problem stringlengths 10 5.15k | answer stringlengths 0 1.23k |
|---|---|
The proportion of top-quality products at this enterprise is $31 \%$. What is the most likely number of top-quality products in a randomly selected batch of 75 products? | 23 |
The diagonal lengths of a rhombus are 24 units and 10 units. What is the area of the rhombus, in square units? | 120 |
What is the largest three-digit multiple of 9 whose digits' sum is 18? | 990 |
Matt's protein powder is 80% protein. He weighs 80 kg and wants to eat 2 grams of protein per kilogram of body weight each day. How much protein powder does he need to eat per week? | He wants to eat 80*2=<<80*2=160>>160 grams of protein a day
That means he needs 160/.8=<<160/.8=200>>200 grams of protein per day
So he needs 200*7=<<200*7=1400>>1400 grams per week
#### 1400 |
If $-2x - 7 = 7x + 2$, what is the value of $x$? | -1 |
For a natural number \( N \), if at least seven out of the nine natural numbers from 1 to 9 are factors of \( N \), \( N \) is called a "seven-star number." What is the smallest "seven-star number" greater than 2000? | 2016 |
Given a parabola $y^2 = 2px$ ($p > 0$) and a hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ ($a > 0, b > 0$) share a common focus $F$, and point $A$ is the intersection point of the two curves. If $AF \perp x$-axis, find the eccentricity of the hyperbola. | \sqrt{2} + 1 |
Mrs. Watson is grading 120 final exams from her American History class. On Monday, she grades 60% of the exams. On Tuesday, she grades 75% of the remaining exams. On Wednesday, how many more exams does she need to grade? | On Monday, Mrs. Watson grades 120 x 60% = <<120*60*.01=72>>72 final exams.
At the end of Monday, she has 120 - 72 = <<120-72=48>>48 exams left to grade.
On Tuesday, she grades 48 x 75% = <<48*75*.01=36>>36 final exams.
On Wednesday, she has 48 - 36 = <<48-36=12>>12 exams left to grade.
#### 12 |
Find the remainder when $x^5-x^4-x^3+x^2+x$ is divided by $(x^2-4)(x+1)$. | -8x^2+13x+20 |
Junior and Carlson ate a barrel of jam and a basket of cookies, starting and finishing at the same time. Initially, Junior ate the cookies and Carlson ate the jam, then (at some point) they switched. Carlson ate both the jam and the cookies three times faster than Junior. What fraction of the jam did Carlson eat, given that they ate an equal amount of cookies? | 9/10 |
It takes 60 grams of paint to paint a cube on all sides. How much paint is needed to paint a "snake" composed of 2016 such cubes? The beginning and end of the snake are shown in the diagram, and the rest of the cubes are indicated by ellipses. | 80660 |
Simplify $(22a+60b)+(10a+29b)-(9a+50b).$ | 23a+39b |
To get to work, Ryan bikes one day a week takes the bus three times a week and gets a ride from a friend once a week. It takes him thirty minutes to bike to work. The bus takes ten minutes longer since it stops at every bus stop for other passengers. His friend driving him is the fastest and cuts two-thirds off his biking time. How many minutes does he spend every week commuting to work? | The bus to Ryan’s work takes 10 minutes longer than his bike ride, so it takes 10 + 30 = <<10+30=40>>40 minutes.
Ryan takes the bus 3 times a week, so the bus rides take 40 * 3 = <<40*3=120>>120 minutes a week.
His friend cuts two-thirds off his biking time, which is 2/3*30 = <<2/3*30=20>>20 minutes a week.
This means his friend takes 30-20 = <<30-20=10>>10 minutes to drive him.
Therefore, Ryan spends 120 + 10 + 30 = <<120+10+30=160>>160 minutes every week commuting to work.
