problem stringlengths 10 5.15k | answer stringlengths 0 1.23k |
|---|---|
Given that $F\_1$ and $F\_2$ are the left and right foci of the ellipse $(E)$: $\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1 (a > b > 0)$, $M$ and $N$ are the endpoints of its minor axis, and the perimeter of the quadrilateral $MF\_1NF\_2$ is $4$, let line $(l)$ pass through $F\_1$ intersecting $(E)$ at points $A$ and $B$ with $|AB|=\frac{4}{3}$.
1. Find the maximum value of $|AF\_2| \cdot |BF\_2|$.
2. If the slope of line $(l)$ is $45^{\circ}$, find the area of $\triangle ABF\_2$. | \frac{2}{3} |
What is the value of $27^3 + 9(27^2) + 27(9^2) + 9^3$? | 46656 |
If $2^a+2^b=3^c+3^d$, the number of integers $a,b,c,d$ which can possibly be negative, is, at most: | 0 |
Lyra has an $80 budget for a week to spend on food. If she bought 1 bucket of fried chicken that costs $12 and 5 pounds of beef that cost $3 per pound. How much is left on her weekly budget? | Five pounds of beef cost $3 x 5 = $<<3*5=15>>15.
So Lyra spent a total of $12 + $15 = $<<12+15=27>>27.
Therefore, $80 - $27 = $<<80-27=53>>53 is left on her weekly budget.
#### 53 |
In the rectangular coordinate system $xOy$, the parametric equation of line $l$ is $\begin{cases} x=t \\ y=t+1 \end{cases}$ (where $t$ is the parameter), and the parametric equation of curve $C$ is $\begin{cases} x=2+2\cos \phi \\ y=2\sin \phi \end{cases}$ (where $\phi$ is the parameter). Establish a polar coordinate system with $O$ as the pole and the non-negative semi-axis of the $x$-axis as the polar axis.
(I) Find the polar coordinate equations of line $l$ and curve $C$;
(II) It is known that ray $OP$: $\theta_1=\alpha$ (where $0<\alpha<\frac{\pi}{2}$) intersects curve $C$ at points $O$ and $P$, and ray $OQ$: $\theta_2=\alpha+\frac{\pi}{2}$ intersects line $l$ at point $Q$. If the area of $\triangle OPQ$ is $1$, find the value of $\alpha$ and the length of chord $|OP|$. | 2\sqrt{2} |
Given that the leftmost position can be occupied by student A or B, and the rightmost position cannot be occupied by student A, find the number of different arrangements of the six high-performing students from Class 1, Grade 12. | 216 |
Let $Q$ be a point chosen uniformly at random in the interior of the unit square with vertices at $(0,0), (1,0), (1,1)$, and $(0,1)$. The probability that the slope of the line determined by $Q$ and the point $\left(\frac{1}{2}, \frac{1}{4} \right)$ is greater than or equal to $1$ can be written as $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$. | 41 |
Coach Grunt is preparing the 5-person starting lineup for his basketball team, the Grunters. There are 12 players on the team. Two of them, Ace and Zeppo, are league All-Stars, so they'll definitely be in the starting lineup. How many different starting lineups are possible? (The order of the players in a basketball lineup doesn't matter.) | 120 |
If 700 were expressed as a sum of at least three distinct powers of 2, what would be the least possible sum of the exponents of these powers? | 30 |
In the Cartesian coordinate plane $xOy$, the equation of the ellipse $C$ is $\frac{x^{2}}{9}+\frac{y^{2}}{10}=1$. Let $F$ be the upper focus of $C$, $A$ be the right vertex of $C$, and $P$ be a moving point on $C$ located in the first quadrant. Find the maximum value of the area of the quadrilateral $OAPF$. | \frac{3 \sqrt{11}}{2} |
A mother buys a box of sweets. She kept 1/3 of the sweets and divided the rest between her 3 children. The eldest got 8 sweets while the youngest got half as many. If there are 27 pieces of sweets in the box, how many sweets did the second child gets? | The mother kept 27 x 1/3 = <<27*1/3=9>>9 pieces of sweets.
