problem stringlengths 10 5.15k | answer stringlengths 0 1.23k |
|---|---|
Mary and Jimmy start running from the same spot, but in opposite directions. Mary runs at 5 miles per hour and Jimmy runs at 4 miles per hour. What is the distance between them after 1 hour? | In 1 hour, Mary will have traveled 5 miles/hour * 1 hour = <<5*1=5>>5 miles.
In 1 hour, Jimmy will have traveled 4 miles/hour * 1 hour = <<4*1=4>>4 miles.
Since they are running in opposite directions, the distance between them will be 5 miles + 4 miles = <<5+4=9>>9 miles.
#### 9 |
We consider positive integers $n$ having at least six positive divisors. Let the positive divisors of $n$ be arranged in a sequence $(d_i)_{1\le i\le k}$ with $$1=d_1<d_2<\dots <d_k=n\quad (k\ge 6).$$
Find all positive integers $n$ such that $$n=d_5^2+d_6^2.$$ | 500 |
Caleb and Cynthia are filling up their inflatable pool with water using buckets. They fill their buckets at the spigot and carry them to the pool. Caleb can add 7 gallons from his bucket and Cynthia can add 8 gallons from her bucket each trip. It will take 105 gallons to fill the pool. How many trips will it take for C... | Each trip, they add Caleb’s 7 gallons + Cynthia’s 8 gallons = <<7+8=15>>15 gallons.
Since they are adding 15 gallons every trip, to fill the 105-gallon pool, it would take 105 / 15 = <<105/15=7>>7 trips.
#### 7 |
The centerpieces at Glenda’s wedding reception include a fishbowl containing 2 fish, except for one table that has 3 fish. There are 32 tables. How many fish are there? | There are 32 tables with 2 fish in a bowl on each, so 32 tables x 2 fish = <<32*2=64>>64 fish.
One table has an additional fish, so 64 fish + 1xtra fish = 65 fish.
#### 65 |
At a shop in Japan, women's T-shirts are sold every 30 minutes for $18, and men's T-shirts are sold every 40 minutes for $15. This shop is open from 10 am until 10 pm. How much does the shop earn selling T-shirts per week? | First, the shop is open 12 hours * 60 minutes = <<12*60=720>>720 minutes per day.
So, 720 / 30 = <<720/30=24>>24 women's T-shirts sold per day.
The shop earns 24 * $18 = $<<24*18=432>>432 selling women's T-shirts per day.
Similarly, 720 / 40 = <<720/40=18>>18 men's T-shirts are sold per day.
The shop earns 18 * $15 = $... |
Let set $I=\{1,2,3,4,5,6\}$, and sets $A, B \subseteq I$. If set $A$ contains 3 elements, set $B$ contains at least 2 elements, and all elements in $B$ are not less than the largest element in $A$, calculate the number of pairs of sets $A$ and $B$ that satisfy these conditions. | 29 |
What is $(5^{-2})^0 + (5^0)^3$? | 2 |
Let $S$ denote the value of the sum\[\sum_{n=0}^{668} (-1)^{n} {2004 \choose 3n}\]Determine the remainder obtained when $S$ is divided by $1000$.
| 6 |
If $\mathbf{a}$, $\mathbf{b}$, $\mathbf{c}$, and $\mathbf{d}$ are unit vectors, find the largest possible value of
\[
\|\mathbf{a} - \mathbf{b}\|^2 + \|\mathbf{a} - \mathbf{c}\|^2 + \|\mathbf{a} - \mathbf{d}\|^2 + \|\mathbf{b} - \mathbf{c}\|^2 + \|\mathbf{b} - \mathbf{d}\|^2 + \|\mathbf{c} - \mathbf{d}\|^2.
\] | 16 |
Let $n$ be an integer greater than or equal to $1$. Find, as a function of $n$, the smallest integer $k\ge 2$ such that, among any $k$ real numbers, there are necessarily two of which the difference, in absolute value, is either strictly less than $1 / n$, either strictly greater than $n$. | n^2 + 2 |
Norris saved $29 in September. He saved $25 in October and $31 in November. Then Hugo spent $75 on an online game. How much money does Norris have left? | The total amount of money saved is $29 + $25 + $31 = $<<29+25+31=85>>85.
