problem stringlengths 10 5.15k | answer stringlengths 0 1.23k |
|---|---|
Two circles, both with the same radius $r$ , are placed in the plane without intersecting each other. A line in the plane intersects the first circle at the points $A,B$ and the other at points $C,D$ , so that $|AB|=|BC|=|CD|=14\text{cm}$ . Another line intersects the circles at $E,F$ , respectively $G,H$ so th... | 13 |
If $n = 2^{10} \cdot 3^{14} \cdot 5^{8}$, how many of the natural-number factors of $n$ are multiples of 150? | 980 |
The workers in a factory produce widgets and whoosits. For each product, production time is constant and identical for all workers, but not necessarily equal for the two products. In one hour, $100$ workers can produce $300$ widgets and $200$ whoosits. In two hours, $60$ workers can produce $240$ widgets and $300$ whoo... | 450 |
The matrix for projecting onto a certain line $\ell,$ which passes through the origin, is given by
\[\renewcommand{\arraystretch}{1.5} \begin{pmatrix} \frac{2}{15} & -\frac{1}{15} & -\frac{1}{3} \\ -\frac{1}{15} & \frac{1}{30} & \frac{1}{6} \\ -\frac{1}{3} & \frac{1}{6} & \frac{5}{6} \end{pmatrix} \renewcommand{\arrays... | \begin{pmatrix} 2 \\ -1 \\ -5 \end{pmatrix} |
Michael is baking a cake and needs 6 cups of flour. The only measuring cup he has is the 1/4 cup. He has an 8 cup bag of flour and realizes it would be faster to measure the flour he doesn't need, take it out of the bag, and then dump the rest of the bag into the bowl. How many scoops should he remove? | He needs to remove 2 cups of flour because 8 minus 6 equals <<8-6=2>>2.
He needs to use 8 scoops because 2 divided by .25 equals <<8=8>>8
#### 8 |
ABCD is a square with side of unit length. Points E and F are taken respectively on sides AB and AD so that AE = AF and the quadrilateral CDFE has maximum area. In square units this maximum area is: | \frac{5}{8} |
Given the sequence $\{a\_n\}$ satisfying $a\_1=2$, $a\_2=6$, and $a_{n+2} - 2a_{n+1} + a\_n = 2$, find the value of $\left\lfloor \frac{2017}{a\_1} + \frac{2017}{a\_2} + \ldots + \frac{2017}{a_{2017}} \right\rfloor$, where $\lfloor x \rfloor$ represents the greatest integer not greater than $x$. | 2016 |
Mary tried to improve her health by changing her diet, but her weight bounced like a yo-yo. At first, she dropped a dozen pounds. Then, she added back twice the weight that she initially lost. Then, she dropped three times more weight than she initially had lost. But finally, she gained back half of a dozen pounds. ... | Since Mary first weighed 99 pounds, after she dropped a dozen pounds she weighed 99-12=<<99-12=87>>87 pounds.
Twice the weight she initially lost is 12*2=<<12*2=24>>24 pounds.
So, when she added back twice the weight that she initially lost, her weight jumped to 87+24=<<87+24=111>>111 pounds.
Three times more than she ... |
Given vectors $a = (2, -1, 3)$, $b = (-1, 4, -2)$, and $c = (7, 5, \lambda)$, if vectors $a$, $b$, and $c$ are coplanar, the real number $\lambda$ equals ( ). | \frac{65}{9} |
A regular octagon is inscribed in a circle and another regular octagon is circumscribed about the same circle. What is the ratio of the area of the larger octagon to the area of the smaller octagon? Express your answer as a common fraction. | \frac{4 - 2\sqrt{2}}{2} |
In the quadratic equation $2x^{2}-1=6x$, the coefficient of the quadratic term is ______, the coefficient of the linear term is ______, and the constant term is ______. | -1 |
Which of the following numbers is not an integer? | $\frac{2014}{4}$ |
Extend the square pattern of 8 black and 17 white square tiles by attaching a border of black tiles around the square. What is the ratio of black tiles to white tiles in the extended pattern? | 32/17 |
A fair six-sided die is rolled 3 times. If the sum of the numbers rolled on the first two rolls is equal to the number rolled on the third roll, what is the probability that at least one of the numbers rolled is 2? | $\frac{8}{15}$ |
If $\alpha \in (0, \frac{\pi}{2})$, and $\tan 2\alpha = \frac{\cos \alpha}{2-\sin \alpha}$, calculate the value of $\tan \alpha$. | \frac{\sqrt{15}}{15} |
Johnny TV makes 25 percent more movies than L&J Productions each year. If L&J Productions produces 220 movies in a year, how many movies does the two production companies produce in five years combined? | Johnny TV makes 25/100*220=<<25/100*220=55>>55 more movies than L&J Productions each year.
