Context:
Calculate the work done by the vector field \( \vec{F}(x,y,z) = (2xe^y + 2x)\vec{i} + (x^2e^y)\vec{j} + (3z^2)\vec{k} \) along the curve \( C: \vec{r}(t) = \frac{2t}{1+t^2}\vec{i} + (t-t^3)\vec{j} + \frac{2}{1+t^2}\vec{k} \) from \( t = 0 \) to \( t = 1 \). Use the formula for work done by a vector field along a curve:
\[ W = \int_0^1 \vec{F}(f(t), g(t), h(t)) \cdot \vec{r}\;'(t) \; \operatorname d\!t \]
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Context:
Find the radius of convergence of the series \( \sum_{n=0}^{\infty} \left(\frac{z}{1+z}\right)^n. \)
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Context:
Find the maximum value of \( t \) such that the following integral equals zero:
\[
\int_{-1}^{t}\left[\sin\left(x^{3} - 9x^{2} + 18x\right) + x^{2} -6x + \frac{11}{3}\right] \, \mathrm{d}x = 0
\]
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Context:
Find a number $n$ such that $n$ is a product of three prime numbers and $n$ is a square modulo 389.
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Context:
Evaluate the function \( g(x) \) defined as \( g(x) = \tan \left(\frac{1}{2}\arcsin\left(\frac{2f(x)}{1+(f(x))^2}\right)\right) \), where \( f(x) \) is given by
\[ f(x) = \lim_{n\rightarrow \infty}\frac{x^2+2(x+1)^{2n}}{(x+1)^{2n+1}+x^2+1} \]
with \( n\in \mathbb{N} \). Determine the number of points where \( g(x) \) is not differentiable.
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Context:
Determine the value of $p_i$ that maximizes the probability of obtaining a sum of $k$ when summing $n$ independent Bernoulli random variables $X_1, X_2, \ldots, X_n$ with success probabilities $p_1, p_2, \ldots, p_n$. Provide your answer for $p_i$.
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Context:
Determine the number of ring homomorphisms from \(\mathbb{Z}[i]\) to \(\mathbb{Z}/85\mathbb{Z}\).
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Context:
For a positive real number $r$, consider the regular surface $M_r$ defined by \( M_r = \{(x,y,z) \mid x^2+y^2=z, 0<x, 0<y, z<r^2\} \). Calculate the integral of the Gaussian curvature \( K \) over \( M_r \), i.e., \( \int_{M_r} K \, dA \). Determine the limit of this integral as \( r \) approaches infinity: \( \lim_{r \rightarrow \infty} \int_{M_r} K \, dA \).
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Context:
Evaluate the integral of the function \( f(z) = z^2 + 2z - 3 \) over the ellipse defined by \( |z-2i|+|z+2i|=12 \), using the parametrization \( z(t) = \sqrt{35} \sin t + 6i\cos t \).
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Context:
Find the conditional expectation $E[X|X>Y]$ where $X$ and $Y$ are independent standard normal variables.
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Context:
Determine the number of cosets of the principal congruence subgroup of level 2 in \( \mathbb{P}SL(2,\mathbb{Z}) \) without using the reduction mod 2 map.
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Context:
Determine the degree of the image of a hypersurface $H \subset \mathbb{P}^n$ of degree $b$ under the Veronese map $\nu_{n,d}$. Provide your answer as a single integer.
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Context:
In a cube $ABCD-A'B'C'D'$ with an edge length of $6$, point $E$ is the midpoint of edge $A'B'$, and point $F$ lies on edge $A'D'$ such that $|A'F|=2|D'F|$. Calculate the volume of the solid $A'EF-ABD$. \( \boxed{\text{Volume}} \)
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Context:
Determine the number of regions formed by the intersection of $n$ unit spheres in $\mathbb{R}^3$. Provide your answer as a single number.
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Context:
Find the minimum value of the polynomial \(x^5 + 4x^4 + 3x^3 + 2x^2 + x + 1\) within the interval \((-2, 2)\).
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Context:
Evaluate the integral \( \int_{C}\frac{y^3dx-xy^2dy}{(x^2+y^2)^2} \) over the curve \( C = \frac{x^2}{16} + \frac{y^2}{9} = 1 \), oriented clockwise, using Green's Theorem. Note that the vector field is not defined at the origin.
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Context:
Calculate the norm of the integral operator $B$ defined by $Bf(x) = \int_{\mathbb{R}} k(x,y)f(y) \, dy$, where $k(x,y) = \frac{1}{\sqrt{2\pi}} e^{-\frac{(x-y)^2}{2}}$. Provide your answer as a single value.
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Context:
Find the second degree Taylor polynomial of the function \( g(x,y) = e^{f(x,y) - 5} \sin(y) \) at the point \((1,0)\), where \( f: \mathbb{R}^2 \to \mathbb{R} \) is a \( C^2 \) function with the second degree Taylor polynomial centered at \((1,0)\) given by \( P(x,y) = 4 + x + xy + \frac{y^2}{2} \).
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Context:
Annie takes a 6 question test, with each question having two parts each worth 1 point. On each part, she receives one of nine letter grades {A, B, C, D, E, F, G, H, I} that correspond to a unique numerical score. For each question, she receives the sum of her numerical scores on both parts. She knows that A corresponds to 1, E corresponds to 0.5, and I corresponds to 0.

When she receives her test, she realizes that she got two of each of A, E, and I, and she is able to determine the numerical score corresponding to all 9 markings. If $ n $ is the number of ways she can receive letter grades, what is the exponent of 2 in the prime factorization of $ n $?

2020 CCA Math Bonanza Individual Round #10
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Context:
Unit circles $O_1$ and $O_2$ are tangent to each other and line $L$. Circle $O_n$ is tangent to $O_1$, $O_{n-1}$, and $L$. Find the radius of $O_{20}$.
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Context:
12 Let $A$ be an $m \times m$ matrix. Let $X_{i}$ be the set of entries in the $i$th row and $Y_{j}$ the set of entries in the $j$th column, $1 \leq i, j \leq m$. $A$ is called cool if the sets $X_{1}, \ldots, X_{m}, Y_{1}, \ldots, Y_{m}$ are all different. Determine the smallest value for $n$ so that a cool $2005 \times 2005$ matrix with entries from the set $\{1,2, \ldots, n\}$ exists.

## Solution
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Context:
6. Find the smallest natural number $n$ such that in the decimal representation of $\sqrt{n}$, two nines immediately follow the decimal point.
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Context:
已知平面直角坐标系中的点集
$$
L=\{(x, y) \mid x, y \in\{1,2, \ldots, 100\}\}.
$$
设 $A$ 是由平面上若干个凸多边形组成的集合，满足： $A$ 中每个凸多边形的所有顶点均属于 $L$, 并且 $L$ 中的每个点都恰好是 $A$ 中一个凸多边形的顶点.
求 $A$ 中所有凸多边形的面积之和的最小可能值。
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Context:
Let $P_1$, $P_2$, $P_3$, $P_4$, $P_5$, and $P_6$ be six parabolas in the plane, each congruent to the parabola $y = \frac{x^2}{16}$. The vertices of the six parabolas are evenly spaced around a circle. The parabolas open outward with their axes being extensions of six of the circle's radii. Parabola $P_1$ is tangent to $P_2$, which is tangent to $P_3$, which is tangent to $P_4$, which is tangent to $P_5$, which is tangent to $P_6$, which is tangent to $P_1$. What is the diameter of the circle?
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Context:
In class 8 'G', there are several failing students, but Vovochka is performing the worst. The teachers' council decided that either Vovochka must improve his grades by the end of the term, or he will be expelled. If Vovochka improves his grades, the percentage of failing students in the class will be $24\%$, and if he is expelled, the percentage of failing students will be $25\%$. What is the current percentage of failing students in class 8 'G'?
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Context:
Zkov G.

A bank serves a million customers, the list of whom is known to Ostap Bender. Each has a six-digit PIN code, and different customers have different codes. In one move, Ostap Bender can choose any customer he has not yet chosen and peek at the digits of the code at any $N$ positions (he can choose different positions for different customers). Ostap wants to find out the code of the millionaire Koreiko. For what smallest $N$ can he guarantee to do this?
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Context:
Dr. Alex, a computer scientist, is exploring the technical feasibility of teleportation devices described in a popular sci-fi author's books. In the novel, each teleportation device requires 5 energy crystals to function, and each crystal can only power the device for 3 hours.

Dr. Alex plans to test the devices for a full day (24 hours) to analyze their feasibility. If she has a supply of 50 energy crystals, how many devices can she test for the entire 24-hour period without running out of crystals?
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Context:
Lele's family raises some chickens and ducks. Grabbing any 6 of them, there are at least 2 that are not ducks; Grabbing any 9 of them, there is at least 1 that is a duck. What is the maximum number of chickens and ducks in Lele's family?
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Context:
若使不等式$x ^ { 2 } - 4 x + 3 < 0$和$x ^ { 2 } - 6 x + 8 < 0$同时成立的x值使关于x的不等式$2 x ^ { 2 } - 9 x + a < 0$也成立，求实数a的取值范围.
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Context:
Given a triangle \( \triangle ABC \) with internal angles \( A, B, \) and \( C \), what is the sum of the numerator and denominator when the maximum value of \( (\sin A \cos B + \sin B \cos C + \sin C \cos A)^{2} \) is expressed in its simplest fractional form?
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Context:
We draw a triangle inside a circle with one vertex at the center of the circle and the other two vertices on the circumference of the circle. The angle at the center of the circle measures $75^\circ$.

Next, we draw a second triangle congruent to the first, also with one vertex at the center of the circle and the other vertices on the circumference, rotated $75^\circ$ clockwise from the first triangle so that it shares a side with the first triangle.

We continue this process to draw a third, fourth, and fifth triangle, each rotated $75^\circ$ clockwise from the previous triangle.

