problem stringlengths 10 5.15k | answer stringlengths 0 1.23k |
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For integers $a$ and $b$ consider the complex number \[\frac{\sqrt{ab+2016}}{ab+100}-\left({\frac{\sqrt{|a+b|}}{ab+100}}\right)i\]
Find the number of ordered pairs of integers $(a,b)$ such that this complex number is a real number. | 103 |
A point $Q$ is chosen at random in the interior of equilateral triangle $DEF$. What is the probability that $\triangle DEQ$ has a greater area than each of $\triangle DFQ$ and $\triangle EFQ$? | \frac{1}{3} |
The average age of $5$ people in a room is $30$ years. An $18$-year-old person leaves the room. What is the average age of the four remaining people? | 33 |
Two different natural numbers are selected from the set $\{1, 2, 3, \ldots, 7\}$. What is the probability that the greatest common divisor of these two numbers is one? Express your answer as a common fraction. | \frac{17}{21} |
Let $P$ be a moving point on curve $C_1$, and $Q$ be a moving point on curve $C_2$. The minimum value of $|PQ|$ is called the distance between curves $C_1$ and $C_2$, denoted as $d(C_1,C_2)$. If $C_1: x^{2}+y^{2}=2$, $C_2: (x-3)^{2}+(y-3)^{2}=2$, then $d(C_1,C_2)=$ \_\_\_\_\_\_ ; if $C_3: e^{x}-2y=0$, $C_4: \ln x+\ln 2... | \sqrt{2} |
As shown in the diagram, four small plates \( A, B, C, D \) are arranged in a circular shape, with an unspecified number of candies placed on each plate. In each move, it is allowed to take all candies from 1, 3, or 4 plates, or from 2 adjacent plates. What is the maximum number of different possible amounts of candies... | 13 |
Aaron has four brothers. If the number of Bennett's brothers is two less than twice the number of Aaron's brothers, what is the number of Bennett's brothers? | Twice the number of Aaron's brothers is 4*2=<<4*2=8>>8.
Bennett has 8-2=<<8-2=6>>6 brothers.
#### 6 |
Kim's flight took off from Newark at 10:34 AM and landed in Miami at 1:18 PM. Both cities are in the same time zone. If her flight took $h$ hours and $m$ minutes, with $0 < m < 60$, what is $h + m$? | 46 |
If $\det \mathbf{A} = 5,$ then find $\det (\mathbf{A^3}).$ | 125 |
In Rodrigo's classroom in the morning there are red chairs, yellow chairs, and blue chairs. There are 4 red chairs. There are 2 times as many yellow chairs as red chairs, and there are 2 fewer blue chairs than yellow chairs. In the afternoon, Lisa borrows 3 chairs. How many chairs are left in Rodrigo's classroom? | There are 2 * 4 red chairs = <<2*4=8>>8 yellow chairs.
There are 8 yellow chairs - 2 = <<8-2=6>>6 blue chairs.
Total in the morning there were 4 red + 8 yellow + 6 blue = <<4+8+6=18>>18 chairs.
After Lisa removed chairs there were 18 - 3 chairs = <<18-3=15>>15 chairs remaining.
#### 15 |
What is the largest perfect square factor of 4410? | 441 |
Given the function $f(x)=\cos ( \sqrt {3}x+\phi)- \sqrt {3}\sin ( \sqrt {3}x+\phi)$, find the smallest positive value of $\phi$ such that $f(x)$ is an even function. | \frac{2\pi}{3} |
Determine the number of subsets $S$ of $\{1,2, \ldots, 1000\}$ that satisfy the following conditions: - $S$ has 19 elements, and - the sum of the elements in any non-empty subset of $S$ is not divisible by 20 . | 8 \cdot\binom{50}{19} |
A square has sides of length 3 units. A second square is formed having sides that are $120\%$ longer than the sides of the first square. This process is continued sequentially to create a total of five squares. What will be the percent increase in the perimeter from the first square to the fifth square? Express your an... | 107.4\% |
If the three lines $3y-2x=1$, $x+2y=2$ and $4x-6y=5$ are drawn in the plane, how many points will lie at the intersection of at least two of the three lines? | 2 |
Given that $\tan\left(\alpha + \frac{\pi}{3}\right)=2$, find the value of $\frac{\sin\left(\alpha + \frac{4\pi}{3}\right) + \cos\left(\frac{2\pi}{3} - \alpha\right)}{\cos\left(\frac{\pi}{6} - \alpha\right) - \sin\left(\alpha + \frac{5\pi}{6}\right)}$. | -3 |
The population of Delaware is 974,000. A study showed that there are 673 cell phones per 1000 people in the state. How many cell phones are there in Delaware? | There are 974000 / 1000 = <<974000/1000=974>>974 groups of 1000 people in the state.