The total number of movies that Johnny TV makes in a year is 220+55=<<220+55=275>>275
Together, Johnny TV and L&J Productions make 275+220=<<275+220=495>>495 movies in a year.
In five years, the two companies make 495*5=<<495*5=2... |
Two distinct numbers are selected simultaneously and at random from the set $\{1, 2, 3, 4, 5\}$. What is the probability that the smaller one divides the larger one? Express your answer as a common fraction. | \frac{1}{2} |
Simplify and then evaluate the expression:
$$( \frac {x}{x-1}- \frac {x}{x^{2}-1})÷ \frac {x^{2}-x}{x^{2}-2x+1}$$
where $$x= \sqrt {2}-1$$ | 1- \frac { \sqrt {2}}{2} |
Four pairs of socks in different colors are randomly selected from a wardrobe, and it is known that two of them are from the same pair. Calculate the probability that the other two are not from the same pair. | \frac{8}{9} |
Points \( M \) and \( N \) are taken on the diagonals \( AB_1 \) and \( BC_1 \) of the faces of the parallelepiped \( ABCD A_1 B_1 C_1 D_1 \), and the segments \( MN \) and \( A_1 C \) are parallel. Find the ratio of these segments. | 1:3 |
Three days' temperatures were recorded for Bucyrus, Ohio. The temperatures were -14 degrees Fahrenheit, -8 degrees Fahrenheit, and +1 degree Fahrenheit. What was the average number of degrees (Fahrenheit) in Bucyrus, Ohio on the 3 days recorded? | -14 + (-8) + 1 = <<-14+-8+1=-21>>-21 degrees Fahrenheit
-21/3 = <<-21/3=-7>>-7 degrees Fahrenheit
The average temperature was -7 degrees Fahrenheit.
#### -7 |
To bake $12$ cookies, I use $2$ quarts of milk. There are $2$ pints in a quart. How many pints of milk do I need to bake $3$ cookies? | 1 |
Lawrence runs \(\frac{d}{2}\) km at an average speed of 8 minutes per kilometre.
George runs \(\frac{d}{2}\) km at an average speed of 12 minutes per kilometre.
How many minutes more did George run than Lawrence? | 104 |
Let \( a \in \mathbf{R} \). A complex number is given by \(\omega = 1 + a\mathrm{i}\). A complex number \( z \) satisfies \( \overline{\omega} z - \omega = 0 \). Determine the value of \( a \) such that \(|z^2 - z + 2|\) is minimized, and find this minimum value. | \frac{\sqrt{14}}{4} |
Let $\theta$ be the smallest acute angle for which $\sin \theta,$ $\sin 2 \theta,$ $\sin 3 \theta$ form an arithmetic progression, in some order. Find $\cos \theta.$ | \frac{3}{4} |
Consider a city grid with intersections labeled A, B, C, and D. Assume a student walks from intersection A to intersection B every morning, always walking along the designated paths and only heading east or south. The student passes through intersections C and D along the way. The intersections are placed such that A t... | \frac{15}{77} |
If the square roots of a positive number are $a+2$ and $2a-11$, find the positive number. | 225 |
Ray always takes the same route when he walks his dog. First, he walks 4 blocks to the park. Then he walks 7 blocks to the high school. Finally, he walks 11 blocks to get back home. Ray walks his dog 3 times each day. How many blocks does Ray's dog walk each day? | The number of blocks in each walk around the neighborhood is 4 blocks + 7 blocks + 11 blocks = <<4+7+11=22>>22.
