problem stringlengths 10 5.15k | answer stringlengths 0 1.23k |
|---|---|
Aleesia lost 1.5 pounds each week for 10 weeks. Alexei lost 2.5 pounds each week for 8 weeks. How many pounds did the two friends combine to lose? | Alessia = 1.5 * 10 = <<1.5*10=15>>15 pounds
Alexei = 2.5 * 8 = <<2.5*8=20>>20 pounds
15 + 20 = <<15+20=35>>35 pounds
Together they lost 35 pounds.
#### 35 |
Let $\mathbf{a}$ and $\mathbf{b}$ be two vectors such that
\[\|\mathbf{a} + \mathbf{b}\| = \|\mathbf{b}\|.\]Find the angle between the vectors $\mathbf{a} + 2 \mathbf{b}$ and $\mathbf{a},$ in degrees | 90^\circ |
A class has a group of 7 people, and now 3 of them are chosen to swap seats with each other, while the remaining 4 people's seats remain unchanged. Calculate the number of different rearrangement plans. | 70 |
Annie goes to school. Today is her birthday, so Annie decided to buy some sweets for her colleagues. Every classmate got 2 candies. In the end, Annie got left with 12 candies. If there are 35 people in Annie's class in total, how much did Annie spend on candies, if one candy costs $0.1? | There are 35 people in Annie's class, which means, she has 35 - 1 = <<35-1=34>>34 classmates.
Every classmate got 2 candies, so in total Annie gave out 34 * 2 = <<34*2=68>>68 candies.
In the beginning she had 68 + 12 = <<68+12=80>>80 candies.
One candy costs $0.1, so Annie spend 80 * 0.1 = $<<80*0.1=8>>8 on candies.
##... |
In $\Delta ABC$, $a$, $b$, $c$ are the sides opposite to angles $A$, $B$, $C$ respectively. Given that $c=\frac{7}{2}$, the area of $\Delta ABC$ is $\frac{3\sqrt{3}}{2}$, and $\tan A+\tan B=\sqrt{3}(\tan A\tan B-1)$,
(1) Find the measure of angle $C$;
(2) Find the value of $a+b$. | \frac{11}{2} |
Alicia starts a sequence with $m=3$. What is the fifth term of her sequence following the algorithm: Step 1: Alicia writes down the number $m$ as the first term. Step 2: If $m$ is even, Alicia sets $n= rac{1}{2} m$. If $m$ is odd, Alicia sets $n=m+1$. Step 3: Alicia writes down the number $m+n+1$ as the next term. Step... | 43 |
$A, B, C$ and $D$ are distinct positive integers such that the product $AB = 60$, the product $CD = 60$ and $A - B = C + D$ . What is the value of $A$? | 20 |
An integer between $1000$ and $9999$, inclusive, is called balanced if the sum of its two leftmost digits equals the sum of its two rightmost digits. How many balanced integers are there? | 615 |
Triangles $ABC$ and $AEF$ are such that $B$ is the midpoint of $\overline{EF}.$ Also, $AB = EF = 1,$ $BC = 6,$ $CA = \sqrt{33},$ and
\[\overrightarrow{AB} \cdot \overrightarrow{AE} + \overrightarrow{AC} \cdot \overrightarrow{AF} = 2.\]Find the cosine of the angle between vectors $\overrightarrow{EF}$ and $\overrightar... | \frac{2}{3} |
Evaluate $2000^3-1999\cdot 2000^2-1999^2\cdot 2000+1999^3$ | 3999 |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and $\cos A= \frac{4}{5}$.
