problem stringlengths 10 5.15k | answer stringlengths 0 1.23k |
|---|---|
In rectangle $ABCD$, we have $A=(6,-22)$, $B=(2006,178)$, $D=(8,y)$, for some integer $y$. What is the area of rectangle $ABCD$? | 40400 |
A number $x$ is equal to $6 \cdot 18 \cdot 42$. What is the smallest positive integer $y$ such that the product $xy$ is a perfect cube? | 441 |
The functions $f(x) = x^2-2x + m$ and $g(x) = x^2-2x + 4m$ are evaluated when $x = 4$. What is the value of $m$ if $2f(4) = g(4)$? | 4 |
Jake's neighbors hire him to mow their lawn and plant some flowers. Mowing the lawn takes 1 hour and pays $15. If Jake wants to make $20/hour working for the neighbors, and planting the flowers takes 2 hours, how much should Jake charge (in total, not per hour) for planting the flowers? | First figure out how many hours total Jake works by adding the time spent mowing the lawn to the time spent planting flowers: 1 hour + 2 hours = <<1+2=3>>3 hours
Then figure out how much money Jake makes if he earns $20/hour for three hours by multiplying the time spent by the pay rate: $20/hour * 3 hours = $<<20*3=60>... |
One hundred people attended a party. Fifty percent of the attendees are women, thirty-five percent of them are men, and the rest are children. How many children attended the party? | Out of 100 attendees, 100 x 50/100 = <<100*50/100=50>>50 are women.
While 100 x 35/100 = <<100*35/100=35>>35 are men.
So there were a total of 50 + 35 = <<50+35=85>>85 men and women who attended the party.
Thus, 100 - 85 = <<100-85=15>>15 children attended the party.
#### 15 |
A play was held in an auditorium and its ticket costs $10. An auditorium has 20 rows and each row has 10 seats. If only 3/4 of the seats were sold, how much was earned from the play? | There are 20 x 10 = <<20*10=200>>200 seats in the auditorium.
Only 200 x 3/4 = <<200*3/4=150>>150 seats were sold.
Hence, the earnings from the play is $10 x 150 = $<<10*150=1500>>1500.
#### 1500 |
The longest side of a right triangle is 13 meters and one of the other sides is 5 meters. What is the area and the perimeter of the triangle? Also, determine if the triangle has any acute angles. | 30 |
A cone is formed from a 300-degree sector of a circle of radius 18 by aligning the two straight sides. [asy]
size(110);
draw(Arc((0,0),1,0,300));
draw((1,0)--(0,0)--(.5,-.5*sqrt(3)));
label("18",(.5,0),S); label("$300^\circ$",(0,0),NW);
[/asy] What is the result when the volume of the cone is divided by $\pi$? | 225\sqrt{11} |
One base of a trapezoid is $100$ units longer than the other base. The segment that joins the midpoints of the legs divides the trapezoid into two regions whose areas are in the ratio $2: 3$. Let $x$ be the length of the segment joining the legs of the trapezoid that is parallel to the bases and that divides the trapez... | 181 |
John started weightlifting when he was 16. When he first started he could Clean & Jerk 80 kg and he could Snatch 50 kg. He manages to double his clean and jerk and increase his snatch by 80%. What is his new combined total lifting capacity? | His clean and jerk goes to 80*2=<<80*2=160>>160 kg
His snatch increases by 50*.8=<<50*.8=40>>40 kg
So his snatch is now 50+40=<<50+40=90>>90 kg
So his total is 160+90=<<160+90=250>>250 kg
#### 250 |
A regular triangular prism has a triangle $ABC$ with side $a$ as its base. Points $A_{1}, B_{1}$, and $C_{1}$ are taken on the lateral edges and are located at distances of $a / 2, a, 3a / 2$ from the base plane, respectively. Find the angle between the planes $ABC$ and $A_{1}B_{1}C_{1}$. | \frac{\pi}{4} |
A math teacher randomly selects 3 questions for analysis from a test paper consisting of 12 multiple-choice questions, 4 fill-in-the-blank questions, and 6 open-ended questions. The number of different ways to select questions such that at least one multiple-choice question and at least one open-ended question are sele... | 864 |
Given $a = 1 + 2\binom{20}{1} + 2^2\binom{20}{2} + \ldots + 2^{20}\binom{20}{20}$, and $a \equiv b \pmod{10}$, determine the possible value(s) for $b$. | 2011 |
What is the greatest possible value of $n$ if Juliana chooses three different numbers from the set $\{-6,-4,-2,0,1,3,5,7\}$ and multiplies them together to obtain the integer $n$? | 168 |
Given that a shop advertises everything as "half price in today's sale," and a 20% discount is applied to sale prices, and a promotional offer is available where if a customer buys two items, they get the lesser priced item for free, calculate the percentage off the total original price for both items that the customer... | 20\% |
Let \[\mathbf{N} = \begin{pmatrix} x & y & z \\ y & z & x \\ z & x & y \end{pmatrix}\] be a matrix with real entries such that $\mathbf{N}^3 = \mathbf{I}.$ If $xyz = -1$, find the possible values of $x^3 + y^3 + z^3.$ | -2 |
