017$ .
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a football tournament, \(n\) teams participate. Each team plays against every other team exactly once. The winning team is awarded 2 points, a draw gives -1 point to each team, and the losing team gets 0 points. What is the maximum point difference that can exist between teams that are next to each other in the ranking?
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Context:
Euler's formula states that for a convex polyhedron with $V$ vertices , $E$ edges , and $F$ faces , $V-E+F=2$ . A particular convex polyhedron has $32$ faces, each of which is either a triangle or a pentagon . At each of its $V$ vertices, $T$ triangular faces and $P$ pentagonal faces meet. What is the value of $100P+10T+V$ ?
Please reason step by step, and put your final answer within \boxed{}.


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:
187. For which integer values of $x$ will the number $\frac{5 x+2}{17}$ be an integer?
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Context:
In the cube $ABCD-A_1B_1C_1D_1$, find the angle formed by the skew lines $AC$ and $BC_1$.
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Context:
In triangle \(ABC\), the angle bisector \(AD\) is drawn. It is known that the centers of the incircle of triangle \(ABD\) and the circumcircle of triangle \(ABC\) coincide. Find \(CD\) if \(AC = \sqrt{5} + 1\). The answer should not include the designations of trigonometric functions and their inverses.
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Context:
Starting from the natural numbers beginning with 1, remove multiples of 4, and then group the remaining numbers into sets of three. Following this pattern, determine the group number and position within the group where the number 2022 falls. Express your answer in the format "Group X, Position Y", where X is the group number and Y is the position within the group (1, 2, or 3).
Please reason step by step, and put your final answer within \boxed{}.


