Context:
Tyler has two calculators, both of which initially display zero.  The first calculators has only two buttons, $[+1]$ and $[\times 2]$.  The second has only the buttons $[+1]$ and $[\times 4]$.  Both calculators update their displays immediately after each keystroke.
A positive integer $n$ is called ambivalent if the minimum number of keystrokes needed to display $n$ on the first calculator equals the minimum number of keystrokes needed to display $n$ on the second calculator. Find the sum of all ambivalent integers between $256$ and $1024$ inclusive.
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Context:
18th Putnam 1958 Problem A2 A rough sphere radius R rests on top of a fixed rough sphere radius R. It is displaced slightly and starts to roll off. At what point does it lose contact?
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Context:
Determine the total variation of the two-variable step function given by:
\[ f(x,y) = \begin{cases} 1 & \text{when } 0<x<a,\ 0<y<b; \\\ 0 & \text{otherwise} \end{cases} \]
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Context:
In \(\Delta ABC\), the points \(P(1,1,2)\) and \(Q(4,7,8)\) satisfy \(\frac{AP}{AQ}=\frac{BP}{BQ}= \frac{CP}{CQ}=2\). Find the coordinates of the circumcenter of \(\Delta ABC\).
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Context:
What is the value of the infinite sum $$\sum_{n=1}^{\infty} \dfrac{1}{4n^2+16n+7}$$?
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Context:
Evaluate the limit of the sequence as \( n \to \infty \):

\[ a_n = n\left(\sqrt[n]{ea} - \sqrt[n]{a}\right), \quad n \ge 1, \ a > 0 \]
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Context:
Find the largest integer \( n \) such that for all integers \( b \) that are relatively prime to \( n \), it holds that \( b^{24} \equiv 1 \pmod{n} \).
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Context:
Determine the dimension of the subset of pairs of 2x2 matrices \((c_1, c_2)\) that generate a 2-dimensional Lie algebra, where the commutator \([c_1, c_2] = k_1 c_1 + k_2 c_2\) for some scalars \(k_1\) and \(k_2\).
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Context:
Six people play the following game: They have a cube, initially white. One by one, the players mark an $ X$ on a white face of the cube, and roll it like a die. The winner is the first person to roll an $ X$ (for example, player 1 wins with probability $ \frac {1}{6}$, while if none of players 1-5 win, player 6 will place an $ X$ on the last square and win for sure). What is the probability that the sixth player wins?
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Context:
Sue is traveling from New Orleans to San Francisco with multiple layovers and time zone changes. Her journey starts at New Orleans, where she takes a flight to Atlanta, then flies to Chicago, from there to New York, to Denver, and finally, from Denver to San Francisco. All layover times and flight durations are given as local times.

The New Orleans to Atlanta flight takes 2 hours. Atlanta is in the same time zone as New Orleans. Sue has a 4-hour layover in Atlanta before the flight to Chicago. The Atlanta to Chicago flight takes 5 hours. Chicago is 1 hour behind Atlanta. Sue has a 3-hour layover in Chicago. Her last flight before New York, Chicago to New York, takes 3 hours. New York is 1 hour ahead of Chicago. When she arrives in New York, Sue must wait 16 hours for her next flight.

From New York, Sue first travels to Denver, which takes 6 hours. Denver is 2 hours behind New York. Sue has a 5-hour layover in Denver before her final leg to San Francisco. The flight from Denver to San Francisco takes 4 hours. San Francisco is 1 hour behind Denver.

Calculate the total time Sue took to reach San Francisco from New Orleans, considering the time zones and time differences affecting her travel and layovers along her entire journey.
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Context:
Calculate \( \mathbb{E}\left[\left(\int_{0}^{\infty}g(t)\,dW_t\right)^2\right] \) where \( g(t) = (W_3 - W_2)\mathbf{1}_{[3,4)}(t) + (W_4 - W_2)\mathbf{1}_{[4,6)}(t) \) and \((W_t)_{t \ge 0}\) is a Brownian motion.
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Context:
Find the angle at which the parabola \(y^{2}=2px\) intersects the circle \(\left(x-\frac{p}{2}\right)^{2}+y^{2}=4p^{2}\).
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Context:
11. Given that when $|x| \leqslant 1$, we always have
$$
2 p x^{2}+q x-p+1 \geqslant 0 \text {. }
$$

Find the maximum value of $p+q$.
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Context:
Consider a sequence of positive integers $a_1, a_2, a_3, \ldots$ such that $\gcd(a_m, a_n) > 1$ if and only if $|m - n| = 1$. Determine the sum of the four smallest possible values of $a_2$. \(\boxed{\text{Your answer here}}\)
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Context:
Let  $f(x)=\cos(\cos(\cos(\cos(\cos(\cos(\cos(\cos(x))))))))$ , and suppose that the number  $a$  satisfies the equation  $a=\cos a$ . Express  $f'(a)$  as a polynomial in  $a$ .
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Context:
Evaluate \(\lim \limits_{n \rightarrow \infty} \sum\limits_{k=1}^{n} \arctan \frac{2}{k^{2}}\). The answer is in the form of \(\frac{k}{m}\), where \(\frac{k}{m}\) is a simplified fraction. Please provide the value of \(k + m\).
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Context:
For a given positive integer \( n \geq 2 \), determine the largest positive real number \( \lambda(n) \) such that for any real numbers \( x_1, x_2, \ldots, x_n \), not all of which are positive, the inequality holds:
\[
\prod_{i=1}^{n}\left(x_{i}^{2}-x_{i}+1\right) \geq \lambda(n) \cdot \left(\left(\prod_{i=1}^{n} x_{i}\right)^{2}-\prod_{i=1}^{n} x_{i}+1\right)
\]
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Context:
Determine the dimension of the vector space \( \mathbb{C}[x,y]/I \) over \( \mathbb{C} \), where \( I = \langle x^2 + 4x + 4, xy+x+2y+2, y^3 + 3y^2 + 3y + 1 \rangle \).
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Context:
Compute

$3(1+3(1+3(1+3(1+3(1+3(1+3(1+3(1+3(1+3)))))))))$
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Context:
What is the last two digits of the decimal representation of  \( {9}^{{8}^{{7}^{{.}^{{.}^{{.}^{2}}}}}}  \) ?
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Context:
A rope is folded into 3 equal parts, then folded in half again, and cut in the middle. In total, it can be cut into \_\_\_\_\_\_ segments.
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Context:
For each positive integer $ n$, let $ f(n)$ denote the greatest common divisor of $ n!+1$ and $ (n+1)!$. Find, without proof, a formula for $ f(n)$.
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Context:
Determine the probability that a 4 × 4 square grid becomes a single uniform color (all white or all black) after rotation.
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Context:
Given that \(\det(A) = \frac{1}{24}\), find the value of \(\det\left(\left(\frac{1}{3}A\right)^{-1} - 120 A^*\right)\).
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Context:
Determine the least possible value of $f(1998),$ where $f:\Bbb{N}\to \Bbb{N}$ is a function such that for all $m,n\in {\Bbb N}$,