#### 160 |
Find the maximum constant $k$ such that $\frac{k a b c}{a+b+c} \leqslant (a+b)^{2} + (a+b+4c)^{2}$ holds for all positive real numbers $a, b, c$. | 100 |
Given that $x - \frac{1}{x} = i \sqrt{2},$ find $x^{2187} - \frac{1}{x^{2187}}.$ | i \sqrt{2} |
A ball bounces back up $\frac{2}{3}$ of the height from which it falls. If the ball is dropped from a height of $243$ cm, after how many bounces does the ball first rise less than $30$ cm? | 6 |
Compute the distance between the parallel lines given by
\[\begin{pmatrix} 2 \\ -3 \end{pmatrix} + t \begin{pmatrix} 1 \\ -7 \end{pmatrix}\]and
\[\begin{pmatrix} 1 \\ -5 \end{pmatrix} + s \begin{pmatrix} 1 \\ -7 \end{pmatrix}.\] | \frac{9 \sqrt{2}}{10} |
Katy, Wendi, and Carrie went to a bread-making party. Katy brought three 5-pound bags of flour. Wendi brought twice as much flour as Katy, but Carrie brought 5 pounds less than the amount of flour Wendi brought. How much more flour, in ounces, did Carrie bring than Katy? | Katy's three 5-pound bags of flour weigh 3*5=<<3*5=15>>15 pounds.
Wendi brought twice as much flour as Katy, or 15*2=<<15*2=30>>30 pounds of flour.
Carrie brought 5 pounds less flour than Wendi, or 30-5=<<30-5=25>>25 pounds of flour.
Thus, Carrie brought 25-15=<<25-15=10>>10 pounds of flour more than Katy.
In ounces, the 10 pound difference is 10*16=<<10*16=160>>160 ounces.
#### 160 |
Let $a,$ $b,$ $c,$ be nonzero real numbers such that $a + b + c = 0.$ Find all possible values of
\[\frac{a^2 b^2}{(a^2 - bc)(b^2 - ac)} + \frac{a^2 c^2}{(a^2 - bc)(c^2 - ab)} + \frac{b^2 c^2}{(b^2 - ac)(c^2 - ab)}.\]Enter all possible values, separated by commas. | 1 |
William is taking the 25-question, multiple choice American Mathematics Competition. Each question has five answer choices. William guesses random answers for the last four questions. What is the probability that he will get at least one of these final four questions right? | \frac{369}{625} |
Given that the probability of bus No. 3 arriving at the bus stop within 5 minutes is 0.20 and the probability of bus No. 6 arriving within 5 minutes is 0.60, calculate the probability that the passenger can catch the bus he needs within 5 minutes. | 0.80 |
Mr. Ha owns 5 more silver dollars than Mr. Phung. Mr. Phung has 16 more silver dollars than Mr. Chiu has. If Mr. Chiu has 56 silver dollars, how many silver dollars the three have in total? | Mr. Chiu has 56 silver dollars.
If Mr. Phung has 16 more silver dollars than Mr. Chiu, then his total silver dollars is 56+16 = <<16+56=72>>72
The total number of silver dollars that Mr. Ha owns is 72 +5= <<72+5=77>>77
Combined, together they have 77+72+56 = <<77+72+56=205>>205 silver dollars
#### 205 |
Among 7 students, 6 are to be arranged to participate in social practice activities in two communities on Saturday, with 3 people in each community. How many different arrangements are there? (Answer with a number) | 140 |
If Sally had $20 less, she would have $80. If Jolly has $20 more, she would have $70. How much money do Sally and Jolly have altogether? | Sally has $80 + $20 = $<<80+20=100>>100.
While Jolly has $70 - $20 = $<<70-20=50>>50.
Thus, Sally and Jolly have $100 + $50 = $<<100+50=150>>150 altogether.
#### 150 |
Thirty-nine students from seven classes came up with 60 problems, with students of the same class coming up with the same number of problems (not equal to zero), and students from different classes coming up with a different number of problems. How many students came up with one problem each? | 33 |
Which five-digit numbers are greater in quantity: those not divisible by 5 or those whose first and second digits from the left are not a five? | 72000 |
Find all values of the real number $a$ so that the four complex roots of
\[z^4 - 6z^3 + 11az^2 - 3(2a^2 + 3a - 3) z + 1 = 0\]form the vertices of a parallelogram in the complex plane. Enter all the values, separated by commas. | 3 |
Nathan will roll two six-sided dice. What is the probability that he will roll a number less than three on the first die and a number greater than three on the second die? Express your answer as a common fraction. | \frac{1}{6} |
Suppose that $\alpha$ is inversely proportional to $\beta$. If $\alpha = -3$ when $\beta = -6$, find $\alpha$ when $\beta = 8$. Express your answer as a fraction. | \frac{9}{4} |
Josh went to the shopping center. He bought 9 films and 4 books. He also bought 6 CDs. Each film cost $5, each book cost $4 and each CD cost $3. How much did Josh spend in all? | The cost of the films is 9 × $5 = $<<9*5=45>>45.
