problem stringlengths 10 5.15k | answer stringlengths 0 1.23k |
|---|---|
Find the area of quadrilateral \(ABCD\) if \(AB = BC = 3\sqrt{3}\), \(AD = DC = \sqrt{13}\), and vertex \(D\) lies on a circle of radius 2 inscribed in the angle \(ABC\), where \(\angle ABC = 60^\circ\). | 3\sqrt{3} |
What is the sum of all the integers between -12.1 and 3.3? | -72 |
Tom decides to make lasagna with all his beef. It takes twice as many noodles as beef. He has 10 pounds of beef. He already has 4 pounds of lasagna noodles and the noodles come in 2-pound packages. How many packages does he need to buy? | He needs 10*2=<<10*2=20>>20 pounds of noodles
That means he needs to buy 20-4=<<20-4=16>>16 pounds of noodles
So he needs to buy 16/2=<<16/2=8>>8 packages
#### 8 |
What is the smallest positive integer with exactly 12 positive integer divisors? | 60 |
Suppose that $a$ is a multiple of 4 and $b$ is a multiple of 8. Which of the following statements are true?
A. $a+b$ must be even.
B. $a+b$ must be a multiple of 4.
C. $a+b$ must be a multiple of 8.
D. $a+b$ cannot be a multiple of 8.
Answer by listing your choices in alphabetical order, separated by commas. For example, if you think all four are true, then answer $\text{A,B,C,D}$ | \text{A,B} |
In triangle $ABC$, $\angle C = 90^\circ$, $AC = 4$, and $AB = \sqrt{41}$. What is $\tan B$? | \frac{4}{5} |
An iterative average of the numbers 2, 3, 4, 6, and 7 is computed by arranging the numbers in some order. Find the difference between the largest and smallest possible values that can be obtained using this procedure. | \frac{11}{4} |
Find the number of solutions to the equation $x+y+z=525$ where $x$ is a multiple of 7, $y$ is a multiple of 5, and $z$ is a multiple of 3. | 21 |
On a $300 \times 300$ board, several rooks are placed that beat the entire board. Within this case, each rook beats no more than one other rook. At what least $k$ , it is possible to state that there is at least one rook in each $k\times k$ square ? | 201 |
Given the ellipse $G$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ with an eccentricity of $\frac{\sqrt{6}}{3}$, and its right focus at $(2\sqrt{2}, 0)$. A line $l$ with a slope of $1$ intersects the ellipse $G$ at points $A$ and $B$. An isosceles triangle is formed with $AB$ as the base and $P$ $(-3, 2)$ as the apex.
(1) Find the equation of the ellipse $G$;
(2) Calculate the area of $\triangle PAB$. | \frac{9}{2} |
Pete has to take a 10-minute walk down to the train station and then board a 1hr 20-minute train to LA. When should he leave if he cannot get to LA later than 0900 hours? (24-hr time) | There are 60 minutes in an hour so 1 hour 20 minutes = (60+20) minutes = 80 minutes
He will spend a total of 80+10 = <<80+10=90>>90 minutes
90 minutes is = (60/60) hours and 30 minutes = 1 hour 30 minutes = 0130 in 24-hr time
He has to leave 0130 hours earlier than 0900 i.e. 0900-0130 = 0730 hours
#### 730 |
Find the smallest positive integer $b$ for which $x^2 + bx + 2008$ factors into a product of two polynomials, each having integer coefficients. | 259 |
There are 3 kids waiting for the swings and twice as many kids waiting for the slide. If each kid waits 2 minutes for the swings and 15 seconds for the slide, how many seconds shorter is the shorter wait? | First find the total wait time for the swings in minutes: 3 kids * 2 minutes/kid = <<3*2=6>>6 minutes
Then convert that number to seconds: 6 minutes * 60 seconds/minute = <<6*60=360>>360 seconds
Then find the total number of kids waiting for the slide: 3 kids * 2 = <<3*2=6>>6 kids
Then find the total wait time for the slide: 15 seconds/kid * 6 kids = 90 seconds
Then subtract the total wait for the slide from the total wait for the swings to find the difference: 360 seconds - 90 seconds = <<360-90=270>>270 seconds
#### 270 |
Find the numerical value of
\[\frac{\sin 18^\circ \cos 12^\circ + \cos 162^\circ \cos 102^\circ}{\sin 22^\circ \cos 8^\circ + \cos 158^\circ \cos 98^\circ}.\] | 1 |
Consider the set $\{8, -7, 2, -4, 20\}$. Find the smallest sum that can be achieved by adding three different numbers from this set. | -9 |
What is the greatest common divisor of $654321$ and $543210$? | 3 |
What is the sum of all positive integer divisors of 77? | 96 |
Given an ellipse $$C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$$ with eccentricity $$\frac{\sqrt{3}}{2}$$, and the distance from its left vertex to the line $x + 2y - 2 = 0$ is $$\frac{4\sqrt{5}}{5}$$.
