problem stringlengths 10 5.15k | answer stringlengths 0 1.23k |
|---|---|
Let $\mathbf{A}$ be a $2 \times 2$ matrix, with real entries, such that $\mathbf{A}^3 = \mathbf{0}.$ Find the number of different possible matrices that $\mathbf{A}^2$ can be. If you think the answer is infinite, then enter "infinite". | 1 |
We build a $4 \times 4 \times 4$ cube out of sugar cubes. How many different rectangular parallelepipeds can the sugar cubes determine, if the rectangular parallelepipeds differ in at least one sugar cube? | 1000 |
The four positive integers $a,$ $b,$ $c,$ $d$ satisfy
\[a \times b \times c \times d = 10!.\]Find the smallest possible value of $a + b + c + d.$ | 175 |
Define $a * b$ as $2a - b^2$. If $a * 5 = 9$, what is the value of $a$? | 17 |
For $n$ a positive integer, let $f(n)$ be the quotient obtained when the sum of all positive divisors of $n$ is divided by $n.$ For example, $f(14)=(1+2+7+14)\div 14=\frac{12}{7}$.
What is $f(768)-f(384)?$ | \frac{1}{192} |
Consider the sequence $(a_k)_{k\ge 1}$ of positive rational numbers defined by $a_1 = \frac{2020}{2021}$ and for $k\ge 1$, if $a_k = \frac{m}{n}$ for relatively prime positive integers $m$ and $n$, then
\[a_{k+1} = \frac{m + 18}{n+19}.\]Determine the sum of all positive integers $j$ such that the rational number $a_j$ can be written in the form $\frac{t}{t+1}$ for some positive integer $t$. | 59 |
Juanico is 4 years less than half the age of Gladys. If Gladys will be 40 years old ten years from now, calculate Juanico's age 30 years from now. | If Gladys will be 40 years old ten years from now, she is 40-10= <<40-10=30>>30 years old currently.
Juanico is 4 years less than half the age of Gladys, meaning he is 4 years younger than 1/2*30 = 15 years.
Juanico's age is 15-4 =<<15-4=11>>11 years.
Juanico's age 30 years from now will be 30+11 = <<30+11=41>>41 years.
#### 41 |
A motel bills its customers by charging a flat fee for the first night and then adding on a fixed amount for every night thereafter. If it costs George $\$155$ to stay in the motel for 3 nights and Noah $\$290$ to stay in the motel for 6 nights, how much is the flat fee for the first night? | \$65 |
A jar of peanut butter which is 3 inches in diameter and 4 inches high sells for $\$$0.60. At the same rate, what would be the price for a jar that is 6 inches in diameter and 6 inches high? | \$3.60 |
Find the largest six-digit number in which all digits are distinct, and each digit, except for the extreme ones, is equal either to the sum or the difference of its neighboring digits. | 972538 |
From milk with a fat content of $5\%$, cottage cheese with a fat content of $15.5\%$ is produced, while there remains whey with a fat content of $0.5\%$. How much cottage cheese is obtained from 1 ton of milk? | 0.3 |
Find all real numbers \( p \) such that the cubic equation \( 5x^3 - 5(p+1)x^2 + (71p-1)x + 1 = 66p \) has two roots that are natural numbers. | 76 |
How many roots does the equation
$$
\overbrace{f(f(\ldots f}^{10 \text{ times }}(x) \ldots))+\frac{1}{2}=0
$$
where $f(x)=|x|-1$ have? | 20 |
There are 90 students who have lunch during period 5. Today, two-thirds of the students sat in the cafeteria, while the remainder sat at the covered picnic tables outside. But some yellow-jackets were attracted to their food, and so one-third of the students outside jumped up and ran inside to the cafeteria, while 3 of the students in the cafeteria went outside to see what all the fuss was about. How many students are now in the cafeteria? | Originally, there were 90*2/3=<<90*2/3=60>>60 in the cafeteria.
Originally, there were 90-60=<<90-60=30>>30 outside the cafeteria.