The cost of the books is 4 × $4 = $<<4*4=16>>16.
The cost of the CDs is 6 × $3 = $<<6*3=18>>18.
Josh spent $45 + $16 + $18 = $<<45+16+18=79>>79 in total.
#### 79 |
There are twenty-two people in a waiting room. If three more people arrive, the number of people in the waiting room becomes five times the number of people inside the interview room. How many people are in the interview room? | There are twenty-two people in a waiting room, and if three more people arrive, the number increases to 22+3=<<22+3=25>>25
If three more people arrive, the number of people in the waiting room becomes five times the number of people that are inside the interview room, meaning there are 25/5=5 people inside the interview room
#### 5 |
Pasha wrote down an equation consisting of integers and arithmetic operation signs in his notebook. He then encoded each digit and operation sign in the expression on the left side of the equation with a letter, replacing identical digits or signs with identical letters, and different ones with different letters. He ended up with the equation:
$$
\text { VUZAKADEM = } 2023 .
$$
Create at least one version of the expression that Pasha could have encoded. Allowed arithmetic operations are addition, subtraction, multiplication, and division. Parentheses cannot be used. | 2065 + 5 - 47 |
Five packages are delivered to five houses, one to each house. If these packages are randomly delivered, what is the probability that exactly three of them are delivered to the correct houses? | \frac{1}{12} |
It takes 60 grams of paint to paint a cube on all sides. How much paint is needed to paint a "snake" formed from 2016 such cubes? The beginning and end of the snake are shown in the picture, while the rest of the cubes are replaced by an ellipsis. | 80660 |
Determine the number of ways to arrange the letters of the word "PERCEPTION". | 907200 |
Shelby drives her scooter at a speed of $30$ miles per hour if it is not raining, and $20$ miles per hour if it is raining. Today she drove in the sun in the morning and in the rain in the evening, for a total of $16$ miles in $40$ minutes. How many minutes did she drive in the rain? | 24 |
Natasha has more than $\$1$ but less than $\$10$ worth of dimes. When she puts her dimes in stacks of 3, she has 1 left over. When she puts them in stacks of 4, she has 1 left over. When she puts them in stacks of 5, she also has 1 left over. How many dimes does Natasha have? | 61 |
Last year Mr. Jon Q. Public received an inheritance. He paid $20\%$ in federal taxes on the inheritance, and paid $10\%$ of what he had left in state taxes. He paid a total of $\$10500$ for both taxes. How many dollars was his inheritance? | 37500 |
In the diagram below, $\overline{AB}\parallel \overline{CD}$ and $\angle AXF= 118^\circ$. Find $\angle FYD$.
[asy]
unitsize(1inch);
pair A,B,C,D,X,Y,EE,F;
A = (0,0);
B=(1,0);
C = (0,0.8);
D=(1,0.8);
EE = (0.35,-0.3);
F = (0.8,1.1);
draw(EE--F);
draw(A--B);
draw(C--D);
dot(A);
dot(B);
dot(C);
dot(D);
dot(EE);
dot(F);
label("$E$",EE,S);
label("$F$",F,N);
X = intersectionpoint(A--B,EE--F);
Y = intersectionpoint(C--D,EE--F);
label("$X$",X,NNW);
label("$Y$",Y,NNW);
label("$A$",A,W);
label("$B$",B,E);
label("$C$",C,W);
label("$D$",D,E);
dot(X);
dot(Y);
[/asy] | 62^\circ |
Consider a round table on which $2014$ people are seated. Suppose that the person at the head of the table receives a giant plate containing all the food for supper. He then serves himself and passes the plate either right or left with equal probability. Each person, upon receiving the plate, will serve himself if necessary and similarly pass the plate either left or right with equal probability. Compute the probability that you are served last if you are seated $2$ seats away from the person at the head of the table. | 1/2013 |
Jon’s textbooks weigh three times as much as Brandon’s textbooks. Jon has four textbooks that weigh two, eight, five and nine pounds respectively. How much do Brandon’s textbooks weigh? | Jon’s textbooks weigh 2 + 8 + 5 + 9 = <<2+8+5+9=24>>24 pounds.
Thus, Brandon’s textbooks weigh 24 / 3 = <<24/3=8>>8 pounds.