Ray's dog walks 3 walks × 22 blocks/walk = <<3*22=66>>66 blocks each day.
#### 66 |
Lydia has a small pool she uses to bathe her dogs. When full, the pool holds 60 gallons of water. She fills her pool using the garden hose, which provides water at the rate of 1.6 gallons per minute. Unfortunately, her pool has a small hole that leaks water at a rate of 0.1 gallons per minute. How long will it take... | With a garden hose that fills at a rate of 1.6 gallons per minute, and a hole that leaks at 0.1 gallons per minute, the net fill rate becomes 1.6-0.1=<<1.6-0.1=1.5>>1.5 gallons per minute.
Therefore, to fill a 60-gallon pool, it will take 60/1.5=<<60/1.5=40>>40 minutes.
#### 40 |
Isabella earns $5 an hour babysitting. She babysits 5 hours every day, 6 afternoons a week. After babysitting for 7 weeks, how much money will Isabella have earned? | She earns 5*5=$<<5*5=25>>25 in a day.
In a week she earns 25*6=$<<25*6=150>>150.
In 7 weeks, she earns 150*7=$<<150*7=1050>>1050
#### 1050 |
How many two-digit positive integers are congruent to 1 (mod 3)? | 30 |
Ayen jogs for 30 minutes every day during weekdays. This week on Tuesday, she jogged 5 minutes more and also jogged 25 minutes more on Friday. How many hours, in total, did Ayen jog this week? | Ayen jogged for a total of 30 x 3 = <<30*3=90>>90 minutes on Monday, Wednesday, and Thursday.
She jogged for 30 + 5 = <<30+5=35>>35 minutes on Tuesday.
She also jogged for 30 + 25 = <<30+25=55>>55 minutes on Friday.
So, Ayen jogged for a total of 90 + 35 + 55 = <<90+35+55=180>>180 minutes.
In hours, this is equal to 18... |
If there are four times as many red crayons as blue crayons in a box, and there are 3 blue crayons. How many crayons total are in the box? | Red crayons: 4(3)=12
Total crayons: 12+3=<<12+3=15>>15 crayons
#### 15 |
Given that \( |z| = 2 \) and \( u = \left|z^{2} - z + 1\right| \), find the minimum value of \( u \) (where \( z \in \mathbf{C} \)). | \frac{3}{2} \sqrt{3} |
The Fibonacci sequence $1,1,2,3,5,8,13,21,\ldots$ starts with two 1s, and each term afterwards is the sum of its two predecessors. Which one of the ten digits is the last to appear in the units position of a number in the Fibonacci sequence? | 6 |
Jessica was trying to win a gift card to her favorite store. To win, she had to guess the total number of red & white jelly beans in the bowl of mixed color jelly beans. She figured it would take three bags of jelly beans to fill up the fishbowl. She assumed that each bag of jellybeans had a similar distribution of c... | We know she counted 24 red jelly beans and 18 white jelly beans so 24 + 18 = <<24+18=42>>42 jelly beans
Even though she only bought 1 bag of jellybeans, she thought it would take 3 bags to fill up the bowl. So 3 bags * 42 jellybeans = <<1*42+1*42+1*42=126>>126 total red & white jellybeans
#### 126 |
Let $D(n)$ denote the number of ways of writing the positive integer $n$ as a product
\[n = f_1\cdot f_2\cdots f_k,\]where $k\ge1$, the $f_i$ are integers strictly greater than $1$, and the order in which the factors are listed matters (that is, two representations that differ only in the order of the factors are count... | 112 |
Let $n$ be the largest real solution to the equation
\[\dfrac{4}{x-2} + \dfrac{6}{x-6} + \dfrac{13}{x-13} + \dfrac{15}{x-15} = x^2 - 7x - 6\]
There are positive integers $p, q,$ and $r$ such that $n = p + \sqrt{q + \sqrt{r}}$. Find $p+q+r$. | 103 |
Greg bought a 20-pack of granola bars to eat at lunch for the week. He set aside one for each day of the week, traded three of the remaining bars to his friend Pete for a soda, and gave the rest to his two sisters. How many did each sister get when they split them evenly? | Greg kept 1 granola bar for each of the 7 days of the week, so there were 20 - 7 = <<20-7=13>>13 bars left.
