problem stringlengths 10 5.15k | answer stringlengths 0 1.23k |
|---|---|
Find the sum of all solutions to the equation $(x-6)^2=25$. | 12 |
The value of $ 21!$ is $ 51{,}090{,}942{,}171{,}abc{,}440{,}000$ , where $ a$ , $ b$ , and $ c$ are digits. What is the value of $ 100a \plus{} 10b \plus{} c$ ? | 709 |
In a certain school, there are $3$ times as many boys as girls and $9$ times as many girls as teachers. Using the letters $b, g, t$ to represent the number of boys, girls, and teachers, respectively, then the total number of boys, girls, and teachers can be represented by the expression | \frac{37b}{27} |
If $x=3$, $y=2x$, and $z=3y$, what is the value of $z$? | 18 |
Given two geometric sequences $\{a_n\}$ and $\{b_n\}$, satisfying $a_1=a$ ($a>0$), $b_1-a_1=1$, $b_2-a_2=2$, and $b_3-a_3=3$.
(1) If $a=1$, find the general formula for the sequence $\{a_n\}$.
(2) If the sequence $\{a_n\}$ is unique, find the value of $a$. | \frac{1}{3} |
Given that angle $A$ is an internal angle of a triangle and $\cos A= \frac{3}{5}$, find $\tan A=$ \_\_\_\_\_\_ and $\tan (A+ \frac{\pi}{4})=$ \_\_\_\_\_\_. | -7 |
Let $f(n)$ be the largest prime factor of $n^{2}+1$. Compute the least positive integer $n$ such that $f(f(n))=n$. | 89 |
Given two parallel lines \\(l_{1}\\) and \\(l_{2}\\) passing through points \\(P_{1}(1,0)\\) and \\(P_{2}(0,5)\\) respectively, and the distance between \\(l_{1}\\) and \\(l_{2}\\) is \\(5\\), then the slope of line \\(l_{1}\\) is \_\_\_\_\_\_. | \dfrac {5}{12} |
Calculate the definite integral:
$$
\int_{0}^{\pi / 4} \frac{5 \operatorname{tg} x+2}{2 \sin 2 x+5} d x
$$ | \frac{1}{2} \ln \left(\frac{14}{5}\right) |
The solutions to the equation $(z+6)^8=81$ are connected in the complex plane to form a convex regular polygon, three of whose vertices are labeled $A,B,$ and $C$. What is the least possible area of triangle $ABC$?
Enter your answer in the form $\frac{a \sqrt{b} - c}{d},$ and simplified as usual. | \frac{3 \sqrt{2} - 3}{2} |
An ordered pair $(a, c)$ of integers, each of which has an absolute value less than or equal to 6, is chosen at random. What is the probability that the equation $ax^2 - 3ax + c = 0$ will not have distinct real roots both greater than 2?
A) $\frac{157}{169}$ B) $\frac{167}{169}$ C) $\frac{147}{169}$ D) $\frac{160}{1... | \frac{167}{169} |
A cone is formed from a 270-degree sector of a circle of radius 18 by aligning the two straight sides. What is the result when the volume of the cone is divided by $\pi$? | 60.75\sqrt{141.75} |
What is the smallest prime whose digits sum to $19$? | 199 |
If $\alpha, \beta, \gamma$ are the roots of the equation $x^3 - x - 1 = 0$, find the value of $\frac{1 + \alpha}{1 - \alpha} + \frac{1 + \beta}{1 - \beta} + \frac{1 + \gamma}{1 - \gamma}$. | -7 |
Equilateral $\triangle ABC$ has side length $2$, and shapes $ABDE$, $BCHT$, $CAFG$ are formed outside the triangle such that $ABDE$ and $CAFG$ are squares, and $BCHT$ is an equilateral triangle. What is the area of the geometric shape formed by $DEFGHT$?
