problem stringlengths 10 5.15k | answer stringlengths 0 1.23k |
|---|---|
Given \(0 \leq x_0 < 1\), let
\[
x_n = \left\{ \begin{array}{ll}
2x_{n-1} & \text{if } 2x_{n-1} < 1 \\
2x_{n-1} - 1 & \text{if } 2x_{n-1} \geq 1
\end{array} \right.
\]
for all integers \(n > 0\). Determine the number of initial values of \(x_0\) that satisfy \(x_0 = x_6\). | 64 |
The sum of seven consecutive even numbers is 686. What is the smallest of these seven numbers? Additionally, calculate the median and mean of this sequence. | 98 |
What is the value of $ rac{(20-16) imes (12+8)}{4}$? | 20 |
A company has a total of 60 employees. In order to carry out club activities, a questionnaire survey was conducted among all employees. There are 28 people who like sports, 26 people who like literary and artistic activities, and 12 people who do not like either sports or literary and artistic activities. How many peop... | 22 |
Randomly and without replacement, select three numbers \(a_1, a_2, a_3\) from the set \(\{1, 2, \cdots, 2014\}\). Then, from the remaining 2011 numbers, again randomly and without replacement, select three numbers \(b_1, b_2, b_3\). What is the probability that a brick with dimensions \(a_1 \times a_2 \times a_3\) can ... | 1/4 |
Free Christmas decorations are being given out to families. Each box of decorations contains 4 pieces of tinsel, 1 Christmas tree and 5 snow globes. If 11 families receive a box of decorations and another box is given to the community center, how many decorations have been handed out? | Each box contains 4 tinsel + 1 tree + 5 snow globes = <<4+1+5=10>>10 decorations.
A total of 11 family boxes + 1 community center box = <<11+1=12>>12 boxes have been given out.
So the total number of decorations given out is 10 decorations * 12 boxes = <<10*12=120>>120 decorations.
#### 120 |
Players A and B participate in a two-project competition, with each project adopting a best-of-five format (the first player to win 3 games wins the match, and the competition ends), and there are no ties in each game. Based on the statistics of their previous matches, player A has a probability of $\frac{2}{3}$ of win... | \frac{209}{162} |
A laboratory has $10$ experimental mice, among which $3$ have been infected with a certain virus, and the remaining $7$ are healthy. Random medical examinations are conducted one by one until all $3$ infected mice are identified. The number of different scenarios where the last infected mouse is discovered exactly on t... | 1512 |
A paper equilateral triangle $ABC$ has side length 12. The paper triangle is folded so that vertex $A$ touches a point on side $\overline{BC}$ a distance 9 from point $B$. Find the square of the length of the line segment along which the triangle is folded.
[asy]
import cse5;
size(12cm);
pen tpen = defaultpen + 1.337;... | \frac{59319}{1225} |
Jamir and his two friends Sarah and Julien, go to their school's swimming pool to swim. Jamir swims 20 more meters per day than Sarah, who swims twice the distance Julien swims. They go to the swimming pool the whole week, swimming the same distances as before. If Julien swam 50 meters, what's the combined distance for... | If Julien swam for the whole week, then the total distance in meters that he covered is 50*7 = <<50*7=350>>350 meters
Sarah swam twice the distance covered by Julien, which is 2*50 = <<2*50=100>>100 meters
Sarah's total for the whole week is 100*7 = <<100*7=700>>700 meters
If Jamir swam 20 meters more than Sarah, he co... |
John buys dinner plates and silverware. The silverware cost $20. The dinner plates cost 50% as much as the silverware. How much did he pay for everything? | The dinner plates cost 20*.5=$<<20*.5=10>>10
So he paid 10+20=$<<10+20=30>>30
#### 30 |
In the isosceles triangle \(ABC\) with the sides \(AB = BC\), the angle \(\angle ABC\) is \(80^\circ\). Inside the triangle, a point \(O\) is taken such that \(\angle OAC = 10^\circ\) and \(\angle OCA = 30^\circ\). Find the angle \(\angle AOB\). | 70 |
Tom takes medication to help him sleep. He takes 2 pills every day before bed. He needs to go to the doctor every 6 months to get a new prescription and a visit to the doctor costs $400. The medication costs $5 per pill, but insurance covers 80% of that cost. How much does he pay a year for everything? | He sees the doctor 12/6=<<12/6=2>>2 times a year
So he pays 400*2=$<<400*2=800>>800 for doctor visits
The cost of the pill is 5*2=$<<5*2=10>>10 per night
But insurance covers 10*.8=$<<10*.8=8>>8
So he pays 10-8=$<<10-8=2>>2 per night
So he pays 2*365=$<<2*365=730>>730 a year
SSo his total cost is 800+730=$<<800+730=153... |
Matěj had written six different natural numbers in a row in his notebook. The second number was double the first, the third was double the second, and similarly, each subsequent number was double the previous one. Matěj copied all these numbers into the following table in random order, one number in each cell.
