problem stringlengths 10 5.15k | answer stringlengths 0 1.23k |
|---|---|
People are standing in a circle - there are liars, who always lie, and knights, who always tell the truth. Each of them said that among the people standing next to them, there is an equal number of liars and knights. How many people are there in total if there are 48 knights? | 72 |
Define a positive integer $n$ to be a factorial tail if there is some positive integer $m$ such that the decimal representation of $m!$ ends with exactly $n$ zeroes. How many positive integers less than $1992$ are not factorial tails?
| 396 |
How many numbers with less than four digits (from 0 to 9999) are neither divisible by 3, nor by 5, nor by 7? | 4571 |
Find the number of pairs $(m,n)$ of positive integers with $1\le m<n\le 30$ such that there exists a real number $x$ satisfying \[\sin(mx)+\sin(nx)=2.\] | 63 |
A chair costs 1/7 of the cost of a table. If a table costs $140, how much will it cost to buy a table and 4 chairs? | One chair costs $140 x 1/7 = $<<140*1/7=20>>20.
Four chairs cost $20 x 4 = $<<20*4=80>>80.
So a table and 4 chairs cost $140 + $80 = $<<140+80=220>>220.
#### 220 |
What is the least possible value of
\[(x+2)(x+3)(x+4)(x+5) + 2024\] where \( x \) is a real number?
A) 2022
B) 2023
C) 2024
D) 2025
E) 2026 | 2023 |
Compute: $$\left\lfloor\frac{2005^{3}}{2003 \cdot 2004}-\frac{2003^{3}}{2004 \cdot 2005}\right\rfloor$$ | 8 |
Compute the number of even positive integers $n \leq 2024$ such that $1,2, \ldots, n$ can be split into $\frac{n}{2}$ pairs, and the sum of the numbers in each pair is a multiple of 3. | 675 |
What is the value of
$\displaystyle \left(\left((2+1)^{-1}+1\right)^{-1}+1\right)^{-1}+1$? | \frac{11}{7} |
Compute the exact value of the expression $\left|3 - |9 - \pi^2| \right|$. Write your answer using only integers, $\pi$, and necessary mathematical operations, without any absolute value signs. | 12 - \pi^2 |
John paints a giant mural that is 6m by 3m. The paint costs $4 per square meter. The artist can paint 1.5 square meters per hour and charges $10 per hour. How much does the mural cost? | The mural is 6*3=<<6*3=18>>18 sq meters
So the paint cost 18*4=$<<18*4=72>>72
The artist works for 18/1.5=<<18/1.5=12>>12 hours
He charges 12*10=$<<12*10=120>>120
So the total cost is 120+72=$<<120+72=192>>192
#### 192 |
The average cost of a long-distance call in the USA in $1985$ was
$41$ cents per minute, and the average cost of a long-distance
call in the USA in $2005$ was $7$ cents per minute. Find the
approximate percent decrease in the cost per minute of a long-
distance call. | 80 |
Walter fell from the eighth platform of some scaffolding and fell past David after falling 4 meters. If he fell for an additional three more times that depth before hitting the ground, and the platforms are evenly spaced out in height, what platform was David on? | Walter fell 4 meters and an additional three times that depth which is 3*4 = 12 meters
He fell for a total of 4+12 = <<4+12=16>>16 meters
The eighth platform is 16 meters high and the platforms are evenly spaced so each platform is 16/8 = <<16/8=2>>2 meters high
David was located 4 meters below the 16 meter level which... |
Find the matrix that corresponds to a dilation centered at the origin with scale factor $-3.$ | \begin{pmatrix} -3 & 0 \\ 0 & -3 \end{pmatrix} |
If $\sec x + \tan x = \frac{5}{2},$ then find $\sec x - \tan x.$ | \frac{2}{5} |
From the set $S={1,2,3,...,100}$, three numbers are randomly selected and arranged in ascending order. Find the probability that $50$ is exactly in the middle. | \frac{1}{66} |
A digital watch now displays time in a 24-hour format, showing hours and minutes. Find the largest possible sum of the digits when it displays time in this format, where the hour ranges from 00 to 23 and the minutes range from 00 to 59. | 24 |
Willy is starting a new TV series on Netflix. The TV series has 3 seasons that are each 20 episodes long. If Willy watches 2 episodes a day, how many days will it take for Willy to finish the entire series? | The TV series has a total of 3 * 20 = <<3*20=60>>60 episodes
At a rate of 2 episodes per day, Willy will finish the series in 60 / 2 = 30 days.