The increasing sequence of positive integers $b_1,$ $b_2,$ $b_3,$ $\dots$ follows the rule
\[b_{n + 2} = b_{n + 1} + b_n\]for all $n \ge 1.$ If $b_9 = 544,$ then find $b_{10}.$ | 883 |
The square of an integer is 182 greater than the integer itself. What is the sum of all integers for which this is true? | 1 |
It is raining outside and Bill puts his empty fish tank in his yard to let it fill with rainwater. It starts raining at 1 pm. 2 inches of rainfall in the first hour. For the next four hours, it rains at a rate of 1 inch per hour. It then rains at three inches per hour for the rest of the day. If the fish tank is 18 inc... | At 2 pm there are 2 inches of rain in the tank.
At 6pm, there are 2 + 4 * 1 = <<2+4*1=6>>6 inches of rain in the tank.
That means that there are 18 - 6 = <<18-6=12>>12 inches of tank that still need to be filled as of 6 pm.
It will take 12 / 3 = <<12/3=4>>4 hours to finish filling the tank.
The tank will be filled at 6... |
Compute $20\left(\frac{256}{4} + \frac{64}{16} + \frac{16}{64} + 2\right)$. | 1405 |
The line $12x+5y=60$ forms a triangle with the coordinate axes. What is the sum of the lengths of the altitudes of this triangle? | \frac{281}{13} |
What is the greatest whole number that must be a divisor of the product of any three consecutive positive integers? | 6 |
In triangle $\triangle ABC$, $a+b=11$. Choose one of the following two conditions as known, and find:<br/>$(Ⅰ)$ the value of $a$;<br/>$(Ⅱ)$ $\sin C$ and the area of $\triangle ABC$.<br/>Condition 1: $c=7$, $\cos A=-\frac{1}{7}$;<br/>Condition 2: $\cos A=\frac{1}{8}$, $\cos B=\frac{9}{16}$.<br/>Note: If both conditions ... | \frac{15\sqrt{7}}{4} |
The yearly changes in the population census of a town for four consecutive years are, respectively, 25% increase, 25% increase, 25% decrease, 25% decrease. The net change over the four years, to the nearest percent, is: | -12 |
Piravena must make a trip from $A$ to $B,$ then from $B$ to $C,$ then from $C$ to $A.$ Each of these three parts of the trip is made entirely by bus or entirely by airplane. The cities form a right-angled triangle as shown, with $C$ a distance of $3000\text{ km}$ from $A$ and with $B$ a distance of $3250\text{ km}$ fro... | \$425 |
The polygon enclosed by the solid lines in the figure consists of 4 congruent squares joined edge-to-edge. One more congruent square is attached to an edge at one of the nine positions indicated. How many of the nine resulting polygons can be folded to form a cube with one face missing? | 6 |
A fly is on the edge of a ceiling of a circular room with a radius of 58 feet. The fly walks straight across the ceiling to the opposite edge, passing through the center of the circle. It then walks straight to another point on the edge of the circle but not back through the center. The third part of the journey is str... | 280 |
The image of the point with coordinates $(1,1)$ under the reflection across the line $y=mx+b$ is the point with coordinates $(9,5)$. Find $m+b$. | 11 |
Bob buys four burgers and three sodas for $\$5.00$, and Carol buys three burgers and four sodas for $\$5.40$. How many cents does a soda cost? | 94 |
What is the remainder when $5^{207}$ is divided by 7? | 6 |
A function $f$ has the property that $f(3x-1)=x^2+x+1$ for all real numbers $x$. What is $f(5)$? | 7 |
A cone is inverted and filled with water to 3/4 of its height. What percent of the cone's volume is filled with water? Express your answer as a decimal to the nearest ten-thousandth. (You should enter 10.0000 for $10\%$ instead of 0.1000.) | 42.1875 |
In a plane, four circles with radii $1,3,5,$ and $7$ are tangent to line $\ell$ at the same point $A,$ but they may be on either side of $\ell$. Region $S$ consists of all the points that lie inside exactly one of the four circles. What is the maximum possible area of region $S$? | 65\pi |
For a given positive integer $ k$ denote the square of the sum of its digits by $ f_1(k)$ and let $ f_{n\plus{}1}(k) \equal{} f_1(f_n(k)).$ Determine the value of $ f_{1991}(2^{1990}).$ | 256 |
Lines $l_1^{}$ and $l_2^{}$ both pass through the origin and make first-quadrant angles of $\frac{\pi}{70}$ and $\frac{\pi}{54}$ radians, respectively, with the positive x-axis. For any line $l^{}_{}$, the transformation $R(l)^{}_{}$ produces another line as follows: $l^{}_{}$ is reflected in $l_1^{}$, and the resultin... | 945 |
Twelve points are spaced around a $3 \times 3$ square at intervals of one unit. Two of the 12 points are chosen at random. Find the probability that the two points are one unit apart. | \frac{2}{11} |
Given the function $f(x)=\cos x\cos \left( x+\dfrac{\pi}{3} \right)$.