The sum... | 96 |
Given two dice are thrown in succession, with the six faces of each die labeled with $1$, $2$, $3$, $4$, $5$, and $6$ points, let event $A$ be "$x+y$ is odd" and event $B$ be "$x$, $y$ satisfy $x+y<6$, where $x$ and $y$ are the numbers of points facing up on the two dice when they land on a tabletop. Calculate the cond... | \frac{1}{3} |
At what point does the line $3y-4x=12$ intersect the $x$-axis? Express your answer as an ordered pair. | (-3,0) |
Let \( n = \overline{abc} \) be a three-digit number, where \( a, b, \) and \( c \) are the digits of the number. If \( a, b, \) and \( c \) can form an isosceles triangle (including equilateral triangles), how many such three-digit numbers \( n \) are there? | 165 |
Given an ellipse $E: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \left(a > b > 0\right)$ that passes through the point $(0,1)$, and its eccentricity is $\frac{\sqrt{3}}{2}$.
$(1)$ Find the standard equation of the ellipse $E$;
$(2)$ Suppose a line $l: y = \frac{1}{2}x + m$ intersects the ellipse $E$ at points $A$ and $C$. ... | \frac{\sqrt{10}}{2} |
For $-1<r<1$, let $T(r)$ denote the sum of the geometric series \[20 + 10r + 10r^2 + 10r^3 + \cdots.\] Let $b$ between $-1$ and $1$ satisfy $T(b)T(-b)=5040$. Find $T(b)+T(-b)$. | 504 |
A regular octahedron is formed by joining the midpoints of the edges of a regular tetrahedron. Calculate the ratio of the volume of this octahedron to the volume of the original tetrahedron. | \frac{1}{2} |
$\textbf{Problem 5.}$ Miguel has two clocks, one clock advances $1$ minute per day and the other one goes $15/10$ minutes per day.
If you put them at the same correct time, What is the least number of days that must pass for both to give the correct time simultaneously? | 1440 |
A cake has a shape of triangle with sides $19,20$ and $21$ . It is allowed to cut it it with a line into two pieces and put them on a round plate such that pieces don't overlap each other and don't stick out of the plate. What is the minimal diameter of the plate? | 21 |
Let $z$ and $w$ be complex numbers such that $|z + 1 + 3i| = 1$ and $|w - 7 - 8i| = 3.$ Find the smallest possible value of $|z - w|.$ | \sqrt{185} - 4 |
Jessica has exactly one of each of the first 30 states' new U.S. quarters. The quarters were released in the same order that the states joined the union. The graph below shows the number of states that joined the union in each decade. What fraction of Jessica's 30 coins represents states that joined the union during th... | \frac{1}{6} |
Two quadratic equations with unequal leading coefficients,
$$
(a-1) x^{2} - \left(a^{2}+2\right) x + \left(a^{2}+2a\right) = 0
$$
and
$$
(b-1) x^{2} - \left(b^{2}+2\right) x + \left(b^{2}+2b\right) = 0
$$
(where \(a\) and \(b\) are positive integers), have a common root. Find the value of \(\frac{a^{b} + b^{a}}{a^{-b} ... | 256 |
Two equal parallel chords are drawn $8$ inches apart in a circle of radius $8$ inches. The area of that part of the circle that lies between the chords is: | $32\sqrt{3}+21\frac{1}{3}\pi$ |
Xiao Ming collected 20 pieces of data in a survey, as follows:
$95\ \ \ 91\ \ \ 93\ \ \ 95\ \ \ 97\ \ \ 99\ \ \ 95\ \ \ 98\ \ \ 90\ \ \ 99$
$96\ \ \ 94\ \ \ 95\ \ \ 97\ \ \ 96\ \ \ 92\ \ \ 94\ \ \ 95\ \ \ 96\ \ \ 98$
$(1)$ When constructing a frequency distribution table with a class interval of $2$, how many cla... | 0.4 |
If the consecutive integers from $50$ to $1$ were written as $$5049484746...,$$ what would be the $67^{\text{th}}$ digit to be written? | 1 |
A $6 \times 9$ rectangle can be rolled to form two different cylinders. Calculate the ratio of the larger volume to the smaller volume. Express your answer as a common fraction. | \frac{3}{2} |
Suppose \(a\), \(b\), and \(c\) are real numbers such that:
\[
\frac{ac}{a + b} + \frac{ba}{b + c} + \frac{cb}{c + a} = -12
\]
and
\[
\frac{bc}{a + b} + \frac{ca}{b + c} + \frac{ab}{c + a} = 15.