(Ⅰ) Find the equation of ellipse C;
(Ⅱ) Suppose line $l$ intersects ellipse C at points A and B. If the circle with diameter AB passes through the origin O, investigate whether the distance from point O to line AB is a constant. If so, find this constant; otherwise, explain why;
(Ⅲ) Under the condition of (Ⅱ), try to find the minimum value of the area $S$ of triangle $\triangle AOB$. | \frac{4}{5} |
A class has a total of 54 students. Now, using the systematic sampling method based on the students' ID numbers, a sample of 4 students is drawn. It is known that students with ID numbers 3, 29, and 42 are in the sample. What is the ID number of the fourth student in the sample? | 16 |
If \(a\), \(b\), and \(c\) are positive numbers with \(ab = 24\sqrt[3]{3}\), \(ac = 40\sqrt[3]{3}\), and \(bc = 15\sqrt[3]{3}\), find the value of \(abc\). | 120\sqrt{3} |
Evaluate $(\sqrt[6]{4})^9$. | 8 |
A population consists of $20$ individuals numbered $01$, $02$, $\ldots$, $19$, $20$. Using the following random number table, select $5$ individuals. The selection method is to start from the numbers in the first row and first two columns of the random number table, and select two numbers from left to right each time. If the two selected numbers are not within the population, remove them and continue selecting two numbers to the right. Then, the number of the $4$th individual selected is ______.<br/><table><tbody><tr><td width="84" align="center">$7816$</td><td width="84" align="center">$6572$</td><td width="84" align="center">$0802$</td><td width="84" align="center">$6314$</td><td width="84" align="center">$0702$</td><td width="84" align="center">$4369$</td><td width="84" align="center">$9728$</td><td width="84" align="center">$0198$</td></tr><tr><td align="center">$3204$</td><td align="center">$9234$</td><td align="center">$4935$</td><td align="center">$8200$</td><td align="center">$3623$</td><td align="center">$4869$</td><td align="center">$6938$</td><td align="center">$7481$</td></tr></tbody></table> | 14 |
If $\frac{m}{n}=\frac{4}{3}$ and $\frac{r}{t}=\frac{9}{14}$, the value of $\frac{3mr-nt}{4nt-7mr}$ is: | -\frac{11}{14} |
Let $L_1$ and $L_2$ be perpendicular lines, and let $F$ be a point at a distance $18$ from line $L_1$ and a distance $25$ from line $L_2$ . There are two distinct points, $P$ and $Q$ , that are each equidistant from $F$ , from line $L_1$ , and from line $L_2$ . Find the area of $\triangle{FPQ}$ . | 210 |
If $a$, $b$, and $c$ are positive integers satisfying $ab+c = bc+a = ac+b = 41$, what is the value of $a+b+c$? | 42 |
What is the probability that the same number will be facing up on each of three standard six-sided dice that are tossed simultaneously? Express your answer as a common fraction. | \frac{1}{36} |
$\triangle ABC$ has side lengths $AB=20$ , $BC=15$ , and $CA=7$ . Let the altitudes of $\triangle ABC$ be $AD$ , $BE$ , and $CF$ . What is the distance between the orthocenter (intersection of the altitudes) of $\triangle ABC$ and the incenter of $\triangle DEF$ ? | 15 |
Find the number of solutions to
\[\sin x = \left( \frac{1}{2} \right)^x\]on the interval $(0,100 \pi).$ | 100 |
At the duck park, there are 25 mallard ducks and ten less than twice as many geese as ducks. Then, a small flock of 4 ducks arrived at the park and joined the birds already there. If five less than 15 geese leave the park, how many more geese than ducks remain at the park? | Initially, twice as many geese as ducks is 25*2=<<25*2=50>>50 geese.