Then 30*1/3=<<30*1/3=10>>10 ran inside
The number of students inside the cafeteria grew by 10-3=<<10-3=7>>7.
There are now 60+7=<<60+7=67>>67 students in the cafeteria.
#### 67 |
In triangle $ABC$, medians $AD$ and $CE$ intersect at $P$, $PE=2$, $PD=6$, and $DE=2\sqrt{10}$. Determine the area of quadrilateral $AEDC$. | 54 |
Given Madeline has 80 fair coins. She flips all the coins. Any coin that lands on tails is tossed again. Additionally, any coin that lands on heads in the first two tosses is also tossed again, but only once. What is the expected number of coins that are heads after these conditions? | 40 |
Given the hyperbola \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1(a,b>0)\) with left and right foci as \(F_{1}\) and \(F_{2}\), a line passing through \(F_{2}\) with an inclination angle of \(\frac{\pi}{4}\) intersects the hyperbola at a point \(A\). If the triangle \(\triangle F_{1}F_{2}A\) is an isosceles right triangle, calculate the eccentricity of the hyperbola. | \sqrt{2}+1 |
In the rhombus \(ABCD\), point \(Q\) divides side \(BC\) in the ratio \(1:3\) starting from vertex \(B\), and point \(E\) is the midpoint of side \(AB\). It is known that the median \(CF\) of triangle \(CEQ\) is equal to \(2\sqrt{2}\), and \(EQ = \sqrt{2}\). Find the radius of the circle inscribed in rhombus \(ABCD\). | \frac{\sqrt{7}}{2} |
Given a polynomial \( P(x) \) with integer coefficients. It is known that \( P(1) = 2013 \), \( P(2013) = 1 \), and \( P(k) = k \), where \( k \) is some integer. Find \( k \). | 1007 |
The coefficient of \\(x^4\\) in the expansion of \\((1+x+x^2)(1-x)^{10}\\). | 135 |
Two congruent squares, $ABCD$ and $EFGH$, each with a side length of $20$ units, overlap to form a $20$ by $35$ rectangle $AEGD$. Calculate the percentage of the area of rectangle $AEGD$ that is shaded. | 14\% |
Find $x$, such that $4^{\log_7x}=16$. | 49 |
In parallelogram $ABCD$, $AB = 38$ cm, $BC = 3y^3$ cm, $CD = 2x +4$ cm, and $AD = 24$ cm. What is the product of $x$ and $y$? | 34 |
Christian is twice as old as Brian. In eight more years, Brian will be 40 years old. How old will Christian be in eight years? | If in eight years Brian will be 40 years old, he is currently 40-8 = <<40-8=32>>32 years old.
Since Christian is twice as old as Brian, he is 2*32 = <<2*32=64>>64 years old.
In 8 years, Christian will be 64+8 = <<64+8=72>>72 years old.