Norris has $85 - $75 = $<<85-75=10>>10 left.
#### 10 |
What is the largest positive integer that is not the sum of a positive integral multiple of $42$ and a positive composite integer?
| 215 |
What is the sum of all two-digit positive integers whose squares end with the digits 01? | 199 |
When $\sqrt[4]{2^7\cdot3^3}$ is fully simplified, the result is $a\sqrt[4]{b}$, where $a$ and $b$ are positive integers. What is $a+b$? | 218 |
Given that the angles A, B, C of triangle ABC correspond to the sides a, b, c respectively, and vectors $\overrightarrow {m}$ = (a, $- \sqrt {3}b$) and $\overrightarrow {n}$ = (cosA, sinB), and $\overrightarrow {m}$ is parallel to $\overrightarrow {n}$.
(1) Find angle A.
(2) If $a = \sqrt{39}$ and $c = 5$, find the are... | \frac{5\sqrt{3}}{2} |
Let positive integers \( a, b, c, d \) satisfy \( a > b > c > d \) and \( a+b+c+d=2004 \), \( a^2 - b^2 + c^2 - d^2 = 2004 \). Find the minimum value of \( a \). | 503 |
In triangle $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. It is known that $4a = \sqrt{5}c$ and $\cos C = \frac{3}{5}$.
$(Ⅰ)$ Find the value of $\sin A$.
$(Ⅱ)$ If $b = 11$, find the area of $\triangle ABC$. | 22 |
Let $\mathcal{P}_{1}, \mathcal{P}_{2}, \mathcal{P}_{3}$ be pairwise distinct parabolas in the plane. Find the maximum possible number of intersections between two or more of the $\mathcal{P}_{i}$. In other words, find the maximum number of points that can lie on two or more of the parabolas $\mathcal{P}_{1}, \mathcal{P... | 12 |
A man can commute either by train or by bus. If he goes to work on the train in the morning, he comes home on the bus in the afternoon; and if he comes home in the afternoon on the train, he took the bus in the morning. During a total of $x$ working days, the man took the bus to work in the morning $8$ times, came home... | 16 |
Find the number of positive integers less than $1000$ that can be expressed as the difference of two integral powers of $2.$ | 50 |
A game involves jumping to the right on the real number line. If $a$ and $b$ are real numbers
and $b > a$, the cost of jumping from $a$ to $b$ is $b^3-ab^2$. For what real numbers
$c$ can one travel from $0$ to $1$ in a finite number of jumps with total cost exactly $c$? | 1/3 < c \leq 1 |
Five points, no three of which are collinear, are given. Calculate the least possible value of the number of convex polygons whose some corners are formed by these five points. | 16 |
The value of $1.000 + 0.101 + 0.011 + 0.001$ is: | 1.113 |
Solve the following cryptarithm ensuring that identical letters correspond to identical digits:
$$
\begin{array}{r}
\text { К O Ш К A } \\
+ \text { К O Ш К A } \\
\text { К O Ш К A } \\
\hline \text { С О Б А К А }
\end{array}
$$ | 50350 |
At a state contest, 21 Mathletes stay in the same hotel. Each Mathlete gets his/her own room and the room numbers are 1 through 21. When all the Mathletes have arrived, except the ones staying in rooms 12 and 13, what is the median room number of the other 19 Mathletes? | 10 |
Let $a$ and $b$ be nonzero complex numbers such that $a^2 + ab + b^2 = 0.$ Evaluate
\[\frac{a^9 + b^9}{(a + b)^9}.\] | -2 |
In $\triangle{ABC}, AB=10, \angle{A}=30^\circ$ , and $\angle{C=45^\circ}$. Let $H, D,$ and $M$ be points on the line $BC$ such that $AH\perp{BC}$, $\angle{BAD}=\angle{CAD}$, and $BM=CM$. Point $N$ is the midpoint of the segment $HM$, and point $P$ is on ray $AD$ such that $PN\perp{BC}$. Then $AP^2=\dfrac{m}{n}$, where ... | 77 |
An ellipse has a major axis of length 12 and a minor axis of 10. Using one focus as a center, an external circle is tangent to the ellipse. Find the radius of the circle. | \sqrt{11} |
Given three members of a group -- Alice, Bob, and Carol -- in how many ways can these three be chosen to be the three officers (president, secretary, and treasurer) of the group, assuming no person holds more than one job? | 6 |
Square $EFGH$ has sides of length 4. A point $P$ on $EH$ is such that line segments $FP$ and $GP$ divide the square’s area into four equal parts. Find the length of segment $FP$.
