problem stringlengths 10 5.15k | answer stringlengths 0 1.23k |
|---|---|
The area of the region bounded by the graph of \[x^2+y^2 = 3|x-y| + 3|x+y|\] is $m+n\pi$, where $m$ and $n$ are integers. What is $m + n$? | 54 |
Let $a\oplus b=3a+4b$ for all real numbers $a$ and $b$. Find $3\oplus 1$. | 13 |
Let $ABC$ be a triangle with $\angle BAC = 60^{\circ}$. Let $AP$ bisect $\angle BAC$ and let $BQ$ bisect $\angle ABC$, with $P$ on $BC$ and $Q$ on $AC$. If $AB + BP = AQ + QB$, what are the angles of the triangle? | \angle B=80^{\circ},\angle C=40^{\circ} |
Jack has a stack of books that is 12 inches thick. He knows from experience that 80 pages is one inch thick. If he has 6 books, how many pages is each one on average? | There are 960 pages because 80 x 12 = <<80*12=960>>960
Each book is 160 pages because 960 / 6 = <<960/6=160>>160
#### 160 |
In the $xy$-plane, consider the L-shaped region bounded by horizontal and vertical segments with vertices at $(0,0)$, $(0,3)$, $(3,3)$, $(3,1)$, $(5,1)$ and $(5,0)$. The slope of the line through the origin that divides the area of this region exactly in half is | \frac{7}{9} |
Tyrone empties his piggy bank and finds two $1 bills, a $5 bill, 13 quarters, 20 dimes, 8 nickels, and 35 pennies. How much money does Tyrone have? | He has $2 in $1 bills because 2 times 1 equals <<2*1=2>>2.
He has $5 in $5 bills because 1 times 5 equals <<5*1=5>>5.
He has $3.25 in quarters because 13 times .25 equals <<13*.25=3.25>>3.25
He has $2 in dimes because 20 times .1 equals <<20*.1=2>>2.
He has $.40 in nickels because 8 times .05 equals .4.
He has $.35 in ... |
Given that the line $2mx+ny-4=0$ passes through the point of intersection of the function $y=\log _{a}(x-1)+2$ where $a>0$ and $a\neq 1$, find the minimum value of $\frac{1}{m}+\frac{4}{n}$. | 3+2\sqrt{2} |
The number $27,000,001$ has exactly four prime factors. Find their sum. | 652 |
It takes 15 men working steadily 4 days to dig the foundation for a new apartment. How many days would it have taken 25 men working at the same rate to dig the foundation? Express your answer as a decimal to the nearest tenth. | 2.4 |
A train leaves station K for station L at 09:30, while another train leaves station L for station K at 10:00. The first train arrives at station L 40 minutes after the trains pass each other. The second train arrives at station K 1 hour and 40 minutes after the trains pass each other. Each train travels at a constant s... | 10:50 |
Integers less than $4010$ but greater than $3000$ have the property that their units digit is the sum of the other digits and also the full number is divisible by 3. How many such integers exist? | 12 |
A line with a slope of 2 and a line with a slope of -4 each have a $y$-intercept of 6. What is the distance between the $x$-intercepts of these lines? | \frac{9}{2} |
What is the matrix $\mathbf{M}$ that performs the transformation which sends square $ABCD$ to square $A'B'C'D'$? (In particular, $A$ goes to $A',$ and so on.)
