problem stringlengths 10 5.15k | answer stringlengths 0 1.23k |
|---|---|
For positive integers $n$, let $L(n)$ be the largest factor of $n$ other than $n$ itself. Determine the number of ordered pairs of composite positive integers $(m, n)$ for which $L(m) L(n)=80$. | 12 |
Calculate $(-1)^{53} + 3^{(2^3 + 5^2 - 4!)}$. | 19682 |
A tournament among 2021 ranked teams is played over 2020 rounds. In each round, two teams are selected uniformly at random among all remaining teams to play against each other. The better ranked team always wins, and the worse ranked team is eliminated. Let $p$ be the probability that the second best ranked team is eli... | 674 |
Evaluate $\left|{-4+\frac{7}{6}i}\right|$. | \frac{25}{6} |
Find all the solutions to
\[\frac{1}{x^2 + 11x - 8} + \frac{1}{x^2 + 2x - 8} + \frac{1}{x^2 - 13x - 8} = 0.\]Enter all the solutions, separated by commas. | 8,1,-1,-8 |
Let \( f(x) = x^2 + px + q \). It is known that the inequality \( |f(x)| > \frac{1}{2} \) has no solutions on the interval \([2, 4]\). Find \( \underbrace{f(f(\ldots f}_{2017}\left(\frac{5-\sqrt{11}}{2}\right)) \ldots) \). Round your answer to the nearest hundredth if necessary. | 4.16 |
Let $\omega$ be a circle, and let $ABCD$ be a quadrilateral inscribed in $\omega$. Suppose that $BD$ and $AC$ intersect at a point $E$. The tangent to $\omega$ at $B$ meets line $AC$ at a point $F$, so that $C$ lies between $E$ and $F$. Given that $AE=6, EC=4, BE=2$, and $BF=12$, find $DA$. | 2 \sqrt{42} |
Find \[\left|\left(\frac 23+\frac 56i\right)^8\right|\] | \frac{2825761}{1679616} |
A flagpole is originally $5$ meters tall. A hurricane snaps the flagpole at a point $x$ meters above the ground so that the upper part, still attached to the stump, touches the ground $1$ meter away from the base. What is $x$? | 2.4 |
Given a right prism $ABC-A_{1}B_{1}C_{1}$, where $AB=3$, $AC=4$, and $AB \perp AC$, $AA_{1}=2$, find the sum of the surface areas of the inscribed sphere and the circumscribed sphere of the prism. | 33\pi |
The base of a triangle is 20; the medians drawn to the lateral sides are 18 and 24. Find the area of the triangle. | 288 |
Suppose we flip four coins simultaneously: a penny, a nickel, a dime, and a quarter. What is the probability that the penny and dime both come up the same? | \dfrac{1}{2} |
Determine the number of ways to arrange the letters of the word "SUCCESS". | 420 |
Let $P_1,P_2,\dots,P_n$ be $n$ distinct points over a line in the plane ($n\geq2$). Consider all the circumferences with diameters $P_iP_j$ ($1\leq{i,j}\leq{n}$) and they are painted with $k$ given colors. Lets call this configuration a ($n,k$)-cloud.
For each positive integer $k$, find all the positive integers $n$ s... | n \geq 2^k + 1 |
What is the remainder when $2^{87} +3$ is divided by $7$? | 4 |
What is the product of $\frac{1}{5}$ and $\frac{3}{7}$ ? | \frac{3}{35} |
Compute $\dbinom{14}{11}$. | 364 |
The $24^\text{th}$ day of a particular month is a Saturday. On what day of the week was the first day of that month? | \text{Thursday} |
Given the exponential function y=f(x) whose graph passes through the point $\left( \frac{1}{2}, \frac{\sqrt{2}}{2} \right)$, find the value of $\log_2 f(2)$. | -2 |
What is the number of degrees in the smaller angle formed by the hour and minute hands of a clock at 8:15? Express your answer as a decimal to the nearest tenth.
[asy]
size(200);
draw(Circle((0,0),5),linewidth(1.2));
pair[] mins;
for(int i = 0; i < 60; ++i){
mins[i] = 4.5*dir(-6*i + 90);
dot(mins[i]);
}
for(int i = 1... | 157.5 |
If $i^2=-1$, then $(1+i)^{20}-(1-i)^{20}$ equals | 0 |
Given that a hyperbola shares common foci $F_1$ and $F_2$ with the ellipse $\dfrac {x^{2}}{9}+ \dfrac {y^{2}}{25}=1$, and their sum of eccentricities is $2 \dfrac {4}{5}$.