Therefore, ten less than twice as many geese as ducks is 50-10=<<50-10=40>>40 geese.
Then, 4 ducks arrived at the park, resulting in 25+4=<<4+25=29>>29 ducks.
Five less than 15 geese is 15-5=<<15-5=10>>10 geese.
Thus, if 5 less than 15 geese leave, then 40-10=30 geese remain.
Thus, there are 30-29=<<30-29=1>>1 more geese than ducks remaining at the park.
#### 1 |
Given the hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 (a > 0, b > 0)$ with its right focus $F$, draw a line perpendicular to the $x$-axis passing through $F$, intersecting the two asymptotes at points $A$ and $B$, and intersecting the hyperbola at point $P$ in the first quadrant. Denote $O$ as the origin. If $\overrightarrow{OP} = λ\overrightarrow{OA} + μ\overrightarrow{OB} (λ, μ \in \mathbb{R})$, and $λ^2 + μ^2 = \frac{5}{8}$, find the eccentricity of the hyperbola. | \frac{2\sqrt{3}}{3} |
In triangle $ABC,$ point $D$ is on $\overline{BC}$ with $CD = 2$ and $DB = 5,$ point $E$ is on $\overline{AC}$ with $CE = 1$ and $EA = 3,$ $AB = 8,$ and $\overline{AD}$ and $\overline{BE}$ intersect at $P.$ Points $Q$ and $R$ lie on $\overline{AB}$ so that $\overline{PQ}$ is parallel to $\overline{CA}$ and $\overline{PR}$ is parallel to $\overline{CB}.$ It is given that the ratio of the area of triangle $PQR$ to the area of triangle $ABC$ is $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$. | 901 |
Find the smallest positive real number \( r \) with the following property: For every choice of 2023 unit vectors \( \mathbf{v}_1, \mathbf{v}_2, \ldots, \mathbf{v}_{2023} \in \mathbb{R}^2 \), a point \( \mathbf{p} \) can be found in the plane such that for each subset \( S \) of \(\{1, 2, \ldots, 2023\}\), the sum
\[ \sum_{i \in S} \mathbf{v}_i \]
lies inside the disc \(\left\{ \mathbf{x} \in \mathbb{R}^2 : \| \mathbf{x} - \mathbf{p} \| \leq r \right\}\). | 2023/2 |
Evaluate \[(3a^3 - 7a^2 + a - 5)(4a - 6)\] for $a = 2$. | -14 |
Reading from left to right, a sequence consists of 6 X's, followed by 24 Y's, followed by 96 X's. After the first \(n\) letters, reading from left to right, one letter has occurred twice as many times as the other letter. What is the sum of the four possible values of \(n\)? | 135 |
The operation $\odot$ is defined as $a \odot b = a + \frac{3a}{2b}$. What is the value of $8 \odot 6$? | 10 |
The perimeter of triangle $APM$ is $180$, and angle $PAM$ is a right angle. A circle of radius $20$ with center $O$ on line $\overline{AP}$ is drawn such that it is tangent to $\overline{AM}$ and $\overline{PM}$. Let $\overline{AM} = 2\overline{PM}$. Find the length of $OP$. | 20 |
Tim rides his bike back and forth to work for each of his 5 workdays. His work is 20 miles away. He also goes for a weekend bike ride of 200 miles. If he can bike at 25 mph how much time does he spend biking a week? | He bikes 20*2=<<20*2=40>>40 miles each day for work
So he bikes 40*5=<<40*5=200>>200 miles for work
That means he bikes a total of 200+200=<<200+200=400>>400 miles for work
So he bikes a total of 400/25=<<400/25=16>>16 hours
#### 16 |
For how many values of $a$ is it true that:
(1) $a$ is a positive integer such that $a \le 50$.