(1) Find the value of $\sin ^{2} \frac{B+C}{2}+\cos 2A$;
(2) If $b=2$, the area of $\triangle ABC$ is $S=3$, find $a$. | \sqrt{13} |
In the Cartesian coordinate system $xOy$, with $O$ as the pole and the non-negative half-axis of the $x$-axis as the polar axis, a polar coordinate system is established. The polar coordinates of point $P$ are $(3, \frac{\pi}{4})$. The parametric equation of curve $C$ is $\rho=2\cos (\theta- \frac{\pi}{4})$ (with $\the... | \frac{\sqrt{10}-1}{2} |
Triangle $ABC$ has vertices with coordinates $A(2,3),$ $B(7,8),$ and $C(-4,6)$. The triangle is reflected about line $L$. The image points are $A'(2,-5),$ $B'(7,-10),$ and $C'(-4,-8)$. What is the equation of line $L$? | y = -1 |
Christy and her friend Tanya go to Target to buy some face moisturizer and body lotions. Christy spends twice as much as Tanya, who pays 50$ for two face moisturizers each and 60$ per body lotion, buying four of them. How much money did they spend together in total? | The total amount Tanya spent on face moisturizers is 50*2 = $<<50*2=100>>100
If she bought four body lotions, the total cost is 4*60 = $<<4*60=240>>240
Tanya spent a total of 240+100 = $<<240+100=340>>340 in the store.
If Christy spent twice this amount, the total was 2*340 = $ 680
The total amount of money they both s... |
In the Cartesian coordinate system $xOy$, it is known that the line $l_1$ is defined by the parametric equations $\begin{cases}x=t\cos \alpha\\y=t\sin \alpha\end{cases}$ (where $t$ is the parameter), and the line $l_2$ by $\begin{cases}x=t\cos(\alpha + \frac{\pi}{4})\\y=t\sin(\alpha + \frac{\pi}{4})\end{cases}$ (where ... | 2\sqrt{2} |
The sum of two numbers is $19$ and their difference is $5$. What is their product? | 84 |
Lyra bought a pair of shoes at a 20% discount. If she paid $480, how much was the original price of the pair of shoes? | Lyra only paid $480 for the pair of shoes which is only 100% - 20% = 80% of the original price.
So let x be the original price.
Then 0.8x = $480
Thus x = $480 / 0.8 = $<<480/0.8=600>>600
#### 600 |
The electricity price in Coco's town is $0.10 per kW. Coco's new oven has a consumption rate of 2.4 kWh (kilowatt-hours). How much will Coco pay for using his oven only if he used it for a total of 25 hours last month? | If Coco used his oven 25 hours last month and its energy consumption is 2.4 kW per hour, then we consumed a total of 25*2.4= <<25*2.4=60>>60 kW on his oven last month
If the price of electricity is $0.10 per kW, then 60 kW would cost = $0.10*60 = $<<0.10*60=6>>6
#### 6 |
James decides to buy a new bed and bed frame. The bed frame is $75 and the bed is 10 times that price. He gets a deal for 20% off. How much does he pay for everything? | The bed cost 75*10=$<<75*10=750>>750
So everything cost 750+75=$<<750+75=825>>825
He gets 825*.2=$<<825*.2=165>>165 off
So that means he pays 825-165=$<<825-165=660>>660
#### 660 |
Dexter went to the mall and saw that Apple products are on sale. He wants to buy an iPhone 12 with a 15% discount and an iWatch with a 10% discount. The price tag shows that an iPhone 12 costs $800 while an iWatch costs $300. Upon check out, he will receive a further 2% cashback discount. How much would the items cost ... | The discount for an iPhone 12 is $800 x 15/100 = $<<800*15/100=120>>120.
So, the iPhone will now cost $800 - $120 = $<<800-120=680>>680.
The discount for an iWatch is $300 x 10/100 = $<<300*10/100=30>>30.
So, the iWatch will now cost $300 - $30 = $<<300-30=270>>270.