Context:
A point is randomly thrown onto the segment [6, 11], and let \( k \) be the resulting value. Find the probability that the roots of the equation \( \left(k^{2}-2k-15\right)x^{2}+(3k-7)x+2=0 \) satisfy the condition \( x_{1} \leq 2x_{2} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the hyperbola $C$: $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$, its right latus rectum intersects with the two asymptotes at points $A$ and $B$, and its right focus is $F$. If $\triangle ABF$ is an equilateral triangle, then the eccentricity of the hyperbola $C$ is _______.
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Context:
有多少个正整数 $a$ 满足：对任意正整数 $k$ ，存在正整数 $n$ ，使 $a n+2016$ 为某个正整数的 $k$ 次方？
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Context:
Find all three-digit numbers such that the square of the number is equal to the sum of their digits raised to the power of 5.
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Context:
Let $N$ be the number of distinct rearrangements of the 34 letters in "SUPERCALIFRAGILISTICEXPIALIDOCIOUS". How many positive factors does $N$ have?
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Context:
Aiming for the center, you throw 100 darts at a dartboard, and 25 of those darts hit within 15 mm from the center. Approximate the standard deviation of your dart throws using this information.
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Context:
Given a convex quadrilateral \(ABCD\) in which \(\angle BAC=20^\circ\), \(\angle CAD=60^\circ\), \(\angle ADB=50^\circ\), and \(\angle BDC=10^\circ\). Find \(\angle ACB\).
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Context:
Let \( f:\Bbb{R}\to\Bbb{R} \) be a continuous function defined by \( xf(x) = e^x-1 \). Find \( \lim _{n\to \infty }nf^{\left(n\right)}\left(x\right) \).
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Context:
If the function $f(x) = (3m-1)x^m$ is a power function, then $m$ equals \_\_\_\_\_\_.
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Context:
Evaluate the integral \( \int_0^{2\pi}\frac{\cos(2\theta)}{5+4\cos(\theta)}\, d\theta \) using the residue theorem.
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Context:
Find all functions \( f: \mathbf{N} \rightarrow \mathbf{N} \) such that for every \( n \in \mathbf{N} \), \( f(n) < f(n+1) \), and \( f(f(n)) = n + 2k \), where \( k \in \mathbf{N} \).
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Context:
Determine the number of pairs of vectors \( \vec{V}_1 = a_1 \mathbf{i} + b_1 \mathbf{j} + c_1 \mathbf{k} \) and \( \vec{V}_2 = a_2 \mathbf{i} + b_2 \mathbf{j} + c_2 \mathbf{k} \) such that \( \vec{V}_1 \) and \( \vec{V}_2 \) are perpendicular, where all components \( a, b, c \) are elements of \( \{-2, -1, 1, 2\} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the sum $$ \sum_{k=1}^{n} (-1)^{k+1} \binom{n-1}{k} k. $$
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Context:
Into how many non-overlapping tetrahedrons can a cube be divided?
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Context:
Determine the values of \(\alpha\) for which the improper integral \(\iint_{\Bbb R^2}\frac{dxdy}{(1+x^2+xy+y^2)^\alpha}\) converges.
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Context:
Twelve candidates for mayor are participating in a televised debate. At some point, one of them says, "So far, we've lied once." A second then says, "Now it's twice." A third exclaims, "Three times now," and so on, up to the twelfth who claims that before him, they lied twelve times. The presenter then stops the discussion. Given that at least one of the candidates correctly stated how many times lies had been told before their turn to speak, determine how many candidates lied in total.
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Context:
In each cell of a strip of length 100, there is a token. You can swap any two adjacent tokens for 1 ruble, and you can also swap any two tokens that have exactly three tokens between them for free. What is the minimum amount of rubles required to rearrange the tokens in reverse order?
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Context:
On the height \( BH \) of triangle \( ABC \), a point \( D \) is marked. The line \( AD \) intersects the side \( BC \) at point \( E \), and the line \( CD \) intersects the side \( AB \) at point \( F \). Points \( G \) and \( J \) are the projections of \( F \) and \( E \) onto side \( AC \) respectively. The area of triangle \( HEJ \) is twice the area of triangle \( HFG \). In what proportion does the height \( BH \) divide segment \( FE \)?
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Context:
Hooligan Vasya likes to run on the escalator in the subway. He runs down twice as fast as he runs up. If the escalator is not working, it takes Vasya 6 minutes to run up and down. If the escalator is moving downward, it takes him 13.5 minutes to run up and down. How many seconds will it take Vasya to run up and down the escalator if it is moving upward? (The escalator always moves at a constant speed.)
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Context:
In country Z, 18% of the people do not have a university diploma but have the job of their choice. Some percent of the people who do not have the job of their choice have a university diploma. If 40% of the people have the job of their choice and 37% of the people have a university diploma, what percent of the people who do not have the job of their choice have a university diploma?
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Context:
In a volleyball tournament for the Euro-African cup, there were nine more teams from Europe than from Africa. Each pair of teams played exactly once, and the European teams won exactly nine times as many matches as the African teams overall. Determine the maximum number of matches that a single African team could have won.
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Context:
Consider a Markov chain with parameters $\alpha,\beta\in(0,1)$, where the variables $X_0,X_1,\dots$ are in the interval $[0,1]$. Given $X_n$, the value of $X_{n+1}$ is either \[ \frac{\alpha X_n}{\alpha X_n + \beta (1-X_n)} \] with probability $\alpha X_n+\beta (1-X_n)$, or \[ \frac{(1-\alpha)X_n}{(1-\alpha) X_n + (1-\beta) (1-X_n)} \] with probability $(1-\alpha)X_n+(1-\beta) (1-X_n)$. If $X_0=0.01$, what is the probability that for a large $n$, $X_n>0.99$?
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Context:
Find the non-negative integer value of $a$ such that the following limit holds:
\[ \lim_{x \to 1 }\left\{ \frac{-ax+\sin(x-1)+a}{x+\sin(x-1)-1} \right\}^{\frac{1-x}{1-\sqrt{x}}} = \frac{1}{4} \]