#### 30 |
Given that $M(2,5)$ is the midpoint of $\overline{AB}$ and $A(3,1)$ is one endpoint, what is the product of the coordinates of point $B$? | 9 |
Oleg drew an empty $50 \times 50$ table and wrote a number above each column and to the left of each row. It turned out that all 100 written numbers are different, with 50 of them being rational and the remaining 50 irrational. Then, in each cell of the table, he wrote the sum of the numbers written next to its row and... | 1250 |
The yearly changes in the population census of a city for five consecutive years are, respectively, 20% increase, 10% increase, 30% decrease, 20% decrease, and 10% increase. Calculate the net change over these five years, to the nearest percent. | -19\% |
A rectangular flowerbed in the city park is 4 meters wide. Its length is 1 meter less than twice its width. The government wants to fence the flowerbed. How many meters of fence are needed? | Twice the width is 4 x 2 = <<4*2=8>>8 meters.
The length of the rectangular flowerbed is 8 - 1= <<8-1=7>>7 meters.
Since the rectangular flower bed has 2 equal lengths, then it needs 7 x 2 = <<7*2=14>>14 meters of fence.
The rectangular bed has also 2 equal widths, so it needs 4 x 2 = <<8=8>>8 meters of fence.
Therefor... |
John skateboarded for 10 miles and then walked another 4 miles to the park. He then skated all the way back home. How many miles has John skateboarded in total? | John traveled 10 + 4 = <<10+4=14>>14 miles to the park.
His round trip would take 14 x 2 = <<14*2=28>>28 miles.
If you subtract the 4 miles he walked then John traveled 28 - 4 = <<28-4=24>>24 miles.
#### 24 |
Bert made 12 sandwiches for his trip. On the first day, he ate half of the sandwiches he made. The next day he ate 2 sandwiches less. How many sandwiches does Bert have left after these two days? | On the first day, Bert ate 12 / 2 = <<12/2=6>>6 sandwiches.
The second day he ate 6 - 2 = <<6-2=4>>4 sandwiches.
So in total Bert is left with 12 - 6 - 4 = <<12-6-4=2>>2 sandwiches.
#### 2 |
Three merchants - Sosipatra Titovna, Olympiada Karpovna, and Poliksena Uvarovna - sat down to drink tea. Olympiada Karpovna and Sosipatra Titovna together drank 11 cups, Poliksena Uvarovna and Olympiada Karpovna drank 15 cups, and Sosipatra Titovna and Poliksena Uvarovna drank 14 cups. How many cups of tea did all thre... | 20 |
Convert the point $\left( 5, \frac{3 \pi}{2} \right)$ in polar coordinates to rectangular coordinates. | (0,-5) |
Point D is from AC of triangle ABC so that 2AD=DC. Let DE be perpendicular to BC and AE
intersects BD at F. It is known that triangle BEF is equilateral. Find <ADB? | 90 |
Since the 40th president launched his reelection campaign today, he has raised $10,000 in campaign funds. His friends raised 40% of this amount and his family raised 30% of the remaining amount. The rest of the funds are from his own savings for the election. How much did he save for the presidency? | The total amount of money raised from friends contribution is 40/100*$10000 = $<<40/100*10000=4000>>4000
Minus his friend's contribution, the 40th president has raised $10000-$4000 = $6000
The family raised 30/100*$6000 = $<<30/100*6000=1800>>1800
If the family raised $1800, the savings the 40th president had for the c... |
Last year Jessica paid $1000 for rent, $200 for food, and $100 for car insurance each month. This year her rent goes up by 30%, food costs increase by 50%, and the cost of her car insurance triples because she was at fault in an accident. How much more does Jessica pay for her expenses over the whole year compared to l... | First find the increase in rent by multiplying last year's rent by 30%: $1000 * .3 = $<<1000*.3=300>>300
Then find the food cost increase by multiplying last year's costs by 50%: $200 * .5 = $<<200*.5=100>>100
Then find the new car insurance price by multiplying last year's price by 3: $100 * 3 = $<<100*3=300>>300
Then... |
The number of 4-digit integers with distinct digits, whose first and last digits' absolute difference is 2, is between 1000 and 9999. | 840 |
For what base-6 digit $d$ is $2dd5_6$ divisible by the base 10 number 11? (Here $2dd5_6$ represents a base-6 number whose first digit is 2, whose last digit is 5, and whose middle two digits are both equal to $d$). | 4 |
If $p(x) = x^4 - 3x + 2$, then find the coefficient of the $x^3$ term in the polynomial $(p(x))^3$. | -27 |
Timothy and Theresa go to the movies very often. Timothy went to the movies 7 more times in 2010 that he did in 2009. In 2009, Timothy went to the movies 24 times. In 2010 Theresa went to see twice as many movies as Timothy did, but in 2009, she only saw half as many as he did. How many movies did Timothy and Theresa g... | In 2010 Timothy watched 24+7 = <<24+7=31>>31 movies.
