problem stringlengths 10 5.15k | answer stringlengths 0 1.23k |
|---|---|
What is the base four equivalent of $123_{10}$? | 1323_{4} |
Crestwood Elementary School has an active four-square league, consisting of ten players, including Justin and Tim. Each day at recess, the ten players split into two four-square games, each with five players in no relevant order. Over the course of a semester, each possible match-up of five players occurs once. How many times did Justin play in the same game as Tim? | 56 |
Add 78.1563 to 24.3981 and round to the nearest hundredth. | 102.55 |
How many sets of two or more consecutive positive integers have a sum of $15$? | 2 |
Given that in a class of 36 students, more than half purchased notebooks from a store where each notebook had the same price in cents greater than the number of notebooks bought by each student, and the total cost for the notebooks was 990 cents, calculate the price of each notebook in cents. | 15 |
There are 30 people in a room, 60\% of whom are men. If no men enter or leave the room, how many women must enter the room so that 40\% of the total number of people in the room are men? | 15 |
Find the maximum number of points $X_{i}$ such that for each $i$, $\triangle A B X_{i} \cong \triangle C D X_{i}$. | 4 |
Solve for $x$: $x+2x = 400-(3x+4x)$. | 40 |
Find the value(s) of $x$ and $z$ such that $10xz - 15z + 3x - \frac{9}{2} = 0$ is true for all values of $z$. | \frac{3}{2} |
Abe can paint the room in 15 hours, Bea can paint 50 percent faster than Abe, and Coe can paint twice as fast as Abe. Abe begins to paint the room and works alone for the first hour and a half. Then Bea joins Abe, and they work together until half the room is painted. Then Coe joins Abe and Bea, and they work together until the entire room is painted. Find the number of minutes after Abe begins for the three of them to finish painting the room. | 334 |
Find the smallest three-digit number \(n\) such that if the three digits are \(a\), \(b\), and \(c\), then
\[ n = a + b + c + ab + bc + ac + abc. \] | 199 |
Suppose $d\not=0$. We can write $\left(12d+13+14d^2\right)+\left(2d+1\right)$, in the form $ad+b+cd^2$, where $a$, $b$, and $c$ are integers. Find $a+b+c$. | 42 |
A standard die is rolled six times. What is the probability that the product of all six rolls is odd? Express your answer as a common fraction. | \frac{1}{64} |
Brittany, Alex, and Jamy all share 600 marbles divided between them in the ratio 3:5:7. If Brittany gives Alex half of her marbles, what's the total number of marbles that Alex has? | The total ratio representing the number of marbles is 3+5 +7 = <<3+5+7=15>>15
From the ratio, the fraction representing the number of marbles that Brittany has is 3/15, which is equal to 3/15*600 = <<3/15*600=120>>120 marbles.
Alex has 5/15*600 = <<5/15*600=200>>200 marbles.
If Brittany gives half of her marbles to Alex, Alex receives 1/2*120 = 60 marbles.
After receiving 60 marbles from Brittany, Alex has 200+60 = <<200+60=260>>260 marbles.
#### 260 |
If $x^2+x+4 = y - 4$ and $x = -7$, what is the value of $y$? | 50 |
Points $A$ and $C$ lie on a circle centered at $O$, each of $\overline{BA}$ and $\overline{BC}$ are tangent to the circle, and $\triangle ABC$ is equilateral. The circle intersects $\overline{BO}$ at $D$. What is $\frac{BD}{BO}$? | \frac{1}{2} |
A man born in the first half of the nineteenth century was $x$ years old in the year $x^2$. He was born in: | 1806 |
A dot is marked at each vertex of a triangle $A B C$. Then, 2,3 , and 7 more dots are marked on the sides $A B, B C$, and $C A$, respectively. How many triangles have their vertices at these dots? | 357 |
Given the quadratic function $f(x)=x^{2}-x+k$, where $k\in\mathbb{Z}$, if the function $g(x)=f(x)-2$ has two distinct zeros in the interval $(-1, \frac{3}{2})$, find the minimum value of $\frac{[f(x)]^{2}+2}{f(x)}$. | \frac{81}{28} |
A venture capital firm is considering investing in the development of a new energy product, with estimated investment returns ranging from 100,000 yuan to 10,000,000 yuan. The firm is planning to establish a reward scheme for the research team: the reward amount $y$ (in units of 10,000 yuan) increases with the investment return $x$ (in units of 10,000 yuan), with the maximum reward not exceeding 90,000 yuan, and the reward not exceeding 20% of the investment return.