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Context:
Determine the number of relatively compact connected components of the set \( X \setminus K \), where \( X = \{(x,y) : x^2 + y^2 < 5\} \) and \( K = \{(x,y) : 1 \leq x^2 + y^2 \leq 2 \text{ or } 3 \leq x^2 + y^2 \leq 4\} \).
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Context:
If we write every day of 2014 in the form of an eight-digit number, for example, 20140125 represents January 25, 2014, how many eight-digit numbers have the digits '1', '2', '0' appearing the same number of times?
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Context:
Determine the conditional probability $P(x > 0 \mid x > y)$, given that $x$ and $y$ are independent variables following a standard Gaussian distribution $N(0,1)$.
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Context:
Given that the prime factorization of a positive integer \( A \) can be written as \( A = 2^{\alpha} \times 3^{\beta} \times 5^{\gamma} \), where \( \alpha, \beta, \gamma \) are natural numbers. If half of \( A \) is a perfect square, one-third of \( A \) is a perfect cube, and one-fifth of \( A \) is a fifth power of some natural number, determine the minimum value of \( \alpha + \beta + \gamma \).
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Context:
数列$\left\{a_n\right\}$ 满足 $a_1=1$，前 $n$ 项和为 $S_n$，且$S_{n+1}=4a_n+2$。求 $a_{2013}$ 的值。假设答案可以表示为$x \times 2^{y}$的形式，其中$y$是指数，且$2$为底数，请计算$x + y$的值。
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Context:
On a $5 \times 5$ board, two players alternately mark numbers on empty cells. The first player always marks 1's, the second 0's. One number is marked per turn, until the board is filled. For each of the nine $3 \times 3$ squares, the sum of the nine numbers on its cells is computed. Denote by \( A \) the maximum of these sums. How large can the first player make \( A \), regardless of the responses of the second player?
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Context:
For some positive integer $n$, consider the board $n\times n$. On this board you can put any rectangles with sides along the sides of the grid. What is the smallest number of such rectangles that must be placed so that all the cells of the board are covered by distinct numbers of rectangles (possibly $0$)? The rectangles are allowed to have the same sizes.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of ideals $A$ in the ring $R$, where $R$ is the integral closure of $\mathbb{Z}$ in $K$ and $K=\mathbb{Q}[\sqrt{-6}]$, such that the norm $N(A) = 2^2 \cdot 11^3$. Given that $2R=\mathfrak{p}^2$ and $11R=\mathfrak{q}_1\mathfrak{q}_2$ for suitable distinct prime ideals $\mathfrak{p}, \mathfrak{q}_1, \mathfrak{q}_2$, find the number of such ideals $A$. Assume $A=\mathfrak{p}^a\mathfrak{q}_1^b\mathfrak{q}_2^c\mathfrak{r}$, where $a,b,c \in \mathbb{N}$ and $\mathfrak{r}$ is coprime with $\mathfrak{p}\mathfrak{q}_1\mathfrak{q}_2$. The norm condition implies $2^2 \cdot 11^3 = N(\mathfrak{p}^a\mathfrak{q}_1^b\mathfrak{q}_2^c\mathfrak{r}) = 2^{2a}11^{b+c}N(\mathfrak{r})$. Determine the values of $a$, $b$, and $c$ that satisfy this condition.
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Context:
Jim starts with a positive integer $ n$ and creates a sequence of numbers. Each successive number is obtained by subtracting the largest possible integer square less than or equal to the current number until zero is reached. For example, if Jim starts with $ n=55$, then his sequence contains $ 5$ numbers:
$\begin{aligned}
&55\\
55-7^2=&\ 6\\
6-2^2=&\ 2\\
2-1^2=&\ 1\\
1-1^2=&\ 0
\end{aligned}$
Let $ N$ be the smallest number for which Jim's sequence has 8 numbers. What is the units digit of $ N$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
The tax on a commodity is diminished by 24%, but its consumption is increased by some percentage. The decrease percent in the revenue derived from it is 14.88%. What is the percentage by which the consumption is increased?
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Context:
Find the slope of the curve at the point (1, 1) formed by the intersection of the surface $z = x^2 + y^2$ and the plane $x + y = 2$. What is the slope at this point?
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Context:
Let $[x]$ denote the greatest integer less than or equal to $x$. Determine the value of $\sum_{k=1}^{2008}\left[\frac{2008 k}{2009}\right]$.
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Context:
Given a function $f:[0,\infty)\to\mathbb{R}$, where $f$ is an increasing function, and its primitive $F$ satisfies $F(0)=0$ and $F(x+y)\leq F(x)+F(y)$ for all $x,y \geq 0$, determine the function $f$.
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Context:
In a triangle inscribed in a circle, perpendicular lines are drawn from a point inside the circle to each side of the triangle. Determine the maximum area of the triangle formed by the intersections of these perpendicular lines with the sides of the original triangle.
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Context:
Cédric needs to write down a phrase in French. He knows that there are $3$ words in the phrase and that the letters appear in the order: \[cesontoiseaux.\] He does not remember what the words are and which letters have what accents in the phrase. If $n$ is the number of possible phrases that he could write down, then determine the number of distinct primes in the prime factorization of $n$.
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Context:
设$n$为正整数，找到一组数$a、b$，使得$\frac{b}{a}$为既约分数，并且满足$a=3n+1, b=5n+2$。假设答案的形式为\frac{m*n+k}{n*m+l}，请计算m+k+l的值。
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Context:
Determine the order of growth for the infinite series \(\sum_{i=1}^{k}n^{\frac{2^i-1}{2^i}}\) as \(k\) approaches infinity, where \(n\) is a constant.
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Context:
The angles in a particular triangle are in arithmetic progression, and the side lengths are $4, 5, x$. The sum of the possible values of x equals $a+\sqrt{b}+\sqrt{c}$ where $a, b$, and $c$ are positive integers. What is $a+b+c$?
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Context:
6 Let $X$ be a 56-element set. Find the smallest positive integer $n$, such that for any 15 subsets of $X$, if the union of any 7 of them has at least $n$ elements, then there must exist 3 of these 15 subsets whose intersection is non-empty.
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Context:
$ f: \mathbb N \times \mathbb Z \rightarrow \mathbb Z$ satisfy the given conditions