In 2010 Theresa watched 31*2 = <<31*2=62>>62 movies.
In 2009 Theresa watched 24/2 = <<24/2=12>>12 movies.
In both 2009 and 2010 Timothy and Theresa watched 24+31+62+12 = <<24+31+62+12=129>>129 movies.
#### 129 |
A $9\times 7$ rectangle is tiled with tiles of the two types: L-shaped tiles composed by three unit squares (can be rotated repeatedly with $90^\circ$) and square tiles composed by four unit squares.
Let $n\ge 0$ be the number of the $2 \times 2 $ tiles which can be used in such a tiling. Find all the values of $n$. | 0 \text{ and } 3 |
Let $n$ be a positive integer. If $a\equiv (3^{2n}+4)^{-1}\pmod{9}$, what is the remainder when $a$ is divided by $9$? | 7 |
James spends 10 minutes downloading a game, half as long installing it, and triple that combined amount of time going through the tutorial. How long does it take before he can play the main game? | First divide the download time in half to find the install time: 10 minutes / 2 = <<10/2=5>>5 minutes
Then add that amount of time to the download time: 5 minutes + 10 minutes = <<5+10=15>>15 minutes
Then triple that amount of time to find the tutorial time: 15 minutes * 3 = <<15*3=45>>45 minutes
Then add that time to ... |
Out of three hundred eleventh-grade students, 77% received excellent and good grades on the first exam, 71% on the second exam, and 61% on the third exam. What is the minimum number of participants who received excellent and good grades on all three exams? | 27 |
What is the largest positive integer that is not the sum of a positive integral multiple of $42$ and a positive composite integer?
| 215 |
A robotic grasshopper jumps 1 cm to the east, then 2 cm to the north, then 3 cm to the west, then 4 cm to the south. After every fourth jump, the grasshopper restarts the sequence of jumps. After a total of $n$ jumps, the position of the grasshopper is 162 cm to the west and 158 cm to the south of its original position... | 22 |
How much money should I invest at an annually compounded interest rate of $5\%$ so that I have $\$500,\!000$ in ten years? Express your answer as a dollar value rounded to the nearest cent. | \$306,\!956.63 |
If three different lines $x+y=1$, $x-y=1$, and $ax+y=1$ cannot form a triangle, then the value of the real number $a$ is. | -1 |
Let $A B C$ be a triangle with $A B=5, B C=4$, and $C A=3$. Initially, there is an ant at each vertex. The ants start walking at a rate of 1 unit per second, in the direction $A \rightarrow B \rightarrow C \rightarrow A$ (so the ant starting at $A$ moves along ray $\overrightarrow{A B}$, etc.). For a positive real numb... | \frac{47}{24} |
If circle $$C_{1}: x^{2}+y^{2}+ax=0$$ and circle $$C_{2}: x^{2}+y^{2}+2ax+ytanθ=0$$ are both symmetric about the line $2x-y-1=0$, then $sinθcosθ=$ \_\_\_\_\_\_ . | -\frac{2}{5} |
In Perfectville, the streets are all $20$ feet wide and the blocks they enclose are all squares of side length $400$ feet, as shown. Sarah runs around the block on the $400$-foot side of the street, while Sam runs on the opposite side of the street. How many more feet than Sarah does Sam run for every lap around the ... | 160 |
If $y<0$, find the range of all possible values of $y$ such that $\lceil{y}\rceil\cdot\lfloor{y}\rfloor=110$. Express your answer using interval notation. | (-11, -10) |
Given the following propositions:
(1) The graph of the function $y=3^{x} (x \in \mathbb{R})$ is symmetric to the graph of the function $y=\log_{3}x (x > 0)$ with respect to the line $y=x$;
(2) The smallest positive period of the function $y=|\sin x|$ is $2\pi$;
(3) The graph of the function $y=\tan (2x+\frac{\pi}{3}... | (1)(3)(4) |
Reynald is the head of the varsity department, and he bought 145 balls. Twenty were soccer balls. There were five more basketballs than the soccer balls. Twice the number of soccer balls were tennis balls. There were ten more baseballs than the soccer balls, and the rest were volleyballs. How many were volleyballs? | There were 20 + 5 = <<20+5=25>>25 basketballs.