These groups will have 974 * 673 = <<974*673=655502>>655502 cell phones.
#### 655502 |
Find the smallest fraction which, when divided by each of the fractions \(\frac{21}{25}\) and \(\frac{14}{15}\), results in natural numbers. | 42/5 |
Given a square \(ABCD\) with side length 1, determine the maximum value of \(PA \cdot PB \cdot PC \cdot PD\) where point \(P\) lies inside or on the boundary of the square. | \frac{5}{16} |
Amaya scored 20 marks fewer in Maths than she scored in Arts. She also got 10 marks more in Social Studies than she got in Music. If she scored 70 in Music and scored 1/10 less in Maths, what's the total number of marks she scored in all the subjects? | The total marks Amaya scored more in Music than in Maths is 1/10 * 70 = <<1/10*70=7>>7 marks.
So the total marks she scored in Maths is 70 - 7 = <<70-7=63>>63 marks.
If she scored 20 marks fewer in Maths than in Arts, then he scored 63 + 20 = <<63+20=83>>83 in Arts.
If she scored 10 marks more in Social Studies than in... |
Karlee has 100 grapes and 3/5 as many strawberries as grapes. Giana and Ansley, two of her friends, come visiting, and she gives each of them 1/5 of each fruit. How many fruits is Karlee left with in total? | If Karlee has 3/5 times as many strawberries as grapes, she has 3/5 * 100 grapes = <<3/5*100=60>>60 strawberries.
When her friends come over, she gives each of them 1/5 * 100 grapes = <<1/5*100=20>>20 grapes.
In total, she gives out to her two friends 2 friends * 20 grapes/friend = <<2*20=40>>40 grapes.
She's left with... |
The wristwatch is 5 minutes slow per hour; 5.5 hours ago, it was set to the correct time. It is currently 1 PM on a clock that shows the correct time. How many minutes will it take for the wristwatch to show 1 PM? | 30 |
A tetrahedron with four equilateral triangular faces has a sphere inscribed within it and a sphere circumscribed about it. For each of the four faces, there is a sphere tangent externally to the face at its center and to the circumscribed sphere. A point $P$ is selected at random inside the circumscribed sphere. The pr... | .2 |
The coefficient of $x^3$ in the expansion of $(x^2-x-2)^4$ is __________ (fill in the answer with a number). | -40 |
Let $S$ be the locus of all points $(x,y)$ in the first quadrant such that $\dfrac{x}{t}+\dfrac{y}{1-t}=1$ for some $t$ with $0<t<1$ . Find the area of $S$ . | 1/6 |
Let $D$, $E$, and $F$ be constants such that the equation \[\frac{(x+E)(Dx+36)}{(x+F)(x+9)} = 3\] has infinitely many solutions for $x$. For these values of $D$, $E$, and $F$, it turns out that there are only finitely many values of $x$ which are not solutions to the equation. Find the sum of these values of $x$. | -21 |
A right triangle has sides of lengths 5 cm and 11 cm. Calculate the length of the remaining side if the side of length 5 cm is a leg of the triangle. Provide your answer as an exact value and as a decimal rounded to two decimal places. | 9.80 |
The expression $64x^6-729y^6$ can be factored as $(ax+by)(cx^2+dxy+ey^2)(fx+gy)(hx^2+jxy+ky^2)$. If $a$, $b$, $c$, $d$, $e$, $f$, $g$, $h$, $j$, and $k$ are all integers, find their sum. | 30 |
In quadrilateral $ABCD$, $\angle A = 120^\circ$, and $\angle B$ and $\angle D$ are right angles. Given $AB = 13$ and $AD = 46$, find the length of $AC$. | 62 |
Given that the integer is a 4-digit positive number with four different digits, the leading digit is not zero, the integer is a multiple of 5, 7 is the largest digit, and the first and last digits of the integer are the same, calculate the number of such integers. | 30 |