#### 8 |
In the center of a circular field, there is a geologist's cabin. From it extend 6 straight roads, dividing the field into 6 equal sectors. Two geologists start a journey from their cabin at a speed of 4 km/h each on a randomly chosen road. Determine the probability that the distance between them will be at least 6 km after one hour. | 0.5 |
Two quadrilaterals are considered the same if one can be obtained from the other by a rotation and a translation. How many different convex cyclic quadrilaterals are there with integer sides and perimeter equal to 32?
$\textbf{(A)}\ 560 \qquad \textbf{(B)}\ 564 \qquad \textbf{(C)}\ 568 \qquad \textbf{(D)}\ 1498 \qquad \textbf{(E)}\ 2255$
| 568 |
A town is holding a fireworks display for New Year’s Eve. They light enough fireworks to show the full year then light enough fireworks to write “HAPPY NEW YEAR” in the sky. They then light another 50 boxes of fireworks as part of the display. Each box of fireworks contains 8 fireworks. If it takes 6 fireworks to display a number and 5 fireworks to display a letter, how many fireworks are lit during the display? | Writing out the full year uses 4 numbers so this takes 4 numbers * 6 fireworks per number = <<4*6=24>>24 fireworks.
“HAPPY NEW YEAR” contains 12 letters so this takes 12 letters * 5 fireworks per letter = <<12*5=60>>60 fireworks.
There is therefore a total of 24 + 60 = <<24+60=84>>84 fireworks lit at the start of the display.
From the boxes, they lit another 50 boxes * 8 fireworks per box = <<50*8=400>>400 fireworks.
So in total, the whole display uses 84 + 400 = <<84+400=484>>484 fireworks.
#### 484 |
Al gets the disease algebritis and must take one green pill and one pink pill each day for two weeks. A green pill costs $1 more than a pink pill, and Al's pills cost a total of $546 for the two weeks. How much does one green pill cost? | $20 |
In triangle $PQR$, $PQ = 4$, $PR = 8$, and $\cos \angle P = \frac{1}{10}$. Find the length of angle bisector $\overline{PS}$. | 4.057 |
The first $30$ numbers of a similar arrangement are shown below. What would be the value of the $50^{\mathrm{th}}$ number if the arrangement were continued?
$\bullet$ Row 1: $3,$ $3$
$\bullet$ Row 2: $6,$ $6,$ $6,$ $6$
$\bullet$ Row 3: $9,$ $9,$ $9,$ $9,$ $9,$ $9$
$\bullet$ Row 4: $12,$ $12,$ $12,$ $12,$ $12,$ $12,$ $12,$ $12$ | 21 |
Isabelle works in a hotel and runs a bubble bath for each customer who enters the hotel. There are 13 rooms for couples and 14 single rooms. For each bath that is run, Isabelle needs 10ml of bubble bath. If every room is filled to maximum capacity, how much bubble bath, in millilitres, does Isabelle need? | In the rooms for couples, there are 13 rooms * 2 people per room = <<13*2=26>>26 people.
So in total, there are 26 people in couples' rooms + 14 people in single rooms = <<26+14=40>>40 people in the hotel.
As each person gets a bubble bath, Isabelle will need a total of 40 baths * 10ml of bubble bath per bath = <<40*10=400>>400ml of bubble bath.
#### 400 |
Darrel has 76 quarters, 85 dimes, 20 nickels and 150 pennies. If he drops all of his money into a coin-counting machine, they will convert his change into dollars for a 10% fee. How much will he receive after the 10% fee? | He has 76 quarters so that’s 76*.25 = $<<76*.25=19.00>>19.00
He has 85 dimes so that’s 85*.10 = $<<85*.10=8.50>>8.50
He has 20 nickels so that’s 20*.05 = $<<20*.05=1.00>>1.00
He has 150 pennies so that’s 150*.01 = $<<150*.01=1.50>>1.50
All total he has 19+8.5+1+1.5 = $<<19+8.5+1+1.5=30.00>>30.00
They charge a 10% fee to count and convert his change into dollars so they will charge him .10*30 = $3.00
He had $30.00 and they charged him $3.00 in fees so he now has 30-3 = $<<30-3=27.00>>27.00
#### 27 |
Sam is stocking up on canned tuna. He got 9 cans at the supermarket, and had 5 coupons for 25 cents off a single can per coupon. He paid $20 and got $5.50 in change. How many cents does a can of tuna cost? | His total without the coupons was $20 - $5.50 = $<<20-5.5=14.50>>14.50
He had 5 coupons that each saved him 25 cents, so the full price of all the tuna is $14.50 + 5*$0.25 =$15.75.