1. Analyze whether the function $y=\frac{x}{150}+1$ meets the company's requirements for the reward function model and explain the reasons.
2. If the company adopts the function model $y=\frac{10x-3a}{x+2}$ as the reward function model, determine the minimum positive integer value of $a$. | 328 |
The coefficient of the $x^2$ term in the expansion of $(1+x)(1+ \sqrt{x})^5$ is \_\_\_\_\_\_. | 15 |
Natalie has a copy of the unit interval $[0,1]$ that is colored white. She also has a black marker, and she colors the interval in the following manner: at each step, she selects a value $x \in[0,1]$ uniformly at random, and (a) If $x \leq \frac{1}{2}$ she colors the interval $[x, x+\frac{1}{2}]$ with her marker. (b) If $x>\frac{1}{2}$ she colors the intervals $[x, 1]$ and $[0, x-\frac{1}{2}]$ with her marker. What is the expected value of the number of steps Natalie will need to color the entire interval black? | 5 |
In $\vartriangle ABC$, the lengths of the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, given that $c=2$, $C=\dfrac{\pi }{3}$.
(1) If the area of $\vartriangle ABC$ is equal to $\sqrt{3}$, find $a$ and $b$;
(2) If $\sin B=2\sin A$, find the area of $\vartriangle ABC$. | \dfrac{2 \sqrt{3}}{3} |
In the diagram, all rows, columns, and diagonals have the sum 12. What is the sum of the four corner numbers? | 16 |
John needs to catch a train. The train arrives randomly some time between 2:00 and 3:00, waits for 20 minutes, and then leaves. If John also arrives randomly between 2:00 and 3:00, what is the probability that the train will be there when John arrives? | \frac{5}{18} |
Given that $a$ and $b$ are coefficients, and the difference between $ax^2+2xy-x$ and $3x^2-2bxy+3y$ does not contain a quadratic term, find the value of $a^2-4b$. | 13 |
Two cards are dealt at random from two standard decks of 104 cards mixed together. What is the probability that the first card drawn is an ace and the second card drawn is also an ace? | \dfrac{7}{1339} |
Shelly makes braided keychains for her friends at school. Each keychain takes 12 inches of thread to braid. This year, she made six friends in classes and half that number from after-school clubs. She wants to make each of them a keychain. How many inches of thread does Shelly need? | Shelly made 6 /2 = <<6/2=3>>3 friends from after-school clubs.
She made 6 + 3 = <<6+3=9>>9 new friends at school this year.
Shelly will need 9 * 12 = <<9*12=108>>108 inches of thread to make each friend a keychain.
#### 108 |
Let $S_n$ be the sum of the first $n$ terms of a geometric sequence $\{a_n\}$, where $a_n > 0$. If $S_6 - 2S_3 = 5$, then the minimum value of $S_9 - S_6$ is ______. | 20 |
Three adults whose average weight is 140 pounds went first in the elevator. Two children whose average weight is 64 pounds also went inside. If an elevator sign reads “Maximum weight 600 pounds.", what is the maximum weight of the next person to get in the elevator so that it will not be overloaded? | The sum of the weights of the three adults is 140 x 3 = <<140*3=420>>420 pounds.
The sum of the weight of the two children is 64 x 2 = <<64*2=128>>128 pounds.