#### 72 |
What is the smallest positive integer $n$ such that $\frac{n}{n+150}$ is equal to a terminating decimal? | 10 |
Triangle $ABC$ has $AB=21$, $AC=22$ and $BC=20$. Points $D$ and $E$ are located on $\overline{AB}$ and $\overline{AC}$, respectively, such that $\overline{DE}$ is parallel to $\overline{BC}$ and contains the center of the inscribed circle of triangle $ABC$. Then $DE=m/n$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
| 923 |
The sum of all of the digits of the integers from 1 to 2008 is: | 28054 |
Tom plants a tree that is 1 year old and 5 feet tall. It gains 3 feet per year. How old is it when it is 23 feet tall? | It has grown 23-5=<<23-5=18>>18 feet
So it is 18/3=<<18/3=6>>6 years older
That means it is 6+1=<<6+1=7>>7 years old
#### 7 |
In the game of rock-paper-scissors-lizard-Spock, rock defeats scissors and lizard, paper defeats rock and Spock, scissors defeats paper and lizard, lizard defeats paper and Spock, and Spock defeats rock and scissors. If three people each play a game of rock-paper-scissors-lizard-Spock at the same time by choosing one of the five moves at random, what is the probability that one player beats the other two? | \frac{12}{25} |
Let $a$, $b$, $c$, $d$, and $e$ be positive integers with $a+b+c+d+e=2510$. Let $N$ be the largest of the sums $a+b$, $b+c$, $c+d$, and $d+e$. What is the smallest possible value of $N$? | 1255 |
The constant term in the expansion of (1+x)(e^(-2x)-e^x)^9. | 84 |
A motorist left point A for point D, covering a distance of 100 km. The road from A to D passes through points B and C. At point B, the GPS indicated that 30 minutes of travel time remained, and the motorist immediately reduced speed by 10 km/h. At point C, the GPS indicated that 20 km of travel distance remained, and the motorist immediately reduced speed by another 10 km/h. (The GPS determines the remaining time based on the current speed of travel.) Determine the initial speed of the car if it is known that the journey from B to C took 5 minutes longer than the journey from C to D. | 100 |
Compute the sum of the squares of the roots of the equation \[x\sqrt{x} - 8x + 9\sqrt{x} - 2 = 0,\] given that all of the roots are real and nonnegative. | 46 |
If angle $A$ lies in the second quadrant and $\sin A = \frac{3}{4},$ find $\cos A.$ | -\frac{\sqrt{7}}{4} |
Given right triangle $ABC$ with $\angle A = 90^\circ$, $AC = 3$, $AB = 4$, and $BC = 5$, point $D$ is on side $BC$. If the perimeters of $\triangle ACD$ and $\triangle ABD$ are equal, calculate the area of $\triangle ABD$. | \frac{12}{5} |
What is the largest $5$ digit integer congruent to $17 \pmod{26}$? | 99997 |
If the line $y=kx+t$ is a tangent line to the curve $y=e^x+2$ and also a tangent line to the curve $y=e^{x+1}$, find the value of $t$. | 4-2\ln 2 |
9 seeds are divided among three pits, labeled A, B, and C, with each pit containing 3 seeds. Each seed has a 0.5 probability of germinating. If at least one seed in a pit germinates, then that pit does not need to be replanted; if no seeds in a pit germinate, then that pit needs to be replanted.
(Ⅰ) Calculate the probability that pit A does not need to be replanted;
(Ⅱ) Calculate the probability that exactly one of the three pits does not need to be replanted;
(Ⅲ) Calculate the probability that at least one pit needs to be replanted. (Round to three decimal places). | 0.330 |
In the expansion of $(x+1)^{42}$, what is the coefficient of the $x^2$ term? | 861 |
Consider all non-empty subsets of the set \( S = \{1, 2, \cdots, 10\} \). A subset is called a "good subset" if the number of even numbers in the subset is not less than the number of odd numbers. How many "good subsets" are there? | 637 |
A right isosceles triangle is inscribed in a triangle with a base of 30 and a height of 10 such that its hypotenuse is parallel to the base of the given triangle, and the vertex of the right angle lies on this base. Find the hypotenuse. | 12 |
Define $\#N$ by the formula $\#N = .5(N) + 1$. Calculate $\#(\#(\#58))$. | 9 |
Vasya has three cans of paint of different colors. In how many different ways can he paint a fence of 10 boards such that any two adjacent boards are of different colors, and all three colors are used? | 1530 |
Mr. Callen bought 10 paintings at $40 each and 8 wooden toys at $20 each from the crafts store to resell at a profit. However, when he sold the items, the selling price of a painting was 10% less and the selling price of a hat 15% less. Calculate the total loss Mr. Callen made from the sale of the items. | If he sold a painting at a 10% loss, then he made a 10/100*$40 = $<<10/100*40=4>>4 loss on each painting.
since he bought 10 paintings, the total loss he made from selling the paintings is 10*$4 = $<<10*4=40>>40
He also made a loss of 15/100*20 = $<<15/100*20=3>>3 loss from selling each wooden toy.