A) $2\sqrt{3}$
B) $3$
C) $2\sqrt{5}$
D) $4$
E) $2\sqrt{7}$ | 2\sqrt{5} |
In the convex pentagon $ABCDE$, $\angle A = \angle B = 120^{\circ}$, $EA = AB = BC = 2$, and $CD = DE = 4$. Calculate the area of $ABCDE$. | 7\sqrt{3} |
Pedro must choose two irreducible fractions, each with a positive numerator and denominator such that:
- The sum of the fractions is equal to $2$ .
- The sum of the numerators of the fractions is equal to $1000$ .
In how many ways can Pedro do this?
| 200 |
The skeletal structure of circumcircumcircumcoronene, a hydrocarbon with the chemical formula $\mathrm{C}_{150} \mathrm{H}_{30}$, is shown below. Each line segment between two atoms is at least a single bond. However, since each carbon (C) requires exactly four bonds connected to it and each hydrogen $(\mathrm{H})$ req... | 267227532 |
Assuming that the clock hands move without jumps, determine how many minutes after the clock shows 8:00 will the minute hand catch up with the hour hand. | 43 \frac{7}{11} |
How many positive integers less than 500 are congruent to 7 (mod 13)? | 38 |
A rectangular prism has 6 faces, 12 edges, and 8 vertices. If a new pyramid is added using one of its rectangular faces as the base, calculate the maximum value of the sum of the exterior faces, vertices, and edges of the resulting shape after the fusion of the prism and pyramid. | 34 |
In the finals of a beauty contest among giraffes, there were two finalists: the Tall one and the Spotted one. There are 135 voters divided into 5 districts, each district is divided into 9 precincts, and each precinct has 3 voters. Voters in each precinct choose the winner by majority vote; in a district, the giraffe t... | 30 |
Point $P$ is on the $y$-axis with $y$-coordinate greater than 0 and less than 100. A circle is drawn through $P, Q(4,4)$ and $O(0,0)$. How many possible positions for $P$ are there so that the radius of this circle is an integer? | 66 |
Let $[r,s]$ denote the least common multiple of positive integers $r$ and $s$. Find the number of ordered triples $(a,b,c)$ of positive integers for which $[a,b] = 1000$, $[b,c] = 2000$, and $[c,a] = 2000$. | 70 |
Given that $|\vec{a}|=1$, $|\vec{b}|=2$, and $(\vec{a}+\vec{b})\cdot \vec{b}=3$, find the angle between $\vec{b}$ and $\vec{a}$. | \frac{2\pi}{3} |
Mr Cruz went to his doctor to seek advice on the best way to gain weight. His doctor told him to include more proteins in his meals and live a generally healthy lifestyle. After a month of following his doctor's advice, Mr Cruz had a weight gain of 20 pounds. He gained 30 more pounds in the second month after more heal... | Mr Cruz's total weight gain in the two months is 20 + 30 = <<20+30=50>>50 pounds.
If he initially weighed 70 pounds, his total weight after the two months is 70 + 50 = <<70+50=120>>120 pounds
#### 120 |
Will's mom gave him $74 to go shopping. He bought a sweater for $9, a T-shirt for $11 and a pair of shoes for $30. He then returned his shoes for a 90% refund. How much money does Will have left? | Will paid $30 for his shoes but then returned them and got 90% of the money back, so he lost 10% of the money, which means 10/100 * $30 = $<<10/100*30=3>>3 is gone.