He traded 3 to his friend Pete, so there were 13 - 3 = 10 bars left.
His 2 sisters split the rest evenly, so each got 10 / 2 = <<10/2=5>>5 granola bars.
#### 5 |
For any two non-zero plane vectors $\overrightarrow \alpha$ and $\overrightarrow \beta$, a new operation $\odot$ is defined as $\overrightarrow \alpha ⊙ \overrightarrow \beta = \frac{\overrightarrow \alpha • \overrightarrow \beta}{\overrightarrow \beta • \overrightarrow \beta}$. Given non-zero plane vectors $\overright... | \frac{2}{3} |
Compute $\sqrt{(31)(30)(29)(28)+1}$. | 869 |
**p4.** What is gcd $(2^6 - 1, 2^9 - 1)$ ?**p5.** Sarah is walking along a sidewalk at a leisurely speed of $\frac12$ m/s. Annie is some distance behind her, walking in the same direction at a faster speed of $s$ m/s. What is the minimum value of $s$ such that Sarah and Annie spend no more than one second within... | \frac{21}{16} |
Given that $α$ is an internal angle of a triangle, and $\sin α + \cos α = \frac{1}{5}$.
$(1)$ Find the value of $\tan α$;
$(2)$ Find the value of $\frac{\sin(\frac{3π}{2}+α)\sin(\frac{π}{2}-α)\tan^{3}(π-α)}{\cos(\frac{π}{2}+α)\cos(\frac{3π}{2}-α)}$. | -\frac{4}{3} |
Simplify $\frac{10a^3}{55a^2}$ when $a=3$. | \frac{6}{11} |
A box holds 2 dozen doughnuts. If the family ate 8 doughnuts, how many doughnuts are left? | Two dozens of doughnuts are equal to 2 x 12 = <<2*12=24>>24 doughnuts.
Since 8 doughnuts were eaten, therefore 24 - 8 = <<24-8=16>>16 doughnuts are left.
#### 16 |
Determine the share of the Japanese yen in the currency structure of the National Wealth Fund (NWF) as of 01.12.2022 using one of the following methods:
First method:
a) Find the total amount of NWF funds placed in Japanese yen as of 01.12.2022:
\[ J P Y_{22} = 1388.01 - 41.89 - 2.77 - 309.72 - 554.91 - 0.24 = 478.4... | -12.6 |
The graph of $y = f(x)$ is shown below.
[asy]
unitsize(0.5 cm);
real func(real x) {
real y;
if (x >= -3 && x <= 0) {y = -2 - x;}
if (x >= 0 && x <= 2) {y = sqrt(4 - (x - 2)^2) - 2;}
if (x >= 2 && x <= 3) {y = 2*(x - 2);}
return(y);
}
int i, n;
for (i = -5; i <= 5; ++i) {
draw((i,-5)--(i,5),gray(0.7));
... | \text{E} |
Find the maximum $y$-coordinate of a point on the graph of $r = \sin 2 \theta.$ | \frac{4 \sqrt{3}}{9} |
On the lateral edges \(AA_1\), \(BB_1\), and \(CC_1\) of a triangular prism \(ABC A_1 B_1 C_1\), points \(M\), \(N\), and \(P\) are located respectively such that \(AM: AA_1 = B_1N: BB_1 = C_1P: CC_1 = 3:4\). On the segments \(CM\) and \(A_1N\), points \(E\) and \(F\) are located respectively such that \(EF \parallel ... | 1/3 |
If a function $f(x)$ satisfies both (1) for any $x$ in the domain, $f(x) + f(-x) = 0$ always holds; and (2) for any $x_1, x_2$ in the domain where $x_1 \neq x_2$, the inequality $\frac{f(x_1) - f(x_2)}{x_1 - x_2} < 0$ always holds, then the function $f(x)$ is called an "ideal function." Among the following three functi... | (3) |
Ivy baked 20 cupcakes in the morning and fifteen more cupcakes in the afternoon than in the morning. How many cupcakes did she bake? | In the afternoon, she baked 20 + 15 = <<20+15=35>>35 cupcakes more cupcakes.