\]
Compute the value of:
\[
\frac{a}{a + b} + \frac{b}{b + c} + \frac{c}{c + a}.
\] | -12 |
Arrange 1, 2, 3, a, b, c in a row such that letter 'a' is not at either end and among the three numbers, exactly two are adjacent. The probability is $\_\_\_\_\_\_$. | \frac{2}{5} |
Let $D$ be a regular ten-sided polygon with edges of length 1. A triangle $T$ is defined by choosing three vertices of $D$ and connecting them with edges. How many different (non-congruent) triangles $T$ can be formed? | 8 |
There are 20 students in a class. In total, 10 of them have black hair, 5 of them wear glasses, and 3 of them both have black hair and wear glasses. How many of the students have black hair but do not wear glasses? | 7 |
Given $\cos\alpha =\frac{\sqrt{5}}{5}$ and $\sin\beta =\frac{3\sqrt{10}}{10}$, with $0 < \alpha$, $\beta < \frac{\pi}{2}$, determine the value of $\alpha +\beta$. | \frac{3\pi}{4} |
The number 770 is written on a blackboard. Melody repeatedly performs moves, where a move consists of subtracting either 40 or 41 from the number on the board. She performs moves until the number is not positive, and then she stops. Let $N$ be the number of sequences of moves that Melody could perform. Suppose $N=a \cd... | 318 |
Let $(1+2x)^2(1-x)^5 = a + a_1x + a_2x^2 + \ldots + a_7x^7$, then $a_1 - a_2 + a_3 - a_4 + a_5 - a_6 + a_7 =$ ? | -31 |
$\triangle ABC$ and $\triangle DBC$ share $BC$. $AB = 5\ \text{cm}$, $AC = 12\ \text{cm}$, $DC = 8\ \text{cm}$, and $BD = 20\ \text{cm}$. What is the least possible integral number of centimeters in $BC$?
[asy]
size(100); import graph; currentpen = fontsize(10pt);
pair B = (0,0), C = (13,0), A = (-5,7), D = (16,10);
... | 13 |
Paco uses a spinner to select a number from 1 through 5, each with equal probability. Manu uses a different spinner to select a number from 1 through 10, each with equal probability. What is the probability that the product of Manu's number and Paco's number is less than 30? Express your answer as a common fraction. | \frac{41}{50} |
Pablo’s mother agrees to pay him one cent for every page he reads. He plans to save the money for some candy. Pablo always checks out books that are exactly 150 pages. After reading his books, he went to the store and bought $15 worth of candy and had $3 leftover. How many books did Pablo read? | He gets $1.5 per book because .01 x 150 = <<.01*150=1.5>>1.5
He earned $18 because 15 + 3 = <<15+3=18>>18
He read 12 books because 18 / 1.5 = <<18/1.5=12>>12
#### 12 |
Evaluate the value of $3^2 \times 4 \times 6^3 \times 7!$. | 39191040 |
A one-way ticket costs $2. A 30-day pass costs $50. What's the minimum number of rides you will need to take every month so that the 30-day pass is strictly cheaper per ride? | For the 30-day pass to be cheaper, then the average fare should be cheaper than the $2 one-way ticket.