There were 2 x 20 = <<2*20=40>>40 tennis balls.
And, there were 20 + 10 = <<20+10=30>>30 baseballs.
So, there were 20 + 25 + 40 + 30 = 115 soccer balls, basketballs, tennis balls and baseball altogether. Therefore, Reynald bought 1... |
Ian had twenty roses. He gave six roses to his mother, nine roses to his grandmother, four roses to his sister, and he kept the rest. How many roses did Ian keep? | Ian gave a total of 6 + 9 + 4 = <<6+9+4=19>>19 roses.
Therefore, Ian kept 20 - 19 = <<20-19=1>>1 rose.
#### 1 |
Allie's making guacamole for a party. Each batch requires 4 avocados and serves about 6 people. If 42 people are going to be at the party including her, how many avocados does she need? | Each batch serves 6 people, and there are 42 people coming to the party, so she needs to make 42 / 6 = <<42/6=7>>7 batches.
Each batch requires 4 avocados, so she will need 4 * 7 = <<4*7=28>>28 avocados.
#### 28 |
Define the derivative of the $(n-1)$th derivative as the $n$th derivative $(n \in N^{*}, n \geqslant 2)$, that is, $f^{(n)}(x)=[f^{(n-1)}(x)]'$. They are denoted as $f''(x)$, $f'''(x)$, $f^{(4)}(x)$, ..., $f^{(n)}(x)$. If $f(x) = xe^{x}$, then the $2023$rd derivative of the function $f(x)$ at the point $(0, f^{(2023)}(... | -\frac{2023}{2024} |
The graph of $y^2 + 2xy + 40|x|= 400$ partitions the plane into several regions. What is the area of the bounded region? | 800 |
What is the maximum number of colours that can be used to paint an $8 \times 8$ chessboard so that every square is painted in a single colour, and is adjacent , horizontally, vertically but not diagonally, to at least two other squares of its own colour?