What is the distance between (-2,4) and (3,-8)? | 13 |
The coefficients of the polynomial
\[a_{10} x^{10} + a_9 x^9 + a_8 x^8 + \dots + a_2 x^2 + a_1 x + a_0 = 0\]are all integers, and its roots $r_1,$ $r_2,$ $\dots,$ $r_{10}$ are all integers. Furthermore, the roots of the polynomial
\[a_0 x^{10} + a_1 x^9 + a_2 x^8 + \dots + a_8 x^2 + a_9 x + a_{10} = 0\]are also $r_1,$... | 11 |
For $a>0$ , let $f(a)=\lim_{t\to\+0} \int_{t}^1 |ax+x\ln x|\ dx.$ Let $a$ vary in the range $0 <a< +\infty$ , find the minimum value of $f(a)$ . | \frac{\ln 2}{2} |
Starting with a list of three numbers, the “*Make-My-Day*” procedure creates a new list by replacing each number by the sum of the other two. For example, from $\{1, 3, 8\}$ “*Make-My-Day*” gives $\{11, 9, 4\}$ and a new “*MakeMy-Day*” leads to $\{13, 15, 20\}$ . If we begin with $\{20, 1, 8\}$ , what is the maxi... | 19 |
Two cars, Car A and Car B, travel towards each other from cities A and B, which are 330 kilometers apart. Car A starts from city A first. After some time, Car B starts from city B. The speed of Car A is $\frac{5}{6}$ of the speed of Car B. When the two cars meet, Car A has traveled 30 kilometers more than Car B. Determ... | 55 |
Given that $21^{-1} \equiv 17 \pmod{53}$, find $32^{-1} \pmod{53}$, as a residue modulo 53. (Give a number between 0 and 52, inclusive.) | 36 |
Sharon’s vacation rental has a Keurig coffee machine. She will be there for 40 days. She has 3 cups of coffee (3 coffee pods) every morning. Her coffee pods come 30 pods to a box for $8.00. How much will she spend on coffee to last her for the entire vacation? | She'll be on vacation for 40 days and drinks 3 cups of coffee/pods per day for a total of 40*3 = <<40*3=120>>120 coffee pods
Her coffee pods come 30 in a box and she needs 120 coffee pods so that’s 120/30 = <<120/30=4>>4 boxes
Each box costs $8.00 and she needs 4 boxes so that’s 8*4 = $<<8*4=32.00>>32.00
#### 32 |
Given the geometric sequence $\{a_n\}$, $a_3$ and $a_7$ are the extreme points of the function $f(x) = \frac{1}{3}x^3 + 4x^2 + 9x - 1$. Calculate the value of $a_5$. | -3 |
Jefferson hires a carriage to go to the church. It is 20 miles away. The horse can go 10 miles per hour. It cost $30 per hour plus a flat fee of $20. How much did he pay for the carriage? | He paid for 20/10=<<20/10=2>>2 hours
That means the hourly fee was 30*2=$<<30*2=60>>60
So the total cost was 60+20=$<<60+20=80>>80
#### 80 |
Exactly two-fifths of NBA players who signed up for a test are aged between 25 and 35 years. If three-eighths of them are older than 35, and a total of 1000 players signed up, how many players are younger than 25 years? | Two-fifths of 1000 players is (2/5)*1000 = <<(2/5)*1000=400>>400 players
Three eights of 1000 players is (3/8)*1000 = <<(3/8)*1000=375>>375 players
The total number of players 25 or older is 400+375 = <<400+375=775>>775
The number of players younger than 25 years is 1000-775 = <<1000-775=225>>225 players
#### 225 |
Consider a sphere inscribed in a right cone with the base radius of 10 cm and height of 40 cm. The radius of the inscribed sphere can be expressed as $b\sqrt{d} - b$ cm. Determine the value of $b+d$. | 19.5 |
There are 50 passengers on a bus. At the first stop, 16 more passengers get on the bus. On the other stops, 22 passengers get off the bus and 5 passengers more get on the bus. How many passengers are there on the bus in total at the last station? | There are 50+16=<<50+16=66>>66 passengers at first.