Therefore, she baked a total of 20 + 35 = <<20+35=55>>55 cupcakes.
#### 55 |
If $\mathbf{a}$, $\mathbf{b}$, $\mathbf{c}$, and $\mathbf{d}$ are unit vectors, then find the largest possible value of
\[
\|\mathbf{a} - \mathbf{b}\|^2 + \|\mathbf{a} - \mathbf{c}\|^2 + \|\mathbf{a} - \mathbf{d}\|^2 + \|\mathbf{b} - \mathbf{c}\|^2 + \|\mathbf{b} - \mathbf{d}\|^2 + \|\mathbf{c} - \mathbf{d}\|^2.
\] | 16 |
The three points $(3,-5)$, $(-a + 2, 3)$, and $(2a+3,2)$ lie on the same line. What is $a$? | \frac{-7}{23} |
Paul earns $12.50 for each hour that he works. He then has to pay 20% for taxes and fees. After working 40 hours, Paul receives his paycheck. If he spends 15% of his paycheck on gummy bears, how much, in dollars, does he have left? | His pay before taxes and fees is $12.50 x 40 hours = $<<12.5*40=500>>500.
He pays $500 * 0.20 = $<<500*0.20=100>>100 for taxes and fees.
Then he had $500 - $100 = $<<500-100=400>>400 remaining money.
He takes out $400 * 0.15 = $<<400*0.15=60>>60 for his gummy bears.
So he's left with $400 - $60 = $<<400-60=340>>340.
##... |
There are 3 different balls to be placed into 5 different boxes, with at most one ball per box. How many methods are there? | 60 |
An insect lives on the surface of a regular tetrahedron with edges of length 1. It wishes to travel on the surface of the tetrahedron from the midpoint of one edge to the midpoint of the opposite edge. What is the length of the shortest such trip? (Note: Two edges of a tetrahedron are opposite if they have no common en... | 1 |
The line $y=-\frac{5}{3}x+15$ crosses the $x$-axis at $P$ and the $y$-axis at $Q$. Point $T(r,s)$ is on the line segment $PQ$. If the area of $\triangle POQ$ is twice the area of $\triangle TOP$, what is the value of $r+s$? | 12 |
A ball travels on a parabolic path in which the height (in feet) is given by the expression $-16t^2+32t+15$, where $t$ is the time after launch. What is the maximum height of the ball, in feet? | 31 |
A right circular cylinder with its diameter equal to its height is inscribed in a right circular cone. The cone has diameter 10 and altitude 12, and the axes of the cylinder and cone coincide. Find the radius of the cylinder. Express your answer as a common fraction. | \frac{30}{11} |
Rachel is writing an essay. She writes 1 page every 30 minutes. She spends 45 minutes researching the topic. She writes a total of 6 pages. Then she spends 75 minutes editing her essay. Altogether, how many hours did she spend completing the essay? | She spends this long researching and editing her essay 45 min + 75 min = <<45+75=120>>120 min.
Converting the minutes to hours for researching and editing is 120 / 60 = <<120/60=2>>2 hours.
She spends this long writing the essay 6 x 30 min = <<6*30=180>>180 min.