Since he bought 8 wooden toys, the total loss he made was 3*8 = $<<8*3=24>>24
In total, Mr. Callen made a loss of $40+$24 = $<<40+24=64>>64 from the sales of the items.
#### 64 |
Given that $n$ is a positive integer, find the minimum value of $|n-1| + |n-2| + \cdots + |n-100|$. | 2500 |
The sequence starts at 2,187,000 and each subsequent number is created by dividing the previous number by 3. What is the last integer in this sequence? | 1000 |
Given the function $f(x) = x^{2-m}$ is defined on the interval $[-3-m, m^2-m]$ and is an odd function, then $f(m) = $ ? | -1 |
Given the real numbers \( x \) and \( y \) that satisfy
\[ x + y = 3 \]
\[ \frac{1}{x + y^2} + \frac{1}{x^2 + y} = \frac{1}{2} \]
find the value of \( x^5 + y^5 \). | 123 |
A trauma hospital uses a rectangular piece of white cloth that is 60m long and 0.8m wide to make triangular bandages with both base and height of 0.4m. How many bandages can be made in total? | 600 |
Each week Jaime saves $50. Every two weeks she spends $46 of her savings on a nice lunch with her mum. How long will it take her to save $135? | Jamie saves 50 * 2 = $<<50*2=100>>100 every two weeks.
After the lunch with her mum, Jamie saves $100 - $46 = $<<100-46=54>>54 every two weeks.
Jamie saves $54 / 2 = $<<54/2=27>>27 per week.
Jamie will need 135 / 27 = <<135/27=5>>5 weeks to save enough.
#### 5 |
The number
\[\text{cis } 75^\circ + \text{cis } 83^\circ + \text{cis } 91^\circ + \dots + \text{cis } 147^\circ\]is expressed in the form $r \, \text{cis } \theta$, where $r > 0$ and $0^\circ \le \theta < 360^\circ$. Find $\theta$ in degrees. | 111^\circ |
Let $a_1,$ $a_2,$ $\dots$ be a sequence of positive real numbers such that
\[a_n = 11a_{n - 1} - n\]for all $n > 1.$ Find the smallest possible value of $a_1.$ | \frac{21}{100} |
Mário completed 30 hours in an extra math course. On days when he had class, it lasted only 1 hour, and it took place exclusively in the morning or in the afternoon. Additionally, there were 20 afternoons and 18 mornings without classes during the course period.
a) On how many days were there no classes?
b) How many days did the course last? | 34 |
A $40$ feet high screen is put on a vertical wall $10$ feet above your eye-level. How far should you stand to maximize the angle subtended by the screen (from top to bottom) at your eye? | 10\sqrt{5} |
In triangle $ABC,$ $E$ lies on $\overline{AC}$ such that $AE:EC = 2:1,$ and $F$ lies on $\overline{AB}$ such that $AF:FB = 1:4.$ Let $P$ be the intersection of $\overline{BE}$ and $\overline{CF}.$
[asy]
unitsize(0.8 cm);
pair A, B, C, D, E, F, P;
A = (1,4);
B = (0,0);
C = (6,0);
E = interp(A,C,2/3);
F = interp(A,B,1/5);
P = extension(B,E,C,F);
draw(A--B--C--cycle);
draw(B--E);
draw(C--F);
label("$A$", A, N);
label("$B$", B, SW);
label("$C$", C, SE);
label("$E$", E, NE);
label("$F$", F, W);
label("$P$", P, S);
[/asy]
Then
\[\overrightarrow{P} = x \overrightarrow{A} + y \overrightarrow{B} + z \overrightarrow{C},\]where $x,$ $y,$ and $z$ are constants such that $x + y + z = 1.$ Enter the ordered triple $(x,y,z).$ | \left( \frac{4}{13}, \frac{1}{13}, \frac{8}{13} \right) |
Grandpa is twice as strong as Grandma, Grandma is three times as strong as Granddaughter, Granddaughter is four times as strong as Doggie, Doggie is five times as strong as Cat, and Cat is six times as strong as Mouse. Grandpa, Grandma, Granddaughter, Doggie, and Cat together with Mouse can pull up the Turnip, but without Mouse they can't. How many Mice are needed so that they can pull up the Turnip on their own? | 1237 |
Compute
$$
\lim _{h \rightarrow 0} \frac{\sin \left(\frac{\pi}{3}+4 h\right)-4 \sin \left(\frac{\pi}{3}+3 h\right)+6 \sin \left(\frac{\pi}{3}+2 h\right)-4 \sin \left(\frac{\pi}{3}+h\right)+\sin \left(\frac{\pi}{3}\right)}{h^{4}}
$$ | \frac{\sqrt{3}}{2} |
Compared to the amount of water she drank, Carla drank three times as much soda minus 6 ounces. If she drank 54 ounces of liquid total, how much water did she drink? | Let s be the amount of soda Carla drank and w be the amount of water she drank. We know that s + w = 54 and s = 3w - 6. Substituting the second equation into the first, we get 3w - 6 + w = 54.
