problem stringlengths 10 5.15k | answer stringlengths 0 1.23k |
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The school decided to add 20% to the gym budget. In a normal month, they could buy 15 dodgeballs for $5 each if they spent the entire budget on that. If they only buy softballs instead, which cost $9 each, how many can they buy with the new budget? | The old budget was $75 because 15 x 5 = <<15*5=75>>75
They are adding $15 to the budget because 75 x .2 = <<75*.2=15>>15
The new budget is $90 because 75 + 15 = <<75+15=90>>90
They can buy 10 softballs on the new budget because 90 / 9 = <<90/9=10>>10
#### 10 |
Carl is hosting an open house for his new business. He knows 50 people will show up and hopes that another 40 people will randomly show up. He’s created 10 extravagant gift bags for the first 10 people who visit his shop. He’s made average gift bags for 20 people but needs to make enough for everyone who visits to h... | He knows 50 people are showing up and he’s hoping 40 more people will drop in for a total of 50+40 = <<50+40=90>>90 people
He’s made 10 extravagant gift bags and 20 average bags for a total of 10+20 = <<10+20=30>>30 bags
He hopes 90 people will show up and has already made 30 bags so he needs to make 90-30 = <<90-30=60... |
A set of teams held a round-robin tournament in which every team played every other team exactly once. Every team won $10$ games and lost $10$ games; there were no ties. How many sets of three teams $\{A, B, C\}$ were there in which $A$ beat $B$, $B$ beat $C$, and $C$ beat $A$? | 385 |
For $k > 0$, let $I_k = 10\ldots 064$, where there are $k$ zeros between the $1$ and the $6$. Let $N(k)$ be the number of factors of $2$ in the prime factorization of $I_k$. What is the maximum value of $N(k)$?
$\textbf{(A)}\ 6\qquad \textbf{(B)}\ 7\qquad \textbf{(C)}\ 8\qquad \textbf{(D)}\ 9\qquad \textbf{(E)}\ 10$
| 7 |
Express $\frac{31}{2\cdot5^6}$ as a terminating decimal. | 0.000992 |
Given that the complex number $z\_1$ satisfies $((z\_1-2)(1+i)=1-i)$, the imaginary part of the complex number $z\_2$ is $2$, and $z\_1z\_2$ is a real number, find $z\_2$ and $|z\_2|$. | 2 \sqrt {5} |
The roots of the equation $x^2+kx+5 = 0$ differ by $\sqrt{61}$. Find the greatest possible value of $k$. | 9 |
The quantity $\tan 7.5^\circ$ can be expressed in the form
\[\tan 7.5^\circ = \sqrt{a} - \sqrt{b} + \sqrt{c} - d,\]where $a \ge b \ge c \ge d$ are positive integers. Find $a + b + c + d.$ | 13 |
Let $a_{n+1} = \frac{4}{7}a_n + \frac{3}{7}a_{n-1}$ and $a_0 = 1$ , $a_1 = 2$ . Find $\lim_{n \to \infty} a_n$ . | 1.7 |
Let $x$ be a real number, $x > 1.$ Compute
\[\sum_{n = 0}^\infty \frac{1}{x^{2^n} - x^{-2^n}}.\] | \frac{1}{x - 1} |
Cozy the Cat and Dash the Dog are going up a staircase with a certain number of steps. However, instead of walking up the steps one at a time, both Cozy and Dash jump. Cozy goes two steps up with each jump (though if necessary, he will just jump the last step). Dash goes five steps up with each jump (though if necessar... | 13 |
In the Cartesian coordinate system $xOy$, with the origin as the pole and the positive half-axis of the $x$-axis as the polar axis, the polar coordinate equation of the curve $C_{1}$ is $\rho \cos \theta = 4$.
$(1)$ Let $M$ be a moving point on the curve $C_{1}$, point $P$ lies on the line segment $OM$, and satisfies... | 2 + \sqrt{3} |
Compute $\sin 510^\circ$. | \frac{1}{2} |
In the book "Nine Chapters on the Mathematical Art," a tetrahedron with all four faces being right-angled triangles is called a "biēnào." Given that tetrahedron $ABCD$ is a "biēnào," $AB\bot $ plane $BCD$, $BC\bot CD$, and $AB=\frac{1}{2}BC=\frac{1}{3}CD$. If the volume of this tetrahedron is $1$, then the surface area... | 14\pi |
Mark buys a Magic card for $100, which then triples in value. How much profit would he make selling it? | The value went up to 100*3=$<<100*3=300>>300
So he gained 300-100=$<<300-100=200>>200
#### 200 |
Every day, while going to school, Shane drives a total of 10 miles. Calculate the total distance he drives in an 80 day semester from home to school and back. | To go to school and back, Shane drives 2*10 = <<2*10=20>>20 miles per day.