A) $3\sqrt{3} - 1$
B) $3\sqrt{3} - 2$
C) $3\sqrt{3} + 2$
D) $4\sq... | 3\sqrt{3} - 2 |
5 people are standing in a row for a photo, among them one person must stand in the middle. There are ways to arrange them. | 24 |
Given sets $A=\{1,2,3,4,5\}$, $B=\{0,1,2,3,4\}$, and a point $P$ with coordinates $(m,n)$, where $m\in A$ and $n\in B$, find the probability that point $P$ lies below the line $x+y=5$. | \dfrac{2}{5} |
Suppose that all four of the numbers \[3 - 2\sqrt{2}, \; -3-2\sqrt{2}, \; 1+\sqrt{7}, \; 1-\sqrt{7}\]are roots of the same nonzero polynomial with rational coefficients. What is the smallest possible degree of the polynomial? | 6 |
How many integers between 100 and 300 are multiples of both 5 and 7, but are not multiples of 10? | 3 |
Each of two boxes contains both black and white marbles, and the total number of marbles in the two boxes is $25.$ One marble is taken out of each box randomly. The probability that both marbles are black is $27/50,$ and the probability that both marbles are white is $m/n,$ where $m$ and $n$ are relatively prime positi... | 26 |
There are 6 forks in the cutlery drawer. There are 9 more knives than forks, and there are twice as many spoons as knives and half as many teaspoons as forks. After 2 of each cutlery is added to the drawer, how many pieces of cutlery are there in all? | There are 6 + 9 = <<6+9=15>>15 knives
There are 15 x 2 = <<15*2=30>>30 spoons
There are 6 / 2 = <<6/2=3>>3 teaspoons
There are 6 + 15 + 30 + 3 = <<6+15+30+3=54>>54 pieces before adding more cutlery.
2 of each 4 types of cutlery are added, and therefore 2 x 4 = <<2*4=8>>8 pieces added.
In total there are 54 + 8 = <<54+8... |
From a group of $3$ orthopedic surgeons, $4$ neurosurgeons, and $5$ internists, a medical disaster relief team of $5$ people is to be formed. How many different ways can the team be selected such that there is at least one person from each specialty? | 590 |
Building one birdhouse requires 7 planks and 20 nails. If 1 nail costs $0.05, and one plank costs $3, what is the cost, in dollars, to build 4 birdhouses? | The cost of the planks for one birdhouse is 7 * 3 = $<<7*3=21>>21.
And the nails are a cost of 20 * 0.05 = $<<20*0.05=1>>1 for each birdhouse.
So to build one birdhouse one will need 21 + 1 = $<<21+1=22>>22.
So the cost of building 4 birdhouses is at 4 * 22 = $<<4*22=88>>88.
#### 88 |
A package of candy has 3 servings with 120 calories each. John eats half the package. How many calories did he eat? | There were 3*120=<<3*120=360>>360 calories in the package
So he ate 360/2=<<360/2=180>>180 calories
#### 180 |
A family had 10 eggs, but the mother used 5 of them to make an omelet. Then, 2 chickens laid 3 eggs each. How many eggs does the family have now? | There were 10 eggs but 5 of them were used so, there are 10 - 5 = <<10-5=5>>5 eggs.
Then the two chickens laid 2 * 3 = <<2*3=6>>6 eggs in total.
Thus, the family has 5 + 6 = <<5+6=11>>11 eggs now.
#### 11 |
Simplify the expression $(-\frac{1}{343})^{-2/3}$. | 49 |
Given the function $f(x)=\frac{1}{x+1}$, point $O$ is the coordinate origin, point $A_{n}(n,f(n))$ where $n \in \mathbb{N}^{*}$, vector $\overrightarrow{a}=(0,1)$, and $\theta_{n}$ is the angle between vector $\overrightarrow{OA}_{n}$ and $\overrightarrow{a}$. Compute the value of $\frac{\cos \theta_{1}}{\sin \theta_{1... | \frac{2016}{2017} |
For a designer suit, Daniel must specify his waist size in centimeters. If there are $12$ inches in a foot and $30.5$ centimeters in a foot, then what size should Daniel specify, in centimeters, if his waist size in inches is $34$ inches? (You may use a calculator on this problem; answer to the nearest tenth.) | 86.4 |
Evaluate the expression \[ \frac{a+3}{a+1} \cdot \frac{b-2}{b-3} \cdot \frac{c + 9}{c+7} , \] given that $c = b-11$, $b = a+3$, $a = 5$, and none of the denominators are zero. | \frac{1}{3} |