Will spent $9 + $11 + $3 = $<<9+11+3=23>>23.
Will started with $74 - $23 that he spent = $<<74-23=51>>51 remaining.
#### 51 |
Adults made up $\frac5{12}$ of the crowd of people at a concert. After a bus carrying $50$ more people arrived, adults made up $\frac{11}{25}$ of the people at the concert. Find the minimum number of adults who could have been at the concert after the bus arrived. | 154 |
The number of two-digit numbers that can be formed using the digits 0, 1, 2, 3, 4 without repeating any digit must be calculated. | 16 |
The function $f(x)$ satisfies $f(1) = 1$ and
\[f(x + y) = 3^y f(x) + 2^x f(y)\]for all real numbers $x$ and $y.$ Find the function $f(x).$ | 3^x - 2^x |
Given the function $f(x)=-3x^2+6x$, let ${S_n}$ be the sum of the first $n$ terms of the sequence ${{a_n}}$. The points $(n, {S_n})$ (where $n \in \mathbb{N}^*$) lie on the curve $y=f(x)$.
(I) Find the general formula for the terms of the sequence ${{a_n}}$.
(II) If ${b_n}={(\frac{1}{2})^{n-1}}$ and ${c_n}=\frac{{a_n... | \frac{1}{2} |
Express $361_9 + 4C5_{13}$ as a base 10 integer, where $C$ denotes the digit whose value is 12 in base 13. | 1135 |
Find $w$, such that $5^65^w=25$. | -4 |
Graphs of several functions are shown below. Which functions have inverses?
[asy]
unitsize(0.5 cm);
picture[] graf;
int i, n;
real funce(real x) {
return(x^3/40 + x^2/20 - x/2 + 2);
}
for (n = 1; n <= 5; ++n) {
graf[n] = new picture;
for (i = -5; i <= 5; ++i) {
draw(graf[n],(i,-5)--(i,5),gray(0.7));
... | \text{B,C} |
A printer prints 17 pages per minute. How many minutes will it take to print 200 pages? Express your answer to the nearest whole number. | 12 |
If $\|\mathbf{v}\| = 4,$ then find $\mathbf{v} \cdot \mathbf{v}.$ | 16 |
Find the integer $n,$ $-90 \le n \le 90,$ such that $\sin n^\circ = \sin 604^\circ.$ | -64 |
Find $x$ such that $\log_x 49 = \log_2 32$. | 7^{2/5} |
A square is inscribed in another square such that its vertices lie on the sides of the first square, and its sides form angles of $60^{\circ}$ with the sides of the first square. What fraction of the area of the given square is the area of the inscribed square? | 4 - 2\sqrt{3} |
Given the function $f\left(x\right)=x^{3}+ax^{2}+bx-4$ and the tangent line equation $y=x-4$ at point $P\left(2,f\left(2\right)\right)$.<br/>$(1)$ Find the values of $a$ and $b$;<br/>$(2)$ Find the extreme values of $f\left(x\right)$. | -\frac{58}{27} |
Given vectors $\overrightarrow{a}$, $\overrightarrow{b}$, $\overrightarrow{c}$ with pairwise angles of $60^\circ$, and $|\overrightarrow{a}|=|\overrightarrow{b}|=|\overrightarrow{c}|=1$, find $|\overrightarrow{a}+\overrightarrow{b}-\overrightarrow{c}|$. | \sqrt{2} |
A right pyramid has a square base with side length 10 cm. Its peak is 12 cm above the center of its base. What is the total surface area of the pyramid, in square centimeters? | 360 |
Given $\sin\theta + \cos\theta = \frac{3}{4}$, where $\theta$ is an angle of a triangle, find the value of $\sin\theta - \cos\theta$. | \frac{\sqrt{23}}{4} |
Given the function $f(x)=a^{2}\sin 2x+(a-2)\cos 2x$, if its graph is symmetric about the line $x=-\frac{\pi}{8}$, determine the maximum value of $f(x)$. | 4\sqrt{2} |
Bryan starts exercising at home during quarantine. To start, he decides to do 3 sets of 15 push-ups each. Near the end of the third set, he gets tired and does 5 fewer push-ups. How many push-ups did he do in total? | In total, Bryan would have done 3 sets * 15 push-ups/set = <<3*15=45>>45 push-ups.