Converting the minutes to hours for writing the essay is ... |
In the six-digit number $1 A B C D E$, each letter represents a digit. Given that $1 A B C D E \times 3 = A B C D E 1$, calculate the value of $A+B+C+D+E$. | 26 |
Li Shuang rides a bike from location $A$ to location $B$ at a speed of 320 meters per minute. On the way, due to a bike malfunction, he pushes the bike and continues walking for 5 minutes to a location 1800 meters from $B$ to repair the bike. Fifteen minutes later, he resumes riding towards $B$ at 1.5 times his origina... | 72 |
Given two unit vectors $a$ and $b$, and $|a-2b| = \sqrt{7}$, then the angle between $a$ and $b$ is ______. | \frac{2\pi}{3} |
Given acute angles $ \alpha $ and $ \beta $ satisfy $ \sin \alpha =\frac{\sqrt{5}}{5},\sin (\alpha -\beta )=-\frac{\sqrt{10}}{10} $, then $ \beta $ equals \_\_\_\_\_\_\_\_\_\_\_\_. | \frac{\pi}{4} |
In $\triangle XYZ$, the medians $\overline{XU}$ and $\overline{YV}$ intersect at right angles. If $XU = 18$ and $YV = 24$, find the area of $\triangle XYZ$. | 288 |
Let $w$ and $z$ be complex numbers such that $|w+z|=2$ and $|w^2+z^2|=18$. Find the smallest possible value of $|w^3+z^3|$. | 50 |
Pete walks backwards three times faster than Susan walks forwards, and Tracy does one-handed cartwheels twice as fast as Susan walks forwards. But Pete can walk on his hands only one quarter the speed that Tracy can do cartwheels. If Pete walks on his hands at 2 miles per hour, how fast can Pete walk backwards, in mi... | If Pete walks on hands at 1/4 what Tracy can do 1-handed cartwheels, then Tracy can do 1-handed cartwheels at 2*4=8 miles per hour.
Since Tracy does 1-handed cartwheels twice the rate of Susan's forward walking, then Susan walks at 8/2=<<8/2=4>>4 miles per hour.
Pete walks backwards 3x faster than Susan walks forward, ... |
Let $x$ and $y$ be complex numbers such that
\[\frac{x + y}{x - y} + \frac{x - y}{x + y} = 1.\]Find
\[\frac{x^4 + y^4}{x^4 - y^4} + \frac{x^4 - y^4}{x^4 + y^4}.\] | \frac{41}{20} |
James has a total of 66 dollars in his piggy bank. He only has one dollar bills and two dollar bills in his piggy bank. If there are a total of 49 bills in James's piggy bank, how many one dollar bills does he have? | 32 |
There are 70 cookies in a jar. If there are only 28 cookies left after a week, and Paul took out the same amount each day, how many cookies did he take out in four days? | Paul took out a total of 70-28 = <<70-28=42>>42 cookies in a week.
Paul took out 42/7 = <<42/7=6>>6 cookies out of the jar each day.
Over four days, Paul took out 6*4 = <<6*4=24>>24 cookies from the jar.
#### 24 |
Michael bought 6 crates of egg on Tuesday. He gave out 2 crates to Susan, who he admires and bought another 5 crates on Thursday. If one crate holds 30 eggs, how many eggs does he have now? | He had 6 crates and then gave out 2 so he now has 6-2 = <<6-2=4>>4 crates left
He bought an additional 5 crates for a total of 4+5 = <<4+5=9>>9 crates
Each crate has 30 eggs so he has 30*9 = <<30*9=270>>270 eggs
#### 270 |
Simplify $\dfrac{5+12i}{2-3i}$. Your answer should be of the form $a+bi$, where $a$ and $b$ are both real numbers and written as improper fractions (if necessary). | -2+3i |
A farmer has 52 cows. Each cow gives 5 liters of milk a day. How many liters of milk does the farmer get in a week? | The amount of milk one cow gives in a week: 5 liters/day * 7 days/week = <<5*7=35>>35 liters/week
The amount of milk 52 cows give in a week: 52 cows * 35 liters/week/cow= <<52*35=1820>>1820 liters/week
#### 1820 |
Let \(\alpha\) and \(\beta\) be angles such that
\[
\frac{\cos^2 \alpha}{\cos \beta} + \frac{\sin^2 \alpha}{\sin \beta} = 2,
\]
Find the sum of all possible values of
\[
\frac{\sin^2 \beta}{\sin \alpha} + \frac{\cos^2 \beta}{\cos \alpha}.