Nine tiles are numbered $1, 2, 3, \cdots, 9,$ respectively. Each of three players randomly selects and keeps three of the tiles, and sums those three values. The probability that all three players obtain an odd sum is $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$
| 17 |
Let $S_n=1-2+3-4+\cdots +(-1)^{n-1}n$, where $n=1,2,\cdots$. Then $S_{17}+S_{33}+S_{50}$ equals: | 1 |
Sarah is leading a class of $35$ students. Initially, all students are standing. Each time Sarah waves her hands, a prime number of standing students sit down. If no one is left standing after Sarah waves her hands $3$ times, what is the greatest possible number of students that could have been standing before her ... | 31 |
A clock takes $7$ seconds to strike $9$ o'clock starting precisely from $9:00$ o'clock. If the interval between each strike increases by $0.2$ seconds as time progresses, calculate the time it takes to strike $12$ o'clock. | 12.925 |
Sector $OAB$ is a quarter of a circle of radius 3 cm. A circle is drawn inside this sector, tangent at three points as shown. What is the number of centimeters in the radius of the inscribed circle? Express your answer in simplest radical form. [asy]
import olympiad; import geometry; size(100); defaultpen(linewidth(0.8... | 3\sqrt{2}-3 |
Let $n \ge 3$ be an integer. Rowan and Colin play a game on an $n \times n$ grid of squares, where each square is colored either red or blue. Rowan is allowed to permute the rows of the grid, and Colin is allowed to permute the columns of the grid. A grid coloring is $orderly$ if:
no matter how Rowan permutes the rows ... | \[ 2 \cdot n! + 2 \] |
A scale drawing of a park shows that one inch represents 800 feet. A line segment in the drawing that is 4.75 inches long represents how many feet? | 3800 |
Consider the set $M=\{1,2,3,...,2020\}.$ Find the smallest positive integer $k$ such that for any subset $A$ of $M$ with $k$ elements, there exist $3$ distinct numbers $a,b,c$ from $M$ such that $a+b, b+c$ and $c+a$ are all in $A.$ | 1011 |
If the system of equations
\begin{align*}
2x-y&=a,\\
3y-6x &=b.
\end{align*}has a solution, find $\frac{a}{b},$ assuming $b \neq 0.$ | -\frac{1}{3} |
Two numbers \( x \) and \( y \) satisfy the equation \( 26x^2 + 23xy - 3y^2 - 19 = 0 \) and are respectively the sixth and eleventh terms of a decreasing arithmetic progression consisting of integers. Find the common difference of this progression. | -3 |
Suppose $a_{1}, a_{2}, \ldots, a_{100}$ are positive real numbers such that $$a_{k}=\frac{k a_{k-1}}{a_{k-1}-(k-1)}$$ for $k=2,3, \ldots, 100$. Given that $a_{20}=a_{23}$, compute $a_{100}$. | 215 |
What is the positive difference between the sum of the first 20 positive even integers and the sum of the first 15 positive odd integers? | 195 |
How many integers $n$ are there such that $3 \leq n \leq 10$ and $121_n$ (the number written as $121$ in base $n$) is a perfect square? | 8 |
In triangle $ABC,$ $D,$ $E,$ and $F$ are points on sides $\overline{BC},$ $\overline{AC},$ and $\overline{AB},$ respectively, so that $BD:DC = CE:EA = AF:FB = 1:2.$
[asy]
unitsize(0.8 cm);
pair A, B, C, D, E, F, P, Q, R;
A = (2,5);
B = (0,0);
C = (7,0);
D = interp(B,C,1/3);
E = interp(C,A,1/3);
F = interp(A,B,1/3);
... | \frac{1}{7} |
The cross below is made up of five congruent squares. The perimeter of the cross is $72$ . Find its area.
[asy]
import graph;
size(3cm);
pair A = (0,0);
pair temp = (1,0);
pair B = rotate(45,A)*temp;
pair C = rotate(90,B)*A;
pair D = rotate(270,C)*B;
pair E = rotate(270,D)*C;
pair F = rotate(90,E)*D;
pair G = rotate(2... | 180 |
In the trapezoid $ABCD$, $CD$ is three times the length of $AB$. Given that the area of the trapezoid is $30$ square units, determine the area of $\triangle ABC$.