The total cost of the iPhone and iWatch is $680 + $27... |
If five points are given on a plane, then by considering all possible triples of these points, 30 angles can be formed. Denote the smallest of these angles by $\alpha$. Find the maximum value of $\alpha$. | 36 |
What is the greatest number of consecutive non-negative integers whose sum is $120$? | 15 |
Triangle $A B C$ has incircle $\omega$ which touches $A B$ at $C_{1}, B C$ at $A_{1}$, and $C A$ at $B_{1}$. Let $A_{2}$ be the reflection of $A_{1}$ over the midpoint of $B C$, and define $B_{2}$ and $C_{2}$ similarly. Let $A_{3}$ be the intersection of $A A_{2}$ with $\omega$ that is closer to $A$, and define $B_{3}$... | 14/65 |
What are the rightmost three digits of $5^{1993}$? | 125 |
If point $P$ is any point on the curve $y=x^{2}-\ln x$, then the minimum distance from point $P$ to the line $y=x-2$ is ____. | \sqrt{2} |
Mitchell has 30 pencils. He has 6 more pencils than Antonio. How many pencils does Mitchell and Antonio have together? | Antonio has 30-6 = <<30-6=24>>24 pencils
Antonio and Mitchell have 24+30 = <<24+30=54>>54 pencils together.
#### 54 |
For how many pairs of consecutive integers in the set $\{1100, 1101, 1102, \ldots, 2200\}$ is no carrying required when the two integers are added? | 1100 |
James gets a cable program. The first 100 channels cost $100 and the next 100 channels cost half that much. He splits it evenly with his roommate. How much did he pay? | The next 100 channels cost 100/2=$<<100/2=50>>50
So the total cost is 100+50=$<<100+50=150>>150
So he pays 150/2=$<<150/2=75>>75
#### 75 |
Compute the value of \[M = 50^2 + 48^2 - 46^2 + 44^2 + 42^2 - 40^2 + \cdots + 4^2 + 2^2 - 0^2,\] where the additions and subtractions alternate in triplets. | 2600 |
Compute the sum of the series:
\[ 2(1 + 2(1 + 2(1 + 2(1 + 2(1 + 2(1 + 2(1 + 2))))))) \] | 510 |
Given real numbers $x$ and $y$ satisfying $x^{2}+y^{2}-4x-2y-4=0$, find the maximum value of $x-y$. | 1+3\sqrt{2} |
Let P be a moving point on the line $3x+4y+3=0$, and through point P, two tangents are drawn to the circle $C: x^2+y^2-2x-2y+1=0$, with the points of tangency being A and B, respectively. Find the minimum value of the area of quadrilateral PACB. | \sqrt{3} |
Container I holds 8 red balls and 4 green balls; containers II and III each hold 2 red balls and 4 green balls. A container is selected at random and then a ball is randomly selected from that container. What is the probability that the ball selected is green? Express your answer as a common fraction. | \frac{5}{9} |
Allen and Yang want to share the numbers \(1,2,3,4,5,6,7,8,9,10\). How many ways are there to split all ten numbers among Allen and Yang so that each person gets at least one number, and either Allen's numbers or Yang's numbers sum to an even number? | 1022 |
The six-digit number $20210A$ is prime for only one digit $A.$ What is $A?$ | 9 |
In a club election, 10 officer positions are available. There are 20 candidates, of which 8 have previously served as officers. Determine how many different slates of officers include at least one of the past officers. | 184,690 |
How many natural numbers greater than 10 but less than 100 are relatively prime to 21? | 51 |
Calculate the line integral
$$
\int_{L} \frac{y}{3} d x - 3 x d y + x d z
$$
along the curve \( L \), which is given parametrically by
$$
\begin{cases}
x = 2 \cos t \\
y = 2 \sin t \\
z = 1 - 2 \cos t - 2 \sin t
\end{cases}
\quad \text{for} \quad 0 \leq t \leq \frac{\pi}{2}
$$ | 2 - \frac{13\pi}{3} |
Clara is selling boxes of cookies to raise money for a school trip. There are 3 different types for sale. The first type has 12 cookies per box. The second type has 20 cookies per box, and the third type has 16 cookies per box. If Clara sells 50 boxes of the first type, 80 boxes of the second type, and 70 boxes of the ... | Clara sells 600 cookies of the first type because 12*50 = <<12*50=600>>600.
She sells 1600 cookies of the second type because 20*80=<<20*80=1600>>1600.
She sells 1120 cookies of the third type because 16*70 = <<16*70=1120>>1120.