(A Shapovalov) | 16 |
Find the inverse of the matrix
\[\begin{pmatrix} 2 & 3 \\ -1 & 7 \end{pmatrix}.\]If the inverse does not exist, then enter the zero matrix. | \begin{pmatrix} 7/17 & -3/17 \\ 1/17 & 2/17 \end{pmatrix} |
Determine all positive integers relatively prime to all the terms of the infinite sequence \[ a_n=2^n+3^n+6^n -1,\ n\geq 1. \] | 1 |
Three fair six-sided dice, each numbered 1 through 6 , are rolled. What is the probability that the three numbers that come up can form the sides of a triangle? | 37/72 |
A robot colors natural numbers starting from 1 in ascending order according to the following rule: any natural number that can be expressed as the sum of two composite numbers is colored red; those that do not meet this requirement are colored yellow. For example, 23 can be expressed as the sum of two composite numbers... | 2001 |
A set containing three real numbers can be represented as $\{a, \frac{b}{a}, 1\}$, and also as $\{a^2, a+b, 0\}$. Find the value of $a^{2003} + b^{2004}$. | -1 |
For a natural number \( N \), if at least five out of the nine natural numbers \( 1 \) through \( 9 \) can divide \( N \) evenly, then \( N \) is called a "Five Sequential Number." What is the smallest "Five Sequential Number" greater than 2000? | 2004 |
The number
\[e^{7\pi i/60} + e^{17\pi i/60} + e^{27 \pi i/60} + e^{37\pi i /60} + e^{47 \pi i /60}\]is expressed in the form $r e^{i \theta}$, where $0 \le \theta < 2\pi$. Find $\theta$. | \dfrac{9\pi}{20} |
Find the $3 \times 3$ matrix $\mathbf{M}$ such that
\[\mathbf{M} \mathbf{v} = -4 \mathbf{v}\]for all three-dimensional vectors $\mathbf{v}.$ | \begin{pmatrix} -4 & 0 & 0 \\ 0 & -4 & 0 \\ 0 & 0 & -4 \end{pmatrix} |
What is the measure, in degrees, of one interior angle of a regular hexagon? | 120 |
Doug constructs a square window using $8$ equal-size panes of glass. The ratio of the height to width for each pane is $5 : 2$, and the borders around and between the panes are $2$ inches wide. In inches, what is the side length of the square window? | 26 |
If $\det \mathbf{M} = -2,$ then find $ \det (\mathbf{M}^4).$ | 16 |
Find the smallest constant $N$, such that for any triangle with sides $a, b,$ and $c$, and perimeter $p = a + b + c$, the inequality holds:
\[
\frac{a^2 + b^2 + k}{c^2} > N
\]
where $k$ is a constant. | \frac{1}{2} |
The equations $x^3 + Ax + 10 = 0$ and $x^3 + Bx^2 + 50 = 0$ have two roots in common. Then the product of these common roots can be expressed in the form $a \sqrt[b]{c},$ where $a,$ $b,$ and $c$ are positive integers, when simplified. Find $a + b + c.$ | 12 |
In the Cartesian coordinate system $xOy$, with the origin $O$ as the pole and the non-negative half-axis of the $x$-axis as the polar axis, a polar coordinate system is established. It is known that the polar equation of curve $C$ is $\rho^{2}= \dfrac {16}{1+3\sin ^{2}\theta }$, and $P$ is a moving point on curve $C$, ... | 2 \sqrt {2}+4 |
A ray passing through the focus $F$ of the parabola $y^2 = 4x$ intersects the parabola at point $A$. Determine the equation of the line to which a circle with diameter $FA$ must be tangent. | y=0 |
Given any point $P$ on the ellipse $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1\; \; (a > b > 0)$ with foci $F\_{1}$ and $F\_{2}$, if $\angle PF\_1F\_2=\alpha$, $\angle PF\_2F\_1=\beta$, $\cos \alpha= \frac{ \sqrt{5}}{5}$, and $\sin (\alpha+\beta)= \frac{3}{5}$, find the eccentricity of this ellipse. | \frac{\sqrt{5}}{7} |
A number like 45132 is called a "wave number," which means the tens and thousands digits are both larger than their respective neighboring digits. What is the probability of forming a non-repeating five-digit "wave number" using the digits 1, 2, 3, 4, 5? | \frac{1}{15} |
Every hour past noon shadows from a building stretch an extra 5 feet, starting at zero at noon. How long are the shadows from the building 6 hours past noon in inches? | If the shadows lengthen by 5 feet per hour, this means that in 6 hours the shadows would have lengthened from zero to 5*6=<<5*6=30>>30 feet.
Since there are 12 inches in every foot, this means the building's shadow would be 30*12= <<30*12=360>>360 inches in length.
#### 360 |
In a plane with a Cartesian coordinate system, there are 16 grid points \((i, j)\), where \(0 \leq i \leq 3\) and \(0 \leq j \leq 3\). If \(n\) points are selected from these 16 points, there will always exist 4 points among the \(n\) points that are the vertices of a square. Find the minimum value of \(n\). | 11 |
Eight students participate in a pie-eating contest. The graph shows the number of pies eaten by each participating student. Sarah ate the most pies and Tom ate the fewest. Calculate how many more pies than Tom did Sarah eat and find the average number of pies eaten by all the students.