Then 66 - 22 = <<66-22=44>>44 passengers remain.
Since 5 more passengers get on the bus there are 44 + 5 = <<44+5=49>>49 passengers on the bus.
#### 49 |
Given two points $A(-2,0)$ and $B(0,2)$, and point $C$ is any point on the circle $x^{2}+y^{2}-2x=0$, find the minimum area of $\triangle ABC$. | 3 - \sqrt{2} |
How many integers $N$ less than $1000$ can be written as the sum of $j$ consecutive positive odd integers from exactly 5 values of $j\ge 1$?
| 15 |
Compute $\begin{pmatrix} 2 & 0 \\ 5 & -3 \end{pmatrix} \begin{pmatrix} 8 & -2 \\ 1 & 1 \end{pmatrix}.$ | \begin{pmatrix} 16 & -4 \\ 37 & -13 \end{pmatrix} |
In trapezoid $ABCD$, sides $\overline{AB}$ and $\overline{CD}$ are parallel, $\angle A = 2\angle D$, and $\angle C = 3\angle B$. Find $\angle A$. | 120^\circ |
Jessica was half her mother's age when her mother died. If her mother were alive now, ten years later, she would have been 70. How old is Jessica currently? | If Jessica's mother would be alive today, ten years later after her death, she would be 70, meaning she died at 70-10=<<60=60>>60
If Jessica was half her mother's age when her mother died, she was 1/2*60=30 years old.
Currently, ten years later, Jessica is 10+30=<<10+30=40>>40 years old.
#### 40 |
Amir is 8 kg heavier than Ilnur, and Daniyar is 4 kg heavier than Bulat. The sum of the weights of the heaviest and lightest boys is 2 kg less than the sum of the weights of the other two boys. All four boys together weigh 250 kg. How many kilograms does Amir weigh? | 67 |
Given that Lauren has 4 sisters and 7 brothers, and her brother Lucas has S sisters and B brothers. Find the product of S and B. | 35 |
Big Joe is the tallest player on the basketball team. He is one foot taller than Ben, who is one foot taller than Larry, who is one foot taller than Frank, who is half a foot taller than Pepe. If Pepe is 4 feet six inches tall, how tall is Big Joe, in feet? | Four feet six inches is the same as 4.5 feet tall.
If Frank is 0.5 feet taller than Pepe, then he is 4.5+0.5=<<0.5+4.5=5>>5 feet tall.
If Larry is one foot taller than Frank, then he is 5+1=6 feet tall.
If Ben is one foot taller than Larry, then he is 6+1=<<6+1=7>>7 feet tall.
If Big Joe is one foot taller than Ben, th... |
Let the operation $x*y$ be defined as $x*y = (x+1)(y+1)$. The operation $x^{*2}$ is defined as $x^{*2} = x*x$. Calculate the value of the polynomial $3*(x^{*2}) - 2*x + 1$ when $x=2$. | 32 |
Given a circle is inscribed in a triangle with side lengths $9, 12,$ and $15$. Let the segments of the side of length $9$, made by a point of tangency, be $u$ and $v$, with $u<v$. Find the ratio $u:v$. | \frac{1}{2} |
The NASA Space Shuttle transports material to the International Space Station at a cost of $\$22,\!000$ per kilogram. What is the number of dollars in the cost of transporting a 250 g control module? | 5500 |
Let $a$, $b$, and $c$ be positive real numbers. Find the minimum value of
\[
\frac{(a^2 + 4a + 4)(b^2 + 4b + 4)(c^2 + 4c + 4)}{abc}.
\] | 64 |
How many ways can you color red 16 of the unit cubes in a 4 x 4 x 4 cube, so that each 1 x 1 x 4 cuboid (and each 1 x 4 x 1 and each 4 x 1 x 1 cuboid) has just one red cube in it? | 576 |
During a space experiment conducted by astronauts, they must implement a sequence of 6 procedures. Among them, Procedure A can only occur as the first or the last step, and Procedures B and C must be adjacent when conducted. How many different sequences are there to arrange the experiment procedures? | 96 |
The school band is having a car wash to raise money. Their goal is to collect $150. So far they have earned $10 each from three families and $5 each from 15 families. How much more money do they have to earn to reach their goal? | The school band earned $10/family * 3 families = $<<10*3=30>>30 from three families.