He bought 9 cans of tuna, so each can costs $15.75 / 9 = $<<15.75/9=1.75>>1.75
As 1 dollar has 100 cents 1.75 * 100 = <<1.75*100=175>>175 cents
#### 175 |
Given a function $f(x)$ such that for any $x$, $f(x+2)=f(x+1)-f(x)$, and $f(1)=\log_3-\log_2$, $f(2)=\log_3+\log_5$, calculate the value of $f(2010)$. | -1 |
We have 10 springs, each originally $0.5 \mathrm{~m}$ long with a spring constant of $200 \mathrm{~N}/\mathrm{m}$. A mass of $2 \mathrm{~kg}$ is hung on each spring, and the springs, along with the masses, are hung in a series. What is the length of the resulting chain? (Neglect the mass of the springs.) | 10.39 |
What is the coefficient of \(x^3y^5\) in the expansion of \(\left(\frac{2}{3}x - \frac{4}{5}y\right)^8\)? | -\frac{458752}{84375} |
Given a triangle \( ACE \) with a point \( B \) on segment \( AC \) and a point \( D \) on segment \( CE \) such that \( BD \) is parallel to \( AE \). A point \( Y \) is chosen on segment \( AE \), and segment \( CY \) is drawn, intersecting \( BD \) at point \( X \). If \( CX = 5 \) and \( XY = 3 \), what is the ratio of the area of trapezoid \( ABDE \) to the area of triangle \( BCD \)? | 39/25 |
On Tuesday, Liza had $800 in her checking account. On Wednesday, she paid her rent, $450. On Thursday, she deposited her $1500 paycheck. On Friday, she paid her electricity and internet bills which were $117 and $100, respectively. Then on Saturday, she paid her phone bill, $70. How much money is left in Liza's account? | On Wednesday, Liza had $800 - $450 = $<<800-450=350>>350 left in her account.
After depositing on Thursday, she had $350 + $1500 = $<<350+1500=1850>>1850 in her account.
On Friday, she paid a total of $117 + $100 = $<<117+100=217>>217 for her electricity and internet bills.
So, $1850 - $217 = $<<1850-217=1633>>1633 was left on her account on Friday.
After paying her phone bill, Liza has $1633 - $70 = $<<1633-70=1563>>1563 left in her account.
#### 1563 |
How many positive integer divisors of $2004^{2004}$ are divisible by exactly 2004 positive integers?
| 54 |
Bob picked 450 apples for his entire family to eat. There are 33 children in his family, and each of them ate 10 apples. If every adult ate 3 apples apiece, how many adults are in his family? | The 33 children ate a total of 33 * 10 = <<33*10=330>>330 apples.
There were 450 - 330 = <<450-330=120>>120 apples left for the adults to eat.
If every adult ate 3 of the 120 apples left, then there were 120 / 3 = <<120/3=40>>40 adults in the family.
#### 40 |
James buys 3 dirt bikes for $150 each and 4 off-road vehicles for $300 each. It also cost him $25 to register each of these. How much did he pay for everything? | The dirtbikes cost 3*150=$<<3*150=450>>450
The off-road vehicles cost 300*4=$<<300*4=1200>>1200
He had to register 3+4=<<3+4=7>>7 vehicles
That means registration cost 25*7=$<<25*7=175>>175
So the total cost of everything was 450+1200+175=$<<450+1200+175=1825>>1825
#### 1825 |
Lana had 8 blank pages left in her binder, but she knew she would need more for her next class. Duane took half of the 42 pages in his binder out and gave them to her. How many pages does Lana have in her binder after adding Duane’s? | Duane gave Lana 42 / 2 = <<42/2=21>>21 pages.
After adding Duane’s, Lana has 21 + 8 = <<21+8=29>>29 pages in her binder.