\[f\left( n^{2}f(m)\right) =m\left( f(n)\right) ^{2}. \]
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Context:
a and b are two multiples of a certain number, and q is the set of consecutive integers between a and b, inclusive. If q contains 10 multiples of that number, there are 19 multiples of 7 in q. What is that certain number?
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Context:
Using $n^{3}$ unit cubes, a cube with edge length $n$ is formed $(n \in \mathbf{N}, n \geqslant 2)$. A line is said to pass through a unit cube if it passes through any point inside the unit cube. What is the maximum number of unit cubes that a single line can pass through?
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Context:
在平面直角坐标系中，甲从原点出发，到点 (4,4)；乙从点 (2,0) 出发，到点 (6,4). 已知两人同时出发，且每次向上或向右走一步. 则甲、乙的行走路线没有公共点的概率为__________.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值
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Context:
Evaluate the limit: \[ I = \lim_{t\to0^{+}}\lim_{x\to+\infty}\dfrac{\displaystyle\int_{0}^{\sqrt{t}}dx\int_{x^2}^{t}\sin{y^2}dy}{\left[\left(\dfrac{2}{\pi}\arctan{\dfrac{x}{t^2}}\right)^x-1\right]\arctan{t^{\frac{3}{2}}}} \]
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Context:
Two three-letter strings, $aaa$ and $bbb$ , are transmitted electronically. Each string is sent letter by letter. Due to faulty equipment, each of the six letters has a $1/3$ chance of being received incorrectly, as an $a$ when it should have been a $b$ , or as a $b$ when it should be an $a$ . However, whether a given letter is received correctly or incorrectly is independent of the reception of any other letter. Let $S_a^{}$ be the three-letter string received when $aaa$ is transmitted and let $S_b^{}$ be the three-letter string received when $bbb$ is transmitted. Let $p$ be the probability that $S_a^{}$ comes before $S_b^{}$ in alphabetical order. When $p$ is written as a fraction in lowest terms , what is its numerator ?
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Context:
Let $n \geq 2$ be a positive integer. Each cell of an $n \times n$ board is colored red or blue. We place dominoes on the board, each covering two cells. We call a domino plain if it lies on two red or two blue cells, and colorful if it lies on one red and one blue cell. Find the largest positive integer $k$ with the following property: no matter how the red/blue coloring of the board is done, it is always possible to place $k$ non-overlapping dominoes on the board that are either all plain or all colorful.
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Context:
Determine the ring to which \( R[X,Y,Z,T]/(X-Y^2,T-Y^4,T^3-Z) \) is isomorphic after substituting \( X=Y^2 \).
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Context:
Let  $\triangle ABC$  be a triangle with side length  $BC= 4\sqrt{6}$ . Denote  $\omega$  as the circumcircle of  $\triangle{ABC}$ . Point  $D$  lies on  $\omega$  such that  $AD$  is the diameter of  $\omega$ . Let  $N$  be the midpoint of arc  $BC$  that contains  $A$ .  $H$  is the intersection of the altitudes in  $\triangle{ABC}$  and it is given that  $HN = HD= 6$ . If the area of  $\triangle{ABC}$  can be expressed as  $\frac{a\sqrt{b}}{c}$ , where  $a,b,c$  are positive integers with  $a$  and  $c$  relatively prime and  $b$  not divisible by the square of any prime, compute  $a+b+c$ .
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Context:
Evaluate the limit of the sum \( \sum_{n=1}^\infty \frac{\cos(n)}{n^k} \) as \( k \to 0^+ \) using the polylogarithm function.
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Context:
A piece of iron wire with a length of $80cm$ is randomly cut into three segments. Calculate the probability that each segment has a length of no less than $20cm$.
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Context:
Consider the set of digits \( \{1, 2, 3, \ldots, 8\} \). How many permutations of these digits have the property that the product of any five consecutive digits is divisible by 5?
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Context:
A train 110 m long traveling at 60 kmph takes 16.7986561075114 seconds to cross a bridge of a certain length. How long is the bridge in meters?
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Context:
Determine the structure of the quotient group $\mathbb{Z}^2/((m,n)\mathbb{Z})$, where $m$ and $n$ are integers greater than one and coprime.
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Context:
Let  $n\ge2$  be an integer and let  $x_1,x_2,\ldots,x_n$  be real numbers. Consider  $N=\binom n2$  sums  $x_i+x_j$ ,  $1\le i<j\le n$ , and denote them by  $y_1,y_2,\ldots,y_N$  (in an arbitrary order). For which  $n$  are the numbers  $x_1,x_2,\ldots,x_n$  uniquely determined by the numbers  $y_1,y_2,\ldots,y_N$ ?
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Context:
Determine the dimension of the cut space for an undirected graph, given that the cut space is a vector space over \( \mathbb{Z}_2 \).
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Context:
Determine the number of homomorphisms from the group \(C_5 \times C_4 \times C_4\) onto \(C_{10}\).
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Context:
A family of 5 members has an average age of some years. If the age of the youngest member is 10 years, and the average age of the family at the time of the birth of the youngest member was 12.5 years, what is the current average age of the family?
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Context:
Let $A=\mathbb{C}[x,y]/(x^3-y^2)$, and consider the maximal ideal $m=(\bar{x} - 1, \bar{y} - 1)$ of $A$. Determine the minimal number of generators, $\mu(mA_m)$, for the ideal $mA_m$. Provide your answer as a single integer.
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Context:
Given a regular triangular pyramid \( PABC \) (with \( P \) as the apex) with a base side length \( a \) and lateral edge \( b \) ( \( b > a \)). A sphere lies above the plane of the base \( ABC \), touches this plane at point \( A \), and also touches the lateral edge \( PB \). Find the radius of the sphere.
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Context:
Find the value of $x$ that satisfies the equation $\log(x) = 2x$. Express your answer as a decimal rounded to three decimal places.
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Context:
Given the graph of $y=\sqrt{2}\sin 3x$, calculate the horizontal shift required to obtain the graph of $y=\sin 3x+\cos 3x$.
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Context:
Calculate the line integral of the vector field \( F = (2x\sin(\pi y)-e^z, \pi x^2\cos(\pi y)-3e^z, -xe^z) \) along the curve \( C \) from \((0,0,0)\) to \((1,1,\ln(2))\), where the curve is defined by the conditions \( z=\ln(1+x) \) and \( y=x \).
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Context:
Let $S_{k}=1^{k}+2^{k}+3^{k}+…+n^{k}$, when $k=1$, $2$, $3…$, observe the following equations:
$S_{1}= \frac {1}{2}n^{2}+ \frac {1}{2}n$, $S_{2}= \frac {1}{3}n^{3}+ \frac {1}{2}n^{2}+ \frac {1}{6}n$, $S_{3}= \frac {1}{4}n^{4}+ \frac {1}{2}n^{3}+ \frac {1}{4}n^{2}$,
$S\;\_{4}= \frac {1}{5}n^{5}+ \frac {1}{2}n^{4}+ \frac {1}{3}n^{3}- \frac {1}{30}n$, $S\_{5}= \frac {1}{6}n^{6}+An^{5}+Bn^{4}- \frac {1}{12}n^{2}$, $…$,
It can be inferred that $A-B=$ \_\_\_\_\_\_.
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Context:
Find the kernel of the derivative operator $D$ for polynomials in $F[x]$, where $F$ is a ring with characteristic $p > 0$. What is the set of polynomials that are mapped to zero by this operator?
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Context:
Given that point $P$ is the intersection point of the lines $l_{1}$: $mx-ny-5m+n=0$ and $l_{2}$: $nx+my-5m-n=0$ ($m$,$n\in R$, $m^{2}+n^{2}\neq 0$), and point $Q$ is a moving point on the circle $C$: $\left(x+1\right)^{2}+y^{2}=1$, calculate the maximum value of $|PQ|$.
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Context:
We have an empty equilateral triangle with length of a side $l$. We put the triangle, horizontally, over a sphere of radius $r$. Clearly, if the triangle is small enough, the triangle is held by the sphere. Which is the distance between any vertex of the triangle and the centre of the sphere (as a function of $l$ and $r$)?
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Context:
What is the computational complexity of finding the $n$-th derivative of $f(x)$, where $f(x)$ is defined as the product of two functions $g(x)$ and $g(1-x)$?
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Context:
Determine the order of growth for the binomial coefficient \( {}^n C_k \) when approximated by a polynomial. Consider the case where \( n \gg k \) and provide the general order of growth for \( {}^n C_k \).
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Context:
What is the official name for an additive abelian group equipped with a multiplication that is distributive over addition but not necessarily associative?
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Context:
Determine the minimum degree \( n \) for which there exists a polynomial \( P(x) \) of degree not exceeding \( n \) such that the creative potential \( c_i = P(a_i) \) of all 6 selected candidates is strictly greater than that of the remaining 7 candidates, given that the scores \( a_1, a_2, \ldots, a_{13} \) are distinct.
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Context:
How many times can the trigonometric series $\sum_{k=1}^{\infty} \frac{\cos k x}{k^{4}}$ be differentiated term by term?
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Context:
12. Chocolate Coupons. A chocolate candy factory places a coupon in each box of candies. For every ten coupons collected, a customer receives a free box of candies. What fraction of the cost of a box does one coupon represent?
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Context:
Let \(f\) be a function defined on the interval \((a,b)\) such that \(f' + f^2 + 1 \ge 0\). Given that \(\lim_{x\to a}f(x)=\infty\) and \(\lim_{x\to b}f(x)=-\infty\), determine the possible interval for \(b-a\).
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Context:
Evaluate the double integral \( \iint_{D} f(x,y) \, dA \) over the region \( D \) bounded by the lines \( x=0 \), \( y=0 \), and \( x+y=1 \), where \( f(x,y) \) is defined as:

\[f(x,y) = \begin{cases} x^2 & x \geq y \\ y^2 & y > x \end{cases}\]
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Context:
Evaluate the limit, if it exists:
\[\lim_{x\rightarrow\infty}\Delta^x_{k=1}\sin\left(\frac{k\pi}{2x}\right)\]
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Context:
Determine the complexity of the recurrence relation \( T(n) = \sqrt{2n}T(\sqrt{2n} )+ n^2 \).
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Context:
Evaluate the limit of the integral: $$\lim_{n\to\infty}\int_0^{2n\pi}\frac{(n+x)\sin(x/n)}{x(1+x)^2}\,dx$$
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Context:
Let $(X_i, P_i)$, with $i$ being an integer, be an independent random variable such that $X_i|P_i$ ~ $Bernoulli(P_i)$ and $P_i$ ~ $Beta(\alpha, \beta)$. What is the variance of $Y = \sum_{i=1}^n X_i$?
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Context:
Evaluate the limit using the Stolz-Cesàro theorem:
\[ \lim_{n\to \infty} \frac{2021(1^{2020}+2^{2020}+3^{2020}+\ldots+n^{2020}) - n^{2021}}{2021(1^{2019}+2^{2019}+3^{2019}+\ldots+n^{2019})} = L \]
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Context:
Provide an example of an absolutely convergent series \( \sum_{n=0}^\infty |r_n| \) with \( r_n \in \mathbb{Q} \) such that \( \sum_{n=0}^\infty r_n \) converges to a real number that is not rational, i.e., \( \sum_{n=0}^\infty r_n \in \mathbb{R} \setminus \mathbb{Q} \).
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Context:
Calculate the line integral \( \int_{C} \vec{F} \cdot \vec{n} \, ds \) where \( \vec{F}(x, y) = x^{10} \vec{i} + (3x - 10x^9y) \vec{j} \) and \( \vec{n} \) is the unit normal vector to the curve \( C \). The curve \( C \) is defined by \( x^2 + y^2 = 1 \) and is restricted to the first quadrant. Provide the value of the integral.
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Context:
a, b, and c started a business with different capitals. At the end of the year, the profit share of b is Rs. 1800, and the difference between the profit shares of a and c is Rs. 720. If a invested Rs. 8000 and b invested Rs. 10000, how much did c invest?
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Context:
Evaluate the limit: \[ \lim_{x\to 0}\left(\frac{1}{(\sin^{-1}(x))^2} - \frac{1}{x^2}\right) \] without using L'Hôpital's rule.
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Context:
Find the value of \( y(\frac{1}{2}) \) for the function \( y(x) \) that minimizes the integral \( \int_{0}^{1} e^{-(y'-x)} + (1+y)y' \, dx \), given that \( y(0) = 0 \) and \( y(1) = \frac{1}{2} \).
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Context:
There is a strip \( 1 \times 99 \) divided into 99 cells of size \( 1 \times 1 \), which are colored alternately in black and white. It is allowed to repaint all the cells of any checkerboard rectangle \( 1 \times k \) at the same time. What is the minimum number of repaintings required to make the entire strip a single color?
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Context:
Let $T: V \rightarrow V$ be a linear operator with the minimal polynomial $m(x) = (x-1)x^8$. Find the minimal polynomial of $T^3$. \( \text{Provide your answer as a polynomial.} \)
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Context:
Find the value of $a+b+c$ given the limit:
\[\lim_{x \to \infty}\frac{a(2x^3-x^2)+b(x^3+5x^2-1)-c(3x^3+x^2)}{a(5x^4-x)-bx^4+c(4x^4+1)+2x^2+5x}=1\]
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Context:
Let $ABCDEF$ be a regular hexagon and let point $O$ be the center of the hexagon. How many ways can you color these seven points either red or blue such that there doesn’t exist any equilateral triangle with vertices of all the same color?
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Context:
If the one-variable quadratic equation $x^{2}-4x+1-t=0$ (where $t$ is a real number) has solutions in the range $0 \lt x \lt \frac{7}{2}$, then the range of values for $t$ is ______.
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Context:
Calculate the surface area generated by revolving the polar curve \( r = \sqrt{\cos 2\theta} \) around the y-axis, with the limits of integration from \( \theta = 0 \) to \( \theta = \frac{\pi}{4} \).
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Context:
Find the smallest natural number \( n \) such that the group \( \mathbb{Z}_{6} \times \mathbb{Z}_{20} \times \mathbb{Z}_{45} \) is isomorphic to a direct product of \( n \) cyclic groups.
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Context:
Let $ABC$ be a triangle with $\angle BAC = 90^\circ$. Let $D$, $E$, and $F$ be the feet of the altitude, angle bisector, and median from $A$ to $BC$, respectively. If $DE = 3$ and $EF = 5$, compute the length of $BC$.
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Context:
Determine the dimension of the affine algebraic set \(X = V(f_1, f_2, f_3) \subseteq \mathbb{A}^6\), where:
\[f_1 = x_1x_5 - x_4x_2, \quad f_2 = x_1x_6 - x_4x_3, \quad f_3 = x_2x_6 - x_5x_3.\]
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Context:
Determine the supremum of \(|f(1)|\) for functions \(f\colon \Bbb C\setminus\{0\}\to \Bbb C\) that are analytic and satisfy \(|f(z)|\le 5|z|^{−\sqrt2}\).
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Context:
There are $n \geq 2$ coins, each with a different positive integer value. Call an integer $m$ sticky if some subset of these $n$ coins have total value $m$. We call the entire set of coins a stick if all the sticky numbers form a consecutive range of integers. Compute the minimum total value of a stick across all sticks containing a coin of value 100.
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Context:
已知复数 $z$ 满足 $|z|=3$. 则 $\left|\frac{z^{2}+1}{z^{2}+\mathrm{i} z+z+\mathrm{i}}\right|$ 的最大值为原始答案的形式为\(\dfrac{m\sqrt{n}+\sqrt{k}}{q}\)，请给出m + n + k+q的值。
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Context:
Consider the system of equations

\[
8x - 6y = c,
\]
\[
12y - 18x = d.
\]