[asy]
size(200);
import graph;
pair Z=(0,0), A=(2,3), B=(-3,2), C=(-4,1), D=(-1,-4);
Label f;
f.p=fontsize(6);
xaxis(-1.5,1.5,Ticks(f, 1.0));
yaxis(-0.5,... | \begin{pmatrix} 1 & -1 \\ 1 & \phantom -1 \end{pmatrix} |
A 92-digit natural number \( n \) has its first 90 digits given: from the 1st to the 10th digit are ones, from the 11th to the 20th are twos, and so on, from the 81st to the 90th are nines. Find the last two digits of \( n \), given that \( n \) is divisible by 72. | 36 |
How many ordered pairs $(a,b)$ such that $a$ is a positive real number and $b$ is an integer between $2$ and $200$, inclusive, satisfy the equation $(\log_b a)^{2017}=\log_b(a^{2017})?$ | 597 |
In triangle \( \triangle ABC \), \( AB = \sqrt{2} \), \( AC = \sqrt{3} \), and \( \angle BAC = 30^\circ \). Let \( P \) be an arbitrary point in the plane containing \( \triangle ABC \). Find the minimum value of \( \mu = \overrightarrow{PA} \cdot \overrightarrow{PB} + \overrightarrow{PB} \cdot \overrightarrow{PC} + \o... | \frac{\sqrt{2}}{2} - \frac{5}{3} |
In a positive geometric sequence $\{a_{n}\}$, it is known that $a_{1}a_{2}a_{3}=4$, $a_{4}a_{5}a_{6}=8$, and $a_{n}a_{n+1}a_{n+2}=128$. Find the value of $n$. | 16 |
In rectangle $ABCD,$ $P$ is a point on side $\overline{BC}$ such that $BP = 9$ and $CP = 27.$ If $\tan \angle APD = 2,$ then find $AB.$ | 27 |
Simplify: $\frac{2^{n+4} - 2(2^n)}{2(2^{n+3})}$. Express your answer as a common fraction. | \frac{7}{8} |
Find the domain of the function $f(x) = \tan(\arccos(x^2)).$ | [-1,0) \cup (0,1] |
Let $z_1$, $z_2$, $z_3$, $\dots$, $z_{12}$ be the 12 zeroes of the polynomial $z^{12} - 2^{36}$. For each $j$, let $w_j$ be one of $z_j$ or $iz_j$. Find the maximum possible value of the real part of
\[\sum_{j = 1}^{12} w_j.\] | 16 + 16 \sqrt{3} |
Given that $\overrightarrow{a}=(2,3)$, $\overrightarrow{b}=(-4,7)$, and $\overrightarrow{a}+\overrightarrow{c}=\overrightarrow{0}$, find the projection of $\overrightarrow{c}$ on the direction of $\overrightarrow{b}$. | -\frac{\sqrt{65}}{5} |
Let $p$, $q$, and $r$ be solutions of the equation $x^3 - 6x^2 + 11x = 14$.
Compute $\frac{pq}{r} + \frac{qr}{p} + \frac{rp}{q}$. | -\frac{47}{14} |
Find $3x^2 y^2$ if $x$ and $y$ are integers such that $y^2 + 3x^2 y^2 = 30x^2 + 517$.
| 588 |
Cindy tosses 5 dimes into the wishing pond. Eric flips 3 quarters into the pond. Garrick throws in 8 nickels. Ivy then drops 60 pennies in. If Eric dips his hands into the water and pulls out a quarter, how much money, in cents, did they put into the pond? | Cindy puts in 10 cents x 5 = <<10*5=50>>50 cents.
Eric puts in 3 x 25 cents = <<3*25=75>>75 cents.
Garrick puts in 8 x 5 cents = <<8*5=40>>40 cents.
Ivy puts in 60 x 1 cent = <<60*1=60>>60 cents.
After Eric pulls out a quarter, he has still left in the pond, 75 - 25 = <<75-25=50>>50 cents.
Together, they put into the p... |
Factor $(x^2 + 3x + 2)(x^2 + 7x + 12) + (x^2 + 5x - 6)$ as the product of two non-constant polynomials. | (x^2 + 5x + 2)(x^2 + 5x + 9) |
The states of Sunshine and Prairie have adopted new license plate configurations. Sunshine license plates are formatted "LDDLDL" where L denotes a letter and D denotes a digit. Prairie license plates are formatted "LDDLDD". Assuming all 10 digits and 26 letters are equally likely to appear in their respective positions... | 10816000 |
Given the equation $3x^{2}-4=-2x$, find the quadratic coefficient, linear coefficient, and constant term. | -4 |
In triangle \( ABC \), angle \( B \) is right. The midpoint \( M \) is marked on side \( BC \), and there is a point \( K \) on the hypotenuse such that \( AB = AK \) and \(\angle BKM = 45^{\circ}\). Additionally, there are points \( N \) and \( L \) on sides \( AB \) and \( AC \) respectively, such that \( BC = CL \) ... | 1:2 |
If the digits of a natural number can be divided into two groups such that the sum of the digits in each group is equal, the number is called a "balanced number". For example, 25254 is a "balanced number" because $5+2+2=4+5$. If two adjacent natural numbers are both "balanced numbers", they are called a pair of "twin b... | 1099 |
The ratio of boys to girls in a math class is 5:8. How many girls are in the class if the total number of students in the class is 260? | If the ratio of boys to girls in the math class is 5:8, the total ratio of the number of boys and girls in the math class is 5+8 = <<5+8=13>>13.