So the total weight of the 5 people who are in the elevator is 420 + 128 = <<420+128=548>>548 pounds.
This would mean that the next person's weight must not exceed 600 - 548 = <<600-548=52>>52 pounds.
#### 52 |
The graph of $y=\frac{5x^2-9}{3x^2+5x+2}$ has vertical asymptotes at $x = a$ and $x = b$. Find $a + b$. | -\frac{5}{3}. |
Equilateral $\triangle ABC$ has side length $\sqrt{111}$. There are four distinct triangles $AD_1E_1$, $AD_1E_2$, $AD_2E_3$, and $AD_2E_4$, each congruent to $\triangle ABC$, with $BD_1 = BD_2 = \sqrt{11}$. Find $\sum_{k=1}^4(CE_k)^2$. | 677 |
If $2^8=16^x$, find $x$. | 2 |
Let $\mathcal{A}$ be the set of finite sequences of positive integers $a_1,a_2,\dots,a_k$ such that $|a_n-a_{n-1}|=a_{n-2}$ for all $3\leqslant n\leqslant k$ . If $a_1=a_2=1$ , and $k=18$ , determine the number of elements of $\mathcal{A}$ . | 1597 |
Let $a,$ $b,$ and $t$ be real numbers such that $a + b = t.$ Find, in terms of $t,$ the minimum value of $a^2 + b^2.$ | \frac{t^2}{2} |
Let $x$, $y$, and $z$ be real numbers greater than $1$, and let $z$ be the geometric mean of $x$ and $y$. The minimum value of $\frac{\log z}{4\log x} + \frac{\log z}{\log y}$ is \_\_\_\_\_\_. | \frac{9}{8} |
Let $p$, $q$, and $r$ be the distinct roots of the polynomial $x^3 - 15x^2 + 50x - 60$. It is given that there exist real numbers $A$, $B$, and $C$ such that \[\dfrac{1}{s^3 - 15s^2 + 50s - 60} = \dfrac{A}{s-p} + \dfrac{B}{s-q} + \frac{C}{s-r}\]for all $s\not\in\{p,q,r\}$. What is $\tfrac1A+\tfrac1B+\tfrac1C$?
A) 133
B) 134
C) 135
D) 136
E) 137 | 135 |
For the pair of positive integers \((x, y)\) such that \(\frac{x^{2}+y^{2}}{11}\) is an integer and \(\frac{x^{2}+y^{2}}{11} \leqslant 1991\), find the number of such pairs \((x, y)\) (where \((a, b)\) and \((b, a)\) are considered different pairs if \(a \neq b\)). | 131 |
Compute \(\arccos(\cos 8.5)\). All functions are in radians. | 2.217 |
Alice starts with the number 0. She can apply 100 operations on her number. In each operation, she can either add 1 to her number, or square her number. After applying all operations, her score is the minimum distance from her number to any perfect square. What is the maximum score she can attain? | 94 |
In triangle $CAT$, we have $\angle{ACT}=\angle{ATC}$ and $\angle{CAT}=36^\circ$. If $\overline{TR}$ bisects $\angle{ATC}$, then how many degrees is $\angle{CRT}$? [asy]
/* AMC8 2000 #13 Problem */
draw((0,0)--(.5,1.75)--(1,0)--cycle);
draw((1,0)--(.15,.5));
label("$R$", (.15,.5), W);
label("$C$", (0,0), SW);
label("$T$", (1,0), SE);
label("$A$", (.5,1.75), N);
[/asy] | 72^\circ |
Find the maximum value of the expression \((\sin 3x + \sin 2y + \sin z)(\cos 3x + \cos 2y + \cos z)\).