So there were 27 - 9 = <<27-9=18>>18 sweets that were divided between the 3 children.
The youngest child got 8/2 = <<8/2=4>>4 sweets.
So the eldest and the youngest got a total of 8 + 4 = <<8+4=12>>12 sweets.
Therefore, the second child gets 18 - 12 = <<18-12=6>>6 sweets.
#### 6 |
Allie and Betty play a game where they take turns rolling a standard die. If a player rolls $n$, she is awarded $f(n)$ points, where \[f(n) = \left\{
\begin{array}{cl} 6 & \text{ if }n\text{ is a multiple of 2 and 3}, \\
2 & \text{ if }n\text{ is only a multiple of 2}, \\
0 & \text{ if }n\text{ is not a multiple of 2}.
\end{array}
\right.\]Allie rolls the die four times and gets a 5, 4, 1, and 2. Betty rolls and gets 6, 3, 3, and 2. What is the product of Allie's total points and Betty's total points? | 32 |
In triangle $PQR$, $PQ = 8$, $QR = 15$, $PR = 17$, and $QS$ is the angle bisector. Find the length of $QS$. | \sqrt{87.04} |
Evaluate $\lfloor\sqrt{17}\rfloor^2$. | 16 |
What is the largest integer \( k \) whose square \( k^2 \) is a factor of \( 10! \)? | 720 |
Lionel walked 4 miles. Esther walked 975 yards and Niklaus walked 1287 feet. How many combined feet did the friends walk? | Lionel = 4 * 5280 = <<4*5280=21120>>21120 feet
Esther = 975 * 3 = <<975*3=2925>>2925 feet
Together = 21120 + 2925 + 1287 = <<21120+2925+1287=25332>>25332 feet
Together the friends walked 25332 feet.
#### 25332 |
Given vectors $\overrightarrow{a} = (4\cos \alpha, \sin \alpha)$, $\overrightarrow{b} = (\sin \beta, 4\cos \beta)$, and $\overrightarrow{c} = (\cos \beta, -4\sin \beta)$, where $\alpha, \beta \in \mathbb{R}$ and neither $\alpha$, $\beta$, nor $\alpha + \beta$ equals $\frac{\pi}{2} + k\pi, k \in \mathbb{Z}$:
1. Find the maximum value of $|\overrightarrow{b} + \overrightarrow{c}|$.
2. When $\overrightarrow{a}$ is parallel to $\overrightarrow{b}$ and perpendicular to $(\overrightarrow{b} - 2\overrightarrow{c})$, find the value of $\tan \alpha + \tan \beta$. | -30 |
Given vectors $\overrightarrow{a}=( \sqrt {3}\sin x,m+\cos x)$ and $\overrightarrow{b}=(\cos x,-m+\cos x)$, and the function $f(x)= \overrightarrow{a}\cdot \overrightarrow{b}$
(1) Find the analytical expression for the function $f(x)$;
(2) When $x\in\left[-\frac{\pi}{6}, \frac{\pi}{3}\right]$, the minimum value of $f(x)$ is $-4$. Find the maximum value of the function $f(x)$ and the corresponding value of $x$ in this interval. | -\frac{3}{2} |
Given that the area of $\triangle ABC$ is $2 \sqrt {3}$, $BC=2$, $C=120^{\circ}$, find the length of side $AB$. | 2 \sqrt {7} |
In convex hexagon $ABCDEF$, all six sides are congruent, $\angle A$ and $\angle D$ are right angles, and $\angle B, \angle C, \angle E,$ and $\angle F$ are congruent. The area of the hexagonal region is $2116(\sqrt{2}+1).$ Find $AB$. | 46 |
What is the product of the numerator and the denominator when $0.\overline{009}$ is expressed as a fraction in lowest terms? | 111 |
Two-thirds of all the animals on my street are cats and the rest are dogs. If there are 300 animals on the street, what's the total number of legs of dogs that are on the street? | The number of cats on the street is 2/3*300 = <<2/3*300=200>>200
If there are 300 animals on the street, and 200 are cats, there are 300-200 = <<300-200=100>>100 dogs.
