Context:
Find the largest positive integer  $n$  with the following property: there are rectangles  $A_1, ... , A_n$  and  $B_1,... , B_n,$  on the plane , each with sides parallel to the axis of the coordinate system, such that the rectangles  $A_i$  and  $B_i$  are disjoint for all  $i \in \{1,..., n\}$ , but the rectangles  $A_i$  and  $B_j$  have a common point for all  $i, j \in \{1,..., n\}$ ,  $i \ne j$ .

*Note: By points belonging to a rectangle we mean all points lying either in its interior, or on any of its sides, including its vertices*
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Context:
The sum of Jim's weight and Bob's weight is 180 pounds. If you subtract Jim's weight from Bob's weight, you get half of Bob's weight. How many pounds does Bob weigh?
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Context:
Given that the positive real numbers \(a, b, c\) satisfy the equation \(a^{2}+4 b^{2}+9 c^{2}=4 b+12 c-2\), find the minimum value of \(\frac{1}{a}+\frac{2}{b}+\frac{3}{c}\).
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Context:
Given the function
$$
f(x) = \left|8x^3 - 12x - a\right| + a
$$

The maximum value of this function on the interval \([0, 1]\) is 0. Find the maximum value of the real number \(a\).
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Context:
Let \( A \subseteq \{0, 1, 2, \cdots, 29\} \) such that for any integers \( k \) and any numbers \( a \) and \( b \) (possibly \( a = b \)), the expression \( a + b + 30k \) is not equal to the product of two consecutive integers. Determine the maximum possible number of elements in \( A \).
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Context:
Determine the maximum possible value of $F(x)$ for functions $F(x)$ satisfying $F(x)=f(x)^2+f'(x)^2$ such that $|f(x)| \le 2$ and $f(x)$ is thrice differentiable.
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Context:
Let \( n = \overline{abc} \) be a three-digit number, where \( a, b, \) and \( c \) are the digits of the number. If \( a, b, \) and \( c \) can form an isosceles triangle (including equilateral triangles), how many such three-digit numbers \( n \) are there?
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Context:
Let $k$ be the product of every third positive integer from $2$ to $2006$, that is, \[ k = 2 \cdot 5 \cdot 8 \cdot 11 \cdots 2006. \] Find the number of zeros at the right end of the decimal representation for $k$.
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Context:
The function $f: \mathbb{Z}^{2} \rightarrow \mathbb{Z}$ satisfies - $f(x, 0)=f(0, y)=0$, and - $f(x, y)=f(x-1, y)+f(x, y-1)+x+y$ for all nonnegative integers $x$ and $y$. Find $f(6,12)$.
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Context:
Call a $7$-digit telephone number $d_1d_2d_3-d_4d_5d_6d_7$ memorable if the prefix sequence $d_1d_2d_3$ is exactly the same as either of the sequences $d_4d_5d_6$ or $d_5d_6d_7$ (possibly both). Assuming that each $d_i$ can be any of the ten decimal digits $0, 1, 2, \ldots, 9$, the number of different memorable telephone numbers is
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Context:
Given a differentiable function $f(x)$ defined on the interval $(0,+\infty)$, with its derivative denoted by $f'(x)$, and satisfying $xf'(x) + 2f(x) > 0$, find the solution set of the inequality $\frac{(x+2016)f(x+2016)}{5} < \frac{5f(5)}{x+2016}$. Express your answer in interval notation.
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Context:
Compute the telescoping sum $\sum_{n=2}^{\infty}\frac 12 (\frac{1}{4.6…(3n-2)} - \frac{1}{4.6…(3n+1)})$.
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Context:
Given a trapezoid \(ABCD\) with bases \(AD = a\) and \(BC = b\). Points \(M\) and \(N\) lie on sides \(AB\) and \(CD\) respectively, with the segment \(MN\) parallel to the bases of the trapezoid. Diagonal \(AC\) intersects this segment at point \(O\). Find \(MN\), given that the areas of triangles \(AMO\) and \(CNO\) are equal.
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Context:
A regular pentagon with area \(\sqrt{5}+1\) is printed on paper and cut out. The five vertices of the pentagon are folded into the center of the pentagon, creating a smaller pentagon. What is the area of the new pentagon? The original answer is in the form of \(n - \sqrt{m}\). Please find the value of m + n.
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Context:
A thin piece of wood of uniform density in the shape of an equilateral triangle with side length $3$ inches weighs $12$ ounces. A second piece of the same type of wood, with the same thickness, also in the shape of an equilateral triangle, has a side length of $5$ inches. Find the weight, in ounces, of the second piece that is closest to an integer value. If the weight is w ounces, please provide the value of 10w.
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Context:
A square flag features a green cross of uniform width, and a yellow square in the center, against a white background. The cross is symmetric with respect to each of the diagonals of the square. Suppose the entire cross (including the green arms and the yellow center) occupies 49% of the area of the flag. If the yellow center itself takes up 4% of the area of the flag, what percent of the area of the flag is green?
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Context:
If some percentage of (x - y) = 20% of (x + y), then y is 42.857142857142854% of x. What is the percentage of (x - y)?
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Context:
We spill the content of a box of paprika-flavored and a box of bear's garlic-flavored, eight-slice Maci cheese onto a table, then put the 16 small cheese slices (with their labels facing up) back into the boxes. In how many different ways can we do this if arrangements that can be transformed into each other by rotation are not considered different, but the two boxes are distinguished from each other?
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Context:
There were some roses and 3 orchids in the vase. Jessica cut some more roses and orchids from her flower garden. There are now 12 roses and 2 orchids in the vase. There are 10 more roses than orchids in the vase now. How many roses were there in the vase initially?
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Context:
In $\triangle ABC$, $\sin^{2} A - \sin^{2} C = (\sin A - \sin B) \sin B$. Find the measure of angle $C$.
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Context:
Let  $ S \equal{} \{1,2,\ldots,2008\}$ . For any nonempty subset  $ A\in S$ , define  $ m(A)$  to be the median of  $ A$  (when  $ A$  has an even number of elements,  $ m(A)$  is the average of the middle two elements). Determine the average of  $ m(A)$ , when  $ A$  is taken over all nonempty subsets of  $ S$ .
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Context:
Let a,b,c, and d be real numbers such that \[a^2 + 3b^2 + \frac{c^2+3d^2}{2} = a + b + c+d-1.\] Find $1000a + 100b + 10c + d$.
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Context:
Let a constant \( a \in \mathbf{R} \) and the function
\[ f(x) = (a - x)|x| \]

have an inverse function \( f^{-1}(x) \). If the inequality
\[ f^{-1}\left(x^{2} + m\right) < f(x) \]

holds for all \( x \in [-2,2] \), find the range of values for the real number \( m \).
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Context:
Line segments drawn from the vertex opposite the hypotenuse of a right triangle to the points trisecting the hypotenuse have lengths $\sin x$ and $\cos x$, where $x$ is a real number such that $0<x<\frac{\pi}{2}$. The length of the hypotenuse is
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Context:
If $\sqrt[3]{0.3}\approx 0.6694$ and $\sqrt[3]{3}\approx 1.442$, then $\sqrt[3]{300}\approx$____.
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Context:
A fat coin is one which, when tossed, has a $2 / 5$ probability of being heads, $2 / 5$ of being tails, and $1 / 5$ of landing on its edge. Mr. Fat starts at 0 on the real line. Every minute, he tosses a fat coin. If it's heads, he moves left, decreasing his coordinate by 1; if it's tails, he moves right, increasing his coordinate by 1. If the coin lands on its edge, he moves back to 0. If Mr. Fat does this ad infinitum, what fraction of his time will he spend at 0?
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Context:
Congcong performs a math magic trick by writing the numbers $1, 2, 3, 4, 5, 6, 7$ on the blackboard and lets others select 5 of these numbers. The product of these 5 numbers is then calculated and told to Congcong, who guesses the chosen numbers. If when it is Benben's turn to select, Congcong cannot even determine whether the sum of the 5 chosen numbers is odd or even, what is the product of the 5 numbers chosen by Benben?
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Context:
Two distinct numbers \(a\) and \(b\) are chosen randomly from the set \(\{2, 2^2, 2^3, \cdots, 2^{25}\}\). Find the probability that \(\log_a b\) is an integer. The original answer is in \(\frac{k}{m}\) format, where the fraction is in simplest form. Please provide the value of \(k + m\).
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Context:
Jesse and Mia are competing in a week long race. They have one week to run 30 miles. On the first three days Jesse averages (2/3) of a mile. On day four she runs a certain distance. Mia averages 3 miles a day over the first 4 days. The average of their average that they have to run over the final three days is 6 miles. How many miles did Jesse run on day four?
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Context:
Let $S = \{1, 2, \ldots, 100\}$. Consider a partition of $S$ into $S_1, S_2, \ldots, S_n$ for some $n$, i.e., $S_i$ are nonempty, pairwise disjoint sets, and $S = \bigcup_{i=1}^n S_i$. Let $a_i$ be the average of the elements of the set $S_i$. Define the score of this partition by:
\[
\frac{a_1 + a_2 + \ldots + a_n}{n}.
\]

Among all $n$ and partitions of $S$, determine the minimum possible score.
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Context:
In a village of 2011 dwarfs, each dwarf is either wearing a red hat or a blue hat. Dwarfs with red hats always tell the truth, while dwarfs with blue hats can lie. Additionally, they can change the color of their hats at any time. One day, each dwarf encountered every other dwarf exactly once, and each one claimed that the other was wearing a blue hat. What is the minimum number of times the dwarfs changed the color of their hats that day?
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Context:
On the first day, Johnny spent 3 hours working on a job that paid $7 per hour, 2 hours working on a job that paid $10 an hour, and 4 hours working on a job that paid $12 an hour. On the second day, his pay rate for the first job increased by $2, for the second job it decreased by $1, and the pay rate for the third job stayed the same. For the next three days (days 3, 4, and 5), his pay rates changed as follows: on day 3, the pay rates for the first two jobs returned to their initial rates, and the third job pay increased by $3; on day 4, the first job pay decreased by $3, the second job pay increased by $2, and the third job pay remained the same as day 3; and on day 5, the pay rates for all jobs returned to their initial rates.