In 80 days of a semester, he drives for 80*20 = <<80*20=1600>>1600 miles
#### 1600 |
Given the set $H$ defined by the points $(x,y)$ with integer coordinates, $2\le|x|\le8$, $2\le|y|\le8$, calculate the number of squares of side at least $5$ that have their four vertices in $H$. | 14 |
Given vectors $\overrightarrow{a}=(1,-2)$ and $\overrightarrow{b}=(3,4)$, find the projection of vector $\overrightarrow{a}$ onto the direction of vector $\overrightarrow{b}$. | -1 |
Let \(A B C\) be an acute triangle with circumcenter \(O\) such that \(A B=4, A C=5\), and \(B C=6\). Let \(D\) be the foot of the altitude from \(A\) to \(B C\), and \(E\) be the intersection of \(A O\) with \(B C\). Suppose that \(X\) is on \(B C\) between \(D\) and \(E\) such that there is a point \(Y\) on \(A D\) s... | \frac{96}{41} |
Real numbers $x$ and $y$ have an arithmetic mean of 18 and a geometric mean of $\sqrt{92}$. Find $x^2+y^2$. | 1112 |
In the expansion of $\left(a - \dfrac{1}{\sqrt{a}}\right)^7$ the coefficient of $a^{-\frac{1}{2}}$ is: | -21 |
In the grid made up of $1 \times 1$ squares, four digits of 2015 are written in the shaded areas. The edges are either horizontal or vertical line segments, line segments connecting the midpoints of adjacent sides of $1 \times 1$ squares, or the diagonals of $1 \times 1$ squares. What is the area of the shaded portion ... | $47 \frac{1}{2}$ |
What is the height of Jack's house, in feet, if the house casts a shadow 56 feet long at the same time a 21-foot tree casts a shadow that is 24 feet long? Express your answer to the nearest whole number. | 49 |
Millie, Monica, and Marius are taking subjects for school. Millie takes three more subjects than Marius, who takes 4 subjects more than Monica. If Monica took 10 subjects, how many subjects all take altogether? | If Monica took ten subjects, Marius took 10+4 = <<10+4=14>>14 subjects.
Marius and Monica have 14+10 = <<14+10=24>>24 subjects which they've taken.
Since Millie is taking three more subjects than Marius, she is taking 14+3 = <<14+3=17>>17 subjects
Together, the three have taken 24+17 = <<24+17=41>>41 subjects.
#### 41 |
John builds a toy bridge to support various weights. It needs to support 6 cans of soda that have 12 ounces of soda. The cans weigh 2 ounces empty. He then also adds 2 more empty cans. How much weight must the bridge hold up? | The weight of soda was 6*12=<<6*12=72>>72 ounces
It had to support the weight of 6+2=<<6+2=8>>8 empty cans
The weight of the empty cans is 8*2=<<8*2=16>>16 ounces
So it must support 72+16=<<72+16=88>>88 ounces
#### 88 |
Tom, Tim, and Paul are collecting photos of cars. Paul has 10 photos more than Tim. Tim has one hundred photos less than the total amount of photos which is 152. How many photos does Tom have? | Tim has 152 photos - 100 photos = <<152-100=52>>52 photos.
When Tim has 52 photos, then Paul has 52 + 10 photos = <<52+10=62>>62 photos.
Tim and Paul have together 52 photos + 62 photos = <<52+62=114>>114 photos.
That leaves Tom with 152 photos - 114 photos = <<152-114=38>>38 photos.
#### 38 |
In the diagram, points $A$, $B$, $C$, $D$, $E$, and $F$ lie on a straight line with $AB=BC=CD=DE=EF=3$. Semicircles with diameters $AF$, $AB$, $BC$, $CD$, $DE$, and $EF$ create a shape as depicted. What is the area of the shaded region underneath the largest semicircle that exceeds the areas of the other semicircles co... | \frac{45}{2}\pi |
Given a triangle $ABC$ with sides $a$, $b$, $c$, and area $S$ satisfying $S=a^{2}-(b-c)^{2}$, and $b+c=8$.