Since a dog has four legs, the total number of legs the 100 dogs have is 100*4 = <<100*4=400>>400 legs.
#### 400 |
A diagonal of a polygon is a segment joining two nonconsecutive vertices of the polygon. How many diagonals does a regular octagon have? | 20 |
Find the number that becomes a perfect square either by adding 5 or by subtracting 11. | 20 |
When three standard dice are tossed, the numbers $a, b, c$ are obtained. Find the probability that the product of these three numbers, $abc$, equals 8. | \frac{7}{216} |
How many different prime numbers are factors of $N$ if
$\log_2 ( \log_3 ( \log_5 (\log_ 7 N))) = 11?$ | 1 |
In triangle $ABC$, $AX = XY = YB = \frac{1}{2}BC$ and $AB = 2BC$. If the measure of angle $ABC$ is 90 degrees, what is the measure of angle $BAC$? | 22.5 |
A kite has sides $15$ units and $20$ units meeting at a right angle, and its diagonals are $24$ units and $x$ units. Find the area of the kite. | 216 |
Two different prime numbers between $4$ and $18$ are chosen. When their sum is subtracted from their product, which of the following numbers could be obtained? $$
\text A. \ \ 21 \qquad \text B. \ \ 60 \qquad \text C. \ \ 119 \qquad \text D. \ \ 180 \qquad \text E. \ \ 231
$$ | 119 |
If a certain number of cats ate a total of 999,919 mice, and all cats ate the same number of mice, how many cats were there in total? Additionally, each cat ate more mice than there were cats. | 991 |
If a pot can hold 2 cakes at a time and it takes 5 minutes to cook both sides of a cake, calculate the minimum time it will take to cook 3 cakes thoroughly. | 15 |
Given \( x_{i} \geq 0 \) for \( i = 1, 2, \cdots, n \) and \( \sum_{i=1}^{n} x_{i} = 1 \) with \( n \geq 2 \), find the maximum value of \( \sum_{1 \leq i \leq j \leq n} x_{i} x_{j} (x_{i} + x_{j}) \). | \frac{1}{4} |
Let $A = (-3, 0),$ $B=(-2,1),$ $C=(2,1),$ and $D=(3,0).$ Suppose that point $P$ satisfies \[PA + PD = PB + PC = 8.\]Then the $y-$coordinate of $P,$ when simplified, can be expressed in the form $\frac{-a + b \sqrt{c}}{d},$ where $a,$ $b,$ $c,$ $d$ are positive integers. Find $a + b + c + d.$ | 35 |
A certain set of integers is assigned to the letters of the alphabet such that $H=10$. The value of a word is the sum of its assigned letter values. Given that $THIS=50$, $HIT=35$ and $SIT=40$, find the value of $I$. | 15 |
Marissa is serving her kids lunch. Each kid gets a burger with 400 calories and 5 carrot sticks with 20 calories each, and some cookies with 50 calories each. Marissa wants each kid to eat a total of 750 calories for lunch. How many cookies does each kid get? | First figure out how many total calories come from the carrots by multiplying the calories per carrot by the number of carrots: 20 calories/carrot * 5 carrots = <<20*5=100>>100 calories
Then subtract the calories the kids have already eaten from the goal number of calories to find out how many more total calories they should eat: 750 calories - 400 calories - 100 calories = <<750-400-100=250>>250 calories
Then divide the total number of calories by the calories per cookie to find the number of cookies each kid gets: 250 calories / 50 calories/cookie = <<250/50=5>>5 cookies
#### 5 |
Let the width and length of the pan be $w$ and $l$ respectively. If the number of interior pieces is twice the number of perimeter pieces, then find the greatest possible value of $w \cdot l$. | 294 |