If this system has a solution \((x, y)\) where both \(x\) and \(y\) are nonzero, find the value of \(\frac{c}{d}\), assuming \(d\) is nonzero.
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Context:
Determine the smallest positive value of $x,$ in degrees, for which
\[\tan (x + 100^{\circ}) = \tan (x + 50^{\circ}) \tan x \tan (x - 50^{\circ}).\]
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Context:
Calculate the value of the Laplace operator \( \Delta\left(\frac{1}{r^2}\right) \) where \( r = |x| = \sqrt{x_1^2 + x_2^2 + x_3^2} \).
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Context:
A scientist begins an experiment with a cell culture that starts with some integer number of identical cells. After the first second, one of the cells dies, and every two seconds from there another cell will die (so one cell dies every odd-numbered second from the starting time). Furthermore, after exactly 60 seconds, all of the living cells simultaneously split into two identical copies of itself, and this continues to happen every 60 seconds thereafter. After performing the experiment for awhile, the scientist realizes the population of the culture will be unbounded and quickly shuts down the experiment before the cells take over the world. What is the smallest number of cells that the experiment could have started with?
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Context:
From 6 students of different heights, 5 are chosen to stand in a row from left to right for a photo. The requirement is that the students standing in even positions must be taller than those in adjacent odd positions. Calculate the total number of different possible photos that can be taken under this arrangement.
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Context:
A circle having radius \( r_1 \) centered at point \( N \) is tangent to a circle of radius \( r_2 \) centered at \( M \). Let \( l \) and \( j \) be the two common external tangent lines to the two circles. A circle centered at \( P \) with radius \( r_2 \) is externally tangent to circle \( N \) at the point at which \( l \) coincides with circle \( N \), and line \( k \) is externally tangent to \( P \) and \( N \) such that points \( M, N \), and \( P \) all lie on the same side of \( k \). For what ratio \( r_1 / r_2 \) are \( j \) and \( k \) parallel?
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Context:
Evaluate the integral \( \oint_{|z| = 4} \frac{e^{1/(1 - z)}}{z^2} \, dz \) using the Residue Theorem.
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Context:
Let \( \{x\} \) denote the fractional part of \( x \). For example, \( \{5.5\} = 0.5 \). Find the smallest prime \( p \) such that the inequality \[ \sum_{n=1}^{p^2}\left\{\frac{n^p}{p^2}\right\} > 2016 \] holds.
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Context:
Evaluate the line integral \( \int_C x \, dy - y \, dx \) where \( C \) consists of a straight line segment from \((-2, 0)\) to \((0, 0)\), another straight line segment from \((0, 0)\) to \((0, -2)\), and the counterclockwise portion of a circle with radius 2, centered at the origin, starting from \((0, -2)\) and ending at \((-2, 0)\).
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Context:
Find the number of pairs \((n, C)\) of positive integers such that \(C \leq 100\) and \(n^2 + n + C\) is a perfect square.
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Context:
A natural number  $k > 1$  is called *good* if there exist natural numbers  $$ a_1 < a_2 < \cdots < a_k $$  such that  $$ \dfrac{1}{\sqrt{a_1}} + \dfrac{1}{\sqrt{a_2}} + \cdots +  \dfrac{1}{\sqrt{a_k}} = 1 $$  .

Let  $f(n)$  be the sum of the first  $n$  *[good* numbers,  $n \geq$  1. Find the sum of all values of  $n$  for which  $f(n+5)/f(n)$  is an integer.
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Context:
Determine the minimum number of coin tosses needed to test if the probability of landing on heads is at most 0.05 with at least 95% confidence.
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Context:
Given a convex figure in the Cartesian plane that is symmetric with respect to both axes, a rectangle $A$ is constructed inside it with maximum area. The rectangle is then enlarged with its center at the center of the rectangle and a ratio of lambda such that it covers the convex figure. Determine the smallest value of lambda that ensures this condition for all such convex figures.
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Context:
Evaluate the limit of the integral as \( n \to \infty \):

\[ I_n = \int_0^3 \sin\left(\frac{\pi}{3} \sin\left(\frac{\pi}{3} \sin \left(\frac{\pi}{3} \dots \sin\left(\frac{\pi}{3} x\right) \dots \right)\right)\right)dx \]

where the sine function is nested \( n \) times.
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Context:
Let $A = (0,0)$, $B=(-1,-1)$, $C=(x,y)$, and $D=(x+1,y)$, where $x > y$ are positive integers. Suppose points $A$, $B$, $C$, and $D$ lie on a circle with radius $r$. Denote by $r_1$ and $r_2$ the smallest and second smallest possible values of $r$. Compute $r_1^2 + r_2^2$.
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Context:
In a group, given that $aba = b^2$ and the order of $a$ is 2, determine the order of the element $b$.
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Context:
Given a rearrangement of the numbers from 1 to $n$, each pair of consecutive elements $a$ and $b$ of the sequence can be either increasing (if $a<b$ ) or decreasing (if $b<a$ ). How many rearrangements of the numbers from 1 to $n$ have exactly two increasing pairs of consecutive elements?
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Context:
The towns in one country are connected with bidirectional airlines, where at least one direction is paid. In a trip from town $A$ to town $B$, there are exactly 22 routes that are free. Find the least possible number of towns in the country.
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Context:
Find a solution to the congruence equation $x^2 \equiv 7 \pmod{787}$, where the solution $x$ is between 1 and 786.
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Context:
已知x∈[0,2π]，且$2 \cos x \leq \vert \sqrt { 1 + \sin 2 x } - \sqrt { 1 - \sin 2 x } \vert \leq$ $\sqrt { 2 } .$求x的取值范围.
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Context:
Find the area of the region enclosed by the curves \( r = 2 + \sin (3 \theta) \) and \( r = 4 - \cos (3 \theta) \) using iterated integrals.
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Context:
A ray starting from point $A(-4,1)$ reflects off the line $l_{1}: x-y+3=0$ and the reflected ray passes through point $B(-3,2)$. Find the slope of the line containing the reflected ray.
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Context:
Given seven distinct letters A, B, C, D, E, F, and G, determine the number of ways to form seven sets of three letters each such that each letter appears with every other letter exactly once. What is the total number of such combinations?
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Context:
The difference between $a$ and $5$ is greater than twice $a$. Express this as an inequality.
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Context:
Simplify the product
$$
p = (10 + 1) (10^2 + 1) ((10^2)^2 + 1) ((10^2)^4 + 1) ((10^2)^8 + 1)...((10^2)^{1024} +1)
$$
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Context:
设a，b，c，d是正整数，满足$a ^ { 5 } =$ $ b ^ { A } ,$ $c ^ { 3 } = d ^ { 2 }$，且$c - a = 1 9$，求$d - b .$
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Context:
Given a positive integer $n (n \geqslant 2)$. From the numbers $1, 2, \cdots, 3n$, select $m$ numbers such that there must be four distinct numbers $a, b, c, d$ among them, which satisfy $a = b + c + d$. Find the minimum value of $m$.
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Context:
A cyclist traveled from point A to point B, stayed there for 30 minutes, and then returned to A. On the way to B, he overtook a pedestrian, and met him again 2 hours later on his way back. The pedestrian arrived at point B at the same time the cyclist returned to point A. How much time did it take the pedestrian to travel from A to B if his speed is four times less than the speed of the cyclist?
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Context:
如果:
(1) $a 、 b 、 c 、 d$ 都属于 $\{1,2,3,4\}$;
(2) $a \neq b, b \neq c, c \neq d, d \neq a$;
(3) $a$ 是 $a 、 b 、 c 、 d$ 中的最小值.
那么，可以组成的不同的四位数 $\overline{a b c d}$ 的个数是 $\qquad$.
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Context:
On the radius \( AO \) of a circle with center \( O \), a point \( M \) was chosen. On one side of \( AO \) on the circle, points \( B \) and \( C \) were chosen such that \(\angle AMB = \angle OMC = \alpha\). Find the length of \( BC \), given that the radius of the circle is \( 12 \) and \(\sin \alpha = \frac{\sqrt{35}}{6}\).
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Context:
Determine for which values of \( p \geq 1 \) the functional \( F(f) = \int_{0}^{1}f(x^2)dx \) is bounded in \( L_p[0,1] \).
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Context:
If $\forall x\geqslant 0$, the inequality $xe^{x}+ae^{x}\ln \left(x+1\right)+1\geqslant e^{x}(x+1)^{a}$ always holds, then the minimum value of the real number $a$ is ______.
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Context:
The Denali Star train starts from Anchorage and travels towards Fairbanks at a certain speed. After some time, another train, Glacier Discovery, starts at Fairbanks on a parallel track and moves towards Anchorage at a speed of 70 mph. Both the Denali Star and Glacier Discovery trains have a length of 1/6 miles each. After the trains meet, the faster train takes 9.999999999999998 seconds to overtake the slower one. What is the speed of the Denali Star train?
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Context:
Given that $f(x)$ is a function with its first derivative $f'(x) = f^2(x) + x$, determine how many times $f(x)$ is differentiable, assuming it is differentiable at least once.
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Context:
Determine the value of \(\lambda\) such that the number of quadratic polynomials \(ax^2+2bx+c\) meeting the following criteria is equal to \(1000\lambda\): (i) \(a, b, c\) are distinct integers, (ii) \(a, b, c \in \{1, 2, 3, \ldots, 2001, 2002\}\), and (iii) \(x+1\) is a factor of \(ax^2+2bx+c\).
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Context:
Determine the period of the function \( f(x) = \sin(3\pi x)\cos(3\pi[x]) \), where \([x]\) is the greatest integer function and \(\{x\}\) is the fractional part function.
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Context:
Let $P(x)$ be a polynomial with real coefficients such that $P(12)=20$ and \[ (x-1) \cdot P(16x) = (8x-1) \cdot P(8x) \] holds for all real numbers $x$. Compute the remainder when $P(2014)$ is divided by $1000$.
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Context:
Evaluate the limit \( \lim_{n\to\infty} n\int_{1}^{2} \frac{1}{x^2(1+x^n)}dx \).
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Context:
Evaluate the limit: $$\lim\limits_{x\to{-2}}(2\sqrt{-1-x}-1)^{\frac{1}{4^{-x}-16}}$$ without using l'Hôpital's rule, derivatives, or Taylor series.
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Context:
Evaluate the integral \( \int dz^{*}dz \exp\bigg[-(z^{*}z-cz^{*}-zd^{*})\bigg] \) where \( z,z^{*},c,d^{*} \in \mathbb{C} \).
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Context:
In triangle $ABC$, the angle bisectors $\overline{AD}$, $\overline{BE}$, and $\overline{CF}$ intersect at the incenter $I$. Given that $DI = 3$, $BD = 4$, and $CI = 5$, find the area of triangle $ABC$. \( \text{Express your answer in square units.} \)
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Context:
John decides to buy utensils. They come in 30 packs with an equal number of knives, forks, and spoons. However, for an upcoming event, John requires a specific ratio of knives, forks, and spoons which is 2:3:5, and he wants to have at least 50 spoons at the event. How many packs does he need to buy to satisfy both the ratio and spoon requirements?
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Context:
Given $y = \frac{(m^2 - m)x}{m^2 + 1}$ is a linear function of $x$, then $m$ equals to __.
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Context:
The integers from $1$ to $25$, inclusive, are randomly placed into a $5 \times 5$ grid such that in each row, the numbers are increasing from left to right. If the columns from left to right are numbered $1, 2, 3, 4,$ and $5$, then the expected column number of the entry $23$ can be written as $\frac{a}{b}$ where $a$ and $b$ are relatively prime positive integers. Find $a + b$.
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Context:
Evaluate the limit: \[ \lim_{n\to\infty} \left( \frac{\Gamma (n+1,n)}{n \Gamma (n)} - e^{-n} \right) \]
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Context:
Two concentric circles have radii \(1\) and \(2\). Two points on the outer circle are chosen independently and uniformly at random. Find the probability that the chord joining the two points intersects the inner circle. Express your answer in the form \(\frac{k}{m}\) where \(\frac{k}{m}\) is a simplified fraction, and give the value of \(k + m\).
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Context:
Find the remainder when \(20!\) is divided by 487.
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Context:
Calculate the expected number of coin tosses required to obtain the sequence H-T-H-T-T-H with a fair coin.
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Context:
Let $n \geqslant 3$ be an integer and let