(1) Find the smallest positive period of $f(x)$;
(2) In $\triangle ABC$, angles $A$, $B$, $C$ correspond to sides $a$, $b$, $c$, respectively. If $f(C)=-\dfrac{1}{4}$, $a=2$, and the area of $\triangle ABC$ is $2\sqrt{3}$, find the value of side le... | 2 \sqrt {3} |
As shown in the figure below, a circular park consists of an outer-ring path for walkers (white) and a ring-shaped flower garden (gray) surrounding a central circular fountain (black). The walking path is six feet wide in all places, the garden ring is eight feet wide in all places, and the fountain has a diameter of 1... | 38 |
Find the distance between the foci of the hyperbola $x^2 - 4x - 9y^2 - 18y = 45.$ | \frac{40}{3} |
The sum of two positive integers is 50 and their difference is 12. What is the value of the positive difference of the squares of the integers? | 600 |
Elsa and her sister watch an opera show every year at Central City Opera. Last year, opera seating cost $85 per ticket. This year, it costs $102. What is the percent increase in the cost of each ticket? | The ticket price increase is $102 - $85 = $<<102-85=17>>17
So there is 17/85 x 100% = 20% increase of price.
#### 20 |
Given that a rectangle with length $3x$ inches and width $x + 5$ inches has the property that its area and perimeter have equal values, what is $x$? | 1 |
It is known that the equation $ |x - 1| + |x - 2| +... + |x - 2001| = a$ has exactly one solution. Find $a$ . | 1001000 |
Given four points $P, A, B, C$ on a sphere, if $PA$, $PB$, $PC$ are mutually perpendicular and $PA=PB=PC=1$, calculate the surface area of this sphere. | 3\pi |
A sequence of three real numbers forms an arithmetic progression with a first term of 9. If 2 is added to the second term and 20 is added to the third term, the three resulting numbers form a geometric progression. What is the smallest possible value for the third term of the geometric progression? | 1 |
Given that the terminal side of angle $\alpha$ passes through the point $(-3, 4)$, then $\cos\alpha=$ _______; $\cos2\alpha=$ _______. | -\frac{7}{25} |
Let $ABCD$ be a regular tetrahedron, and let $O$ be the centroid of triangle $BCD$. Consider the point $P$ on $AO$ such that $P$ minimizes $PA+2(PB+PC+PD)$. Find $\sin \angle PBO$. | \frac{1}{6} |
A line has a slope of $-7$ and contains the point $(3,0)$. The equation of this line can be written in the form $y = mx+b$. What is the value of $m+b$? | 14 |
Four planes divide space into $n$ parts at most. Calculate $n$. | 15 |
The polynomial $x^{3}-3 x^{2}+1$ has three real roots $r_{1}, r_{2}$, and $r_{3}$. Compute $\sqrt[3]{3 r_{1}-2}+\sqrt[3]{3 r_{2}-2}+\sqrt[3]{3 r_{3}-2}$. | 0 |
In the third season of "China Poetry Conference", there were many highlights under the theme of "Life has its own poetry". In each of the ten competitions, there was a specially designed opening poem recited in unison by a hundred people under the coordination of lights and dances. The poems included "Changsha Spring i... | 144 |
In △ABC, B = $$\frac{\pi}{3}$$, AB = 8, BC = 5, find the area of the circumcircle of △ABC. | \frac{49\pi}{3} |
How many square units are in the area of the parallelogram with vertices at (0, 0), (6, 0), (2, 8) and (8, 8)? | 48 |
Calculate using your preferred method!