The area of rectangle $ABCD$ with vertices $A$(0, 0), $B$(0, 4), $C$($x$, 4) and $D$($x$, 0) is 28 square units. If $x > 0$, what is the value of $x$? | 7 |
Engineer Sergei received a research object with a volume of approximately 200 monoliths (a container designed for 200 monoliths, which was almost completely filled). Each monolith has a specific designation (either "sand loam" or "clay loam") and genesis (either "marine" or "lake-glacial" deposits). The relative frequency (statistical probability) that a randomly chosen monolith is "sand loam" is $\frac{1}{9}$. Additionally, the relative frequency that a randomly chosen monolith is "marine clay loam" is $\frac{11}{18}$. How many monoliths with lake-glacial genesis does the object contain if none of the sand loams are marine? | 77 |
A cyclist initially traveled at a speed of 20 km/h. After covering one-third of the distance, the cyclist looked at the clock and decided to increase the speed by 20%. With the new speed, the cyclist traveled the remaining part of the distance. What is the average speed of the cyclist? | 22.5 |
If $a$, $b$, and $c$ are natural numbers, and $a < b$, $a + b = 719$, $c - a = 915$, then the largest possible value of $a + b + c$ is. | 1993 |
The sum of a positive number and its square is 156. What is the number? | 12 |
An equilateral triangle and a square have the same perimeter of 12 inches. What is the ratio of the side length of the triangle to the side length of the square? Express your answer as a common fraction. | \frac{4}{3} |
First, a number \( a \) is randomly selected from the set \(\{1,2,3, \cdots, 99,100\}\), then a number \( b \) is randomly selected from the same set. Calculate the probability that the last digit of \(3^{a} + 7^{b}\) is 8. | \frac{3}{16} |
Every June 1, an ecologist takes a census of the number of wrens in a state park. She noticed that the number is decreasing by $40\%$ each year. If this trend continues, in what year will the census show that the number of wrens is less than $10\%$ of what it was on June 1, 2004? | 2009 |
Let $x,$ $y,$ $z$ be positive real numbers such that $xyz = 8.$ Find the minimum value of $x + 2y + 4z.$ | 12 |
A metal bar at a temperature of $20^{\circ} \mathrm{C}$ is placed in water at a temperature of $100^{\circ} \mathrm{C}$. After thermal equilibrium is established, the temperature becomes $80^{\circ} \mathrm{C}$. Then, without removing the first bar, another identical metal bar also at $20^{\circ} \mathrm{C}$ is placed in the water. What will be the temperature of the water after thermal equilibrium is established? | 68 |
For how many positive integers $n$, with $n \leq 100$, is $n^{3}+5n^{2}$ the square of an integer? | 8 |
A truck travels due west at $\frac{3}{4}$ mile per minute on a straight road. At the same time, a circular storm, whose radius is $60$ miles, moves southwest at $\frac{1}{2}\sqrt{2}$ mile per minute. At time $t=0$, the center of the storm is $130$ miles due north of the truck. Determine the average time $\frac{1}{2}(t_1 + t_2)$ during which the truck is within the storm circle, where $t_1$ is the time the truck enters and $t_2$ is the time the truck exits the storm circle. | 208 |