$ a）$ $ f（0,0）=1$ , $ f（0,1）=1$ ,

$ b）$ $ \forall k \notin \left\{0,1\right\}$ $ f（0，k）=0$ and

$ c）$ $ \forall n \geq 1$ and $ k$ , $ f（n，k）=f（n-1，k）+f（n-1，k-2n）$

find the sum $ \displaystyle\sum_{k=0}^{\binom{2009}{2}}f(2008,k)$
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Context:
Let  $a$ ,  $b$ ,  $c$ , and  $d$  be positive real numbers such that
\[a^2 + b^2 - c^2 - d^2 = 0 \quad \text{and} \quad a^2 - b^2 - c^2 + d^2 = \frac{56}{53}(bc + ad).\]
Let  $M$  be the maximum possible value of  $\tfrac{ab+cd}{bc+ad}$ . If  $M$  can be expressed as  $\tfrac{m}{n}$ , where  $m$  and  $n$  are relatively prime positive integers, find  $100m + n$ .

*Proposed by Robin Park*
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Context:
At a baking contest, five different types of cakes were evaluated: unicorn, witch, dragon, mermaid, and fairy. The votes were distributed in the following manner:

- The unicorn cake received three times as many votes as the witch cake.
- The dragon cake received 40% more votes than the witch cake.
- The mermaid cake received 7 fewer votes than the dragon cake, and twice as many votes as the fairy cake.
- The fairy cake received 5 votes less than the witch cake.