(2) the quadratic equation $x^2 + (2a+1)x + a^2 = 0$ has two integer solutions? | 6 |
What is the sum of the digits of the base $7$ representation of $2019_{10}$? | 15 |
Let $n \in \mathbb{N}^*$, $a_n$ be the sum of the coefficients of the expanded form of $(x+4)^n - (x+1)^n$, $c=\frac{3}{4}t-2$, $t \in \mathbb{R}$, and $b_n = \left[\frac{a_1}{5}\right] + \left[\frac{2a_2}{5^2}\right] + ... + \left[\frac{na_n}{5^n}\right]$ (where $[x]$ represents the largest integer not greater than the real number $x$). Find the minimum value of $(n-t)^2 + (b_n + c)^2$. | \frac{4}{25} |
Find a quadratic with rational coefficients and quadratic term $x^2$ that has $\sqrt{3}-2$ as a root. | x^2+4x+1 |
At a conference, there are only single women and married men with their wives. The probability that a randomly selected woman is single is $\frac{3}{7}$. What fraction of the people in the conference are married men? | \frac{4}{11} |
Ten boys brought 15 popsicle sticks each. Twelve girls brought 12 popsicle sticks each. How many fewer popsicle sticks did the girls bring than the boys? | The boys brought a total of 10 x 15 = <<10*15=150>>150 popsicle sticks.
The girls brought a total of 12 x 12 = <<12*12=144>>144 popsicle sticks.
Therefore, the girls brought 150 - 144 = <<150-144=6>>6 fewer popsicle sticks than the boys.
#### 6 |
What code will be produced for this message in the new encoding where the letter А is replaced by 21, the letter Б by 122, and the letter В by 1? | 211221121 |
$ABCDEFGH$ shown below is a cube with volume 1. Find the volume of pyramid $ABCH$.
[asy]
import three;
triple A,B,C,D,EE,F,G,H;
A = (0,0,0);
B = (1,0,0);
C = (1,1,0);
D= (0,1,0);
EE = (0,0,1);
F = B+EE;
G = C + EE;
H = D + EE;
draw(B--C--D);
draw(B--A--D,dashed);
draw(EE--F--G--H--EE);
draw(A--EE,dashed);
draw(B--F);
draw(C--G);
draw(D--H);
label("$A$",A,S);
label("$B$",B,W);
label("$C$",C,S);
label("$D$",D,E);
label("$E$",EE,N);
label("$F$",F,W);
label("$G$",G,SW);
label("$H$",H,E);
[/asy] | \frac16 |
Given the function $f(x)=\sin^2x+\sin x\cos x$, when $x=\theta$ the function $f(x)$ attains its minimum value, find the value of $\dfrac{\sin 2\theta+2\cos \theta}{\sin 2\theta -2\cos 2\theta}$. | -\dfrac{1}{3} |
One-half of a pound of mangoes costs $0.60. How many pounds can Kelly buy with $12? | One pound of mangoes cost $0.60 x 2 = $<<0.60*2=1.20>>1.20.
So Kelly can buy $12/$1.20 = <<12/1.20=10>>10 pounds of mangoes.
#### 10 |
Using the numbers 0, 1, 2, 3, 4, 5 to form unique three-digit numbers, determine the total number of even numbers that can be formed. | 52 |
Phillip's mother asked him to go to the supermarket to buy some things and gave him $95, so he spent $14 on oranges, $25 on apples and $6 on candy. How much money does he have left? | If we add everything Phillip bought, we will have: $14 + $25 + $6 = $<<14+25+6=45>>45
He spent $45, and we know that he had $95 dollars, so now we have to subtract: $95 - $45 = $<<95-45=50>>50
#### 50 |
One hundred concentric circles with radii $1, 2, 3, \dots, 100$ are drawn in a plane. The interior of the circle of radius $1$ is colored red, and each region bounded by consecutive circles is colored either red or green, with no two adjacent regions the same color. The ratio of the total area of the green regions to the area of the circle of radius $100$ can be expressed as $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$ | 301 |
In a magical swamp there are two species of talking amphibians: toads, whose statements are always true, and frogs, whose statements are always false. Four amphibians, Brian, Chris, LeRoy, and Mike live together in this swamp, and they make the following statements.
Brian: "Mike and I are different species."
Chris: "LeRoy is a frog."
LeRoy: "Chris is a frog."
Mike: "Of the four of us, at least two are toads."