#### 29 |
There is an angle $\theta$ in the range $0^\circ < \theta < 45^\circ$ which satisfies
\[\tan \theta + \tan 2 \theta + \tan 3 \theta = 0.\]Calculate $\tan \theta$ for this angle. | \frac{1}{\sqrt{2}} |
Given a geometric sequence $\{a_n\}$ with $a_1=1$, $0<q<\frac{1}{2}$, and for any positive integer $k$, $a_k - (a_{k+1}+a_{k+2})$ is still an element of the sequence, find the common ratio $q$. | \sqrt{2} - 1 |
The arithmetic mean of an odd number of consecutive odd integers is $y$. Find the sum of the smallest and largest of the integers in terms of $y$. | 2y |
Given an arithmetic sequence $\{a\_n\}$, where $a\_1+a\_2=3$, $a\_4+a\_5=5$.
(I) Find the general term formula of the sequence.
(II) Let $[x]$ denote the largest integer not greater than $x$ (e.g., $[0.6]=0$, $[1.2]=1$). Define $T\_n=[a\_1]+[a\_2]+…+[a\_n]$. Find the value of $T\_30$. | 175 |
What is $2a+3b$, where $a=2-i$ and $b=-1+i$? | 1+i |
Bob has planted corn in his garden, and it has just started to sprout. A week after planting it, it had grown 2 inches. The next week, its height increased by twice as much as it had the first week. In the third week, it grew 4 times as much as it did the week before. How tall are the corn plants now? | The second week it grew twice as much as the first week, so 2 * 2 inches = <<2*2=4>>4 inches.
The third week it grew 4 times as much as in the second week, so 4 * 4 inches = <<4*4=16>>16 inches.
In total, it grew 2 inches + 4 inches + 16 inches = <<2+4+16=22>>22 inches.
#### 22 |
Find $x,$ given that $x$ is nonzero and the numbers $\{x\},$ $\lfloor x \rfloor,$ and $x$ form an arithmetic sequence in that order. (We define $\{x\} = x - \lfloor x\rfloor.$) | \tfrac32 |
Consider a 9x9 chessboard where the squares are labelled from a starting square at the bottom left (1,1) increasing incrementally across each row to the top right (9,9). Each square at position $(i,j)$ is labelled with $\frac{1}{i+j-1}$. Nine squares are chosen such that there is exactly one chosen square in each row and each column. Find the minimum product of the labels of the nine chosen squares. | \frac{1}{362880} |
Given the function $f(x)=2 \sqrt {3}\sin \frac {ωx}{2}\cos \frac {ωx}{2}-2\sin ^{2} \frac {ωx}{2}(ω > 0)$ with a minimum positive period of $3π$.
(I) Find the interval where the function $f(x)$ is monotonically increasing.
(II) In $\triangle ABC$, $a$, $b$, and $c$ correspond to angles $A$, $B$, and $C$ respectively, with $a < b < c$, $\sqrt {3}a=2c\sin A$, and $f(\frac {3}{2}A+ \frac {π}{2})= \frac {11}{13}$. Find the value of $\cos B$. | \frac {5 \sqrt {3}+12}{26} |
A 20-quart container is fully filled with water. Five quarts are removed and replaced with pure antifreeze liquid. Then, five quarts of the mixture are removed and replaced with pure antifreeze. This process is repeated three more times (for a total of five times). Determine the fractional part of the final mixture that is water. | \frac{243}{1024} |
Nine copies of a certain pamphlet cost less than $10.00 while ten copies of the same pamphlet (at the same price) cost more than $11.00. How much does one copy of this pamphlet cost? | $1.11 |
One focus of the ellipse $\frac{x^2}{2} + y^2 = 1$ is at $F = (1,0).$ There exists a point $P = (p,0),$ where $p > 0,$ such that for any chord $\overline{AB}$ that passes through $F,$ angles $\angle APF$ and $\angle BPF$ are equal. Find $p.$
[asy]
unitsize(2 cm);
pair A, B, F, P;
path ell = xscale(sqrt(2))*Circle((0,0),1);
F = (1,0);
A = (sqrt(2)*Cos(80),Sin(80));
B = intersectionpoint(interp(A,F,0.1)--interp(A,F,5),ell);
P = (2,0);
draw(ell);
draw(A--B);
draw(A--P--B);
draw(F--P);
dot("$A$", A, N);
dot("$B$", B, SE);
dot("$F$", F, SW);
dot("$P$", P, E);
[/asy] | 2 |
Find the sum of all values of $x$ such that $2^{x^2-3x-2} = 4^{x - 4}$. | 5 |
What is the largest three-digit integer $n$ that satisfies $$55n\equiv 165\pmod{260}~?$$ | 991 |
Given the function $f(x)=2\sin x( \sqrt {3}\cos x+\sin x)-2$.