$$
M=\left\{\frac{a_{1}+a_{2}+\cdots+a_{k}}{k}: 1 \leqslant k \leqslant n \text { and } 1 \leqslant a_{1}<\cdots<a_{k} \leqslant n\right\}
$$

be the set of the arithmetic means of the elements of all non-empty subsets of $\{1,2, \ldots, n\}$. Find $\min \{|a-b|: a, b \in M$ with $a \neq b\}$.

## Proposed by Romania
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Context:
A rectangle inscribed in a triangle has its base coinciding with the base $b$ of the triangle. If the altitude of the triangle is $h$, and the altitude $x$ of the rectangle is half the base of the rectangle, then:
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Context:
A quadrilateral \(ABCD\) is inscribed in a circle. At point \(C\), a tangent line \(\ell\) to this circle is drawn. A circle \(\omega\) passes through points \(A\) and \(B\) and is tangent to the line \(\ell\) at point \(P\). The line \(PB\) intersects segment \(CD\) at point \(Q\). Find the ratio \(\frac{BC}{CQ}\), given that \(B\) is the point of tangency to the circle \(\omega\).
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Context:
Determine the number of real solutions to the equation \(2^{\sin x} + 2^{\cos x} = 2^{3x^2 - 2x^3 - 1}\).
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Context:
Sergei chooses two different natural numbers $a$ and $b$. He writes four numbers in a notebook: $a$, $a+2$, $b$, and $b+2$. He then writes all six pairwise products of the numbers in the notebook on the blackboard. Let $S$ be the number of perfect squares on the blackboard. Find the maximum value of $S$.
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Context:
For a real number $y$, find the maximum value of
\[
\frac{y^6}{y^{12} + 3y^9 - 9y^6 + 27y^3 + 81}.
\]
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Context:
Josh gets together with 7 friends. Everyone including him puts 5 dollars into a pot. First place gets a certain percentage of the money. Second and third place split the rest. Third place gets $4. What percentage of the money does first place get?
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Context:
A small class of nine boys are to change their seating arrangement by drawing their new seat numbers from a box. After the seat change, what is the probability that there is only one pair of boys who have switched seats with each other and only three boys who have unchanged seats?
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Context:
Three identical cones with vertex \( A \) and an apex angle of \(\frac{\pi}{3}\) touch each other externally. Each of them touches a fourth cone with vertex at point \( A \) internally. Find the apex angle of the fourth cone. (The apex angle of a cone is defined as the angle between its generatrices in an axial section.)
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\boxed{\frac{\pi}{3} + 2 \arcsin \frac{1}{\sqrt{3}}}


Context:
Evaluate the integral \( \iint _R \frac{y^2}{x} \, dx \, dy \) over the region \( R \) in the \( xy \)-plane bounded by \( y=x^2 \), \( y=4x^2 \), \( xy=1 \), and \( xy=5 \).
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Context:
When $m$ is such that the function $y = (m+2)x + 4x - 5$ is a linear function.
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Context:
Evaluate the limit \( \lim_{x\to 0^+} \frac{1}{x}\int_0^x \frac{f(t)}{t}\, dt \), where \( f \) is integrable on \([0,1]\) and \( \lim_{x\to 0} f(x)=L \).
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Context:
Given a circle O: x^2 + y^2 = 1 and a point P(-1, sqrt(3)), where A and B are two moving points on circle O, find the maximum value of angle APB. Express your answer in radians.
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Context:
Evaluate the definite integral: \[ \int_{0}^{\frac{\pi}{2}}{\frac{dx}{\sqrt{\cos(x)}\cdot\sin(x)}} \]
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Context:
The slant height of a cone is \( a \), and the distance from the vertex of the cone to the center of the inscribed sphere is \( b \). Find the angle between the slant height and the base plane.
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Context:
8. $\frac{\tan 96^{\circ}-\tan 12^{\circ}\left(1+\frac{1}{\sin 6^{\circ}}\right)}{1+\tan 96^{\circ} \tan 12^{\circ}\left(1+\frac{1}{\sin 6^{\circ}}\right)}=$ $\qquad$
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Context:
Evaluate the definite integral: $$\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\frac{\cos^5(x)}{1+\sin^3(x)}\,dx$$
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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  $$ \max\{a+b-c,a-b+c,-a+b+c\}. $$
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Context:
How many roots does the polynomial $z^6 - 5z^4 + 8z - 1$ have in the domain $|z| < 2$?
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Context:
Bill is laying power cable for a new neighborhood. There are going to be 18 east-west streets that are 2 miles long and 10 north-south streets that are 4 miles long. It takes 5 miles of cable to electrify 1 mile of street due to underground wiring and redundancy measures. The cable costs vary, with east-west cables costing $2500/mile and north-south cables costing $3500/mile. Additional costs include $5000 for each of the 20 intersections. If there are also 5 streets that require a higher grade cable priced at $4000/mile, 3 of these streets being east-west and 2 being north-south, what is the total cost of cable for the neighborhood?
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Context:
A circle with radius \( r \) passes through the vertices \( A \) and \( B \) of triangle \( ABC \) and intersects side \( BC \) at point \( D \). Find the radius of the circle passing through points \( A \), \( D \), and \( C \), given that \( AB = c \) and \( AC = b \).
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Context:
Determine the smallest natural number $k$ such that the series \( \sum\limits_{n=1}^\infty\frac{(n!)^{k-1}(27n)^n}{(kn)!e^n} \) converges.
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Context:
Given the ellipse $\dfrac{x^2}{1-k} + \dfrac{y^2}{2+k} = 1$, if its focus lies on the $x$-axis, determine the range of values for $k$.
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Context:
Determine the Lebesgue measure of the set $A$ within the interval [0,1], where $A$ consists of numbers whose continued fraction expansions contain an infinite subsequence such that each term is divisible by its predecessor.
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Context:
Determine the number of points within the interval $[0, 6\pi]$ where the function $f(x)$, defined as $f(x) = \tan x$ for rational $x$ and $f(x) = x^2 + 1$ for irrational $x$, is continuous.
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Context:
Find the residue at infinity for the function \( f(z)=\frac{\sin(\pi z)}{z^2(z-1)} \) by using the substitution \( z=\frac{1}{w} \) and evaluating the limit. What is the correct residue at infinity for \( f(z) \)?
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Context:
Find the coordinates of point \( A \) that is equidistant from points \( B \) and \( C \).