(Possible points: 15) | 4.5 |
Natural numbers \( a, b, c \) are chosen such that \( a < b < c \). It is also known that the system of equations \( 2x + y = 2025 \) and \( y = |x - a| + |x - b| + |x - c| \) has exactly one solution. Find the minimum possible value of \( c \). | 1013 |
Solve for $x$: $$5^{x + 4} = 125^x.$$ | x = 2 |
A noodle shop offers classic specialty noodles to customers, who can either dine in at the shop (referred to as "dine-in" noodles) or purchase packaged fresh noodles with condiments (referred to as "fresh" noodles). It is known that the total price of 3 portions of "dine-in" noodles and 2 portions of "fresh" noodles is 31 yuan, and the total price of 4 portions of "dine-in" noodles and 1 portion of "fresh" noodles is 33 yuan.
$(1)$ Find the price of each portion of "dine-in" noodles and "fresh" noodles, respectively.
$(2)$ In April, the shop sold 2500 portions of "dine-in" noodles and 1500 portions of "fresh" noodles. To thank the customers, starting from May 1st, the price of each portion of "dine-in" noodles remains unchanged, while the price of each portion of "fresh" noodles decreases by $\frac{3}{4}a\%$. After analyzing the sales and revenue in May, it was found that the sales volume of "dine-in" noodles remained the same as in April, the sales volume of "fresh" noodles increased by $\frac{5}{2}a\%$ based on April, and the total sales of these two types of noodles increased by $\frac{1}{2}a\%$ based on April. Find the value of $a$. | \frac{40}{9} |
A piece of iron wire with a length of $80cm$ is randomly cut into three segments. Calculate the probability that each segment has a length of no less than $20cm$. | \frac{1}{16} |
Suppose $a,$ $b,$ and $c$ are real numbers such that
\[\frac{ac}{a + b} + \frac{ba}{b + c} + \frac{cb}{c + a} = -9\]and
\[\frac{bc}{a + b} + \frac{ca}{b + c} + \frac{ab}{c + a} = 10.\]Compute the value of
\[\frac{b}{a + b} + \frac{c}{b + c} + \frac{a}{c + a}.\] | 11 |
The average years of experience of three employees, David, Emma, and Fiona, at a company is 12 years. Five years ago, Fiona had the same years of experience as David has now. In 4 years, Emma's experience will be $\frac{3}{4}$ of David's experience at that time. How many years of experience does Fiona have now? | \frac{183}{11} |
If $A=20^{\circ}$ and $B=25^{\circ}$, then the value of $(1+\tan A)(1+\tan B)$ is | 2 |
Consider a three-person game involving the following three types of fair six-sided dice. - Dice of type $A$ have faces labelled $2,2,4,4,9,9$. - Dice of type $B$ have faces labelled $1,1,6,6,8,8$. - Dice of type $C$ have faces labelled $3,3,5,5,7,7$. All three players simultaneously choose a die (more than one person can choose the same type of die, and the players don't know one another's choices) and roll it. Then the score of a player $P$ is the number of players whose roll is less than $P$ 's roll (and hence is either 0,1 , or 2 ). Assuming all three players play optimally, what is the expected score of a particular player? | \frac{8}{9} |
How many irreducible fractions with numerator 2015 exist that are less than \( \frac{1}{2015} \) and greater than \( \frac{1}{2016} \)? | 1440 |
The number $(2^{48}-1)$ is exactly divisible by two numbers between $60$ and $70$. These numbers are | 63,65 |
What is the smallest three-digit multiple of 13? | 104 |
Given the function $f(x)=\sin \omega x\cos \omega x- \sqrt{3}\cos^2\omega x+ \frac{\sqrt{3}}{2} (\omega > 0)$, the two adjacent axes of symmetry of its graph are $\frac{\pi}{2}$.
$(1)$ Find the equation of the axis of symmetry for the function $y=f(x)$.