The base of the fifth triangle will intersect the base of the first triangle. What is the degree measure of the obtuse angle formed by this intersection?
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Context:
A teacher grades students' tests by using a certain ratio to subtract the number of incorrect responses from the number of correct responses. Student A answers each of the 100 questions on her test and receives a score of 73. She answered 91 questions correctly. What is the ratio of the number of incorrect responses subtracted to the number of correct responses?
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Context:
Knights, who always tell the truth, and liars, who always lie, live on an island. One day, 30 inhabitants of this island sat around a round table. Each of them said one of two phrases: "My neighbor on the left is a liar" or "My neighbor on the right is a liar." What is the minimum number of knights that can be at the table?
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Context:
Let  $AXYZB$  be a regular pentagon with area  $5$  inscribed in a circle with center  $O$ .  Let  $Y'$  denote the reflection of  $Y$  over  $\overline{AB}$  and suppose  $C$  is the center of a circle passing through  $A$ ,  $Y'$  and  $B$ . Compute the area of triangle  $ABC$ .
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Context:
Given that the graph of the function $y=e^{-2x+1}$ is symmetric with respect to a certain line $l$ to the graph of the function $y=\frac{{\ln(-x-1)-3}}{2}$, if $P$ and $Q$ are two moving points on their graphs, then the minimum distance between these two points is ______.
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Context:
Task 2. Find the number of ways to color all natural numbers from 1 to 20 in blue and red such that both colors are used and the product of all red numbers is coprime with the product of all blue numbers.
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Context:
实数x、y满足$x ^ { 2 } + y ^ { 2 } = 20 $，则xy+8x+y的最大值是
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Context:
What would the 25th number be in a numeric system where the base is five?
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Context:
Some people know each other in a group of people, where \u0022knowing\u0022 is a symmetric relation. For a person, we say that it is $social$ if it knows at least $20$ other persons and at least $2$ of those $20$ know each other. For a person, we say that it is $shy$ if it doesn\u0027t know at least $20$ other persons and at least $2$ of those $20$ don\u0027t know each other. Find the maximal number of people in that group, if we know that group doesn\u0027t have any $social$ nor $shy$ persons.
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Context:
Given a line $l$ with a slope of $2$ intersects the hyperbola $C$: $\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$ ($a > 0, b > 0$) at points $A$ and $B$. If point $P(3,1)$ is the midpoint of $AB$, calculate the eccentricity of the hyperbola $C$. Express your answer as a single value.
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Context:
Triangle  $ABC$  with  $\measuredangle A = 90^\circ$  has incenter  $I$ . A circle passing through  $A$  with center  $I$  is drawn, intersecting  $\overline{BC}$  at  $E$  and  $F$  such that  $BE < BF$ . If  $\tfrac{BE}{EF} = \tfrac{2}{3}$ , then  $\tfrac{CF}{FE} = \tfrac{m}{n}$ , where  $m$  and  $n$  are relatively prime positive integers. Find  $m+n$ .

*Proposed by Lewis Chen*
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Context:
In the expansion of $\left(x+1\right)\left(2x+1\right)\cdots \left(nx+1\right)$ (where $n$ is a positive integer), find the coefficient of the linear term in $x$. Express your answer as a combination formula (e.g., C_n^k).
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Context:
Compute\[\log_2\left(\prod_{a=1}^{2015}\prod_{b=1}^{2015}\left(1+e^{2\pi iab/2015}\right)\right)\]Here $i$ is the imaginary unit (that is, $i^2=-1$).
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Context:
A piece of rope is randomly cut into two parts. Calculate the probability that the longer segment is at least x times the shorter segment. Express your answer as a fraction or ratio involving x.
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Context:
Calculate the phase margin \( \phi \) of the second-order system given by the transfer function:
\[ F(j\omega) = \frac{1}{(1-\frac{\omega^2}{\omega_f^2}) + j\omega\frac{2\xi}{\omega_f}} \]
where \( \xi < 1 \). Find \( \phi = \text{Arg}(F(j\omega)) \) for \( \omega = 2\omega_f \) and \( \xi = 0.25 \).
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Context:
10. (20 points) Point $A$ lies on the line $y=\frac{15}{8} x-8$, and point $B$ lies on the parabola $y=x^{2}$. What is the minimum length of segment $A B$?
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Context:
Find the value of \( n \) that is the closest solution to the equation:
\[
\frac{(4.554 \times 10^{9})!(4.6 \times 10^9 - n)!}{(4.554 \times 10^{9} -n)!(4.6 \times 10^9)!} \approx 0.997
\]
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Context:
In the pyramid \( SABC \), the product of the edge lengths for each of the four faces is equal to the same number. The length of the height of the pyramid, dropped from \( S \) to the face \( ABC \), is \( 2 \sqrt{\frac{102}{55}} \), and the angle \( CAB \) is \( \arccos \left(\frac{1}{6} \sqrt{\frac{17}{2}}\right) \). Find the volume of the pyramid \( SABC \) given that

\[
|SA|^{2} + |SB|^{2} - 5|SC|^{2} = 60
\]
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Context:
Determine the quotient ring $T_2(\mathbb{Z_4})/I$, where $I = \left\{ \begin{bmatrix}0&b\\0&c\end{bmatrix}: b,c \in \mathbb{Z_{4}}\right\}$ is an ideal of the ring of upper triangular matrices $T_2(\mathbb{Z_4})$.
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Context:
How many distinct bracelets can be created using 12 beads, with 3 red, 4 green, and 5 blue beads, considering rotation and reflection symmetry? Use the Dihedral group formula to determine this number.
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Context:
Consider the function $$f(x) =1+3x \int^{x^2}_{2x} e^{-(t)^{1/2}}dt$$ What is the first order Taylor polynomial around $x=2$?
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Context:
Given the parabola $y^{2}=2x$, and $A$, $B$ are two points on the parabola, the perpendicular bisector of segment $AB$ intersects the $x$-axis at point $P(x_{0},0)$. The range of $x_{0}$ is _______. (Express the range as an interval)
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Context:
Given the first-order partial differential equation (PDE) $xu_x + (x^2 + y)u_y = u$ for all $x, y \in \mathbb{R}$, where $u(2, y) = y - 4$, find the value of $u(1, 2)$.
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Context:
Given that the random variable $\xi$ follows the normal distribution $N(1, 4)$, if $P(\xi > 4) = 0.1$, then $P(-2 \leq \xi \leq 4)$ equals _______.
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Context:
Let \(a\) and \(b\) be two natural numbers. There are \(ab\) pieces of candy, which are to be packaged into several bags in such a way that the candies can be divided into \(a\) parts, each with \(b\) pieces, and also can be divided into \(b\) parts, each with \(a\) pieces. Find the minimum number of bags needed.
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Context:
Marge had a winning lottery ticket for $20000. She paid 55% of her winnings towards taxes. She then used a half of the remaining amount to pay off a part of her mortgage. Wanting to secure her future, she put a third of what was left into a retirement fund. Afterwards, she decided to contribute a quarter of the remaining amount to her children’s college fund. Eventually, she also put $1200 in savings and invested half of her savings in the stock market. How many dollars does she have left for fun?
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Context:
In the land of Draconia, there live red, green, and blue dragons. Each dragon has three heads, each of which always tells the truth or always lies. Moreover, each dragon has at least one head that tells the truth. One day, 530 dragons sat around a round table, and each one said:

- 1st head: "The dragon to my left is green."
- 2nd head: "The dragon to my right is blue."
- 3rd head: "There is no red dragon next to me."

What is the maximum number of red dragons that could have been at the table?
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Context:
Find all real functions \( f(x) \) that are defined and continuous on the interval \((-1, 1)\) and satisfy the functional equation:
\[ f(x+y) = \frac{f(x) + f(y)}{1 - f(x) f(y)} \quad \left( x, y, x+y \in (-1,1) \right). \]
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Context:
Given \( x = -2272 \), \( y = 10^3 + 10^2 c + 10 b + a \), and \( z = 1 \), which satisfy the equation \( a x + b y + c z = 1 \), where \( a \), \( b \), \( c \) are positive integers and \( a < b < c \). Find \( y \).
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Context:
Find the area of the region bounded by the curves $xy=4$, $xy=8$, $xy^3=5$, and $xy^3=15$. Express your answer in terms of natural logarithms.
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Context:
Let \( f:\mathbb{R} \to \mathbb{R} \) be a differentiable function defined by \( f(x) = x + \int_0^1 (xy + x^2)f(y) \, dy \). Find the value of \( \int_0^1 f(x) \, dx \).
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Context:
Below is a "right triangular number array":
\[
\frac{1}{4},
\]
\[
\frac{1}{2}, \quad \frac{1}{4}
\]
\[
\frac{3}{4}, \quad \frac{3}{8}, \quad \frac{3}{16}
\]
Each column forms an arithmetic sequence, starting from the third row, each row forms a geometric sequence with a constant common ratio. Let the number in the \(i^\text{th}\) row and \(j^\text{th}\) column be \(a_{i j}\) (\(i \geqslant j, i, j \in \mathbf{N}_{+}\)), find \(a_{n 3}\).
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Context:
设 $P-A B C D$ 与 $Q-A B C D$ 为两个正四棱雉, 且 $\angle P A Q=90^{\circ}$, 点 $M$ 在线段 $A C$ 上, 且 $C M=3 A M$. 将异面直线 $P M, Q B$ 所成的角记为 $\theta$ ，则 $\cos \theta$ 的最大可能值为 $\qquad$.  原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值
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Context:
Find the value of \( \sqrt{2+\sqrt{5}-\sqrt{6-3\sqrt{5}+\sqrt{14-6\sqrt{5}}}} \).
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Context:
Find the asymptote of the function \( g(x) = \frac{x^2 + f(x)}{x + 1 + 2f(x)} \), given that \( \lim_{x \to \infty} x^n f(x) = 0 \) for any \( n \in \mathbb{N} \).
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Context:
Find a value of $R > 0$ such that for all complex numbers $z$ with $|z| > R$, the inequality $|z^{10} + 100z + 9| > 1$ holds.
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Context:
Determine the dual group of the multiplicative group of positive real numbers, denoted by $\mathbb{R}_{>0}^*$.
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Context:
Evaluate the limit: \[ \lim_{n \rightarrow \infty} \int^{n}_{0} \left(1+\frac{x}{n}\right)^{-n} \log\left(2+ \cos\left(\frac{x}{n}\right)\right)\ \mathrm{d}x. \]
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Context:
Find the locus of the centers of circles with a constant radius that move while touching the faces of a trihedral angle, where all plane angles are right angles.
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Context:
When the integer $({\sqrt{3} + 5})^{103} - ({\sqrt{3} - 5})^{103}$ is divided by 9, what is the remainder?
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Context:
The number $2.29^{\star \star} N$ is an integer. Its representation in base $b$ is 777. Find the smallest positive integer $b$ such that $N$ is a perfect fourth power.
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Context:
Determine the number of points on the curve defined by the equation \(x^3 + x + 1 = y^2\) over the finite field \(\mathbb{F}_{125}\). Given that the curve has 9 points over \(\mathbb{F}_5\), and using Hasse's theorem, the number of points over \(\mathbb{F}_{125}\) must be a multiple of 9. The possible orders are 108, 117, 126, 135, and 144. Find the correct number of points.
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Context:
Quadrilateral $ABCD$ is inscribed in a circle with the following properties:
- $\angle BAC = 70^{\circ}$
- $\angle ADB = 40^{\circ}$
- $AD = 4$
- $BC = 6$

What is the length of $AC$?
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Context:
Given the cubic equation $2x^3+8x^2-42x-188$, with roots $a, b, c$, evaluate $|a^2b+a^2c+ab^2+b^2c+c^2a+bc^2|$.
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Context:
A finite set of integers is called bad if its elements add up to 2010. A finite set of integers is a Benelux-set if none of its subsets is bad. Determine the smallest integer \( n \) such that the set \( \{502, 503, 504, \ldots, 2009\} \) can be partitioned into \( n \) Benelux-sets.