Combining like terms, we get 4w - 6 = 54
Adding 6 to both sides, we get 4w = 60
Dividing both sides by 4, we get w = 15
#### 15 |
Noelle needs to follow specific guidelines to earn homework points: For each of the first ten homework points she wants to earn, she needs to do one homework assignment per point. For each homework point from 11 to 15, she needs two assignments; for each point from 16 to 20, she needs three assignments and so on. How many homework assignments are necessary for her to earn a total of 30 homework points? | 80 |
What is the remainder when $2001 \cdot 2002 \cdot 2003 \cdot 2004 \cdot 2005$ is divided by 19? | 11 |
If 260 were expressed as a sum of at least three distinct powers of 2, what would be the least possible sum of the exponents of these powers? | 10 |
The equation $y = -16t^2 + 80t$ describes the height (in feet) of a projectile launched from the ground at 80 feet per second. At what $t$ will the projectile reach 36 feet in height for the first time? Express your answer as a decimal rounded to the nearest tenth. | 0.5 |
The average age of the 10 females in a choir is 30 years. The average age of the 15 males in the same choir is 35 years. What is the average age, in years, of the 25 people in the choir? | 33 |
If $z$ is a complex number such that
\[
z + z^{-1} = 2\sqrt{2},
\]
what is the value of
\[
z^{100} + z^{-100} \, ?
\] | -2 |
The students in Mrs. Reed's English class are reading the same $760$-page novel. Three friends, Alice, Bob and Chandra, are in the class. Alice reads a page in $20$ seconds, Bob reads a page in $45$ seconds and Chandra reads a page in $30$ seconds. Chandra and Bob, who each have a copy of the book, decide that they can save time by `team reading' the novel. In this scheme, Chandra will read from page $1$ to a certain page and Bob will read from the next page through page $760,$ finishing the book. When they are through they will tell each other about the part they read. What is the last page that Chandra should read so that she and Bob spend the same amount of time reading the novel? | 456 |
If $a$ and $b$ are positive numbers such that $a^b=b^a$ and $b=9a$, then the value of $a$ is | \sqrt[4]{3} |
A new train goes $20\%$ farther than an older train in the same amount of time. During the time it takes the older train to go 200 miles, how many miles can the newer train complete? | 240 |
Farmer James wishes to cover a circle with circumference $10 \pi$ with six different types of colored arcs. Each type of arc has radius 5, has length either $\pi$ or $2 \pi$, and is colored either red, green, or blue. He has an unlimited number of each of the six arc types. He wishes to completely cover his circle without overlap, subject to the following conditions: Any two adjacent arcs are of different colors. Any three adjacent arcs where the middle arc has length $\pi$ are of three different colors. Find the number of distinct ways Farmer James can cover his circle. Here, two coverings are equivalent if and only if they are rotations of one another. In particular, two colorings are considered distinct if they are reflections of one another, but not rotations of one another. | 93 |
Let's consider two fictional states, Sunland and Moonland, which have different license plate formats. Sunland license plates have the format LLDDLLL (where 'L' stands for a letter and 'D' for a digit), while Moonland license plates have the format LLLDDD. Assuming all 10 digits and all 26 letters are equally likely to appear in their respective positions, calculate how many more license plates can Sunland issue than Moonland. | 1170561600 |
The sides of a triangle have lengths $11, 15,$ and $k,$ where $k$ is a positive integer. For how many values of $k$ is the triangle obtuse? | 13 |
Let the function $f(x) = 2\cos^2x + 2\sqrt{3}\sin x\cos x + m$.