A sphere is inscribed in a right cone with base radius $12$ cm and height $24$ cm, as shown. The radius of the sphere can be expressed as $a\sqrt{c} - a$ cm. What is the value of $a + c$? [asy]
import three; size(120); defaultpen(linewidth(1)); pen dashes = linetype("2 2") + linewidth(1);
currentprojection = orthographic(0,-1,0.16);
void drawticks(triple p1, triple p2, triple tickmarks) {
draw(p1--p2); draw(p1 + tickmarks-- p1 - tickmarks); draw(p2 + tickmarks -- p2 - tickmarks);
}
real r = 6*5^.5-6;
triple O = (0,0,0), A = (0,0,-24);
draw(scale3(12)*unitcircle3); draw((-12,0,0)--A--(12,0,0)); draw(O--(12,0,0),dashes);
draw(O..(-r,0,-r)..(0,0,-2r)..(r,0,-r)..cycle);
draw((-r,0,-r)..(0,-r,-r)..(r,0,-r)); draw((-r,0,-r)..(0,r,-r)..(r,0,-r),dashes);
drawticks((0,0,2.8),(12,0,2.8),(0,0,0.5));
drawticks((-13,0,0),(-13,0,-24),(0.5,0,0));
label("$12$", (6,0,3.5), N); label("$24$",(-14,0,-12), W);
[/asy] | 11 |
What is the smallest positive integer with exactly 16 positive divisors? | 120 |
Compute
\[\cos^6 0^\circ + \cos^6 1^\circ + \cos^6 2^\circ + \dots + \cos^6 90^\circ.\] | \frac{229}{8} |
Let $x_1,$ $x_2,$ $\dots,$ $x_{100}$ be real numbers such that $x_1 + x_2 + \dots + x_{100} = 1$ and
\[\frac{x_1}{1 - x_1} + \frac{x_2}{1 - x_2} + \dots + \frac{x_{100}}{1 - x_{100}} = 1.\]Find
\[\frac{x_1^2}{1 - x_1} + \frac{x_2^2}{1 - x_2} + \dots + \frac{x_{100}^2}{1 - x_{100}}.\] | 0 |
On an exam there are 5 questions, each with 4 possible answers. 2000 students went on the exam and each of them chose one answer to each of the questions. Find the least possible value of $n$ , for which it is possible for the answers that the students gave to have the following property: From every $n$ students there are 4, among each, every 2 of them have no more than 3 identical answers. | 25 |
Calculate the value of $\sin 135^{\circ}\cos (-15^{\circ}) + \cos 225^{\circ}\sin 15^{\circ}$. | \frac{1}{2} |
What is the remainder when $x^4-7x^3+9x^2+16x-13$ is divided by $x-3$? | 8 |
Two cards are chosen at random from a standard 52-card deck. What is the probability that the first card is a spade and the second card is an ace? | \frac{1}{52} |
Given a geometric sequence with positive terms $\{a_n\}$ and a common ratio of $2$, if $a_ma_n=4a_2^2$, then the minimum value of $\frac{2}{m}+ \frac{1}{2n}$ equals \_\_\_\_\_\_. | \frac{3}{4} |
Matt is playing basketball. In the first quarter, he made 2-point shots four times and 3-point shots twice. How many points did he score for that quarter? | His total score for the 2-point shots is 2 x 4 = <<2*4=8>>8.
His total score for the 3-point shots is 3 x 2 = <<3*2=6>>6.
His total score for the first quarter is 8 + 6 = <<8+6=14>>14.
#### 14 |
Find $x$ if
\[1 + 5x + 9x^2 + 13x^3 + \dotsb = 85.\] | \frac{4}{5} |
Given an ellipse with the equation \\(\\dfrac{x^{2}}{a^{2}}+\\dfrac{y^{2}}{b^{2}}=1(a > b > 0)\\) and an eccentricity of \\(\\dfrac{\\sqrt{3}}{2}\\). A line $l$ is drawn through one of the foci of the ellipse, perpendicular to the $x$-axis, and intersects the ellipse at points $M$ and $N$, with $|MN|=1$. Point $P$ is located at $(-b,0)$. Point $A$ is any point on the circle $O:x^{2}+y^{2}=b^{2}$ that is different from point $P$. A line is drawn through point $P$ perpendicular to $PA$ and intersects the circle $x^{2}+y^{2}=a^{2}$ at points $B$ and $C$.