Johnny also has work-related expenses, which include a daily $5 lunch and a weekly $20 transportation cost. Additionally, he must pay 15% taxes on his total earnings for the week. Calculate Johnny's net income for the week considering his pay rates, taxes, and work-related expenses.
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Context:
A convex broken line of length \( d \) rotates around a straight line connecting its ends. What can the area of the surface of rotation obtained in this way be?
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Context:
已知圆内接四边形 ABCD 的边长分别为 $\mathrm{AB}=1, \mathrm{BC}=2, \mathrm{CD}=3, \mathrm{DA}=4$. 求四边形 ABCD 外接圆的半径。原始答案为 $\frac{\sqrt{m}}{n}$ 的形式，请给出 m + n 的值。
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Context:
There are three solutions with different percentages of alcohol. If you mix them in a ratio of $1:2:3$, you get a $20\%$ solution. If you mix them in a ratio of $5:4:3$, you will get a solution with $50\%$ alcohol content. What percentage of alcohol will the solution contain if equal amounts of the original solutions are mixed?
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Context:
Jeanette is practicing her juggling. Each week she can juggle 2 more objects than the week before. She practices 3 times a week, and with each practice session, she can juggle 1 extra object than the previous session that week. If she starts out juggling 3 objects and practices for 5 weeks, how many objects can she juggle by the end of the 5th week?
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Context:
In the following diagram, two lines that meet at a point \( A \) are tangent to a circle at points \( B \) and \( C \). The line parallel to \( AC \) passing through \( B \) meets the circle again at \( D \). Join the segments \( CD \) and \( AD \). Suppose \( AB = 49 \) and \( CD = 28 \). Determine the length of \( AD \).
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Context:
Evaluate the limit of the double integral as \( n \to \infty \):

\[ \lim_{n \to \infty} \int_{x=-\infty}^{x=\infty} \int_{y=x-\frac{1}{n}}^{y=x+\frac{1}{n}} dy\ dx \]
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Context:
Find all real values of $y$ that satisfy $\frac{1}{y(y+2)} - \frac{1}{(y+2)(y+4)} < \frac{1}{4}$.
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Context:
求最大的正整数 E，使得存在一个有 60 个顶点、E 条边的简单图 G，并且可以将 G 的边红蓝二染色，满足没有长度是 3 或 5 的同色圈.
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Context:
Evaluate the limit \( I = \lim_{x \to 0^{+}} \frac{\int_{0}^{x^2} t \sin{(\sqrt{x^2-t^2})} \, dt}{x^3} \).
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Context:
Find the sum of the series \( \sum_{k=1}^{n} (-1)^{k+1} \frac{1}{k} \binom{n}{k} \) for \( n \ge 1 \).
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Context:
Let \(p,\) \(q,\) and \(r\) be constants, and suppose that the inequality \[\frac{(x-p)(x-q)}{x-r} \le 0\] is true if and only if either \(x < -6\) or \(|x-30| \le 2.\) Given that \(p < q,\) find the value of \(p + 2q + 3r.\)
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Context:
2. Given the line $L: x+y-9=0$ and the circle $M: 2 x^{2}+2 y^{2}-8 x-8 y-1=0$, point $A$ is on line $L$, $B, C$ are two points on circle $M$, in $\triangle A B C$, $\angle B A C=45^{\circ}, A B$ passes through the center of circle $M$, then the range of the x-coordinate of point $A$ is $\qquad$
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Context:
there is a church tower 150 feet tall and another catholic tower at a distance of 350 feet from it which is 200 feet tall . there is one each bird sitting on top of both the towers . they fly at a constant speed and time to reach a grain in b / w the towers at the same time . at what distance from the church is the grain ?
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Context:
Given $\angle A = 55^\circ$, the complement of $\angle A$ is ____ degrees.
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Context:
There are 4 puppies and 3 dogs at a camp. Each dog eats a certain amount of food compared to a puppy, but each puppy eats three times as often as a dog. If a dog eats 4 pounds of food three times a day, the total amount of food the dogs and puppies eat in a day is 108 pounds. What is the ratio of the amount of food a dog eats to the amount a puppy eats?
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Context:
Find the dimension of the quotient ring $\mathbb{C}[x,y]/\left\langle x^3-y^2, x^2y-y^3\right\rangle$ as a vector space over $\mathbb{C}$.
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Context:
The product of four different positive integers is equal to $11^4$. What is the sum of these four integers?
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Context:
Point \( K \) lies on side \( AB \) of triangle \( ABC \) with a \(120^\circ\) angle at vertex \( C \). Circles are inscribed in triangles \( AKC \) and \( BKC \) with centers \( O \) and \( Q \), respectively. Find the radius of the circumcircle of triangle \( OQC \), given that \( OK = 6 \) and \( KQ = 7 \).
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Context:
Given that the random variable X follows a normal distribution X ~ N(1, 4), and P(X ≤ 0) = 0.1, calculate the probability P(0 < X < 2). Express your answer as a decimal value between 0 and 1, rounded to 1 decimal place if necessary.
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Context:
A reduction in the price of petrol by 15% enables a motorist to buy 7 gallons more for $300. Find the original price of petrol?
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Context:
Given that \( A \) and \( B \) are two points on the surface of a sphere with a radius of 5, and \( AB = 8 \). Planes \( O_1AB \) and \( O_2AB \) are perpendicular to each other and pass through \( AB \). The intersections of these planes with the sphere create cross-sections \(\odot O_1\) and \(\odot O_2\). Let the areas of \(\odot O_1\) and \(\odot O_2\) be denoted as \( S_1 \) and \( S_2 \) respectively. Solve for \( S_1 + S_2 \).
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Context:
Evaluate the integral:
$$\int_{0}^{\frac{\pi}{2}} x\ln(\tan(x))dx $$
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Context:
Consider the set $\Omega$ consisting of all bijections $\phi$ from $\mathbb{N} \cup \{0\}$ to $\mathbb{Q}$. Each bijection $\phi$ can be visualized as a "staircase" in which, for any $x \in \mathbb{R}^+$, $x$ maps to $\phi(\lfloor x \rfloor)$. The largest step in this staircase is given by $\sup_{n \geq 0} |\phi(n+1)-\phi(n)|$. Determine the smallest possible value for this largest step, formally expressed as:
\[ \inf_{\phi \in \Omega} \sup_{n \geq 0} |\phi(n+1)-\phi(n)| \ \ ? \]
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Context:
On a table, there are 100 piles of pebbles, with 1, 2, ..., up to 100 pebbles in each pile respectively. In one step, you can reduce any number of selected piles, provided that you remove the same number of pebbles from each selected pile.

What is the minimum number of steps needed to remove all the pebbles from the table?
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Context:
Given proposition p: The solution set of the equation $4x^2-2ax+2a+5=0$ has at most two subsets, and proposition q: $1-m \leq x \leq 1+m$, $m>0$. If $\neg p$ is a necessary but not sufficient condition for $\neg q$, find the range of the real number $m$.
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Context:
Given $|\vec{a}|=1$, $|\vec{b}|=6$, and $\vec{a}\cdot(\vec{b}-\vec{a})=2$, calculate the angle between $\vec{a}$ and $\vec{b}$.
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Context:
对任意正实数 $a$ ，记函数 $f(x)=|\lg x|$ 在 $[a,+\infty)$ 上的最小值为 $m_{a}$, 函数 $g(x)=\sin \frac{\pi x}{2}$ 在 $[0, a]$ 上的最大值为 $M_{a}$. 若 $M_{a}-m_{a}=\frac{1}{2}$, 求 $a$ 的所有可能值。已知原始答案形式为 \frac{m}{n} 或 \sqrt{m}，请给出 m + n 的最大值。
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Context:
When Vitya was a first grader, he had a set of 12 number cards: two cards with the number 1, two cards with the number 2, and so on up to the number 6. Vitya placed them on the table in a random order from left to right, and then removed the first 1, the first 2, the first 3, and so on. For example, if Vitya initially had the sequence 434653625112, the resulting sequence would be 436512. What is the probability that the sequence left on the table is 123456?
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Context:
Johnny buys some packs of colored pencils for his class. Each pack has a red, yellow, and green pencil inside. When he gets home he notices that 3 of the packs have two extra red pencils inside. Johnny bought 21 red colored pencils. How many packs of colored pencils did Johnny buy?
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Context:
The function $y=|\log_{2}x|$ is meaningful and not monotonic in the interval $(k-1, k+1)$. Find the range of values for $k$. Express your answer as an interval in standard notation.
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Context:
A train leaves its departure station with 6 coal cars, some iron cars, and 2 wood cars. Between each station, there are 6 miles, and it takes 25 minutes to travel between them. At each station, the train can deposit up to 2 coal cars, 3 iron cars, and 1 wood car. It takes 100 minutes to deliver all the cars. How many iron cars does the train have initially?
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Context:
A goes to place X and B goes to place Y. After A travelled 180 km more than B, they together meet at a point between X and Y. They together had their lunch for half an hour. After that, A took some time to reach X and B took 4.5 hrs to reach Y. The distance between X and Y is 900 km. How long did A take to reach X after lunch?
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Context:
From a 12 × 12 grid, a 4 × 4 square has been cut out, located at the intersection of horizontals from the fourth to the seventh and the same verticals. What is the maximum number of rooks that can be placed on this board such that no two rooks attack each other, given that rooks do not attack across the cut-out cells?
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Context:
Find the greatest integer less than or equal to $\sum_{k=1}^{2^{1983}} k^{\frac{1}{1983} -1}$.
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Context:
Let $A_{12}$ denote the answer to problem $12$.  There exists a unique triple of digits $(B,C,D)$ such that $10>A_{12}>B>C>D>0$ and \[\overline{A_{12}BCD}-\overline{DCBA_{12}}=\overline{BDA_{12}C},\] where $\overline{A_{12}BCD}$ denotes the four digit base $10$ integer.  Compute $B+C+D$.
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Context:
What is the limit of the sequence $S_n$ defined as follows? $$S_{n}=\sum_{i=1}^n \left(\frac{{n \choose i}}{2^{in}}\sum_{j=0}^i {i \choose j}^{n+1} \right)$$
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Context:
If the acute angle \(\alpha\) satisfies \(\frac{1}{\sqrt{\tan \frac{\alpha}{2}}}=\sqrt{2 \sqrt{3}} \sqrt{\tan 10^{\circ}}+\sqrt{\tan \frac{\alpha}{2}}\), then the measure of the angle \(\alpha\) in degrees is \(\qquad\)
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Context:
Find the largest integer  $n$  for which  $2^n$  divides \[ \binom 21 \binom 42 \binom 63 \dots \binom {128}{64}. \]*Proposed by Evan Chen*
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Context:
In triangle \( \triangle ABC \), given that \( \sin A = 10 \sin B \sin C \) and \( \cos A = 10 \cos B \cos C \), find the value of \( \tan A \).
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Context:
A circle with a radius of 2 is inscribed in triangle \(ABC\) and touches side \(BC\) at point \(D\). Another circle with a radius of 4 touches the extensions of sides \(AB\) and \(AC\), as well as side \(BC\) at point \(E\). Find the area of triangle \(ABC\) if the measure of angle \(\angle ACB\) is \(120^{\circ}\).
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Context:
Let point P $(x, 2)$ be a point on the terminal side of angle $\alpha$, and it satisfies $\sin\alpha= \frac{2}{3}$. Then, the value of $x$ is \_\_\_\_\_\_.
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Context:
Consider the system of equations:

\[
8x - 6y = a,
\]
\[
12y - 18x = b.
\]
If there's a solution $(x, y)$ where both $x$ and $y$ are nonzero, determine $\frac{a}{b}$, assuming $b$ is nonzero.
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Context:
对于任何集合 $S$, 以 $|S|$ 表示集合 $S$ 的元素个数, 以 $n(S)$ 表示集合 $S$ 的子集的个数（包括空集与 $S$ 本身）。如果 $A, B$ 和 $C$ 是三个集合，满足条件：

$$
n(A)+n(B)+n(C)=n(A \cup B \cup C), \quad|A|=|B|=100,
$$

求 $|A \cap B \cap C|$ 的最小可能值.
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Context:
What is the correct relationship between the number 2 and the set {0, 1, 2}? Express your answer using the appropriate symbol (e.g., ∈, ⊆, =).
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Context:
Sacha wants to cut a \(6 \times 7\) rectangle into squares that all have integer length sides. What is the smallest number of squares he could obtain?
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Context:
Rotating an isosceles trapezoid around its axis of symmetry forms a \_\_\_\_\_\_.
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Context:
8. Given the set $A=\{1,2, \cdots, 104\}, S$ is a subset of $A$. If $x \in S$, and at the same time $x-1 \notin S$ and $x+1 \notin S$, then $x$ is called an "isolated point" of $S$. The number of all 5-element subsets of $A$ with no "isolated points" is . $\qquad$
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Context:
The ancient Chinese mathematical text, "The Nine Chapters on the Mathematical Art," presents the following problem: "There is a gold bar that is 5 feet long. When 1 foot is cut from the thicker end, it weighs 4 pounds; when 1 foot is cut from the thinner end, it weighs 2 pounds. What is the weight of each successive foot?" Assume that the gold bar's thickness and weight decrease uniformly from the thicker end to the thinner end. Let M be the total weight of the gold bar. If the bar is divided into 10 equal segments, let ai denote the weight of the i-th segment, where a1 < a2 <... < a10. If 48ai = 5M, then find the value of i. Express your answer as a single integer.
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Context:
Calculate the residue of the function $\frac{z^9}{z^{10} -1}$ at infinity using the formula for the residue at infinity involving the function evaluated at the reciprocal of the variable.
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Context:
Let  $ABC$  be a triangle.  The incircle  $\omega$  of  $\triangle ABC$ , which has radius  $3$ , is tangent to  $\overline{BC}$  at  $D$ .  Suppose the length of the altitude from  $A$  to  $\overline{BC}$  is  $15$  and  $BD^2 + CD^2 = 33$ .  What is  $BC$ ?
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Context:
Alberto, Bernardo, and Carlos participated in a race, each running at a constant speed throughout the entire distance. When Alberto crossed the finish line, Bernardo and Carlos were 36 meters and 46 meters behind him, respectively. When Bernardo crossed the finish line, Carlos was 16 meters behind him. Calculate the length of the track in meters.
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Context:
Two trains of different lengths are running towards each other on parallel lines at 42 kmph and 30 kmph respectively. The trains will be clear of each other in 21.998240140788738 seconds from the moment they meet. If the first train is 120 m long, how long is the second train?
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Context:
A mathematician left point A for point B. After some time, a physicist left point A as well. The physicist caught up with the mathematician after 20 km, and without stopping, continued to point B and then turned back. They met for the second time 20 km away from point B. Each then continued in their respective directions. Upon reaching points A and B respectively, they turned back and headed towards each other again. At how many kilometers from point A will they meet for the third time, if the distance between points A and B is 100 km?
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Context:
Haylee has 3 dozen guppies in her aquarium. Jose has half as many guppies as the square root of Haylee's guppies in his tank. Charliz has one-third of what Jose has in his tank, but Charliz sells 1.5 guppies to a neighbor. Nicolai has 4 times as many guppies as Charliz in his pond, minus 2 guppies. Alice has 5 more guppies than Nicolai in her fishbowl, while Bob has half the guppies of ((Jose's guppies)^2)/3 + (Charliz's guppies after selling) in his aquarium. Sara has 3i guppies (where i is an imaginary unit) in her experimental tank. Cameron has 2^(Haylee's dozens of guppies) guppies in his fish tank. How many guppies do these eight friends have altogether, excluding imaginary guppies?
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Context:
If the function $y=f(x)$ defined on $R$ satisfies that there exist a finite number of non-zero independent variable values $x_{0}$ such that $f(-x_{0})=-f(x_{0})$, then $y=f(x)$ is called a class odd function. If the function $y=x^{4}+(a^{2}-1)x^{2}+a\sin x$ is a class odd function, then the range of real number $a$ is ____.
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Context:
During Bill's counting process, 6 toddlers are taken to the playground by another teacher, and 4 new toddlers join the daycare midway through his counting. Additionally, Bill double-counts 9 toddlers and misses 5 who are hiding. If Bill thinks he counted 28 toddlers, considering the added difficulties, how many are there really?
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Context:
Bill and Ted head to the river to throw stuff into the water. Bill throws 6 more sticks into the river than Ted does, but Ted tosses twice as many rocks into the river as Bill. If Bill tossed 21 objects into the river, how many sticks did Ted toss into the river?
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Context:
When the expression \(\left[(1+x)\left(1+2 x^{3}\right)\left(1+4 x^{9}\right)\left(1+8 x^{27}\right)\left(1+16 x^{81}\right)\left(1+32 x^{243}\right)\left(1+64 x^{729}\right)\right]^{2}\) is expanded and simplified, what is the coefficient of \(x^{2003}\)? Express your answer as a power of 2.
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Context:
The median $A D$ and the angle bisector $C E$ of the right triangle $A B C \left(\angle B=90^{\circ}\right)$ intersect at point $M$. Find the area of triangle $A B C$ if $C M =8$, and $M E =5$.
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Context:
If $yz:zx:xy = 1:2:3$, find the ratio $\frac{x}{yz}:\frac{y}{zx}$. The original answer is in the format m:n, please give the value of m + n.
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Context:
Let $ABC$ be an isosceles triangle with the apex at $A$. Let $M$ be the midpoint of segment $[BC]$. Let $D$ be the symmetric point of $M$ with respect to the segment $[AC]$. Let $x$ be the angle $\widehat{BAC}$. Determine, as a function of $x$, the value of the angle $\widehat{MDC}$.
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Context:
Arrange the digits from 0 to 9 into a ten-digit number where the first digit is not zero and all digits are unique. How many such ten-digit numbers are divisible by 99?
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Context:
Find the number of perfect squares of five digits whose last two digits are equal.
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Context:
If angle $\alpha = 2$, then $\alpha$ is an angle in the \_\_\_\_\_\_ quadrant.
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Context:
3 people run for president. John manages to capture 150 votes. James captures a certain percentage of the remaining vote. There were 1150 people voting, and the third guy got 150 more votes than John. What percentage of the remaining vote did James capture?
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Context:
Two circles touch each other externally at point $K$. On their common internal tangent, a point $P$ is marked such that $K P = 14$. Through point $P$, two secants are drawn to the circles such that one of them cuts a chord $AB = 45$ on the first circle, and the other cuts a chord $CD = 21$ on the second circle, with point $A$ lying between points $B$ and $P$, and point $C$ lying between points $D$ and $P$. Find the ratio $BC: AD$.
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Context:
In how many ways can 6 people be seated at a round table, considering the ways different if at least some of the people get new neighbors?
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Context:
Problem 6. Calculate the maximum number of distinct real roots that a polynomial $P$ can have, which satisfies the following property: the product of two distinct roots of $P$ is still a root of $P$.
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Context:
Given $f\left(x\right)=-\ln x+\frac{a}{x}-ex+4$, $g\left(x\right)=\frac{1}{3}{x^3}-{x^2}+2$. If for all $x_{1}\in \left(0,1\right]$, there exists $x_{2}\in \left[-1,1\right]$, such that $g(x_{2})\geqslant f(x_{1})$, then the range of $a$ is ____.
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Context:
In rectangle $PQRS$, $PQ=8$ and $QR=6$. Points $A$ and $B$ lie on $\overline{PQ}$, points $C$ and $D$ lie on $\overline{QR}$, points $E$ and $F$ lie on $\overline{RS}$, and points $G$ and $H$ lie on $\overline{SP}$ so that $AP=BQ<4$ and the convex octagon $ABCDEFGH$ is equilateral. The length of a side of this octagon can be expressed in the form $k+m\sqrt{n}$, where $k$, $m$, and $n$ are integers and $n$ is not divisible by the square of any prime. What is $k+m+n$?
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Context:
设 $0<\alpha<\pi, \pi<\beta<2 \pi$. 若对任意的 $x \in \mathbf{R}$ ，都有
$$
\cos (x+\alpha)+\sin (x+\beta)+\sqrt{2} \cos x=0
$$
恒成立，试求 $\alpha, \beta$ 的值。答案为 $\frac{m\pi}{n}$ 的形式，请给出 $m+n$ 的值。
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Context:
Holly needs to take 2 insulin pills per day, 3 blood pressure pills per day, and twice as many anticonvulsants as blood pressure pills each day. She takes calcium supplements every other day, consuming three times as many calcium pills as insulin pills. Also, she needs to take one multivitamin once every three days, and anxiety pills, which are taken once every four days. How many pills does Holly take in a week?
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Context:
Petya bought one cake, two cupcakes and three bagels, Apya bought three cakes and a bagel, and Kolya bought six cupcakes. They all paid the same amount of money for purchases. Lena bought two cakes and two bagels. And how many cupcakes could be bought for the same amount spent to her?
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Context:
A natural number $n$ is called individual if $n$ has no primitive roots and at least $4$ squares are needed so that it is expressed as a sum of squares. How many natural numbers $n$ between $2$ and $60$ are there that are individual?
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Context:
Determine the number of solutions for the Diophantine equation $x^4 - 1785y^2 = 1$ using Cohn's Theorem.
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Context:
Let \(ABCD\) be a rectangle with \(AB = 8\) and \(AD = 20\). Two circles of radius 5 are drawn with centers in the interior of the rectangle - one tangent to \(AB\) and \(AD\), and the other passing through both \(C\) and \(D\). What is the area inside the rectangle and outside of both circles?
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Context:
Given the function $f(x)=\sin 3x$, where $\frac{f(x_{i})}{x_{i}}=m$, $x_{i}\in[-2π,2π]$, $i=1$, $2$, $...n$, $n∈\mathbb{N}^{*}$, find the maximum value of $n$.
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Context:
Let $f$ be a continuously differentiable function on $[0, 1]$ and $m \in \mathbb{N}$. Let $A = f(1)$ and let $B=\int \limits_{0}^1 x^{-\frac{1}{m}}f(x)dx$. Calculate $$\lim \limits_{n \to \infty} n\left(\int \limits_{0}^1 f(x)dx-\sum \limits_{k=1}^n \left(\frac{k^m}{n^m}-\frac{(k-1)^m}{n^m}\right)f\left(\frac{(k-1)^m}{n^m}\right)\right)$$in terms of $A$ and $B$.
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Context:
For a bijective function $g : \mathbb{R} \to \mathbb{R}$, we say that a function $f : \mathbb{R} \to \mathbb{R}$ is its superinverse if it satisfies the following identity:
\[(f \circ g)(x) = g^{-1}(x)\]
where $g^{-1}$ is the inverse of $g$. Given:
\[g(x) = x^3 + 9x^2 + 27x + 81\]
and $f$ is its superinverse, find:
\[|f(-289)|\]
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Context:
For an upcoming holiday, the weather forecast indicates a probability of $30\%$ chance of rain on Monday and a $60\%$ chance of rain on Tuesday. Moreover, once it starts raining, there is an additional $80\%$ chance that the rain will continue into the next day without interruption. Calculate the probability that it rains on at least one day during the holiday period. Express your answer as a percentage.
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Context:
60. In what ratio does a plane, parallel to two skew edges of a triangular pyramid and dividing one of the other edges in the ratio $2: 1$, divide the volume of the pyramid?
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Context:
Find all real values of $x$ that satisfy $\frac{1}{x+1} + \frac{3}{x+7} \ge \frac23.$ (Give your answer in interval notation.)
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Context:
How many ways are there to insert plus signs (+) between the digits of 1111111111111111 (fifteen 1's) so that the result will be a multiple of 30?
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Context:
there are 5 boxes in a cargo . the weight of the 1 st box is 200 kg , the weight of the 2 nd box is 20 % higher than the third box , whose weight is 25 % higher than the 1 st box weight . the 4 th box which weighs 350 kg is 30 % lighter than the 5 th box . find the difference in average weight of the 4 heaviest boxes and the 4 lightest boxes .
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Context:
Two circles with radii $15$ and $95$ have their centers $d$ units apart. The external tangents to the circles intersect at an angle of $60$ degrees. What is the distance $d$ between the centers of the circles?
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Context:
Find the number of positive integers less than 1,000,000 that are divisible by some perfect cube greater than 1.
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Context:
Let $a_1, a_2, \ldots, a_{2005}$ be real numbers such that $$\begin{array}{ccccccccccc}
a_1\cdot 1 &+ &a_2 \cdot 2 &+& a_3 \cdot 3 &+ &\cdots& + &a_{2005} \cdot 2005 &=& 0 \\
a_1\cdot 1^2 &+& a_2\cdot 2^2 &+& a_3 \cdot 3^2 &+ &\cdots& + & a_{2005} \cdot 2005^2 &=& 0 \\
a_1 \cdot 1^3 &+& a_2 \cdot 2^3 &+& a_3 \cdot 3^3 &+ &\cdots& + & a_{2005} \cdot 2005^3 &=& 0 \\ \vdots&&\vdots&&\vdots&&&&\vdots&&\vdots \\
a_1\cdot 1^{2004} &+& a_2\cdot 2^{2004} &+& a_3\cdot 3^{2004} &+ &\cdots& + &a_{2005} \cdot 2005^{2004} &=& 0
\end{array}$$and $$\begin{array}{ccccccccccc}
a_1 \cdot 1^{2005}& +& a_2\cdot 2^{2005} &+& a_3\cdot 3^{2005} &+ &\cdots& + &a_{2005} \cdot 2005^{2005} &=& 1.
\end{array}$$What is the value of $a_1$?
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Context:
Let \(ABCD\) be a quadrilateral with an inscribed circle \(\omega\). Let \(I\) be the center of \(\omega\), and suppose \(IA = 12\), \(IB = 16\), \(IC = 14\), and \(ID = 11\). Let \(M\) be the midpoint of segment \(AC\). Compute \(\frac{IM}{IN}\), where \(N\) is the midpoint of segment \(BD\).
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Context:
Let $x$ be a real number between 0 and $\tfrac{\pi}{2}$ for which the function  $3\sin^2 x + 8\sin x \cos x + 9\cos^2 x$ obtains its maximum value, $M$. Find the value of $M + 100\cos^2x$.
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Context:
Let \(x, y, z\) be positive real numbers such that \(xyz = 1\). Find the maximum value of
\[
\frac{x^2y}{x+y} + \frac{y^2z}{y+z} + \frac{z^2x}{z+x}.
\]
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Context:
12.172. A trapezoid is inscribed in a circle. The larger base of the trapezoid forms an angle $\alpha$ with a lateral side, and an angle $\beta$ with a diagonal. Find the ratio of the area of the circle to the area of the trapezoid.
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Context:
For each integer $k\ge2$, the decimal expansions of the numbers $1024, 1024^2, \dots, 1024^k$ are concatenated, in that order, to obtain a number $X_k$. (For example, $X_2 = 10241048576$.) If $ \frac{X_n}{1024^n}$ is an odd integer, find the smallest possible value of $n$, where $n\ge2$ is an integer.
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Context:
Find all the common points of the graphs