The fraction that represents the number of boys in the class is 5/13, and since there are 260 students in the class, the number of boys in the class is 5/13*260 = <<5/13*260=1... |
$p$ and $q$ are primes such that the numbers $p+q$ and $p+7 q$ are both squares. Find the value of $p$. | 2 |
Find $A+B$ (in base 10), given the following addition problem \[ \begin{array}{c@{}c@{\;}c@{}c@{}c@{}c}& & & 4 & A & B_{6}\\ &+& & & 4 & 1_{6}\\ \cline{2-6}& & & 5 & 3 & A_{6}\\ \end{array} \] | 9 |
Given a complex number $z$ satisfying $z+ \bar{z}=6$ and $|z|=5$.
$(1)$ Find the imaginary part of the complex number $z$;
$(2)$ Find the real part of the complex number $\dfrac{z}{1-i}$. | \dfrac{7}{2} |
A right circular cone is sliced into three pieces by planes parallel to its base, each piece having equal height. The pieces are labeled from top to bottom; hence the smallest piece is at the top and the largest at the bottom. Calculate the ratio of the volume of the smallest piece to the volume of the largest piece. | \frac{1}{27} |
A day can be evenly divided into 86,400 periods of 1 second; 43,200 periods of each 2 seconds; or in many other ways. In total, how many ways are there to divide a day into $n$ periods of $m$ seconds, where $n$ and $m$ are positive integers? | 96 |
Given a quadratic function $f\left(x\right)=ax^{2}+bx+c$, where $f\left(0\right)=1$, $f\left(1\right)=0$, and $f\left(x\right)\geqslant 0$ for all real numbers $x$. <br/>$(1)$ Find the analytical expression of the function $f\left(x\right)$; <br/>$(2)$ If the maximum value of the function $g\left(x\right)=f\left(x\righ... | m = 2 |
For a certain natural number $n$, $n^2$ gives a remainder of 4 when divided by 5, and $n^3$ gives a remainder of 2 when divided by 5. What remainder does $n$ give when divided by 5? | 3 |
The height $BL$ of the rhombus $ABCD$, dropped perpendicular to the side $AD$, intersects the diagonal $AC$ at point $E$. Find $AE$ if $BL = 8$ and $AL:LD = 3:2$. | 3\sqrt{5} |
Given that $x > 0$, $y > 0$, and $x + 2y = 2$, find the minimum value of $xy$. | \frac{1}{2} |
In a box, there are 100 balls of different colors: 28 red balls, 20 green balls, 12 yellow balls, 20 blue balls, 10 white balls, and 10 black balls. How many balls must be drawn randomly from the box to ensure that at least 15 of them are of the same color? | 75 |
Solve for $x>0$ in the following arithmetic sequence: $1^2, x^2, 3^2, \ldots$. | \sqrt{5} |
The times between $7$ and $8$ o'clock, correct to the nearest minute, when the hands of a clock will form an angle of $84^{\circ}$ are: | 7: 23 and 7: 53 |
For any real numbers \( a \) and \( b \), the inequality \( \max \{|a+b|,|a-b|,|2006-b|\} \geq C \) always holds. Find the maximum value of the constant \( C \). (Note: \( \max \{x, y, z\} \) denotes the largest among \( x, y, \) and \( z \).) | 1003 |
In the diagram, rectangle $PQRS$ is divided into three identical squares. If $PQRS$ has perimeter 120 cm, what is its area, in square centimeters? [asy]
size(4cm);
pair p = (0, 1); pair q = (3, 1); pair r = (3, 0); pair s = (0, 0);
draw(p--q--r--s--cycle);
draw(shift(1) * (p--s)); draw(shift(2) * (p--s));
label("$... | 675 |
Two cubic dice are thrown in succession, where \\(x\\) represents the number shown by the first die, and \\(y\\) represents the number shown by the second die.