If the number of votes for the witch cake was 12, determine the total number of votes cast during the contest.
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Context:
The eccentricity of the ellipse $\frac {x^{2}}{9}+ \frac {y^{2}}{4+k}=1$ is $\frac {4}{5}$. Find the value of $k$.
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Context:
Given the parabola $C$: $y^{2}=6x$ with its focus $F$, $B$ is a point on the directrix of $C$, and $A$ is an intersection point of line $BF$ and $C$. If $\overrightarrow{FB}=3\overrightarrow{FA}$, find the distance from point $A$ to the origin.
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Context:
Calculate the power over a period for a square wave represented by its Fourier series. Given the voltage function:
\[ v(t) = \frac{4V}{\pi} \left( \sin(\omega t) + \frac{1}{3}\sin(3\omega t) + \frac{1}{5}\sin(5\omega t) + \ldots \right) \]
where \( \omega \) and \( V \) are constants, find the power \( P \) over a period \( T \) using the formula:
\[ P = \frac{1}{T} \int_{0}^{T} \frac{v(t)^2}{R} \, dt \]
where \( R \) is a constant resistance.
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Context:
Vasya thought of a natural number not greater than 100. Fedya asks Vasya questions of the type "What is the quotient of the division of the thought number by \( m \) ?", where \( m \) is a natural number, not smaller than 2, and Fedya chooses \( m \) himself. What is the minimum number of such questions Fedya needs to ask to guarantee determining the number thought by Vasya?
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Context:
Rational numbers $a$ and $b$ are chosen at random among all rational numbers in the interval $[0,2)$ that can be written as fractions $\frac{n}{d}$ where $n$ and $d$ are integers with $1 \le d \le 5$. What is the probability that
<cmath>(\text{cos}(a\pi)+i\text{sin}(b\pi))^4</cmath>
is a real number?
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Context:
43. (FIN 1) Evaluate
$$ S=\sum_{k=1}^{n} k(k+1) \cdots(k+p), $$
where $n$ and $p$ are positive integers.
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Context:
A parking lot has a row of eight parking spaces numbered from 1 to 8. Now, 2 trucks and 2 cars are to be parked at the same time, with each parking space accommodating at most one vehicle. If vehicles of the same type must be parked in adjacent parking spaces, there are a total of ______ parking arrangements.
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Context:
Let \( a \) be a given complex number satisfying \( 0 < \left\vert a\right\vert < 2 \) and let \( \gamma \) be the unit circle of radius 3 oriented in the positive sense. Determine the value of the contour integral \( \oint_{\gamma}{\Re\left(z\right) \over z - a}\,\mathrm{d}z \) in terms of \( a \).
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Context:
In the convex pentagon $ABCDE$, the angles $\angle A$ and $\angle C$ are both $90^{\circ}$. The sides $AB$ and $AE$ are equal, as well as the sides $BC$ and $CD$. The length of the diagonal $AC$ is $1$. Find the area of the pentagon.
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Context:
[ Triangle inequality (miscellaneous). ] [ Minimum or maximum distance (length).]

Petya bought a "Constructor" set, which contained 100 sticks of different lengths. The instructions for the "Constructor" state that any three sticks from the set can form a triangle. Petya decided to test this statement by forming triangles with the sticks. The sticks are arranged in the constructor in ascending order of length. What is the
minimum number of checks (in the worst case) that Petya needs to make to prove or disprove the statement in the instructions?
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Context:
Arrange the 5 natural numbers from 1 to 5 in a row from left to right, in such a way that starting from the third number, each number is either the sum or difference of the previous two numbers. How many valid arrangements are there? Express your answer as a single integer.
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Context:
Circles with radii \( r \) and \( R \) touch each other externally at point \( K \). A line touches these circles at different points \( A \) and \( B \).