\( A(0, y, 0) \)

\( B(-2, 4, -6) \)

\( C(8, 5, 1) \)
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Context:
Determine the time complexity of an algorithm that uses Stirling's approximation to compute the hyperfactorial function for large input values $m$.
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Context:
Peter has 25 cards, each printed with a different integer from 1 to 25. He wishes to place N cards in a single row so that the numbers on every adjacent pair of cards have a prime factor in common. What is the largest possible value of N? Express your answer as a whole number.
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Context:
Given the function $f(x)= \frac {1}{x+2}$, point $O$ is the origin, point $A_{n}(n,f(n))$ where $n \in N^{*}$, vector $ \overrightarrow{i}=(0,1)$, $\theta_{n}$ is the angle between vector $ \overrightarrow{OA_{n}}$ and $ \overrightarrow{i}$. Find the range of real number $t$ such that $\frac {\cos \theta_{1}}{\sin \theta_{1}}+ \frac {\cos \theta_{2}}{\sin \theta_{2}}+ \frac {\cos \theta_{3}}{\sin \theta_{3}}+…+ \frac {\cos \theta_{n}}{\sin \theta_{n}} < t$ always holds.
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Context:
Reginald is selling apples. He usually sells each apple for $1.25, but he decides to offer a promotional discount for customers who buy in bulk. The discount offer is: if a customer buys 5 apples, they get 1 additional apple for free. He plans to use his profits to help repair his bike. His bike cost $80 and the repairs cost 25% of what his bike cost. After the repairs are paid for, 1/5 of the money he'd earned remains. Given that all customers purchasing apples from Reginald took advantage of the promotional discount, how many apples did he sell?
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Context:
In a compartment, any group of at least 3 passengers has a unique common friend. If friendship is mutual and no one is their own friend, determine the maximum number of friends that any single person in the compartment can have.
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Context:
Let \(ABC\) be a triangle with \(AB=2021, AC=2022\), and \(BC=2023\). Compute the minimum value of \(AP+2BP+3CP\) over all points \(P\) in the plane.
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Context:
Evaluate the limit: \[ \lim_{n\rightarrow\infty}\left(\sum\limits_{r=0}^{n}{\left(\frac{\binom{n}{r}}{n^r}\cdot\frac{1}{r+3}\right)}\right) \]
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Context:
Determine the cardinality of the set of all injective functions from the natural numbers to the real numbers.
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Context:
Compute the volume of the solid bounded by the surfaces: \( z = x^2 \), \( z = x + 2 \), \( y + z = 5 \), and \( y = -1 \).
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Context:
Evaluate the limit \(\lim\limits_{m \to \infty}\int_0^1 |\ln^m x\ln^n(1+x)|{\rm d}x\) where \(n \in \mathbb{N^+}\).
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Context:
Consider two numerical sequences $X_n$ and $Y_n$. The sequence $X_n$ consists of random integers from one to six for each $n$. Define $Y_1 = X_1$, and for $n \geq 2$, $$Y_n = X_n + \frac{1}{Y_{n-1}}.$$ Determine the probability $P$ that $$\frac{1+\sqrt{3}}{2} \leq Y_n \leq 1+ \sqrt{3}$$ as $n$ approaches infinity.
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Context:
Estimate an upper bound for \( \lim_{N \to \infty} \text{Pr}\left(\frac{Y_N}{N} \leq \alpha\right) \) where \( Y_N = \max_{1 \leq i \leq N} \left|\sum_{j=1}^{i} X_j\right| \) and \( X_1, X_2, \ldots, X_N \) are independent random variables with \( \text{Pr}(X_i = 1) = \text{Pr}(X_i = -1) = 1/2 \), for \( 0 < \alpha < 1 \).
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Context:
Find the functional derivative of the differential entropy with respect to the probability density function, i.e., determine \( \frac{\delta F[p]}{\delta p(x)} \), where \( F[p] = \int_X p(x)\ln(p(x))dx \) and \( p(x) \) is a probability density function.
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Context:
Evaluate the limit \( \lim_{n \to \infty} n^2 \int_0^1 \frac{dx}{(1+x^2)^n} \).
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Context:
Determine the number of distinct diagonalizable matrices of size $9 \times 9$ with real entries that satisfy the equation $A^2 = I$. Provide your answer as a single integer.
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Context:
Find a group $G$ that contains an element $x$ of order 5 and an element $y$ of order 6, such that the order of the product $xy$ is less than 30.
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Context:
Find the gradient \( \nabla_W J \) of the function \( J(W) = \left\| W^T W - I \right\|_F^2 \), where \( W \) is a matrix and \( \left\| \cdot \right\|_F \) denotes the Frobenius norm.
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Context:
Evaluate the limit as \( n \to \infty \) of the expression:
\[
\frac{e^{1/n}}{n^2} + 2\frac{e^{2/n}}{n^2} + 3\frac{e^{3/n}}{n^2} + \ldots + n \frac{e^{n/n}}{n^2}.
\]
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Context:
Find the value of \( p \) such that the function \( f(x,y) = \frac{1}{(x^2+y^3-2)^p} \) is integrable over the region defined by \( 1 \leq x \leq 2 \) and \( 1 \leq y \leq 3 \).
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Context:
Determine the number of self-intersections in a regular \((2018 ; 25)\)-star, where \(2018\) and \(25\) are coprime, \(2018 \geq 5\), and \(25 < \frac{2018}{2}\).
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Context:
fractions are corrected by using each of the numbers $1,2,...,26$ once.Example:$\frac{12}{5},\frac{18}{26}.... $
What is the maximum number of fractions which are integers?
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Context:
Determine the number of conjugacy classes in a group of order $625$ if its center has order $25$. Provide your answer as a single integer.
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Context:
Two decks, each containing 36 cards, were placed on a table. The first deck was shuffled and placed on top of the second deck. For each card in the first deck, the number of cards between it and the same card in the second deck was counted. What is the sum of these 36 numbers?
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Context:
Let $a_1, a_2, \ldots, a_{2017}$ be the 2017 distinct complex numbers which satisfy $a_i^{2017} = a_i + 1$ for $i = 1, 2, \ldots, 2017$. Compute $$\sum_{i=1}^{2017} \frac{a_i}{a_i^2 + 1}.$$
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Context:
Evaluate the integral \( \int \int \int_A \frac{dxdydz}{(x+y+1)^2} \) where \( A=\{(x,y,z):0<x<1, 0<x+y<1, 0<z(x+y+1)<1\} \).
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Context:
Find a real number $x$ such that $x^5 - x - 1 = 0$.
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Context:
Determine the number of roots of the equation \(4\cos(e^x) = 2^x + 2^{-x}\).
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Context:
Evaluate the integral \( \int_0^\infty \frac{\sin x(1-\cos x)}{x^2} \, dx \).
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Context:
In a cube with edges of length 2 units, point Q is the intersection of the diagonals of one face. Calculate the length of QR. Express your answer as a radical.
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Context:
In a restaurant, clients arrive following a homogeneous Poisson process with an average rate of 10 clients per hour. Each client independently has a 20% probability of eating at the restaurant and an 80% probability of taking away. On average, how many clients should be expected before one eats at the restaurant?
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Context:
Let $N \subset 2^X$. Let $M$ be the $\sigma$-algebra generated by $N$ and let $M'$ be the algebra generated by $N$. What is the relationship between $M$ and $M'$?
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Context:
Find four consecutive odd numbers, none of which are divisible by 3, such that their sum is divisible by 5. What is the smallest possible value of this sum?
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Context:
Given real numbers $x$ and $y$ that satisfy the equation $\frac{1}{x} + \frac{2}{y} = \sqrt{xy}$, find the minimum value of $xy$.
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Context:
Evaluate the integral \( \int_{-1}^{2} (x^3 - 2) \, d[x] \), where \([x]\) is the floor function.
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Context:
Evaluate the integral \( \int_0^{\frac{\pi}{2}} \log\left(\frac{1+a\cos(x)}{1-a\cos(x)}\right)\frac{1}{\cos(x)}dx \) for \( |a| \le 1 \).
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Context:
Dave walks to school and averages 85 steps per minute, with each step being 80 cm long. It now takes him 15 minutes to get to school. Jack, walking the same route to school, takes steps that are 72 cm long and averages 104 steps per minute. Find the time it takes Jack to reach school.
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Context:
In a game with 111,400 rounds numbered from 0 to 111,400, a player has won 3,000 rounds. For each winning round, the last three digits are noted (e.g., Round 89,345 becomes "345"). What is the expected number of three-digit endings, from "000" to "999", that do not appear among these 3,000 winning rounds?
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Context:
Calculate the variational derivative of the expression $-f\nabla^2f$ with respect to $f$. Express your answer as $\frac{\delta(-f\nabla^2f)}{\delta f}$. What is the result?
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Context:
Dan is holding one end of a 26-inch long piece of light string that has a heavy bead on it with each hand (so that the string lies along two straight lines). If he starts with his hands together and keeps his hands at the same height, how far does he need to pull his hands apart so that the bead moves upward by 8 inches?
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Context:
Evaluate the contour integral \(\oint_{|z| = 3} \frac{z^3e^{1/z}}{z^3+1}dz\) using the Residue Theorem.
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Context:
Three uncoordinated aliens launch a 3-day attack on 4 galaxies. Each day, each of the three aliens chooses a galaxy uniformly at random from the remaining galaxies and destroys it. They make their choice simultaneously and independently, so two aliens could destroy the same galaxy. If the probability that every galaxy is destroyed by the end of the attack can be expressed as \(\frac{m}{n}\) for relatively prime positive integers \(m,n\), what is \(m+n\)?
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Context:
Determine the number of positive integers, each with exactly \(2^{2005}\) digits, where each digit is either 7 or 8, such that among any two chosen integers, at most half of their corresponding digits are the same.
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Context:
Let \( F: \mathbb{R}^3 \rightarrow \mathbb{R}^3 \) be a linear transformation such that \( F(0,0,1) = (2,-1,\alpha) \). The vectors \( (1,0,-1) \) and \( (0,1,0) \) are eigenvectors of \( F \) with associated eigenvalues \( 3 \) and \( -9 \) respectively. Determine the value of \( \alpha \) given that \( -2 \) is also an eigenvalue of \( F \).
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Context:
Find the smallest integer \( n \) such that a square can be decomposed into \( n \) non-overlapping triangles, all of whose angles are acute.
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Context:
A necklace consists of 100 beads of red, blue, and green colors. It is known that among any five consecutive beads, there is at least one blue bead, and among any seven consecutive beads, there is at least one red bead. What is the maximum number of green beads that can be in this necklace? (The beads in the necklace are arranged cyclically, meaning the last one is adjacent to the first one.)
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Context:
Calculate the value of $S_{10} + I_{10}$, where $S_n = \sum_{i=1}^n \frac{1}{i}$ and $I_n = \int_{1}^{n} \frac{x - \lfloor x \rfloor}{x^2} \, dx$. Given that $I_{10} = \ln(10) - \int_{1}^{10} \frac{\lfloor x \rfloor}{x^2} \, dx$, evaluate the integral $\int_{1}^{10} \frac{\lfloor x \rfloor}{x^2} \, dx$ to find $S_{10} + I_{10}$. Provide your answer as a single numerical value.
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Context:
Let $A$ be a $6 \times 6$ matrix with characteristic polynomial $x(x+1)^2(x-1)^3$. Find the dimension of the centralizer $\gamma = \{B \in M_6(\mathbb{R}) : AB = BA\}$.
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Context:
Lucas and Monk play chess, and the first to win a match receives a book as a reward. Each match can result in a win, loss, or tie. The probability of there being $k$ matches until someone wins is $\frac{1}{2^{k+2}}$, where $k=1,2,...$. Lucas' chance of winning the book is twice that of Monk. What is the probability that Monk will win the book?
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Context:
Determine the distribution of the second greatest value among three independent random variables X1, X2, and X3, each uniformly distributed over the interval (0,1).
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Context:
Let  $a$  be real numbers. Given a matrix  $A=\left(\begin{array}{cc} a & 1-a  1-a & a \end{array}\right).$  Let  $\left(\begin{array}{c} x_0  y_0 \end{array}\right)=\left(\begin{array}{c} 1  0 \end{array}\right),$  and the sequences  $\{x_n\},\ \{y_n\}$  are defined by the following equation.