(1) Find the smallest positive period of the function $f(x)$ and its intervals of monotonic decrease;
(2) If $x \in \left[0, \frac{\pi}{2}\right]$, does there exist a real number $m$ such that the range of the function $f(x)$ is exactly $\left[\frac{1}{2}, \frac{7}{2}\right]$? If it exists, find the value of $m$; if not, explain why. | \frac{1}{2} |
Let $U$ be a positive integer whose only digits are 0s and 1s. If $Y = U \div 18$ and $Y$ is an integer, what is the smallest possible value of $Y$? | 61728395 |
It is possible to arrange eight of the nine numbers $2, 3, 4, 7, 10, 11, 12, 13, 15$ in the vacant squares of the $3$ by $4$ array shown on the right so that the arithmetic average of the numbers in each row and in each column is the same integer. Exhibit such an arrangement, and specify which one of the nine numbers must be left out when completing the array.
[asy]
defaultpen(linewidth(0.7));
for(int x=0;x<=4;++x)
draw((x+.5,.5)--(x+.5,3.5));
for(int x=0;x<=3;++x)
draw((.5,x+.5)--(4.5,x+.5));
label(" $1$ ",(1,3));
label(" $9$ ",(2,2));
label(" $14$ ",(3,1));
label(" $5$ ",(4,2));[/asy] | 10 |
The total number of books on four shelves, with 400 books each, is the same as the distance in miles that Karen bikes back to her home from the library. Calculate the total distance that Karen covers if she bikes from her home to the library and back. | If each shelf has 400 books, then the four shelves have 4*400 = 1600 books.
Since the total number of books on the shelves is the same as the distance that Karen bikes back home, biking to and from the library takes 1600+1600 = <<1600+1600=3200>>3200 miles
#### 3200 |
If $\mathbf{a},$ $\mathbf{b},$ and $\mathbf{c}$ are vectors such that $\mathbf{a} \cdot \mathbf{b} = -3,$ $\mathbf{a} \cdot \mathbf{c} = 4,$ and $\mathbf{b} \cdot \mathbf{c} = 6,$ then find
\[\mathbf{b} \cdot (7 \mathbf{c} - 2 \mathbf{a}).\] | 48 |
Simplify and evaluate the following expressions:
\\((1){{(\\dfrac{9}{4})}^{\\frac{1}{2}}}{-}{{({-}2017)}^{0}}{-}{{(\\dfrac{27}{8})}^{\\frac{2}{3}}}\\)
\\((2)\\lg 5+{{(\\lg 2)}^{2}}+\\lg 5\\bullet \\lg 2+\\ln \\sqrt{e}\\) | \frac{3}{2} |
In the expansion of $(\frac{3}{{x}^{2}}+x+2)^{5}$, the coefficient of the linear term in $x$ is ____. | 200 |
The longer leg of a right triangle is $1$ foot shorter than twice the length of the shorter leg. The area of the triangle is $60$ square feet. What is the length of the hypotenuse, in feet? | 17 |
In a kindergarten's junior group, there are two identical small Christmas trees and five children. The teachers want to divide the children into two circles around each tree, with at least one child in each circle. The teachers distinguish the children but do not distinguish the trees: two such divisions into circles are considered the same if one can be obtained from the other by swapping the trees (along with their respective circles) and rotating each circle around its tree. In how many ways can the children be divided into the circles? | 50 |
Smaug the dragon hoards 100 gold coins, 60 silver coins, and 33 copper coins. If each silver coin is worth 8 copper coins, and each gold coin is worth 3 silver coins, what is the total value of Smaug's hoard expressed as a number of copper coins? | First figure out how many silver coins the 100 gold coins are worth by multiplying the number of gold coins by the exchange rate between gold and silver: 100 gold * 3 silver/gold = <<100*3=300>>300 silver