Find the area of triangle \( A K B \).
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Context:
There are $2012$ backgammon checkers (stones, pieces) with one side black and the other side white. These checkers are arranged in a line such that no two consecutive checkers are the same color. At each move, we choose two checkers and flip them along with all the checkers between them. What is the minimum number of moves required to make all checkers the same color?
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Context:
Given that the sides of $\triangle ABC$ are $13$, $14$, and $15$. There are 4 circles $O$, $O_1$, $O_2$, and $O_3$ with the same radius $r$ placed inside $\triangle ABC$. Circle $O_1$ is tangent to sides $AB$ and $AC$, circle $O_2$ is tangent to sides $BA$ and $BC$, circle $O_3$ is tangent to sides $CB$ and $CA$, and circle $O$ is tangent to circles $O_1$, $O_2$, and $O_3$. Find $r$.
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Context:
定义在$R$上的函数$y=f\left( x-1 \right)$的图像关于$\left( 1,0 \right)$对称，且当$x\in \left( -\infty ,0 \right)$时，$f\left( x \right)+x{f}'\left( x \right)<0$（其中${f}'\left( x \right)$是$f\left( x \right)$的导函数），若$a=\left( {{3}^{0.3}} \right)\cdot f\left( {{30.3}} \right),b=\left( \text{lo}{{\text{g}}_{\pi }}3 \right)\cdot f\left( \text{lo}{{\text{g}\pi }}3 \right),$ $c=\left( \text{lo}{{\text{g}}_{3}}\frac{1}{9} \right)\cdot f\left( \text{lo}{{\text{g}3}}\frac{1}{9} \right)$，
则$a,b,c$的大小关系是________．
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Context:
Place eight spheres, each with a radius of 1, in two layers inside a cylinder. Ensure that each sphere is tangent to its four neighboring spheres, as well as tangent to one base and the lateral surface of the cylinder. Determine the height of the cylinder.
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Context:
A group of \( n \) friends participated in a math contest consisting of eight short-answer problems \( S_1, S_2, S_3, S_4, S_5, S_6, S_7, S_8 \) and four full-solution problems \( F_1, F_2, F_3, F_4 \). Each person in the group correctly solved exactly 11 out of the 12 problems.