$(2)$ If the zeros of the function $y=f(x)- \frac{1}{3}$ in the interval $(0,\pi)$ are $x_{1}$ and $x_{2}$, find the value of $\cos (x_{1}-x_{2})$. | \frac{1}{3} |
Kimberly borrows $1000$ dollars from Lucy, who charged interest of $5\%$ per month (which compounds monthly). What is the least integer number of months after which Kimberly will owe more than twice as much as she borrowed? | 15 |
Our school's girls volleyball team has 14 players, including a set of 3 triplets: Alicia, Amanda, and Anna. In how many ways can we choose 6 starters if exactly two of the triplets are in the starting lineup? | 990 |
Given $995 + 997 + 999 + 1001 + 1003 = 5100 - N$, determine $N$. | 100 |
The cafeteria in a certain laboratory is open from noon until 2 in the afternoon every Monday for lunch. Two professors eat 15 minute lunches sometime between noon and 2. What is the probability that they are in the cafeteria simultaneously on any given Monday? | 15 |
Given the set $A=\{m+2, 2m^2+m\}$, if $3 \in A$, then the value of $m$ is \_\_\_\_\_\_. | -\frac{3}{2} |
Compute
\[\sin^2 6^\circ + \sin^2 12^\circ + \sin^2 18^\circ + \dots + \sin^2 174^\circ.\] | \frac{31}{2} |
A triangle with interior angles $60^{\circ}, 45^{\circ}$ and $75^{\circ}$ is inscribed in a circle of radius 2. What is the area of the triangle? | 3 + \sqrt{3} |
Maria subtracts 2 from the number 15, triples her answer, and then adds 5. Liam triples the number 15, subtracts 2 from his answer, and then adds 5. Aisha subtracts 2 from the number 15, adds 5 to her number, and then triples the result. Find the final value for each of Maria, Liam, and Aisha. | 54 |
When $\frac{1}{2222}$ is expressed as a decimal, what is the sum of the first 60 digits after the decimal point? | 108 |
Melissa sells a coupe for $30,000 and an SUV for twice as much. If her commission is 2%, how much money did she make from these sales? | First find the total cost of the SUV: $30,000 * 2 = $<<30000*2=60000>>60,000
Then add the cost of the coupe to find the total cost of the cars: $60,000 + $30,000 = $<<60000+30000=90000>>90,000
Multiplying that amount by Melissa's commission rate, we find her earnings are $90,000 * 2% = $<<90000*2*.01=1800>>1800
#### 1800 |
Angle PQR is a right angle. The three quadrilaterals shown are squares. The sum of the areas of the three squares is 338 square centimeters. What is the number of square centimeters in the area of the largest square?
[asy]
draw((0,0)--(12,0)--(0,5)--cycle);
dot((0,0));
dot((12,0));
dot((0,5));
draw((0,0)--(0,5)--(-5,5)--(-5,0)--cycle);
draw((0,0)--(0,-12)--(12,-12)--(12,0));
draw((0,5)--(5,17)--(17,12)--(12,0)--cycle);
label("$P$",(0,5),NW);
label("$Q$",(0,0),SE);
label("$R$",(12,0),E);
[/asy] | 169 |
Find the sum of all positive real solutions $x$ to the equation \[2\cos2x \left(\cos2x - \cos{\left( \frac{2014\pi^2}{x} \right) } \right) = \cos4x - 1,\]where $x$ is measured in radians. | 1080 \pi |
In a circle of radius $42$, two chords of length $78$ intersect at a point whose distance from the center is $18$. The two chords divide the interior of the circle into four regions. Two of these regions are bordered by segments of unequal lengths, and the area of either of them can be expressed uniquely in the form $m\pi-n\sqrt{d},$ where $m, n,$ and $d$ are positive integers and $d$ is not divisible by the square of any prime number. Find $m+n+d.$
| 378 |
A finite non-empty set of integers is called $3$ -*good* if the sum of its elements is divisible by $3$ . Find the number of $3$ -good subsets of $\{0,1,2,\ldots,9\}$ . | 351 |
A building has seven rooms numbered 1 through 7 on one floor, with various doors connecting these rooms. The doors can be either one-way or two-way. Additionally, there is a two-way door between room 1 and the outside, and there is a treasure in room 7. Design the arrangement of rooms and doors such that:
(a) It is possible to enter room 1, reach the treasure in room 7, and return outside.