[asy]
defaultpen(linewidth(1pt)+... | 4.5 |
The product of two inches and the circumference of a circle, in inches, is equal to the circle's area. What is the length of the radius of the circle, in inches? | 4 |
Jason has to drive home which is 120 miles away. If he drives at 60 miles per hour for 30 minutes, what speed does he have to average for the remainder of the drive to get there in exactly 1 hour 30 minutes? | Jason drives 60 miles per hour * 0.5 hours = <<60*0.5=30>>30 miles initially
He needs to drive an additional 120 miles total - 30 miles driven = <<120-30=90>>90 more miles
He has 1.5 hours - 0.5 hours = <<1.5-0.5=1>>1 hour to drive the remaining distance
He must average 90 miles per hour to drive the remaining 90 miles... |
20 points were marked inside a square and connected with non-intersecting segments to each other and to the vertices of the square, such that the square was divided into triangles. How many triangles were formed? | 42 |
A class has 50 students, and their scores in a math test $\xi$ follow a normal distribution $N(100, 10^2)$. It is known that $P(90 \leq \xi \leq 100) = 0.3$. Estimate the number of students scoring above 110. | 10 |
Find all nonnegative integer solutions $(x,y,z,w)$ of the equation\[2^x\cdot3^y-5^z\cdot7^w=1.\] | (1, 1, 1, 0), (2, 2, 1, 1), (1, 0, 0, 0), (3, 0, 0, 1) |
Three distinct diameters are drawn on a unit circle such that chords are drawn as shown. If the length of one chord is \(\sqrt{2}\) units and the other two chords are of equal lengths, what is the common length of these chords? | \sqrt{2-\sqrt{2}} |
Extend the definition of the binomial coefficient to $C_x^m = \frac{x(x-1)\dots(x-m+1)}{m!}$ where $x\in\mathbb{R}$ and $m$ is a positive integer, with $C_x^0=1$. This is a generalization of the binomial coefficient $C_n^m$ (where $n$ and $m$ are positive integers and $m\leq n$).
1. Calculate the value of $C_{-15}^3$.
... | \sqrt{2} |
All triangles have the same value, and all circles have the same value. What is the sum of three circles? \begin{align*}
\Delta + \bigcirc + \Delta + \bigcirc + \Delta&= 21\\
\bigcirc + \Delta+\bigcirc+\Delta+\bigcirc &= 19\\
\bigcirc + \bigcirc + \bigcirc &= \ ?
\end{align*} | 9 |
John decides to trade in his stereo system. His old system cost 250 dollars and he got 80% of the value for it. He then buys a system that costs $600 that he gets a 25% discount on. How much money came out of his pocket? | He got 250*.8=$<<250*.8=200>>200
He gets a 600*.25=$<<600*.25=150>>150 discount on the second system
So it cost 600-150=$<<600-150=450>>450
So he was 450-200=$<<450-200=250>>250 out of pocket
#### 250 |
If $\tan (\alpha+ \frac{\pi}{3})=2 \sqrt {3}$, find the value of $\tan (\alpha- \frac{2\pi}{3})$ and $2\sin^{2}\alpha-\cos^{2}\alpha$. | -\frac{43}{52} |
Squares of integers that are palindromes (i.e., they read the same left-to-right and right-to-left) are an interesting subject of study. For example, the squares of $1, 11, 111,$ and $1111$ are $1, 121, 12321,$ and $1234321$ respectively, and all these numbers are palindromes. This rule applies to any number of ones up... | 698896 |
If Ella rolls a standard six-sided die until she rolls the same number on consecutive rolls, what is the probability that her 10th roll is her last roll? Express your answer as a decimal to the nearest thousandth. | .039 |
Find $|3-2i|\cdot |3+2i|$. | 13 |
There is a committee composed of eight women and two men. When they meet, they sit in a row---the women in indistinguishable rocking chairs and the men on indistinguishable stools. How many distinct ways are there for me to arrange the eight chairs and two stools for a meeting? | 45 |
Tim has some cans of soda. Jeff comes by, and takes 6 cans of soda from Tim. Tim then goes and buys another half the amount of soda cans he had left. If Tim has 24 cans of soda in the end, how many cans of soda did Tim have at first? | Let x be the number of cans Tim has at first.