We create an \( 8 \times 4 \) table. Inside the square located in the \( i \)-th row and \( j \)-th column, we write down the number of people who correctly solved both problem \( S_i \) and \( F_j \). If the 32 entries in the table sum to 256, what is the value of \( n \)?
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Context:
A positive integer $n$ is loose if it has six positive divisors and satisfies the property that any two positive divisors $a<b$ of $n$ satisfy $b \geq 2 a$. Compute the sum of all loose positive integers less than 100.
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Context:
If there is a 95% confidence that events A and B are related, determine the condition that the test statistic K^2 must satisfy. Express your answer as an inequality involving K^2 and the specific numerical value.
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Context:
Find the value of \(2 - 13 \cos 2\alpha + \sin^{-1}(2\alpha)\), given that \(\operatorname{ctg} \alpha = -\frac{1}{5}\).
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Context:
A triangle with vertices $(6, 5)$, $(8, -3)$, and $(9, 1)$ is reflected about the line $x=8$ to create a second triangle.  What is the area of the union of the two triangles?
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Context:
Find the sum of the infinite series $\frac{1}{2\cdot 4}+\frac{1\cdot3}{2\cdot4\cdot6}+\frac{1\cdot3\cdot5}{2\cdot4\cdot6\cdot8}+\dots$
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Context:
Find the smallest four-digit number SEEM for which there is a solution to the puzzle MY + ROZH = SEEM. (The same letters correspond to the same digits, different letters - different.)
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Context:
Consider the graph in 3-space of
\[0 = xyz(x + y)(y + z)(z + x)(x - y)(y - z)(z - x).\]
This graph divides 3-space into \(N\) connected regions. What is \(N\)?
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Context:
Two equilateral triangles are contained in a square whose side length is \(2\sqrt{3}\). The bases of these triangles are the opposite sides of the square, and their intersection is a rhombus. The area of the rhombus is in the form \(k\sqrt{3} - m\). Please find the value of k + m.
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Context:
Given an infinite geometric sequence \{a\_n}(n∈N*) with the first term a\_1=1 and common ratio q=13\*, find the limit of the sum of the first n terms S\_n, i.e., limn→∞Sn=\_\_\_\_\_\_.
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Context:
Randomly select two points on a line segment of length 3 to divide it into three segments. Calculate the probability that exactly two of these segments are longer than 1. Express your answer as a fraction in simplest form.
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Context:
How many distinct ways can we color the faces of a tetrahedron using four different colors, taking into account the symmetries of the shape (orientation-preserving only)? Apply Burnside's orbit-counting theorem, where $G$ is the symmetry group of the tetrahedron, $X_g$ is the number of colorings fixed by element $g \in G$, and $r$ is the number of orbits.
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Context:
Let \( A, B, C, \) and \( D \) be four points lying in this order on a circle. Assume that there is a point \( K \) on the line \( AB \) such that \( BD \) bisects the line \( KC \) and \( AC \) bisects the line \( KD \). Determine the smallest possible value that \( \left|\frac{AB}{CD}\right| \) can take.
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Context:
已知复数$z _ { 1 } , z _ { 2 }$满足$\vert z _ { 1 } \vert \geq 1 ,$,$\vert z _ { 2 } \vert \geq \frac { 3 } { 2 }$.则复数$i ^ { 2 0 0 1 } z _ { 1 } + i ^ { 1 9 9 9 } z _ { 2 } + 2 z _ { 1 } z _ { 2 }$的模长的最小值是.
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Context:
A small sphere with a radius of 1 is placed inside a closed cubical box with an edge length of 6. If the box is arbitrarily turned, find the volume of the space that the sphere cannot reach.
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Context:
已知 a_k 为 \log_2k, \log_3k, \cdots, \log_{2018}k 中整数项的个数，则 \ds{k=1}{2018}a_k=__________.
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Context:
A kite has sides $15$ units and $20$ units meeting at a right angle, and its diagonals are $24$ units and $x$ units. Find the area of the kite.
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Context:
Given a hyperbola E with vertices A and B, and a point M on E forming an isosceles triangle ABM with a 120° vertex angle, calculate the eccentricity of E. Express your answer as a numerical value, including the square root symbol if necessary.
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Context:
Problem 6. A row of new recruits stood facing the sergeant. On the command "left," some turned left, while the rest turned right. It turned out that six times more soldiers were looking at the back of their neighbor than in the face. Then, on the command "about face," everyone turned in the opposite direction. Now, seven times more soldiers were looking at the back of their neighbor than in the face. How many soldiers are in the row?

$[8$ points] (A. V. Shipovalov)
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Context:
1. (YUG) Find all real numbers $x \in[0,2 \pi]$ such that
$$
2 \cos x \leq|\sqrt{1+\sin 2 x}-\sqrt{1-\sin 2 x}| \leq \sqrt{2} \text {. }
$$
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Context:
How much work is required to slam a solid wooden door with a mass of $24 \mathrm{~kg}$ so that its edge hits the door frame at a speed of $5 \mathrm{~m} / \mathrm{s}$?
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Context:
What is the minimum value of the expression $$\left\lfloor\frac{a+b+c}{d}\right\rfloor+\left\lfloor\frac{a+b+d}{c}\right\rfloor+\left\lfloor\frac{a+d+c}{b}\right\rfloor+\left\lfloor\frac{d+b+c}{a}\right\rfloor$$ where $a, b, c,$ and $d$ are positive real numbers?
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Context:
已知集合$A = \{ x \vert x = a _ { 0 } + a _ { 1 } \times 8 + a _ { 2 } \times 8 ^ { 2 } + a _ { 3 } \times 8 ^ { 3 } \}$，其中