$$
y=8 \cos \pi x \cdot \cos 2 \pi x \cdot \cos ^{2} 4 \pi x \quad \text { and } \quad y=\cos 11 \pi x
$$

with abscissas belonging to the interval $[0, 1]$. In your answer, provide the sum of the abscissas of the found points.
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Context:
In the Cartesian coordinate system, the graph of the quadratic function $y=ax^{2}+bx+c\left(a \lt 0\right)$ passes through the points $\left(m,b\right)$ and $\left(m+1,a\right)$. When $b\geqslant a$ and $m \lt 0$, the graph of the quadratic function $y=ax^{2}+bx+c$ has a maximum value of $-2$. The maximum value of $a$ is ______.
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Context:
All faces of a triangular pyramid are congruent isosceles triangles, and the height of the pyramid coincides with the height of one of its lateral faces. Find the volume of the pyramid if the distance between the longest opposite edges is 1.
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Context:
When the Absent-Minded Scientist comes up with a genius idea, he writes it down on a piece of paper but immediately realizes that the idea is not genius, crumples the paper and throws it under the table where there are two trash bins. The scientist misses the first bin with probability \( p > 0.5 \), and he misses the second bin with the same probability. Over the course of a morning, the scientist throws five crumpled genius ideas under the table. Find the probability that each bin contains at least one of the morning's ideas.
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Context:
记 $F(x)=\int_{0}^{x}(1+\frac{x-t}{1!}+\frac{(x-t)^{2}}{1!}+\cdots+\frac{(x-t)^{n-1}}{(n-1)!}) e^{n t} d t \quad$当 $n=4$, 求 $\left.\frac{d^{4} F(x)}{d x^{4}}\right|_{x=0}$
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Context:
43. If $a, b, c, d$ are the lengths of the four sides of a quadrilateral, and satisfy $a^{4}+b^{4}+c^{4}+d^{4}=4 a b c d$, then this quadrilateral must be . $\qquad$
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Context:
Rectangular tiles each of size 25 cm by 16 cm must be laid horizontally on a rectangular floor of length 180 cm and a certain width, such that the tiles do not overlap and they are placed with edges jutting against each other on all edges. A tile can be placed in any orientation so long as its edges are parallel to the edges of the floor. No tile should overshoot any edge of the floor. The maximum number of tiles that can be accommodated on the floor is 54. What is the width of the floor?
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Context:
The radian measure of -150° is equal to what fraction of π.
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Context:
Determine the order of the pole of the function \( \frac{1}{(e^{2z}-e^{-2z}+z^4-4)^2} \) at \( z=0 \).
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Context:
Pete was assigned several physics problems and several math problems at home. All the problems solved by Pete constitute 5% of all the physics problems and 20% of all the math problems. What percentage of the total number of problems did Pete solve?
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Context:
Compute the sum of the series:

\[ 5(1+5(1+5(1+5(1+5(1+5(1+5(1+5(1+5(1+5(1+5(1+5)))))))))) \]
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Context:
What percent of the palindromes between 1000 and 2000 contain at least one 3 or 5, except in the first digit?
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Context:
In a sequence of 2003 circles, each circle is tangent to the sides of a $60^{\circ}$ angle $\angle M O N$, and each adjacent pair of circles is externally tangent. If the radius of the smallest circle is 1, determine the radius of the 2003rd circle in ascending order of radii.
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Context:
The expression $10y^2 - 51y + 21$ can be written as $(Cy - D)(Ey - F)$, where $C, D, E, F$ are integers. What is $CE + C$?
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Context:
Let $\mathbb{N}_{>1}$ denote the set of positive integers greater than 1. Let $f: \mathbb{N}_{>1} \rightarrow \mathbb{N}_{>1}$ be a function such that $f(mn)=f(m)f(n)$ for all $m, n \in \mathbb{N}_{>1}$. If $f(101!)=101$!, compute the number of possible values of $f(2020 \cdot 2021)$.
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Context:
When you add 124, each digit is the same number. The sum of the hundreds and ones digits of this number is 5, and the tens digit is 3. If each of these three digit numbers is different, find the three digit number.
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Context:
Radii of five concentric circles  $\omega_0,\omega_1,\omega_2,\omega_3,\omega_4$  form a geometric progression with common ratio  $q$  in this order. What is the maximal value of  $q$  for which it's possible to draw a broken line  $A_0A_1A_2A_3A_4$  consisting of four equal segments such that  $A_i$  lies on  $\omega_i$  for every  $i=\overline{0,4}$ ?