(A partition of a set \( S \) into \( n \) subsets is a collection of \( n \) pairwise disjoint subsets of \( S \), the union of which equals \( S \).)
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Context:
Arthur, Bob, and Carla each choose a three-digit number. They each multiply the digits of their own numbers. Arthur gets 64, Bob gets 35, and Carla gets 81. Then, they add corresponding digits of their numbers together. The total of the hundreds place is 24, that of the tens place is 12, and that of the ones place is 6. What is the difference between the largest and smallest of the three original numbers?
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Context:
In triangle ABC, where \( \angle A = 70^\circ \), the angle bisector AE intersects side AC at point D and side BD at point H. Given the ratios \( \frac{AH}{HE} = \frac{3}{1} \) and \( \frac{BH}{HD} = \frac{5}{3} \), find the measure of \( \angle C \).
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Context:
A car traveling at a certain constant speed takes 5 seconds longer to travel 1 km than it would take to travel 1 km at a certain speed. The car is traveling at 80 km/hour. What is the speed it is being compared to?
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Context:
Example 10. Solve the equation

$$
x \int_{0}^{x} y(t) d t=(x+1) \int_{0}^{x} t y(t) d t, \quad x>0
$$
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Context:
9.5. If there are several piles of stones on the table, it is considered that there are many stones on the table if one can find 50 piles and number them from 1 to 50 such that the first pile has at least one stone, the second pile has at least two stones, ..., the fiftieth pile has at least fifty stones. Suppose initially there are 100 piles, each with 100 stones. Find the largest \( n \leqslant 10000 \) such that after removing any \( n \) stones from the initial piles, there will still be many stones on the table. (When removing stones, a pile does not split into several.) (D. Khramov)
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Context:
设a,b,c,d>0，且a+b+c+d=3，求

$\frac { a ^ { 3 } } { b + c + d } + \frac { b ^ { 3 } } { a + c + d } + \frac { c ^ { 3 } } { a + b + d } + \frac { d ^ { 3 } } { a + b + c }$的最小值.
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Context:
Determine the image of the set \( \mathcal{G} \) under the map that takes a function \( g(x) \) in \( \mathcal{G} \) to \( \int_{0}^{2}g(x) \, dx \), where \( \mathcal{G} \) is the subset of real-valued continuous functions on \([0, 2]\) such that \( 0 \leq g(x) \leq 1 \) for all \( x \) in \([0, 2]\) and \( \liminf_{n\rightarrow \infty}\int_{0}^{2}g(x)^{n}dx >0 \).
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Context:
Evaluate the integral: $$\int_{-\infty}^\infty\exp\left(-\frac{(x^2-13x-1)^2}{611x^2}\right)\ dx$$
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Context:
A particle is located on the coordinate plane at $(5,0)$. Define a ''move'' for the particle as a counterclockwise rotation of $\frac{\pi}{4}$ radians about the origin followed by a translation of $10$ units in the positive $x$-direction.  Find the particle's position after $150$ moves.
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Context:
Determine the value of \( p \) that minimizes the expression \( S = \frac{p-1}{p^n-1}\sum_{i=1}^n(n-i) p^{i-1} \) for the maximum possible value of \( n \).
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Context:
A subset of $\left\{1,2,3,\ldots,2017,2018\right\}$ has the property that none of its members are $5$ times another. What is the maximum number of elements that such a subset could have?
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Context:
Find the sum of all roots of the equation \((\sin x + \cos x)^{(1 + \sin 2x)} = 2\) for \(x \in [-2\pi, 4\pi]\).
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Context:
1. Given that $x$ satisfies $\log _{5 x} 2 x=\log _{625 x} 8 x$, then the value of $\log _{2} x$ is
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Context:
An HMMT party has $m$ MIT students and $h$ Harvard students for some positive integers $m$ and $h$, For every pair of people at the party, they are either friends or enemies. If every MIT student has 16 MIT friends and 8 Harvard friends, and every Harvard student has 7 MIT enemies and 10 Harvard enemies, compute how many pairs of friends there are at the party.
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Context:
In the sequence \{a_n\}, a_1 = 0, a_2 = 2, and for any k in the set of positive integers, a_{2k-1}, a_{2k}, a_{2k+1} form an arithmetic sequence, and a_{2k}, a_{2k+1}, a_{2k+2} form a geometric sequence with common ratio q_k. Calculate q_{10}. Express your answer as a fraction.
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Context:
Determine the fundamental group of the complement of a circle in 3D space, \( \Bbb{R}^3 \setminus S^1 \).
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Context:
A company stipulates that a retired employee can receive a pension each year, which is directly proportional to the square root of the number of years they have worked. If they work $a$ more years, their pension increases by $p$ dollars, and if they work $b$ more years ($b \neq a$), their pension increases by $q$ dollars. The annual pension of the employee is (expressed in terms of $a$, $b$, $p$, $q$) ______ dollars.
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\boxed{\frac{a q^2 - b p^2}{2(bp - aq)}}


Context:
A number is upwards if its digits in base 10 are nondecreasing when read from left to right. Compute the number of positive integers less than $10^6$ that are both upwards and multiples of 11.
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Context:
Given that \( A^4 = -1.6A^2 - 0.64I \), find \( \lim_{m \to \infty} A^m \).
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Context:
Determine the smallest integer $n$ such that the error in approximating $\pi$ using the formula $\pi = 4\arctan(1/2) + 4\arctan(1/3)$ with $n$ terms of the series expansion is less than $10^{-8}$.
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Context:
Determine the minimum number of straight lines required to construct exactly 100 squares.
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Context:
Determine the order of $25+3017\mathbb{Z}$ in $(\mathbb{Z}/3017\mathbb{Z})^\times$. Note that $3017 = 7 \cdot 431$.
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Context:
Let \( x < {0.1} \) be a positive real number. Let the foury series be
 \( 4 + {4x} + 4{x}^{2} + 4{x}^{3} + \ldots \) , and let the fourier series be
 \( 4 + {44x} + {444}{x}^{2} + {4444}{x}^{3} + \ldots \) . Suppose that the sum of the fourier
 series is four times the sum of the foury series. Compute \( x \) .
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Context:
Find $ \lim\limits_{x\rightarrow \infty} \left(\int_0^x \sqrt{1+e^{2t}}\ dt-e^x\right)$.
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Context:
The director of the school decided to take a photo of the 2008 graduates. He arranged the students in parallel rows, all with the same number of students, but this arrangement was too wide for the camera's field of view. To solve this problem, the director noticed that it was enough to remove one student from each row and place them in a new row. This arrangement did not please the director because the new row had 4 students fewer than the others. He then decided to remove one more student from each row and place them in the new row he created, and found that this way all the rows had the same number of students, and finally took the photo. How many students appeared in the photo?
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Context:
Find the first three 7-adic digits of a root of the polynomial \(x^3 - 1\) in \(\mathbb{Z}_7\) that is not equal to 1.
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Context:
There are  $110$  guinea pigs for each of the  $110$  species, arranging as a  $110\times 110$  array. Find the maximum integer  $n$  such that, no matter how the guinea pigs align, we can always find a column or a row of  $110$  guinea pigs containing at least  $n$  different species.
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Context:
Determine the order of $k$ such that the expression \( \sum_{i=0}^{k} \frac{i}{2n - i} \) converges to a limit independent of $n$ as $n \to \infty$, given that $k = cn$ does not work.
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Context:
Let $p(x)=2+4x+3x^{2}+5x^{3}+3x^{4}+4x^{5}+2x^{6}$. For $k$ with $0<k<5$, define $$I_{k}=\int_{0}^{\infty} \frac{x^{k}}{p(x)} d x$$ For which $k$ is $I_{k}$ smallest?
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Context:
Calculate the number of subgroups of $A = \mathbb{Z}_{360} \oplus \mathbb{Z}_{150} \oplus \mathbb{Z}_{75} \oplus \mathbb{Z}_{3}$ that are isomorphic to $\mathbb{Z}_{5} \oplus \mathbb{Z}_{5}$.
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Context:
For a positive integer $n$, let $f(n)$ be the sum of the positive integers that divide at least one of the nonzero base $10$ digits of $n$. For example, $f(96) = 1 + 2 + 3 + 6 + 9 = 21$. Find the largest positive integer $n$ such that for all positive integers $k$, there is some positive integer $a$ such that $f^k(a) = n$, where $f^k(a)$ denotes $f$ applied $k$ times to $a$.
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Context:
What is the remainder when \(2,468,135,790\) is divided by \(99\)?
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Context:
在 \(\triangle A B C\) 中, 已知 \(\angle C=90^{\circ}, \angle B=30^{\circ}\), \(A C=2, M\) 是 \(A B\) 的中点，将 \(\triangle A C M\) 沿 \(C M\) 翻折，使 \(A 、 B\) 两点间的距离为 \(2 \sqrt{2}\)。求三棱锥 \(A-B C M\) 的体积 \(V\)。答案的形式为\(\frac{k \sqrt{m}}{n}\)，请给出k+m+n的值。
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Context:
Find the minimal polynomial of \(\beta^7\) over \(\mathbb{F}_2\), where \(\beta\) is an element of \(\mathbb{F}_{16}\) with minimal polynomial \(x^4 + x + 1\) over \(\mathbb{F}_2\).
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Context:
The odometer of a family car shows \(15,951\) miles. The driver noticed that this number is palindromic: it reads the same backward as forwards. "Curious," the driver said to himself, "it will be a long time before that happens again." Surprised, he saw his third palindromic odometer reading (not counting \(15,951\)) exactly five hours later. How many miles per hour was the car traveling in those 5 hours (assuming speed was constant)?
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Context:
The integers  $ 1,2,...,n$  are placed in order so that each value is either strictly bigger than all the preceding values or is strictly smaller than all preceding values. In how many ways can this be done?
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Context:
3. Find the maximum number of rectangles of dimensions $1 \times 10 \sqrt{2}$ that can be obtained from a rectangle of dimensions $50 \times 90$, if cutting along lines parallel to the sides of the given rectangle is allowed.