100 - 54 - 46
234 - (134 + 45)
125 × 7 × 8
15 × (61 - 45)
318 ÷ 6 + 165. | 218 |
Matilda has a summer job delivering newspapers. She earns \$6.00 an hour plus \$0.25 per newspaper delivered. Matilda delivers 30 newspapers per hour. How much money will she earn during a 3-hour shift? | \$40.50 |
John decides to take up illustration. He draws and colors 10 pictures. It takes him 2 hours to draw each picture and 30% less time to color each picture. How long does he spend on all the pictures? | The coloring takes 2*.3=<<2*.3=.6>>.6 hours less than the drawing
So it takes 2-.6=<<2-.6=1.4>>1.4 hours per picture
So it takes him 2+1.4=<<2+1.4=3.4>>3.4 hours per picture
That means it takes 3.4*10=<<3.4*10=34>>34 hours
#### 34 |
We wrote an even number in binary. By removing the trailing $0$ from this binary representation, we obtain the ternary representation of the same number. Determine the number! | 10 |
Jeff will pick a card at random from ten cards numbered 1 through 10. The number on this card will indicate his starting point on the number line shown below. He will then spin the fair spinner shown below (which has three congruent sectors) and follow the instruction indicated by his spin. From this new point he wi... | \frac{31}{90} |
Lee can make 18 cookies with two cups of flour. How many cookies can he make with three cups of flour? | 27 |
The mean, median, and mode of the $7$ data values $60, 100, x, 40, 50, 200, 90$ are all equal to $x$. What is the value of $x$? | 90 |
Let \( a_{1}, a_{2}, \cdots, a_{n} \) be an arithmetic sequence, and it is given that
$$
\sum_{i=1}^{n}\left|a_{i}+j\right|=2028 \text{ for } j=0,1,2,3.
$$
Find the maximum value of the number of terms \( n \). | 52 |
A younger brother leaves home and walks to the park at a speed of 4 kilometers per hour. Two hours later, the older brother leaves home and rides a bicycle at a speed of 20 kilometers per hour to catch up with the younger brother. How long will it take for the older brother to catch up with the younger brother? | 0.5 |
What is the smallest positive value of $m$ so that the equation $10x^2 - mx + 420 = 0$ has integral solutions? | 130 |
Estimate the product $(.331)^3$. | 0.037 |
In the Cartesian coordinate system $xOy$, the parametric equation of curve $C_1$ is $$\begin{cases} x=4t^{2} \\ y=4t \end{cases}$$ (where $t$ is the parameter). Taking the origin $O$ as the pole and the positive $x$-axis as the polar axis, and using the same unit length, the polar equation of curve $C_2$ is $$ρ\cos(θ+ ... | 16 |
If $(2,12)$ and $(8,3)$ are the coordinates of two opposite vertices of a rectangle, what is the sum of the $x$-coordinates of the other two vertices? | 10 |
During a commercial break in the Super Bowl, there were three 5-minute commercials and eleven 2-minute commercials. How many minutes was the commercial break? | The 5-minute commercials were 3 * 5 = <<3*5=15>>15 minutes in total.
The 2-minute commercials were 2 * 11 - 22 minutes in total.
The commercial break was 15 + 22 = <<15+22=37>>37 minutes long.
#### 37 |
Bill is trying to decide whether to make blueberry muffins or raspberry muffins. Blueberries cost $5.00 per 6 ounce carton and raspberries cost $3.00 per 8 ounce carton. If Bill is going to make 4 batches of muffins, and each batch takes 12 ounces of fruit, how much money would he save by using raspberries instead of b... | First find how many ounces of fruit Bill will need total: 12 ounces/batch * 4 batches = <<12*4=48>>48 ounces
Then find how many cartons of blueberries Bill would need: 48 ounces / 6 ounces/carton = <<48/6=8>>8 cartons
Then multiply that number by the cost per carton to find the total cost of the blueberries: 8 cartons ... |
Your math friend Steven rolls five fair icosahedral dice (each of which is labelled $1,2, \ldots, 20$ on its sides). He conceals the results but tells you that at least half of the rolls are 20. Assuming that Steven is truthful, what is the probability that all three remaining concealed dice show $20 ?$ | \frac{1}{58} |
There were 63 Easter eggs in the yard. Hannah found twice as many as Helen. How many Easter eggs did Hannah find? | Hannah found twice as many as Helen, so there are 2+1 = <<2+1=3>>3 units of eggs
There were 63 Easter eggs total and there are 3 units of eggs so 63/3 = <<63/3=21>>21 easter eggs per unit
So if Helen found 21 eggs and Hannah found twice as many, then Hannah found 2*21 = 42 Easter eggs
#### 42 |
Lena is making a collage with pictures of all her closest friends and newspaper clippings that are about their interests. She has found three clippings for each friend’s pictures. It takes her six drops of glue to stick down each clipping. Lena has already glued her seven closest friends’ pictures. How many drops of gl... | Lena has 3 clippings per friend, so she has 3 * 7 = <<3*7=21>>21 clippings to glue.