1. If point $P( \sqrt {3},-1)$ is on the terminal side of angle $α$, find the value of $f(α)$.
2. If $x∈[0, \frac {π}{2}]$, find the minimum value of $f(x)$. | -2 |
$A B C D$ is a cyclic quadrilateral in which $A B=4, B C=3, C D=2$, and $A D=5$. Diagonals $A C$ and $B D$ intersect at $X$. A circle $\omega$ passes through $A$ and is tangent to $B D$ at $X . \omega$ intersects $A B$ and $A D$ at $Y$ and $Z$ respectively. Compute $Y Z / B D$. | \frac{115}{143} |
Let $\bold{v} = \begin{pmatrix} 5 \\ -3 \end{pmatrix}$ and $\bold{w} = \begin{pmatrix} 11 \\ -2 \end{pmatrix}$. Find the area of the parallelogram with vertices $\bold{0}$, $\bold{v}$, $\bold{w}$, and $\bold{v} + \bold{w}$. | 23 |
Eliza can iron a blouse in 15 minutes and a dress in 20 minutes. If she spends 2 hours ironing blouses and 3 hours ironing dresses, how many pieces of clothes did she iron? | Since 60 minutes equal 1 hour, in 2 hours, Eliza ironed blouses for 2 x 60 = <<2*60=120>>120 minutes.
Eliza can iron 120 / 15 = <<120/15=8>>8 blouses.
Since 60 minutes equal 1 hour, in 3 hours, Eliza ironed dresses for 3 x 60 = <<3*60=180>>180 minutes.
Eliza can iron 180 / 20 = <<180/20=9>>9 dresses.
In total, Eliza ironed 8 + 9 = <<8+9=17>>17 pieces of clothes.
#### 17 |
How many positive integers less than 60 have an even number of positive divisors? | 52 |
The sequence $3, 8, 13, a, b, 33$ is arithmetic. What is the sum of values $a$ and $b$? | 41 |
What is the largest integer that must divide the product of any 5 consecutive integers? | 60 |
Let $\mathbf{D}$ be a matrix representing a dilation with scale factor $k > 0,$ and let $\mathbf{R}$ be a matrix representing a rotation about the origin by an angle of $\theta$ counter-clockwise. If
\[\mathbf{R} \mathbf{D} = \begin{pmatrix} 8 & -4 \\ 4 & 8 \end{pmatrix},\]then find $\tan \theta.$ | \frac{1}{2} |
As a prank, Tim decides to steal Nathan's fork at dinner, but so he doesn't get caught, he convinces other people to do it for him. On Monday, he convinces Joe to do it. On Tuesday, he could get either Ambie or John to do it. On Wednesday, he can't convince any of those three people to do it, but there are five other people he could convince to do it. On Thursday, none of those five will do it, nor the first three, but there are four others who are willing. Finally, on Friday, Tim does it himself. How many different combinations of people could Tim involve in the prank? | 40 |
A disk with radius $1$ is externally tangent to a disk with radius $5$. Let $A$ be the point where the disks are tangent, $C$ be the center of the smaller disk, and $E$ be the center of the larger disk. While the larger disk remains fixed, the smaller disk is allowed to roll along the outside of the larger disk until the smaller disk has turned through an angle of $360^\circ$. That is, if the center of the smaller disk has moved to the point $D$, and the point on the smaller disk that began at $A$ has now moved to point $B$, then $\overline{AC}$ is parallel to $\overline{BD}$. Then $\sin^2(\angle BEA)=\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. | 58 |
What is the 100th digit after the decimal point of the decimal representation of 1/7? | 8 |
A list of $3042$ positive integers has a unique mode, which occurs exactly $15$ times. Calculate the least number of distinct values that can occur in the list. | 218 |
Two positive integers \(m\) and \(n\) are chosen such that \(m\) is the smallest positive integer with only two positive divisors, and \(n\) is the largest integer less than 200 that has exactly four positive divisors. What is \(m+n\)? | 192 |
Let $a/b$ be the probability that a randomly chosen positive divisor of $12^{2007}$ is also a divisor of $12^{2000}$ , where $a$ and $b$ are relatively prime positive integers. Find the remainder when $a+b$ is divided by $2007$ . | 79 |
Dennis wants to buy 4 pairs of pants from the store which cost $110.00 each with a 30% discount and he also wants to buy 2 pairs of socks which cost $60.00 with a 30% discount. How much money will Dennis have to spend in total after he bought all the items he wants after the discount? | Dennis wants to buy 4 pairs of pants which cost 110x4= $<<110*4=440>>440.