(Yugoslavia)
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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:
In an ellipse \(\Gamma\), \(F\) is a focus, and \(A\) and \(B\) are two vertices. Given that \(|FA|=3\) and \(|FB|=2\), determine all possible values of \(|AB|\).
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Context:
Let  $n = 2188 = 3^7+1$  and let  $A_0^{(0)}, A_1^{(0)}, ..., A_{n-1}^{(0)}$  be the vertices of a regular  $n$ -gon (in that order) with center  $O$  . For  $i = 1, 2, \dots, 7$  and  $j=0,1,\dots,n-1$ , let  $A_j^{(i)}$  denote  the centroid of the triangle \[ \triangle A_j^{(i-1)} A_{j+3^{7-i}}^{(i-1)} A_{j+2 \cdot 3^{7-i}}^{(i-1)}. \] Here the subscripts are taken modulo  $n$ . If \[ \frac{|OA_{2014}^{(7)}|}{|OA_{2014}^{(0)}|} = \frac{p}{q} \] for relatively prime positive integers  $p$  and  $q$ , find  $p+q$ .

*Proposed by Yang Liu*
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Context:
Determine the minimum degree \( n \) of the Taylor polynomial centered at \( \pi/2 \) that approximates \( \sin(1) \) with an error of less than \( 10^{-17} \).
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Context:
What is the term for a structure similar to a semiring, but where associativity is not required for multiplication?
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Context:
What is the probability of having 2 adjacent white balls or 2 adjacent blue balls in a random arrangement of 3 red, 2 white, and 2 blue balls? Express your answer as a simplified fraction.
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Context:
In a football tournament with 15 teams, each team played every other team exactly once. A victory awarded 3 points, a draw awarded 1 point, and a loss awarded 0 points. After the tournament, it was found that 6 teams each scored at least $N$ points. What is the maximum integer value of $N$?
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Context:
Find the variance of the cumulative distribution function of a normal distribution, where $X \sim N(\mu,\sigma^2)$ and $\Phi(x)$ is the cumulative distribution function of this normal distribution.
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Context:
Find the volume of the region bounded by the planes $4x+2y+4z=6$, $y=x$, $x=0$, and $z=0$.
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Context:
In triangle $A B C \angle C A B=75^{\circ}, \angle A B C=45^{\circ}$. On side $C A$ a point $K$ is taken, and on side $C B$ a point $M$, $C K: A K=3: 1$.

Find $K M: A B$, if this ratio is less than $3 / 4$, and line $M K$ cuts off from triangle $A B C$ a triangle similar to it.
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Context:
In acute $\triangle ABC,$ let $I$ denote the incenter and suppose that line $AI$ intersects segment $BC$ at a point $D.$ Given that $AI=3, ID=2,$ and $BI^2+CI^2=64,$ compute $BC^2.$

[i]Proposed by Kyle Lee[/i]
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Context:
Given that \( \frac{\cos x+\cos y+\cos z}{\cos(x+y+z)}=\frac{\sin x+\sin y+\sin z}{\sin(x+y+z)}=k \), find the value of \( \cos(x+y)+\cos(y+z)+\cos(z+x) \).
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Context:
In triangle ABC, the magnitudes of the sides are a, b, and c, as shown, with c ≤ b ≤ a. Through interior point P and the vertices A, B, C, lines are drawn meeting the opposite sides in A', B', C', respectively. Let s = AA' + BB' + CC'. Find the upper bound for s for all positions of point P. Express your answer as a single expression involving a, b, and c.
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Context:
Given that $A$, $B$, and $C$ are angles of a triangle, and the equations $5\sin(A) + 12\cos(B) = 15$ and $12\sin(B) + 5\cos(A) = 2$ hold, find the measure of angle $C$.
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Context:
$240$ students are participating a big performance show. They stand in a row and face to their coach. The coach askes them to count numbers from left to right, starting from $1$ . (Of course their counts be like $1,2,3,...$)The coach askes them to remember their number and do the following action:
First, if your number is divisible by $3$  then turn around.
Then, if your number is divisible by $5$  then turn around.
Finally, if your number is divisible by $7$  then turn around.
What is the number of the $66^{\text{th}}$ student counting from left who is face to coach?
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Context:
Over the course of a week, Trevor was tasked to collect fresh eggs from the family's 8 chickens. Each chicken laid a different number of eggs everyday. On Monday, Gertrude laid 4 eggs, Blanche laid 3 eggs, Nancy laid 2 eggs, Martha laid 3 eggs, Ophelia laid 5 eggs, Penelope laid 1 egg, Quinny laid 3 eggs, and Rosie laid 4 eggs. On Tuesday, Gertrude laid 2 more eggs than she did on Monday, Blanche laid the same number of eggs as she did on Monday, Nancy laid 1 fewer egg than she did on Monday, Martha laid 1 more egg than she did on Monday, Ophelia laid 2 fewer eggs than she did on Monday, Penelope laid 2 more eggs than she did on Monday, Quinny laid 2 less eggs than she did on Monday, and Rosie didn't lay any eggs. On Wednesday, each chicken laid the same number of eggs as they did on Monday. Each day, Trevor ate 2 eggs for breakfast. On Monday and Wednesday, he accidentally dropped 3 eggs each day. On Tuesday, he sold half of the collected eggs. How many eggs did Trevor have by the end of Wednesday?
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Context:
Evaluate the limit: $$\lim_{x \to 0^{+}} \left[\frac{\exp(\sin x \log x) - \exp(x \log \sin x)}{x^3} + \frac{\log x}{6}\right]$$
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Context:
A $15 \times 15$ chessboard is completely covered using $3 \times 3$ and $2 \times 2$ plates. The plates do not overlap, and they do not extend beyond the edges of the chessboard. Determine the minimum number of $3 \times 3$ plates required to achieve this coverage.
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Context:
Determine the number of generalized eigenvectors required to construct the Jordan canonical form for a $4 \times 4$ matrix with a single eigenvalue and two linearly independent eigenvectors.
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Context:
Among the 40 students, the average score excluding Jimin and Jungkook's scores is 79. If Jimin and Jungkook scored 98 and 100 respectively, what is the average score of Jimin's class?
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Context:
For an integer $x \geq 1$, let $p(x)$ be the least prime that does not divide $x$, and define $q(x)$ to be the product of all primes less than $p(x)$. In particular, $p(1) = 2$. For $x$ having $p(x) = 2$, define $q(x) = 1$. Consider the sequence $x_{0}, x_{1}, x_{2}, \ldots$ defined by $x_{0} = 1$ and
$$x_{n+1} = \frac{x_{n} p(x_{n})}{q(x_{n})}$$

for $n \geq 0$. Find all $n$ such that $x_{n} = 1995$.
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Context:
Let $\omega (n)$ denote the number of prime divisors of the integer $n>1$.
Find the least integer $k$ such that the inequality $2^{\omega (n) } \leq k \cdot  n^{\frac 1 4}$ holds for all $n > 1.$
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Context:
Determine the values of \( t \in \mathbb{R} \) for which the function \( f_t: \mathbb{R}^3 \rightarrow \mathbb{R}^3 \) defined by the following mappings is a linear transformation:
\[
f_t\left(\begin{pmatrix} 1 \\ 1 \\ -1 \end{pmatrix}\right)=\begin{pmatrix} -1 \\ 2 \\ -4-t \end{pmatrix}, \quad
f_t\left(\begin{pmatrix} 0 \\ 1 \\ t \end{pmatrix}\right)=\begin{pmatrix} 1 \\ 2 \\ -1 \end{pmatrix}, \quad
f_t\left(\begin{pmatrix} 0 \\ 1 \\ 1 \end{pmatrix}\right)=\begin{pmatrix} 7 \\ 2 \\ 2t+7 \end{pmatrix}
\]
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Context:
Given vector $\vec{b}=(\frac{1}{2}, \frac{\sqrt{3}}{2})$, and $\vec{a}\cdot \vec{b}=\frac{1}{2}$, calculate the projection of vector $\vec{a}$ in the direction of vector $\vec{b}$.
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Context:
A circle of radius 6 is drawn centered at the origin. How many squares of side length 1 and integer coordinate vertices intersect the interior of this circle?
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Context:
Given points $A(0,0,2)$ and $B(3,4,1)$ in the coordinate system $Oxyz$, find the minimum value of $AX + BY$ where $X$ and $Y$ are points in the $Oxy$ plane such that the distance $XY = 1$. Provide your answer as a single number.
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Context:
Calculate the value of the integral \( \int_{0}^{0.4}\sqrt{1+x^4} \, dx \) such that the absolute error is less than or equal to \( \frac{0.4^9}{72} \) using the alternating series expansion.
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Context:
Given \( x_{i} \in \mathbf{R}, x_{i} \geq 0 \) for \( i=1,2,3,4,5 \), and \( \sum_{i=1}^{5} x_{i} = 1 \), find the minimum value of \(\max \left\{ x_{1} + x_{2}, x_{2} + x_{3}, x_{3} + x_{4}, x_{4} + x_{5} \right\} \).
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Context:
Suppose $v_1, ..., v_n$ are linearly independent vectors in $\mathbb{R}^m$. Let $w$ be a linear combination of these vectors such that $w = a_1v_1 + ... + a_nv_n$. If the vectors $v_1, w, v_3, ..., v_n$ are linearly independent, what can we conclude about $a_2$?
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Context:
A  $3 \times 3$  grid of unit cells is given. A *snake of length  $k$* is an animal which occupies an ordered  $k$ -tuple of cells in this grid, say  $(s_1, \dots, s_k)$ . These cells must be pairwise distinct, and  $s_i$  and  $s_{i+1}$  must share a side for  $i = 1, \dots, k-1$ . After being placed in a finite  $n \times n$  grid, if the snake is currently occupying  $(s_1, \dots, s_k)$  and  $s$  is an unoccupied cell sharing a side with  $s_1$ , the snake can *move* to occupy  $(s, s_1, \dots, s_{k-1})$  instead. The snake has *turned around* if it occupied  $(s_1, s_2, \dots, s_k)$  at the beginning, but after a finite number of moves occupies  $(s_k, s_{k-1}, \dots, s_1)$  instead.

