Context:
Given the function $f(x) = \lg \frac{\sum_{i=1}^{n-1} i^{x} + n^{x}a}{n}$, where $a \in \mathbb{R}$, for any positive integer $n \geqslant 3$, if the inequality $f(x) > (x-1)\lg n$ has a solution in the interval $[1, +\infty)$, then the range of values for the real number $a$ is \_\_\_\_\_\_.
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Context:
A right triangle \(ABC\) is divided by the altitude \(CD\), drawn to the hypotenuse, into two triangles \(BCD\) and \(ACD\). The radii of the circles inscribed in triangles \(BCD\) and \(ACD\) are 4 cm and 3 cm, respectively. Determine the distance between their centers.
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Context:
Find the greatest integer less than or equal to \(\sum_{k=1}^{2^{13a}} k^{1/1983 - 1}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In $\triangle ABC$, $AB=6$, $AC=8$, $BC=10$, and $D$ is the midpoint of $\overline{BC}$. What is the sum of the radii of the circles inscribed in $\triangle ADB$ and $\triangle ADC$?
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Context:
Let $S = \{1,2,\cdots,2013\}$. Let $N$ denote the number of $9$-tuples of sets $(S_1, S_2, \dots, S_9)$ such that $S_{2n-1}, S_{2n+1} \subseteq S_{2n} \subseteq S$ for $n=1,2,3,4$. Find the remainder when $N$ is divided by $1000$.
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Context:
Let $P=\left\{x\mid x+2\geqslant x^{2}\right\}$, $Q=\left\{x\in \mathbb{N}\mid |x|\leqslant 3\right\}$. Calculate the intersection of sets P and Q, and express your answer as a set of numbers.
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Context:
Find all  $t\in \mathbb Z$  such that: exists a function  $f:\mathbb Z^+\to \mathbb Z$  such that: $f(1997)=1998$  $\forall x,y\in \mathbb Z^+ , \text{gcd}(x,y)=d : f(xy)=f(x)+f(y)+tf(d):P(x,y)$
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Context:
In the plane rectangular coordinate system \(xOy\), \(F\) is the focus of the parabola \(\Gamma: y^2 = 2px (p > 0)\). Point \(B\) is on the \(x\)-axis, and is to the right of point \(F\). Point \(A\) is on \(\Gamma\), and \(|AF| = |BF|\). The second intersection points of the line \(AF\) and the line \(AB\) with \(\Gamma\) are \(M\) and \(N\) respectively. If \(\angle AMN = 90^\circ\), find the slope of the line \(AF\).
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Context:
There are three piles of stones. Each time A moves one stone from one pile to another, A earns a reward from B. The reward is equal to the difference in the number of stones between the pile to which A moves the stone and the pile from which A takes the stone. If this difference is negative, A must pay B that amount instead (if A doesn’t have enough money, A can owe it temporarily). At a certain moment, all the stones are in the piles where they initially started. Determine the maximum possible amount of money A can have by that moment.
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Context:
The sum \( \sum_{n=0}^{2016\cdot2017^2}2018^n \) can be represented uniquely in the form \( \sum_{i=0}^{\infty}a_i\cdot2017^i \) for nonnegative integers \( a_i \) less than 2017. Compute \( a_0+a_1 \).
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Context:
Find the sum of the $x$-coordinates of the solutions to the system of equations $y=|x^2-8x+12|$ and $y=4-x$.
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Context:
Let  $m,n > 2$  be even integers. Consider a board of size  $m \times n$  whose every cell is colored either black or white. The Guesser does not see the coloring of the board but may ask the Oracle some questions about it. In particular, she may inquire about two adjacent cells (sharing an edge) and the Oracle discloses whether the two adjacent cells have the same color or not. The Guesser eventually wants to find the parity of the number of adjacent cell-pairs whose colors are different. What is the minimum number of inquiries the Guesser needs to make so that she is guaranteed to find her answer?(Czech Republic)
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Context:
$A$，$B$，$C$为$\Delta ABC$的三个内角，且$ctg\frac{A}{2}+ctg\frac{B}{2}+ctg\frac{C}{2}-2(ctgA+ctgB+ctgC)\ge T$.则${{T}_{\max }}=$_____.
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Context:
Find the value of $\displaystyle \int^{\pi}_{0}\cos^4(x+\sin 3x)dx$.
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Context:
A wooden cube has edges of length 3 meters. Square holes, of side one meter, centered in each face are cut through to the opposite face. The edges of the holes are parallel to the edges of the cube. Calculate the entire surface area of the cube, including the inside surfaces exposed by the holes. Express your answer in square meters.
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Context:
Given a hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 (a > 0, b > 0)$ with left and right foci $F\_1$, $F\_2$, and a focal distance of $2c (c > 0)$. If a parabola $y^{2} = 4cx$ intersects the hyperbola in the first quadrant at point $M$, and $|MF\_1| = 4c$, find the eccentricity of the hyperbola.
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Context:
Consider an inifinte sequence  $x_1, x_2,\dots$  of positive integers such that, for every integer  $n\geq 1$ :

- If  $x_n$  is even,  $x_{n+1}=\dfrac{x_n}{2}$ ;
- If  $x_n$  is odd,  $x_{n+1}=\dfrac{x_n-1}{2}+2^{k-1}$ , where  $2^{k-1}\leq x_n<2^k$ .

Determine the smaller possible value of  $x_1$  for which  $2020$  is in the sequence.
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Context:
$在\triangle ABC中，内角A，B，C的对边分别为a，b，c，且(\sin B-\sin C)^2=\sin^2A-\sin B\sin C.$
求角A；
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Context:
It is well-known that the $n^{\text{th}}$ triangular number can be given by the formula $n(n+1)/2$. A Pythagorean triple of $\textit{square numbers}$ is an ordered triple $(a,b,c)$ such that $a^2+b^2=c^2$. Let a Pythagorean triple of $\textit{triangular numbers}$ (a PTTN) be an ordered triple of positive integers $(a,b,c)$ such that $a\leq b<c$ and
\[\dfrac{a(a+1)}2+\dfrac{b(b+1)}2=\dfrac{c(c+1)}2.\]
For instance, $(3,5,6)$ is a PTTN ($6+15=21$). Here we call both $a$ and $b$ $\textit{legs}$ of the PTTN. Find the smallest natural number $n$ such that $n$ is a leg of $\textit{at least}$ six distinct PTTNs.
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Context:
Integrate $\int \dfrac{\ln(x)}{(1-x^2)^{3/2}}\text{d}x$
Please reason step by step, and put your final answer within \boxed{}.

\boxed{\frac{x\ln(x)}{\sqrt{1-x^{2}}} - \arcsin(x) + c}


Context:
设对于任意实数x，不等式$\vert x+1\vert+\vert x+a\vert+\vert x+3\vert>5$恒成立，则参数a的取值范围为
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Context:
Given that the function f(x) defined on the set of real numbers ℝ satisfies f(x+1) = 1/2 + √(f(x) - f^2(x)), find the maximum value of f(0) + f(2017).
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Context:
Rhona wrote down a list of nine multiples of ten: 10, 20, 30, 40, 50, 60, 70, 80, 90. She then deleted some of these multiples so that the product of the remaining multiples was a square number. What is the least number of multiples that she could have deleted? Provide your answer as a whole number.
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Context:
A shepherd uses 15 segments of fencing, each 2 meters long, to form a square or rectangular sheep pen with one side against a wall. What is the maximum area of the sheep pen in square meters?
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Context:
Compute $\lim_{z\to 0}z^3\Gamma(z)\Gamma(z-1)\Gamma(z-2)$.
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Context:
Determine the number of non-isomorphic groups of order 36 that have a normal subgroup of order 4.
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Context:
Given the cross-section of cone PQ is an equilateral triangle and plane α is parallel to PQ, find the eccentricity of the hyperbola on which the curve obtained by intersecting the cone's lateral surface with plane α lies. Express your answer as a numerical value, including the square root symbol if necessary.
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Context:
Find the smallest positive real number \(\alpha\), such that

\[ \frac{x+y}{2} \geq \alpha \sqrt{x y}+(1-\alpha) \sqrt{\frac{x^{2}+y^{2}}{2}} \]

for all positive real numbers \(x\) and \(y\).
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Context:
Determine the value of the following limit using real analysis methods without complex analysis:

$$\lim_{b \to 0}\int _0^{\infty}\frac{\left(x^2-1\right)\cos \left(bx\right)}{\left(x^2+1\right)^2}dx$$
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Context:
Let  $a$  be the sum of the numbers: $99 \times 0.9$  $999 \times 0.9$  $9999 \times 0.9$  $\vdots$  $999\cdots 9 \times 0.9$ where the final number in the list is  $0.9$  times a number written as a string of  $101$  digits all equal to  $9$ .
Find the sum of the digits in the number  $a$ .
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Context:
Farmer John has a flexible fence of length $L$ and two straight walls that intersect at a corner perpendicular to each other. He knows that if he doesn't use any walls, he can enclose a maximum possible area of $A_0$. When he uses one of the walls or both walls, he gets a maximum area of $A_1$ and $A_2$, respectively. If $n=\frac{A_1}{A_0}+\frac{A_2}{A_1}$, find $\lfloor 1000n \rfloor$.
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Context:
The edge of cube $A B C D A_{1} B_{1} C_{1} D_{1}$ is equal to 1. Construct a cross-section of the cube by a plane that has the maximum perimeter.
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Context:
The distance between two cells of an infinite chessboard is defined as the minimum nuber to moves needed for a king for move from one to the other.One the board are chosen three cells on a pairwise distances equal to  $ 100$ . How many cells are there that are on the distance  $ 50$  from each of the three cells?
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Context:
The sequence of integers $a_1, a_2, \dots$ is defined as follows:
- $a_1 = 1$.
- For $n > 1$, $a_{n+1}$ is the smallest integer greater than $a_n$ such that $a_i + a_j \neq 3a_k$ for any $i, j, k$ from $\{1, 2, \dots, n+1\}$, where $i, j, k$ are not necessarily different.