(b) The minimum number of steps required to achieve this (each step involving walking through a door) is as large as possible. | 14 |
Given that the sum of three numbers, all equally likely to be $1$, $2$, $3$, or $4$, drawn from an urn with replacement, is $9$, calculate the probability that the number $3$ was drawn each time. | \frac{1}{13} |
Let \( x, y, z \) be positive real numbers such that \( xyz = 3 \). Compute the minimum value of
\[ x^2 + 4xy + 12y^2 + 8yz + 3z^2. \] | 162 |
Given the equation $2x + 3k = 1$ with $x$ as the variable, if the solution for $x$ is negative, then the range of values for $k$ is ____. | \frac{1}{3} |
The values of $k$ for which the equation $2x^2-kx+x+8=0$ will have real and equal roots are: | 9 and -7 |
The right focus of the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ (with $a>0$, $b>0$) is $F$, and $B$ is a point on the left branch of the hyperbola. The line segment $BF$ intersects with one asymptote of the hyperbola at point $A$, and it is given that $(\vec{OF} - \vec{OB}) \cdot \vec{OA} = 0$ and $2\vec{OA} = \vec{OB} + \vec{OF}$ (where $O$ is the origin). Find the eccentricity $e$ of the hyperbola. | \sqrt{5} |
The product of several distinct positive integers is divisible by ${2006}^{2}$ . Determine the minimum value the sum of such numbers can take. | 228 |
The largest prime factor of 101101101101 is a four-digit number $N$. Compute $N$. | 9901 |
Let $\triangle ABC$ be a right triangle with $B$ as the right angle. A circle with diameter $AC$ intersects side $BC$ at point $D$. If $AB = 18$ and $AC = 30$, find the length of $BD$. | 14.4 |
What is the smallest positive integer $n$ such that $2n$ is a perfect square and $3n$ is a perfect cube? | 72 |
A beam of light is emitted from point $P(1,2,3)$, reflected by the $Oxy$ plane, and then absorbed at point $Q(4,4,4)$. The distance traveled by the light beam is ______. | \sqrt{62} |
Let $[x]$ denote the greatest integer less than or equal to the real number $x$. If $n$ is a positive integer, then
$$
\sum_{n=1}^{2014}\left(\left[\frac{n}{2}\right]+\left[\frac{n}{3}\right]+\left[\frac{n}{6}\right]\right)=
$$ | 2027091 |
Define a sequence by \( a_0 = \frac{1}{3} \) and \( a_n = 1 + (a_{n-1} - 1)^3 \). Compute the infinite product \( a_0 a_1 a_2 \dotsm \). | \frac{3}{5} |
It is known that there exists a natural number \( N \) such that \( (\sqrt{3}-1)^{N} = 4817152 - 2781184 \cdot \sqrt{3} \). Find \( N \). | 16 |
The probability of snow for each of the next three days is $\frac{3}{4}$. What is the probability that it will not snow at all during the next three days? Express your answer as a common fraction. | \frac{1}{64} |
Find the fraction \(\frac{p}{q}\) with the smallest possible natural denominator for which \(\frac{1}{2014} < \frac{p}{q} < \frac{1}{2013}\). Enter the denominator of this fraction in the provided field. | 4027 |
I have 10 distinguishable socks in my drawer: 4 white, 4 brown, and 2 blue. In how many ways can I choose a pair of socks, provided that I get two socks of different colors? | 32 |
Given the function $y=x^2+10x+21$, what is the least possible value of $y$? | -4 |
In triangle $PQR,$ $\angle Q = 30^\circ,$ $\angle R = 105^\circ,$ and $PR = 4 \sqrt{2}.$ Find $QR.$ | 8 |
Six consecutive integers are written on a blackboard. When one of them is erased, the sum of the remaining five integers is 2012. What is the sum of the digits of the integer that was erased? | 7 |
How many zeros are at the end of the product \( s(1) \cdot s(2) \cdot \ldots \cdot s(100) \), where \( s(n) \) denotes the sum of the digits of the natural number \( n \)? | 19 |
How many integers from 1 to 16500
a) are not divisible by 5;
b) are not divisible by either 5 or 3;
c) are not divisible by either 5, 3, or 11? | 8000 |
Calculate how many numbers from 1 to 30030 are not divisible by any of the numbers between 2 and 16. | 5760 |
If $X$, $Y$ and $Z$ are different digits, then the largest possible $3-$digit sum for
$\begin{array}{ccc} X & X & X \ & Y & X \ + & & X \ \hline \end{array}$
has the form | $YYZ$ |
Simplify and rationalize the denominator: $$\frac{1}{1+ \frac{1}{\sqrt{3}+1}}.$$ | \sqrt{3}-1 |
Let $f(x)$ be an odd function. Is $f(f(x))$ even, odd, or neither?