Subtracting the push-ups he didn't do in the third set, Bryan did 45 push-ups - 5 push-ups = <<45-5=40>>40 push-ups.
#### 40 |
Find all nonnegative integers $a, b, c$ such that
$$\sqrt{a} + \sqrt{b} + \sqrt{c} = \sqrt{2014}.$$ | (0, 0, 2014) |
On a long straight section of a two-lane highway where cars travel in both directions, cars all travel at the same speed and obey the safety rule: the distance from the back of the car ahead to the front of the car behind is exactly one car length for every 10 kilometers per hour of speed or fraction thereof. Assuming ... | 200 |
There is a set of points \( M \) on a plane and seven different circles \( C_{1}, C_{2}, \cdots, C_{7} \). Circle \( C_{7} \) passes through exactly 7 points in \( M \), circle \( C_{6} \) passes through exactly 6 points in \( M \), and so on, with circle \( C_{1} \) passing through exactly 1 point in \( M \). Determin... | 12 |
Zebadiah has 3 red shirts, 3 blue shirts, and 3 green shirts in a drawer. Without looking, he randomly pulls shirts from his drawer one at a time. What is the minimum number of shirts that Zebadiah has to pull out to guarantee that he has a set of shirts that includes either 3 of the same colour or 3 of different colou... | 5 |
Carmen takes a long bike ride on a hilly highway. The graph indicates the miles traveled during the time of her ride. What is Carmen's average speed for her entire ride in miles per hour? | 5 |
The greatest common divisor (GCD) and the least common multiple (LCM) of 45 and 150 are what values? | 15,450 |
There is a unique triplet of positive integers \((a, b, c)\) such that \(a \leq b \leq c\) and
$$
\frac{25}{84}=\frac{1}{a}+\frac{1}{a b}+\frac{1}{a b c}.
$$
Determine \(a + b + c\). | 17 |
In a square array of 25 dots arranged in a 5x5 grid, what is the probability that five randomly chosen dots will be collinear? Express your answer as a common fraction. | \frac{2}{8855} |
A soccer team has 16 members. We need to select a starting lineup including a goalkeeper, a defender, a midfielder, and two forwards. However, only 3 players can play as a goalkeeper, 5 can play as a defender, 8 can play as a midfielder, and 4 players can play as forwards. In how many ways can the team select a startin... | 1440 |
Convert the quinary (base-5) number $234_{(5)}$ to a decimal (base-10) number, then convert it to a binary (base-2) number. | 1000101_{(2)} |
A local farm is famous for having lots of double yolks in their eggs. One carton of 12 eggs had five eggs with double yolks. How many yolks were in the whole carton? | The carton had 12 - 5 = <<12-5=7>>7 eggs with one yolk.
It had 5 eggs with double yolks, which added * 2 = <<5*2=10>>10 yolks.
Thus, there were 7 + 10 = <<7+10=17>>17 yolks in the whole carton.
#### 17 |
Wang Hong's father deposited 20,000 yuan in the bank for a fixed term of three years, with an annual interest rate of 3.33%. How much money, including the principal and interest, can Wang Hong's father withdraw at the end of the term? | 21998 |
Find the smallest solution to the equation \[\lfloor x^2 \rfloor - \lfloor x \rfloor^2 = 17.\] | 7\sqrt2 |
A train takes 2 hours longer to go an equal distance of a car. If the car and the train leave station A at the same time, and the car reaches station B 4.5 hours later, calculate the combined time the car and the train take to reach station B. | Since the train takes 2 hours longer to go an equal distance as the car, to reach station B, the train took 4.5+2 = 6.5 hours.
Together, the train and the car take a total of 6.5+4.5 = <<6.5+4.5=11>>11 hours to travel from station A to station B.