Define $a_{2004}$.
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Context:
Determine the interval where the function y = -x^2 is monotonically increasing. Express your answer as an interval.
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Context:
Compute $\left\lceil\sum_{k=2018}^{\infty}\frac{2019!-2018!}{k!}\right\rceil$. The notation $\left\lceil x\right\rceil$ denotes the least integer $n$ such that $n\geq x$.
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Context:
$$\frac{2}{\sqrt{4-3\sqrt[4]{5}+2\sqrt{5}-\sqrt[4]{125}}}=a+\sqrt[b]{c}$$
Find the value of $a+b+c$.
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Context:
In square \( ABCD \), diagonals \( AC \) and \( BD \) intersect at point \( O \). Point \( K \) is the midpoint of side \( AB \). Points \( M \) and \( N \) are chosen on sides \( AD \) and \( BC \) respectively, such that rays \( OK \), \( OM \), and \( ON \) divide the square into three regions of equal area. In what ratio does point \( M \) divide side \( AD \)?
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Context:
1. The numbers from 1 to 2150 are written on a board. Every minute, each number undergoes the following operation: if the number is divisible by 100, it is divided by 100; if it is not divisible by 100, 1 is subtracted from it. Find the largest number on the board after 87 minutes.
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Context:
Given two circles: $x^{2}+y^{2}+2ax+a^{2}-9=0$ and $x^{2}+y^{2}-4by-1+4b^{2}=0$, they have three shared tangents. If $a\in R$, $b\in R$, and $ab\neq 0$, determine the minimum value of $\frac{4}{a^{2}}+\frac{1}{b^{2}}$. Express your answer as a single number.
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Context:
Eight red boxes and eight blue boxes are randomly placed in four stacks of four boxes each. The probability that exactly one of the stacks consists of two red boxes and two blue boxes is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
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Context:
11. Given positive integers $a, b, c, d$ satisfy $a^{2}+2 a=c d+11 c+d+10, b^{2}-2 b=c d-11 c+d-12$, then the sum of all possible values of $d$ is $\qquad$.
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Context:
Given the function f(x) = sin(ωx + φ) where ω > 0 and |φ| < π/2, it is known that the smallest positive period is 4π, and f(π/3) = 1. Find the coordinates of a symmetry center of f(x). Express your answer as an ordered pair (x, y), where x is a specific numerical value and y is 0.
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Context:
Let \( C_1 \) and \( C_2 \) be two concentric circles, with \( C_2 \) inside \( C_1 \). Let \( A \) be a point on \( C_1 \) and \( B \) a point on \( C_2 \) such that the line segment \( AB \) is tangent to \( C_2 \). Let \( C \) be the second point of intersection of the line segment \( AB \) with \( C_1 \), and let \( D \) be the midpoint of \( AB \). A line passing through \( A \) intersects \( C_2 \) at points \( E \) and \( F \) such that the perpendicular bisectors of \( DE \) and \( CF \) intersect at a point \( M \) on the line segment \( AB \). Determine the ratio \( AM / MC \).
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Context:
Three speed skaters have a friendly "race" on a skating oval. They all start from the same point and skate in the same direction, but with different speeds that they maintain throughout the race. The slowest skater does $1$ lap per minute, the fastest one does $3.14$ laps per minute, and the middle one does $L$ laps a minute for some $1 < L < 3.14$. The race ends at the moment when all three skaters again come together to the same point on the oval (which may differ from the starting point.) Determine the number of different choices for $L$ such that exactly $117$ passings occur before the end of the race.
Note: A passing is defined as when one skater passes another one. The beginning and the end of the race when all three skaters are together are not counted as passings.
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Context:
"渐升数"是指每个数字比其左边的数字大的正整数, 如 34689 , 已知有 $C_9^5=126$ 个五位
"渐升数", 若把这些数按从小到大的顺序排列, 则第 100 个数是 $\qquad$
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Context:
There are originally 5 different books placed side by side on a bookshelf. Now, if we want to insert 3 different books, one at a time, into the gaps formed by the existing books, how many different insertion methods are there in total? Express your answer as a numerical value.
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Context:
Evaluate the integral $$\iiint_K e^z \, dx\, dy\, dz$$ where $K$ is the solid ball defined by $$K = \{x^2 + y^2 + z^2 \leq 1\}$$
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Context:
Given the hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ (where $a > 0, b > 0$), a tangent line $FM$ to the circle $x^{2} + y^{2} = a^{2}$ is drawn from the right focus $F$ of the hyperbola, intersecting the y-axis at point $P$. If $M$ is the midpoint of segment $FP$, find the eccentricity of the hyperbola.
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Context:
Let $m \ge 3$ be an integer and let $S = \{3,4,5,\ldots,m\}$. Find the smallest value of $m$ such that for every partition of $S$ into two subsets, at least one of the subsets contains integers $a$, $b$, and $c$ (not necessarily distinct) such that $ab = c$. <b>Note</b>: a partition of $S$ is a pair of sets $A$, $B$ such that $A \cap B = \emptyset$, $A \cup B = S$.
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Context:
Siva has the following expression, which is missing operations:
$$\frac{1}{2} \_ \frac{1}{4} \_ \frac{1}{8} \_ \frac{1}{16} \_ \frac{1}{32}.$$

For each blank, he flips a fair coin:
- If it comes up heads, he fills it with a plus.
- If it comes up tails, he fills it with a minus.

Afterwards, he computes the value of the expression. He then repeats the entire process with a new set of coin flips and operations.

If the probability that the positive difference between his computed values is greater than $\frac{1}{2}$ can be expressed as $\frac{a}{b}$ for relatively prime positive integers $a$, $b$, then find $a + b$.
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Context:
Compute the decimal expansion of \(\sqrt{\pi}\). Your score will be \(\min (23, k)\), where \(k\) is the number of consecutive correct digits immediately following the decimal point in your answer.
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Context:
A fly and $\mathrm{k}$ spiders move on a $2019 \times 2019$ grid. On their turn, the fly can move 1 square and the $k$ spiders can each move 1 square. What is the minimal $k$ for which the spiders are sure to catch the fly.
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Context:
Let $t$ represent all real values. Consider the curve formed by the set of points $(x, y)$ defined by the equations:
\[
x = t^3 - 3t + 1,
\]
\[
y = t^4 - 4t^2 + 4.
\]
Determine the ordered pair $(x,y)$ where the curve crosses itself.
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Context:
Given Tom paid $180, Dorothy paid $200, Sammy paid $240, and Alice paid $280, and they agreed to split the costs evenly, calculate the amount that Tom gave to Sammy minus the amount that Dorothy gave to Alice.
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Context:
Two positive integers $p,q \in \mathbf{Z}^{+}$ are given. There is a blackboard with $n$ positive integers written on it. A operation is to choose two same number $a,a$ written on the blackboard, and replace them with $a+p,a+q$. Determine the smallest $n$ so that such operation can go on infinitely.
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Context:
What is the largest $2$-digit prime factor of the integer $n = {300 \choose 150}$?
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Context:
Given a semicircle O whose diameter lies along the base AB of a trapezoid ABCD, and BC, CD, DA are tangent to the semicircle, find the length of AB. The lengths of BC and DA are 2 and 3, respectively. Express your answer as a whole number.
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Context:
Let $M=\{1,2,3,\ldots,1995\}$, and let $A$ be a subset of $M$ that satisfies the condition: if $x \in A$, then $15x \notin A$. The maximum number of elements in $A$ is ____.
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Context:
Zebadiah has 3 red shirts, 3 blue shirts, and 3 green shirts in a drawer. Without looking, he randomly pulls shirts from his drawer one at a time. What is the minimum number of shirts that Zebadiah must pull out to guarantee that he has either 3 shirts of the same color or 3 shirts of different colors? Express your answer as a whole number.
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Context:
Let $\triangle ABC$ be a triangle with $AB = 13$, $BC = 14$, and $CA = 15$. The altitude from $A$ intersects $BC$ at $D$.

Let $\omega_1$ and $\omega_2$ be the incircles of $\triangle ABD$ and $\triangle ACD$, respectively. The common external tangent of $\omega_1$ and $\omega_2$ (other than $BC$) intersects $AD$ at $E$. Compute the length of $AE$.
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Context:
It is known that the quadratic equations $a x^{2} + b x + c = 0$ and $b x^{2} + c x + a = 0$ (where $a$, $b$, and $c$ are non-zero numbers) have a common root. Find this root.
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Context:
## Task 1

The members of a youth brigade helped to prepare apartments for elderly citizens. The value amounted to 1200 Marks. They helped with cable installation. This value amounted to 2100 Marks. In the operation, they saved materials. This value amounted to 1500 Marks.