Find the largest integer  $k$  such that one can place some snake of length  $k$  in a  $3 \times 3$  grid which can turn around.
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Context:
In the equation "Xiwangbei jiushi hao $\times$ 8 = Jiushihao Xiwangbei $\times$ 5", different Chinese characters represent different digits. The six-digit even number represented by "Xiwangbei jiushi hao" is ____.
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Context:
Find the number \( S \) such that if \( a + b + c + d = S \) and \( \frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{d} = S \) (where \( a, b, c, d \) are different from zero and one), then \( \frac{1}{a - 1} + \frac{1}{b - 1} + \frac{1}{c - 1} + \frac{1}{d - 1} = S \).
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Context:
Let $(X_n)_{n \in \mathbb{N}}$ be a sequence of independent random variables, each uniformly distributed over the interval [0, 1]. Define $Y_n = (\prod^n_{i=1} X_i)^{\frac{1}{n}}$. Use the Central Limit Theorem to approximate $\mathbb{P}(Y_{50} \geq \mathbb{E}(Y_{50}))$. Compute this probability.
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Context:
Compute the integral \( \int_C i^* \alpha \) where \( C \) is a smooth, closed curve in \( \mathbb{R}^2 \) that does not pass through the origin, \( \alpha = \frac{xdy-ydx}{x^2 + y^2} \), and \( i: C \to \mathbb{R}^2 \setminus \{(0,0)\} \) is the inclusion.
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Context:
There were 3 candidates in an election. Two candidates received 4136 and 7636 votes respectively. The winning candidate got a certain percentage of the total votes in that election. If the winning candidate got 49.69230769230769% of the total votes, how many votes did the winning candidate receive?
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Context:
For a given number of digits \(n\), determine the expected number of places in the decimal expansion of \(\pi\) where the first \(n\) digits repeat.
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Context:
There are three boats, A, B, and C, in a pond. Boat A can carry 3 people, boat B can carry 2 people, and boat C can carry 1 person. Now, there are 2 adults and 2 children who need to be divided among these boats. For safety reasons, children must be accompanied by an adult to board the boat. Calculate the total number of ways they can be divided among these boats.
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Context:
Determine the smallest integer \(d\) such that every simple finite planar graph can be oriented so that the out-degree of each vertex is at most \(d\).
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Context:
三棱雉$P-ABC$中，三个侧面与底面所成角相等，三个侧面的面积分别为$3、4、5$，且底面面积为6。已知三棱雉$P-ABC$的外接球的表面积的原始答案形式为 $\frac{m\pi}{n}$，请给出 m + n 的值。
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Context:
Using the 0.618 method to find the optimal addition amount for an experiment, if the current optimal range is $[628, 774]$ and the better point is 718, then the value of the addition point for the experiment at this time is ___.
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Context:
Given a complex number $z=a+(a-1)i$ ($a\in \mathbb{R}$, $i$ is the imaginary unit), find the value of $\int_{0}^{a}{\left( \sqrt{1-{x}^{2}}+x \right)dx}$.
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Context:
Calculate the limit: $$\lim_{n\rightarrow\infty}\left(\frac{\prod\limits_{r=1}^n\left(2 +\frac{r}{n^2}\right)}{e}\right)^n$$
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Context:
Compute the limit of the recursive sequence defined by $x_0 = a$, $x_1 = b$, and for $n \geq 1$:

$$x_{n+1} = \left(1 - \frac{1}{2n}\right)x_n + \frac{1}{2n}x_{n-1}.$$

Find the value of $\lim_{n \to \infty} x_n$, if it exists.
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Context:
Find the value of the series \( \sum_{n=1}^{\infty} \left( \frac{2}{2n+1} - \frac{1}{n+1} \right) \).
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Context:
Consider a  $5 \times 5$  grid of squares. Vladimir colors some of these squares red, such that the centers of any four red squares do  $\textbf{not}$  form an axis-parallel rectangle (i.e. a rectangle whose sides are parallel to those of the squares). What is the maximum number of squares he could have colored red?
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Context:
Each edge of a graph with $15$ vertices is colored either red or blue in such a way that no three vertices are pairwise connected with edges of the same color. Determine the largest possible number of edges in the graph.
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Context:
Find the first four non-zero terms of the Maclaurin series for the complex function $f(z) = e^{\frac{1}{1-z}}$, centered at $z_0=0$.
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Context:
What is the smallest \( k \) for which one can mark \( k \) cells on a \( 12 \times 12 \) board such that any placement of a four-cell figure \(\square \square\) on the board will touch at least one marked cell? (The figure can be rotated and flipped.)
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Context:
Find the residue of the function \( \frac{1}{e^{e^{\frac{1}{z}}}} \).
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Context:
若集合 $X \subseteq\{1,2, \cdots, n\}$ 满足: 在 $X$ 的任意五元子集 $A$ 中都可以找到两个元素 $a 、 b(a<b)$ 使得 $a \mid b$, 则记 $|X|$ 的最大可能值是 $f(n)$ （ $|X|$ 表示集合 $X$ 中的元素个数）。求$f(2023)+f(3000)+f(10000)$.
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Context:
Given the power series representation of a function $f(x)$ for $x \in (0,1)$:

$$f(x) = x + \frac{2}{1\cdot3}x^3 + \frac{2\cdot4}{1\cdot3\cdot5}x^5 + \frac{2\cdot4\cdot6}{1\cdot3\cdot5\cdot7}x^7 + \cdots$$

If $f\left(\frac{1}{2}\right)$ equals $\frac{\pi}{a\sqrt{b}}$, where $a$ and $b$ are real numbers, what is the absolute value of $a+b$?
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Context:
Find the number of 20-tuples of integers $x_{1}, \ldots, x_{10}, y_{1}, \ldots, y_{10}$ with the following properties: - $1 \leq x_{i} \leq 10$ and $1 \leq y_{i} \leq 10$ for each $i$; - $x_{i} \leq x_{i+1}$ for $i=1, \ldots, 9$; - if $x_{i}=x_{i+1}$, then $y_{i} \leq y_{i+1}$.
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Context:
Determine the upper bound for the length of the shortest word that can distinguish two non-equivalent NFAs, one with $n$ states and the other with $k$ states.
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Context:
What is the total variation of the Dirac delta function \( \delta(x) \)?
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Context:
Find the smallest natural $k$ such that among any $k$ distinct and pairwise coprime naturals smaller than $2018$, a prime can be found.
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Context:
A pizza parlor has six different toppings. How many different one- and two-topping pizzas can you order?
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Context:
In how many ways can 8 distinguishable people, consisting of 2 Americans, 2 Canadians, 2 Mexicans, and 2 Jamaicans, be seated around a round table such that at least two pairs of countrymen are seated together? Consider rotations as identical.
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Context:
In a school with $n$ students, each student has exactly 2023 friends. Additionally, any two students who are not friends have exactly 2022 mutual friends. Determine all possible values of $n$.
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Context:
There are 4 walls in a room, each with different dimensions: the first wall is 10ft by 8ft, the second wall is 10ft by 9ft, the third wall is 15ft by 8ft, and the fourth wall is 15ft by 9ft. The room includes two windows, each with dimensions 3ft by 4ft, and a door with dimensions 3ft by 7ft. The cost of painting each square foot of area is $2.

Now, the room is being modified, and its length, breadth and height are each increased by 50%. In the new room, an additional door of the same size as before is added, and two additional windows of the same size are also added. What is the cost to paint the new room?
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Context:
Given that $P$ is a moving point on the parabola $y^{2}=4x$, and $Q$ is a moving point on the circle $x^{2}+(y-4)^{2}=1$, the minimum value of the sum of the distance from point $P$ to point $Q$ and the distance from point $P$ to the directrix of the parabola is ______.
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Context:
Two semicircles, each with radius \(\sqrt{2}\), are tangent to each other. If \( AB \parallel CD \), determine the length of segment \( AD \).
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Context:
We say that a $2023$-tuple of nonnegative integers $(a_1,\ldots,a_{2023})$ is \textit{sweet} if the following conditions hold:

- $a_1+a_2+\ldots+a_{2023}=2023$
- $\frac{a_1}{2}+\frac{a_2}{2^2}+\ldots+\frac{a_{2023}}{2^{2023}}\le 1$

Determine the greatest positive integer $L$ so that $a_1+2a_2+\ldots+2023a_{2023}\ge L$ holds for every sweet $2023$-tuple $(a_1,\ldots,a_{2023})$.
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Context:
Find the distance between the vector \( v_2 = \begin{pmatrix} 9 \\ 0 \\ 0 \end{pmatrix} \) and the subspace \( U = (\text{span}(v))^\perp \), where \( v = \begin{pmatrix} \frac{1}{3} \\ \frac{2}{3} \\ \frac{2}{3} \end{pmatrix} \).
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Context:
The vertices and midpoints of the sides of a regular decagon (thus a total of 20 points marked) are noted.

How many triangles can be formed with vertices at the marked points?
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Context:
Find the Fourier transform of the Laplacian operator in polar coordinates, given by \( \nabla^2 = \frac{\partial^2}{\partial r^2} + \frac{1}{r}\frac{\partial}{\partial r} \). Assume no angular dependence.
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Context:
Find the volume of the solid defined by the inequality $3x^2 + 2y^2 + z^2 \leq 6$. Express your answer in terms of $
abla$. (Note: $
abla$ is a placeholder for the correct unit or constant factor, if needed.)
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Context:
A large cube has an edge length of \(2x^2 + 1\). A smaller cube has a volume of \(x^2 - 1\). When the large cube is filled with smaller cubes of the same size, the remaining space has a volume that is 4 cubic units less than the volume of the smaller cube. What is the edge length of the larger cube?
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Context:
设 $A$ 是一个 $2 \times 2$ 的可逆矩阵, 且方阵中每个位置的元素在模 3 同余系 $\mathrm{Z} / 3 Z$ 中取值。那么满足 $A^{50} \equiv I_{2}(\bmod 3)$ 方阵 $A$ 的个数为 ？
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Context:
Let $\mathfrak{g}$ be a finite-dimensional Lie algebra, and let $\mathfrak{g}^{(i)}$ denote the derived series of $\mathfrak{g}$, with $\mathfrak{g}^{(0)} = \mathfrak{g}$. Let $\operatorname{rad}(\mathfrak{g})$ be the solvable radical of $\mathfrak{g}$. There exist minimal natural numbers $n$ and $m$ such that $\mathfrak{g}^{(n)} = \mathfrak{g}^{(n+1)}$ and $\operatorname{rad}(\mathfrak{g})^{(m)} = 0$. Determine the relationship between the numbers $n$ and $m$. \(\text{Hint: Consider the properties of solvable and nilpotent Lie algebras.}\)
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Context:
Three positive reals $x$, $y$, and $z$ satisfy the following equations:

1. $x^2 + y^2 = 9$
2. $y^2 + yz + z^2 = 16$
3. $x^2 + \sqrt{3}xz + z^2 = 25$

Find the value of $2xy + xz + \sqrt{3}yz$.
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Context:
On the segment \( AB \) of length 10, a circle \(\omega\) is constructed with \( AB \) as the diameter. A tangent is drawn to \(\omega\) at point \( A \), on which a point \( K \) is chosen. A line through \( K \) different from \( AK \) is drawn, which touches the circle \(\omega\) at point \( C \). The altitude \(CH\) of triangle \(ABC\) intersects the segment \( BK \) at point \( L \). Given that \( BH: AH = 1:4 \), find the area of triangle \(CKL\).
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Context:
Given that \( m, n \in \mathbf{Z} \), and \( 1 \leq m \leq 99 \), \( 1 \leq n \leq 99 \), how many ordered pairs \((m, n)\) are there such that \((m+n)^{2} + 3m + n\) is a perfect square?
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Context:
How can we simplify $\sum_{i=0}^n(-1)^i { n \choose i } (n - i)^3$?
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Context:
如果平面 \(\alpha, \beta\), 直线 \(m, n\), 点 \(A, B\) 满足: \(\alpha \parallel \beta, m \subset \alpha, n \subset \beta, A \in \alpha, B \in \beta\)，且 \(AB\) 与 \(\alpha\) 所成的角为 \(\frac{\pi}{4}\), \(m \perp AB\), \(n\) 与 \(AB\) 所成的角为 \(\frac{\pi}{3}\)，那么 \(m\) 与 \(n\) 所成的角大小为 \(\frac{k}{m}\)，请计算k+m的值。
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Context:
Example 6. Find the flux of the vector field $\mathbf{a} = \mathbf{i} - \mathbf{j} + x y z \mathbf{k}$ through the circle $S$, obtained by the intersection of the sphere $x^{2} + y^{2} + z^{2} \leqslant R^{2}$ with the plane $y = x$. Take the side of the circle facing the positive part of the $O x$ axis.
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Context:
A square is drawn inside a rectangle. The ratio of the width of the rectangle to a side of the square is $2:1$. The ratio of the rectangle's length to its width is $2:1$. What percent of the rectangle's area is inside the square?
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Context:
Frist Campus Center is located $1$ mile north and $1$ mile west of Fine Hall. The area within $5$ miles of Fine Hall that is located north and east of Frist can be expressed in the form $\frac{a}{b} \pi - c$, where $a, b, c$ are positive integers and $a$ and $b$ are relatively prime. Find $a + b + c$.
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Context:
Evaluate the integral \( I = \int_{-\infty}^\infty \frac{x \sin(\pi x)}{x^2 + 2x + 5} \, dx. \)
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Context:
What are the possible values of $a$ and $b$ in $\mathbb{R}$ such that the following limit is equal to $1$?

$$ \lim_{n\rightarrow\infty}\left( \sqrt[3]{an^{3}+bn^{2}+1}-\log _{5}\left( 3^{n}+4^{n}+5^{n}\right) -\sqrt[4]{n^{4}+1}\right) =1 $$
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Context:
Determine the range of values of the real number $c > 0$ for which the series \( \sum_{n=1}^{\infty}\frac{1}{n^{c+\cos n}} \) converges.
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Context:
If \(\lim_{x \to \infty}\frac{a(2x^3-x^2)+b(x^3+5x^2-1)-c(3x^3+x^2)}{a(5x^4-x)-bx^4+c(4x^4+1)+2x^2+5x}=1\), find the value of \(a+b+c\).
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Context:
If $f''(x_{0})=2$, find the value of $\lim\limits_{k\to 0} \frac{f(x_{0}-k)-f(x_{0})}{2k}$. Express your answer as a single numerical value.
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Context:
Determine the domain of convergence for the series \( \sum_{n=0}^{\infty} (-1)^n x^{(n+1)^2} \).
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Context:
Determine the intersection of all the kernels of irreducible characters of a finite group $G$ that contain a normal subgroup $H$.
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Context:
Find the number of imaginary roots of the equation:
\[ \frac{A_1^2}{x-a_1} + \frac{A_2^2}{x-a_2} + \cdots + \frac{A_n^2}{x-a_n} = k \]
where \(a_1, a_2, \ldots, a_n, A_1, A_2, \ldots, A_n, k\) are all real numbers.
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Context:
Given the ellipse \(\frac{x^{2}}{4}+\frac{y^{2}}{3}=1\) with the left and right foci \(F_{1}\) and \(F_{2}\) respectively, a line \(l\) passes through the right focus and intersects the ellipse at points \(P\) and \(Q\). Find the maximum area of the inscribed circle of \(\triangle F_{1}PQ\).
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Context:
Determine the asymptotic behavior of the integral \( \int_{0}^{1}\exp\left(2\pi i n p(x)\right)\,dx \) as \( n \to \infty \), where \( p(x) \) is a polynomial of degree \( d \).
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Context:
18.14 People are participating in a round-robin Japanese chess tournament. Each person plays against 13 others, with no draws in the matches. Find the maximum number of "circular triples" (where each of the three participants wins against one and loses to another) in the tournament.
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Context:
Determine the number of isomorphism classes of countably infinite models of the empty theory over one binary relation, also known as the number of distinct unlabeled countable graphs.
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Context:
In the given configuration, circle ω touches circle Ω internally at point P. The center O of Ω is outside ω. Diameter XY of Ω is tangent to ω, with PY > PX. The line segment PY intersects ω at point Z, and it is given that YZ = 2PZ. Determine the magnitude of angle PYX in degrees.
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Context:
Determine the number of steps, Tn, required to solve the Towers of Hanoi problem with n disks, where all disks start on pillar A and must be moved to pillar C. It is not allowed to move a disk directly from pillar A to pillar C; all disks must pass through pillar B. What is Tn?
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Context:
Let's call a number \( \mathrm{X} \) "20-supporting" if for any 20 real numbers \( a_{1}, \ldots, a_{20} \), whose sum is an integer, there is at least one number for which \( \left|a_{i}-\frac{1}{2}\right| \geq X \).

Find the largest 20-supporting \( X \), rounded to the nearest thousandth by standard mathematical rules.
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Context:
Starting at 6 a.m., cars, buses, and motorcycles arrive at a highway toll booth according to independent Poisson processes. Cars arrive about once every 5 minutes. Buses arrive about once every 10 minutes. Motorcycles arrive about once every 30 minutes. Calculate the probability that the seventh motorcycle arrives within 45 minutes of the third motorcycle.
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Context:
Let $a, b, c$ be non-negative real numbers satisfying $a \le b + c$. Determine the positive real values of $n$ for which the inequality $a^n \le b^n + c^n$ holds.
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Context:
Determine the appropriate center for the Laurent series expansion of \( f(z) = \frac{z^2+1}{z(z-3)} \) in the regions \(0<|z|<3\) and \(3<|z|< \infty\).
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Context:
Evaluate the expression \[\frac{x^1\cdot x^2\cdot x^3\cdots x^{20}}{x^2\cdot x^4 \cdot x^6 \cdots x^{30}}\] when $x = 3$.
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Context:
Erin is watching a TV mini series of Pride and Prejudice. There are 6 episodes that are each of a certain length. If Erin watches all of the episodes in one sitting, one after the other with no breaks, she will spend 5 hours watching the series. How long is each episode?
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Context:
正 2015 边形 $A_{1} A_{2} \cdots A_{2015}$ 内接于单位圆 $O$, 任取它的两个不同顶点 $A_{i}, A_{j}$,则 $\left|\overrightarrow{O A_{i}}+\overrightarrow{O A_{j}}\right| \geq 1$ 的概率为 $\qquad$.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值
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Context:
The tangents of the angles of a triangle are natural numbers. What can they be equal to?
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Context:
Let \(H\) be the intersection of altitudes in triangle \(ABC\). If \(\angle B = \angle C = \alpha\) and \(O\) is the center of the circle passing through \(A\), \(H\), and \(C\), then find \(\angle HOC\) in terms of \(\alpha\) and express your answer in degrees. The original answer is in the form \(k^\circ - m\alpha\), please give the value of \(k + m\).
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Context:
A and B take turns marking numbers from 1 to 17. The rules are: a number that is already marked, twice a marked number, or half a marked number cannot be marked again. The player who cannot mark a number loses. Now, A has marked 8 first. To ensure victory, what number should B mark next?
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Context:
Find the limit of the following expression as \( s \to 1 \):
\[ \lim_{s \to 1} \frac{D(s)}{\zeta(s)} \]
where \( D(s) = e^{i \theta} + \frac{e^{i \theta/2}}{2^s} + \frac{e^{i \theta / 3}}{3^s} + \dots \) is a Dirichlet series.
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Context:
Determine the number of unique ways to rearrange 16 people into 4 groups of 4, such that no two people who were initially in the same group are placed together in any new arrangement.
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Context:
已知x,y∈(0,+∞)，且$\frac { 1 9 } { x } + \frac { 9 8 } { y } = 1$，则x+y的最小值是多少？
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Context:
Evaluate the definite integral: $$\int\limits_{0}^{2}\left(\sqrt{1+x^3}+\sqrt[3]{x^2+2x}\:\right)dx$$
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Context:
From the set of three-digit numbers that do not contain the digits 0, 6, 7, 8, or 9, several numbers were written on paper in such a way that no two numbers can be obtained from each other by swapping two adjacent digits. What is the maximum number of such numbers that could have been written?
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Context:
The function $f(x)= \frac{1}{x}-\log_{2} \frac{1+ax}{1-x}$ is an odd function, then the real number $a=$ ______.
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Context:
3. (1990 Vietnam Olympiad) Let $a, b, c \in \mathbf{R}^{+}$, and $a b c + a + c = b$, determine the maximum value of $p = \frac{2}{a^{2}+1} - \frac{2}{b^{2}+1} + \frac{3}{c^{2}+1}$.
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Context:
Find the inverse of $\overline{x+1}$ in the ring $\mathbb{Q}[x]/(x^2+x+1)$. Express your answer in terms of the equivalence class in this quotient ring.
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Context:
Given that the sum of the first $n$ terms of the sequence $\{a_{n}\}$ is $S_{n}$, and $a_{1}=4$, $a_{n}+a_{n+1}=4n+2$ for $n\in \mathbb{N}^{*}$, calculate the maximum value of $n$ that satisfies $S_{n} \lt 2023$.
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Context:
Find the derivative of the trace function \( \text{Tr} \left( X|e^{itK} - X|^3 \right) \) with respect to \( t \) at \( t = 0 \), where \( K \) is a Hermitian matrix and \( X \) is a positive matrix.
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Context:
Let \(X\) be the set of positive integers that contain distinct digits in their decimal representation. For each \(n \in X\), let \(A_{n}\) denote the set of numbers that can be formed by permuting the digits of \(n\), and let \(d_{n}\) denote the greatest common divisor of the elements of \(A_{n}\). Determine the maximum value of \(d_{n}\).
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Context:
IX OM - II - Task 2

Six equal disks are placed on a plane in such a way that their centers lie at the vertices of a regular hexagon with a side equal to the diameter of the disks. How many rotations will a seventh disk of the same size make while rolling externally on the same plane along the disks until it returns to its initial position?
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Context:
Evaluate the limit: $$\lim_{x\rightarrow\infty}\left(\frac{p\left(x\right)}{q\left(x\right)}\right)^{\frac{1}{\ln x}}$$ where $q(x)$ is a polynomial of degree $m$ with leading coefficient $b_m$.
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Context:
In triangle \(ABC\), a median \(AM\) is drawn. Find the angle \(AMC\) if the angles \(BAC\) and \(BCA\) are \(45^\circ\) and \(30^\circ\) respectively.
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Context:
Sandy purchases a perpetuity-immediate with annual payments starting at 100, increasing by 10 each year. Danny purchases a perpetuity-due with annual payments of 180. If the present values of both perpetuities are equal at an effective interest rate i > 0, calculate i.
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Context:
John adopts a dog from a shelter. The dog ends up having health problems and this requires 3 vet appointments, which cost $400 each. After the first appointment, John paid $100 for pet insurance. He paid $660 in total. What percentage of the subsequent visits did the pet insurance cover?
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Context:
For four different integers, all pairwise sums and pairwise products were calculated.