\\((1)\\) Find the probability that point \\(P(x,y)\\) lies on the line \\(y=x-1\\);
\\((2)\\) Find the probability that point \\(P(x,y)\\) satisfies \\(y^{... | \dfrac{17}{36} |
Jonah’s five cousins are visiting and there are four identical rooms for them to stay in. If any number of cousins can occupy any room, how many different ways can the cousins be arranged among the rooms? | 51 |
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 |
Robert, a sales agent, earns a basic salary of $1250 per month and, 10% commission on his monthly sales. Last month, his total sales were $23600. He allocated 20% of his total earnings to savings and the rest of the money to his monthly expenses. How much were his monthly expenses last month? | Robert earned a commission of $23600 x 10/100 = $<<23600*10/100=2360>>2360.
So, he earned a total of $1250 + $2360 = $<<1250+2360=3610>>3610.
And, he saved $3610 x 20/100 = $<<3610*20/100=722>>722.
Therefore, his total expenses last month was $3610 - $722= $<<3610-722=2888>>2888
#### 2888 |
Find the largest possible number in decimal notation where all the digits are different, and the sum of its digits is 37. | 976543210 |
In a national park, the number of redwoods is 20% more than the number of pines. If there are 600 pines in the national park, calculate the total number of pines and redwoods that are there. | If there are 600 pines, there are 20/100*600 = <<600*20/100=120>>120 more redwoods than pine trees in the national park.
In total, there are 120+600 = <<120+600=720>>720 redwoods in the national park.
Altogether, there are 720+600 = <<720+600=1320>>1320 pines and redwoods in the national park.
#### 1320 |
At 9:00, a pedestrian set off on a journey. An hour later, a cyclist set off from the same starting point. At 10:30, the cyclist caught up with the pedestrian and continued ahead, but after some time, the bicycle broke down. After repairing the bike, the cyclist resumed the journey and caught up with the pedestrian aga... | 100 |
What is the smallest four-digit number that is divisible by $33$? | 1023 |
There are two rows of seats, with 6 seats in the front row and 7 seats in the back row. Arrange seating for 2 people in such a way that these 2 people cannot sit next to each other. Determine the number of different seating arrangements. | 134 |
Given that $C_{n}^{4}$, $C_{n}^{5}$, and $C_{n}^{6}$ form an arithmetic sequence, find the value of $C_{n}^{12}$. | 91 |
Leah and Jackson run for 45 minutes on a circular track. Leah runs clockwise at 200 m/min in a lane with a radius of 40 meters, while Jackson runs counterclockwise at 280 m/min in a lane with a radius of 55 meters, starting on the same radial line as Leah. Calculate how many times they pass each other after the start. | 72 |
Menkara has a $4 \times 6$ index card. If she shortens the length of one side of this card by $1$ inch, the card would have area $18$ square inches. What would the area of the card be in square inches if instead she shortens the length of the other side by $1$ inch? | 20 |
Erin is watching a TV mini series of Pride and Prejudice. There are 6 episodes that are each 50 minutes long. If Erin watches all of the episodes in one sitting, one after the other with no breaks, how many hours will she spend watching the series? | The total minutes in the series is 6 * 50 = <<6*50=300>>300 minutes
Since there are 60 minutes in an hour, Erin will spend 300 / 60 = <<300/60=5>>5 hours watching the series
#### 5 |
If $\sin x = 3 \cos x,$ then what is $\sin x \cos x$? | \frac{3}{10} |
Three fair, standard six-sided dice are rolled. What is the probability that the sum of the numbers on the top faces is 18? Express your answer as a common fraction. | \frac{1}{216} |
Given positive real numbers $a$ and $b$ satisfying $a+b=2$, the minimum value of $\dfrac{1}{a}+\dfrac{2}{b}$ is ______. | \dfrac{3+2 \sqrt{2}}{2} |
In Morse code, each symbol is represented by a sequence of dashes and dots. How many distinct symbols can be represented using sequences of 1, 2, 3, or 4 total dots and/or dashes? | 30 |
Tom's cat needs an expensive surgery. He has had pet insurance for 24 months that cost $20 per month. The procedure cost $5000 but the insurance covers all but 20% of this. How much money did he save by having insurance? | The insurance cost 24*20=$<<24*20=480>>480
With insurance he pays 5000*.2=$<<5000*.2=1000>>1000 for the procedure
So he paid 1000+480=$<<1000+480=1480>>1480
So he saved 5000-1480=$<<5000-1480=3520>>3520
#### 3520 |
Given points \( A(3,1) \) and \( B\left(\frac{5}{3}, 2\right) \), and the four vertices of quadrilateral \( \square ABCD \) are on the graph of the function \( f(x)=\log _{2} \frac{a x+b}{x-1} \), find the area of \( \square ABCD \). | \frac{26}{3} |
Consider a sequence $F_0=2$ , $F_1=3$ that has the property $F_{n+1}F_{n-1}-F_n^2=(-1)^n\cdot2$ . If each term of the sequence can be written in the form $a\cdot r_1^n+b\cdot r_2^n$ , what is the positive difference between $r_1$ and $r_2$ ?