#### 11 |
Given the equations:
\[
\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 4 \quad \text{and} \quad \frac{a}{x} + \frac{b}{y} + \frac{c}{z} = 3,
\]
find the value of \(\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2}\). | 16 |
Let \(a,\ b,\ c,\ d\) be real numbers such that \(a + b + c + d = 10\) and
\[ab + ac + ad + bc + bd + cd = 20.\]
Find the largest possible value of \(d\). | \frac{5 + \sqrt{105}}{2} |
How many two-digit positive integers have at least one 7 as a digit? | 18 |
The side $AB$ of triangle $ABC$ is divided into $n$ equal parts (with division points $B_0 = A, B_1, B_2, \ldots, B_n = B$), and the side $AC$ of this triangle is divided into $n+1$ equal parts (with division points $C_0 = A, C_1, C_2, \ldots, C_{n+1} = C$). The triangles $C_i B_i C_{i+1}$ are shaded. What fraction of... | \frac{1}{2} |
Una rolls 8 standard 6-sided dice simultaneously and calculates the product of the 8 numbers obtained. What is the probability that the product is divisible by 8?
A) $\frac{273}{288}$
B) $\frac{275}{288}$
C) $\frac{277}{288}$
D) $\frac{279}{288}$ | \frac{277}{288} |
Cagney can frost a cupcake every 20 seconds and Lacey can frost a cupcake every 30 seconds. Working together, how many cupcakes can they frost in 5 minutes? | 25 |
Given the function $f(x)=\frac{1}{2}x^{2}+(2a^{3}-a^{2})\ln x-(a^{2}+2a-1)x$, and $x=1$ is its extreme point, find the real number $a=$ \_\_\_\_\_\_. | -1 |
Write the product of the digits of each natural number from 1 to 2018 (for example, the product of the digits of the number 5 is 5; the product of the digits of the number 72 is \(7 \times 2=14\); the product of the digits of the number 607 is \(6 \times 0 \times 7=0\), etc.). Then find the sum of these 2018 products. | 184320 |
Emily bought 9 packs of candy necklaces to give her classmates at school for Valentine’s Day. Each pack had 8 candy necklaces in it. Emily opened one pack at a time. After her classmates took as many as they wanted, there were 40 candy necklaces left. How many packs did Emily open for her classmates? | Emily bought 9 packs of candy necklaces * 8 candy necklaces in a pack = <<9*8=72>>72 candy necklaces.
Her classmates took 72 candy necklaces – 40 candy necklaces left = <<72-40=32>>32 candy necklaces
Since each pack contained 8 necklaces, this means 32 / 8 = <<32/8=4>>4 packs of candy necklaces were opened.
#### 4 |
There exist several solutions to the equation $1+\frac{\sin x}{\sin 4 x}=\frac{\sin 3 x}{\sin 2 x}$ where $x$ is expressed in degrees and $0^{\circ}<x<180^{\circ}$. Find the sum of all such solutions. | 320^{\circ} |
Given point O is the circumcenter of ∆ABC, and |→BA|=2, |→BC|=6, calculate →BO⋅→AC. | 16 |
Mandy has three $20 bills while Manny has two $50 bills. If both of them decide to exchange their bills for $10 bills, how many more $10 bills than Mandy will Manny have? | Mandy has a total of $20 x 3 = $<<20*3=60>>60.
So, she will have $60/$10 = <<60/10=6>>6 $10 bills.
Manny has a total of $50 x 2 = $<<50*2=100>>100.
So, he will have $100/$10 = <<100/10=10>>10 $10 bills.
Hence, Manny has 10 - 6 = <<10-6=4>>4 more $10 bills than Mandy.