$(1)$ Find $\cos A$;
$(2)$ Find the maximum value of $S$. | \frac{64}{17} |
Given the function $f(x)=(2-a)(x-1)-2\ln x$ $(a\in \mathbb{R})$.
(Ⅰ) If the tangent line at the point $(1,g(1))$ on the curve $g(x)=f(x)+x$ passes through the point $(0,2)$, find the interval where the function $g(x)$ is decreasing;
(Ⅱ) If the function $y=f(x)$ has no zeros in the interval $\left(0, \frac{1}{2}\right... | 2-4\ln 2 |
How many integers fall between $\sqrt7$ and $\sqrt{77}$ on a number line? | 6 |
Find one third of 7.2, expressed as a simplified fraction or a mixed number. | 2 \frac{2}{5} |
The quadratic polynomial \( f(x) = a x^{2} + b x + c \) has exactly one root, and the quadratic polynomial \( f(3x + 2) - 2f(2x - 1) \) also has exactly one root. Find the root of the polynomial \( f(x) \). | -7 |
Let $b$ and $c$ be real numbers. If the polynomial $x^2+bx+c$ has exactly one real root and $b=c+1$, find the value of the product of all possible values of $c$. | 1 |
In an arcade game, the "monster" is the shaded sector of a circle of radius $1$ cm, as shown in the figure. The missing piece (the mouth) has central angle $60^\circ$. What is the perimeter of the monster in cm? | \frac{5}{3}\pi + 2 |
Eugene, Brianna, and Katie are going on a run. Eugene runs at a rate of 4 miles per hour. If Brianna runs $\frac{2}{3}$ as fast as Eugene, and Katie runs $\frac{7}{5}$ as fast as Brianna, how fast does Katie run? | \frac{56}{15} |
The number of significant digits in the measurement of the side of a square whose computed area is $1.1025$ square inches to the nearest ten-thousandth of a square inch is: | 5 |
The fictional country of Isoland uses a 6-letter license plate system using the same 12-letter alphabet as the Rotokas of Papua New Guinea (A, E, G, I, K, O, P, R, T, U, V). Design a license plate that starts with a vowel (A, E, I, O, U), ends with a consonant (G, K, P, R, T, V), contains no repeated letters and does n... | 151200 |
Three cards are dealt at random from a standard deck of 52 cards. What is the probability that the first card is a 4, the second card is a $\clubsuit$, and the third card is a 2? | \frac{1}{663} |
Positive numbers \(a\), \(b\), and \(c\) satisfy the following equations:
\[ a^{2} + a b + b^{2} = 1 \]
\[ b^{2} + b c + c^{2} = 3 \]
\[ c^{2} + c a + a^{2} = 4 \]
Find \(a + b + c\). | \sqrt{7} |
In triangle $ABC$, $BC = 8$. The length of median $AD$ is 5. Let $M$ be the largest possible value of $AB^2 + AC^2$, and let $m$ be the smallest possible value. Find $M - m$. | 0 |
Anya, Vanya, Dania, Sanya, and Tanya were collecting apples. It turned out that each of them collected an integer percentage of the total number of apples, and all these percentages are different and greater than zero. What is the minimum number of apples that could have been collected? | 20 |
Jennie makes quilts. She can make 7 quilts with 21 yards of material. How many yards of material would be required to make 12 quilts? | The amount of material that Jennie needs per quilts is 21 yards / 7 quilts = <<21/7=3>>3 yards per quilt.
In order to make 12 quilts, Jennie will need 12 quilts * 3 yards = <<12*3=36>>36 yards.