| \frac{\sqrt{17}}{2} |
Define a set of integers "spacy" if it contains no more than one out of any three consecutive integers. How many subsets of $\{1, 2, 3, \dots, 10\}$, including the empty set, are spacy? | 60 |
In a singing contest, a Rooster, a Crow, and a Cuckoo were contestants. Each jury member voted for one of the three contestants. The Woodpecker tallied that there were 59 judges, and that the sum of votes for the Rooster and the Crow was 15, the sum of votes for the Crow and the Cuckoo was 18, and the sum of votes for ... | 13 |
Determine all six-digit numbers \( p \) that satisfy the following properties:
(1) \( p, 2p, 3p, 4p, 5p, 6p \) are all six-digit numbers;
(2) Each of the six-digit numbers' digits is a permutation of \( p \)'s six digits. | 142857 |
A teen age boy wrote his own age after his father's. From this new four place number, he subtracted the absolute value of the difference of their ages to get $4,289$ . The sum of their ages was | 59 |
**Q14.** Let be given a trinagle $ABC$ with $\angle A=90^o$ and the bisectrices of angles $B$ and $C$ meet at $I$ . Suppose that $IH$ is perpendicular to $BC$ ( $H$ belongs to $BC$ ). If $HB=5 \text{cm}, \; HC=8 \text{cm}$ , compute the area of $\triangle ABC$ . | 40 |
To estimate the consumption of disposable wooden chopsticks, in 1999, a sample of 10 restaurants from a total of 600 high, medium, and low-grade restaurants in a certain county was taken. The daily consumption of disposable chopstick boxes in these restaurants was as follows:
0.6, 3.7, 2.2, 1.5, 2.8, 1.7, 1.2, 2.1, 3... | 7260 |
Sally sews 4 shirts on Monday, 3 shirts on Tuesday, and 2 shirts on Wednesday. Each shirt has 5 buttons. How many buttons does Sally need to sew all the shirts? | Total number of shirts Sally sews is 4 + 3 + 2 = <<4+3+2=9>>9 shirts.
The total number of buttons she needs is 9 x 5 = <<9*5=45>>45 buttons.
#### 45 |
Let $a \oslash b = (\sqrt{2a+b})^3$. If $4 \oslash x = 27$, find the value of $x$. | 1 |
Before starting to paint, Bill had $130$ ounces of blue paint, $164$ ounces of red paint, and $188$ ounces of white paint. Bill painted four equally sized stripes on a wall, making a blue stripe, a red stripe, a white stripe, and a pink stripe. Pink is a mixture of red and white, not necessarily in equal amounts. When ... | 114 |
Aaron is gathering can lids to take to the recycling center. He uses 3 equal-sized boxes of canned tomatoes and adds the lids he gets to the 14 can lids he already has. He is now taking 53 can lids to the recycling center. How many cans lids did he get from each box? | Calculating the difference between the can lids Aaron has now and the amount that he had initially shows there were 53 – 14 = <<53-14=39>>39 can lids in the boxes of canned tomatoes.