<details><summary>thanks </summary>Thanks to the user Vlados021 for translating the problem.</details>
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Context:
One of the angles of a trapezoid is $60^{\circ}$. Find the ratio of its bases if it is known that a circle can be inscribed in this trapezoid and also circumscribed around this trapezoid.
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Context:
In the game Deal or No Deal, participants choose a box at random from a set of $26,$ one containing each of the following values: \begin{tabular}{|c|c|}\hline\$.01&\$1,000\\\hline\$1&\$5,000\\\hline\$5&\$10,000\\\hline\$10&\$25,000\\\hline\$25&\$50,000\\\hline\$50&\$75,000\\\hline\$75&\$100,000\\\hline\$100&\$200,000\\\hline\$200&\$300,000\\\hline\$300&\$400,000\\\hline\$400&\$500,000\\\hline\$500&\$750,000\\\hline\$750&\$1,000,000\\\hline\end{tabular} After choosing a box, participants eliminate other boxes by opening them, showing the amount of money in the box to the crowd, and then removing that box (and its money!) from the game. What is the minimum number of boxes a participant needs to eliminate in order to have a half chance of holding at least $\$100,\!000$ as his or her chosen box?
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Context:
记 $F(x)=\int_{0}^{x}\left(1+\frac{x-t}{1!}+\frac{(x-t)^{2}}{1!}+\cdots+\frac{(x-t)^{n-1}}{(n-1)!}\right) e^{n t} d t \quad$ 当 $n=4$, 求 $\left.\frac{d^{4} F(x)}{d x^{4}}\right|_{x=0}$
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Context:
Let  $S={1,2,\ldots,100}$ . Consider a partition of  $S$  into  $S_1,S_2,\ldots,S_n$  for some  $n$ , i.e.  $S_i$  are nonempty, pairwise disjoint and  $\displaystyle S=\bigcup_{i=1}^n S_i$ . Let  $a_i$  be the average of elements of the set  $S_i$ . Define the score of this partition by
\[\dfrac{a_1+a_2+\ldots+a_n}{n}.\]

Among all  $n$  and partitions of  $S$ , determine the minimum possible score.
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Context:
In the game of Guess the Card, two players each have a  $\frac{1}{2}$  chance of winning and there is exactly one winner. Sixteen competitors stand in a circle, numbered  $1,2,\dots,16$  clockwise. They participate in an  $4$ -round single-elimination tournament of Guess the Card. Each round, the referee randomly chooses one of the remaining players, and the players pair off going clockwise, starting from the chosen one; each pair then plays Guess the Card and the losers leave the circle. If the probability that players  $1$  and  $9$  face each other in the last round is  $\frac{m}{n}$  where  $m,n$  are positive integers, find  $100m+n$ .

*Proposed by Evan Chen*
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Context:
For how many numbers  $n$  does  $2017$  divided by  $n$  have a remainder of either  $1$  or  $2$ ?
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Context:
【Question 13】
A positive integer with $n+3$ digits $144 \cdots 430(n$ 4's$)$, is a multiple of 2015, the smallest positive integer $n$ is $\qquad$
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Context:
Given a regular hexagon \( A B C D E F \) with side length \( 10 \sqrt[4]{27} \), find the area of the union of triangles \( ACE \) and \( BDF \).
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Context:
A  $4\times4\times4$  cube is composed of  $64$  unit cubes. The faces of  $16$  unit cubes are to be coloured red. A colouring is called interesting if there is exactly  $1$  red unit cube in every  $1\times1\times 4$  rectangular box composed of  $4$  unit cubes. Determine the number of interesting colourings.
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Context:
Evaluate the infinite sum: $$\sum_{r=1}^{\infty}\frac{2^{2r+1}\cdot (r)!\cdot (r+1)!}{(2r+3)!}$$
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Context:
Let $M$ be the maximum possible value of $x_1x_2+x_2x_3+\cdots +x_5x_1$ where $x_1, x_2, \dots, x_5$ is a permutation of $(1,2,3,4,5)$ and let $N$ be the number of permutations for which this maximum is attained. Evaluate $M+N$.
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Context:
Compute the number of ways there are to assemble $2$ red unit cubes and $25$ white unit cubes into a $3 \times 3 \times 3$ cube such that red is visible on exactly $4$ faces of the larger cube. (Rotations and reflections are considered distinct.)
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Context:
Given that the positive integer \( n \) does not exceed 2000 and can be expressed as the sum of at least 60 consecutive positive integers, how many such \( n \) exist?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Three trucks need to transport \(k\) full barrels, \(k\) half-full barrels, and \(k\) empty barrels such that each truck is equally loaded and each truck carries the same number of barrels. In how many ways can this be done for \(k=7\), considering the trucks and barrels with the same fullness indistinguishable?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In an isosceles right triangle, the radius of the inscribed circle is 2.

Find the distance from the vertex of the acute angle to the point where the inscribed circle touches the leg opposite to this angle.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The square root of a two-digit number is expressed as an infinite decimal fraction, the first four digits of which (including the integer part) are the same. Find this number without using tables.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the smallest positive integer \( m \) such that for any 14-partition \( A = \{1, 2, \cdots, m\} \) into subsets \( A_1, A_2, \cdots, A_{14} \), there always exists a subset \( A_i \) (with \( 1 < i < 14 \)) containing two elements \( a \) and \( b \) where \( b < a \leq \frac{4}{3}b \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( S = \{1, 2, \cdots, 1990\} \). If the sum of all the numbers in a 31-element subset of \( S \) is a multiple of 5, it is called a "good subset". Find the number of good subsets of \( S \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Carl decided to fence his rectangular flowerbed using 24 fence posts, including one on each corner. He placed the remaining posts spaced exactly 3 yards apart along the perimeter of the bed. The bed’s longer side has three times as many posts compared to the shorter side, including the corner posts. Calculate the area of Carl’s flowerbed, in square yards.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A quadrilateral is inscribed in a circle. If an angle is inscribed into each of the four segments outside the quadrilateral, the sum of these four angles, expressed in degrees, is:
Please reason step by step, and put your final answer within \boxed{}.


Context:
A circle is tangent to two parallel lines and their secant. The segment of the secant enclosed between the parallel lines is divided by the tangent point in the ratio $1:3$. At what angle does the secant intersect each of the parallel lines?
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the covariance of the process $X(t) = \int_0^t B(u)\,du$, where $B$ is a standard Brownian motion? Specifically, calculate $E[X(t)X(s)]$ for $0<s<t<\infty$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
For how many triples of positive integers $(a, b, c)$ with $1 \leq a, b, c \leq 5$ is the quantity $(a+b)(a+c)(b+c)$ not divisible by $4$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
At a certain middle school, there are 110 teachers who know both English and Russian. According to the statistics, there are 75 teachers who know English and 55 teachers who know Russian. How many teachers at this school know English but not Russian?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the limit of \( \frac{\sin_{n+1}x}{\sin_nx} \) as \( x \to 0 \), where \( \sin_n \) is defined as in the original question.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( X \) be a countably infinite (or larger) set with the cofinite topology. For every \( x \in X \), does there exist a family \( \xi \subset \tau \) such that \( \{ x \} = \bigcap \xi \)? If so, determine the cardinality of \( \xi \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Rectangles $R_1$ and $R_2,$ and squares $S_1,\,S_2,\,$ and $S_3,$ shown below, combine to form a rectangle that is 3322 units wide and 2020 units high. What is the side length of $S_2$ in units?

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


Context:
Suppose we are given seven points that are equally spaced around a circle.  If $P$, $Q$, and $R$ are chosen to be any three of these points, then how many different possible values are there for $m\angle PQR$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let  $ABCD$  be a parallelogram that is not rhombus. We draw the symmetrical half-line of  $(DC$  with respect to line  $BD$ . Similarly we draw the symmetrical half- line of  $(AB$  with respect to  $AC$ . These half- lines intersect each other in  $P$ . If  $\frac{AP}{DP}= q$  find the value of  $\frac{AC}{BD}$  in function of  $q$ .
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of natural number triples \((a, b, p)\) with \(a > b > 0\) such that \[ \left(\sum_{k=1}^n k^a\right)^p = \sum_{k=1}^n k^b \] holds for all natural numbers \(n\). Identify these triples.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $ ABCD$ be an isosceles trapezoid with $ \overline{AD}\parallel{}\overline{BC}$ whose angle at the longer base $ \overline{AD}$ is $ \dfrac{\pi}{3}$. The diagonals have length $ 10\sqrt {21}$, and point $ E$ is at distances $ 10\sqrt {7}$ and $ 30\sqrt {7}$ from vertices $ A$ and $ D$, respectively. Let $ F$ be the foot of the altitude from $ C$ to $ \overline{AD}$. The distance $ EF$ can be expressed in the form $ m\sqrt {n}$, where $ m$ and $ n$ are positive integers and $ n$ is not divisible by the square of any prime. Find $ m \plus{} n$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given angles $\alpha$ and $\beta$ in a triangle, where $\alpha, \beta \in (0, \pi)$, determine if $\alpha = \beta$ given that:

$$\left(\frac{\sin \alpha}{\sin \beta}\right)^{100}= \frac{\sin^2 \frac {\alpha}{2}}{\sin^2 \frac{\beta}{2}}$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
For a positive integer $n$, let $g(n) = \left[ \displaystyle \frac{2
024}{n} \right]$. Find the value of$$\sum_{n = 1}^{2024}\left(1 - (-1)^{g(n)}\right)\phi(n).$$

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


Context:
Let  $S$  be a real number. It is known that however we choose several numbers from the interval  $(0, 1]$  with sum equal to  $S$ , these numbers can be separated into two subsets with the following property: The sum of the numbers in one of the subsets doesn’t exceed 1 and the sum of the numbers in the other subset doesn’t exceed 5.

Find the greatest possible value of  $S$ .
Please reason step by step, and put your final answer within \boxed{}.


Context:
Ang, Ben, and Jasmin each have $5$ blocks, colored red, blue, yellow, white, and green; and there are $5$ empty boxes. Each of the people randomly and independently of the other two people places one of their blocks into each box. The probability that at least one box receives $3$ blocks all of the same color is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m + n ?$
Please reason step by step, and put your final answer within \boxed{}.


Context:
若点M到x轴的距离是它到y轴距离的2倍，则点M的轨迹方程是？
Please reason step by step, and put your final answer within \boxed{}.


Context:
Regular tetrahedron $ABCD$ has center $O$ and side length $1$. Points $A'$, $B'$, $C'$, and $D'$ are defined by reflecting $A$, $B$, $C$, and $D$ about $O$. Compute the volume of the polyhedron with vertices $ABCDA'B'C'D'$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A frog sitting at the point $(1, 2)$ begins a sequence of jumps, where each jump is parallel to one of the coordinate axes and has length $1$, and the direction of each jump (up, down, right, or left) is chosen independently at random. The sequence ends when the frog reaches a side of the square with vertices $(0,0), (0,4), (4,4),$ and $(4,0)$. What is the probability that the sequence of jumps ends on a vertical side of the square$?$

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


Context:
Consider an inverse system of modules over an integral domain $R$ defined by the maps $A_{i+1} \to A_i, a \mapsto r_{i+1} a$, where $r_i \neq 0,1$. Determine if the inverse limit of this system is zero.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Points \( M, N, \) and \( K \) are located on the lateral edges \( A A_{1}, B B_{1}, \) and \( C C_{1} \) of the triangular prism \( A B C A_{1} B_{1} C_{1} \) such that \( A M : A A_{1} = 1 : 2, B N : B B_{1} = 1 : 3, \) and \( C K : C C_{1} = 1 : 4 \). Point \( P \) belongs to the prism. If the volume of the prism is 16, find the maximum possible volume of the pyramid \( M N K P \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine if the function \( g(x) = \int_{0}^{\infty} t^{x^2+1} e^{-3t} \, dt \) can be expressed as a power series in a neighborhood of zero. If it can, find the radius of convergence of this series.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine all values of $a$ for which the function \[ h(x) = \begin{cases} x^3 - x^2, & \text{if } x \in \mathbb{Q} \\ 2x - 2x^2, & \text{otherwise} \end{cases} \] is differentiable.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the largest integer  $n\geq 3$  for which the edges of the complete graph on  $n$  vertices
can be assigned pairwise distinct non-negative integers such that the edges of every triangle have numbers which form an arithmetic progression.
Please reason step by step, and put your final answer within \boxed{}.