[asy]
draw((0,0)--(1,4)--(7,4)--(12,0)--cycle);
draw((1,4)--(0,0));
label("$A$",(1,4),NW);
label("$B$",(7,4),NE);
label("$C$",(12,0),E);
label("$D$",(0,0),W... | 7.5 |
Elroy decides to enter a walk-a-thon and wants to make sure he ties last year's winner's cash collection. Last year, walkers earned $4 a mile. This year walkers earn $2.75 a mile. If last year's winner collected $44, how many more miles will Elroy walk than last year's winner to collect the same amount of money? | Last year's winner walked 11 miles because 44 / 4 = <<44/4=11>>11
Elroy has to walk 16 miles to collect $44 because 44 / 2.75 = <<44/2.75=16>>16
Elroy will walk 5 more miles because 16 - 11 = <<16-11=5>>5
#### 5 |
The function $f(x)$ satisfies
\[f(x - y) = f(x) f(y)\]for all real numbers $x$ and $y,$ and $f(x) \neq 0$ for all real numbers $x.$ Find $f(3).$ | 1 |
The surface of a clock is circular, and on its circumference, there are 12 equally spaced points representing the hours. Calculate the total number of rectangles that can have these points as vertices. | 15 |
Four families visit a tourist spot that has four different routes available for exploration. Calculate the number of scenarios in which exactly one route is not visited by any of the four families. | 144 |
Given in the polar coordinate system, circle $C$: $p=2\cos (\theta+ \frac {\pi}{2})$ and line $l$: $\rho\sin (\theta+ \frac {\pi}{4})= \sqrt {2}$, point $M$ is a moving point on circle $C$. Find the maximum distance from point $M$ to line $l$. | \frac {3 \sqrt {2}}{2}+1 |
Let $u$ and $v$ be real numbers satisfying the inequalities $2u + 3v \le 10$ and $4u + v \le 9.$ Find the largest possible value of $u + 2v$. | 6.1 |
What is the area of the quadrilateral formed by the points of intersection of the circle \(x^2 + y^2 = 16\) and the ellipse \((x-3)^2 + 4y^2 = 36\). | 14 |
Let $f(x)=\frac{3x^2+5x+8}{x^2-x+4}$ and $g(x)=x-1$. Find $f(g(x))+g(f(x))$, evaluated when $x=1$. | 5 |
Nine balls numbered $1, 2, \cdots, 9$ are placed in a bag. These balls differ only in their numbers. Person A draws a ball from the bag, the number on the ball is $a$, and after returning it to the bag, person B draws another ball, the number on this ball is $b$. The probability that the inequality $a - 2b + 10 > 0$ ho... | 61/81 |
In \(\triangle ABC\), \(AC = AB = 25\) and \(BC = 40\). From \(D\), perpendiculars are drawn to meet \(AC\) at \(E\) and \(AB\) at \(F\), calculate the value of \(DE + DF\). | 24 |
Find the sum of the squares of the solutions to
\[\left| x^2 - x + \frac{1}{2008} \right| = \frac{1}{2008}.\] | \frac{1003}{502} |
Find the number of cubic centimeters in the volume of the cylinder formed by rotating a rectangle with side lengths 8 cm and 16 cm about its longer side. Express your answer in terms of \(\pi\). | 256\pi |
Peter needs to buy birdseed to last a week. He knows that each parakeet eats 2 grams a day. His parrots eat 14 grams a day. His finches eat half of what a parakeet eats. If he has 3 parakeets, 2 parrots and 4 finches, how many grams of birdseed does he need to buy? | His parakeets eat 6 grams a day, because three parakeets times two grams each equals 6 grams.
His parrots will eat 28 grams a day because two parrots times 14 grams equals <<2*14=28>>28 grams.
Each finch will eat 1 gram a day, because they eat half of what a parakeet does and 1/2 of 2 equals one.