Thus, Clara sells 3300 cookies total because 600+1600+1120=3320 cookies.
#### 3320 |
Given the function $f\left(x\right)=4^{x}+m\cdot 2^{x}$, where $m\in R$.
$(1)$ If $m=-3$, solve the inequality $f\left(x\right) \gt 4$ with respect to $x$.
$(2)$ If the minimum value of the function $y=f\left(x\right)+f\left(-x\right)$ is $-4$, find the value of $m$. | -3 |
For any positive integers \( n \) and \( k \) (where \( k \leqslant n \)), let \( f(n, k) \) denote the number of positive integers that do not exceed \( \left\lfloor \frac{n}{k} \right\rfloor \) and are coprime with \( n \). Determine the value of \( f(100, 3) \). | 14 |
Find the total number of solutions to the equation $(a-b)(a+b)+(a-b)(c)=(a-b)(a+b+c)=2012$ where $a, b, c$ are positive integers. | 1755 |
A cone has a volume of $2592\pi$ cubic inches and the vertex angle of the vertical cross section is 90 degrees. What is the height of the cone? Express your answer as a decimal to the nearest tenth. | 20.0 |
The real value of $x$ such that $64^{x-1}$ divided by $4^{x-1}$ equals $256^{2x}$ is: | -\frac{1}{3} |
Evaluate: $(2^2)^3$. | 64 |
Jay has decided to save money from his paycheck every week. He has decided that he will increase the amount he saves each week by 10 dollars. If he started by saving 20 dollars this week, how much will he have saved in a month from now? | In one week he will have saved 20+10=<<20+10=30>>30 dollars
In two weeks he will have saved 30+10=<<30+10=40>>40 dollars
In three weeks he will have saved 40+10=<<40+10=50>>50 dollars
At the end of week four, he will have saved 50+10=<<50+10=60>>60 dollars
#### 60 |
Compute the sum of the number $10 - \sqrt{2018}$ and its radical conjugate. | 20 |
\( S \) is the set of all ordered tuples \((a, b, c, d, e, f)\) where \(a, b, c, d, e, f\) are integers and \(a^2 + b^2 + c^2 + d^2 + e^2 = f^2\). Find the largest \( k \) such that \( k \) divides \( a b c d e f \) for all elements in \( S \). | 24 |
What is the result of $120 \div (6 \div 2 \times 3)$? | \frac{120}{9} |
The units digit of $3^{1001} 7^{1002} 13^{1003}$ is | 3 |
In Yang's number theory class, Michael K, Michael M, and Michael R take a series of tests. Afterwards, Yang makes the following observations about the test scores:
(a) Michael K had an average test score of $90$ , Michael M had an average test score of $91$ , and Michael R had an average test score of $92$ .
(b) M... | 413 |
It takes Mina 90 seconds to walk down an escalator when it is not operating, and 30 seconds to walk down when it is operating. Additionally, it takes her 40 seconds to walk up another escalator when it is not operating, and only 15 seconds to walk up when it is operating. Calculate the time it takes Mina to ride down t... | 69 |
Alicia has 20 gumballs. Pedro has that many gumballs plus an additional number of gumballs equal to three times the number Alicia has. They put their gumballs in a bowl, and later Pedro takes out 40% of the gumballs to eat. How many gumballs are remaining in the bowl? | If Pedro has three times more gumballs than Alicia, he has 3 * 20 gumballs = <<3*20=60>>60 gumballs more