Context:
All the numbers $1, 2, 3, 4, 5, 6, 7, 8, 9$ are written in a $3\times3$ array of squares, one number in each square, in such a way that if two numbers of consecutive then they occupy squares that share an edge. The numbers in the four corners add up to $18$. What is the number in the center?

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Context:
Determine whether the Kullback-Leibler divergence between the standard Gaussian distribution \( f(x) = \frac{1}{\sqrt{2\pi}}\exp{(-\frac{x^2}{2})} \) and the standard Cauchy distribution \( g(x) = \frac{1}{\pi(1+x^2)} \) is finite.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A burger at Ricky C's weighs $120$ grams, of which $30$ grams are filler. What percent of the burger is not filler?

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


Context:
Let $m \ge 3$ be an integer and let $S = \{3,4,5,\ldots,m\}$. Find the smallest value of $m$ such that for every partition of $S$ into two subsets, at least one of the subsets contains integers $a$, $b$, and $c$ (not necessarily distinct) such that $ab = c$.

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Context:
In a group $G$, given elements $x, a, y$ such that $xay = a^{-1}$, determine if $yax = a^{-1}$ must also hold.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the largest integer $r$ and the smallest integer $q$ such that $\exp(r) < \int_{1}^{2} \exp(x^3 + x) \, dx < \exp(q)$. Provide your answer as a pair $(r, q)$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Carlos Montado was born on Saturday, November 9, 2002.  On what day of the week will Carlos be 706 days old? (Spell out the entire day of the week.)
Please reason step by step, and put your final answer within \boxed{}.


Context:
When $x$ is added to both the numerator and denominatorof the fraction $\frac{a}{b},a\neb,b\ne0$ (Error compiling LaTeX. Unknown error_msg),the value of the fraction is changed to $\frac{c}{d}$. Then $x$ equals:

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


Context:
Determine whether the ring \(\mathbb{Z}[x]/(23,x-2)\) is an integral domain or isomorphic to another ring.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given two sequences of events $\{A_n\}$ and $\{B_n\}$ with $\mathbb{P}(\liminf_n A_n) = \mathbb{P}(\limsup_n B_n) = 1$, determine if $\mathbb{P}(\limsup_n A_n \cap B_n) = 1$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine whether the subspace of $[0,1]^\mathbb{N}$, consisting of monotone (not necessarily strictly) real sequences with values in $[0,1]$ and equipped with the product topology, is connected.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine whether the paths $p \ast f \ast \bar{p}$ and $h \ast f \ast \bar{h}$ are path homotopic, where $p(s) = (1, s)$, $f(s) = (e^{2\pi is}, 1)$, and $h(s) = (e^{2\pi is}, s)$ for $s \in [0, 1]$. Here, $g \ast k$ denotes the path $\sigma(s) = \begin{cases} g(2s), & s \in [0, 1/2] \\ k(2s-1), & s \in [1/2, 1] \end{cases}$, and $\bar{g}(s)$ denotes the path $g(1-s)$. Answer "yes" if they are path homotopic and "no" otherwise.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $G:[0,\infty)\to \mathbb{R}^n$ be a continuous and differentiable function such that $G(0) = 0\in\mathbb{R}^n$. Suppose there exists a constant $c>0$ such that for all $t>0$, the inequality $$||DG(t)||\leq c||G(t)||$$ holds, where $DG(t)$ is the derivative of $G$ and $||\cdot||$ denotes the Euclidean norm. Does this imply that $G(t) \equiv 0$ for all $t \geq 0$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find all real numbers  $x,y,z$  such that satisfied the following equalities at same time: $\sqrt{x^3-y}=z-1 \wedge \sqrt{y^3-z}=x-1\wedge \sqrt{z^3-x}=y-1$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine whether the following sum approaches zero as \( n \to \infty \): \[ \sum_{k=1}^{n}\frac{k^{1-n}}{n(n-1)}. \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $ABCDE$ be a convex pentagon with $AB\parallel CE$, $BC\parallel AD$, $AC\parallel DE$, $\angle ABC=120^\circ$, $AB=3$, $BC=5$, and $DE=15$. Given that the ratio between the area of triangle $ABC$ and the area of triangle $EBD$ is $m/n$, where $m$ and $n$ are relatively prime positive integers, find $m+n$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine all integers $n$ for which the polynomial $P(x) = 3x^3-nx-n-2$ can be written as the product of two non-constant polynomials with integer coeffcients.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Bianca has a rectangle whose length and width are distinct primes less than 100.  Let  $P$  be the perimeter of her rectangle, and let  $A$  be the area of her rectangle.  What is the least possible value of  $\frac{P^2}{A}$ ?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the second derivative of the function \( f(x) = \sec(3x)\sqrt{324\cos^2(3x) + 396 + 121\sec^2(3x)} \). Then, evaluate the second derivative at \( x = \frac{\pi}{12} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a financial market with a bond $A$, a stock $S$, and call options $f$ on the stock, consider two times $0$ and $T$ with two possible outcomes for $S$ at time $T$. The values are: $A(0)=100$, $A(T)=120$, $S(0)=50$, $S^u(T)=70$, $S^d(T)=30$. Construct a portfolio $\\phi$ of shares of $S$ and call options $f$ with strike $K=50$ and expiration $T$ such that $V(T)=120$ regardless of the stock's movement. Determine the price of the call option at time $0$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $\Phi_1$ and $\Phi_2$ be cumulative probability distribution functions with domain $[L, \infty)$, $L\geq 0$, both distributions having the same expectation $\mu$ and the same second moment. If the variance of the distribution behind $\Phi_2$ is higher than that of $\Phi_1$, determine whether the following inequality holds:

$$G'-G \geq 0$$

where:

$$G=1-\frac{1}{\mu}\int_L^\infty \left(1- \Phi_1(x)\right)^2 \, \mathrm{d} x$$

$$G'=1-\frac{1}{\mu}\int_L^\infty \left(1- \frac{1}{2}\left(\Phi_1(x)+\Phi_2(x)\right)\right)^2 \, \mathrm{d} x$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
The internal angles of quadrilateral $ABCD$ form an arithmetic progression. Triangles $ABD$ and $DCB$ are similar with $\angle DBA = \angle DCB$ and $\angle ADB = \angle CBD$. Moreover, the angles in each of these two triangles also form an arithemetic progression. In degrees, what is the largest possible sum of the two largest angles of $ABCD$?

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


Context:
What is the area of the triangle formed by the lines $y=5$, $y=1+x$, and $7=1-x$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine whether the language \( L_G = \{ H \mid G \text{ contains a copy of } H \} \) for some fixed graph \( G \) is in \( P \) or is \( NPC \). Assume \( P \neq NP \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the work done by a gas in the cycle $1-2-3-4-5-6-1$, given that $p_{0}=10^{5} \Pi a$ and $V_{0}=1$. Note that states 1 and 3 coincide, as do states 4 and 6.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $V$ be a real $d$-dimensional vector space with $1<k<d$. Given a basis $\{v_i\}$ for $V$ and an induced basis for $\bigwedge^k V$ consisting of $v_{i_1} \wedge \dots \wedge v_{i_k}$, suppose there exists another basis $\{w_i\}$ such that each $v_{i_1} \wedge \dots \wedge v_{i_k}$ can be expressed as $\lambda_{i_1,\ldots,i_k}w_{j_1} \wedge \dots \wedge w_{j_k}$, where $\lambda_{i_1,\ldots,i_k} \in \mathbb R$ are non-zero scalars. Determine whether there exists a permutation $\sigma \in S_d$ such that $\text{span}(v_i)=\text{span}(w_{\sigma(i)})$. Provide your answer as 'yes' or 'no'.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $z_1,$ $z_2,$ $z_3,$ and $z_4$ be the four distinct complex solutions of the equation
\[
  z^4 - 6z^2 + 8z + 1 = -4(z^3 - z + 2)i.
\]Find the sum of the six pairwise distances between $z_1,$ $z_2,$ $z_3,$ and $z_4$ in the complex plane.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Students Arn, Bob, Cyd, Dan, Eve, and Fon are arranged in that order in a circle. They start counting: Arn first, then Bob, and so forth. When the number contains a 7 as a digit (such as 47) or is a multiple of 7 that person leaves the circle and the counting continues. Who is the last one present in the circle?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine whether the solution to the recurrence relation \( T(n) = T\left(\frac{1}{8n}\right) + 8 \) is infinite.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Is the largest singular value of a matrix $A$, denoted as $\sigma_1$, the solution to the optimization problem \( \max_{\|x\|\leq1,\|y\|\leq1} \quad x^TAy \) where $A \in \mathbb{R}^{m \times n}$, $x \in \mathbb{R}^{m \times 1}$, and $y \in \mathbb{R}^{n \times 1}$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $ABC$ be at triangle with incircle $\Gamma$. Let $\Gamma_1$, $\Gamma_2$, $\Gamma_3$ be three circles inside $\triangle ABC$ each of which is tangent to $\Gamma$ and two sides of the triangle and their radii are $1,4,9$. Find the radius of $\Gamma$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Is the number\[
\left( 1 + \frac12 \right) \left( 1 + \frac14 \right) \left( 1 + \frac16 \right)\cdots\left( 1 + \frac{1}{2018} \right)
\]greater than, less than, or equal to $50$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the Maclaurin series expansion for the function \( f(z) \) that satisfies \( f(z)e^{f(z)}= z \). Determine the radius of convergence of this series using the Lagrange inversion formula.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine whether the function \( f(z) = \mathfrak{Re}(z) \mathfrak{Im}(z) \) is pointwise differentiable at \( z=0 \) using the limit definition:
\[ \lim_{z\to 0} \frac{f(z) - f(0)}{z}. \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $C \subset \mathbb{R}$ be an infinite and perfect set. Suppose there exists a family of compact sets $\{Q_k\}_{k\in \mathbb{N}}$ such that $Q_{k+1} \subseteq Q_k$ and $Q_k \cap C$ is infinite for all $k$. Given that $Q := \bigcap_{k\in \mathbb{N}} Q_k \neq \emptyset$, is it true that $Q \subseteq C$ or at least $Q \cap C \neq \emptyset$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Suppose $A, R, S$, and $T$ all denote distinct digits from $1$ to $9$. If $\sqrt{STARS} = SAT$, what are $A, R, S$, and $T$?