Then add the value of the gold in silver to the number of silver coins to find the total value of the gold and silver expressed in silver coins: 300 silver + 60 silver = <<300+60=360>>360 silver
Now multiply that value by the exchange rate between silver and copper to express its value in terms of copper coins: 360 silver * 8 copper/silver = <<360*8=2880>>2880 copper
Then add the value of the gold and silver expressed in copper coins (the value from the last step) to the number of copper coins to find the total value of the hoard: 2880 + 33 = <<2880+33=2913>>2913
#### 2913 |
Find the point of intersection of the asymptotes of the graph of
\[y = \frac{x^2 - 4x + 3}{x^2 - 4x + 4}.\] | (2,1) |
Given that \( x - \frac{1}{x} = \sqrt{3} \), find \( x^{2048} - \frac{1}{x^{2048}} \). | 277526 |
Kanga labelled the vertices of a square-based pyramid using \(1, 2, 3, 4,\) and \(5\) once each. For each face, Kanga calculated the sum of the numbers on its vertices. Four of these sums equaled \(7, 8, 9,\) and \(10\). What is the sum for the fifth face? | 13 |
Find a 4-digit perfect square, knowing that the number formed by the first two digits is one more than the number formed by the last two digits. | 8281 |
John and his two brothers decide to split the cost of an apartment. It is 40% more expensive than John's old apartment which costs $1200 per month. How much does John save per year by splitting the apartment compared to living alone? | His old apartment cost 1200*12=$<<1200*12=14400>>14,400 per year
His new apartment 1200*1.4=$<<1200*1.4=1680>>1680 per month
That means it cost 1680/3=$<<1680/3=560>>560 per month
So it cost 560*12=$<<560*12=6720>>6720
So he saves 14400-6720=$<<14400-6720=7680>>7680
#### 7680 |
Ted needs to purchase 5 bananas and 10 oranges. If bananas cost $2 each and oranges cost $1.50 each. How much money does Ted need to purchase 5 bananas and 10 oranges? | Ted needs 5 bananas and they each cost $2. To purchase the bananas, Ted needs 5*2= <<5*2=10>>10 dollars
Because Ted needs 10 oranges and they each cost $1.50, Ted needs 10*1.5= <<10*1.5=15>>15 dollars to pay for the oranges.
All together Ted needs $10 for the bananas and $15 for the oranges, so combined Ted needs 10+15= <<10+15=25>>25 dollars to purchase all the produce.
#### 25 |
In triangle ABC, medians AD and BE intersect at centroid G. The midpoint of segment AB is F. Given that the area of triangle GFC is l times the area of triangle ABC, find the value of l. | \frac{1}{3} |
In $\triangle ABC$, it is known that $\cos C+(\cos A- \sqrt {3}\sin A)\cos B=0$.
(1) Find the measure of angle $B$;
(2) If $\sin (A- \frac {π}{3})= \frac {3}{5}$, find $\sin 2C$. | \frac {24+7 \sqrt {3}}{50} |
It takes David 10 minutes to wash 4 windows. David's house has 64 windows. How many minutes will it take David to wash all of the windows? | It takes 10 minutes to wash 4 windows and he has 64 windows so that breaks down to 64/4 = 16 units
It takes 10 minutes to wash a unit of windows and he has 16 units so 10*16 = <<10*16=160>>160 minutes
#### 160 |
What is the first year after 2000 for which the sum of the digits is 15? | 2049 |
Let $X \sim B(4, p)$, and $P(X=2)=\frac{8}{27}$, find the probability of success in one trial. | \frac{2}{3} |
Compute the following expression:
\[
\frac{(1 + 15) \left( 1 + \dfrac{15}{2} \right) \left( 1 + \dfrac{15}{3} \right) \dotsm \left( 1 + \dfrac{15}{17} \right)}{(1 + 17) \left( 1 + \dfrac{17}{2} \right) \left( 1 + \dfrac{17}{3} \right) \dotsm \left( 1 + \dfrac{17}{15} \right)}.