\] | \sqrt{2} |
How many solutions does the equation
$$
\{x\}^{2}=\left\{x^{2}\right\}
$$
have in the interval $[1, 100]$? ($\{u\}$ denotes the fractional part of $u$, which is the difference between $u$ and the largest integer not greater than $u$.) | 9901 |
A rectangle has a length of $\frac{3}{5}$ and an area of $\frac{1}{3}$. What is the width of the rectangle? | \\frac{5}{9} |
Pegs are put in a board $1$ unit apart both horizontally and vertically. A rubber band is stretched over $4$ pegs as shown in the figure, forming a quadrilateral. Its area in square units is
[asy]
int i,j;
for(i=0; i<5; i=i+1) {
for(j=0; j<4; j=j+1) {
dot((i,j));
}}
draw((0,1)--(1,3)--(4,1)--(3,0)--cycle, linewidth(0.7... | 6 |
Let $ABC$ be a triangle. There exists a positive real number $k$, such that if the altitudes of triangle $ABC$ are extended past $A$, $B$, and $C$, to $A'$, $B'$, and $C'$, as shown, such that $AA' = kBC$, $BB' = kAC$, and $CC' = kAB$, then triangle $A'B'C'$ is equilateral.
[asy]
unitsize(0.6 cm);
pair[] A, B, C;
pa... | \frac{1}{\sqrt{3}} |
Xiao Ming must stand in the very center, and Xiao Li and Xiao Zhang must stand together in a graduation photo with seven students. Find the number of different arrangements. | 192 |
Given triangle \( ABC \). On the side \( AC \), which is the largest in the triangle, points \( M \) and \( N \) are marked such that \( AM = AB \) and \( CN = CB \). It turns out that angle \( NBM \) is three times smaller than angle \( ABC \). Find \( \angle ABC \). | 108 |
Find the number of ordered quadruples $(a,b,c,d)$ of real numbers such that
\[\begin{pmatrix} a & b \\ c & d \end{pmatrix}^{-1} = \renewcommand{\arraystretch}{1.5} \begin{pmatrix} \frac{1}{a} & \frac{1}{b} \\ \frac{1}{c} & \frac{1}{d} \end{pmatrix} \renewcommand{\arraystretch}{1}.\] | 0 |
A circle rests in the interior of the parabola with equation $y = x^2,$ so that it is tangent to the parabola at two points. How much higher is the center of the circle than the points of tangency? | \frac{1}{2} |
Let $f(x) = Ax - 2B^2$ and $g(x) = Bx$, where $B \neq 0$. If $f(g(1)) = 0$, what is $A$ in terms of $B$? | 2B |
In right triangle $ABC$ the hypotenuse $\overline{AB}=5$ and leg $\overline{AC}=3$. The bisector of angle $A$ meets the opposite side in $A_1$. A second right triangle $PQR$ is then constructed with hypotenuse $\overline{PQ}=A_1B$ and leg $\overline{PR}=A_1C$. If the bisector of angle $P$ meets the opposite side in $P_... | \frac{3\sqrt{5}}{4} |
Given the function $f(x)=a\cos (x+\frac{\pi }{6})$, its graph passes through the point $(\frac{\pi }{2}, -\frac{1}{2})$.