In total, Pedro has 20 gumballs + 60 gumballs = <<20+60=80>>80 gumballs
When they put their gumballs together, the total number of gumballs in the bowl becomes 20 gumballs + 80 gumballs = <<20+80=100>>100 gumballs
P... |
Four pens and three pencils cost $\$2.24$. Two pens and five pencils cost $\$1.54$. No prices include tax. In cents, what is the cost of a pencil? | 12 |
Determine the area of a triangle with side lengths 7, 7, and 5. | \frac{5\sqrt{42.75}}{2} |
Express as a fraction in lowest terms: $0.\overline{1} + 0.\overline{01}$ | \frac{4}{33} |
The product of two 2-digit numbers is $4536$. What is the smaller of the two numbers? | 54 |
Let $p,$ $q,$ $r,$ $s$ be real numbers such that $p +q + r + s = 8$ and
\[pq + pr + ps + qr + qs + rs = 12.\]Find the largest possible value of $s.$ | 2 + 3 \sqrt{2} |
Find the center of the circle with equation $9x^2-18x+9y^2+36y+44=0.$ | (1,-2) |
Let \( p, q, r \) be the roots of the polynomial \( x^3 - 8x^2 + 14x - 2 = 0 \). Define \( t = \sqrt{p} + \sqrt{q} + \sqrt{r} \). Find \( t^4 - 16t^2 - 12t \). | -8 |
A soccer ball takes twenty minutes to inflate. Alexia and Ermias are inflating balls, with Alexia inflating 20 balls and Ermias inflating 5 more balls than Alexia. Calculate the total time in minutes they took to inflate all the soccer balls. | To inflate a soccer ball, it takes 20 minutes, and if Alexia inflated 20 soccer balls, she took 20*20 = <<20*20=400>>400 minutes.
Ermias has 5 more soccer balls than Alexia, meaning she has a total of 20+5 = <<5+20=25>>25 soccer balls.
If each soccer ball takes 20 minutes to inflate, Ermias took 25*20 = <<25*20=500>>50... |
By the time Anne is two times as old as Emile, Emile will be six times as old as Maude. If Maude will be 8 years old, how old will Anne be? | If Maude's age is 8 by the time Anne's age is four times Emile's age, Emile will be six times as old as Maude, which totals 6*8 = 48 years.
If Emile's age is 48 years old by the time Anne's age is twice her number, Anne will be 2*48 = <<48*2=96>>96 years.
#### 96 |
What is the number of square units in the area of the hexagon below?
[asy]
unitsize(0.5cm);
defaultpen(linewidth(0.7)+fontsize(10));
dotfactor = 4;
int i,j;
for(i=0;i<=4;++i)
{
for(j=-3;j<=3;++j)
{
dot((i,j));
}
}
for(i=1;i<=4;++i)
{
draw((i,-1/3)--(i,1/3));
}
for(j=1;j<=3;++j)
{
draw((-1/3,j)--(1/3,j));
... | 18 |
In the diagram, there are several triangles formed by connecting points in a shape. If each triangle has the same probability of being selected, what is the probability that a selected triangle includes a vertex marked with a dot? Express your answer as a common fraction.
[asy]
draw((0,0)--(2,0)--(1,2)--(0,0)--cycle,l... | \frac{1}{2} |
Celia runs twice as fast as Lexie. If it takes Lexie 20 minutes to run a mile, how long, in minutes, will it take Celia to 30 miles? | If Lexie takes 20 minutes to run a mile, Celia takes 20/2 = <<20/2=10>>10 minutes to run the same distance since she is twice as fast.
Celia takes 10 minutes to run one mile so it will take 10*30 = <<10*30=300>>300 minutes to run 30 miles.