After Jeff comes by, Tim has x-6 cans left.
Tim buys another (x-6)/2 cans.
x-6+(x-6)/2=24
2*x-12+x-6=48
3*x-18=48
3*x=66
x=<<22=22>>22
#### 22 |
In triangle $ABC,$ $a = 7,$ $b = 9,$ and $c = 4.$ Let $I$ be the incenter.
[asy]
unitsize(0.8 cm);
pair A, B, C, D, E, F, I;
B = (0,0);
C = (7,0);
A = intersectionpoint(arc(B,4,0,180),arc(C,9,0,180));
I = incenter(A,B,C);
draw(A--B--C--cycle);
draw(incircle(A,B,C));
label("$A$", A, N);
label("$B$", B, SW);
label(... | \left( \frac{7}{20}, \frac{9}{20}, \frac{1}{5} \right) |
Solve for $x$: $4x^{1/3}-2 \cdot \frac{x}{x^{2/3}}=7+\sqrt[3]{x}$. | 343 |
One necklace is worth $34. Bob decided to buy one for his wife. But, he also bought a book, which is $5 more expensive than the necklace. Before he went shopping, Bob set a limit and decided not to spend more than $70. How many dollars over the "limit" did Bob spend? | The book is $5 more expensive than the necklace, which means the book costs 34 + 5 = $<<34+5=39>>39.
Both products which Bob wants to buy cost in total 39 + 34 = $<<39+34=73>>73.
So Bob will spend 73 - 70 = $<<73-70=3>>3 over the limit.
#### 3 |
How many points does one have to place on a unit square to guarantee that two of them are strictly less than 1/2 unit apart? | 10 |
Let $A B C$ be a triangle with $A B=13, B C=14, C A=15$. Let $O$ be the circumcenter of $A B C$. Find the distance between the circumcenters of triangles $A O B$ and $A O C$. | \frac{91}{6} |
Consider a dark square in an array of unit squares, part of which is shown. The first ring of squares around this center square, which is at a distance of 1 unit from the center, contains 8 unit squares. The second ring, at a distance of 2 units from the center, contains 16 unit squares. If the pattern of increasing di... | 400 |
Given two lines $l_{1}$: $mx+2y-2=0$ and $l_{2}$: $5x+(m+3)y-5=0$, if $l_{1}$ is parallel to $l_{2}$, determine the value of $m$. | -5 |
Consider the non-decreasing sequence of positive integers
\[1,2,2,3,3,3,4,4,4,4,5,5,5,5,5,\cdots\]
in which the $n^{th}$ positive integer appears $n$ times. The remainder when the $1993^{rd}$ term is divided by $5$ is | 3 |
The graph of $y = ax^2 + bx + c$ is shown, where $a$, $b$, and $c$ are integers. The vertex of the parabola is at $(-2, 3)$, and the point $(1, 6)$ lies on the graph. Determine the value of $a$. | \frac{1}{3} |
In the rectangular coordinate system $xoy$, the parametric equations of the curve $C$ are $x=3\cos \alpha$ and $y=\sin \alpha$ ($\alpha$ is the parameter). In the polar coordinate system with the origin as the pole and the positive semi-axis of $x$ as the polar axis, the polar equation of the line $l$ is $\rho \sin (\t... | \frac{18\sqrt{2}}{5} |
In triangle $\triangle ABC$, $\cos C=\frac{2}{3}$, $AC=4$, $BC=3$, calculate the value of $\tan B$. | 4\sqrt{5} |
Mark started the day with 14 buttons. His friend Shane gave him 3 times that amount of buttons. Then his other friend Sam asked if he could have half of Mark’s buttons. How many buttons did Mark end up with? | Shane gave Mark 14*3=<<14*3=42>>42 buttons
After that Mark had 42+14=<<42+14=56>>56 buttons
Then Mark gave 56/2=<<56/2=28>>28 buttons to Shane
Which left Mark with 28 buttons
#### 28 |
An infinite sequence of circles is composed such that each circle has a decreasing radius, and each circle touches the subsequent circle and the two sides of a given right angle. The ratio of the area of the first circle to the sum of the areas of all subsequent circles in the sequence is | $(16+12 \sqrt{2}): 1$ |
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