His finches will eat 4... |
Below is a portion of the graph of a function, $y=f(x)$:
[asy]
import graph; size(8cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-3.25,xmax=5.25,ymin=-3.25,ymax=4.25;
pen cqcqcq=rgb(0.75,0.75,0.75);
/*grid*/ pen gs=linewidth(0.7)+cqcqcq+linetype("2 2"); real gx=1,gy... | 2 |
Tony has $87. He needs to buy some cheese, which costs $7 a pound and a pound of beef that costs $5 a pound. After buying the beef and his cheese, he has $61 left. How many pounds of cheese did he buy? | He spent $26 because 87 - 61 = <<87-61=26>>26
He spend $21 on cheese because 26 -5 = <<26-5=21>>21
He bought 3 pounds of cheese because 21 / 7 = <<21/7=3>>3
#### 3 |
What is the smallest positive integer that has exactly eight distinct positive factors? | 24 |
Given the sequence $\{a_k\}_{k=1}^{11}$ of real numbers defined by $a_1=0.5$, $a_2=(0.51)^{a_1}$, $a_3=(0.501)^{a_2}$, $a_4=(0.511)^{a_3}$, and in general,
$a_k=\begin{cases}
(0.\underbrace{501\cdots 01}_{k+1\text{ digits}})^{a_{k-1}} & \text{if } k \text{ is odd,} \\
(0.\underbrace{501\cdots 011}_{k+1\text{ digits}})... | 30 |
In right triangle $ABC$, we have $\sin A = \frac{3}{5}$ and $\sin B = 1$. Find $\sin C$. | \frac{4}{5} |
The number of distinct pairs $(x,y)$ of real numbers satisfying both of the following equations:
\[x=x^2+y^2\] \[y=2xy\]
is | 4 |
Kevin has four red marbles and eight blue marbles. He arranges these twelve marbles randomly, in a ring. Determine the probability that no two red marbles are adjacent. | \frac{7}{33} |
Find the remainder when $r^{13} + 1$ is divided by $r - 1$. | 2 |
Experts and Viewers play "What? Where? When?" until one side wins six rounds—the first to win six rounds wins the game. The probability of the Experts winning a single round is 0.6, and there are no ties. Currently, the Experts are losing with a score of $3:4$. Find the probability that the Experts will still win. | 0.4752 |
A four-digit number $2\Box\Box5$ is divisible by $45$. How many such four-digit numbers are there? | 11 |
Find the smallest integer $n \geq 5$ for which there exists a set of $n$ distinct pairs $\left(x_{1}, y_{1}\right), \ldots,\left(x_{n}, y_{n}\right)$ of positive integers with $1 \leq x_{i}, y_{i} \leq 4$ for $i=1,2, \ldots, n$, such that for any indices $r, s \in\{1,2, \ldots, n\}$ (not necessarily distinct), there ex... | 8 |
Find all values of $z$ such that $z^4 - 4z^2 + 3 = 0$. Enter all the solutions, separated by commas. | -\sqrt{3},-1,1,\sqrt{3} |
Bryan bought 5 t-shirts and 4 pairs of pants for $1500. If a t-shirt costs $100, how much does each pair of pants cost? | The 5 t-shirts cost $100*5=$<<100*5=500>>500.
The 4 pairs of pants cost $1500-$500=$1000.
Each pair of pants costs $1000/4=$<<1000/4=250>>250.
#### 250 |
A 40 meters rope was cut into 2 parts in the ratio of 2:3. How long is the shorter part? | The rope was cut into 2 + 3 = <<2+3=5>>5 parts.
So each part is 40/5 = <<40/5=8>>8 meters long.
Since the shorter part is consists of 2 parts, then it is 8 x 2 = <<8*2=16>>16 meters long.
#### 16 |
The operation $\otimes$ is defined for all nonzero numbers by $a \otimes b = \frac{a^{2}}{b}$. Determine $[(1 \otimes 2) \otimes 3] - [1 \otimes (2 \otimes 3)]$. | -\frac{2}{3} |
How many non-congruent triangles with perimeter 7 have integer side lengths? | 2 |
Four identical isosceles triangles $A W B, B X C, C Y D$, and $D Z E$ are arranged with points $A, B, C, D$, and $E$ lying on the same straight line. A new triangle is formed with sides the same lengths as $A X, A Y,$ and $A Z$. If $A Z = A E$, find the largest integer value of $x$ such that the area of this new triang... | 22 |
Each of the $2001$ students at a high school studies either Spanish or French, and some study both. The number who study Spanish is between $80$ percent and $85$ percent of the school population, and the number who study French is between $30$ percent and $40$ percent. Let $m$ be the smallest number of students who cou... | 298 |
Circle $B$ has its center at $(-6, 2)$ and a radius of $10$ units. What is the sum of the $y$-coordinates of the two points on circle $B$ that are also on the $y$-axis? | 4 |
Simplify the expression $20(x+y)-19(y+x)$ for all values of $x$ and $y$. | x+y |
Given the ellipse $C$: $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1 (a > b > 0)$ with a focal length of $2\sqrt{2}$, and passing through the point $A(\frac{3}{2}, -\frac{1}{2})$.