How many of these amphibians are frogs? | 3 |
Given the four equations:
$\textbf{(1)}\ 3y-2x=12 \qquad\textbf{(2)}\ -2x-3y=10 \qquad\textbf{(3)}\ 3y+2x=12 \qquad\textbf{(4)}\ 2y+3x=10$
The pair representing the perpendicular lines is: | \text{(1) and (4)} |
Caroline has 40 pairs of socks. She loses 4 pairs of socks at the laundromat. Of the remaining pairs of socks, she donates two-thirds to the thrift store. Then she purchases 10 new pairs of socks, and she receives 3 new pairs of socks as a gift from her dad. How many pairs of socks does Caroline have in total? | After losing some socks, Caroline has 40 - 4 = <<40-4=36>>36 pairs of socks
Caroline donates 36 * (2/3) = <<36*(2/3)=24>>24 pairs of socks to the thrift store
After donating some socks, Caroline has 36 - 24 = <<36-24=12>>12 pairs of socks remaining
After getting new socks, Caroline has 12 + 10 + 3 = <<12+10+3=25>>25 pairs of socks
#### 25 |
A triangle has area $30$, one side of length $10$, and the median to that side of length $9$. Let $\theta$ be the acute angle formed by that side and the median. What is $\sin{\theta}$? | \frac{2}{3} |
In acute triangle $ABC$, $\sin A=\frac{2\sqrt{2}}{3}$. Find the value of $\sin^{2}\frac{B+C}{2}+\cos (3\pi -2A)$. | \frac{13}{9} |
Two cards are dealt at random from a standard deck of 52 cards (13 hearts, 13 clubs, 13 spades, and 13 diamonds). What is the probability that the first card is a 6 and the second card is a Queen? | \dfrac{4}{663} |
Exactly at noon, Anna Kuzminichna looked out the window and saw Klava, the rural store salesperson, going on break. At two minutes past noon, Anna Kuzminichna looked out the window again, and there was still no one in front of the closed store. Klava was gone for exactly 10 minutes, and when she returned, she found Ivan and Foma in front of the store, with Foma evidently arriving after Ivan. Find the probability that Foma had to wait for the store to open for no more than 4 minutes. | 0.75 |
Ali is collecting bottle caps. He has 125 bottle caps. He has red ones and green ones. If he has 50 red caps, what percentage of caps are green? | He has 75 green caps because 125 - 50 = <<125-50=75>>75
The proportion of caps that are green is .6 because 75 / 125 = <<75/125=.6>>.6
The percentage that are green is 60 because .6 x 100% = <<60=60>>60%
#### 60 |
There are enough provisions in a castle to feed 300 people for 90 days. After 30 days, 100 people leave the castle. How many more days are left until all the food runs out? | After 30 days, there will be enough food left to sustain 300 people for 90 days – 30 days = 60 days.
After the 100 people leave, there will be 300-100 = <<300-100=200>>200 people left.
The 200 people will eat 200/300 = 2/3 as much food as the original group of people in the castle.
The 60 days' worth of food will last this smaller group for 60 days / (2/3) = <<60/(2/3)=90>>90 more days.
#### 90 |
For some constants $x$ and $a$, the third, fourth, and fifth terms in the expansion of $(x + a)^n$ are 84, 280, and 560, respectively. Find $n.$ | 7 |
Given the circle \(\Gamma: x^{2} + y^{2} = 1\) with two intersection points with the \(x\)-axis as \(A\) and \(B\) (from left to right), \(P\) is a moving point on circle \(\Gamma\). Line \(l\) passes through point \(P\) and is tangent to circle \(\Gamma\). A line perpendicular to \(l\) passes through point \(A\) and intersects line \(BP\) at point \(M\). Find the maximum distance from point \(M\) to the line \(x + 2y - 9 = 0\). | 2\sqrt{5} + 2 |
In how many ways can Michael choose 3 out of 8 math classes to take? | 56 |
$\triangle DEF$ is inscribed inside $\triangle ABC$ such that $D,E,F$ lie on $BC, AC, AB$, respectively. The circumcircles of $\triangle DEC, \triangle BFD, \triangle AFE$ have centers $O_1,O_2,O_3$, respectively. Also, $AB = 23, BC = 25, AC=24$, and $\stackrel{\frown}{BF} = \stackrel{\frown}{EC},\ \stackrel{\frown}{AF} = \stackrel{\frown}{CD},\ \stackrel{\frown}{AE} = \stackrel{\frown}{BD}$. The length of $BD$ can be written in the form $\frac mn$, where $m$ and $n$ are relatively prime integers. Find $m+n$.