\] | 496 |
The distance a dog covers in 3 steps is the same as the distance a fox covers in 4 steps and the distance a rabbit covers in 12 steps. In the time it takes the rabbit to run 10 steps, the dog runs 4 steps and the fox runs 5 steps. Initially, the distances between the dog, fox, and rabbit are as shown in the diagram. When the dog catches up to the fox, the rabbit says: "That was close! If the dog hadn’t caught the fox, I would have been caught by the fox after running $\qquad$ more steps." | 40 |
The space station, Lupus-1, is an enormous spacecraft made up of three identical cylindrical structures that house the living quarters for the crew. The three cylindrical structures are linked together by a series of tunnels that hold them together and allow the crew to move between cylinders. Each cylindrical structure contains 12 bedrooms, 7 bathrooms, and several kitchens. If the entire space station has 72 rooms, how many kitchens are there on the entire space station? | If there are 3 identical cylindrical structures, with a combined total of 72 rooms, then each individual cylindrical structure contains 72/3=<<72/3=24>>24 rooms.
In each 24-room cylindrical structure, there are 24-12-7=<<24-12-7=5>>5 kitchens.
Thus, the entire, three-cylinder space station contains a total of 3*5=<<3*5=15>>15 kitchens.
#### 15 |
A deck of 60 cards, divided into 5 suits of 12 cards each, is shuffled. In how many ways can we pick three different cards in sequence? (Order matters, so picking card A, then card B, then card C is different from picking card B, then card A, then card C.) | 205320 |
James decides to replace his car. He sold his $20,000 car for 80% of its value and then was able to haggle to buy a $30,000 sticker price car for 90% of its value. How much was he out of pocket? | He sold his car for 20000*.8=$<<20000*.8=16000>>16,000
He bought the new car for 30,000*.9=$<<30000*.9=27000>>27,000
That means he was out of pocket 27,000-16,000=$<<27000-16000=11000>>11,000
#### 11000 |
Micah can type 20 words per minute and Isaiah can type 40 words per minute. How many more words can Isaiah type than Micah in an hour? | There are 60 minutes in an hour.
Micah can type 20 x 60 = <<20*60=1200>>1200 words in an hour.
Isaiah can type 40 x 60 = <<2400=2400>>2400 words in an hour.
Isaiah can type 2400 - 1200 = <<2400-1200=1200>>1200 words more than Micah in an hour.
#### 1200 |
Let \( f(x) = \frac{1}{x^3 + 3x^2 + 2x} \). Determine the smallest positive integer \( n \) such that
\[ f(1) + f(2) + f(3) + \cdots + f(n) > \frac{503}{2014}. \] | 44 |
There are two types of products, A and B, with profits of $p$ ten-thousand yuan and $q$ ten-thousand yuan, respectively. Their relationship with the invested capital $x$ ten-thousand yuan is: $p= \frac{1}{5}x$, $q= \frac{3}{5} \sqrt{x}$. Now, with an investment of 3 ten-thousand yuan in managing these two products, how much capital should be allocated to each product in order to maximize profit, and what is the maximum profit? | \frac{21}{20} |
Jacob loves to build things. In Jacob's toy bin there are 18 red blocks. There are 7 more yellow blocks than red blocks. There are also 14 more blue blocks than red blocks. How many blocks are there in all? | The number of yellow blocks is 18 blocks + 7 blocks = <<18+7=25>>25 blocks.
The number of blue blocks is 18 blocks + 14 blocks = <<18+14=32>>32 blocks.
The total number of red, yellow, and blue blocks is 18 blocks + 25 blocks + 32 blocks = <<18+25+32=75>>75 blocks.
#### 75 |
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