#### 300 |
A set of $n$ people participate in an online video basketball tournament. Each person may be a member of any number of $5$-player teams, but no two teams may have exactly the same $5$ members. The site statistics show a curious fact: The average, over all subsets of size $9$ of the set of $n$ participants, of the numbe... | 557 |
If $\log_{k}{x} \cdot \log_{5}{k} = 3$, then $x$ equals: | 125 |
The teacher asks Bill to calculate $a-b-c$, but Bill mistakenly calculates $a-(b-c)$ and gets an answer of 11. If the correct answer was 3, what is the value of $a-b$? | 7 |
John's hair grows 1.5 inches every month. Every time it gets to 9 inches long he cuts it down to 6 inches. A haircut costs $45 and he gives a 20% tip. How much does he spend on haircuts a year? | He cuts off 9-6=<<9-6=3>>3 inches when he gets a haircut
That means he needs to cut it every 3/1.5=<<3/1.5=2>>2 months
So he gets 12/2=<<12/2=6>>6 haircuts a year
He gives a 45*.2=$<<45*.2=9>>9 tip
So that means each hair cut cost 45+9=$<<45+9=54>>54
So he pays 54*6=$<<54*6=324>>324 a year
#### 324 |
Given that Chloe's telephone numbers have the form $555-ab-cdef$, where $a$, $b$, $c$, $d$, $e$, and $f$ are distinct digits, in descending order, and are chosen between $1$ and $8$, calculate the total number of possible telephone numbers that Chloe can have. | 28 |
Find the volume of the region in space defined by
\[ |z + x + y| + |z + x - y| \leq 10 \]
and \(x, y, z \geq 0\). | 62.5 |
Knights, who always tell the truth, and liars, who always lie, live on an island. One day, 30 inhabitants of this island sat around a round table. Each of them said one of two phrases: "My neighbor on the left is a liar" or "My neighbor on the right is a liar." What is the minimum number of knights that can be at the t... | 10 |
Four girls — Mary, Alina, Tina, and Hanna — sang songs in a concert as trios, with one girl sitting out each time. Hanna sang $7$ songs, which was more than any other girl, and Mary sang $4$ songs, which was fewer than any other girl. How many songs did these trios sing? | 7 |
Given a sequence of natural numbers $\left\{x_{n}\right\}$ defined by:
$$
x_{1}=a, x_{2}=b, x_{n+2}=x_{n}+x_{n+1}, \quad n=1,2,3,\cdots
$$
If an element of the sequence is 1000, what is the minimum possible value of $a+b$? | 10 |
Kiana has two older twin brothers. The product of their three ages is 128. What is the sum of their three ages? | 18 |
A positive number is called $n$-primable if it is divisible by $n$ and each of its digits is a one-digit prime number. How many 3-primable positive integers are there that are less than 1000? | 28 |
Olivia's insurance premium starts out at $50/month. It goes up 10% for every accident and $5/month for every ticket. If she gets in one accident and gets 3 tickets, what's her new insurance premium? | First find the increase due to the accident: $50 * 10% = $<<50*10*.01=5>>5
Then find the total increase due to the tickets: $5/ticket * 3 tickets = $<<5*3=15>>15
Then add both increases to the base price to find the new price: $50 + $5 + $15 = $<<50+5+15=70>>70
#### 70 |
A line in the plane is called strange if it passes through \((a, 0)\) and \((0, 10-a)\) for some \(a\) in the interval \([0,10]\). A point in the plane is called charming if it lies in the first quadrant and also lies below some strange line. What is the area of the set of all charming points? | 50/3 |
Determine how many five-letter words can be formed such that they begin and end with the same letter and the third letter is always a vowel. | 87880 |
When $5^{35}-6^{21}$ is evaluated, what is the units (ones) digit? | 9 |
Determine the number of solutions to the equation
\[\tan (7 \pi \cos \theta) = \cot (3 \pi \sin \theta)\] where $\theta \in (0, 3\pi).$ | 90 |
Find the sum of the $x$-coordinates of the solutions to the system of equations $y=|x^2-8x+12|$ and $y=4-x$. | 16 |
Find the smallest integer $n$ such that an $n\times n$ square can be partitioned into $40\times 40$ and $49\times 49$ squares, with both types of squares present in the partition, if
a) $40|n$ ; b) $49|n$ ; c) $n\in \mathbb N$ . | 1960 |
The function $f(x)$ satisfies
\[f(x + y) = f(x) + f(y)\]for all real numbers $x$ and $y,$ and $f(4) = 5.$ Find $f(5).$ | \frac{25}{4} |
Given a pyramid $P-ABC$ where $PA=PB=2PC=2$, and $\triangle ABC$ is an equilateral triangle with side length $\sqrt{3}$, the radius of the circumscribed sphere of the pyramid $P-ABC$ is _______. | \dfrac{\sqrt{5}}{2} |
In writing the integers from 10 through 99 inclusive, how many times is the digit 7 written? | 19 |
In city "N", there are 10 horizontal and 12 vertical streets. A pair of horizontal and a pair of vertical streets form the rectangular boundary of the city, while the rest divide it into blocks shaped like squares with a side length of 100 meters. Each block has an address consisting of two integers \((i, j)\), \(i = 1... | 14 |
What is the modular inverse of $11$, modulo $1000$?