Enter "odd", "even", or "neither". | \text{odd} |
Ten standard 6-sided dice are rolled. What is the probability that exactly one of the dice shows a 1? Express your answer as a decimal rounded to the nearest thousandth. | 0.323 |
The vertices of a square are the centers of four circles as shown below. Given each side of the square is 6cm and the radius of each circle is $2\sqrt{3}$cm, find the area in square centimeters of the shaded region. [asy]
fill( (-1,-1)-- (1,-1) -- (1,1) -- (-1,1)--cycle, gray);
fill( Circle((1,1), 1.2), white);
fill( Circle((-1,-1), 1.2), white);
fill( Circle((1,-1),1.2), white);
fill( Circle((-1,1), 1.2), white);
draw( Arc((1,1),1.2 ,180,270));
draw( Arc((1,-1),1.2,90,180));
draw( Arc((-1,-1),1.2,0,90));
draw( Arc((-1,1),1.2,0,-90));
draw( (-1,-1)-- (1,-1) -- (1,1) -- (-1,1)--cycle,linewidth(.8));
[/asy] | 36 - 12\sqrt{3} - 4\pi |
If $64$ is divided into three parts proportional to $2$, $4$, and $6$, the smallest part is: | $10\frac{2}{3}$ |
Determine the distance in feet between the 5th red light and the 23rd red light, where the lights are hung on a string 8 inches apart in the pattern of 3 red lights followed by 4 green lights. Recall that 1 foot is equal to 12 inches. | 28 |
Grace is looking to plant some lettuce in her raised bed garden. Her raised bed is comprised of 2 large beds on top with 2 medium beds on the bottom. The top bed can hold 4 rows of lettuce with 25 seeds being sown per row. The medium bed can house 3 rows with 20 seeds being sown per row. How many seeds can Grace plant in all four beds of her raised bed garden? | A large bed can hold 4 rows with 25 seeds per row, 4 * 25=<<4*25=100>>100 seeds per large bed
100 seeds per large bed and there are 2 beds, 100 * 2= <<100*2=200>>200 seeds needed in total for both large beds.
A medium bed can hold 3 rows with 20 seeds sown per row, 3 * 20=<<3*20=60>>60 seeds per medium bed.
60 seeds per medium bed and there are 2 medium beds, 60 * 2=<<60*2=120>>120 seeds needed in total for both medium beds.
200 seeds needed for the large beds combined with 120 seeds needed for the medium beds comes to 200 +120= <<200+120=320>>320 seeds needed to plant all four beds of the raised garden bed.
#### 320 |
Line $l_1$ has equation $3x - 2y = 1$ and goes through $A = (-1, -2)$. Line $l_2$ has equation $y = 1$ and meets line $l_1$ at point $B$. Line $l_3$ has positive slope, goes through point $A$, and meets $l_2$ at point $C$. The area of $\triangle ABC$ is $3$. What is the slope of $l_3$? | \tfrac34 |
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