\[ \left(\begin{array}{c} x_n  y_n \end{array}\right)=A \left(\begin{array}{c} x_{n-1}  y_{n-1} \end{array}\right),\ \ n=1,2,\ \cdots \]

then find the necessray and sufficient condition for  $a$  such that the sequence  $\{x_n\}$  converges.
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Context:
已知复数数列 $\left\{z_{n}\right\}$ 满足 $z_{1}=1, z_{n+1}=\overline{z_{n}}+1+n \mathrm{i}$ ，其中 $n=1,2, 3\cdots$ ，其中 i 是虚数单位, $\overline{z_{n}}$ 表示 $z_{n}$ 的共轭复数, 则 $z_{2015}$ 的实部值为 $\qquad$。
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Context:
Given 2022 lines in the plane, such that no two are parallel and no three are concurrent. We denote \( E \) as the set of their intersection points. We want to assign a color to each point in \( E \) such that any two points on the same line, whose connecting segment does not contain any other point from \( E \), have different colors. What is the minimum number of colors required to achieve this coloring?
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Context:
Determine the size of the conjugacy class of the permutation $(123)(456)$ in the alternating group $A_7$.
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Context:
将 $1,2,3,4,5,6,7,8$ 分别放在正方体的八个顶点上，使得每一个面上的任意三个数之和均不小于 10 。求每一个面上四个数之和的最小值。
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Context:
Compute the flux of the vector field \( \vec{F} \) through the surface \( S \), where \( \vec{F} : U \subseteq \mathbb{R}^3 \to \mathbb{R}^3 \) is given by:
\[
\vec{F} = \left( \frac{x}{(x^2 + y^2 + z^2)^{3/2}}, \frac{y}{(x^2 + y^2 + z^2)^{3/2}}, \frac{z}{(x^2 + y^2 + z^2)^{3/2}} \right)
\]
where \( U = \mathbb{R}^3 \setminus \{(0, 0, 0)\} \). The surface \( S \) is a sphere of radius 12345 centered at the origin, with a small section chopped off from the top and replaced by a flat disk, ensuring \( S \) remains closed. Assume \( S \) is oriented with an outward-pointing normal. Calculate the flux of \( \vec{F} \) through \( S \).
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Context:
A regular  $2008$ -gon is located in the Cartesian plane such that  $(x_1,y_1)=(p,0)$  and  $(x_{1005},y_{1005})=(p+2,0)$ , where   $p$  is prime and the vertices, \[(x_1,y_1),(x_2,y_2),(x_3,y_3),\cdots,(x_{2008},y_{2008}),\]
are arranged in counterclockwise order. Let \begin{align*}S&=(x_1+y_1i)(x_3+y_3i)(x_5+y_5i)\cdots(x_{2007}+y_{2007}i),T&=(y_2+x_2i)(y_4+x_4i)(y_6+x_6i)\cdots(y_{2008}+x_{2008}i).\end{align*} Find the minimum possible value of  $|S-T|$ .
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Context:
Find the constant-coefficient linear differential equation of the smallest possible order that has \( y(x) = x^3 \) as a solution.
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Context:
Given a power function $y = f(x)$ whose graph passes through the point $(2, \frac{\sqrt{2}}{2})$, if $f(m) = 2$, then $m = \_\_\_\_\_$.
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Context:
Calculate the Euler characteristic of a topological space formed by joining two projective planes, each with two punctures, along their punctures.
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Context:
Elbert and Yaiza each draw $10$ cards from a $20$-card deck with cards numbered $1,2,3,\dots,20$. Then, starting with the player with the card numbered $1$, the players take turns placing down the lowest-numbered card from their hand that is greater than every card previously placed. When a player cannot place a card, they lose and the game ends.

Given that Yaiza lost and $5$ cards were placed in total, compute the number of ways the cards could have been initially distributed. (The order of cards in a player’s hand does not matter.)
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Context:
In the plane, two points $A$ and $B$ are given, with the distance between them being $d$. Construct a square such that points $A$ and $B$ lie on its boundary, in a way that minimizes the sum of the distances from point $A$ to the vertices of the square. What is this minimum sum?
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Context:
Find the closed-form expression for the limit:

$$\lim_{n\to \infty}\left(\frac{1}{ \Gamma\left(\displaystyle \frac{1}{1}\right)}+ \frac{1}{ \Gamma\left( \displaystyle \frac{1}{2}\right)}+ \cdots + \frac{ 1}{ \Gamma \left( \displaystyle \frac{1}{ n}\right) }- \log\left( \Gamma\left(\displaystyle\frac{1}{n}\right)\right)\right),$$

where the sequence is known as the Gammaharmonic series.
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Context:
Students in a class participate in a tree-planting activity. There are three different types of saplings, labeled A, B, and C, in the nursery. They are to be planted in a row of 5 pits, with the condition that the same type of sapling cannot be adjacent, and only type A saplings can be planted in the first and the fifth pits. Calculate the total number of planting methods.
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Context:
Let \( n > 0 \) be an integer. We have a two-pan balance and \( n \) weights, with masses \( 2^{0}, 2^{1}, \ldots, 2^{n-1} \). We want to place these \( n \) weights onto the balance one by one in such a way that the right pan is never heavier than the left pan. At each step, we choose one of the weights that has not yet been placed on the balance and place it either in the left pan or the right pan, continuing until all the weights have been placed. Determine the number of ways to do this.
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\boxed{1 \cdot 3 \cdot 5 \cdot \ldots \cdot (2n - 1)}


Context:
By mistakenly subtracting 20 from a number gives 52, when 4 should have divided by a number. How much do you get when you calculate it correctly?
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Context:
Calculate the fundamental group of $S^3 - \gamma$, where $\gamma$ is a circle in $\mathbb{R}^3$ and $S^3 = \mathbb{R}^3 \cup \{\infty\}$ is the one-point compactification of $\mathbb{R}^3$. Provide your answer as a group.
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Context:
Mark on the line with red all points of the form $81x + 100y$, where $x, y$ are natural numbers, and paint the remaining integer points blue. Find a point on the line such that any symmetrical integer points relative to it are painted in different colors.
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Context:
Line \( AB \) is the common perpendicular segment of skew lines \( a \) and \( b \). Point \( A \) is on line \( a \) and point \( B \) is on line \( b \). Given that \( AB = 2 \) and the angle between the skew lines \( a \) and \( b \) is \( 30^\circ \), and \( AP = 4 \) on line \( a \), find the distance from point \( P \) to line \( b \).
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Context:
Find the minimum page size $s \in \Bbb N$ such that the inequality \( \frac{i + l}{1 + \lfloor \frac{i}{s} \rfloor} \leq s \) holds, given that $i > l$ and $\{i, l\} \in \Bbb N$. What is the minimum value of $s$?
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Context:
Evaluate the expression \( I(a) = \lim_{\Omega \to \mathbb{R}^3} \, \int_\Omega \nabla \left( \frac{1}{|x-a|} - \frac{1}{|x|} \right) dV \), where \(x\) and \(a\) are vectors in \(\mathbb{R}^3\). Determine the value of \(I(a)\) or explain why the evaluation fails.
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Context:
Evaluate \( \mathop{\lim }\limits_{{x \rightarrow  {1}^{ - }}}\mathop{\prod }\limits_{{n = 0}}^{\infty }{\left( \frac{1 + {x}^{n + 1}}{1 + {x}^{n}}\right) }^{{x}^{n}} \) .
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Context:
Evaluate the contour integral \( \frac{1}{2 \pi i} \int_{|z| = 1} \frac{6z^{98}}{23z^{99}-2z^{81}+z^4-7} \, dz \) using the residue theorem.
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Context:
Compute the residue of the function \( \frac{(\sin(z))^2}{(\sinh(z))^5} \) at \( z = 0 \).
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Context:
Given the polynomial \( f(x) = \frac{x^3}{\alpha} + \frac{x^2}{\beta} + \frac{x}{\gamma} + \frac{1}{\delta} \) where \( \alpha, \beta > 0 \), and the condition \( |f(x)| \leq 1 \) for \( |x| \leq 1 \), determine the maximum value of \( \frac{1}{\alpha} + \frac{1}{\beta} + \left| \frac{1}{\gamma} \right| + \left| \frac{1}{\delta} \right| \).
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Context:
Find the largest value of \( k \) such that the \( k \)-th absolute moment \( E[|X|^k] \) is finite for a random variable \( X \) with the cumulative distribution function \( F_X(x) = \frac{1}{1+e^{-x}} \) for \( x \geq 0 \).
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Context:
Given a convex quadrilateral $ABCD$, points $A_1$, $B_1$, $C_1$, and $D_1$ are located on the extensions of sides $AB$, $BC$, $CD$, and $DA$, respectively, such that $|BB_1|=|AB|$, $|CC_1|=|BC|$, $|DD_1|=|DC|$, and $|AA_1|=|DA|$. Express the area of quadrilateral $A_1B_1C_1D_1$ in terms of the area of quadrilateral $ABCD$.
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Context:
Given a positive integer \( n \), let \( G \) be a simple, undirected graph with \( 2n + 1 \) vertices such that there is no cycle of length greater than 3. Find the maximum number of edges in \( G \).
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Context:
For which values of \(a\) does the linear transformation \(L: \mathbb{R}^3 \rightarrow \mathbb{R}^3\) satisfy \(v_1, v_2 \in \ker(L)\) and \(L(v_3) = (1, 1, 1)\), where \(v_1 = (1, a, a^2)\), \(v_2 = (a^2, 1, a)\), and \(v_3 = (a, a^2, 1)\)?
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Context:
For an \( n \times  n \) matrix \( A \) , let \( {X}_{i} \) be the set of entries in row \( i \) , and \( {Y}_{j} \) the set of entries in column \( j \) , \( 1 \leq  i,j \leq  n \) . We say that \( A \) i
 golden if \( {X}_{1},\ldots ,{X}_{n},{Y}_{1},\ldots ,{Y}_{n} \) are distinct sets. Find the least integer \( n \) such that there exists a \( {2004} \times  {2004} \) golden
 matrix with entries in the set \( \{ 1,2,\ldots ,n\} \) .
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Context:
Calculate the flux of the vector field \( F(x,y,z) = (2x+y-z, x-2y+z, x+y-2z) \) through the surface defined by the equation \( |x+y-z| + |x-2y+z| + |x+y-2z| = 2 \).
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Context:
Calculate the integral \( I = \int_0^1 |f(x) - f^{-1}(x)| \, dx \), where \( f(x) = x e^{x^2 - 1} \) and \( f: [0, 1] \to [0, 1] \).
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Context:
Consider a cross-shaped grid of size 2017, where each cell initially contains a 1. A transformation \( \pi \) is defined as selecting any 3x3 sub-cross of the grid and multiplying all the cells inside by -1. Determine the minimum number of \( \pi \) transformations required to change every cell in the grid to -1.
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Context:
Calculate the exterior derivative \(d\omega(X,Y)\), where \(\omega = 2x\,dx \wedge dy + y^2\,dx \wedge dz\) is a differential form, and \(X = x^2y\frac{\partial}{\partial y} + x\frac{\partial}{\partial z}\), \(Y = x\frac{\partial}{\partial y}\) are vector fields.
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Context:
Evaluate the integral \( \int\limits_{t,t'}{\frac{dt\,dt'}{\|x(t)-y(t')\|}} \) where \( x(t) \) and \( y(t') \) are curves traversing the boundary of \([0,1]^2\) in \( \mathbb{R}^2 \) counterclockwise.
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Context:
A crazy physicist has discovered a new particle called an omon. He has a machine that takes two omons of mass $a$ and $b$ and entangles them. This process destroys the omon with mass $a$, preserves the one with mass $b$, and creates a new omon whose mass is $\frac{1}{2}(a+b)$. The physicist can then repeat the process with the two resulting omons, choosing which omon to destroy at every step. The physicist initially has two omons whose masses are distinct positive integers less than $1000$. What is the maximum possible number of times he can use his machine without producing an omon whose mass is not an integer?
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Context:
Let $S_n$ and $T_n$ be the respective sums of the first $n$ terms of two arithmetic series. If $S_n:T_n=(7n+1):(4n+27)$ for all $n$, the ratio of the eleventh term of the first series to the eleventh term of the second series is:
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Context:
Find the number of permutations \( a_1, a_2, \ldots, a_{10} \) of the numbers \( 1, 2, \ldots, 10 \), such that \( a_{i+1} \) is not less than \( a_i - 1 \) for \( i = 1, 2, \ldots, 9 \).
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Context:
Evaluate the double summation and provide a closed formula for it: $$\sum_{i=1}^b \sum_{j=i-b}^{i-1} j$$ where the evaluation considers different cases based on the relationship between \(i\) and \(b\).
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Context:
On a $4 \times 4 \times 3$ rectangular parallelepiped, vertices $A$, $B$, and $C$ are adjacent to vertex $D$. The perpendicular distance from $D$ to the plane containing $A$, $B$, and $C$ is closest to
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the internal angle that guarantees two isosceles triangles are similar. Express your answer as a single angle value in degrees, including the degree symbol (°).
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Context:
If the maximum and minimum values of the function $f\left( x \right)=\frac{3\cdot {{e}^{\left| x-1 \right|}}-\sin \left( x-1 \right)}{{{e}^{\left| x-1 \right|}}}$ on the interval $\left[ -3,5 \right]$ are $p$ and $q$ respectively, then find the value of $p+q$. Express your answer as a single number.
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Context:
A garden fence, similar to the one shown in the picture, had in each section (between two vertical posts) the same number of columns, and each vertical post (except for the two end posts) divided one of the columns in half. When we absentmindedly counted all the columns from end to end, counting two halves as one whole column, we found that there were a total of 1223 columns. We also noticed that the number of sections was 5 more than twice the number of whole columns in each section.