She needs 6 drops of glue per clipping, so she will need 6 * 21 = <<6*21=126>>126 drops of glue.
#### 126 |
For all complex numbers $z$, let \[f(z) = \left\{
\begin{array}{cl}
z^{2}&\text{ if }z\text{ is not real}, \\
-z^2 &\text{ if }z\text{ is real}.
\end{array}
\right.\]Find $f(f(f(f(1+i))))$. | -256 |
Given that \( A \) and \( B \) are two points on the surface of a sphere with a radius of 5, and \( AB = 8 \). Planes \( O_1AB \) and \( O_2AB \) are perpendicular to each other and pass through \( AB \). The intersections of these planes with the sphere create cross-sections \(\odot O_1\) and \(\odot O_2\). Let the ar... | 41 \pi |
Among 6 internists and 4 surgeons, there is one chief internist and one chief surgeon. Now, a 5-person medical team is to be formed to provide medical services in rural areas. How many ways are there to select the team under the following conditions?
(1) The team includes 3 internists and 2 surgeons;
(2) The team i... | 191 |
Farmer Brown fed 7 chickens and 5 sheep. How many total legs were there among the animals he fed? | The chickens had 7*2=<<7*2=14>>14 legs.
The sheep had 5*4=<<5*4=20>>20 legs.
There were 14+20=<<14+20=34>>34 legs.
#### 34 |
What is $4+10\div2-2\cdot3$? | 3 |
In ten years, I'll be twice my brother's age. The sum of our ages will then be 45 years old. How old am I now? | Let X be my age now.
In ten years, I'll be X+<<+10=10>>10 years old.
In ten years, my brother will be (X+10)*1/2 years old.
In ten years, (X+10) + (X+10)*1/2 = 45.
So (X+10)*3/2 = 45.
X+10 = 45 * 2/3 = 30.
X = 30 - 10 = <<30-10=20>>20 years old.
#### 20 |
A point $(x,y)$ is a distance of 15 units from the $x$-axis. It is a distance of 13 units from the point $(2,7)$. It is a distance $n$ from the origin. Given that $x>2$, what is $n$? | \sqrt{334 + 4\sqrt{105}} |
A polygon \(\mathcal{P}\) is drawn on the 2D coordinate plane. Each side of \(\mathcal{P}\) is either parallel to the \(x\) axis or the \(y\) axis (the vertices of \(\mathcal{P}\) do not have to be lattice points). Given that the interior of \(\mathcal{P}\) includes the interior of the circle \(x^{2}+y^{2}=2022\), find... | 8 \sqrt{2022} |
Let $a$ and $b$ be positive real numbers, with $a > b.$ Compute
\[\frac{1}{ba} + \frac{1}{a(2a - b)} + \frac{1}{(2a - b)(3a - 2b)} + \frac{1}{(3a - 2b)(4a - 3b)} + \dotsb.\] | \frac{1}{(a - b)b} |
Find all pairs of positive integers $m, n$ such that $9^{|m-n|}+3^{|m-n|}+1$ is divisible by $m$ and $n$ simultaneously. | (1, 1) \text{ and } (3, 3) |
Determine all real numbers $x$ for which the function $$g(x) = \frac{1}{2+\frac{1}{1+\frac{1}{x-1}}}$$ is undefined, and find their sum. | \frac{4}{3} |
Find $2.5-0.32.$ | 2.18 |
A cube 4 units on each side is composed of 64 unit cubes. Two faces of the larger cube that share an edge are painted blue, and the cube is disassembled into 64 unit cubes. Two of the unit cubes are selected uniformly at random. What is the probability that one of two selected unit cubes will have exactly two painted f... | \frac{1}{14} |
A function $f:\{1,2,3,4,5\} \rightarrow\{1,2,3,4,5\}$ is said to be nasty if there do not exist distinct $a, b \in\{1,2,3,4,5\}$ satisfying $f(a)=b$ and $f(b)=a$. How many nasty functions are there? | 1950 |
Given square $ABCD$, points $E$ and $F$ lie on $\overline{AB}$ so that $\overline{ED}$ and $\overline{FD}$ bisect $\angle ADC$. Calculate the ratio of the area of $\triangle DEF$ to the area of square $ABCD$. | \frac{1}{4} |