| 14 |
Sangho uploaded a video to a website where viewers can vote that they like or dislike a video. Each video begins with a score of $0$, and the score increases by $1$ for each like vote and decreases by $1$ for each dislike vote. At one point Sangho saw that his video had a score of $90$, and that $65\%$ of the votes cast on his video were like votes. How many votes had been cast on Sangho's video at that point? | 300 |
For how many integers $n$ between 1 and 20 (inclusive) is $\frac{n}{42}$ a repeating decimal? | 20 |
Find the limit of the function:
\[
\lim _{x \rightarrow 1}\left(\frac{x+1}{2 x}\right)^{\frac{\ln (x+2)}{\ln (2-x)}}
\] | \sqrt{3} |
Calculate the sum of all integers greater than 4 and less than 21. | 200 |
Sally and Bob have made plans to go on a trip at the end of the year. They both decide to work as babysitters and save half of what they've earned for their trip. If Sally makes $6 per day and Bob makes $4 per day, how much money will they both have saved for their trip after a year? | Saly saves 1/2 * $6/day = $<<1/2*6=3>>3/day.
Since each year have 365 days, the total amount of money Sally will save in a year is $3/day * 365 days/year = $<<3*365=1095>>1095/year
Bob saves 1/2 * $4/day = $<<1/2*4=2>>2/day.
The total amount of money Bob will have saved in a year is $2/day * 365 days/year = $<<2*365=730>>730/year
In total, Sally and Bob would have saved $730 + $1095 = $<<730+1095=1825>>1825
#### 1825 |
Reginald is selling apples. He sells each apple for $1.25. He plans to use his profits to help repair his bike. His bike cost $80 and the repairs cost 25% of what his bike cost. After the repairs are paid for, 1/5 of the money he'd earned remains. How many apples did he sell? | The bike repairs cost $20 because 80 x .25 = <<80*.25=20>>20
Call the amount he earns e. We know that e - 20 = 1/5e
Therefore 5e - 100 = e
Therefore, 4e = 100
Therefore, e = <<25=25>>25
Therefore, he sold 20 apples because 25 / 1.25 = <<25/1.25=20>>20
#### 20 |
What is the sum of all numbers in the following square matrix?
$$
\begin{array}{l}
1, 2, 3, \ldots, 98, 99, 100 \\
2, 3, 4, \ldots, 99, 100, 101 \\
3, 4, 5, \ldots, 100, 101, 102 \\
\ldots \\
100, 101, 102, \ldots, 197, 198, 199
\end{array}
$$ | 1000000 |
A chord of a circle is perpendicular to a radius at the midpoint of the radius. The ratio of the area of the larger of the two regions into which the chord divides the circle to the smaller can be expressed in the form $\frac{a\pi+b\sqrt{c}}{d\pi-e\sqrt{f}},$ where $a, b, c, d, e,$ and $f$ are positive integers, $a$ and $e$ are relatively prime, and neither $c$ nor $f$ is divisible by the square of any prime. Find the remainder when the product $abcdef$ is divided by 1000. | 592 |
Draw a perpendicular line from the left focus $F_1$ of the ellipse $\frac {x^{2}}{a^{2}}+ \frac {y^{2}}{b^{2}}=1 (a > b > 0)$ to the $x$-axis meeting the ellipse at point $P$, and let $F_2$ be the right focus. If $\angle F_{1}PF_{2}=60^{\circ}$, calculate the eccentricity of the ellipse. | \frac{\sqrt{3}}{3} |
Determine the exact value of the series
\[\frac{1}{5 + 1} + \frac{2}{5^2 + 1} + \frac{4}{5^4 + 1} + \frac{8}{5^8 + 1} + \frac{16}{5^{16} + 1} + \dotsb.\] | \frac{1}{4} |
What is the least positive integer value of $x$ such that $(2x)^2 + 2\cdot 37\cdot 2x + 37^2$ is a multiple of 47? | 5 |
Let the roots of the polynomial $f(x) = x^6 + 2x^3 + 1$ be denoted as $y_1, y_2, y_3, y_4, y_5, y_6$. Let $h(x) = x^3 - 3x$. Find the product $\prod_{i=1}^6 h(y_i)$. | 676 |
In a circle, parallel chords of lengths 2, 3, and 4 determine central angles of $\alpha$, $\beta$, and $\alpha + \beta$ radians, respectively, where $\alpha + \beta < \pi$. If $\cos \alpha$, which is a positive rational number, is expressed as a fraction in lowest terms, what is the sum of its numerator and denominator? | 49 |
Determine the number of times and the positions in which it appears $\frac12$ in the following sequence of fractions: $$ \frac11, \frac21, \frac12 , \frac31 , \frac22 , \frac13 , \frac41,\frac32,\frac23,\frac14,..., \frac{1}{1992} $$ | 664 |
An ellipse $\frac {x^{2}}{a^{2}}+ \frac {y^{2}}{b^{2}}=1 (a>b>0)$ has its two foci and the endpoints of its minor axis all lying on the circle $x^{2}+y^{2}=1$. A line $l$ (not perpendicular to the x-axis) passing through the right focus intersects the ellipse at points A and B. The perpendicular bisector of segment AB intersects the x-axis at point P.