If the 30-day pass costs $50, then I should take at least $50/$2=<<50/2=25>>25 rides to pay the same as a one-way ticket
However, given that we want the 30-day pass per ride to be strictly cheaper we should ride at le... |
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 |
Nissa is calculating a jail sentence for a man who was convicted of burglary. The base sentence is 1 year of jail for every $5,000 of goods stolen. The burglar is going to have his sentence length increased by 25% because this is his third offense, plus 2 additional years for resisting arrest. How many years total is t... | First find how many years the base sentence will be: $40,000 / 1 year/$5,000 = <<40000/5000=8>>8 years
Then find how long the increase for the third offense is: 8 years * 25% = <<8*25*.01=2>>2 years
Then add the third-offense increase and the sentence for resisting arrest to the base sentence: 8 years + 2 years + 2 yea... |
Rewrite $\sqrt[3]{2^6\cdot3^3\cdot11^3}$ as an integer. | 132 |
How much money did you make if you sold 220 chocolate cookies at $1 per cookie and 70 vanilla cookies at $2 per cookie? | You sold 220 chocolate cookies at $1 per cookie earning 220 x 1 = $<<220*1=220>>220
You sold 70 vanilla cookies at $2 per cookie earning 70 x 2 =$<<70*2=140>>140.
In total you earned 220 + 140 = $<<220+140=360>>360
#### 360 |
Simplify the following expression: \[2x+3x^2+1-(6-2x-3x^2).\] | 6x^2+4x-5 |
Find $3^{\frac{1}{3}} \cdot 9^{\frac{1}{9}} \cdot 27^{\frac{1}{27}} \cdot 81^{\frac{1}{81}} \dotsm.$ | \sqrt[4]{27} |
Ray has 95 cents in nickels. If Ray gives 25 cents to Peter, and twice as many cents to Randi as he gave to Peter, how many nickels does Ray have left? | Ray gave 25*2 = <<25*2=50>>50 cents to Randi.
Ray has 95-25-50 = <<95-25-50=20>>20 cents in nickels left.
Ray has 20/5 = <<20/5=4>>4 nickels.
#### 4 |
The matrix
\[\begin{pmatrix} a & 3 \\ -8 & d \end{pmatrix}\]is its own inverse, for some real numbers $a$ and $d.$ Find the number of possible pairs $(a,d).$ | 2 |
Given that $a-b=5$ and $a^2+b^2=35$, find $a^3-b^3$. | 200 |
Let $\mathbf{P}$ be the matrix for projecting onto the vector $\begin{pmatrix} -3 \\ -2 \end{pmatrix}.$ Find $\mathbf{P}^{-1}.$
If the inverse does not exist, then enter the zero matrix. | \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix} |
The number of math problems that Marvin practiced today is three times as many as the number of problems he solved yesterday. His friend, Arvin, has practiced twice as many math problems on each day. How many math problems have they practiced altogether if Marvin solved 40 math problems yesterday? | If Marvin solved 40 math problems yesterday, he has solved 40*3 = <<40*3=120>>120 math problems today.
The total number of math problems that Marvin has practiced is 120+40 = <<120+40=160>>160
Arvin has practiced twice as many as Marvin, so has practiced 160 * 2 = <<160*2=320>>320 problems.
Together, Marvin and Arvin h... |
In the diagram, points \(B\), \(C\), and \(D\) lie on a line. Also, \(\angle ABC = 90^\circ\) and \(\angle ACD = 150^\circ\). The value of \(x\) is: | 60 |
What is the largest positive multiple of $12$ that is less than $350?$ | 348 |
Given that $n$ is a positive integer, and $4^7 + 4^n + 4^{1998}$ is a perfect square, then one value of $n$ is. | 3988 |
Determine the number of distinct odd numbers that can be formed by rearranging the digits of the number "34396". | 36 |
Let
\[x^8 - 98x^4 + 1 = p(x) q(x),\]where $p(x)$ and $q(x)$ are monic, non-constant polynomials with integer coefficients. Find $p(1) + q(1).$ | 4 |
Let $a_{1}, a_{2}, \ldots$ be a sequence defined by $a_{1}=a_{2}=1$ and $a_{n+2}=a_{n+1}+a_{n}$ for $n \geq 1$. Find $$\sum_{n=1}^{\infty} \frac{a_{n}}{4^{n+1}}$$ | \frac{1}{11} |
Ben rolls four fair 20-sided dice, and each of the dice has faces numbered from 1 to 20. What is the probability that exactly two of the dice show an even number? | \frac{3}{8} |
The floor plan of a castle wall is a regular pentagon with a side length of \( a (= 100 \text{ m}) \). The castle is patrolled by 3 guards along paths from which every point can see the base shape of the castle wall at angles of \( 90^\circ, 60^\circ, \) and \( 54^\circ \) respectively. Calculate the lengths of the pat... | 5.83 |
A four-digit integer $m$ and the four-digit integer obtained by reversing the order of the digits of $m$ are both divisible by 63. If $m$ is also divisible by 11, what is the greatest possible value of $m$? | 9702 |
Brady worked 6 hours every day in April. He worked 5 hours every day in June and 8 hours every day in September. What is the average amount of hours that Brady worked per month in those 3 months? | April = 6 * 30 = <<6*30=180>>180 hours
June = 5 * 30 = <<5*30=150>>150 hours
September = 8 * 30 = <<8*30=240>>240 hours
Total hours = 180 + 150 + 240 = <<180+150+240=570>>570 hours
570 hours/3 = <<570/3=190>>190 hours
He averaged working 190 hours per month.