Let $ f(x) = x^3 + x + 1$. Suppose $ g$ is a cubic polynomial such that $ g(0) = - 1$, and the roots of $ g$ are the squares of the roots of $ f$. Find $ g(9)$. | 899 |
In the year 2001, the United States will host the International Mathematical Olympiad. Let $I$, $M$, and $O$ be distinct positive integers such that the product $I\cdot M\cdot O=2001$. What is the largest possible value of the sum $I+M+O$? | 671 |
The witch Gingema cast a spell on a wall clock so that the minute hand moves in the correct direction for five minutes, then three minutes in the opposite direction, then five minutes in the correct direction again, and so on. How many minutes will the hand show after 2022 minutes, given that it pointed exactly to 12 o... | 28 |
In a polar coordinate system, the midpoint of the line segment whose endpoints are $\left( 8, \frac{5 \pi}{12} \right)$ and $\left( 8, -\frac{3 \pi}{12} \right)$ is the point $(r, \theta).$ Enter $(r, \theta),$ where $r > 0$ and $0 \le \theta < 2 \pi.$ | \left( 4, \frac{\pi}{12} \right) |
Given that 216 sprinters enter a 100-meter dash competition, and the track has 6 lanes, determine the minimum number of races needed to find the champion sprinter. | 43 |
Johann has $64$ fair coins. He flips all the coins. Any coin that lands on tails is tossed again. Coins that land on tails on the second toss are tossed a third time. What is the expected number of coins that are now heads? | 56 |
The volume of a hemisphere is $\frac{500}{3}\pi$. What is the total surface area of the hemisphere including its base? Express your answer in terms of $\pi$. | 3\pi \times 250^{2/3} |
A roadwork company is paving a newly constructed 16-mile road. They use a mixture of pitch and gravel to make the asphalt to pave the road. Each truckloads of asphalt uses two bags of gravel and five times as many bags of gravel as it does barrels of pitch to make. It takes three truckloads of asphalt to pave each mile... | On the second day, the company paved 4 * 2 - 1 = <<4*2-1=7>>7 miles.
The company has 16 - 7 - 4 = <<16-7-4=5>>5 miles of road remaining to pave.
They will need 3 * 5 = <<3*5=15>>15 truckloads of asphalt to pave 5 miles of road.
For 15 truckloads, they will need 15 * 2 = <<15*2=30>>30 bags of gravel.
Thus, the company w... |
The polynomial $x^3 - 2004 x^2 + mx + n$ has integer coefficients and three distinct positive zeros. Exactly one of these is an integer, and it is the sum of the other two. How many values of $n$ are possible? | 250500 |
Let $p, q, r$ be primes such that $2 p+3 q=6 r$. Find $p+q+r$. | 7 |
What is the volume of the region in three-dimensional space defined by the inequalities $|x|+|y|+|z|\le1$ and $|x|+|y|+|z-1|\le1$? | \frac{1}{6} |
The sequence $\{a\_n\}$ is a geometric sequence with the first term $a\_1=4$, and $S\_3$, $S\_2$, $S\_4$ form an arithmetic sequence.
(1) Find the general term formula of the sequence $\{a\_n\}$;
(2) If $b\_n=\log \_{2}|a\_n|$, let $T\_n$ be the sum of the first $n$ terms of the sequence $\{\frac{1}{b\_n b\_{n+1}}\}$. ... | \frac{1}{16} |
Mia is "helping" her mom pick up $30$ toys that are strewn on the floor. Mia’s mom manages to put $3$ toys into the toy box every $30$ seconds, but each time immediately after those $30$ seconds have elapsed, Mia takes $2$ toys out of the box. How much time, in minutes, will it take Mia and her mom to put all $30$ toys... | 14 |
A nine-joint bamboo tube has rice capacities of 4.5 *Sheng* in the lower three joints and 3.8 *Sheng* in the upper four joints. Find the capacity of the middle two joints. | 2.5 |
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