$(1)$ Find the standard equation of the hyperbola;
$(2)$ Let $P$ be a point of intersection between the hyperbola and the ellipse, calculate $\c... | - \dfrac {1}{7} |
Helen cuts her lawn starting in March and finishes in October. Her lawn mower uses 2 gallons of gas every 4th time she cuts her lawn. For March, April, September and October, she only cuts her law 2 times per month. In May, June, July and August, she has to cut her lawn 4 times per month. How many gallons of gas wi... | For March, April, September and October, she cuts her lawn 2 a month. There are 4 months so she cuts her yard 2*4 = <<2*4=8>>8 times
For May, June, July and August, she cuts her lawn 4 times a month. There are 4 months so she cuts her yard 4*4 = <<4*4=16>>16 times
During the entire 8 month period, she cuts her lawn 8+1... |
For a positive integer $n,$ let
\[f(n) = \frac{1}{2^n} + \frac{1}{3^n} + \frac{1}{4^n} + \dotsb.\]Find
\[\sum_{n = 2}^\infty f(n).\] | 1 |
Calculate $(-1)^{45} + 2^{(3^2+5^2-4^2)}$. | 262143 |
Given Elmer’s new car provides a 70% better fuel efficiency and uses a type of fuel that is 35% more expensive per liter, calculate the percentage by which Elmer will save money on fuel costs if he uses his new car for his journey. | 20.6\% |
Let $m$ be the smallest positive, three-digit integer congruent to 5 (mod 11). Let $n$ be the smallest positive, four-digit integer congruent to 5 (mod 11). What is $n-m$? | 902 |
A curry house sells curries that have varying levels of spice. Recently, a lot of the customers have been ordering very mild curries and the chefs have been having to throw away some wasted ingredients. To reduce cost and food wastage, the curry house starts monitoring how many ingredients are actually being used and c... | The curry house previously bought 3 peppers per very spicy curry * 30 very spicy curries = <<3*30=90>>90 peppers for very spicy curries.
They also bought 2 peppers per spicy curry * 30 spicy curries = <<2*30=60>>60 peppers for spicy curries.
They also bought 1 pepper per mild curry * 10 mild curries = <<1*10=10>>10 pep... |
Given the parabola $y^2 = 4x$, a line passing through point $P(4, 0)$ intersects the parabola at points $A(x_1, y_1)$ and $B(x_2, y_2)$. Find the minimum value of $y_1^2 + y_2^2$. | 32 |
An angle $x$ is chosen at random from the interval $0^\circ < x < 90^\circ.$ Let $p$ be the probability that the numbers $\sin^2 x, \cos^2 x,$ and $\sin x \cos x$ are not the lengths of the sides of a triangle. Given that $p = d/n,$ where $d$ is the number of degrees in $\text{arctan}$ $m$ and $m$ and $n$ are positive ... | 92 |
Given that $a$, $b$, and $c$ represent the sides opposite to angles $A$, $B$, and $C$ of $\triangle ABC$, respectively, and $\overrightarrow{m}=(a,−\sqrt {3}b)$, $\overrightarrow{n}=(\sin B,\cos A)$, if $a= \sqrt {7}$, $b=2$, and $\overrightarrow{m} \perp \overrightarrow{n}$, then the area of $\triangle ABC$ is ______. | \frac{3\sqrt{3}}{2} |
Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfy $|\overrightarrow{a}|=1$, $|\overrightarrow{b}|=2$, and $\overrightarrow{a} \cdot \overrightarrow{b} = -\sqrt{3}$, calculate the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$. | \frac{5\pi}{6} |
For natural numbers, when $P$ is divided by $D$, the quotient is $Q$ and the remainder is $R$. When $Q$ is divided by $D'$, the quotient is $Q'$ and the remainder is $R'$. Then, when $P$ is divided by $DD'$, the remainder is: | $R+R'D$ |
Given a random variable $\xi$ that follows the normal distribution $N(0, \sigma^2)$, if $P(\xi > 2) = 0.023$, calculate $P(-2 \leq \xi \leq 2)$. | 0.954 |
Let $f(x)=2x+1$. Find the sum of all $x$ that satisfy the equation $f^{-1}(x)=f(x^{-1})$. | 3 |
A journey is divided into three segments: uphill, flat, and downhill. The ratio of the lengths of these segments is $1: 2: 3$, and the ratio of the times taken to walk each segment is $4: 5: 6$. Given that the person’s speed uphill is 3 km/h and the total distance is 50 km, how much time does the person take to complet... | 10.4166666667 |
Renaldo drove 15 kilometers. Ernesto drove 7 kilometers more than one-third of Renaldo's distance. How many kilometers did the 2 men drive in total? | Renaldo = <<15=15>>15 km
Ernesto = (1/3) * 15 + 7 = <<(1/3)*15+7=12>>12 km
Total = 15 + 12 = <<27=27>>27 km
Renaldo and Ernesto drove a total of 27 kilometers.