As the boxes are the same size, splitting these can lids equally shows that there were 39 / 3 = <<39/3=13>>13 can lids in each box.
#### ... |
If $y=\frac{12x^4+4x^3+9x^2+5x+3}{3x^4+2x^3+8x^2+3x+1}$, at what value of $y$ will there be a horizontal asymptote? | 4 |
Find the distance from the point $(1,-1,2)$ to the line passing through $(-2,2,1)$ and $(-1,-1,3).$ | \sqrt{5} |
A cone is formed from a 300-degree sector of a circle of radius 18 by aligning the two straight sides. [asy]
size(110);
draw(Arc((0,0),1,0,300));
draw((1,0)--(0,0)--(.5,-.5*sqrt(3)));
label("18",(.5,0),S); label("$300^\circ$",(0,0),NW);
[/asy] What is the result when the volume of the cone is divided by $\pi$? | 225\sqrt{11} |
Given vectors \(\boldsymbol{a}\), \(\boldsymbol{b}\), and \(\boldsymbol{c}\) such that
\[
|a|=|b|=3, |c|=4, \boldsymbol{a} \cdot \boldsymbol{b}=-\frac{7}{2},
\boldsymbol{a} \perp \boldsymbol{c}, \boldsymbol{b} \perp \boldsymbol{c}
\]
Find the minimum value of the expression
\[
|x \boldsymbol{a} + y \boldsymbol{b} + (... | \frac{4 \sqrt{33}}{15} |
Let $ABC$ be a triangle with incenter $I$ and $AB = 1400$ , $AC = 1800$ , $BC = 2014$ . The circle centered at $I$ passing through $A$ intersects line $BC$ at two points $X$ and $Y$ . Compute the length $XY$ .
*Proposed by Evan Chen* | 1186 |
How many natural numbers greater than 6 but less than 60 are relatively prime to 15? | 29 |
Starting at $(0,0),$ an object moves in the coordinate plane via a sequence of steps, each of length one. Each step is left, right, up, or down, all four equally likely. Let $p$ be the probability that the object reaches $(2,2)$ in six or fewer steps. Given that $p$ can be written in the form $m/n,$ where $m$ and $n$ a... | 67 |
Joan is answering English and Math exams with 30 and 15 questions, respectively. The English exam is to be completed in 1 hour and the Math exam in 1.5 hours. If Joan wants to spend an equal amount of time on each question, how many more minutes does she have in answering each question on the Math exam than the English... | Joan is given 1 x 60 = <<1*60=60>>60 minutes to answer the English exam.
So Joan will spend 60/30 = <<60/30=2>>2 minutes for each English question.
She is given 1.5 x 60 = <<1.5*60=90>>90 minutes to answer the Math exam.
So, she will spend 90/15 = <<90/15=6>>6 minutes for each Math question.
Hence, Joan has 6 - 2 = <<6... |
A covered rectangular soccer field of length 90 meters and width 60 meters is being designed. It must be illuminated by four floodlights, each hung at some point on the ceiling. Each floodlight illuminates a circle with a radius equal to the height at which it is hung. Determine the minimum possible height of the ceili... | 27.1 |
A rectangle has dimensions $4$ and $2\sqrt{3}$. Two equilateral triangles are contained within this rectangle, each with one side coinciding with the longer side of the rectangle. The triangles intersect, forming another polygon. What is the area of this polygon?
A) $2\sqrt{3}$
B) $4\sqrt{3}$
C) $6$
D) $8\sqrt{3}$ | 4\sqrt{3} |
Carla needs to bring water to her animals. Each horse needs twice as much water as a pig, and the chickens drink from one tank that needs 30 gallons. How many gallons of water does Carla need to bring if she has 8 pigs and 10 horses and each pig needs 3 gallons of water? | First figure out how much water all the pigs need: 8 pigs * 3 gallons/pig = <<8*3=24>>24 gallons
Then figure out how much water one horse needs by multiplying a pig's needs by 2: 3 gallons * 2 = <<3*2=6>>6 gallons
Now find how much water all the horses need: 10 horses * 6 gallons/horse = <<10*6=60>>60 gallons
Finally, ... |
Points are marked on a circle, dividing it into 2012 equal arcs. From these, $k$ points are chosen to construct a convex $k$-gon with vertices at the chosen points. What is the maximum possible value of $k$ such that this polygon has no parallel sides? | 1509 |
The complement of an angle is $5^{\circ}$ more than four times the angle. What is the number of degrees in the measure of the angle? | 17^\circ |
Triangle $PQR$ has vertices $P = (4,0)$, $Q = (0,4)$, and $R$, where $R$ is on the line $x + y = 8$ and also on the line $y = 2x$. Find the area of $\triangle PQR$.