#### 190 |
The positive integers \( r \), \( s \), and \( t \) have the property that \( r \times s \times t = 1230 \). What is the smallest possible value of \( r + s + t \)? | 52 |
Given that $x$, $y$, $z \in \mathbb{R}$, if $-1$, $x$, $y$, $z$, $-3$ form a geometric sequence, calculate the value of $xyz$. | -3\sqrt{3} |
Each row of a seating arrangement seats 7 or 8 people. Forty-six people are to be seated. How many rows seat exactly 8 people if every seat is occupied? | 4 |
Given a prime $p$ and an integer $a$, we say that $a$ is a $\textit{primitive root} \pmod p$ if the set $\{a,a^2,a^3,\ldots,a^{p-1}\}$ contains exactly one element congruent to each of $1,2,3,\ldots,p-1\pmod p$.
For example, $2$ is a primitive root $\pmod 5$ because $\{2,2^2,2^3,2^4\}\equiv \{2,4,3,1\}\pmod 5$, and th... | 8 |
In a Cartesian coordinate plane, call a rectangle $standard$ if all of its sides are parallel to the $x$ - and $y$ - axes, and call a set of points $nice$ if no two of them have the same $x$ - or $y$ - coordinate. First, Bert chooses a nice set $B$ of $2016$ points in the coordinate plane. To mess with Bert... | 2015 |
With all angles measured in degrees, the product $\prod_{k=1}^{45} \csc^2(2k-1)^\circ=m^n$, where $m$ and $n$ are integers greater than 1. Find $m+n$.
| 91 |
In the convex quadrilateral \(ABCD\): \(AB = AC = AD = BD\) and \(\angle BAC = \angle CBD\). Find \(\angle ACD\). | 60 |
A piece of cheese is located at $(12,10)$ in a coordinate plane. A mouse is at $(4,-2)$ and is running up the line $y=-5x+18$. At the point $(a,b)$ the mouse starts getting farther from the cheese rather than closer to it. What is $a + b$? | 10 |
Reese joined a Pinterest group where members contributed an average of 10 pins per day. The group owner deleted older pins at the rate of 5 pins per week per person. If the group has 20 people and the total number of pins is 1000, how many pins would be there after Reese had been a member for a month? | In a day, the total number of new pins is 10*20= <<10*20=200>>200.
There will be 30*200 = <<30*200=6000>>6000 new pins after a month.
The total number with old pins present is 6000+1000 = <<6000+1000=7000>>7000 pins
If the owner deletes five pins every week, the total number of deleted pins in a week is 5*20 = <<5*20=1... |
Severus Snape, the potions professor, prepared three potions, each in an equal volume of 400 ml. The first potion makes the drinker smarter, the second makes them more beautiful, and the third makes them stronger. To ensure the effect of any potion, it is sufficient to drink at least 30 ml of that potion. Snape intend... | 60 |
The force needed to loosen a bolt varies inversely with the length of the handle of the wrench used. A wrench with a handle length of 9 inches requires 375 pounds of force to loosen a certain bolt. A wrench of 15 inches will require how many pounds of force to loosen the same bolt? | 225 |
Four girls and eight boys came for a class photograph. Children approach the photographer in pairs and take a joint photo. Among how many minimum photos must there necessarily be either a photo of two boys, a photo of two girls, or two photos with the same children? | 33 |
In July 1861, $366$ inches of rain fell in Cherrapunji, India. What was the average rainfall in inches per hour during that month? | \frac{366}{31 \times 24} |
$44 \times 22$ is equal to | $88 \times 11$ |
For how many positive integers $x$ is $x^2 + 6x + 9$ between 20 and 40? | 2 |
If four different positive integers $m$, $n$, $p$, $q$ satisfy $(7-m)(7-n)(7-p)(7-q)=4$, find the value of $m+n+p+q$. | 28 |
Suppose that $wz = 12-8i$, and $|w| = \sqrt{13}$. What is $|z|$? | 4 |
The number of integers \(N\) from 1 to 1990 for which \(\frac{N^{2}+7}{N+4}\) is not a reduced fraction is: | 86 |
Evaluate $\left\lfloor |{-34.1}|\right\rfloor$. | 34 |
Given vectors $\overrightarrow {a}$ = (4, -7) and $\overrightarrow {b}$ = (3, -4), find the projection of $\overrightarrow {a}$ - $2\overrightarrow {b}$ in the direction of $\overrightarrow {b}$. | -2 |
My grandpa has 10 pieces of art, including 3 prints by Escher. If he hangs the pieces of art in a row in a random order, what is the probability that all three pieces by Escher will be placed consecutively? | \dfrac{1}{15} |
Say that a polynomial with real coefficients in two variables, $x,y$, is \emph{balanced} if
the average value of the polynomial on each circle centered at the origin is $0$.