How many Marks did the youth brigade bill for?
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Context:
Given a function $f(x) = (m^2 - m - 1)x^{m^2 - 2m - 1}$ which is a power function and is increasing on the interval $(0, \infty)$, find the value of the real number $m$.
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Context:
In the dihedral angle $\alpha - E F - \beta $, $AE \subset \alpha, BF \subset \beta$, and $AE \perp EF, BF \perp EF$. Given $EF = 1$, $AE = 2$, and $AB = \sqrt{2}$, find the maximum volume of the tetrahedron $ABEF$.
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Context:
An ant walks inside a 18 cm by 150 cm rectangle. The ant's path follows straight lines which always make angles of 45° to the sides of the rectangle. The ant starts from a point X on one of the shorter sides. The first time the ant reaches the opposite side, it arrives at the midpoint. Calculate the distance, in centimeters, from X to the nearest corner of the rectangle.
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Context:
An insect lives on the surface of a regular tetrahedron with edges of length 1. It wishes to travel on the surface of the tetrahedron from the midpoint of one edge to the midpoint of the opposite edge. What is the length of the shortest such trip? (Note: Two edges of a tetrahedron are opposite if they have no common endpoint.)
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Context:
3. Petrov lists the odd numbers: $1,3,5, \ldots, 2013$, while Vasechkin lists the even numbers: $2,4, \ldots, 2012$. Each of them calculated the sum of all digits of all their numbers and told the excellent student Masha. Masha subtracted Vasechkin's result from Petrov's result. What did she get?
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Context:
Consider two parallel lines $a$ and $b$. The circles $C, C_1$ are tangent to each other and to the line $a$. The circles $C, C_2$ are tangent to each other and to the line $b$. The circles $C_1, C_2$ are tangent to each other, with radii $R_1 = 9$ and $R_2 = 16$. What is the radius $R$ of the circle $C$?
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Context:
设 $\left\{a_{n}\right\}$ 为等差数列， $\left\{b_{n}\right\}$ 为等比数列，且 $b_{1}=a_{1}{ }^{2}, b_{2}=a_{2}{ }^{2}, b_{3}=a_{3}{ }^{2}\left(a_{1}<a_{2}\right)$ ，又 $\lim \limits_{n \rightarrow+\infty}\left(b_{1}+b_{2}+\cdots+b_{n}\right)=\sqrt{2}+1$. 试求 $\left\{a_{n}\right\}$ 的2倍首项与公差的和.
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Context:
6.2. On the sides $AB$ and $AC$ of triangle $ABC$, points $E$ and $F$ are taken, respectively. Segments $BF$ and $CE$ intersect at point $D$. Find the area of triangle $AEF$, if the areas of triangles $BCD$, $BDE$, and $CDF$ are $1$, $\frac{1}{2}$, and $\frac{1}{6}$, respectively.
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Context:
In a 6-deck game of cards (each deck contains 52 cards), what is the probability of drawing a spade and specifically the King of Diamonds when drawing two cards consecutively without replacement?
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Context:
There has been successive increases of 15% and then 10% in the price of gas from the previous month. By what percentage should a driver reduce gas consumption so that the expenditure does not change?
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Context:
A line intersects a circle with radius \( R \) at points \( A \) and \( B \) such that the measure of arc \( \overset{\frown}{AB} = 45^\circ \). Additionally, a line perpendicular to the diameter \( AM \) of the circle and passing through its center intersects the first line at point \( D \). The line passing through point \( B \) perpendicular to the diameter \( AM \) intersects \( AM \) at point \( C \). Find the area of the trapezoid \( OCBD \).
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Context:
Let $ABCD$ be an isosceles trapezoid with bases $AB=92$ and $CD=19$. Suppose $AD=BC=x$ and a circle with center on $\overline{AB}$ is tangent to segments $\overline{AD}$ and $\overline{BC}$. If $m$ is the smallest possible value of $x$, then $m^2$=
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Context:
A certain number of men are equal to as many women as are equal to 8 boys. All of them earn Rs. 105 only. Men's wages are Rs. 7. How many men are there?
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Context:
A straight line is drawn on an 8 x 8 chessboard. What is the largest possible number of the unit squares with interior points on the line?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let points $A = (0,0)$, $B = (2,4)$, $C = (6,6)$, and $D = (8,0)$. Quadrilateral $ABCD$ is cut into two pieces by a line passing through $A$ and intersecting $\overline{CD}$ such that the area above the line is twice the area below the line. This line intersects $\overline{CD}$ at a point $\left(\frac{p}{q}, \frac{r}{s}\right)$, where these fractions are in lowest terms. Calculate $p + q + r + s$.
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Context:
A local college is offering German lessons and currently has 8 students enrolled. For the first month, they run an advertising campaign that increases enrollment by 30%, with a fourth of these new students dropping out within the first week. Meanwhile, a nearby school hosts a cultural exchange festival and 5 international students decide to join the class, but 10% of them leave within two weeks. In the second month, the class holds an open house event which increases enrollment by 25% of the students already enrolled, but 3 of these new students dropped out due to scheduling conflicts.

During the third month, the class receives a boost in enrollment when a high school's language club decides to join the course. This adds an enrollment boom that's equal to 50% of the total current students, but 20% of them eventually quit due to the difficulty level. By the end of the third month, two-thirds of all remaining students graduate.

The college plans to run one last advertising campaign in the fourth month, with an expected 20% increase in enrollment. However, they also predict within this month that 30% of the new enrollees will not be able to keep up with the course material and drop out, while 10% of all the remaining students will ultimately graduate. How many students are expected to still be enrolled in the German lessons at the end of the fourth month?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Three sectors of a circle are removed from a regular hexagon to form a shaded shape. Each sector has a perimeter of 18 mm. Calculate the perimeter, in millimeters, of the shaded shape formed.
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Context:
A boy runs 4.8 miles in 33 minutes. What is his speed in kilometers per hour?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A circle has its center at $(2,0)$ with a radius of 2, and another circle has its center at $(5,0)$ with a radius of 1. A line is tangent to both circles in the first quadrant. The $y$-intercept of this line is closest to:
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Context:
In a New Year's cultural evening of a senior high school class, there was a game involving a box containing 6 cards of the same size, each with a different idiom written on it. The idioms were: 意气风发 (full of vigor), 风平浪静 (calm and peaceful), 心猿意马 (restless), 信马由缰 (let things take their own course), 气壮山河 (majestic), 信口开河 (speak without thinking). If two cards drawn randomly from the box contain the same character, then it's a win. The probability of winning this game is ____.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $\alpha$ be an algebraic element over a field, $F$, with minimal polynomial $p(t) = t^3 + t + 1$. Determine a non-zero polynomial $q(t) \in F[t]$ such that $q(\alpha^2) = 0$.
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Context:
Given point $A(a-1,2a-4)$, if the point is symmetric about the $y$-axis in the second quadrant, then the range of $a$ is ______.
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Context:
A teacher received letters on Monday through Friday with counts of $10$, $6$, $8$, $5$, $6$ respectively. Calculate the variance (${s^{2}} =$) of this data set.
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Context:
Square \( ABCD \) has a side length of 12 inches. A segment \( AE \) is drawn where \( E \) is on side \( DC \) and \( DE \) is 5 inches long. The perpendicular bisector of \( AE \) intersects \( AE, AD, \) and \( BC \) at points \( M, P, \) and \( Q \) respectively. The ratio of the segments \( PM \) to \( MQ \) is:
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $n$ points be given arbitrarily in the plane, no three of them collinear. Let us draw segments between pairs of these points. What is the minimum number of segments that can be colored red in such a way that among any four points, three of them are connected by segments that form a red triangle?
Please reason step by step, and put your final answer within \boxed{}.


Context:
When a plane intersects a cube, what type of polygon cannot be formed as a result of the intersection? Express your answer as the name of the polygon.
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Context:
Andrew and Blair are bored in class and decide to play a game. They pick a pair $(a, b)$ with $1 \le a, b \le 100$. Andrew says the next number in the geometric series that begins with $a,b$ and Blair says the next number in the arithmetic series that begins with $a,b$. For how many pairs $(a, b)$ is Andrew's number minus Blair's number a positive perfect square?
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Context:
Let $n$ be a positive integer such that $\lfloor\sqrt{n}\rfloor - 2$ divides $n - 4$ and $\lfloor\sqrt{n}\rfloor + 2$ divides $n + 4$. Find the greatest such $n$ less than $1000$. (Note: $\lfloor x \rfloor$ refers to the greatest integer less than or equal to $x$).
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Context:
A black bishop and a white king are placed randomly on a  $2000 \times 2000$  chessboard (in distinct squares). Let  $p$  be the probability that the bishop attacks the king (that is, the bishop and king lie on some common diagonal of the board). Then  $p$  can be expressed in the form  $\tfrac{m}{n}$ , where  $m$  and  $n$  are relatively prime positive integers. Compute  $m$ .

*Proposed by Ahaan Rungta*
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Context:
Consider the polynomial
\[ P(x)=\prod\limits_{k=0}^{10}(x^{2^k}+2^k)=(x+1)(x^2+2)(x^4+4)\cdots (x^{1024}+1024) \]
The coefficient of \( x^{2012} \) is equal to \( 2^a \). What is the value of \( a \)?
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Context:
For a Poisson process with rate $\lambda$, calculate $\mathbb{E}(T_2\mid N_1=1)$, where $T_2$ is the time of the second arrival.
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Context:
If $f(x)$ is a differentiable function satisfying $f(x+y)=f(x)+f(y)-(e^{-x}-1)(e^{-y}-1)+1$ and $f(0)=-1$, find the value of $f'(0)$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $n$ be a positive integer. In an $n\times n$ garden, a fountain is to be built with $1\times 1$ platforms covering the entire garden. Ana places all the platforms at a different height. Afterwards, Beto places water sources in some of the platforms. The water in each platform can flow to other platforms sharing a side only if they have a lower height. Beto wins if he fills all platforms with water.
Find the least number of water sources that Beto needs to win no matter how Ana places the platforms.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given $(2x-1)^{2015} = a_{0} + a_{1}x + a_{2}x^{2} + \ldots + a_{2015}x^{2015}$ ($x \in \mathbb{R}$), evaluate the expression $\frac {1}{2}+ \frac {a_{2}}{2^{2}a_{1}}+ \frac {a_{3}}{2^{3}a_{1}}+\ldots+ \frac {a_{2015}}{2^{2015}a_{1}}$.
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Context:
Let $$f_{n}(x)=\dfrac{1}{n}(\sin{x}+\sin{(2x)}+\sin{(3x)}+\cdots+\sin{(nx)}),x\in R,n\in N$$ Let $$a_{n}=\max_{x\in R}{(f_{n}(x))}$$ Find this limit $$\lim_{n\to\infty}a_{n}$$
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Context:
Al and Bill play a game involving a fair six-sided die. The die is rolled until either a number less than $5$ is rolled on consecutive tosses, or a number greater than $4$ is rolled on consecutive tosses. Al wins if the last roll is a $5$ or $6$. Bill wins if the last roll is a $2$ or lower. Let $m$ and $n$ be relatively prime positive integers such that $\frac{m}{n}$ is the probability that Bill wins. Find the value of $m+n$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the number of ordered triples of nonnegative integers $(a, b, c)$ that satisfy:
\[
(ab + 1)(bc + 1)(ca + 1) = 84.
\]
Please reason step by step, and put your final answer within \boxed{}.