#### 27 |
If $x$ is an integer, find the largest integer that always divides the expression \[(12x + 2)(8x + 14)(10x + 10)\] when $x$ is odd. | 40 |
For a positive integer \( n \), let \( s(n) \) denote the sum of its digits, and let \( p(n) \) denote the product of its digits. If the equation \( s(n) + p(n) = n \) holds true, then \( n \) is called a coincidence number. What is the sum of all coincidence numbers? | 531 |
The larger of two consecutive odd integers is three times the smaller. What is their sum? | 4 |
Tom needs to lower a rope down 6 stories. One story is 10 feet. The only rope being sold is 20 feet long but you lose 25% when lashing them together. How many pieces of rope will he need to buy? | He needs 10*6=<<10*6=60>>60 feet
He loses 20*.25=<<20*.25=5>>5 feet each time
So he gets 20-5=<<20-5=15>>15 feet from each piece
That means he needs 60/15=<<60/15=4>>4 pieces of rope
#### 4 |
While entertaining his younger sister Alexis, Michael drew two different cards from an ordinary deck of playing cards. Let $a$ be the probability that the cards are of different ranks. Compute $\lfloor 1000a\rfloor$ . | 941 |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively, and it is given that $\sqrt{3}b\cos A=a\cos B$.
$(1)$ Find the angle $A$;
$(2)$ If $a= \sqrt{2}$ and $\frac{c}{a}= \frac{\sin A}{\sin B}$, find the perimeter of $\triangle ABC$. | 3 \sqrt{2} |
A chemistry student conducted an experiment: starting with a bottle filled with syrup solution, the student poured out one liter of liquid, refilled the bottle with water, then poured out one liter of liquid again, and refilled the bottle with water once more. As a result, the syrup concentration decreased from 36% to ... | 1.2 |
Compute $\frac{x^{10} - 32x^5 + 1024}{x^5 - 32}$ when $x=8$. | 32768 |
A binary operation $\diamondsuit$ has the properties that $a\,\diamondsuit\, (b\,\diamondsuit \,c) = (a\,\diamondsuit \,b)\cdot c$ and that $a\,\diamondsuit \,a=1$ for all nonzero real numbers $a, b,$ and $c$. (Here $\cdot$ represents multiplication). Find the solution to the equation $2016 \,\diamondsuit\, (6\,\diamon... | \frac{25}{84} |
The graph of the function $f(x)=\sqrt{3}\cos 2x-\sin 2x$ can be obtained by translating the graph of the function $f(x)=2\sin 2x$ by an unspecified distance. | \dfrac{\pi}{6} |
Given the function $f(x)= \begin{cases} a+\ln x,x > 0 \\ g(x)-x,x < 0\\ \end{cases}$, which is an odd function, and $g(-e)=0$, find the value of $a$. | -1-e |
A polyhedron has 12 faces and is such that:
(i) all faces are isosceles triangles,
(ii) all edges have length either \( x \) or \( y \),
(iii) at each vertex either 3 or 6 edges meet, and
(iv) all dihedral angles are equal.