(1) Find the equation of the ellipse;
(2) Investigate whether the ratio $\frac {|AB|}{|PF|}$ is a constant value. If it is, find this constant value. If not, explain why. | 2 \sqrt {2} |
Walnuts and hazelnuts were delivered to a store in $1 \mathrm{~kg}$ packages. The delivery note only mentioned that the shipment's value is $1978 \mathrm{Ft}$, and its weight is $55 \mathrm{~kg}$. The deliverers remembered the following:
- Walnuts are more expensive;
- The prices per kilogram are two-digit numbers, and one can be obtained by swapping the digits of the other;
- The price of walnuts consists of consecutive digits.
How much does $1 \mathrm{~kg}$ of walnuts cost? | 43 |
Find the mass of the plate $D$ with surface density $\mu = \frac{x^2}{x^2 + y^2}$, bounded by the curves
$$
y^2 - 4y + x^2 = 0, \quad y^2 - 8y + x^2 = 0, \quad y = \frac{x}{\sqrt{3}}, \quad x = 0.
$$ | \pi + \frac{3\sqrt{3}}{8} |
In the diagram, $O$ is the center of a circle with radii $OP=OQ=5$. What is the perimeter of the shaded region?
[asy]
size(100);
import graph;
label("$P$",(-1,0),W); label("$O$",(0,0),NE); label("$Q$",(0,-1),S);
fill(Arc((0,0),1,-90,180)--cycle,mediumgray);
draw(Arc((0,0),1,-90,180));
fill((0,0)--(-1,0)--(0,-1)--cycle,white);
draw((-1,0)--(0,0)--(0,-1));
draw((-.1,0)--(-.1,-.1)--(0,-.1));
[/asy] | 10 + \frac{15}{2}\pi |
Given that $\sin A+\sin B=1$ and $\cos A+\cos B=3 / 2$, what is the value of $\cos (A-B)$? | 5/8 |
Let $A, B, C$ be points in that order along a line, such that $A B=20$ and $B C=18$. Let $\omega$ be a circle of nonzero radius centered at $B$, and let $\ell_{1}$ and $\ell_{2}$ be tangents to $\omega$ through $A$ and $C$, respectively. Let $K$ be the intersection of $\ell_{1}$ and $\ell_{2}$. Let $X$ lie on segment $\overline{K A}$ and $Y$ lie on segment $\overline{K C}$ such that $X Y \| B C$ and $X Y$ is tangent to $\omega$. What is the largest possible integer length for $X Y$? | 35 |
Find the sum of all positive real solutions $x$ to the equation \[2\cos2x \left(\cos2x - \cos{\left( \frac{2014\pi^2}{x} \right) } \right) = \cos4x - 1,\]where $x$ is measured in radians. | 1080 \pi |
Let a, b be positive integers such that $5 \nmid a, b$ and $5^5 \mid a^5+b^5$ . What is the minimum possible value of $a + b$ ? | 25 |
Find the sum of the roots of the equation \[(2x^3 + x^2 - 8x + 20)(5x^3 - 25x^2 + 19) = 0.\] | \tfrac{9}{2} |
A palindrome is an integer that reads the same forward and backward, such as 1221. What percent of the palindromes between 1000 and 2000 contain at least one digit 7? | 10\% |
Evaluate $|7-24i|$. | 25 |
Given that $x$ and $y$ are positive integers, and $x^2 - y^2 = 53$, find the value of $x^3 - y^3 - 2(x + y) + 10$. | 2011 |
The number of games won by five cricket teams is displayed in a chart, but the team names are missing. Use the clues below to determine how many games the Hawks won:
1. The Hawks won fewer games than the Falcons.
2. The Raiders won more games than the Wolves, but fewer games than the Falcons.
3. The Wolves won more than 15 games.
The wins for the teams are 18, 20, 23, 28, and 32 games. | 20 |
Susan wants to throw a party for her mom. She is planning on having 30 guests. For dinner she is making a recipe that makes 2 servings each. Each batch of the recipe calls for 4 potatoes and 1 teaspoon of salt. A potato costs $.10 and a container of salt costs $2 at the supermarket. If each container of salt has 5 teaspoons, how much money will Susan spend on food? | Susan needs to make 30/2=<<30/2=15>>15 batches of the recipe.
Therefore, she will need 15*4=<<15*4=60>>60 potatoes.
She will also need 1*15=<<1*15=15>>15 teaspoons of salt.
Thus she will have to buy 15/5 =<<15/5=3>>3 containers of salt.
Therefore, she will spend 60*$.1=$6 on potatoes and 3*$2=$6 on salt.
Thus, she will spend $6+$6=$<<6+6=12>>12 on food.
#### 12 |
In triangle $ABC,$ $AB = 3,$ $AC = 6,$ and $\cos \angle A = \frac{1}{8}.$ Find the length of angle bisector $\overline{AD}.$ | 3 |
A $\textit{palindrome}$ is a positive integer which reads the same forward and backward, like $12321$ or $4884$.
How many $4$-digit palindromes are there? | 90 |
When Julia divides her apples into groups of nine, ten, or eleven, she has two apples left over. Assuming Julia has more than two apples, what is the smallest possible number of apples in Julia's collection? | 200 |
The total in-store price for a laptop is $299.99. A radio advertisement offers the same laptop for five easy payments of $55.98 and a one-time shipping and handling charge of $12.99. Calculate the amount of money saved by purchasing the laptop from the radio advertiser. | 710 |
A bag contains 5 red, 6 green, 7 yellow, and 8 blue jelly beans. A jelly bean is selected at random. What is the probability that it is blue? | \frac{4}{13} |
In the Cartesian coordinate system $xOy$, the sum of distances from point $P$ to the two points $(0, -\sqrt{3})$ and $(0, \sqrt{3})$ equals $4$. Let the trajectory of point $P$ be $C$.
$(1)$ Write the equation of $C$;
$(2)$ Suppose the line $y=kx+1$ intersects $C$ at points $A$ and $B$. For what value of $k$ is $\overrightarrow{OA} \bot \overrightarrow{OB}$? What is the value of $|\overrightarrow{AB}|$ at this time? | \frac{4\sqrt{65}}{17} |
Five people, named A, B, C, D, and E, stand in a row. If A and B must be adjacent, and B must be to the left of A, what is the total number of different arrangements? | 24 |
Consider the L-shaped region formed by three unit squares joined at their sides, as shown below. Two points $A$ and $B$ are chosen independently and uniformly at random from inside the region. The probability that the midpoint of $\overline{AB}$ also lies inside this L-shaped region can be expressed as $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$ [asy] unitsize(2cm); draw((0,0)--(2,0)--(2,1)--(1,1)--(1,2)--(0,2)--cycle); draw((0,1)--(1,1)--(1,0),dashed); [/asy] | 035 |
If the system of inequalities about $x$ is $\left\{{\begin{array}{l}{-2({x-2})-x＜2}\\{\frac{{k-x}}{2}≥-\frac{1}{2}+x}\end{array}}\right.$ has at most $2$ integer solutions, and the solution to the one-variable linear equation about $y$ is $3\left(y-1\right)-2\left(y-k\right)=7$, determine the sum of all integers $k$ that satisfy the conditions. | 18 |
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