#### 36 |
James runs a TV show and there are 5 main characters and 4 minor characters. He pays the minor characters $15,000 each episode. He paid the major characters three times as much. How much does he pay per episode? | He pays the minor characters 15000*4=$<<15000*4=60000>>60,000
The major characters each get 15000*3=$<<15000*3=45000>>45,000
So he pays the main characters 45,000*5=$<<45000*5=225000>>225,000
So in total he pays 225,000+60,000=$<<225000+60000=285000>>285,000
#### 285,000 |
Colston knows that his teacher loves drinking coffee and one day wants to see how much she drinks each week. He sees that she has a 20-ounce thermos and when she makes her coffee she pours a 1/2 cup of milk in, and then fills the coffee to the top. She does this twice a day. After the five-day school week, he decides t... | 1/2 a cup = 4 ounces
Each thermos contains 16 ounces of coffee because 20 - 4 = <<20-4=16>>16
She drinks 32 ounces a day because 2 x 16 = <<2*16=32>>32
She drinks 160 ounces a week because 32 x 5 = <<32*5=160>>160
She starts drinking only 40 ounces because 160 / 4 = <<160/4=40>>40
#### 40 |
In the vertices of a unit square, perpendiculars are erected to its plane. On them, on one side of the plane of the square, points are taken at distances of 3, 4, 6, and 5 from this plane (in order of traversal). Find the volume of the polyhedron whose vertices are the specified points and the vertices of the square. | 4.5 |
In an isosceles triangle \(ABC\) with \(\angle B\) equal to \(30^{\circ}\) and \(AB = BC = 6\), the altitude \(CD\) of triangle \(ABC\) and the altitude \(DE\) of triangle \(BDC\) are drawn.
Find \(BE\). | 4.5 |
What is the smallest positive integer $n$ such that all the roots of $z^4 - z^2 + 1 = 0$ are $n^{\text{th}}$ roots of unity? | 12 |
Seven people stand in a row. (Write out the necessary process, and use numbers for the answers)
(1) How many ways can person A and person B stand next to each other?
(2) How many ways can person A and person B stand not next to each other?
(3) How many ways can person A, person B, and person C stand so that no tw... | 4320 |
Mrs. Fredrickson has 80 chickens where 1/4 are roosters and the rest are hens. Only three-fourths of those hens lay eggs. How many chickens does Mr. Fredrickson have that do not lay eggs? | Mrs. Fredrickson has 80 x 1/4 = <<80*1/4=20>>20 roosters.
So he has 80 - 20 = <<80-20=60>>60 hens.
Only 60 x 3/4 = <<60*3/4=45>>45 of the hens lay eggs.
Thus, 60 - 45 = <<60-45=15>>15 hens do not lay eggs.
Therefore, Mr. Fredrickson has a total of 15 hens + 20 roosters = <<15+20=35>>35 chickens that do not lay eggs.
##... |
The numbers \(1, 2, 3, \ldots, 400\) are written on 400 cards. Two players, \(A\) and \(B\), play the following game:
1. In the first step, \(A\) takes 200 cards for themselves.
2. \(B\) then takes 100 cards from both the remaining 200 cards and the 200 cards that \(A\) has, totaling 200 cards for themselves, and leav... | 20000 |
Micheal decided to take some piano lessons. One lesson costs $30 and lasts for 1.5 hours. How much will Micheal need to pay for 18 hours of lessons? | One lesson lasts 1.5 hours, so 18 hours of them is 18 / 1.5 = <<18/1.5=12>>12 lessons.
If one lesson costs $30, then 12 lessons cost 12 * 30 = $<<12*30=360>>360.
#### 360 |
If $\log_M{N}=\log_N{M}$, $M \ne N$, $MN>0$, $M \ne 1$, $N \ne 1$, then $MN$ equals: | 1 |
Brian has a 20-sided die with faces numbered from 1 to 20, and George has three 6-sided dice with faces numbered from 1 to 6. Brian and George simultaneously roll all their dice. What is the probability that the number on Brian's die is larger than the sum of the numbers on George's dice? | \frac{19}{40} |
Find the maximum value of
\[\frac{x + 2y + 3}{\sqrt{x^2 + y^2 + 1}}\]over all real numbers $x$ and $y.$ | \sqrt{14} |
To make a yellow score mixture, Taylor has to combine white and black scores in the ratio of 7:6. If she got 78 yellow scores, what's 2/3 of the difference between the number of black and white scores she used? | The total ratio representing the yellow scores that Taylor got is 7+6=<<7+6=13>>13
The difference in the ratio between the number of black and white scores Taylor used is 7-6=<<7-6=1>>1
The fraction representing the difference in the ratio between the number of black and white scores Taylor used is 1/13, representing 1... |
An urn contains 101 balls, exactly 3 of which are red. The balls are drawn one by one without replacement. On which draw is it most likely to pull the second red ball? | 51 |
A point $P$ is chosen uniformly at random in the interior of triangle $ABC$ with side lengths $AB = 5$ , $BC = 12$ , $CA = 13$ . The probability that a circle with radius $\frac13$ centered at $P$ does not intersect the perimeter of $ABC$ can be written as $\frac{m}{n}$ where $m, n$ are relatively pri... | 61 |
Let $f(x)$ be a polynomial with real, nonnegative coefficients. If $f(6) = 24$ and $f(24) = 1536,$ find the largest possible value of $f(12).$ | 192 |
The sum of four different positive integers is 100. The largest of these four integers is $n$. What is the smallest possible value of $n$? | 27 |
Using the digits 0, 1, 2, 3, 4, and 5, form six-digit numbers without repeating any digit.