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Context:
Liquid $X$ does not mix with water. Unless obstructed, it spreads out on the surface of water to form a circular film $0.1$cm thick. A rectangular box measuring $6$cm by $3$cm by $12$cm is filled with liquid $X$. Its contents are poured onto a large body of water. What will be the radius, in centimeters, of the resulting circular film?

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


Context:
Let $x \in \mathbb{R}^+$ and define the operation $x \diamond 1 = x$ and $x \diamond (n+1) = x^{x \diamond n}$ for $n \in \mathbb{N}$. For example, $2 \diamond 3 = 2^{2^2} = 16$. Given a unique positive real number $t$ such that $t^t = 10^{10^{10}}$, it is known that $10^{10} > t$. Let $k$ be the unique positive real number such that $k \diamond 50 = 10 \diamond 51$, and $s$ be the unique positive real number such that $s \diamond 51 = 10 \diamond 52$. Determine if $k > s$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
26th BMO 1990 Problem 2 The fractional part { x } of a real number is defined as x - [x]. Find a positive real x such that { x } + { 1/x } = 1 (*). Is there a rational x satisfying (*)?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Define a common chord between two intersecting circles to be the line segment connecting their two intersection points. Let $\omega_1,\omega_2,\omega_3$ be three circles of radii $3, 5,$ and $7$, respectively. Suppose they are arranged in such a way that the common chord of $\omega_1$ and $\omega_2$ is a diameter of $\omega_1$, the common chord of $\omega_1$ and $\omega_3$ is a diameter of $\omega_1$, and the common chord of $\omega_2$ and $\omega_3$ is a diameter of $\omega_2$. Compute the square of the area of the triangle formed by the centers of the three circles.
Please reason step by step, and put your final answer within \boxed{}.


Context:
You have a 4-digit whole number that is a perfect square.
 Another number is built adding 1 to the unit’s digit,
 subtracting 1 from the ten's digit,adding 1 to the hundred's
 digit and subtracting 1 from the ones digit of one thousanc
 If the number you get is also a perfect square, find the
 original number. It's unique?
Please reason step by step, and put your final answer within \boxed{}.


Context:
p5. Hao has $32$ ounces of pure lemonade in his juice box. On the first day, he drinks an ounce and then refills the juice box with an ounce of Hater-ade and mixes the mixture thoroughly. On the second day, he drinks $2$ oz of the juice and then refills the juice box with $2$ oz of Hater-ade. On the third day, he drinks $3$ oz and then refills the juice box with $3$ oz of Hater-ade, and so on. Hao stops when the juice box has no more lemonade in it. How many ounces of Hater-ade did Hao drink?

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


Context:
Determine the points at which the function \( f: \mathbb{R} \to \mathbb{R} \) defined by \( f(x) = \sin x \) if \( x \) is rational, and \( f(x) = 0 \) otherwise, is continuous.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider a power series \(\sum_{k=0}^\infty a_k x^k\) with a radius of convergence of 1. Assume that \(\sum_{k=0}^\infty a_k = \infty\). Determine whether \(\lim_{x\to 1^-} \sum_{k=0}^\infty a_k x^k = \infty\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
The polynomial $f(x)=x^3-4\sqrt{3}x^2+13x-2\sqrt{3}$ has three real roots, $a$, $b$, and $c$. Find
$$\max\{a+b-c,a-b+c,-a+b+c\}.$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine if the only solution to the differential equation \( \frac{dy}{dx}=8y^2x^3 \) with the initial condition \( y(2)=0 \) is the constant solution \( y=0 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider an infinite double random array $\{X_{ij}:i\geq 1, j\geq 1\}$ where all elements are independent, identically distributed (i.i.d.) with mean zero and unit variance. Define $Y_{i,n}=\frac{1}{n}\sum_{j=1}^nX_{ij}$. By the strong law of large numbers, we know that for any $i\geq 1$, $Y_{i,n}\to 0$ a.s. and $Y_{i,n}^2\to 0$ a.s. as $n\to\infty$. Does this imply that $\frac{1}{n}\sum_{i=1}^nY_{i,n}^2\to 0$ a.s.?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A traveler token is placed on the central tile of a 19-tile hexagonal grid. Each turn, the traveler moves to an adjacent tile based on a six-sided die roll, with all directions equally probable. The game ends when the traveler exits the board. Calculate the expected number of turns until the game concludes.
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Context:
Suppose $X(k)$ is a $k \times m$ matrix with elements $x_{ij}(k)$. If \(\lim_{k \to \infty} \frac{X'(k)X(k)}{k}=R,\) where $R$ is positive semi-definite, determine if the following holds: \(\max_{i=1, \ldots, k} x_i'(X'(k)X(k))^{-1}x_i =0, \text{ as } k \to \infty.\)
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Context:
Let $x,y$ be complex numbers such that $\dfrac{x^2+y^2}{x+y}=4$ and $\dfrac{x^4+y^4}{x^3+y^3}=2$. Find all possible values of $\dfrac{x^6+y^6}{x^5+y^5}$.
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Context:
All twelve points on the circle are at equal distances. The only marked point inside is the center of the circle. Determine which part of the whole circle in the picture is filled in.
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Context:
Determine the cardinality of the set formed by the union of all Cartesian products of the natural numbers with itself, i.e., the set of all sequences of natural numbers of any finite length. Is the cardinality of this set aleph-one or higher?
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Context:
Determine whether the set of computable real numbers has a positive Hausdorff dimension.
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Context:
Determine whether the limit \( \lim_{x \to 0} x \cdot (\log(x))^\alpha = 0 \) holds for every \( \alpha > 0 \).
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Context:
Let F be a countable field and B an infinite set. Let \((F^B)_0\) be the set of all functions with finite support from F to B. Determine whether \(|(F^B)_0| = |B|\).
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Context:
Find all solutions $a, b, c, d, e$ if it is known that they represent distinct
digits and satisfy the following:
$\begin{array}{c}& a & b & c & d \\+ & a & c & a & c \\ \hline c & d & e & b & c \\\end{array}$
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Context:
Suppose that for real $x,y,z,t$ the following equalities hold:$\{x+y+z\}=\{y+z+t\}=\{z+t+x\}=\{t+x+y\}=1/4$.
Find all possible values of $\{x+y+z+t\}$.(Here$\{x\}=x-[x]$)
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Context:
Jerry begins his English homework at $07:39$ a.m. At $07:44$ a.m., he has finished $2.5\%$ of his homework. Subsequently, for every five minutes that pass, he completes three times as much homework as he did in the previous five minute interval. If Jerry finishes his homework at $AB : CD$ a.m., what is $A + B + C + D$? For example, if he finishes at $03:14$ a.m., $A + B + C + D = 0 + 3 + 1 + 4$.

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Context:
Determine whether the first-order predicate formula \((A \land B) \Rightarrow C\) is valid, where:

- \(A = \forall a (\neg P(a,a))\)
- \(B = \forall a \forall b \forall c (P(a,b)\wedge P(b,c)\Rightarrow P(a,c))\)
- \(C = \forall a \forall b (P(a,b)\Rightarrow \neg P(b,a))\)

A formula is valid if there is no truth assignment that makes it false.
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Context:
Determine whether the integral \( \int_0^2 \frac{5\,dx}{\sqrt{2x-x^2}} \) converges or diverges.
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Context:
Find the number of second-degree polynomials $f(x)$ with integer coefficients and integer zeros for which $f(0)=2010.$
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Context:
Let  $O$  be the centre of the square  $ABCD$ . Let  $P,Q,R$  be respectively on the segments  $OA,OB,OC$  such that  $OP=3,OQ=5,OR=4$ . Suppose  $S$  is on  $OD$  such that  $X=AB\cap PQ,Y=BC\cap QR$  and  $Z=CD\cap RS$  are collinear. Find  $OS$ .
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Context:
一个有盖的长方体形状的文具盒的长、宽、高分别是12cm,4cm,3cm,那么它最多能放13 cm长的铅笔.
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Context:
Determine whether the Krull dimension of any commutative semilocal Hilbert ring is equal to zero.
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Context:
Determine the asymptotic value of \(\underset{c \in \text{Coloring}}{\min} \text{ score}(c) = \underset{c \in \text{Coloring}}{\min} \left(\underset{\begin{subarray}{c} B \subset \text{Colors}, \\ |B|\leq n  \end{subarray}}{\max} V_c(B)\right)\). Is it \(\Theta(n^2)\) or \(o(n^2)\)?
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Context:
Evaluate the integral \(\int_0^{2\pi} \frac{\sin^2\theta}{5+4\cos(\theta)}\,d\theta.\) Use the substitution \(\theta=2\arctan t\) to express the integral in terms of \(t\) and find its value.
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Context:
King Midas spent $\frac{100}{x}\%$ of his gold deposit yesterday. He is set to earn gold today. What percentage
of the amount of gold King Midas currently has would he need to earn today to end up with as much
gold as he started?
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Context:
Determine the values of 's' for which the matrix A is positive semi-definite.
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Context:
Let \( f(x), g(x), \) and \( h(x) \) be linear functions. Define \( j(x) = \max\{f(x), g(x), h(x)\} \) and \( k(x) = \min\{f(x), g(x), h(x)\} \). Given the graph of \( y = j(x) \) for \(-3.5 \le x \le 3.5\), determine the value of \( \ell^2 \), where \( \ell \) is the length of the graph of \( y = k(x) \) over the same interval.
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Context:
Is 45 vertices the current record for the smallest triangle-free 6-chromatic graph?
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Context:
Last year a bicycle cost $160 and a cycling helmet $40. This year the cost of the bicycle increased by $5\%$, and the cost of the helmet increased by $10\%$. The percent increase in the combined cost of the bicycle and the helmet is:
$
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Context:
 From the given, find the ratio of the number of triangles to the number of circles inside the rectangle.
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Context:
Find the solutions to
\[\frac{13x - x^2}{x + 1} \left( x + \frac{13 - x}{x + 1} \right) = 42.\]Enter all the solutions, separated by commas.
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Context:
No math tournament exam is complete without a self referencing question. What is the product of
the smallest prime factor of the number of words in this problem times the largest prime factor of the
number of words in this problem
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Context:
Equiangular hexagon \( {ABCDEF} \) has side lengths
 \( {AB} = {CD} = {EF} = 1 \) and \( {BC} = {DE} = {FA} = r \) . The
 area of \( \bigtriangleup  {ACE} \) is \( {70}\% \) of the area of the hexagon. What is the
 sum of all possible values of \( r \) ?