The resulting sums and products were written on the board. What is the smallest number of different numbers that could be on the board?
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Context:
Find the derivative.

$$
y=\frac{\operatorname{sh} x}{2 \operatorname{ch}^{2} x}+\frac{1}{2} \cdot \operatorname{arctg}(\operatorname{sh} x)
$$
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Context:
Determine the Krull dimension of the injective hull $E_R(R/\mathfrak{m})$ of the residue field $R/\mathfrak{m}$ in a Noetherian local ring $(R,\mathfrak{m})$. Provide your answer as a single number.
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Context:
Let $u$ and $v$ be vectors in $\mathbb{R}^n$ such that $u^Tv=0$. Define the matrix $A=uv^T+vu^T \in \mathbb{R}^{n\times n}$. Find an expression for $||A||_2$, the 2-norm of the matrix $A$.
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Context:
A and B play a number-filling game on a $5 \times 5$ grid where A starts and they take turns filling empty squares. Each time A fills a square with a 1, and each time B fills a square with a 0. After the grid is completely filled, the sum of the numbers in each $3 \times 3$ square is calculated, and the highest sum among these is recorded as $A$. A aims to maximize $A$, while B aims to minimize $A$. What is the maximum value of $A$ that A can achieve?
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Context:
Determine the maximum value of the sum
$$S=\sum_{n=1}^{\infty}\frac{n}{2^{n}}\left(a_{1}a_{2}\cdots a_{n}\right)^{1/n}$$
over all sequences $a_{1},a_{2},a_{3},\cdots$ of nonnegative real numbers satisfying
$$\sum_{k=1}^{\infty}a_{k}=1.$$ The original answer is in \frac{m}{n} format, please give the value of m + n.
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Context:
A is traveling at 50 mph and needs to pass B, who is traveling at 40 mph, by gaining 30 feet. Meanwhile, C is 210 feet away from A and is headed towards A at 50 mph. If B and C maintain their speeds, what is the minimum speed increase required for A to pass safely? Express your answer in mph.
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Context:
Along the road connecting Masha and Sasha's houses, there are 17 apple trees and 18 poplar trees. While Masha was walking to visit Sasha, she took pictures of all the trees. Right after the tenth apple tree, the memory on Masha's phone was filled, and she couldn't take pictures of the remaining 13 trees. The next day, when Sasha was walking to visit Masha, starting from the eighth apple tree, he plucked one leaf from each tree. How many leaves did Sasha pluck?
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Context:
Petia has created two polynomials \( f(x) \) and \( g(x) \), each of the form \( ax^2 + bx + c \). Vasya names a number \( t \), and Petia provides him with either \( f(t) \) or \( g(t) \) without specifying which. After \( n \) turns, Vasya must determine one of Petia's polynomials. What is the smallest \( n \) for which Vasya can always succeed in identifying one of the polynomials?
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Context:
In the diagram, \(P Q R S T\) is a pentagon with \(P Q=8\), \(Q R=2\), \(R S=13\), \(S T=13\), and \(T P=8\). Also, \(\angle T P Q=\angle P Q R=90^\circ\). What is the area of pentagon \(P Q R S T\) ?
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Context:
Elisa has 24 science books and other books in mathematics and literature. If Elisa had one more mathematics book, then one-ninth of her books would be in mathematics and one-fourth in literature. If Elisa has fewer than 100 books, how many mathematics books does she have?
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Context:
A palindrome is an integer that reads the same forward and backward, such as 3663. What percent of the palindromes between 100 and 500 contain at least one 5?
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Context:
Let $G$ be a non-abelian group of order $20$. Determine the order of $\operatorname{Aut}(G)$. \(\boxed{?}\)
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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:
设$\frac { 1 } { 1 + \frac { 1 } { 1 + \frac { 1 } { 1 + \cdots } + \frac { 1 } { 1 } } } = \frac { m } { n } ,$

其中m和n是互质的自然数，而等式左边含有1988条分数线，试计算$m ^ { 2 } + m n - n ^ { 2 }$的值.
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Context:
How many not necessarily distinct monic irreducible polynomials are there in the prime factorization of \(\left(x^{8}+x^{4}+1\right)\left(x^{8}+x+1\right)\)? (For instance, \((x+1)^{2}\) has two prime factors.)
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Context:
Determine the values of \( p \) for which the integral \( \int _0 ^\infty \frac {\sin x}{x^p} \, dx \) converges.
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Context:
Let $P$ be the set of points $$\{(x, y) \mid 0 \leq x, y \leq 25, x, y \in \mathbb{Z}\}$$ and let $T$ be the set of triangles formed by picking three distinct points in $P$ (rotations, reflections, and translations count as distinct triangles). Compute the number of triangles in $T$ that have area larger than 300.
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Context:
Determine the prime ideals of the ring of integers \( \mathcal{O}_K \) of the number field \( K = \mathbb{Q}(\sqrt{5}) \) that lie over the prime ideal \( 7\mathbb{Z} \).
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Context:
Find the average value of the function \( f(x,y,z) = xyz \) over the tetrahedron with vertices \((0,0,0), (1,0,0), (1,1,0), \text{and } (1,1,1)\).
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Context:
In a batch of N products containing M defective items, if we randomly select n items and exactly X of them are defective, what is the mathematical expression that represents the maximum value of X in terms of M and n? Express your answer using mathematical notation for minimum or maximum functions if necessary.
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Context:
Calculate the contour integral $\oint_{|z|=2} \frac{dz}{z\sqrt{1-\frac{1}{z^2}}}$ using the main branch of the square root, where $|z|=2$ denotes a circle centered at the origin with radius 2.
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Context:
14. A four-digit number, if a comma is placed between the hundreds and tens place, can be written as two two-digit numbers $(3126 \rightarrow 31,62)$. If the two two-digit numbers have an integer multiple relationship, we call such a four-digit number a "clever number". If you select 4 numbers from $1, 2, 4, 6, 8$ to form a four-digit number, how many "clever numbers" are there? $\qquad$
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Context:
Find the value of $\lambda$ such that the function $f(x) = (\sin(x))^3 + \lambda(\sin(x))^2$ has no local extrema in its domain, which is $x \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$.
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Context:
How to calculate $$\int_{0}^L\int_{a}^{L-a}\frac{\pi}{\cosh(s)}ds\,da$$ where $L$ is a constant?
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Context:
Znayka cut out a semicircle from paper. Neznayka marked a point \( C \) on the diameter \( AB \) of this semicircle and cut out two smaller semicircles with diameters \( AC \) and \( CB \) from Znayka's semicircle. Find the area of the remaining figure, if the length of the chord passing through point \( C \) perpendicular to \( AB \) inside the figure is 8. Round the answer to two decimal places if necessary.
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Context:
Compute the limit \( \lim_{z\to 0}z^3\Gamma(z)\Gamma(z-1)\Gamma(z-2) \) using the functional relation of the Gamma function.
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Context:
Four circles of radius 1 are externally tangent to each other and internally tangent to a larger circle. What is the radius of the large circle? Express your answer as a common fraction in simplest radical form.
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Context:
Let $S=\\{(a,b)|a=1,2,\\dots,n,b=1,2,3\\}$. A [i]rook tour[\/i] of $S$ is a polygonal path made up of line segments connecting points $p_1,p_2,\\dots,p_{3n}$ is sequence such that \r\n\r\n(i) $p_i\\in S,$ \r\n\r\n(ii) $p_i$ and $p_{i+1}$ are a unit distance apart, for $1\\le i\u003C3n,$ \r\n\r\n(iii) for each $p\\in S$ there is a unique $i$ such that $p_i=p.$ \r\n\r\nHow many rook tours are there that begin at $(1,1)$ and end at $(n,1)?$\r\n\r\n(The official statement includes a picture depicting an example of a rook tour for $n=5.$ This example consists of line segments with vertices at which there is a change of direction at the following points, in order: $(1,1),(2,1),(2,2),(1,2), (1,3),(3,3),(3,1),(4,1), (4,3),(5,3),(5,1).$)
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Context:
Let \( p(x) \) be a polynomial of degree strictly less than 100 such that it does not have \( x^3-x \) as a factor. If \( \frac{d^{100}}{d x^{100}} \frac{p(x)}{x^3-x} =\frac{f(x)}{g(x)} \) for some polynomials \( f(x) \) and \( g(x) \), find the smallest possible degree of \( f(x) \).
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Context:
Determine the distribution of the sum of 50 independent and identically distributed uniform random variables on the interval (2, 5).
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Context:
In the Cartesian coordinate plane $(xOy)$, a moving point $P$ passes through the chords $PA$ and $PB$ of the circles $C_{1}$: $x^{2}+y^{2}+2x+2y+1=0$ and $C_{2}$: $x^{2}+y^{2}-4x-6y+9=0$ respectively, where $A$ and $B$ are the points of intersection. If $|PA|=|PB|$, find the minimum value of $|OP|$.
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Context:
Let  $n$  be a positive integer. Every square in a  $n \times n$ -square grid is either white or black.
How many such colourings exist, if every  $2 \times 2$ -square consists of exactly two white and two black squares?
The squares in the grid are identified as e.g. in a chessboard, so in general colourings obtained from each other by rotation are different.
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Context:
Find the closed form expression for the derivative of the determinant with respect to matrix $A$:
\[ \frac{\partial \det(X - XA'(AXA')^{-1}AX)}{\partial A} \] where $X$ is an $n \times n$ full rank matrix and $A$ is an $m \times n$ full rank matrix with $m \le n$. Assume $rank(X) = n$ and $rank(A) = m$. Provide your answer in terms of $X$ and $A$.
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Context:
A certain debt will be paid in 52 installments from January 1 to December 31 of a certain year. Each of the first 8 payments is to be $410; each of the remaining payments is to be a certain amount more than each of the first 8 payments. The average (arithmetic mean) payment that will be made on the debt for the year is $465. What is the difference between the remaining payments and the first 8 payments?
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Context:
A random sample of size 7 is drawn from a distribution with probability density function \( f_{\theta}(x) = \frac{1+x^2}{3\theta(1+\theta^2)} \) for \( -2\theta \le x \le \theta \) and \( x > 0 \), and zero otherwise. The observations are 12, -54, 26, -2, 24, 17, -39. What is the maximum likelihood estimate of \( \theta \)?
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Context:
Let an integer $n > 1$ be given. In the space with orthogonal coordinate system $Oxyz$ we denote by $T$  the set of all points $(x, y, z)$ with $x, y, z$ are integers, satisfying the condition: $1 \leq x, y, z \leq n$. We  paint all the points of $T$ in such a way that: if the point $A(x_0, y_0, z_0)$ is painted then points $B(x_1, y_1, z_1)$ for which $x_1 \leq x_0, y_1 \leq y_0$ and $z_1 \leq z_0$ could not be painted. Find the maximal number of points that we can paint in such a way the above mentioned condition is satisfied.
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\boxed{\left\lfloor \frac{3n^2 + 1}{4} \right\rfloor}