And he also wants to buy 2 pairs of socks that cost 60x2= $<<60*2=120>>120.
The total amount he needs to spend without the discount is 440+120= $<<440+120=560>>560.
The discount he will have amounts to $560x.0.30= $168.00.
So the total amount he needs to spend after deducting the discount from the total amount is 560-168=$<<560-168=392>>392.
#### 392 |
Call a positive integer $n$ $k$-pretty if $n$ has exactly $k$ positive divisors and $n$ is divisible by $k$. For example, $18$ is $6$-pretty. Let $S$ be the sum of positive integers less than $2019$ that are $20$-pretty. Find $\tfrac{S}{20}$.
| 472 |
Three circles, each of radius $3$, are drawn with centers at $(14, 92)$, $(17, 76)$, and $(19, 84)$. A line passing through $(17,76)$ is such that the total area of the parts of the three circles to one side of the line is equal to the total area of the parts of the three circles to the other side of it. What is the absolute value of the slope of this line?
| 24 |
Julian has 400 legos and wants to make lego models of two identical airplanes. If each airplane model requires 240 legos, how many more legos does Julian need? | Two aeroplanes require 240 legos/airplane x 2 airplanes = <<240*2=480>>480 legos.
Julian needs 480 legos - 400 legos = <<480-400=80>>80 more legos
#### 80 |
In triangle $XYZ$, $XY = 15$, $XZ = 35$, $YZ = 42$, and $XD$ is an angle bisector of $\angle XYZ$. Find the ratio of the area of triangle $XYD$ to the area of triangle $XZD$, and find the lengths of segments $XD$ and $ZD$. | 29.4 |
A pyramid is constructed on a $5 \times 12$ rectangular base. Each of the four edges joining the apex to the corners of the rectangular base has a length of $15$. Determine the volume of this pyramid. | 270 |
What is the degree of the polynomial $(3x^2 +11)^{12}$? | 24 |
Shaniqua styles hair. For every haircut she makes $12 and for every style she makes $25. How many dollars would Shaniqua make if she gave 8 haircuts and 5 styles? | Haircuts = 8 * 12 = <<8*12=96>>96
Styles = 5 * 25 = <<5*25=125>>125
Total earned = 96 + 125 = $<<96+125=221>>221
#### 221 |
Bill buys a stock that decreases by $20\%$ on the first day, and then on the second day the stock increases by $30\%$ of its value at the end of the first day. What was the overall percent increase in Bill's stock over the two days? | 4 |
Jason rolls four fair standard six-sided dice. He looks at the rolls and decides to either reroll all four dice or keep two and reroll the other two. After rerolling, he wins if and only if the sum of the numbers face up on the four dice is exactly $9.$ Jason always plays to optimize his chances of winning. What is the probability that he chooses to reroll exactly two of the dice?
**A)** $\frac{7}{36}$
**B)** $\frac{1}{18}$
**C)** $\frac{2}{9}$
**D)** $\frac{1}{12}$
**E)** $\frac{1}{4}$ | \frac{1}{18} |
A mathematical organization is producing a set of commemorative license plates. Each plate contains a sequence of five characters chosen from the four letters in AIME and the four digits in 2007. No character may appear in a sequence more times than it appears among the four letters in AIME or the four digits in 2007. A set of plates in which each possible sequence appears exactly once contains N license plates. Find $\frac{N}{10}$. | 372 |
a) Find all the divisors of the number 572 based on its prime factorization.
b) How many divisors does the number \(572 a^{3} b c\) have if:
I. \(a, b\), and \(c\) are prime numbers greater than 20 and different from each other?
II. \(a = 31\), \(b = 32\), and \(c = 33\)? | 384 |
Before the lesson, Nestor Petrovich wrote several words on the board. When the bell rang for the lesson, he noticed a mistake in the first word. If he corrects the mistake in the first word, the words with mistakes will constitute $24\%$, and if he erases the first word from the board, the words with mistakes will constitute $25\%$. What percentage of the total number of written words were words with mistakes before the bell rang for the lesson? | 28 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.