Find the ratio \( x / y \). | 3/5 |
The overall idea is a common method in mathematical problem-solving. Below is the train of thought for factoring the polynomial $(a^{2}+2a)(a^{2}+2a+2)+1$: Consider "$a^{2}+2a$" as a whole, let $a^{2}+2a=x$, then the expression $=x(x+2)+1=x^{2}+2x+1=(x+1)^{2}$, then restore "$x$" to "$a^{2}+2a$". The solution process i... | 2024 |
One fine summer day, François was looking for Béatrice in Cabourg. Where could she be? Perhaps on the beach (one chance in two) or on the tennis court (one chance in four), it could be that she is in the cafe (also one chance in four). If Béatrice is on the beach, which is large and crowded, François has a one in two c... | \frac{3}{5} |
A haunted house has six windows. In how many ways can Georgie the Ghost enter the house by one window and leave by a different window? | 18 |
Sharon wants to get kitchen supplies. She admired Angela's kitchen supplies which consist of 20 pots, 6 more than three times as many plates as the pots, and half as many cutlery as the plates. Sharon wants to buy half as many pots as Angela, 20 less than three times as many plates as Angela, and twice as much cutlery ... | Angela has 6+3*20=<<6+3*20=66>>66 plates.
Angela has 1/2*66=<<1/2*66=33>>33 cutlery.
Sharon wants to buy 1/2*20=<<1/2*20=10>>10 pots.
Sharon wants to buy 3*66-20=<<3*66-20=178>>178 plates.
Sharon wants to buy 2*33= <<2*33=66>>66 cutlery.
Sharon wants to buy a total of 10+178+66=<<10+178+66=254>>254 kitchen supplies.
##... |
The two squares shown share the same center $O_{}$ and have sides of length 1. The length of $\overline{AB}$ is $43/99$ and the area of octagon $ABCDEFGH$ is $m/n,$ where $m_{}$ and $n_{}$ are relatively prime positive integers. Find $m+n.$
[asy] //code taken from thread for problem real alpha = 25; pair W=dir(225), X=... | 185 |
Given that the equation about $x$, $x^{2}-2a\ln x-2ax=0$ has a unique solution, find the value of the real number $a$. | \frac{1}{2} |
On the coordinate plane, the points \(A(0, 2)\), \(B(1, 7)\), \(C(10, 7)\), and \(D(7, 1)\) are given. Find the area of the pentagon \(A B C D E\), where \(E\) is the intersection point of the lines \(A C\) and \(B D\). | 36 |
Let $A B C D$ be a square of side length 10 . Point $E$ is on ray $\overrightarrow{A B}$ such that $A E=17$, and point $F$ is on ray $\overrightarrow{A D}$ such that $A F=14$. The line through $B$ parallel to $C E$ and the line through $D$ parallel to $C F$ meet at $P$. Compute the area of quadrilateral $A E P F$. | 100 |
In the repeating decimal 0.2017, if the sum of all digits from the $m$-th digit to the $n$-th digit after the decimal point is 2017, find the value of $n$ when $m$ takes the minimum value. | 808 |
The area of each of the four congruent L-shaped regions of this 100-inch by 100-inch square is 3/16 of the total area. How many inches long is the side of the center square? | 50 |
What is the smallest positive multiple of $32$? | 32 |
Compute $\dbinom{8}{0}$. | 1 |
Carson is a night security guard. He's supposed to patrol the outside of a warehouse that's 600 feet long and 400 feet wide. If Carson is supposed to circle the warehouse 10 times, but gets tired and skips 2 times, how far does he walk in one night? | First find the total distance Carson walks along the warehouse's long sides: 600 feet/side * 2 sides = <<600*2=1200>>1200 feet
Then find the total distance Carson walks along the warehouse's short sides: 400 feet/side * 2 sides = <<400*2=800>>800 feet
Add those two amounts to find the total distance walk in one circle:... |
The point $(1,1,1)$ is rotated $180^\circ$ about the $y$-axis, then reflected through the $yz$-plane, reflected through the $xz$-plane, rotated $180^\circ$ about the $y$-axis, and reflected through the $xz$-plane. Find the coordinates of the point now. | (-1,1,1) |
A function $f$ is defined by $f(z) = i\overline{z}$, where $i^2 = -1$ and $\overline{z}$ is the complex conjugate of $z$. How many values of $z$ satisfy both $|z| = 5$ and $f(z) = z$? | 2 |
Suppose that $f(x)$ and $g(x)$ are functions on $\mathbb{R}$ such that the range of $f$ is $[-5,3]$, and the range of $g$ is $[-2,1]$. The range of $f(x) \cdot g(x)$ is $[a,b]$. What is the largest possible value of $b$? | 10 |
The function \(f(x) = 5x^2 - 15x - 2\) has a minimum value when x is negative. | -13.25 |
Given the numbers from $000$ to $999$, calculate how many have three digits in non-decreasing or non-increasing order, including cases where digits can repeat. | 430 |
In triangle $PQR$, $PQ = 8$, $QR = 15$, and $PR = 17$. Point $S$ is the angle bisector of $\angle QPR$. Find the length of $QS$ and then find the length of the altitude from $P$ to $QS$. | 25 |
The value $2^{10} - 1$ is divisible by several prime numbers. What is the sum of these prime numbers? | 26 |
Someone claims that the first five decimal digits of the square root of 51 are the same as the first five significant digits of the square root of 2. Verify this claim. Which rational approximating fraction can we derive from this observation for \(\sqrt{2}\)? How many significant digits does this fraction share with t... | \frac{99}{70} |
Kenneth has $50 to go to the store. Kenneth bought 2 baguettes and 2 bottles of water. Each baguette cost $2 and each bottle of water cost $1. How much money does Kenneth have left? | The cost of the baguettes is 2 × $2 = $<<2*2=4>>4.