The school band earned $15/family * 5 families = $<<15*5=75>>75 from 15 families.
The school band earned $30 + $75 = $<<30+75=105>>105 total.
The school band needs $150 - $105 = $<<150-105=45>>45 more to reach their goal.
#### 45 |
The digits of a four-digit positive integer add up to 14. The sum of the two middle digits is nine, and the thousands digit minus the units digit is one. If the integer is divisible by 11, what is the integer? | 3542 |
What is the smallest possible value of the sum $\lvert x + 2\rvert + \lvert x + 4\rvert + \lvert x + 5\rvert$? | 3 |
A cat has found $432_{9}$ methods in which to extend each of her nine lives. How many methods are there in base 10? | 353 |
The sequence ${a_n}$ satisfies $a_1=1$, $a_{n+1} \sqrt { \frac{1}{a_{n}^{2}}+4}=1$. Let $S_{n}=a_{1}^{2}+a_{2}^{2}+...+a_{n}^{2}$. If $S_{2n+1}-S_{n}\leqslant \frac{m}{30}$ holds for any $n\in\mathbb{N}^{*}$, find the minimum value of the positive integer $m$. | 10 |
Makenna is selling candy for her Softball Team. The box contains 25 milk chocolate bars, 25 dark chocolate bars, 25 milk chocolate with almond bars, and 25 white chocolate bars. What is the percentage of each type of chocolate bar? | Since there are four types of chocolate bars and each type has 25 bars, the total number of chocolate bars is 25*4=<<25*4=100>>100.
Since there is a total of 100 chocolate bars and four types of chocolate, we divide 100/4=<<100/4=25>>25%.
#### 25 |
In $\triangle ABC$ , point $D$ lies on side $AC$ such that $\angle ABD=\angle C$ . Point $E$ lies on side $AB$ such that $BE=DE$ . $M$ is the midpoint of segment $CD$ . Point $H$ is the foot of the perpendicular from $A$ to $DE$ . Given $AH=2-\sqrt{3}$ and $AB=1$ , find the size of $\angle AME$ . | 15 |
Given that in quadrilateral ABCD, $\angle A : \angle B : \angle C : \angle D = 1 : 3 : 5 : 6$, express the degrees of $\angle A$ and $\angle D$ in terms of a common variable. | 144 |
Given that the perimeter of a sector is 10cm, and its area is 4cm<sup>2</sup>, find the radian measure of the central angle $\alpha$. | \frac{1}{2} |
In the figure, $PA$ is tangent to semicircle $SAR$, $PB$ is tangent to semicircle $RBT$, and $SRT$ is a straight line. If arc $AS$ is $58^\circ$ and arc $BT$ is $37^\circ$, then find $\angle APB$, in degrees.
[asy]
import graph;
unitsize(1.5 cm);
pair A, B, P, R, S, T;
pair[] O;
real[] r;
r[1] = 1;
r[2] = 0.8;
S ... | 95^\circ |
Determine the sum of all prime numbers $p$ for which there exists no integer solution in $x$ to the congruence $3(6x+1)\equiv 4\pmod p$. | 5 |
In triangle \\(ABC\\), the sides opposite to angles \\(A\\), \\(B\\), and \\(C\\) are \\(a\\), \\(b\\), and \\(c\\) respectively, and it is given that \\(A < B < C\\) and \\(C = 2A\\).
\\((1)\\) If \\(c = \sqrt{3}a\\), find the measure of angle \\(A\\).