Context:
Let \( K \) be a field and consider \( a \in K \) with \( f(x) = x^n - a \) being an irreducible polynomial. If \( m \in \mathbb{N} \) is a divisor of \( n \), and \( \alpha \) is a root of \( f(x) \) in an extension field of \( K \), determine the minimal polynomial of \( \alpha^m \).
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Context:
Find all squares which have only two non-zero digits, one of them 3.
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Context:
Determine all values of \( p \in \mathbb{R} \) for which \( |x-y|^p \) defines a metric on \( \mathbb{R} \).
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Context:
Consider the set of complex numbers $z$ satisfying $|1+z+z^{2}|=4$. The maximum value of the imaginary part of $z$ can be written in the form $\frac{\sqrt{m}}{n}$, where $m$ and $n$ are relatively prime positive integers. Find the value of $m+n$.
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Context:
Miki has a dozen oranges of the same size and a dozen pears of the same size. Miki uses her juicer to extract 8 ounces of pear juice from 3 pears and 8 ounces of orange juice from 2 oranges. She makes a pear-orange juice blend from an equal number of pears and oranges. What percentage of the blend is pear juice? Express your answer as a percentage (%).
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Context:
Find the fixed field of the automorphism $t \mapsto t-2$ of the field $k(t)$, where the characteristic of $k$ is non-zero.
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Context:
Given the function $f(x)$, its domain is $\mathbb{R}$. When $x < 0$, $f(x) = x^5 - 1$; when $-1 \leq x \leq 1$, $f(-x) = -f(x)$; when $x > 0$, $f(x+1) = f(x)$. Calculate $f(2016)$. Express your answer as a single integer.
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Context:
Evaluate the limit without using L'Hôpital's rule:
\[ \lim_{a\to 0}\frac{\delta_{a}+\delta_{-a}-2\delta}{a^{2}} \]
where \(\delta_{a}\) is the Dirac distribution at the point \(a\).
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Context:
Evaluate the integral \( I = \int_{0}^{\pi/2} \lfloor \cot(x) \rfloor \, dx \).
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Context:
Determine the largest integer \( K \) such that player A can always place at least \( K \) red stones on a 20x20 grid, ensuring that the distance between any two red stones is never equal to \( \sqrt{5} \), regardless of how player B places blue stones on the grid.
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Context:
Given two independent random variables $X$ and $Y$, where $X \sim U[a, b]$ and $Y \sim U[c, d]$, with $a < c < 0$ and $d < b > 0$. How do you calculate the probability $P(X > Y)$?
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Context:
Given the arithmetic sequence $\{a_n\}$, find the maximum number of different arithmetic sequences that can be formed by choosing any 3 distinct numbers from the first 20 terms.
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Context:
Oleg drew an empty 50×50 table and wrote a number above each column and to the left of each row. It turned out that all 100 written numbers are different, 50 of which are rational and the remaining 50 are irrational. Then, in each cell of the table, he recorded the product of the numbers written near its row and its column (a "multiplication table"). What is the maximum number of products in this table that could turn out to be rational numbers?
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Context:
Suppose two independent sequences of Bernoulli trials continue until a success occurs. For sequences A and B, with respective success-probabilities a and b, what is the probability that A is shorter than B?

Here's an example: Suppose that I have two unfair coins. Coin 1 has a probability of coming up heads of 1/3, and coin 2 has a probability of coming up heads of 2/3. I flip both coins at the same time. What is the probability that Coin 1 is heads before coin 2?
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Context:
Find the 2021st derivative of the function \( f(x) = \frac{1+x}{1+x+x^2} \) evaluated at \( x = 0 \).
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Context:
12. Let $A, B$ be subsets of the set $X=\{1,2, \cdots, n\}$. If every number in $A$ is strictly greater than all the numbers in $B$, then the ordered subset pair $(A, B)$ is called "good". Find the number of "good" subset pairs of $X$.
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Context:
Given three points \((a, b), (c, d), (x, y)\) on a circle with radius \(r\) in a rectangular coordinate plane, determine the minimum possible value of the expression:
\[
(ax + by - c)^2 + (bx - ay + d)^2 + (cx + dy + a)^2 + (dx - cy - b)^2.
\]
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Context:
A trailer carries 3, 4, and 5 crates on a trip. Each crate weighs a certain amount. The maximum weight of the crates on a single trip is 6250 kg. What is the minimum weight of each crate?
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Context:
Let  $ABC$  be a triangle with circumcircle  $\Gamma$  and let  $D$  be the midpoint of minor arc  $BC$ . Let  $E, F$  be on  $\Gamma$  such that  $DE \bot AC$  and  $DF \bot AB$ . Lines  $BE$  and  $DF$  meet at  $G$ , and lines  $CF$  and  $DE$  meet at  $H$ . Given that  $AB = 8, AC = 10$ , and  $\angle BAC = 60^\circ$ , find the area of  $BCHG$ .

*Note: this is a modified version of Premier #2*
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Context:
Determine the smallest positive integer $n$ such that in any two-coloring of a complete graph $K_n$, there exist $m$ pairwise disjoint edges of the same color.
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Context:
A graph has 12 vertices, and one of these vertices has a degree of 9. How many different labeled spanning trees can be formed that include this vertex?
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Context:
There was an integer number of cheese heads in the kitchen. During the night, some rats came and ate 10 cheese heads, with each rat eating an equal portion. Afterward, some of the rats got sick from overeating. The next night, the remaining seven rats finished the remaining cheese, but each rat was able to eat only half as much cheese as they did the previous night. How much cheese was originally in the kitchen?
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Context:
Shapovalov A.V.

At a round table, 12 places were prepared for the jury with each place labeled with a name. Nikolai Nikolaevich, who arrived first, absent-mindedly sat not in his own place, but in the next one clockwise. Each subsequent jury member, approaching the table, would take their own place or, if it was already occupied, walk around the table clockwise and sit in the first available place. The resulting seating arrangement of the jury members depends on the order in which they approached the table. How many different seating arrangements can arise?
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Context:
# 7. Problem 7

An angle of 90 degrees is covered by some angles with the same vertex in two or three layers, and the sum of the angles is 290 degrees. Find the measure of the angle covered in three layers.

#
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Context:
Given Laila had a 10-hour work day and attended three different meetings during this period, the first meeting took 50 minutes, the second meeting twice as long as the first, and the third meeting half as long as the second meeting, calculate the percentage of her work day that was spent attending meetings.
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Context:
Find the largest natural number such that all its digits are distinct, and the sum of any two adjacent digits is a square of a natural number.
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Context:
Determine the height of a tower from a 20-meter distant building, given that the angle of elevation to the top of the tower is 30° and the angle of depression to the base of the tower is 45°.
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Context:
Calculate the remainder when $12^{34^{56^{78}}}$ is divided by 90.
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Context:
Define $\lfloor x \rfloor$ as the greatest integer less than or equal to x, and ${x} = x - \lfloor x \rfloor$ as the fractional part of $x$. If $\lfloor x^2 \rfloor =2 \lfloor x \rfloor$ and $\{x^2\} =\frac12 \{x\}$, determine all possible values of $x$.
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Context:
Find an upper bound for the recursive function \( T(n) = T(n-1) + n^2 + n^2 \log n \).
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Context:
A farmer has thrice as many pigs as cows, 7 more cows than goats, twice as many chickens as the total of cows and pigs, and half as many ducks as the sum of goats and chickens. If there are also 5 more sheep than the total number of cows and chickens combined, and the farmer has 346 animals total, how many goats does he have?
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Context:
Given the function $f(x) = \frac{1}{2}x^{2} + a\ln x$, if for any two distinct positive real numbers $x_{1}$, $x_{2}$, it always holds that $\frac{f(x_{1})-f(x_{2})}{x_{1}-x_{2}} > 2$, then the range of the real number $a$ is ______.
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Context:
Determine the number of ways to arrange 10 men and 4 women around a circular table such that no three women sit consecutively.
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Context:
Consider coins with positive real denominations not exceeding 1 . Find the smallest $C>0$ such that the following holds: if we are given any 100 such coins with total value 50 , then we can always split them into two stacks of 50 coins each such that the absolute difference between the total values of the two stacks is at most $C$.
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Context:
How many positive integers at most 420 leave different remainders when divided by each of 5, 6, and 7?
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Context:
Calculate the fundamental group of the space defined by the equation: \[ X=\{(x,y,z)\in\mathbb{R}^3:(x^2+y^2+z^2-100)[(x-10)^2+y^2+z^2-1]=0\}. \]
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Context:
Let $ABC$ be an equilateral triangle with side length $1$. This triangle is rotated by some angle about its center to form triangle $DEF$. The intersection of $ABC$ and $DEF$ is an equilateral hexagon with an area that is $\frac{4}{5}$ the area of $ABC$. The side length of this hexagon can be expressed in the form $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. What is $m+n$?
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Context:
Calculate the flux of the vector field \( \vec{F}(x,y,z) = \langle x, y, 3 \rangle \) out of the region \( T \) bounded by \( z = x^2 + y^2 \) and \( z = 4 \).
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