How many columns were there in each section?
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Context:
The rank of a rational number \( q \) is the unique \( k \) for which \( q=\frac{1}{a_{1}}+\cdots+\frac{1}{a_{k}} \), where each \( a_{i} \) is the smallest positive integer such that \( q \geq \frac{1}{a_{1}}+\cdots+\frac{1}{a_{i}} \). Let \( q \) be the largest rational number less than \( \frac{1}{4} \) with rank 3, and suppose the expression for \( q \) is \( \frac{1}{a_{1}}+\frac{1}{a_{2}}+\frac{1}{a_{3}} \). Find the ordered triple \( \left( a_{1}, a_{2}, a_{3} \right) \).
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Context:
Determine all points \( a \) at which the function \( h(x) \) is differentiable, where:
\[ h(x) = \begin{cases} x^3 - x^2, & \text{if } x \in \mathbb{Q} \\ 2x - 2x^2, & \text{otherwise} \end{cases} \]
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Context:
Determine the difference in the number of solutions to the equation \(z^2 + 1 = 0\) in the tessarine number system compared to the split-complex number system.
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Context:
Provide an example of a continuous function \( f: \mathbb{R} \to [0, \infty) \) that is unbounded, but such that \( f^2 \) is uniformly continuous on \( \mathbb{R} \).
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Context:
A parallelogram with an angle of $60^o$ has $a$ as the longest side and a shortest side $b$. Let's take the perpendiculars down from the vertices of the obtuse angles to the longest diagonal, then it is divided into three equal parts. Determine the ratio $\frac{a}{b}$.
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Context:
In a round robin chess tournament each player plays every other player exactly once. The winner of each game gets  $ 1$  point and the loser gets  $ 0$  points. If the game is tied, each player gets  $ 0.5$  points. Given a positive integer  $ m$ , a tournament is said to have property  $ P(m)$  if the following holds: among every set  $ S$  of  $ m$  players, there is one player who won all her games against the other  $ m\minus{}1$  players in  $ S$  and one player who lost all her games against the other  $ m \minus{} 1$  players in  $ S$ . For a given integer  $ m \ge 4$ , determine the minimum value of  $ n$  (as a function of  $ m$ ) such that the following holds: in every  $ n$ -player round robin chess tournament with property  $ P(m)$ , the final scores of the  $ n$  players are all distinct.
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Context:
Two congruent 30-60-90 are placed so that they overlap partly and their hypotenuses coincide. If the hypotenuse of each triangle is 12, the area common to both triangles is
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Context:
已知二元函数
$f(x, y)$
$=\sqrt{4 y^{2}-12 y+10}+\sqrt{18 x^{2}-18 x+5}+\sqrt{18 x^{2}+4 y^{2}-12 x y+6 x-4 y+1}$
的最小值为 $a$ ，求 $\left[8 a^{2}\right]$ 的值.
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Context:
Let $r = \sqrt{\frac{\sqrt{53}}{2} + \frac{3}{2}}$. There is a unique triple of positive integers $(a, b, c)$ such that $r^{100} = 2r^{98} + 14r^{96} + 11r^{94} - r^{50} + ar^{46} + br^{44} + cr^{40}$. What is the value of $a^{2} + b^{2} + c^{2}$?
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Context:
Let $\mathbb{R}$ be the set of real numbers. Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function such that for all real numbers $x$ and $y$, we have $$f\left(x^{2}\right)+f\left(y^{2}\right)=f(x+y)^{2}-2 x y$$ Let $S=\sum_{n=-2019}^{2019} f(n)$. Determine the number of possible values of $S$.
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Context:
Determine the value of $K$ in the integral equation below, given that $e^2 < 1$:
\[ \int_{0}^{\pi}\frac{x \, dx}{1+ e \sin x}=K\frac{\arccos e}{\sqrt{1-e^{2}}} \]
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Context:
Burrito Bear has a white unit square. She inscribes a circle inside the square and paints it black. She then inscribes a square inside the black circle and paints it white. Burrito repeats this process indefinitely. The total black area can be expressed as $\frac{a\pi+b}{c}$. Find $a+b+c$.
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Context:
The distance \( AB \) is 100 km. Cyclists depart simultaneously from \( A \) and \( B \) towards each other with speeds of 20 km/h and 30 km/h respectively. Along with the first cyclist from \( A \), a fly departs with a speed of 50 km/h. The fly travels until it meets the cyclist from \( B \), then turns around and flies back to meet the cyclist from \( A \), and continues this pattern. How many kilometers will the fly travel in the direction from \( A \) to \( B \) before the cyclists meet?
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Context:
The sides of triangle $EFG$ are in the ratio of $3:4:5$. Segment $FK$ is the angle bisector drawn to the shortest side, dividing it into segments $EK$ and $KG$. What is the length, in inches, of the longer subsegment of side $EG$ if the length of side $EG$ is $15$ inches? Express your answer as a common fraction.
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Context:
Daniel writes over a board, from top to down, a list of positive integer numbers less or equal to 10. Next to each number of Daniel's list, Martin writes the number of times exists this number into the Daniel's list making a list with the same length. If we read the Martin's list from down to top, we get the same
list of numbers that Daniel wrote from top to down. Find the greatest length of the Daniel's list can have.
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Context:
Among all such numbers \( n \) such that any convex 100-gon can be represented as the intersection (i.e., the common part) of \( n \) triangles, find the smallest.
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Context:
A Vandal and a Moderator are editing a Wikipedia article. The article originally is error-free. Each day, the Vandal introduces one new error into the Wikipedia article. At the end of the day, the moderator checks the article and has a $2 / 3$ chance of catching each individual error still in the article. After 3 days, what is the probability that the article is error-free?
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Context:
Given that for all $2\leqslant x\leqslant 3$, $3\leqslant y\leqslant 6$, the inequality $mx^{2}-xy+y^{2}\geqslant 0$ always holds, the minimum value of the real number $m$ is ______.
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Context:
Let $\triangle AEF$ be a triangle with $EF = 20$ and $AE = AF = 21$. Let $B$ and $D$ be points chosen on segments $AE$ and $AF$, respectively, such that $BD$ is parallel to $EF$. Point $C$ is chosen in the interior of $\triangle AEF$ such that quadrilateral $ABCD$ is cyclic. If $BC = 3$ and $CD = 4$, then the ratio of the areas $\frac{[ABCD]}{[AEF]}$ can be written as $\frac{a}{b}$ for relatively prime positive integers $a$ and $b$. Compute $100a + b$.
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Context:
A group of frogs (called an army) is living in a tree. A frog turns green when in the shade and turns yellow when in the sun. Initially, the ratio of green to yellow frogs was $3 : 1$. Then $3$ green frogs moved to the sunny side and $5$ yellow frogs moved to the shady side. Now the ratio is $4 : 1$. What is the difference between the number of green frogs and the number of yellow frogs now?
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Context:
A coloring of all plane points with coordinates belonging to the set  $S=\{0,1,\ldots,99\}$  into red and white colors is said to be *critical* if for each  $i,j\in S$  at least one of the four points  $(i,j),(i + 1,j),(i,j + 1)$  and  $(i + 1, j + 1)$   $(99 + 1\equiv0)$  is colored red. Find the maximal possible number of red points in a critical coloring which loses its property after recoloring of any red point into white.
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Context:
If \( 2A99561 \) is equal to the product when \( 3 \times (523 + A) \) is multiplied by itself, find the digit \( A \).
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Context:
The remainder when a number is divided by 9 is 6. What is the maximum possible value that can be filled in the blank for the remainder?
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Context:
According to analysts' forecasts, next year the economically active population (employed and unemployed) of a certain city will increase by $4\%$, and the number of unemployed will decrease by $9\%$. What percentage of the economically active population next year will the unemployed constitute if this year they made up $5.6\%$?
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Context:
Suppose that $x$ and $y$ are complex numbers such that $x+y=1$ and that $x^{20}+y^{20}=20$. Find the sum of all possible values of $x^{2}+y^{2}$.
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Context:
Given a sample set $a, 3, 5, 7$ with an average of $b$, and $a, b$ are respectively the second and fourth terms of the sequence $\{2^{n-2}\}$ ($n \in \mathbb{N}^*$), find the variance of this sample set.
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Context:
A group of cows and horses are randomly divided into two equal rows. Each animal in one row is directly opposite an animal in the other row. If 75 of the animals are horses and the number of cows opposite cows is 10 more than the number of horses opposite horses, determine the total number of animals in the group.
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Context:
Given that the sequence $\{a\_n\}$ is a non-zero arithmetic sequence, $S\_n$ is the sum of its first $n$ terms, and $a\_n = \sqrt{S_{2n-1}}$ ($n \in \mathbb{N}^*$). If the inequality $\lambda S\_n \geq a\_n - 2016$ holds for any $n \in \mathbb{N}^*$, find the minimum value of the real number $\lambda$.
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Context:
P runs a certain percentage faster than Q, so P gives Q a 300-meter head start. If the race ends in a tie, P ran 1800 meters in the race. What is the percentage by which P runs faster than Q?
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Context:
On a twenty-question test, each correct answer is worth 5 points, each unanswered question is worth 1 point, and each incorrect answer is worth 0 points. Determine the score that is NOT possible to achieve in this test. Express your answer as a single numerical value.
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Context:
On the sides of a general triangle \( A B C \), points \( K, L, M, N, U \) are given:

- Point \( K \) is the midpoint of side \( A C \),
- Point \( U \) is the midpoint of side \( B C \),
- Points \( L \) and \( M \) lie consecutively on segments \( C K \) and \( C U \) such that \( L M \parallel K U \),
- Point \( N \) lies on segment \( A B \) such that \( |A N|:|A B|=3:7 \),
- The ratio of the areas of polygons \( U M L K \) and \( M L K N U \) is 3:7.

Determine the ratio of the lengths of segments \( L M \) and \( K U \).
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Context:
Suppose that $A$ is an abelian group with order $p^5$ where $p$ is a prime number. If there exists a subgroup $N$ such that $A/N\simeq \mathbb{Z}/p^2\mathbb{Z}$, how many possible groups $A$ satisfy this condition?
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Context:
2019 numbers are arranged in a circle. For any two neighboring numbers \( x \) and \( y \), the inequalities \( |x - y| \geq 2 \) and \( x + y \geq 6 \) hold. Find the smallest possible sum of the numbers arranged in the circle.
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Context:
The roots of the polynomial $f(x) = x^8 + x^7 - x^5 - x^4 - x^3 + x + 1$ are all roots of unity. We say that a real number $r \in [0, 1)$ is nice if $e^{2i \pi r} = \cos 2\pi r + i \sin 2\pi r$ is a root of the polynomial $f$ and if $e^{2i \pi r}$ has a positive imaginary part. Let $S$ be the sum of the values of nice real numbers $r$. If $S = \frac{p}{q}$ for relatively prime positive integers $p, q$, find $p + q$.
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Context:
For how many odd integers $k$ between 0 and 100 does the equation $2^{4m^{2}}+2^{m^{2}-n^{2}+4}=2^{k+4}+2^{3m^{2}+n^{2}+k}$ have exactly two pairs of positive integers $(m, n)$ that are solutions?
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Context:
Let \( n > 4 \) be a positive integer. Determine the number of ways to walk from \( (0,0) \) to \( (n, 2) \) using only up and right unit steps such that the path does not meet the lines \( y = x \) or \( y = x - n + 2 \) except at the start and at the end.
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\boxed{\frac{1}{2}\left(n^2 - 5n + 2\right)}


Context:
The inequality $x^2+2x+a\geq -y^2-2y$ holds for any real numbers $x, y$. Find the range of values for the real number $a$ that satisfy this inequality. Express your answer as an inequality involving $a$.
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Context:
What is the smallest number of distinct integers needed so that among them one can select both a geometric progression and an arithmetic progression of length 5?
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Context:
Integers $a$, $b$, $c$ are selected independently and at random from the set $\{ 1, 2, \cdots, 10 \}$, with replacement. If $p$ is the probability that $a^{b-1}b^{c-1}c^{a-1}$ is a power of two, compute $1000p$.
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Context:
Determine the number of elements \( \alpha \in \mathbb{F}_{3^5} \) such that \( \mathbb{F}_{3}(\alpha) = \mathbb{F}_{3^5} \). In other words, find the number of irreducible polynomials of degree 5 over \( \mathbb{F}_{3} \) that divide \( x^{3^5} - x \).
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Context:
Two particles are placed on opposite vertices of the same face of a cube. If they are on the same edge at time \( t \), they remain in the same position at time \( t+1 \). If not, one of the particles is chosen at random and moves to one of its three neighboring vertices with equal probability at time \( t+1 \). What is the expected value of the minimum time until the particles are on the same edge?
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Context:
In a game involving two players, a fair coin is tossed repeatedly until one player wins. Player A wins if the sequence (Z, K, Z) appears, while player B wins if the sequence (K, K, K) appears. Calculate the probability that player A wins the game.
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Context:
Calculate the probability that a list of length \(b^{n/2}\) contains only unique numbers when selecting from a set of \(b^n\) distinct numbers, with replacement, until the list is filled.
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Context:
Determine the highest degree of the polynomial for which the quadrature formula \( \int _{-1}^1\vert x \vert f(x)dx \approx \frac{1}{2}[f(x_0)+f(x_1)] \) is exact, where \( x_0 \) and \( x_1 \) are quadrature points.
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Context:
Find the value of $R$ such that the circle $(x-\sqrt{7})^2 + y^2 = R^2$ and the ellipse $\frac{x^2}{16} + \frac{y^2}{9} = 1$ intersect at exactly one point.
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Context:
A giant panda is transporting bamboo from location A to location B. Each time, the panda can carry 50 bamboo sticks, but it eats 5 bamboo sticks both on the way from A to B and on the way back from B to A. There are currently 200 bamboo sticks at location A. What is the maximum number of bamboo sticks that the panda can transport to location B? Express your answer as a whole number.
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Context:
Given an angle $\theta,$ consider the polynomial
 $$
P(x) = \sin(\theta)x^2+(\cos(\theta)+\tan(\theta)x+1.
$$
Given that $P$ only has one real root, find all possible values of $\sin(\theta).$
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Context:
Solve the equation $x^3 = 2^{-x}$ for $x$.
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Context:
Isaac repeatedly flips a fair coin. Whenever a particular face appears for the $2n+1$th time, for any nonnegative integer $n$, he earns a point. The expected number of flips it takes for Isaac to get $10$ points is $\frac{a}{b}$ for coprime positive integers $a$ and $b$. Find $a + b$.
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Context:
Find the maximum likelihood estimator (MLE) of \( \theta \) for an i.i.d. sample \( X_1, \ldots, X_n \) from a uniform distribution on the interval \((0, \theta)\), where \( \theta > 0 \) is unknown.
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Context:
Evaluate the limit \[ \lim_{x\to \infty} x^2 \left( \sqrt{\frac{x+2}{x}} - \sqrt[3]{\frac{x+3}{x}} \right). \]
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Context:
Determine the maximum number of arcs in a digraph with $n$ vertices that is unilaterally connected but not strongly connected.
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Context:
The points $Q(1,-1), R(-1,0)$ and $S(0,1)$ are three vertices of a parallelogram. What could be the coordinates of the fourth vertex of the parallelogram?
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Context:
187. Elementary arithmetic. Here is one question similar to those that were so popular in Venice (and not only there) in the mid-16th century. Their appearance was largely due to Niccolò Fontana, better known as Tartaglia (the stammerer).

If a quarter of twenty were four, then what would a third of ten be?
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Context:
Determine the maximum expected response time for each geo tag if the system must locate all 4 tigers within 5 minutes with at least 95% probability, given that the waiting times for responses follow an iid exponential distribution.
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Context:
Consider a sequence  $x_1,x_2,\cdots x_{12}$  of real numbers such that  $x_1=1$  and for  $n=1,2,\dots,10$  let \[ x_{n+2}=\frac{(x_{n+1}+1)(x_{n+1}-1)}{x_n}. \] Suppose  $x_n>0$  for  $n=1,2,\dots,11$  and  $x_{12}=0$ . Then the value of  $x_2$  can be written as  $\frac{\sqrt{a}+\sqrt{b}}{c}$  for positive integers  $a,b,c$  with  $a>b$  and no square dividing  $a$  or  $b$ . Find  $100a+10b+c$ .
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Context:
Let $f: \Bbb{Z} \times \Bbb{Z} \to G$ be an epimorphism with $\ker f$ generated by $\langle (3,0),(0,5) \rangle$. Determine the type of abelian group $G$.
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Context:
Find the number of solutions for $x$ in the equation $$(x + (a<<s)-(b<<s))>>s = v$$ where $a, b, s,$ and $v$ are positive integers.
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Context:
An initial solution Y is 45 percent liquid X and 55 percent water. If 5 kilograms of water evaporate from 12 kilograms of solution Y, and 7 kilograms of solution Y are added to the remaining 9 kilograms of liquid, what percent of this new solution is liquid X?
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Context:
A normal to the hyperbola \( \frac{x^2}{4} - \frac{y^2}{1} = 1 \) has equal intercepts on the positive x and y axes. If this normal also touches the ellipse \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), find the value of \( a^2 + b^2 \).
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Context:
Evaluate \( \lim_{n \rightarrow \infty} \int_0^n \left(1 - \frac{x}{n}\right)^n \cos\left(\frac{x}{\sqrt{n}}\right) e^{x/2} \, dx \).
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Context:
Determine the number of distinct terms in the binomial expansion of \( \left(x + \frac{1}{x} + x^2 + \frac{1}{x^2}\right)^{15} \).
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Context:
Evaluate the following limit: $$\lim\limits_{x\to 0}\frac{1}{2x}\int_{0}^{x}\left[1-\cos\left(\frac{2}{u}\right)\right]du$$
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Context:
A room is lit by $3$ bulbs. Bulbs are replaced only when all bulbs burn out. Lifetimes of bulbs are i.i.d exponentially distributed with parameter $\lambda=1$. What fraction of the time is the room only one-third lit?
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