#### 4 |
The graph of the line $x+y=b$ intersects the line segment from $(2,5)$ to $(4,9)$ at its midpoint. What is the value of $b$? | 10 |
(x^2+1)(2x+1)^9=a_0+a_1(x+2)+a_2(x+2)^2+\ldots+a_{11}(x+2)^{11}, calculate the sum of the coefficients a_0 through a_11. | -2 |
Two concentric circles $\omega, \Omega$ with radii $8,13$ are given. $AB$ is a diameter of $\Omega$ and the tangent from $B$ to $\omega$ touches $\omega$ at $D$ . What is the length of $AD$ . | 19 |
31 cars simultaneously started from the same point on a circular track: the first car at a speed of 61 km/h, the second at 62 km/h, and so on up to the 31st car at 91 km/h. The track is narrow, and if one car overtakes another, they collide and both crash out of the race. Eventually, one car remains. What is its speed? | 76 |
Two-thirds of the people in a room are seated in three-fourths of the chairs. The rest of the people are standing. If there are 6 empty chairs, how many people are in the room? | 27 |
Let $k$ be an integer greater than 1. Suppose $a_0 > 0$, and define \[a_{n+1} = a_n + \frac{1}{\sqrt[k]{a_n}}\] for $n > 0$. Evaluate \[\lim_{n \to \infty} \frac{a_n^{k+1}}{n^k}.\] | \left( \frac{k+1}{k} \right)^k |
According to the table below, how many dollars are in the median value of the 59 salaries paid to this company's employees?
\begin{tabular}{|c|c|c|}
\hline
\textbf{Position Title}&\textbf{\# with Title}&\textbf{Salary}\\\hline
President&1&$\$130{,}000$\\\hline
Vice-President&5&$\$90{,}000$\\\hline
Director&10&$\$75{,}... | \$23{,}000 |
If the odd function \( y=f(x) \) defined on \( \mathbf{R} \) is symmetrical about the line \( x=1 \), and when \( 0 < x \leqslant 1 \), \( f(x)=\log_{3}x \), find the sum of all real roots of the equation \( f(x)=-\frac{1}{3}+f(0) \) in the interval \( (0,10) \). | 30 |
Completely factor the expression: $$x^8-256$$ | (x^4+16)(x^2+4)(x+2)(x-2) |
There are 10 cars parked in a mall’s parking lot, each with the same number of customers inside. Each customer only makes 1 purchase. If the sports store makes 20 sales and the music store makes 30 sales, how many customers are in each of the cars? | As each sale represents a customer, there must be a total of 20 sports store sales + 30 music store sales = <<20+30=50>>50 customers.
There is an equal number of customers in every car, so there must be 50 total customers / 10 cars = <<50/10=5>>5 customers in each car.
#### 5 |
What is the smallest possible value of $n$ if a solid cube is made of white plastic and has dimensions $n \times n \times n$, the six faces of the cube are completely covered with gold paint, the cube is then cut into $n^{3}$ cubes, each of which has dimensions $1 \times 1 \times 1$, and the number of $1 \times 1 \time... | 9 |
An auto shop buys tires to replace all the tires on every customer’s car. They buy the tires as soon as cars are brought into the shop. There are four cars in the shop already, and another six customers come into the shop throughout the week. Some of the customers decide they don't want any of the tires changing, and t... | A total of 4 + 6 = <<4+6=10>>10 cars are in the shop throughout the week.
This means the shop buys 10 cars * 4 tires = <<10*4=40>>40 tires.
For each of the customers that only want half the tires changing, there are 4 tires * 0.5 = <<4*0.5=2>>2 tires left over.
Two of the customers only want half the tires changing, so... |
Anya has 4 times as many erasers as Andrea. If Andrea has 4 erasers, how many more erasers does Anya have than Andrea? | Anya has 4 x 4 = <<4*4=16>>16 erasers.
Thus, Anya has 16 - 4 = <<16-4=12>>12 more erasers than Andrea.
#### 12 |
Given an arithmetic-geometric sequence $\{a\_n\}$ with a sum of the first $n$ terms denoted as $S\_n$ and a common ratio of $\frac{3}{2}$.
(1) If $S\_4 = \frac{65}{24}$, find $a\_1$;
(2) If $a\_1=2$, $c\_n = \frac{1}{2}a\_n + bn$, and $c\_2$, $c\_4$, $c\_5$ form an arithmetic sequence, find $b$. | -\frac{3}{16} |
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