Context:
A school held a soccer match, where each team played against every other team. The winning team of each match received $2$ points, the losing team received $0 points, and in case of a draw, both teams received $1$ point each. The team with the highest score would be crowned the champion. It is known that there is a team with the highest score (higher than all other teams), but this team has fewer wins than the other teams. The minimum number of teams participating in the match is ______.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Ryan Alweiss storms into the Fine Hall common room with a gigantic eraser and erases all integers $n$ in the interval $[2, 728]$ such that $3^t$ doesn’t divide $n!$, where $t = \left\lceil \frac{n-3}{2} \right\rceil$. Find the sum of the leftover integers in that interval modulo $1000$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Three cones are standing on their bases on a table, touching each other. The radii of their bases are 32, 48, and 48, and the apex angles are $\frac{\pi}{3}$, $\frac{2 \pi}{3}$, and $\frac{2 \pi}{3}$ respectively (the apex angle of a cone is the angle between its generatrices in an axial section). A sphere is suspended above the table, touching all the cones. It turns out that the center of the sphere is equidistant from the centers of the bases of all the cones. Find the radius of the sphere.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let the function \( f(x) \) have a derivative \( f'(x) \) on \( \mathbf{R} \), such that for any \( x \in \mathbf{R} \), \( f(x) + f(-x) = x^2 \). In the interval \( (0, +\infty) \), \( f'(x) > x \). If \( f(1+a) - f(1-a) \geq 2a \), then the range of the real number \( a \) is ______.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A fault occurred in the transmission line from the substation to the Wuyang Hospital, which cannot be detected from the outside. It requires sending people to climb to the top of the utility poles for inspection. Each inspection can determine whether the fault is at the beginning, middle, or end of the line segment being inspected. If there are a total of 240 utility poles in the transmission line, and each time 2 people are sent to climb the poles, what is the maximum number of inspections needed to ensure the fault location is identified under the optimal plan?
Please reason step by step, and put your final answer within \boxed{}.


Context:
James has 20 pairs of red socks and half as many black socks. He has a certain number of white socks as red and black combined. James has a total of 90 socks combined. What is the ratio of the number of white socks to the number of red and black socks combined?
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the least positive integer m such that, given m integers between 1 and 2023 inclusive, there must exist two of them a, b such that 1 < a/b <= 2? Express your answer as a single integer.
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Context:
In $\triangle ABC$, $\angle A = 30^{\circ}$, $BC = 13$. Given two circles $\gamma_1$ and $\gamma_2$ with radii $r_1$ and $r_2$, respectively, contain point $A$ and touch line $BC$ at points $B$ and $C$. Find the product $r_1r_2$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Emilia wishes to create a basic solution with $7 \%$ hydroxide $(\mathrm{OH})$ ions. She has three solutions of different bases available: $10 \%$ rubidium hydroxide $(\mathrm{Rb}(\mathrm{OH})$ ), $8 \%$ cesium hydroxide $(\mathrm{Cs}(\mathrm{OH})$ ), and $5 \%$ francium hydroxide $(\operatorname{Fr}(\mathrm{OH})$ ). ( $\mathrm{The} \mathrm{Rb}(\mathrm{OH})$ solution has both $10 \% \mathrm{Rb}$ ions and $10 \% \mathrm{OH}$ ions, and similar for the other solutions.) Since francium is highly radioactive, its concentration in the final solution should not exceed $2 \%$. What is the highest possible concentration of rubidium in her solution?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Phil has a coin collection he enjoys adding to. He began with 1,000 international coins his grandparents gave him. Over the next five years, his collection habits varied:

- In the first year, he managed to quadruple his collection.
- In the second year, he collected 7 coins each week.
- In the third year, he collected 3 coins every second day.
- In the fourth year, he found two rare coins every week and added them to his collection.
- However, during the fifth year, a portion of his collection was stolen, resulting in the loss of 40% of his coins.

Phil was not discouraged and continued to grow his collection over the next four years:

- In the sixth year, he added 5 coins to his collection every fourth day.
- In the seventh year, he got a job at a coin shop and added 20 coins to his collection each month.
- In the eighth year, he collected 10 coins per week.
- In the ninth year, unfortunately, Phil misplaced a third of his collection during a move to a new city.

How many coins did Phil have left after losing some in the ninth year?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the vectors $\overrightarrow{a} = (\cos 25^\circ, \sin 25^\circ)$, $\overrightarrow{b} = (\sin 20^\circ, \cos 20^\circ)$, and $\overrightarrow{u} = \overrightarrow{a} + t\overrightarrow{b}$, where $t\in\mathbb{R}$, find the minimum value of $|\overrightarrow{u}|$.
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Context:
The function f(x) is defined as f(x) = $$\frac{1}{3} \sin^4 x + \frac{1}{4} \cos^4 x$$. If f(θ) = $$\frac{1}{7}$$, find the value of tanθ.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The sets \( A \) and \( B \) are bounded subsets of the real line. The set \( A \) is the union of \( n \) disjoint and open intervals, and the set \( B \) is the union of \( m \) disjoint and open intervals. What is the maximum number of disjoint intervals that the set \( A \cap B \) can consist of?
Please reason step by step, and put your final answer within \boxed{}.


Context:
现安排 7 名同学去参加 5 个运动项目, 要求甲、乙两同学不能参加同一个项目，每个项目都有人参加，每人只参加一个项目，则满足上述要求的不同安排方案数为 $\qquad$。（用数字作答）
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Context:
Six children stand in a line outside their classroom. When they enter the classroom, they sit in a circle in random order. There are relatively prime positive integers $m$ and $n$ so that $\tfrac{m}{n}$ is the probability that no two children who stood next to each other in the line end up sitting next to each other in the circle. Find $m + n$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
# 8. Problem 8.10

Pentagon $A B C D E$ is circumscribed around a circle.

Angles $\angle A B C, \angle B A E, \angle C D E$ are each $104^{\circ}$. Find $\angle A D B$. Write the answer in degrees (the number only, without specifying the unit of measurement).

#
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Context:
34. In a triangular prism $A B C A_{1} B_{1} C_{1}{ }^{*}$, two planes are drawn: one passes through the vertices $A, B$ and $C_{1}$, and the other through the vertices $A_{1}, B_{1}$ and $C$. These planes divide the prism into four parts. The volume of the smallest of these parts is $V$. Find the volume of the prism.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A local college is offering German lessons and currently has 8 students enrolled. Through advertising, 12 more became interested but a fourth of these dropped out within a day. Meanwhile, 3 more students from a neighboring school joined the class, but 2 got frustrated and left within a week. The class then rallied to demonstrate how simple the course actually is and increased enrollment by exactly 1.5 times the students already enrolled. However, 4 of them eventually dropped out due to scheduling conflicts, and one week later, 7 more people enrolled.

In the following weeks, 8 students formed a study group outside of class, and as a result, tripled their enrollment from their current study group size. But, 15% of these students got overwhelmed by their workload, and decided to drop the German lessons. By the end of the session, one-third of all remaining students graduated. How many are still enrolled?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A furniture shop owner charges his customers a 20% markup on the cost price of the items. However, the customers are also given a special discount. If a customer purchases a computer table and a chair together, they will receive a 10% discount on the total price. A customer paid Rs. 8400 for a computer table and a chair after the discount. Determine the cost price of both the computer table and the chair before the discount was applied.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Factorize: $2a^{5}-8a=$____.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a chess tournament, 12 people participated. After the tournament concluded, each participant created 12 lists. The first list includes only themselves, the second one includes themselves and those they defeated, the third includes everyone from the second list and those they defeated, and so on. The 12th list includes everyone from the 11th list and those they defeated. It is known that for any participant in the tournament, there is a person in their 12th list who was not in their 11th list. How many drawn games were played in the tournament?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Two players are playing a game; one of them thinks of a set of integers \( x_{1}, x_{2}, \ldots, x_{n} \), where each \( x_i \) can be a single-digit positive or negative number. The second player is allowed to ask for the sum \( a_{1} x_{1} + \ldots + a_{n} x_{n} \), with any chosen set of coefficients \( \left(a_{1}, \ldots, a_{n}\right) \). What is the minimum number of questions needed for the guesser to find out the thought-of set?
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Context:
In an exotic country, the National Bank issues coins that can take any value in the interval  $[0, 1]$ . Find the smallest constant  $c > 0$  such that the following holds, no matter the situation in that country:

*Any citizen of the exotic country that has a finite number of coins, with a total value of no more than  $1000$ , can split those coins into  $100$  boxes, such that the total value inside each box is at most  $c$ .*
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Context:
Three planes intersect pairwise, and their three lines of intersection are parallel to each other. How many distinct parts does this divide the space into? Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Hay's Linens sells hand towels in sets of 23 and bath towels in sets of 29. If the store sold the same number of each set this morning and the total number of towels sold was between 500 and 700, what is the smallest number of each type of towel that the store could have sold?
Please reason step by step, and put your final answer within \boxed{}.


Context:
13. (BUL) ${ }^{\mathrm{IMO} 6}$ Determine the least possible value of $f(1998)$, where $f$ is a function from the set $\mathbb{N}$ of positive integers into itself such that for all $m, n \in \mathbb{N}$,
$$ f\left(n^{2} f(m)\right)=m[f(n)]^{2} $$
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Context:
carl is facing very difficult financial times and can only pay the interest on a $ 10,000 loan he has taken . the bank charges him a quarterly compound rate of 5 % . what is the approximate interest he pays annually ?
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Context:
设a为实数，n为任意给定的自然数，且n≥2，若不等式$1 + 2 ^ { x }$ $+ 3 ^ { x } + \cdots + ( n - 1 ) ^ { x } + n ^ { x } a > 0$.对(-∞,1]上的一切x都成立，则a的取值范围为·
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Context:
Sam invested $10,000 and earned a certain interest compounded for 3 years. He then invested more until he had three times as much invested. The next year, he got a 15% return on his investment. Sam now has $59,616. What was the interest rate on his initial investment?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Two different cubes of the same size are to be painted, with the color of each face being chosen independently and at random to be either black or white. Find the probability that after they are painted, the cubes can be rotated to be identical in appearance. The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.
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Context:
There are 5 pairs of socks in various colors and patterns: 2 red striped, 2 green polka-dotted, 2 blue checked, 2 yellow floral, and 2 purple plaid. A person must wear socks from different pairs, and the mismatched pair must not have the same color or pattern. How many unique mismatched pairs can be formed in this situation?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A number, when divided by 44, gives 432 as quotient and a certain remainder. When dividing the same number by 39, the remainder is 15. What is the remainder when the number is divided by 44?
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Context:
In the Cartesian coordinate system $xOy$, given a circle $C$: $x^{2}+(y-1)^{2}=4$, if one side $AB$ of the equilateral triangle $\triangle PAB$ is a chord of circle $C$, then the maximum value of $PC$ is.
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Context:
Find the smallest positive integer  $n$  with the property that in the set  $\{70, 71, 72,... 70 + n\}$  you can choose two different numbers whose product is the square of an integer.
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Context:
7. Determine the number of 8-digit positive integers such that after deleting any one digit, the remaining 7-digit number is divisible by 7 .
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Context:
The monkey has 100 bananas and its home is 50 meters away. The monkey can carry at most 50 bananas at a time and eats one banana for every meter walked. Calculate the maximum number of bananas the monkey can bring home.
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Context:
The edges of \( K_{2017} \) are each labeled with 1, 2, or 3 such that any triangle has a sum of labels of at least 5. Determine the minimum possible average of all \( \binom{2017}{2} \) labels.
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Context:
A boy rides his bicycle for a total of 60 km. He travels 15 km on flat ground with an average speed of 12 km/hr, then goes uphill for 20 km at an average speed of 8 km/hr. After reaching the top, he travels 10 km downhill at an average speed of 25 km/hr, and finally rides 15 km on flat ground again at an average speed of 18 km/hr. What is his overall average speed for the entire journey?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a circle of radius $R$, a chord $AB$ and a diameter $AC$ are drawn. Chord $PQ$, which is perpendicular to the diameter $AC$, intersects chord $AB$ at point $M$. Given that $AB = a$ and the ratio $PM : MQ = 3$, find $AM$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a bounded sequence \(\left\{a_{n}\right\}\), with
\[
a_{1}=a, \quad a_{n+1}=3 a_{n}^{3}-7 a_{n}^{2}+5 a_{n} \quad (n \in \mathbf{Z}_{+}),
\]
determine the range of values for \(a\).
Please reason step by step, and put your final answer within \boxed{}.