\\((2)\\) If \\(a\\), \\(b\\), and \\(c\\) are three consecutive... | \dfrac{15\sqrt{7}}{4} |
How many zeroes does $10!$ end with, when $10!$ is written in base 9? | 2 |
In hexagon $ABCDEF$, $AC$ and $CE$ are two diagonals. Points $M$ and $N$ divide $AC$ and $CE$ internally such that $\frac{AM}{AC}=\frac{CN}{CE}=r$. Given that points $B$, $M$, and $N$ are collinear, find $r$. | \frac{\sqrt{3}}{3} |
Let $\mathbf{M}$ be a matrix, and let $\mathbf{v}$ and $\mathbf{w}$ be vectors, such that
\[\mathbf{M} \mathbf{v} = \begin{pmatrix} 1 \\ -5 \end{pmatrix} \quad \text{and} \quad \mathbf{M} \mathbf{w} = \begin{pmatrix} 7 \\ 2 \end{pmatrix}.\]Compute $\mathbf{M} (-2 \mathbf{v} + \mathbf{w}).$ | \begin{pmatrix} 5 \\ 12 \end{pmatrix} |
A cuboid has dimensions of 2 units by 2 units by 2 units. It has vertices $P_1, P_2, P_3, P_4, P_1', P_2', P_3', P_4'.$ Vertices $P_2, P_3,$ and $P_4$ are adjacent to $P_1$, and vertices $P_i' (i = 1,2,3,4)$ are opposite to $P_i$. A regular octahedron has one vertex in each of the segments $\overline{P_1P_2}, \overline... | \frac{4\sqrt{2}}{3} |
Suppose \(a\), \(b\), and \(c\) are three positive numbers that satisfy \(abc = 1\), \(a + \frac{1}{c} = 7\), and \(b + \frac{1}{a} = 34\). Find \(c + \frac{1}{b}\). | \frac{43}{237} |
Given the function $f(x)=\sin ^{2}x+a\sin x\cos x-\cos ^{2}x$, and $f(\frac{\pi }{4})=1$.
(1) Find the value of the constant $a$;
(2) Find the smallest positive period and minimum value of $f(x)$. | -\sqrt{2} |
Three balls labeled 1, 2, and 3 are placed in a jar. A ball is drawn from the jar, its number is recorded, and it is then returned to the jar. This process is repeated three times, with each ball having an equal chance of being drawn in each trial. If the sum of the recorded numbers is 6, what is the probability that t... | $\frac{1}{7}$ |
While one lion cub, who is 6 minutes away from the water hole, heads there, another, having already quenched its thirst, heads back along the same road 1.5 times faster than the first. At the same time, a turtle starts towards the water hole along the same road, being 32 minutes away from it. At some point, the first l... | 2.4 |
When a right triangle is rotated about one leg, the volume of the cone produced is $800\pi \;\textrm{ cm}^3$. When the triangle is rotated about the other leg, the volume of the cone produced is $1920\pi \;\textrm{ cm}^3$. What is the length (in cm) of the hypotenuse of the triangle?
| 26 |
Kara mixes cups of sugar and water in the ratio of 1:2 to make one liter of lemonade. Calculate the total number of cups of sugar used if she used 84 cups to make lemonade. | The total ratio for the cups of sugar and water needed to make lemonade is 1+2 = <<1+2=3>>3
In the 3 cups needed to make one liter of lemonade, the fraction representing the cups of sugar needed is 1/3.
If she used 84 cups while making lemonade, she used 1/3*84 = <<84*1/3=28>>28 cups of sugar.
#### 28 |
Lena played video games for 3.5 hours last weekend. Her brother played 17 minutes more than she did. How many minutes together did Lena and her brother play video games last weekend? | Lena = 3.5 hours = 210 minutes
Brother = 210 + 17 = <<210+17=227>>227 minutes
210 + 227 = <<210+227=437>>437 minutes
Together, Lena and her brother played 437 minutes of video games.