(1) Find the equation of the ellipse;
(2) Find the coordinates of a point $P$ on the ellipse $C$ such that its distance to the line $l$: $x+y+... | \sqrt{2} |
Tilly needs to sell 100 bags at $10 per bag to make $300 in profit. How much did she buy each bag for? | Tilly will make 100 x $10 = $<<100*10=1000>>1000 in total
She has spent $1000 - $300 = $<<1000-300=700>>700 on 100 bags
Each bag cost Tilly $700 / 100 = $<<700/100=7>>7
#### 7 |
We have a $100\times100$ garden and we’ve plant $10000$ trees in the $1\times1$ squares (exactly one in each.). Find the maximum number of trees that we can cut such that on the segment between each two cut trees, there exists at least one uncut tree. | 2500 |
The real function $f$ has the property that, whenever $a,$ $b,$ $n$ are positive integers such that $a + b = 2^n,$ the equation
\[f(a) + f(b) = n^2\]holds. What is $f(2002)$? | 96 |
Let $M = 39 \cdot 48 \cdot 77 \cdot 150$. Calculate the ratio of the sum of the odd divisors of $M$ to the sum of the even divisors of $M$. | \frac{1}{62} |
If we let $f(n)$ denote the sum of all the positive divisors of the integer $n$, how many integers $i$ exist such that $1 \le i \le 2010$ and $f(i) = 1 + \sqrt{i} + i$? | 14 |
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 |
Calculate the area of the polygon with vertices at $(2,1)$, $(4,3)$, $(6,1)$, $(4,-2)$, and $(3,4)$. | \frac{11}{2} |
Solve
\[\sqrt{1 + \sqrt{2 + \sqrt{x}}} = \sqrt[3]{1 + \sqrt{x}}.\] | 49 |
Given $|a|=1$, $|b|=2$, and $a+b=(1, \sqrt{2})$, the angle between vectors $a$ and $b$ is _______. | \frac{2\pi}{3} |
Let $g$ be defined by \[g(x) = \left\{
\begin{array}{cl}
x+3 & \text{ if } x \leq 2, \\
x^2 - 4x + 5 & \text{ if } x > 2.
\end{array}
\right.\]Calculate $g^{-1}(1)+g^{-1}(6)+g^{-1}(11)$. | 2 + \sqrt{5} + \sqrt{10} |
The 12 numbers from 1 to 12 on a clock face divide the circumference into 12 equal parts. Using any 4 of these division points as vertices to form a quadrilateral, find the total number of rectangles that can be formed. | 15 |
Given that there are 25 cities in the County of Maplewood, and the average population per city lies between $6,200$ and $6,800$, estimate the total population of all the cities in the County of Maplewood. | 162,500 |
How many of the numbers from the set $\{1,\ 2,\ 3,\ldots,\ 100\}$ have a perfect square factor other than one? | 40 |
Crestview's school colors are purple and gold. The students are designing a flag using three solid-colored horizontal stripes, as shown. Using one or both of the school colors, how many different flags are possible if adjacent stripes may be the same color?
[asy]
size(75);
draw((0,0)--(0,12));
dot((0,12));
draw((0,12)... | 8 |
Triangle $ABC$ has an area 1. Points $E,F,G$ lie, respectively, on sides $BC$, $CA$, $AB$ such that $AE$ bisects $BF$ at point $R$, $BF$ bisects $CG$ at point $S$, and $CG$ bisects $AE$ at point $T$. Find the area of the triangle $RST$. | \frac{7 - 3 \sqrt{5}}{4} |
Solve for $x$: $2^{x-3}=4^{x+1}$ | -5 |
James rents his car out for $20 an hour. He rents it for 8 hours a day 4 days a week. How much does he make a week? | He rents it out for 8*4=<<8*4=32>>32 hours
That means he makes 32*20=$<<32*20=640>>640
#### 640 |
Suppose that $a$ and $b$ are integers such that $$3b = 8 - 2a.$$How many of the first six positive integers must be divisors of $2b + 12$? | 3 |
What is $\frac{2}{5}$ divided by 3? | \frac{2}{15} |
One more than the reciprocal of a particular number is $\frac{7}{3}$. What is the original number expressed as a common fraction? | \frac{3}{4} |
12 balls numbered 1 through 12 are placed in a bin. Joe produces a list of three numbers by performing the following sequence three times: he chooses a ball, records the number, and places the ball back in the bin. How many different lists are possible? | 1728 |
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