\boxed{0 \leqslant a \leqslant \frac{4}{3}}


Context:
In a novel that consists of 530 pages, each page number is printed once, starting from page 1 up to page 530. How many more 3's are printed than 7's throughout the book?
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Context:
Lyka wants to buy a smartphone worth $800 but she only has $200 at the moment. She also has to pay for her monthly $50 gym membership. She plans to save money for the remaining amount she needs for the smartphone and the gym membership over the next four months. However, Lyka expects a raise in her income during this period.

For the first two months, she can save $50 per week, and after the raise, for the next two months, she can save $80 per week.

Given the varying weekly savings and additional expenses, how much will she have saved by the end of the four months?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In an isosceles trapezoid with an acute angle $\alpha$ at the base, a circle constructed on the lateral side as a diameter touches the other lateral side.

In what ratio does it divide the larger base of the trapezoid?
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Context:
Point $D$ is on side $AC$ of triangle $ABC$, $\angle ABD=15^{\circ}$ and  $\angle DBC=50^{\circ}$. What is the measure of angle $BAD$, in degrees?

[asy]draw((-43,0)--(43,0)--(43,40)--cycle);
draw((-4,0)--(43,40));
draw((39,4)--(39,0));
draw((39,4)--(43,4));
draw((-1,36)--(22.5,26),Arrow);
label("$15^{\circ}$",(-1,36),W);
label("$50^{\circ}$",(36,30),S);
label("D",(-4,0),S);
label("A",(-43,0),W);
label("B",(43,40),N);
label("C",(43,0),E);
[/asy]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Mason is cleaning out all the junk in his attic. Initially, the attic had 20% useful items, 10% valuable heirlooms, and 70% junk. Since then, he has given away 4 useful items to his neighbors and has sold 20 valuable items online. Now, his attic has 16 useful items remaining. Considering the new distribution and the items he has given away or sold, how many junk items does his attic currently have?
Please reason step by step, and put your final answer within \boxed{}.


Context:
10 Let $x, y, z \in \mathbf{R}^{+}$, and satisfy $x y z + x + z = y$, find the maximum value of $p = \frac{2}{x^{2}+1} - \frac{2}{y^{2}+1} + \frac{3}{z^{2}+1}$.
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Context:
Given three points \(A, B, C\) forming a triangle with angles \(30^{\circ}\), \(45^{\circ}\), and \(105^{\circ}\). Two of these points are chosen, and the perpendicular bisector of the segment connecting them is drawn. The third point is then reflected across this perpendicular bisector to obtain a fourth point \(D\). This procedure is repeated with the resulting set of four points, where two points are chosen, the perpendicular bisector is drawn, and all points are reflected across it. What is the maximum number of distinct points that can be obtained as a result of repeatedly applying this procedure?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the minimum value for \(a, b > 0\) of the expression

$$
\frac{(3 a b - 6 b + a(1 - a))^{2} + \left(9 b^{2} + 2 a + 3 b(1 - a)\right)^{2}}{a^{2} + 9 b^{2}}
$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Fill the numbers $1, 2, \cdots, 9$ into a $3 \times 3$ grid such that each cell contains one number, each row's numbers increase from left to right, and each column's numbers decrease from top to bottom. How many different arrangements are possible? Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a certain football competition, a victory is worth 3 points, a draw is worth some points, and a defeat is worth 0 points. Each team plays 20 matches. A team scored 14 points after 5 games. The team needs to win at least 6 of the remaining matches to reach the 40-point mark by the end of the tournament. How many points is a draw worth?
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Context:
Art, Bella, Carla, and Dan each bake cookies of different shapes with the same thickness. Art's cookies are rectangles: 4 inches by 6 inches, while Bella's cookies are equilateral triangles with a side length of 5 inches. Carla's cookies are squares with a side length of 4 inches, and Dan's cookies are circles with a radius of 3 inches. If Art makes 18 cookies in one batch, how many cookies will be in one batch of Dan's cookies?
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Context:
Carl's typing speed varies throughout the week. On Monday, Wednesday, and Friday, he types at a speed of 50 words per minute for 5 hours with 10 minutes of rest after every hour. On Tuesday and Thursday, his typing speed decreases to 40 words per minute, and he types for 3 hours with 15 minutes of rest after every hour. On Saturday, he types at a speed of 60 words per minute for 4 hours with no breaks. On Sunday, he rests and does not type at all. Calculate the total number of words Carl types in a week.
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Context:
If $$I_1=\int _0 ^1 \frac {dx}{1+\sqrt {x }}, \hspace{5mm} I_2=\int _0 ^1 \frac {dx}{1+\frac {1}{1+\sqrt {x}}}, \hspace{5mm} I_3=\int _0 ^1 \frac {dx}{1+\frac {1}{1+\frac {1}{1+\sqrt {x}}}} $$ then $$ \lim _{n \to \infty} I_n \hspace{5mm}\text{ is equal to ?} $$
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Context:
On the number line, consider the point $x$ that corresponds to the value $10$. Consider $24$ distinct integer points $y_1, y_2, \ldots, y_{24}$ on the number line such that for all $k$ where $1 \leq k \leq 12$, we have $y_{2k-1}$ is the reflection of $y_{2k}$ across $x$. Find the minimum possible value of \[\sum_{n=1}^{24} (|y_n-1| + |y_n+1|).\]
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Context:
若 $\triangle A B C$ 的内角 $A, B, C$ 满足 $\sin A=\cos B=\tan C$, 求 $\cos ^{3} A+\cos ^{2} A-\cos A$ 的值.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:
Circle \(\omega\) is inscribed in rhombus \(H M_{1} M_{2} T\) so that \(\omega\) is tangent to \(\overline{H M_{1}}\) at \(A\), \(\overline{M_{1} M_{2}}\) at \(I\), \(\overline{M_{2} T}\) at \(M\), and \(\overline{T H}\) at \(E\). Given that the area of \(H M_{1} M_{2} T\) is 1440 and the area of \(E M T\) is 405, find the area of \(A I M E\).
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Context:
Calculate: $\sqrt{8}\div 2{2}^{\frac{1}{2}}=\_\_\_\_\_\_$.
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Context:
A polynomial $p(x)$ is called self-centered if it has integer coefficients and $p(100) = 100.$  If $p(x)$ is a self-centered polynomial, what is the maximum number of integer solutions $k$ to the equation $p(k) = k^3$?
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Context:
Given the equation (z+2i)(z-2i) = 2, solve for z. Express your answer in terms of i (imaginary unit), including the ± symbol if applicable.
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Context:
Given that $F\_1$ is the left focus of the hyperbola $C$: $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1 (a > 0, b > 0)$, point $B$ has coordinates $(0, b)$, and the line $F\_1B$ intersects with the two asymptotes of hyperbola $C$ at points $P$ and $Q$. If $\overrightarrow{QP} = 4\overrightarrow{PF\_1}$, find the eccentricity of the hyperbola $C$.
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Context:
5. The maximum value of the orthogonal projection area of a regular tetrahedron with edge length 1 on a horizontal plane is $\qquad$
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Context:
If $n$ is an odd number, calculate the remainder of $7^n+C_n^17^{n-1}+C_n^27^{n-2}+\ldots+C_n^{n-1}7$ when divided by 9. Express your answer as a single integer.
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Context:
There are $2024$ people, consisting of knights and liars, and some of them are friends. Each person was asked how many friends they have, and the possible answers were $0, 1, \ldots, 2023$. Every knight answered truthfully, while every liar altered the real answer by exactly $1$. What is the minimal number of liars?
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Context:
On an $m × n$ rectangular piece of graph paper there are several squares whose sides are on the vertical and horizontal lines of the paper. It is known that no two squares coincide and no square is situated inside another one. What is the maximal number of such squares? Please provide the value when $m=3,n=4$.
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Context:
Let  $\Omega$  be a circle centered at  $O$ . Let  $ABCD$  be a quadrilateral inscribed in  $\Omega$ , such that  $AB = 12$ ,  $AD = 18$ , and  $AC$  is perpendicular to  $BD$ . The circumcircle of  $AOC$  intersects ray  $DB$  past  $B$  at  $P$ . Given that  $\angle PAD = 90^\circ$ , find  $BD^2$ .
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Context:
Find the largest prime number \( p \) such that there exist positive integers \( a \) and \( b \) satisfying
\[ p = \frac{b}{2} \sqrt{\frac{a - b}{a + b}} .\]
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Context:
Two circles touch internally. A line passing through the center of the larger circle intersects it at points \( A \) and \( D \), and intersects the smaller circle at points \( B \) and \( C \). Find the ratio of the radii of the circles, given that \( AB : BC : CD = 3 : 7 : 2 \).
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Context:
Two circles intersect at the points  $C$  and  $D$ . The straight lines  $CD$  and  $BYXA$  intersect at the point  $Z$ . Moreever, the straight line  $WB$  is tangent to both of the circles. Suppose  $ZX=ZY$  and  $AB\cdot AX=100$ . Find the value of  $BW$ .
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Context:
Let point P be a moving point on the curve $y=x^2$, and let the tangent line to the curve at point P be $l$. A line passing through point P and perpendicular to line $l$ intersects the curve $y=x^2$ at another point Q. The minimum value of PQ is \_\_\_\_\_\_.
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Context:
Let \( p \) be an odd prime number. How many \( p \)-element subsets \( A \) of \(\{1, 2, \ldots, 2p\}\) are there, the sum of whose elements is divisible by \( p \)?
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Context:
## Task A-3.3.