(1) How many such six-digit odd numbers are there?
(2) How many such six-digit numbers are there where the digit 5 is not in the unit place?
(3) How many such six-digit numbers are there where the digits 1 and 2 are not adj... | 408 |
(1) Given $0 < x < \frac{1}{2}$, find the maximum value of $y= \frac{1}{2}x(1-2x)$;
(2) Given $x > 0$, find the maximum value of $y=2-x- \frac{4}{x}$;
(3) Given $x$, $y\in\mathbb{R}_{+}$, and $x+y=4$, find the minimum value of $\frac{1}{x}+ \frac{3}{y}$. | 1+ \frac{ \sqrt{3}}{2} |
A bar of chocolate is made of 10 distinguishable triangles as shown below. How many ways are there to divide the bar, along the edges of the triangles, into two or more contiguous pieces? | 1689 |
A kilogram of pork costs $6 while a kilogram of chicken costs $2 less. How much will a 3-kilogram of chicken and a kilogram of pork cost? | A kilogram of chicken costs $6 - $2 = $<<6-2=4>>4.
Three kilograms of chicken cost $4 x 3 = $<<4*3=12>>12.
So, a 3-kilogram of chicken and a kilogram of pork cost $12 + $6 = $18.
#### 18 |
Ken is the best sugar cube retailer in the nation. Trevor, who loves sugar, is coming over to make an order. Ken knows Trevor cannot afford more than 127 sugar cubes, but might ask for any number of cubes less than or equal to that. Ken prepares seven cups of cubes, with which he can satisfy any order Trevor might make... | 64 |
On the extension of side $AD$ of rectangle $ABCD$ beyond point $D$, point $E$ is taken such that $DE = 0.5 AD$ and $\angle BEC = 30^\circ$.
Find the ratio of the sides of rectangle $ABCD$. | \sqrt{3}/2 |
A cuboctahedron is a polyhedron whose faces are squares and equilateral triangles such that two squares and two triangles alternate around each vertex. What is the volume of a cuboctahedron of side length 1? | 5 \sqrt{2} / 3 |
A plane intersects a right circular cylinder of radius $1$ forming an ellipse. If the major axis of the ellipse of $50\%$ longer than the minor axis, the length of the major axis is
$\textbf{(A)}\ 1\qquad \textbf{(B)}\ \frac{3}{2}\qquad \textbf{(C)}\ 2\qquad \textbf{(D)}\ \frac{9}{4}\qquad \textbf{(E)}\ 3$
| 3 |
John decided to start rowing around a square lake. Each side of the lake is 15 miles. Jake can row at twice the speed he can swim. It takes him 20 minutes to swim 1 mile. How long, in hours, does it take to row the lake? | The perimeter of the lake is 4*15=<<4*15=60>>60 miles
He swims at a speed of 60/20=<<60/20=3>>3 mph
So he rows at a speed of 3*2=<<3*2=6>>6 mph
That means it takes him 60/6=<<60/6=10>>10 hours to row around the lake
#### 10 |
Julie runs the school newspaper. In preparation for printing the next issue of The School News, she bought two boxes of standard paper, each containing 5 packages, with 250 sheets of paper per package. If this issue of The School News uses 25 sheets of paper to print one newspaper, how many newspapers can Julie print ... | Five packages per box, with 250 sheets per package is 5*250=<<5*250=1250>>1250 sheets per box.
Two boxes contain a total of 1250*2=<<1250*2=2500>>2500 sheets.
If each newspaper requires 25 pages, then with 2500 sheets she can print 2500/25=<<2500/25=100>>100 newspapers.
#### 100 |
A "pass level game" has the following rules: On the \( n \)-th level, a die is thrown \( n \) times. If the sum of the points that appear in these \( n \) throws is greater than \( 2^{n} \), then the player passes the level.