(1) Find the standard equation of the ellipse;
(2) Determine whether $|BC|^{2}+|CA|^{2}+|AB|^{2}$ is a constant value. If it is, find that value; if not, explain why. | 26 |
The expression $\dfrac{\sqrt[4]{7}}{\sqrt[3]{7}}$ equals 7 raised to what power? | -\frac{1}{12} |
If $\begin{vmatrix} a & b \\ c & d \end{vmatrix} = 5,$ then find
\[\begin{vmatrix} a - c & b - d \\ c & d \end{vmatrix}.\] | 5 |
What is the largest three-digit multiple of 8 whose digits' sum is 24? | 888 |
Three people, A, B, and C, are taking an elevator from the 1st floor to the 3rd to 7th floors of a mall. Each floor can accommodate at most 2 people getting off the elevator. How many ways are there for them to get off the elevator? | 120 |
Given two circles intersecting at points A(1, 3) and B(m, -1), where the centers of both circles lie on the line $x - y + c = 0$, find the value of $m + c$. | -1 |
In trapezoid \(ABCD\) with bases \(AD\) and \(BC\), the diagonals intersect at point \(E\). The areas of \(\triangle ADE\) and \(\triangle BCE\) are given as 12 and 3, respectively. Find the area of the trapezoid. | 27 |
The expression $y^2+10y+33$ can be written as a combination of a square of a binomial and an integer. Find the integer. | 8 |
A bag has seven apples, eight oranges, and 15 mangoes. Luisa takes out two apples from the bag, and takes out twice as many oranges as apples as she took from the bag. She then takes out 2/3 the number of mangoes from the bag. What is the remaining number of fruits in the bag? | When Luisa takes out two apples from the bag, 7 apples - 2 apples = <<7-2=5>>5 apples remain in the bag.
She also takes 2 apples * 2 oranges/apple = <<2*2=4>>4 oranges from the bag.
The total number of oranges remaining in the bag is 8 oranges - 4 oranges = <<8-4=4>>4 oranges
Additionally, a total of 2/3 * 15 mangoes = <<2/3*15=10>>10 mangoes are also removed from the bag.
The total number of mangoes remaining in the bag is 15 mangoes - 10 mangoes = <<15-10=5>>5 mangoes.
The total number of fruits that Luisa left in the bag is 5 apples + 4 oranges + 5 mangoes = <<5+4+5=14>>14
#### 14 |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are denoted as $a$, $b$, $c$ respectively. It is given that $2b\cos C=a\cos C+c\cos A$.
$(I)$ Find the magnitude of angle $C$;
$(II)$ If $b=2$ and $c= \sqrt {7}$, find $a$ and the area of $\triangle ABC$. | \dfrac {3 \sqrt {3}}{2} |
Indicate the integer closest to the number: \(\sqrt{2012-\sqrt{2013 \cdot 2011}}+\sqrt{2010-\sqrt{2011 \cdot 2009}}+\ldots+\sqrt{2-\sqrt{3 \cdot 1}}\). | 31 |
During intervals, students played table tennis. Any two students played against each other no more than one game. At the end of the week, it turned out that Petya played half, Kolya played a third, and Vasya played a fifth of all the games played during the week. How many games could have been played during the week if it is known that Vasya did not play with either Petya or Kolya? | 30 |
Let $p,$ $q,$ $r,$ and $s$ be the roots of \[x^4 + 10x^3 + 20x^2 + 15x + 6 = 0.\] Find the value of \[\frac{1}{pq} + \frac{1}{pr} + \frac{1}{ps} + \frac{1}{qr} + \frac{1}{qs} + \frac{1}{rs}.\] | \frac{10}{3} |
What is the coefficient of $x^3$ when
$$x^5 - 4x^3 + 3x^2 - 2x + 5$$
is multiplied by
$$3x^2 - 2x + 4$$
and further multiplied by
$$1 - x$$
and the like terms are combined? | -3 |
Brennan was researching his school project and had to download files from the internet to his computer to use for reference. After downloading 800 files, he deleted 70% of them because they were not helpful. He downloaded 400 more files but again realized that 3/5 of them were irrelevant. How many valuable files was he left with after deleting the unrelated files he downloaded in the second round? | The number of non-valuable files Brennan downloaded in the first round is 70/100*800 = <<70/100*800=560>>560 files.