A) $\frac{4}{3}$
B) $\frac{6}{3}$
C) $\frac{8}{3}$
D) $\frac{10}{3}$
E) $\frac{12}{3}$ | \frac{8}{3} |
The circle is divided into 30 equal parts by 30 points on the circle. Randomly selecting 3 different points, what is the probability that these 3 points form an equilateral triangle? | 1/406 |
There are 4 representatives from each of 4 companies at a convention. At the start of the convention, every person shakes hands once with every person except the other representatives from their company. How many handshakes are there? | 96 |
Thirty teams play a tournament in which every team plays every other team exactly once. No ties occur, and each team has a $50 \%$ chance of winning any game it plays. The probability that no two teams win the same number of games is $\frac{p}{q},$ where $p$ and $q$ are relatively prime positive integers. Find $\log_2 ... | 409 |
For real number \( x \), let \( [x] \) denote the greatest integer less than or equal to \( x \). Find the positive integer \( n \) such that \(\left[\log _{2} 1\right] + \left[\log _{2} 2\right] + \left[\log _{2} 3\right] + \cdots + \left[\log _{2} n\right]=1994\). | 312 |
Ryanne is 7 years older than Hezekiah. Together Ryanne's and Hezekiah's ages equal 15 years. How many years old is Hezekiah? | Let H = Hezekiah's age
Ryanne = H + <<+7=7>>7
2H + 7 = 15
2H = 8
H = <<4=4>>4
Hezekiah is 4 years old.
#### 4 |
Cheryl needs 4 cups of basil to make 1 cup of pesto. She can harvest 16 cups of basil from her farm every week for 8 weeks. How many cups of pesto will she be able to make? | She harvests 16 cups of basil every week for 8 weeks for a total of 16*8 = <<16*8=128>>128 cups of basil
She needs 4 cups of basil to make 1 cup of pesto and she has 128 cups of basil so she can make 128/4 = 32 cups of pesto
#### 32 |
Let \( A = (-4, 0) \), \( B = (-1, 2) \), \( C = (1, 2) \), and \( D = (4, 0) \). Suppose that point \( P \) satisfies
\[ PA + PD = 10 \quad \text{and} \quad PB + PC = 10. \]
Find the \( y \)-coordinate of \( P \), when simplified, which can be expressed in the form \( \frac{-a + b \sqrt{c}}{d} \), where \( a, b, c, d... | 35 |
Chad has 100 apples and each apple has different sizes and different price ranges. Each small apple cost $1.5, medium apple cost $2, and big apples cost $3. If Donny bought 6 small and medium apples and also 8 big apples, how much will Donny have to pay for all of it? | Donny was charged $1.5 x 6 = $<<1.5*6=9>>9 for the small apples.
He was charged $2 x 6 = $<<2*6=12>>12 for the medium apples.
And he was also charged $8 x 3 = $<<8*3=24>>24 for the big apples.
Therefore, Donny has to pay $9 + $12 + $24 = $<<9+12+24=45>>45 for all the apples.
#### 45 |
The areas of three squares are 16, 49 and 169. What is the average (mean) of their side lengths? | 8 |
In a circle with center $O$, the measure of $\angle TIQ$ is $45^\circ$ and the radius $OT$ is 12 cm. Find the number of centimeters in the length of arc $TQ$. Express your answer in terms of $\pi$. | 6\pi |
Twelve people arrive at dinner, but the circular table only seats eight. If two seatings, such that one is a rotation of the other, are considered the same, then in how many different ways can we choose eight people, divide them into two groups of four each, and seat each group at two separate circular tables? | 1247400 |
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