The cost of the water is 2 × $1 = $<<2*1=2>>2.
The total cost of the shopping is $4 + $2 = $<<4+2=6>>6.
Kenneth has $50 − $6 = $44 left.
#### 44 |
Given right $\triangle ABC$ with legs $BC=3, AC=4$. Find the length of the shorter angle trisector from $C$ to the hypotenuse: | \frac{12\sqrt{3}-9}{13} |
Crystal has a running course marked out for her daily run. She starts this run by heading due north for one mile. She then runs northeast for one mile, then southeast for one mile. The last portion of her run takes her on a straight line back to where she started. How far, in miles is this last portion of her run? | \sqrt{3} |
In a table containing $A$ columns and 100 rows, natural numbers from 1 to $100 \cdot A$ are written by rows in ascending order, starting from the first row. The number 31 is in the fifth row. In which row is the number 100? | 15 |
The five books "Poetry," "Documents," "Rites," "Changes," and "Spring and Autumn" all have different numbers of pages. The differences in the number of pages between the books are as follows:
1. "Poetry" and "Documents" differ by 24 pages.
2. "Documents" and "Rites" differ by 17 pages.
3. "Rites" and "Changes" differ b... | 34 |
Given the function $y= \sqrt {x^{2}-ax+4}$, find the set of all possible values of $a$ such that the function is monotonically decreasing on the interval $[1,2]$. | \{4\} |
The terms $x, x + 2, x + 4, \dots, x + 2n$ form an arithmetic sequence, with $x$ an integer. If each term of the sequence is cubed, the sum of the cubes is $-1197$. What is the value of $n$ if $n > 3$? | 6 |
Victoria was given a $50 bill by her mother to buy her friends snacks for their group meeting. She bought 2 boxes of pizza that costs $12 each and 2 packs of juice drinks that cost $2 each. How much money should Victoria return to her mother? | Two boxes of pizza amount to $12 x 2 = $<<12*2=24>>24.
Two packs of juice amount to $2 x 2 = $<<2*2=4>>4.
Victoria paid a total of $24 + $4 = $<<24+4=28>>28.
Hence, she should return the $50 - $28 = $<<50-28=22>>22 to her mother.