(1) Find the value of $a$;
(2) If $\sin \theta =\frac{1}{3}, 0 < \theta < \frac{\pi }{2}$, find $f(\theta ).$ | \frac{2\sqrt{6}-1}{6} |
I have a bag with blue marbles and yellow marbles in it. At the moment, the ratio of blue marbles to yellow marbles is 8:5. If I remove 12 blue marbles and add 21 yellow marbles, the ratio will be 1:3. How many blue marbles were in the bag before I removed some? | 24 |
Solve for the sum of all possible values of $x$ when $3^{x^2+4x+4}=9^{x+2}$. | -2 |
In the diagram, $D$ and $E$ are the midpoints of $\overline{AB}$ and $\overline{BC}$ respectively. Determine the area of $\triangle DBC$.
[asy]
size(180); defaultpen(linewidth(.7pt)+fontsize(10pt));
pair A, B, C, D, E, F;
A=(0,6);
B=(0,0);
C=(8,0);
D=(0,3);
E=(4,0);
F=(8/3,2);
draw(E--A--C--D);
draw((-1,0)--(10,0), E... | 12 |
Find all real solutions to $x^3+(x+1)^3+(x+2)^3=(x+3)^3$. Enter all the solutions, separated by commas. | 3 |
Martha needs 4 cups of berries and 2 cups of heavy cream to make 1 quart of ice cream. She wants to make 1 quart of strawberry ice cream and 1 quart of raspberry ice cream. At the farmers market, the 2 cup packages of strawberries are $3.00 each and the 2 cup package of raspberries are $5.00 each. The heavy cream is... | She needs 4 cups of strawberries per quart of ice cream and the strawberries are sold in 2 cup packages. So 4/2 = <<4/2=2>>2 packages of strawberries
The strawberries cost $3.00 per package and she needs 2 so 3*2=$<<3*2=6.00>>6.00
She needs 4 cups of raspberries per quart of ice cream and the raspberries are sold in 2 ... |
If the product $(3x^2 - 5x + 4)(7 - 2x)$ can be written in the form $ax^3 + bx^2 + cx + d$, where $a,b,c,d$ are real numbers, then find $8a + 4b + 2c + d$. | 18 |
There are $522$ people at a beach, each of whom owns a cat, a dog, both, or neither. If $20$ percent of cat-owners also own a dog, $70$ percent of dog-owners do not own a cat, and $50$ percent of people who don’t own a cat also don’t own a dog, how many people own neither type of pet? | 126 |
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 |
What is the difference between the sum of the first 1000 even counting numbers including 0, and the sum of the first 1000 odd counting numbers? | -1000 |
Evaluate $\lfloor\sqrt{80}\rfloor$. | 8 |
How many of the 200 students surveyed said that their favourite food was sandwiches, given the circle graph results? | 20 |
How many triangles are in the figure to the right? [asy]
defaultpen(linewidth(0.7));
pair hexcoords (real over, real upover)
{
return dir(0)*over+dir(60)*upover;
}
real r = 0.3;
int i,j;
for(i=0;i<=2;++i)
{
for(j=0;j<=2-i;++j)
{
draw(hexcoords(i,j)--hexcoords(i+1,j));
draw(hexcoords(i,j)--hexcoords(i,j+1));
draw... | 16 |
Given that the random variable $X$ follows a normal distribution $N(1,4)$, and $P(0 \leq X \leq 2) = 0.68$, find $P(X > 2)$. | 0.16 |
Jezebel needs to buy two dozens of red roses and 3 pieces of sunflowers for a bouquet that she is going to arrange. Each red rose costs $1.50 and each sunflower costs $3. How much will Jezebel pay for all those flowers? | Jezebel needs to buy 2 x 12 = <<2*12=24>>24 red roses.
Twenty-four red roses will cost 24 x $1.50 = $<<24*1.5=36>>36.
Three sunflower will cost 3 x $3 = $<<3*3=9>>9.
Thus, Jezebel will pay $36 + $9 = $<<36+9=45>>45 for all those flowers.
#### 45 |
How many different positive three-digit integers can be formed using only the digits in the set $\{1, 3, 4, 4, 7, 7, 7\}$ if no digit may be used more times than it appears in the given set of available digits? | 43 |
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