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Context:
Each of the $5{ }$ sides and the $5{ }$ diagonals of a regular pentagon are randomly and independently colored red or blue with equal probability. What is the probability that there will be a triangle whose vertices are among the vertices of the pentagon such that all of its sides have the same color?

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Context:
The integers $a$, $b$, $c$ and $d$ are such that $a$ and $b$ are relatively prime, $d\leq 2022$ and $a+b+c+d = ac + bd = 0$. Determine the largest possible value of $d$,
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Context:
Let $a, b, c$ be two-digit, three-digit, and four-digit numbers, respectively. Assume that the sum of all digits of number $a+b$, and the sum of all digits of $b + c$ are all equal to $2$. The largest value of $a + b + c$ is

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Context:
Let $x,y,z$ is positive. Solve:
$\begin{cases}{x\left( {6 - y} \right) = 9}\\
{y\left( {6 - z} \right) = 9}\\
{z\left( {6 - x} \right) = 9}\end{cases}$
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Context:
Suppose $G$ is the Galois group of an irreducible degree 5 polynomial $f \in \mathbb{Q}[x]$ with $|G|=10$. Determine whether $f$ must have precisely 1 or 5 real roots.
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Context:
例4 有1000张编号为000,001，···，999的证件和100个编号为00，01，···，99的盒子．若盒子的号码可以由证件的号码划掉一个数字而得到，则该证件可以放入该盒子中.若选择k个盒子可以装下所有证件，求k的最小值.
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Context:
It takes 60 grams of paint to paint a cube on all sides. How much paint is needed to paint a "snake" composed of 2016 such cubes? The beginning and end of the snake are shown in the diagram, and the rest of the cubes are indicated by ellipses.
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Context:
In the diagram, \( AB \) and \( CD \) intersect at \( E \). If \(\triangle BCE\) is equilateral and \(\triangle ADE\) is a right-angled triangle, what is the value of \( x \)?
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Context:
Find all values of $z$ such that $z^4 - 4z^2 + 3 = 0$.  Enter all the solutions, separated by commas.
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Context:
Simplify the expression \u221a[3]{5+2\u221a{13}} + \u221a[3]{5-2\u221a{13}} and find its exact value.
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Context:
Let $M$ be the smallest positive multiple of $2012$ that has $2012$ divisors.
Suppose $M$ can be written as  $\Pi_{k=1}^{n}p_k^{a_k}$ where the $p_k$’s are distinct primes and the $a_k$’s are positive integers.
Find $\Sigma_{k=1}^{n}(p_k + a_k)$
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Context:
In how many ways can the letters in $\textbf{BEEKEEPER}$ be rearranged so that two or more $\textbf{E}$s do not appear together?
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Context:
A circle of radius $2$ is centered at $O$. Square $OABC$ has side length $1$. Sides $AB$ and $CB$ are extended past $B$ to meet the circle at $D$ and $E$, respectively. What is the area of the shaded region in the figure, which is bounded by $BD$, $BE$, and the minor arc connecting $D$ and $E$?
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Context:
In the cube $ABCD-A_1B_1C_1D_1$, it is known that $M$ is the midpoint of $BB_1$. Calculate the cosine value of the dihedral angle between planes $M-CD_1$ and $A$. Express your answer in simplified form.
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Context:
The sequence $a_0$, $a_1$, $a_2$, $\ldots$ satisfies the recurrence equation:
\[
a_n = 2 a_{n-1} - 2 a_{n - 2} + a_{n - 3}
\]
for every integer $n \ge 3$. If $a_{20} = 1$, $a_{25} = 10$, and $a_{30} = 100$, what is the value of $a_{1331}$?
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Context:
若抛物线 $y=a x^2-1$ 上存在关于 $x+y=0$ 成轴对称的两个点, 试求 $a$ 的取值范围.
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Context:
the least number which when divided by 5 , 6 , 7 and 8 leaves a remainder 3 , but when divided by 9 leaves no remainder , is
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Context:
Given that $b_1, b_2, b_3, b_4$ are positive integers, the polynomial $g(z)=(1-z)^{b_1}\left(1-z^2\right)^{b_2}\left(1-z^3\right)^{b_3}\left(1-z^4\right)^{b_4}$ simplifies to $1-2z$ when terms higher than the 4th degree are ignored. Additionally, $\alpha$ is the largest root of the polynomial $f(x)=x^3-b_4 x^2+b_2$. Find the remainder when $\left[\alpha^{1995}\right]$ is divided by 9, where $[x]$ denotes the greatest integer less than or equal to $x$.
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Context:
Andrew wants to write the numbers 1, 2, 3, 4, 5, 6, and 7 in the circles in the diagram so that the sum of the three numbers joined by each straight line is the same. What number should he write in the top circle? Express your answer as a single integer.
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Context:
A cat ate half of the smallest fish in 30 seconds. In total, it ate 5 fish and took a whole number of minutes to do so (the cat eats fish at a constant 'cell' rate). The picture shows all the fish the cat had. Which fish did the cat not eat?
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Context:
The figure consists of five squares and two right-angled triangles. The areas of three of the squares are 5, 8, and 32, as shown. Find the area of the shaded square. Express your answer as a whole number.
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Context:
A parallelogram has three of its vertices at $(-1,0)$, $(2,4)$ and $(2,-4)$.  What is the positive difference between the greatest possible perimeter and the least possible perimeter of the parallelogram?
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Context:
A cross composed of two identical large squares and two identical small squares is placed inside an even larger square. Calculate the side length of the largest square in centimeters if the area of the cross is \( 810 \mathrm{~cm}^{2} \).
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Context:
In triangle \( \triangle ABC \), \( AM \) and \( CN \) are the medians to sides \( BC \) and \( AB \), respectively, intersecting at point \( O \). \( P \) is the midpoint of \( AC \), and \( MP \) intersects \( CV \) at \( Q \). If the area of \( \triangle OMQ \) is \( n \), calculate the area of \( \triangle ABC \) in terms of \( n \). Express your answer as a multiplier of \( n \).
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Context:
Given the odd function $f(x)$ is increasing on the interval $[3, 7]$ and its minimum value is 5, determine the nature of $f(x)$ and its minimum value on the interval $[-7, -3]$.
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Context:
Given that α is an angle in the second quadrant and |cos(α/3)| = -cos(α/3), determine the quadrant of α/3. Express your answer as the name of the quadrant.
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Context:
The vertex $A$ of parallelogram $A B C D$ is connected by segments with the midpoints of the sides $B C$ and $C D$. One of these segments turned out to be twice as long as the other. Determine whether angle $B A D$ is acute, right, or obtuse.
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Context:
It is easy to place the complete set of ships for the game "Battleship" on a $10 \times 10$ board (see illustration). What is the smallest square board on which this set can be placed? (Recall that according to the rules, ships must not touch each other, even at the corners.)
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Context:
An object in the plane moves from one lattice point to another. At each step, the object may move one unit to the right, one unit to the left, one unit up, or one unit down. If the object starts at the origin and takes a ten-step path, how many different points could be the final point? Write your answer as an integer.
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Context:
Rectangle \(ABCD\) has length 9 and width 5. Diagonal \(AC\) is divided into 5 equal parts at \(W, X, Y\), and \(Z\). Determine the area of the shaded region.
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Context:
Let  $m\geq 4$  and  $n\geq 4$ . An integer is written on each cell of a  $m \times n$  board. If each cell has a number equal to the arithmetic mean of some pair of numbers written on its neighbouring cells, determine the maximum amount of distinct numbers that the board may have.

Note: two neighbouring cells share a common side.
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Context:
地面上有 10 只小鸟在啄食,其中任意 5 只小鸟中至少有 4 只在同一圆上, 问有鸟最多的圆上最少有几只鸟?
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\boxed{有鸟最多的圆周上至少有 9 只鸟.}


Context:
An electric kettle heats water from room temperature $T_{0}=20^{\circ} \mathrm{C}$ to a temperature of $T_{m}=100^{\circ} \mathrm{C}$ in $t=10$ minutes. How long will it take for all the water to boil away, $t_{1}$, if the kettle is not turned off and the automatic shut-off system is malfunctioning? The specific heat capacity of water is $c=4200$ J/kg·K, and the specific heat of vaporization of water is $L=2.3$ MJ/kg. Round the answer to the nearest whole number of minutes.
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Context:
In the diagram, triangles \(ABC\) and \(CBD\) are isosceles. The perimeter of \(\triangle CBD\) is 19, the perimeter of \(\triangle ABC\) is 20, and the length of \(BD\) is 7. Find the length of \(AB\). Express your answer as a whole number.
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Context:
Find the largest $r$ such that $4$ balls each of radius $r$ can be packed into a regular tetrahedron with side length $1$. In a packing, each ball lies outside every other ball, and every ball lies inside the boundaries of the tetrahedron. If $r$ can be expressed in the form $\frac{\sqrt{a}+b}{c}$ where $a, b, c$ are integers such that $\gcd(b, c) = 1$, what is $a + b + c$?
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Context:
Circle $X$ has a radius of $\pi$. Circle $Y$ has a circumference of $8 \pi$. Circle $Z$ has an area of $9 \pi$. List the circles in order from smallest to the largest radius.
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Context:
Find all integers \( k \) for which the values of the fraction

$$
\frac{2 k^{2}+k-8}{k-1}
$$

are integers.
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Context:
One notebook, 3 notepads, and 2 pens cost 98 rubles, while 3 notebooks and a notepad cost 36 rubles less than 5 pens. How much does each item cost if the notebook costs an even number of rubles? (Each of these items costs an integer number of rubles.)
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Context:
What is the mass percentage of Al in Aluminum bromide?
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Context:
A fly is being chased by three spiders on the edges of a regular octahedron. The fly has a speed of  $50$  meters per second, while each of the spiders has a speed of  $r$  meters per second. The spiders choose their starting positions, and choose the fly's starting position, with the requirement that the fly must begin at a vertex. Each bug knows the position of each other bug at all times, and the goal of the spiders is for at least one of them to catch the fly. What is the maximum  $c$  so that for any  $r<c,$   the fly can always avoid being caught?

*Author: Anderson Wang*
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Context:
On an 8 by 8 Battleship game board, a "patychok" figure is placed. What is the minimum number of shots required to ensure hitting at least one "patychok"? (7 points)
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