The number of valuable files Brennan downloaded in the first round is 800-560 = <<800-560=240>>240
When he downloaded 400 new files, there were 3/5*400= <<3/5*400=240>>240 non-useful files, which he deleted again.
The total number of valuable files he downloaded in the second round is 400-240 = <<400-240=160>>160
To write his research, Brennan had 160+240 = <<160+240=400>>400 useful files to reference to write his research.
#### 400 |
What is the least positive integer with exactly $10$ positive factors? | 48 |
Stella’s antique shop has 3 dolls, 2 clocks and 5 glasses for sale. She sells the dolls for $5 each. The clocks are priced at $15 each. The glasses are priced at $4 each. If she spent $40 to buy everything and she sells all of her merchandise, how much profit will she make? | The dolls will sell for 3 * $5 = $<<3*5=15>>15.
The clocks will bring in 2 * $15 = $<<2*15=30>>30.
The glasses will sell for 5 * $4 = $<<5*4=20>>20.
In total, she will bring in $15 + $30 + $20 = $<<15+30+20=65>>65
Her total profit will be $65 - $40 = $<<65-40=25>>25.
#### 25 |
48 blacksmiths need to shoe 60 horses. Each blacksmith takes 5 minutes to make one horseshoe. What is the minimum time they should spend on the job? (Note: A horse cannot stand on two legs.) | 25 |
Susan walked to the market to buy five dozen peaches. To carry them home, she brought two identically-sized cloth bags and a much smaller knapsack. Into the knapsack she placed half as many peaches as she placed in each of the two cloth bags. How many peaches did she put in the knapsack? | 5 dozen peaches is 5*12=<<5*12=60>>60 peaches.
Let "x" be the number of peaches put into the knapsack.
Thus, the total number of peaches is x+(2*x)+(2*x)=60.
Simplifying the expression, we see that 5*x=60.
Dividing each side of the equation by 5, we see that x=12 peaches,
#### 12 |
The average score of 60 students is 72. After disqualifying two students whose scores are 85 and 90, calculate the new average score for the remaining class. | 71.47 |
Through points \( R \) and \( E \), located on sides \( AB \) and \( AD \) of parallelogram \( ABCD \) respectively, where \( AR = \frac{2}{3} AB \) and \( AE = \frac{1}{3} AD \), a line is drawn.
Find the ratio of the area of the parallelogram to the area of the resulting triangle. | 9:1 |
Six bags contain 18, 19, 21, 23, 25, and 34 marbles, respectively. One of the bags contains marbles with cracks, while the remaining five bags contain marbles without cracks. Jenny took three of the bags, and George took two of the other bags, leaving the bag with the cracked marbles. If the number of marbles Jenny received is exactly twice the number of marbles George received, determine the number of marbles in the bag with cracks. | 23 |
The axis cross-section $SAB$ of a cone with an equal base triangle side length of 2, $O$ as the center of the base, and $M$ as the midpoint of $SO$. A moving point $P$ is on the base of the cone (including the circumference). If $AM \perp MP$, then the length of the trajectory formed by point $P$ is ( ). | $\frac{\sqrt{7}}{2}$ |
Find the number of pairs of integers $x, y$ with different parities such that $\frac{1}{x}+\frac{1}{y} = \frac{1}{2520}$ . | 90 |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, with $C= \dfrac {\pi}{3}$, $b=8$. The area of $\triangle ABC$ is $10 \sqrt {3}$.