$a _ { i } \in \{ 0 ,$,1,2,3,4,5,6,7}, i=0,1,2,3

且$a _ { 3 } \neq 0$，若正整数m,n∈A满足m+n=2018，且$m _ { . } > n$，则符合条件的正整数m有＿个（请用数字作答）
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Context:
Two adjacent coins (neighboring by side) weigh \$9\$ grams each, and the remaining coins weigh \$10\$ grams each. What is the minimum number of weighings on a scale, which shows the total weight in grams, needed to identify these two coins?
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Context:
Karcsi has 10 identical balls: 5 are red, 3 are white, and 2 are green. He also has two boxes, one of which can hold 4 balls and the other can hold 6 balls. In how many ways can he distribute the balls between the two boxes? (The arrangement of balls within the boxes does not matter.)
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Context:
Let  $k$  be a positive integer. A sequence of integers  $a_1, a_2, \cdots$  is called  $k$ -pop if the following holds: for every  $n \in \mathbb{N}$ ,  $a_n$  is equal to the number of distinct elements in the set  $\{a_1, \cdots , a_{n+k} \}$ . Determine, as a function of  $k$ , how many  $k$ -pop sequences there are.
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Context:
There are 8 seats in a row. If each seat can only be occupied by 1 person, and 4 people are seated, the number of different ways to arrange the seating such that exactly two empty seats are adjacent is ______ (provide your answer as a number).
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Context:
In triangle $\triangle ABC$, $AB=AC=10cm$, $\angle A=60^{\circ}$, then $BC=$____.
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Context:
An eager young acoustic engineering student is shadowing a seasoned consultant on a project to design a new concert hall. The student is tasked with calculating the optimal number of sound panels needed for a section of the hall to ensure the best acoustics. The hall section is rectangular and measures 60 feet in length and 40 feet in width. Each sound panel covers an area of 5 square feet. If the consultant advises that 20% of the total wall area should be covered with sound panels, how many sound panels does the student need to recommend for this section?
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Context:
The teacher invented a puzzle by replacing the example \(a + b = c\) with the addition of two natural numbers with their digits replaced by letters: identical digits are replaced by identical letters, and different digits by different letters. (For example, if \(a = 23\), \(b = 528\), then \(c = 551\), resulting in the puzzle \(A B + V A \Gamma = VVD\) precisely up to the choice of letters). It turned out that the original example can be uniquely restored from the given puzzle. Find the smallest possible value of the sum \(c\).
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Context:
设平面上有 \(n\) 个点，任取出三点，均能被单位圆覆盖，且存在三点不能被恰好 1 个单位圆覆盖。求可能覆盖这些点的最小圆半径。
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Context:
The administrator accidentally mixed up the keys for 10 rooms. If each key can only open one room, what is the maximum number of attempts needed to match all keys to their corresponding rooms?
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Context:
Given a parabola $y^2=2px$ ($p>0$) with focus $F$, a circle is drawn with $F$ as the center and $p$ as the radius, intersecting the $y$-axis at points $A$ and $B$. Connect $F$ to $A$, intersecting the parabola at point $D$ (which lies on segment $FA$). Extend $FA$ to intersect the parabola's directrix at point $C$. If $|AD|=m$ and $m\in[1,2]$, determine the maximum value of $|FD|\cdot|CD|$.
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Context:
Given a sequence $\{a_n\}$, where $a_1=1$ and $a_{n+1}=3a_n+1$ for $n\in\mathbb{N}^*$, let $b_n= \frac{n}{(3^n-1)\cdot 2^{n-2}}\cdot a_n$, and let $T_n$ be the sum of the first $n$ terms of $\{b_n\}$. If the inequality $2^n\lambda < 2^{n-1}T_n+n$ holds for all $n\in\mathbb{N}_+$, then the range of the real number $\lambda$ is \_\_\_\_\_\_.
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Context:
To participate in the school club tour event, the public welfare club is preparing to purchase watercolor pens and signature pens for $105$ yuan. It is known that the unit prices of watercolor pens and signature pens are $5$ yuan per box and $7$ yuan per box, respectively. Therefore, the club can buy a maximum of ____ boxes of these two types of pens.
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Context:
Each edge of a regular tetrahedron is divided into three equal parts. Through each point of division, two planes are drawn, each parallel to one of the two faces of the tetrahedron that do not pass through this point. Into how many parts do the constructed planes divide the tetrahedron?
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