1. What is the maximum number of levels a person can pass in this game?
2. What is the probab... | \frac{100}{243} |
Mr. Morgan G. Bloomgarten wants to distribute 1,000,000 dollars among his friends. He has two specific rules for distributing the money:
1. Each gift must be either 1 dollar or a power of 7 (7, 49, 343, 2401, etc.).
2. No more than six people can receive the same amount.
How can he distribute the 1,000,000 dollars und... | 1,000,000 |
Find the cross product of $\begin{pmatrix} 2 \\ 0 \\ 3 \end{pmatrix}$ and $\begin{pmatrix} 5 \\ -1 \\ 7 \end{pmatrix}.$ | \begin{pmatrix} 3 \\ 1 \\ -2 \end{pmatrix} |
A flower shop buys a number of roses from a farm at a price of 5 yuan per rose each day and sells them at a price of 10 yuan per rose. If the roses are not sold by the end of the day, they are discarded.
(1) If the shop buys 16 roses in one day, find the profit function \( y \) (in yuan) with respect to the demand \( ... | 16 |
Jenny and Kenny are walking in the same direction, Kenny at 3 feet per second and Jenny at 1 foot per second, on parallel paths that are 200 feet apart. A tall circular building 100 feet in diameter is centered midway between the paths. At the instant when the building first blocks the line of sight between Jenny and K... | 163 |
Two unit squares $S_{1}$ and $S_{2}$ have horizontal and vertical sides. Let $x$ be the minimum distance between a point in $S_{1}$ and a point in $S_{2}$, and let $y$ be the maximum distance between a point in $S_{1}$ and a point in $S_{2}$. Given that $x=5$, the difference between the maximum and minimum possible val... | 472 |
Teams $A$ and $B$ are playing in a basketball league where each game results in a win for one team and a loss for the other team. Team $A$ has won $\frac{2}{3}$ of its games and team $B$ has won $\frac{5}{8}$ of its games. Also, team $B$ has won $7$ more games and lost $7$ more games than team $A.$ How many games has t... | 42 |
Let $P$ be the plane passing through the origin with normal vector $\begin{pmatrix} 1 \\ -2 \\ 1 \end{pmatrix}.$ Find the matrix $\mathbf{P}$ such that for any vector $\mathbf{v},$ $\mathbf{P} \mathbf{v}$ is the projection of $\mathbf{v}$ onto plane $P.$ | \begin{pmatrix} \frac{5}{6} & \frac{1}{3} & -\frac{1}{6} \\ \frac{1}{3} & \frac{1}{3} & \frac{1}{3} \\ -\frac{1}{6} & \frac{1}{3} & \frac{5}{6} \end{pmatrix} |
A frustum of a cone has a lower base radius of 8 inches, an upper base radius of 4 inches, and a height of 5 inches. Calculate its lateral surface area and total surface area. | (80 + 12\sqrt{41})\pi |
Solve for $x$: $2(3^x) = 162$. | 4 |
What is the minimum value of the product $\prod_{i=1}^{6} \frac{a_{i}-a_{i+1}}{a_{i+2}-a_{i+3}}$ given that $\left(a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{6}\right)$ is a permutation of $(1,2,3,4,5,6)$? | 1 |
Given events A, B, and C with respective probabilities $P(A) = 0.65$, $P(B) = 0.2$, and $P(C) = 0.1$, find the probability that the drawn product is not a first-class product. | 0.35 |
For the "Skillful Hands" club, Anton needs to cut several identical pieces of wire (the length of each piece is an integer number of centimeters). Initially, Anton took a piece of wire 10 meters long and was able to cut only 9 required pieces from it. Then Anton took a piece 11 meters long, but it was also only enough ... | 111 |
Express the sum of $0.\overline{123}+0.\overline{0123}+0.\overline{000123}$ as a common fraction. | \frac{123 \times 1000900}{999 \times 9999 \times 100001} |
Place several small circles with a radius of 1 inside a large circle with a radius of 11, so that each small circle is tangentially inscribed in the large circle and these small circles do not overlap. What is the maximum number of small circles that can be placed? | 31 |
A teacher finds that when she offers candy to her class of 30 students, the mean number of pieces taken by each student is 5. If every student takes some candy, what is the greatest number of pieces one student could have taken? | 121 |
To a natural number \( N \), the largest divisor of \( N \) that is less than \( N \) was added, resulting in a power of ten. Find all such \( N \). | 75 |
John jogs at a speed of 4 miles per hour when he runs alone, but runs at 6 miles per hour when he is being dragged by his 100-pound German Shepherd dog. If John and his dog go on a run together for 30 minutes, and then John runs for an additional 30 minutes by himself, how far will John have traveled? | John runs at 6 miles per hour for 0.5 hours, for a distance of 6*0.5=<<6*0.5=3>>3 miles.