Express your answer as an integer from $0$ to $999$, inclusive. | 91 |
Determine the maximum value of $m^2+n^2$, where $m$ and $n$ are integers in the range $1,2,\ldots,1981$ satisfying $(n^2-mn-m^2)^2=1$. | \boxed{987^2+1597^2} |
Let $x_1$ , $x_2$ , …, $x_{10}$ be 10 numbers. Suppose that $x_i + 2 x_{i + 1} = 1$ for each $i$ from 1 through 9. What is the value of $x_1 + 512 x_{10}$ ? | 171 |
Select 5 different letters from the word "equation" to arrange in a row, including the condition that the letters "qu" are together and in the same order. | 480 |
Given points $A(-2,0)$ and $B(2,0)$, the slope of line $PA$ is $k_1$, and the slope of line $PB$ is $k_2$, with the product $k_1k_2=-\frac{3}{4}$.
$(1)$ Find the equation of the locus $C$ for point $P$.
$(2)$ Let $F_1(-1,0)$ and $F_2(1,0)$. Extend line segment $PF_1$ to meet the locus $C$ at another point $Q$. Let poin... | \frac{3}{2} |
Given that $$x^{5}=a_{0}+a_{1}(2-x)+a_{2}(2-x)^{2}+…+a_{5}(2-x)^{5}$$, find the value of $$\frac {a_{0}+a_{2}+a_{4}}{a_{1}+a_{3}}$$. | - \frac {61}{60} |
A convex hexagon \( A_{1} A_{2} \ldots A_{6} \) is circumscribed around a circle \( \omega \) with a radius of 1. Consider three segments that connect the midpoints of the opposite sides of the hexagon. What is the greatest \( r \) for which it can be stated that at least one of these segments is not shorter than \( r ... | \sqrt{3} |
Find the sum of the values of $x$ such that $\cos^3 3x+ \cos^3 5x = 8 \cos^3 4x \cos^3 x$, where $x$ is measured in degrees and $100< x< 200.$ | 906 |
Let the function $f(x)=\begin{cases} 2^{x}+\int_{0}^{1}{3(\sqrt{x}-x^{2})dx} & (x\geqslant 4) \\ f(x+2) & (x < 4) \end{cases}$. Evaluate the expression $f(\log_{2}3)$. | 49 |
A square with sides 8 inches is shown. If $Q$ is a point such that the segments $\overline{QA}$, $\overline{QB}$, $\overline{QC}$ are equal in length, and segment $\overline{QC}$ is perpendicular to segment $\overline{HD}$, find the area, in square inches, of triangle $AQB$. [asy]
pair A, B, C, D, H, Q;
A = (0,0); B= (... | 12 |
Given a circle of radius $\sqrt{13}$, let $A$ be a point at a distance $4 + \sqrt{13}$ from the center $O$ of the circle. Let $B$ be the point on the circle nearest to point $A$. A line passing through the point $A$ intersects the circle at points $K$ and $L$. The maximum possible area for $\triangle BKL$ can be writte... | 146 |
Given a quadrilateral $ABCD$ with $AB = BC =3$ cm, $CD = 4$ cm, $DA = 8$ cm and $\angle DAB + \angle ABC = 180^o$ . Calculate the area of the quadrilateral.
| 13.2 |
Three candles can burn for 30, 40, and 50 minutes respectively (but they are not lit simultaneously). It is known that the three candles are burning simultaneously for 10 minutes, and only one candle is burning for 20 minutes. How many minutes are there when exactly two candles are burning simultaneously? | 35 |
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