On a board of dimensions $20 \times 19$, tiles of dimensions $3 \times 1$ are placed such that they cover exactly three squares of the board, and they do not overlap or touch each other, not even at the corners.

Determine the maximum possible number of $3 \times 1$ tiles on this board.
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Context:
Given an arithmetic-geometric sequence ${a_{n}}$ with the sum of the first $n$ terms denoted as $S_{n}$, and $S_{2}=1$, $S_{4}=3$, find the value of $S_{6}$. Express your answer as a single integer.
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Context:
Let $A B C$ be a triangle with $A B=5, B C=8$, and $C A=7$. Let $\Gamma$ be a circle internally tangent to the circumcircle of $A B C$ at $A$ which is also tangent to segment $B C. \Gamma$ intersects $A B$ and $A C$ at points $D$ and $E$, respectively. Determine the length of segment $D E$.
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Context:
For a set $S \subseteq \mathbb{N}$, define $f(S) = \{\lceil \sqrt{s} \rceil \mid s \in S\}$. Find the number of sets $T$ such that $|f(T)| = 2$ and $f(f(T)) = \{2\}$.
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Context:
How many 3-digit positive integers have the property that the sum of their digits is greater than the product of their digits?
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Context:
At a killers' convention, 1000 participants attended, each receiving a registration number from 1 to 1000. By the end of the convention, it turned out that all the killers except for number 1 were dead. It is known that each killer could only kill killers with larger numbers, and the number of their victims could not exceed their own number. What is the minimum number of killers who could have participated in the killings at the convention?
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Context:
Evaluate the integral $ \int_{-\infty}^{\infty}\frac{x^{2}}{\cosh\left(x\right)}\,{\rm d}x $ without using the change of variables $x = \log(t)$.
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Context:
Among the coordinates  $(x,y)$   $(1\leq x,y\leq 101)$ , choose some points such that there does not exist  $4$  points which form a isoceles trapezium with its base parallel to either the  $x$  or  $y$  axis(including rectangles). Find the maximum number of coordinate points that can be chosen.
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Context:
A point is taken arbitrarily inside a triangle, and lines parallel to the sides of the triangle are drawn through this point. These lines divide the area of the triangle into six parts. What is the area of the given triangle if the areas of the three resulting triangles are given as: $t_{1}, t_{2}, t_{3}$?
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Context:
9. Use the digits $1 \sim 9$ to form three three-digit numbers, each digit used only once, and all three numbers are divisible by 4. Then the largest of these three numbers is at least $\qquad$ .
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Context:
For real numbers $x$ and $y$, a new operation is defined as follows: $x*y = ax + by - 1$, where $a$ and $b$ are constants. If $1*2=4$ and $\left(-2\right)*3=10$, then $a*b = \_\_\_\_\_\_$.
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Context:
In a table with $88$ rows and $253$ columns, each cell is colored either purple or yellow. Suppose that for each yellow cell $c$, the following condition holds:
$$x(c) \cdot y(c) \geq 184,$$
where $x(c)$ is the number of purple cells in the same row as $c$, and $y(c)$ is the number of purple cells in the same column as $c$.
Find the least possible number of cells that are colored purple.
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Context:
An triangle with coordinates $(x_1,y_1)$, $(x_2, y_2)$, $(x_3,y_3)$ has centroid at $(1,1)$. The ratio between the lengths of the sides of the triangle is $3:4:5$. Given that \[x_1^3+x_2^3+x_3^3=3x_1x_2x_3+20\] and \[y_1^3+y_2^3+y_3^3=3y_1y_2y_3+21,\] the area of the triangle can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$?
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Context:
I'm sorry, I misunderstood your previous message. Here's a more difficult version of the problem without generating an answer:

One night, 27 percent of the female officers on a police force were on duty. If 275 police officers were on duty that night and the ratio of male to female officers was 4:3, how many female officers were on the police force?
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Context:
$ABCD$  is a rectangle with  $AB = CD = 2$ . A circle centered at  $O$  is tangent to  $BC$ ,  $CD$ , and  $AD$  (and hence has radius  $1$ ). Another circle, centered at  $P$ , is tangent to circle  $O$  at point  $T$  and is also tangent to  $AB$  and  $BC$ . If line  $AT$  is tangent to both circles at  $T$ , find the radius of circle  $P$ .
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Context:
In a recent election, there were three candidates: Geoff, Laura and Jim. Geoff received 0.5 percent of the 6,000 votes cast, and Laura got twice as many votes as Geoff. The remaining votes went to Jim. To win the election, a candidate needed to receive more votes than any other. If Geoff needed exactly 3,000 more votes to win the election, what is the minimum percentage of total votes Jim had to have to guarantee his victory?
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Context:
$A_1, A_2, \ldots, A_n$ are subsets of a set $S$ with $|S| = 2019$ such that the union of any three of them equals $S$, but the union of any two does not equal $S$. Find the maximum value of $n$.
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Context:
Toner Drum and Celery Hilton are both running for president. A total of $2015$ people cast their vote, giving $60\%$ to Toner Drum. Let $N$ be the number of "representative" sets of the $2015$ voters that could have been polled to correctly predict the winner of the election (i.e., more people in the set voted for Drum than Hilton). Compute the remainder when $N$ is divided by $2017$.
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Context:
The distance from \(A\) to \(B\) is 999 km. Along the road, there are kilometer markers indicating the distances to \(A\) and \(B\): 0।999, 1।998, \(\ldots, 999।0. How many of these markers have only two different digits?
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Context:
How many ring homomorphisms are there from $\Bbb Z \times \Bbb Z\times \Bbb Z$ to $\Bbb Z$?
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Context:
Determine all possible values of the parameter \(a\) such that for all \(x\) and \(y\), the inequality \(2a x^{2} + 2a y^{2} + 4a x y - 2 x y - y^{2} - 2 x + 1 \geq 0\) holds.
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Context:
Find the largest six-digit number in which all digits are distinct, and each digit, except the first and last ones, is either the sum or the difference of its neighboring digits.
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Context:
$\frac{n(n + 1)}{2}$ distinct numbers are arranged at random into $n$ rows. The first row has $1$ number, the second has $2$ numbers, the third has $3$ numbers and so on. Find the probability that the largest number in each row is smaller than the largest number in each row with more numbers.
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Context:
Let $S(N)$ be the number of 1's in the binary representation of an integer $N$, and let $D(N) = S(N + 1) - S(N)$. Compute the sum of $D(N)$ over all $N$ such that $1 \le N \le 2017$ and $D(N) < 0$.
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Context:
Find the value of the algebraic expression $\frac{3 \sqrt{\sqrt{7+\sqrt{48}}}}{2(\sqrt{2}+\sqrt{6})}$.
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Context:
A chess piece called "the four-liner" attacks two vertical and two horizontal squares adjacent to the square on which it stands. What is the maximum number of non-attacking four-liners that can be placed on a $10 \times 10$ board?
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Context:
Let function $G(n)$ denote the number of solutions to the equation $\cos x = \sin nx$ on the interval $[0, 2\pi]$. For each integer $n$ greater than 2, what is the sum $\sum_{n=3}^{100} G(n)$?
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Context:
In the derivation of Von Mangoldt's explicit formula, what is the residue of the integrand at $s=0$?
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Context:
The Binomial Expansion is valid for exponents that are not integers. That is, for all real numbers $x,y$ and $r$ with $|x|>|y|$,
\[(x+y)^r=x^r+rx^{r-1}y+\dfrac{r(r-1)}{2}x^{r-2}y^2+\dfrac{r(r-1)(r-2)}{3!}x^{r-3}y^3 \cdots\]
 What are the first three digits to the right of the decimal point in the decimal representation of $(10^{2002}+1)^{\frac{10}{7}}$?
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Context:
Calculate the limit

$$
\lim _{x \rightarrow 0}\left(\frac{1+x^{2} 2^{x}}{1+x^{2} 5^{x}}\right)^{1 / \sin ^{3} x}
$$
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Context:
In set theory, consider set S = {1, 2, 3, 4, 5} and subset A ⊆ S × S defined as A = {(1, 1), (1, 4), (2, 1), (2, 2), (2, 4), (3, 5), (4, 3), (4, 4), (5, 2)}. Given that B ⊆ S × S is a transitive relation such that A ⊆ B, what is the smallest possible cardinality of B?
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Context:
In the park, there are 10,000 trees planted in a square grid (100 rows of 100 trees each). What is the maximum number of trees that can be cut down so that the following condition is met: if you stand on any stump, you will not see any other stump? (The trees can be considered thin enough.)