Then John runs at 4 miles per hour for 0.5 hours, for a distance of 4*0.5=<<4*0.5=2>>2 miles.
In total, John runs 3+2=<<3+2=5>>5 miles.
#### 5 |
Liz bought a recipe book that cost $6, a baking dish that cost twice as much, five ingredients that cost $3 each, and an apron that cost a dollar more than the recipe book. Collectively, how much in dollars did Liz spend? | The baking dish cost 6*2 = <<6*2=12>>12.
The ingredients cost 5*3 = <<5*3=15>>15.
The apron cost 6+1 = <<6+1=7>>7.
Collectively, Liz spend 6+12+15+7 = <<6+12+15+7=40>>40
#### 40 |
Compute $\sum_{n=2009}^{\infty} \frac{1}{\binom{n}{2009}}$ | \frac{2009}{2008} |
The inclination angle of the line $\sqrt {3}x-y+1=0$ is \_\_\_\_\_\_. | \frac {\pi}{3} |
How many positive divisors of 30! are prime? | 10 |
Solve for $n$: $0.03n + 0.08(20 + n) = 12.6$. | 100 |
A fair die, numbered 1, 2, 3, 4, 5, 6, is thrown three times. The numbers obtained are recorded sequentially as $a$, $b$, and $c$. The probability that $a+bi$ (where $i$ is the imaginary unit) is a root of the equation $x^{2}-2x+c=0$ is $\_\_\_\_\_\_$. | \frac{1}{108} |
Given that 28×15=420, directly write out the results of the following multiplications:
2.8×1.5=\_\_\_\_\_\_、0.28×1.5=\_\_\_\_\_\_、0.028×0.15=\_\_\_\_\_\_. | 0.0042 |
Leon ordered 3 sets of toy organizers for $78 per set and 2 gaming chairs for $83 each. If there is a delivery fee that is 5% of the total sales, how much did Leon pay? | Three sets of toy organizers cost $78 x 3 = $<<78*3=234>>234.
Two gaming chairs cost $83 x 2 = $<<83*2=166>>166
Leon's total orders amount to $234 + $166 = $<<234+166=400>>400.
Then, the delivery fee is $400 x 5/100 = $<<400*5/100=20>>20.
So, Leon had to pay a total of $400 + $20 = $<<400+20=420>>420.
#### 420 |
Pablo has 27 solid $1 \times 1 \times 1$ cubes that he assembles in a larger $3 \times 3 \times 3$ cube. If 10 of the smaller cubes are red, 9 are blue, and 8 are yellow, what is the smallest possible surface area of the larger cube that is red? | 12 |
Calculate the greatest integer less than or equal to $\frac{5^{98} + 2^{104}}{5^{95} + 2^{101}}$. | 125 |
Sophie's aunt gave her $260 to spend on clothes at the mall. She bought 2 shirts that cost $18.50 each and a pair of trousers that cost $63. She then decides to purchase 4 more articles of clothing with her remaining budget. How much money would each item cost if she split the cost of each item evenly? | Sophie spent $18.50 × 2 = $<<18.5*2=37>>37 on the two shirts.
She spent in total $37 + $63 = $<<37+63=100>>100 on clothes so far.
She has $260 - $100 = $<<260-100=160>>160 left to spend on clothes.
Sophie has $160 ÷ 4 = $<<160/4=40>>40 left to spend on each additional item of clothing
#### 40 |
Let the sequence $\{a_n\}$ have a sum of the first $n$ terms denoted by $S_n$. It is known that $4S_n = 2a_n - n^2 + 7n$ ($n \in \mathbb{N}^*$). Find $a_{11}$. | -2 |
The numbers $a_1,$ $a_2,$ $a_3,$ $b_1,$ $b_2,$ $b_3,$ $c_1,$ $c_2,$ $c_3$ are equal to the numbers $1,$ $2,$ $3,$ $\dots,$ $9$ in some order. Find the smallest possible value of
\[a_1 a_2 a_3 + b_1 b_2 b_3 + c_1 c_2 c_3.\] | 214 |
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