The balanced polynomials of degree at most $2009$ form a vector space $V$ over $\mathbb{R}$.
Find the dimension of $V$. | 2020050 |
Find the product of all real values of $r$ for which $\frac{1}{2x}=\frac{r-x}{7}$ has exactly one real solution. | -14 |
On the board, two sums are written:
\[1+22+333+4444+55555+666666+7777777+88888888+999999999\]
\[9+98+987+9876+98765+987654+9876543+98765432+987654321\]
Determine which one is greater (or if they are equal). | 1097393685 |
If $x^{2y}= 4$ and $x = 4$, what is the value of $y$? Express your answer as a common fraction. | \frac{1}{2} |
We have $2022$ $1s$ written on a board in a line. We randomly choose a strictly increasing sequence from ${1, 2, . . . , 2022}$ such that the last term is $2022$ . If the chosen sequence is $a_1, a_2, ..., a_k$ ( $k$ is not fixed), then at the $i^{th}$ step, we choose the first a $_i$ numbers on the line a... | 1012 |
Find the number of six-digit palindromes. | 9000 |
Compute $1-2+3-4+ \dots -98+99$ . | 50 |
Given that \( M \) is a subset of \(\{1, 2, 3, \cdots, 15\}\) such that the product of any 3 distinct elements of \( M \) is not a perfect square, determine the maximum possible number of elements in \( M \). | 11 |
Simplify $\frac{x+1}{3}+\frac{2-3x}{2}$. Express your answer as a single fraction. | \frac{8-7x}{6} |
A square has vertices \( P, Q, R, S \) labelled clockwise. An equilateral triangle is constructed with vertices \( P, T, R \) labelled clockwise. What is the size of angle \( \angle RQT \) in degrees? | 135 |
Given triangle $ ABC$ of area 1. Let $ BM$ be the perpendicular from $ B$ to the bisector of angle $ C$ . Determine the area of triangle $ AMC$ . | \frac{1}{2} |
Suppose functions $g$ and $f$ have the properties that $g(x)=3f^{-1}(x)$ and $f(x)=\frac{24}{x+3}$. For what value of $x$ does $g(x)=15$? | 3 |
Dad says he is exactly 35 years old, not counting weekends. How old is he really? | 49 |
Given vectors $\overrightarrow{m}=(\sin x, -1)$ and $\overrightarrow{n}=(\sqrt{3}\cos x, -\frac{1}{2})$, let $f(x)=(\overrightarrow{m}+\overrightarrow{n})\cdot \overrightarrow{m}$.
(1) Find the analytic expression for $f(x)$ and its intervals of monotonic increase;
(2) Given that $a$, $b$, and $c$ are the sides opposit... | 2\sqrt{3} |
The integers \(a, b,\) and \(c\) satisfy the equations \(a + 5 = b\), \(5 + b = c\), and \(b + c = a\). What is the value of \(b\)? | -10 |
In a group of nine people each person shakes hands with exactly two of the other people from the group. Let $N$ be the number of ways this handshaking can occur. Consider two handshaking arrangements different if and only if at least two people who shake hands under one arrangement do not shake hands under the other ar... | 16 |
$\frac{2}{1-\frac{2}{3}}=$ | 6 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.