#
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Context:
A mischievous child mounted the hour hand on the minute hand's axle and the minute hand on the hour hand's axle of a correctly functioning clock. The question is, how many times within a day does this clock display the correct time?
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Context:
We are given a triangle  $ABC$ . Points  $D$  and  $E$  on the line  $AB$  are such that  $AD=AC$  and  $BE=BC$ , with the arrangment of points  $D - A - B - E$ . The circumscribed circles of the triangles  $DBC$  and  $EAC$  meet again at the point  $X\neq C$ , and the circumscribed circles of the triangles  $DEC$  and  $ABC$  meet again at the point  $Y\neq C$ . Find the measure of  $\angle ACB$  given the condition  $DY+EY=2XY$ .
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Context:
Evaluate the limit $\displaystyle \lim\limits_{n\to\infty}\frac{I_n}{n^n}$, where the sequence $I_n$ is defined as $I_n = \int_0^n x^n\arctan\left(x^\frac{1}{n}\right)\,dx$ for $n \ge 2$.
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Context:
Given two points O(0,0) and Q(a,b), point P1 is the midpoint of the line segment OQ, point P2 is the midpoint of the line segment QP1, P3 is the midpoint of the line segment P1P2, and so on, Pn+2 is the midpoint of the line segment PnPn+1. Find the limit position of point Pn as n approaches infinity. Express your answer as a coordinate pair (x, y).
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\boxed{ \left( \frac{2a}{3}, \frac{2b}{3} \right) }


Context:
Denmark wants to order pizza.  For toppings, he has 3 cheese, 4 meat and 5 vegetable options, one of which is peppers. He can have only one selection from each topping category (one cheese, one meat and one vegetable). However, if he chooses to have pepperoni, he cannot have peppers. How many topping combinations does he have total?
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Context:
How can one solve $$\int_0^\infty \frac{1}{1-t^4} \, dt$$?
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Context:
Two chords \(AB\) and \(CD\) of a circle with center \(O\) each have a length of 10. The extensions of segments \(BA\) and \(CD\) beyond points \(A\) and \(D\) respectively intersect at point \(P\), with \(DP = 3\). The line \(PO\) intersects segment \(AC\) at point \(L\). Find the ratio \(AL : LC\).
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Context:
5 Real numbers $a_{1}, a_{2}, \cdots, a_{n}(n>3)$ satisfy: $a_{1}+a_{2}+\cdots+a_{n} \geqslant n$, and $a_{1}^{2}+a_{2}^{2}+\cdots+$ $a_{n}^{2} \geqslant n^{2}$. Find the minimum value of $\max \left\{a_{1}, a_{2}, \cdots, a_{n}\right\}$. (28th United States of America Mathematical Olympiad)
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Context:
In a plane, there are 7 points, with no three points being collinear. If 18 line segments are connected between these 7 points, then at most how many triangles can these segments form?
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Context:
If integers $m$, $n$, and $k$ satisfy the equation $m^2 + n^2 + 1 = kmn$, what values can $k$ have?
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Context:
There are three equal-radius, tangent circles. What is the area of the lobe between the circles if the radius of each circle is $r$?
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Context:
Given vectors $\overrightarrow {m}=(a,-1)$, $\overrightarrow {n}=(2b-1,3)$ where $a > 0$ and $b > 0$. If $\overrightarrow {m}$ is parallel to $\overrightarrow {n}$, determine the value of $\dfrac{2}{a}+\dfrac{1}{b}$.
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Context:
30 students participated in the mathematical Olympiad. Each of them was given 8 problems to solve. The jury estimated their work with the following rule:
1) Each problem was worth \( k \) points, if it wasn't solved by exactly \( k \) students;
2) Each student received the maximum possible points in each problem or got 0 in it.

Lasha got the least number of points. What's the maximal number of points he could have?

**Remark:** Rule 1 means that if the problem was solved by exactly \( k \) students, then each of them got \( 30 - k \) points in it.
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Context:
Four congruent semicircles are drawn on the surface of a sphere with radius $2$, as shown, creating a close curve that divides the surface into two congruent regions. The length of the curve is $\pi\sqrt{n}$. What is $n$?
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Context:
In how many ways can $n$ square envelopes of different sizes be arranged by inclusion? For instance, with $6$ envelopes $A, B, C, D, E, F$ (listed in decreasing order of their sizes), one way of arranging them would be $F \in C \in B$, $E \in B, D \in A$, where $I \in J$ means envelope $I$ is placed in envelope $J$.
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Context:
Given a function $f(x)=e^{x}+ax-3$ where $a\in \mathbb{R}$, if for any $x_{1}$, $x_{2}\in [1,+\infty)$ and $x_{1} < x_{2}$, the inequality $x_{2}f(x_{1})-x_{1}f(x_{2}) < a(x_{1}-x_{2})$ holds, then the range of values for $a$ is ______.
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Context:
Exercise 6. In 5 boxes are found respectively $402, 403, 404, 405$ and 406 stones. The only allowed operation is to take 4 stones from a pile that has at least 4 stones and to put one in each of the other piles. What is the maximum number of stones that can be in a single pile?
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Context:
5. Points $M, N, P$ are the feet of the altitudes dropped from the vertices of triangle $A B C$ with angles $37.5^{0}, 67.5^{0}, 75^{0}$ to its sides. Find the ratio of the areas of triangles $M N P$ and $A B C$.
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Context:
At the end of the term, Bob wrote his current grades in singing in a row and placed multiplication signs between some of them. The product of the resulting numbers turned out to be 2007. What grade does Bob get in singing for the term? (The music teacher does not give "fails".)
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Context:
Let $ABC$ be a triangle with circumcenter $O$, incenter $I, \angle B=45^{\circ}$, and $OI \parallel BC$. Find $\cos \angle C$.
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Context:
In the Cartesian coordinate system $xOy$, point $P$ is a common point of the graph of the function $y=2\ln x$ and the circle $M$: $(x-3)^{2}+y^{2}=r^{2}$, and they have a common tangent line at point $P$. If the graph of the quadratic function $y=f(x)$ passes through points $O$, $P$, and $M$, then the maximum value of $y=f(x)$ is \_\_\_\_\_\_.
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Context:
The average of the numbers $3$, $4$, $5$, $6$, $a$, $10$, $11$ is $8$. What is the $60$th percentile of this set of data?
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Context:
There are 51 students taking a test with 6 questions, and the scoring rules are as follows: each person starts with 6 points, with 4 points added for each correct answer, 1 point deducted for each incorrect answer, and no change in points for unanswered questions. What is the minimum number of students that must have the same score? Provide your answer as a whole number.
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Context:
Shvecov D.V.

On the hypotenuse $AC$ of the right triangle $ABC$, a point $C_{1}$ is marked such that $BC = CC_{1}$. Then, on the leg $AB$, a point $C_{2}$ is marked such that

$AC_{2} = AC_{1}$; the point $A_{2}$ is defined similarly. Find the angle $AMC$, where $M$ is the midpoint of the segment $A_{2}C_{2}$.
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Context:
If a number is selected at random from the set of all five-digit numbers in which the sum of the digits is equal to 43, what is the probability that this number will be divisible by 11?
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Context:
Problem 9.7. In front of a sweet-tooth lie five boxes of candies: the first box contains 11 candies, the second - 22 candies, the third - 33 candies, the fourth - 44 candies, the fifth - 55 candies. In one move, the sweet-tooth can take four candies from one box and distribute them, one candy to each of the remaining four boxes.

At any point, the sweet-tooth can take the candies from any box and leave. What is the maximum number of candies he can take?
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Context:
Let \(a, b\) be nonzero complex numbers, and \(\frac{a}{b}\) is not a real number. Define:
\[
L_{a, b}=\{r a+s b \mid r, s \in \mathbf{Z}\},
\]
\[
R_{a, b}=\left\{z \mid z \text{ is a nonzero complex number, and } L_{a, b} = L_{z a, z}\right\}.
\]

Find the maximum number of elements in the set \(R_{a, b}\) as \(a, b\) change.
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Context:
The alphabet contains \( n > 1 \) letters; a word is any finite sequence of letters in which any two neighboring letters are different. A word is called good if it is not possible to delete all letters except for four such that the remaining sequence is of the form \( a a b b \), where \( a \) and \( b \) are different letters. Find the maximum possible number of letters in a good word.

(D. Khramiov)
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Context:
A pen and its ink refill together cost $\;\$1.10$. The pen costs $\;\$1$ more than the ink refill. What is the cost of the pen in dollars?
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Context:
设 A 为数集 \{1,2,3,\cdots,2024\} 的 n 元子集，且 A 中的任意两个数既不互素又不存在整除关系. 求 n 的最大值.
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Context:
The pentagon \(ABCDE\) is inscribed around a circle. The angles at its vertices \(A\), \(C\), and \(E\) are \(100^{\circ}\). Find the angle \(ACE\).
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Context:
Company C sells a line of 40 products with an average retail price of $1,800. If none of these products sells for less than $500, and exactly 15 of the products sell for less than $1,400, what is the greatest possible selling price of the most expensive product?
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Context:
1st Irish 1988 Problem 5 Let X = {1, 2, 3, 4, 5, 6, 7}. How many 7-tuples (X 1 , X 2 , X 3 , X 4 , X 5 , X 6 , X 7 ) are there such that each X i is a different subset of X with three elements and the union of the X i is X?
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Context:
Eleven best football teams played one match against each other. Each team scored 1 goal in the first match, 2 goals in the second match, up to 10 goals in the tenth match. What is the maximum number of drawn matches that could have occurred?
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Context:
In the sequence $\{a_n\}$, we have $a_1=1$, and for $n \geqslant 2$, $a_n=a_{n-1}+n$. If the inequality $\frac{λ}{n} > \frac{n+1}{a_n+1}$ is satisfied for any $n \in \mathbb{N}^{*}$, then the range of the real number $λ$ is __________.
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Context:
What is the volume of the region above the cone $z= \sqrt{x^2+y^2}$ and bounded by the spheres $𝑥^2+y^2+𝑧^2=1$ and $𝑥^2+y^2+𝑧^2=4$?
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Context:
There are  $20$  geese numbered  $1-20$  standing in a line. The even numbered geese are standing at the front in the order  $2,4,\dots,20,$  where  $2$  is at the front of the line. Then the odd numbered geese are standing behind them in the order,  $1,3,5,\dots ,19,$  where  $19$  is at the end of the line. The geese want to rearrange themselves in order, so that they are ordered  $1,2,\dots,20$  (1 is at the front), and they do this by successively swapping two adjacent geese. What is the minimum number of swaps required to achieve this formation?