(I) Find the value of $c$;
(II) Find the value of $\cos (B-C)$. | \dfrac {13}{14} |
In triangle $ABC$, $AB=20$ and $AC=11$. The angle bisector of $\angle A$ intersects $BC$ at point $D$, and point $M$ is the midpoint of $AD$. Let $P$ be the point of the intersection of $AC$ and $BM$. The ratio of $CP$ to $PA$ can be expressed in the form $\dfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
| 51 |
Tonya is in a hamburger eating contest. Each hamburger is 4 ounces. Last year the winner ate 84 ounces. How many hamburgers does she have to eat to beat last year's winner? | Last year's winner ate 21 hamburgers because 84 / 4 = <<84/4=21>>21
She needs to eat 22 hamburgers because 21 + 1 = <<21+1=22>>22
#### 22 |
Thirty girls - 13 in red dresses and 17 in blue dresses - were dancing around a Christmas tree. Afterwards, each was asked if the girl to her right was in a blue dress. It turned out that only those who stood between two girls in dresses of the same color answered correctly. How many girls could have answered affirmatively? | 17 |
Among all the roots of
\[z^8 - z^6 + z^4 - z^2 + 1 = 0,\]the maximum imaginary part of a root can be expressed as $\sin \theta,$ where $-90^\circ \le \theta \le 90^\circ.$ Find $\theta.$ | 54^\circ |
Find the sum of all positive divisors of $50$ that are also divisors of $15$. | 6 |
What is half of the absolute value of the difference of the squares of 18 and 16? | 34 |
Let $f(x) = |3\{x\} - 1.5|$, where $\{x\}$ denotes the fractional part of $x$. Find the smallest positive integer $n$ such that the equation \[nf(xf(x)) = 2x\] has at least $1000$ real solutions. | 250 |
Two angles of a triangle measure 30 and 45 degrees. If the side of the triangle opposite the 30-degree angle measures $6\sqrt2$ units, what is the sum of the lengths of the two remaining sides? Express your answer as a decimal to the nearest tenth. | 28.4 |
Anh traveled 75 miles on the interstate and 15 miles on a mountain pass. The speed on the interstate was four times the speed on the mountain pass. If Anh spent 45 minutes driving on the mountain pass, determine the total time of his journey in minutes. | 101.25 |
The area of triangle \( ABC \) is 1. On the rays \( AB \), \( BC \), and \( CA \), points \( B' \), \( C' \), and \( A' \) are marked respectively, such that:
\[ BB' = AB, \quad CC' = 2BC, \quad AA' = 3CA \]
Calculate the area of triangle \( A'B'C' \). | 18 |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $a = 2c \cos A$ and $\sqrt{5} \sin A = 1$, find:
1. $\sin C$
2. $\frac{b}{c}$ | \frac{2\sqrt{5} + 5\sqrt{3}}{5} |
Sidney has 4 kittens and 3 adult cats. She has 7 cans of cat food. Each adult cat eats 1 can of food per day. Each kitten eats 3/4 of a can per day. How many additional cans of food does Sidney need to buy to feed all of her animals for 7 days. | Sidney has 3 adult cats * 1 can of food = <<3*1=3>>3 cans of food per day.
Sidney's adult cats will need 3 cans * 7 days = <<3*7=21>>21 cans for the week.
Sidney has 4 kittens * 3/4 cup of food = <<4*3/4=3>>3 cans of food per day.
Sidney's kittens will need 3 cans * 7 days = <<3*7=21>>21 cans for the week.
Sidney will need to buy 21 cans + 21 cans - 7 cans = <<21+21-7=35>>35 cans.
#### 35 |
Given \((1+x-x^2)^{10} = a_0 + a_1 x + a_2 x^2 + \cdots + a_{20} x^{20}\), find \( a_0 + a_1 + 2a_2 + 3a_3 + \cdots + 20a_{20} \). | -9 |
How many times does the digit 9 appear in the list of all integers from 1 to 1000? | 300 |
How many positive integers less than $500$ can be written as the sum of two positive perfect cubes? | 26 |
Three lathes \( A, B, C \) each process the same type of standard parts at a certain work efficiency. Lathe \( A \) starts 10 minutes earlier than lathe \( C \), and lathe \( C \) starts 5 minutes earlier than lathe \( B \). After lathe \( B \) has been working for 10 minutes, the number of standard parts processed by lathes \( B \) and \( C \) is the same. After lathe \( C \) has been working for 30 minutes, the number of standard parts processed by lathes \( A \) and \( C \) is the same. How many minutes after lathe \( B \) starts will it have processed the same number of standard parts as lathe \( A \)? | 15 |
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