#### 22 |
A regular pentagon \(Q_1 Q_2 \dotsb Q_5\) is drawn in the coordinate plane with \(Q_1\) at \((1,0)\) and \(Q_3\) at \((5,0)\). If \(Q_n\) is the point \((x_n,y_n)\), compute the numerical value of the product
\[(x_1 + y_1 i)(x_2 + y_2 i)(x_3 + y_3 i) \dotsm (x_5 + y_5 i).\] | 242 |
Given the function $f(x)=ax^{2}+bx+c(a,b,c∈R)$, if there exists a real number $a∈[1,2]$, for any $x∈[1,2]$, such that $f(x)≤slant 1$, then the maximum value of $7b+5c$ is _____. | -6 |
Let $x$ and $y$ be real numbers, $y > x > 0,$ such that
\[\frac{x}{y} + \frac{y}{x} = 4.\]Find the value of \[\frac{x + y}{x - y}.\] | \sqrt{3} |
When two fair dice are thrown once each, what is the probability that the upward-facing numbers are different and that one of them shows a 3? | \frac{5}{18} |
In how many ways can 8 people be seated in a row of chairs if three of those people, Alice, Bob, and Charlie, refuse to sit next to each other in any order? | 36000 |
Alia has 2 times as many markers as Austin. Austin has one-third as many markers as Steve does. If Steve has 60 markers, how many does Alia have? | Austin has 60/3=<<60/3=20>>20 markers
Alia has 20*2=<<20*2=40>>40 markers
#### 40 |
What is the intersection point of the line $y = 2x + 5$ and the line perpendicular to it that passes through the point $(5, 5)$? | (1, 7) |
In a sequence, the first term is \(a_1 = 2010\) and the second term is \(a_2 = 2011\). The values of the other terms satisfy the relation:
\[ a_n + a_{n+1} + a_{n+2} = n \]
for all \(n \geq 1\). Determine \(a_{500}\). | 2177 |
Semicircles $POQ$ and $ROS$ pass through the center $O$. What is the ratio of the combined areas of the two semicircles to the area of circle $O$? | \frac{1}{2} |
Ilya takes a triplet of numbers and transforms it following the rule: at each step, each number is replaced by the sum of the other two. What is the difference between the largest and the smallest numbers in the triplet after the 1989th application of this rule, if the initial triplet of numbers was \(\{70, 61, 20\}\)?... | 50 |
The numbers $1,2,\dots,9$ are randomly placed into the $9$ squares of a $3 \times 3$ grid. Each square gets one number, and each of the numbers is used once. What is the probability that the sum of the numbers in each row and each column is odd? | \frac{1}{14} |
For every integers $ a,b,c$ whose greatest common divisor is $n$ , if
\[ \begin{array}{l} {x \plus{} 2y \plus{} 3z \equal{} a}
{2x \plus{} y \minus{} 2z \equal{} b}
{3x \plus{} y \plus{} 5z \equal{} c} \end{array}
\]
has a solution in integers, what is the smallest possible value of positive number $ n$ ? | 28 |
Let $d_n$ be the determinant of the $n \times n$ matrix whose entries, from
left to right and then from top to bottom, are $\cos 1, \cos 2, \dots, \cos
n^2$. Evaluate
$\lim_{n\to\infty} d_n$. | 0 |
Let $F_1 = (0,1)$ and $F_ 2= (4,1).$ Then the set of points $P$ such that
\[PF_1 + PF_2 = 6\]form an ellipse. The equation of this ellipse can be written as
\[\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1.\]Find $h + k + a + b.$ | 6 + \sqrt{5} |
Suppose that $\{b_n\}$ is an arithmetic sequence with $$
b_1+b_2+ \cdots +b_{150}=150 \quad \text{and} \quad
b_{151}+b_{152}+ \cdots + b_{300}=450.
$$What is the value of $b_2 - b_1$? Express your answer as a common fraction. | \frac{1}{75} |
A positive integer is equal to the sum of the squares of its four smallest positive divisors. What is the largest prime that divides this positive integer? | 13 |
Distribute 5 students into three groups: A, B, and C. Group A must have at least two students, while groups B and C must have at least one student each. Determine the number of different distribution schemes. | 80 |
Let \( N \) be the smallest positive integer whose digits have a product of 2000. The sum of the digits of \( N \) is | 25 |
How many multiples of 4 are between 70 and 300? | 57 |
Xiao Ming forgot the last two digits of his WeChat login password. He only remembers that the last digit is one of the letters \\(A\\), \\(a\\), \\(B\\), or \\(b\\), and the other digit is one of the numbers \\(4\\), \\(5\\), or \\(6\\). The probability that Xiao Ming can successfully log in with one attempt is \_\_\_\... | \dfrac{1}{12} |
Find the circulation of the vector field
$$
\vec{a}=\frac{y}{3} \vec{i} 3-3 x \vec{j}+x \vec{k}
$$
along the closed contour $\Gamma$
$$
\left\{\begin{array}{l}
x=2 \cos t \\
y=2 \sin t \\
z=1-2 \cos t - 2 \sin t
\end{array} \quad t \in [0,2\pi]
\right.
$$ | -\frac{52 \pi}{3} |
A truck delivered 4 bags of cement. They are stacked in the truck. A worker can carry one bag at a time either from the truck to the gate or from the gate to the shed. The worker can carry the bags in any order, each time taking the top bag, carrying it to the respective destination, and placing it on top of the existi... | \frac{1}{8} |
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