*Author: Ray Li*
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Context:
Given the parabolic equation \( y^2 = 2px \) with \( p > 0 \) and focus \( F \), a line passing through \( F \) and having an inclination angle \( \theta \) intersects the parabola at points \( A \) and \( B \). Find the maximum area of the triangle \( \triangle ABO \) (where \( O \) is the origin).
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Context:
There are 7 clean sheets of paper on a table. Every minute, Vasya selects any 4 out of them and draws one star on each of the chosen sheets. Vasya wants each sheet to end up with a different number of stars (meaning no two sheets have the same number of stars). What is the minimum total number of stars Vasya will have to draw to achieve this?
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Context:
Given that $|\overrightarrow{a}|=5$, $|\overrightarrow{b}|=3$, and $\overrightarrow{a} \cdot \overrightarrow{b} = -12$, find the projection of vector $\overrightarrow{a}$ on vector $\overrightarrow{b}$.
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Context:
A train of some length moving at a speed of 72 kmph takes 91.9926405887529 seconds to cross another train of length 540 m, moving at 36 kmph in the same direction. What is the length of the first train?
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Context:
For every positive integer $n$, the function $f$ is defined as follows:
- $f(2n+1) = 2f(2n)$
- $f(2n) = f(2n-1) + 1$
- $f(1) = 0$

Determine the remainder when $f(2005)$ is divided by $5$.
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Context:
All the vertices of a regular pyramid $P A B C D$ lie on the lateral surface of a cylinder, the axis of which is perpendicular to the plane $P A B$. Find the radius of the base of the cylinder if $A B = a$.
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Context:
Evaluate the line integral $\displaystyle \int _C\dfrac{1+2z}{1+z}dz$ where $C$ is the smooth curve lying in $\Omega=\{z\in \mathbb C:Im z>0\}$ with initial point $-1+2i$ and final point $1+2i$, given that the Residue Theorem cannot be applied.
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Context:
Let $a, b, c$ be distinct positive integers such that $b+c-a$, $c+a-b$, and $a+b-c$ are all perfect squares. What is the largest possible value of $a+b+c$ smaller than 100?
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Context:
$7 \cdot 32$ Find the smallest natural number $n$, such that every convex 100-gon can be represented as the intersection of $n$ triangles.
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Context:
Determine the number of irreducible submodules isomorphic to $V_4$ in an $\mathfrak{sl}(2,\mathbb{C})$-module $V$ given that the eigenvalues of the action $h$ are 6, 4, 2, 0, -2, -4, and -6.
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Context:
Positive integers $a$, $b$, and $c$ are such that $a<b<c$. Consider the system of equations
\[
2x + y = 2022 \quad \text{and} \quad y = |x-a| + |x-b| + |x-c|
\]
Determine the minimum value of $c$ such that the system has exactly one solution.
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Context:
An assembly line produces, on average, 85% first grade products. How many products need to be sampled so that, with a probability of 0.997, the deviation of the proportion of first grade products from 0.85 in absolute value does not exceed 0.01?
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Context:
设 $n(n \geqslant 3)$ 是正整数，集合 $M=$ $\{1,2, \cdots, 2 n\}$ 。求最小的正整数 $k$, 使得对于 $M$ 的任何一个 $k$ 元子集，其中必有 $4$ 个互不相同的元素之和等于 $4 n+1$ 。请提供当 n = 48 时的 k 的值。
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Context:
According to relevant regulations, the behavior of a motor vehicle driver with a blood alcohol content of 20 mg/100 ml or more is considered drunk driving. Suppose that after drinking, the blood alcohol content is $p_0$ mg/100 ml, and after $x$ hours, the alcohol content drops to $p$ mg/100 ml, satisfying the relationship $$p=p_{0}\cdot e^{rx}$$ (where $r$ is a constant). If a person's blood alcohol content is 89 mg/100 ml after drinking, and 61 mg/100 ml 2 hours later, then this person can drive after \_\_\_\_\_\_ hours. (Round up to the nearest hour)
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Context:
Given that $$32+ \frac {2}{7} = 3 \frac {2}{7}$$, $$33+ \frac {3}{26} = 3 \frac {3}{26}$$, $$34+ \frac {4}{63} = 3 \frac {4}{63}$$, ..., $$32014+ \frac {m}{n} = 2014 \cdot 3 \frac {m}{n}$$, ..., $$32016+ \frac {a}{b} = 2016 \cdot 3 \frac {a}{b}$$, then find the value of $$\frac {b+1}{a^{2}}$$.
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Context:
There are prime numbers $a$, $b$, and $c$ such that the system of equations
$$a \cdot x - 3  \cdot y + 6  \cdot z = 8$$
$$b  \cdot x + 3\frac12  \cdot y + 2\frac13  \cdot z = -28$$
$$c  \cdot x - 5\frac12  \cdot y + 18\frac13  \cdot z = 0$$
has infinitely many solutions for $(x, y, z)$. Find the product $a \cdot b \cdot c$.
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Context:
Ali and Veli go hunting. The probability that each will successfully hit a duck is 1/2 on any given shot. During the hunt, Ali shoots 12 times, and Veli shoots 13 times. Calculate the probability that Veli hits more ducks than Ali. Express your answer as a simplified fraction.
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Context:
Let $f: \mathbb{N} \to \mathbb{R}$ be the function \[ f(n) = \sum_{k=1}^{\infty} \frac{1}{\operatorname{lcm}(k,n)^2}. \] It is well-known that $f(1) = \frac{\pi^2}{6}$. What is the smallest positive integer $m$ such that $m \cdot f(10)$ is the square of a rational multiple of $\pi$?
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Context:
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $C=60^\circ$, $b=\sqrt{6}$, and $c=3$, find the measure of angle $A$.
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Context:
Consider a circle with a radius of 5 cm. If we draw a rectangle of maximum size inside the circle and a parallelogram of maximum size tangent to the circle from outside such that the longer sides of the parallelogram are tangent to the circle and the vertices of the rectangles are touching the shorter sides of the parallelogram, what is the sum of the areas of the rectangle and the parallelogram?
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Context:
On the board, the number 0 is written. Two players take turns appending to the expression on the board: the first player appends a + or - sign, and the second player appends one of the natural numbers from 1 to 1993. The players make 1993 moves each, and the second player uses each of the numbers from 1 to 1993 exactly once. At the end of the game, the second player receives a reward equal to the absolute value of the algebraic sum written on the board. What is the maximum reward the second player can guarantee for themselves?
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Context:
When $x=$____, the expressions $\frac{x-1}{2}$ and $\frac{x-2}{3}$ are opposite in sign.
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Context:
If 6 women take an hour to dig a ditch, then how long should 18 women take to dig a ditch of the same type?
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Context:
What is the $33$-rd number after the decimal point of $(\sqrt{10} + 3)^{2001}$?
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Context:
Given the function f(x) = ax^2 + bx + 3a + b, which is an even function with a domain of [2a, a-1]. Find the value of a+b.
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Context:
Find the last non-zero digit of the product of all natural numbers from 1 to 40.
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Context:
What is the degree of the field extension $K=\mathbb{Q}(\alpha\zeta)$ over $\mathbb{Q}$, where $\alpha =2^{1/5}$ and $\zeta=e^{2i\pi/5}$?
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Context:
If \( x + k \) is a factor of \( 3x^2 + 14x + a \), find \( k \). (\( k \) is an integer.)
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Context:
A rectangle, a square, and an equilateral triangle all have the same perimeter. The length of the side of the square is 9 cm. Find the length of the shorter sides of the rectangle. Express your answer in centimeters.
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Context:
In three-dimensional space, let $S$ be the region of points $(x, y, z)$ satisfying $-1 \leq z \leq 1$. Let $S_{1}, S_{2}, \ldots, S_{2022}$ be 2022 independent random rotations of $S$ about the origin ( $0,0,0$). The expected volume of the region $S_{1} \cap S_{2} \cap \cdots \cap S_{2022}$ can be expressed as $\frac{a \pi}{b}$, for relatively prime positive integers $a$ and $b$. Compute $100 a+b$.
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Context:
A classroom consists of a $5 \times 5$ array of desks, to be filled by anywhere from 0 to 25 students, inclusive. No student will sit at a desk unless either all other desks in its row or all others in its column are filled (or both). Considering only the set of desks that are occupied (and not which student sits at each desk), how many possible arrangements are there?
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Context:
A motorist traveled from point A to point $\mathrm{D}$, which are 100 km apart. The road from A to D passes through points B and C. At point B, the navigator showed that 30 minutes remained to reach the destination, and the motorist immediately reduced his speed by 10 km/h. At point C, the navigator showed that 20 km remained, and the motorist reduced his speed by another 10 km/h. (The navigator determines the remaining time based on the current speed of travel.) Determine the initial speed of the car, given that the travel time from B to C was 5 minutes longer than the travel time from C to D.
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Context:
6.1. We will call a natural number interesting if all its digits, except the first and last, are less than the arithmetic mean of the two adjacent digits. Find the largest interesting number.
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Context:
In a race, A covers the distance in 20 seconds and B in 25 seconds. A beats B by 20 meters. What is the total distance of the race?
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Context:
Given the hyperbola $C:\frac{{{x^2}}}{{{a^2}}}-\frac{{{y^2}}}{{27}}=1({a＞0})$ with left and right foci $F_{1}$ and $F_{2}$ respectively, and one of its asymptotes has a slope of $\frac{π}{3}$. If point $P$ lies on the hyperbola and $|PF_{1}|=7$, then $|PF_{2}|=$____.
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Context:
What is the inverse of the polynomial $2x^2+2$ in the finite field $\mathbb{Z}_3[x]/(x^3+2x^2+2)$?
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Context:
Let $\alpha$ be a real number, and consider the sequence $(a_n)_n$ defined by $a_0 = 1$ and $a_n = \alpha(\alpha-1)\cdots(\alpha - (n-1))$ for $n \geq 1$. Determine the exponential generating function (EGF) for this sequence.
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Context:
If the range of \( y = \log_{2016} (x^2 - ax + 65) \) is \( \mathbb{R}^{+} \), then the range of the value of \( a \) is ________.
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Context:
Place the numbers 1, 2, 3, 4, 2, 6, 7, and 8 at the eight vertices of a cube such that the sum of the numbers on any given face is at least 10. Determine the minimum possible sum of the numbers on each face.
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Context:
The Greene family, composed of Mr. and Mrs. Greene and their three children, went to the amusement park. They spent a total of $45 on discounted admission tickets, which originally cost $50, and spent $13 less than the original ticket cost on food and beverages. Mr. Greene spent an additional $15 on a souvenir for himself, while Mrs. Greene spent double that amount on souvenirs for their children. Each child played games costing $9 each, and the family paid $25 for round-trip transportation. The tax applied on all their purchases, except transportation and admission tickets, was 8%. Calculate the total amount the Greene family spent on their trip to the amusement park after taking into account all expenditures, discounts, and taxes.
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Context:
A cube with side length 2 is cut three times to form various polyhedral pieces. It is first cut through the diagonal planes intersecting at vertex W, then an additional cut is made that passes through the midpoint of one side of the cube and perpendicular to its adjacent face, effectively creating 16 pieces. Consider the piece containing vertex W, which now forms a pyramid with a triangular base. What is the volume of this pyramid?
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Context:
Given skew lines l₁ and l₂ with direction vectors $\overrightarrow {a}$ = (-1, 1, 0) and $\overrightarrow {b}$ = (1, 0, -1) respectively, find the size of the angle formed by skew lines l₁ and l₂.
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