#### 437 |
A spinner has eight congruent sections, each labeled with numbers 1 to 8. Jane and her brother each spin this spinner once. Jane wins if the non-negative difference of their numbers is less than three; otherwise, her brother wins. Determine the probability of Jane winning. Express your answer as a common fraction. | \frac{17}{32} |
Rolling a die twice, let the points shown the first and second times be \(a\) and \(b\) respectively. Find the probability that the quadratic equation \(x^{2} + a x + b = 0\) has two distinct real roots both less than -1. (Answer with a number). | 1/12 |
Given that $\textstyle\binom{2k}k$ results in a number that ends in two zeros, find the smallest positive integer $k$. | 13 |
The measures of angles $A$ and $B$ are both positive, integer numbers of degrees. The measure of angle $A$ is a multiple of the measure of angle $B$, and angles $A$ and $B$ are complementary angles. How many measures are possible for angle $A$? | 11 |
There are 2009 numbers arranged in a circle, each of which is either 1 or -1, and not all numbers are the same. Consider all possible consecutive groups of ten numbers. Compute the product of the numbers in each group of ten and sum these products. What is the maximum possible sum? | 2005 |
The sum of the first 1000 terms of a geometric sequence is 300. The sum of the first 2000 terms is 570. Find the sum of the first 3000 terms. | 813 |
Given the price of Product A was set at 70 yuan per piece in the first year, with an annual sales volume of 118,000 pieces, starting from the second year, the price per piece increased by $$\frac {70\cdot x\%}{1-x\%}$$ yuan due to a management fee, and the annual sales volume decreased by $10^4x$ pieces, calculate the ... | 10 |
If $x \cdot (x+y) = x^2 + 8$, what is the value of $xy$? | 8 |
A chess team has $26$ members. However, only $16$ members attended the last meeting: half of the girls attended but all of the boys attended. How many girls are on the chess team? | 20 |
At a laundromat, it costs $4 for a washer and a quarter for every 10 minutes in the dryer. Samantha does 2 loads of laundry in the wash and then divides all the washed clothes equally into 3 separate dryers. If she lets the dryers run for 40 minutes each, how much does she spend altogether? | Samantha spends $4 x 2 = $<<4*2=8>>8 on washing.
Each dryer costs 25 x 4 = 100 cents = $1
Three dryers cost 1 x 3 = $<<3=3>>3
Altogether, she spends 8 + 3 = $<<8+3=11>>11
#### 11 |
Successive discounts of $10\%$ and $20\%$ are equivalent to a single discount of: | 28\% |
If $a$, $b\in \{-1,1,2\}$, then the probability that the function $f\left(x\right)=ax^{2}+2x+b$ has a zero point is ______. | \frac{2}{3} |
Quadrilateral $ABCD$ is inscribed in a circle with segment $AC$ a diameter of the circle. If $m\angle DAC = 30^\circ$ and $m\angle BAC = 45^\circ$, the ratio of the area of $ABCD$ to the area of the circle can be expressed as a common fraction in simplest radical form in terms of $\pi$ as $\frac{a+\sqrt{b}}{c\pi}$, whe... | 7 |
The sum to infinity of the terms of an infinite geometric progression is 10. The sum of the first two terms is 7. Compute the first term of the progression. | 10\left(1 + \sqrt{\frac{3}{10}}\right) |
The numbers $\log(a^3b^7)$, $\log(a^5b^{12})$, and $\log(a^8b^{15})$ are the first three terms of an arithmetic sequence, and the $12^{\text{th}}$ term of the sequence is $\log{b^n}$. What is $n$? | 112 |
Given that $m$ is a positive integer, and given that $\mathop{\text{lcm}}[40,m]=120$ and $\mathop{\text{lcm}}[m,45]=180$, what is $m$? | 60 |
You know that the Jones family has five children, and the Smith family has three children. Of the eight children you know that there are five girls and three boys. Let $\dfrac{m}{n}$ be the probability that at least one of the families has only girls for children. Given that $m$ and $n$ are relatively prime p... | 67 |
Square A has side lengths each measuring $x$ inches. Square B has side lengths each measuring $4x$ inches. What is the ratio of the area of the smaller square to the area of the larger square? Express your answer as a common fraction. | \frac{1}{16} |
Given any two positive real numbers $x$ and $y$, then $x \diamond y$ is a positive real number defined in terms of $x$ and $y$ by some fixed rule. Suppose the operation $x \diamond y$ satisfies the equations \((x \cdot y) \diamond y=x(y \diamond y)\) and \((x \diamond 1) \diamond x=x \diamond 1\) for all $x, y>0$. Give... | 19 |
Faces $ABC$ and $BCD$ of tetrahedron $ABCD$ meet at an angle of $30^\circ$. The area of face $ABC$ is $120$, the area of face $BCD$ is $80$, and $BC=10$. Find the volume of the tetrahedron.
| 320 |
Given $\sin\alpha + \cos\alpha = \frac{\sqrt{2}}{3}$, where $\alpha \in (0, \pi)$, calculate the value of $\sin\left(\alpha + \frac{\pi}{12}\right)$. | \frac{2\sqrt{2} + \sqrt{3}}{6} |
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