problem stringlengths 10 5.15k | answer stringlengths 0 1.23k |
|---|---|
Given lines $l_{1}$: $\rho\sin(\theta-\frac{\pi}{3})=\sqrt{3}$ and $l_{2}$: $\begin{cases} x=-t \\ y=\sqrt{3}t \end{cases}$ (where $t$ is a parameter), find the polar coordinates of the intersection point $P$ of $l_{1}$ and $l_{2}$. Additionally, three points $A$, $B$, and $C$ lie on the ellipse $\frac{x^{2}}{4}+y^{2}=... | \frac{15}{8} |
In the Cartesian coordinate system $xOy$, the parametric equation of curve $C_1$ is
\[
\begin{cases}
x=4t^2 \\
y=4t
\end{cases}
\]
where $t$ is the parameter. With the origin $O$ as the pole and the positive half-axis of $x$ as the polar axis, a polar coordinate system is established with the same unit length. The pol... | 16 |
A right pyramid has a square base that measures 10 cm on each side. Its peak is 12 cm above the center of its base. What is the sum of the lengths of the pyramid's eight edges? Express your answer to the nearest whole number.
[asy]
size(150);
draw((0,0)--(3,3)--(13,3)--(10,0)--cycle,linewidth(1));
draw((0,0)--(6.5,15)... | 96 |
From Sunday to Thursday, Prudence sleeps 6 hours a night. Friday and Saturday she sleeps for 9 hours a night. She also takes a 1-hour nap on Saturday and Sunday. How much sleep does she get in 4 weeks? | 5 nights a week she sleeps for 6 hours so she sleeps 5*6 = <<5*6=30>>30 hours
2 nights a week she sleeps for 9 hours so she sleeps 2*9 = <<2*9=18>>18 hours
2 days a week she naps for 1 hour so she sleeps 2*1 = <<2*1=2>>2 hours
Over 1 week she sleeps 30+18+2 = <<30+18+2=50>>50 hours
Over 4 weeks she sleeps 4*50 = <<4*50... |
Let $R$ be a unit square region and $n \geq 4$ an integer. A point $X$ in the interior of $R$ is called n-ray partitional if there are $n$ rays emanating from $X$ that divide $R$ into $n$ triangles of equal area. How many points are $100$-ray partitional but not $60$-ray partitional?
$\textbf{(A)}\ 1500 \qquad \textbf{... | 2320 |
A line is parameterized by
\[\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 1 \\ 1 \end{pmatrix} + t \begin{pmatrix} 2 \\ -3 \end{pmatrix}.\]A second line is parameterized by
\[\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 5 \\ -9 \end{pmatrix} + u \begin{pmatrix} 4 \\ 2 \end{pmatrix}.\]Find the point ... | \begin{pmatrix} 7 \\ -8 \end{pmatrix} |
Two unit squares are selected at random without replacement from an $n \times n$ grid of unit squares. Find the least positive integer $n$ such that the probability that the two selected unit squares are horizontally or vertically adjacent is less than $\frac{1}{2015}$. | 90 |
An amoeba reproduces by fission, splitting itself into two separate amoebae. An amoeba reproduces every two days. How many days will it take one amoeba to divide into 16 amoebae? | The amoeba will divide into 1 * 2 = <<1*2=2>>2 amoebae after 2 days.
The amoeba will divide into 2 * 2 = <<2*2=4>>4 amoebae after 4 days.
The amoeba will divide into 4 * 2 = <<4*2=8>>8 amoebae after 6 days.
The amoeba will divide into 8 * 2 = <<8*2=16>>16 amoebae after 8 days.
#### 8 |
Uncle Bradley has a $1000 bill that he wants to change into smaller bills so he could give them to his nieces and nephews. He wants to change 3/10 of the money into $50 bills while the rest into $100 bills. How many pieces of bills will Uncle Bradley have in all? | There will be $1000 x 3/10 = $<<1000*3/10=300>>300 worth of money that will be changed into $50 bills.
Uncle Bradley will have $300/$50 = <<300/50=6>>6 pieces of $50 bills.
He will change $1000 - $300 = $<<1000-300=700>>700 worth of money into $100 bills.
So there will be $700/$100 = <<700/100=7>>7 pieces of $100 bills... |
Given that $a$, $b$, $c$ are the opposite sides of the acute angles $A$, $B$, $C$ of triangle $\triangle ABC$, $\overrightarrow{m}=(3a,3)$, $\overrightarrow{n}=(-2\sin B,b)$, and $\overrightarrow{m} \cdot \overrightarrow{n}=0$.
$(1)$ Find $A$;
$(2)$ If $a=2$ and the perimeter of $\triangle ABC$ is $6$, find the area of... | 6 - 3\sqrt{3} |
Compute without using a calculator: $8!-7!$ | 35,\!280 |
Find the number of all trees planted at a five-foot distance from each other on a rectangular plot of land, the sides of which are 120 feet and 70 feet. | 375 |
The lengths of the sides of a triangle are $\sqrt{3}, \sqrt{4}(=2), \sqrt{5}$.
In what ratio does the altitude perpendicular to the middle side divide it? | 1:3 |
How many positive integers less than $250$ are multiples of $5$, but not multiples of $10$? | 25 |
The expansion of (ax- \frac {3}{4x}+ \frac {2}{3})(x- \frac {3}{x})^{6} is given, and the sum of its coefficients is 16. Determine the coefficient of the x^{3} term in this expansion. | \frac{117}{2} |
We define $|\begin{array}{l}{a}&{c}\\{b}&{d}\end{array}|=ad-bc$. For example, $|\begin{array}{l}{1}&{3}\\{2}&{4}\end{array}|=1\times 4-2\times 3=4-6=-2$. If $x$ and $y$ are integers, and satisfy $1 \lt |\begin{array}{l}{2}&{y}\\{x}&{3}\end{array}| \lt 3$, then the minimum value of $x+y$ is ____. | -5 |
Perry, Dana, Charlie, and Phil played golf together every week. At the end of the season, Perry had won five more games than Dana, but Charlie had won 2 games fewer than Dana. Phil had won 3 games more than Charlie did. If Phil won a total of 12 games, how many more games did Perry win than did Phil? | If Phil had won 3 games more than did Charlie, and Phil won 12 games, then Charlie won 12-3=<<12-3=9>>9 games.
If Charlie had won 2 games fewer than Dana, and Charlie won 9 games, then Dana won 9+2=<<9+2=11>>11 games.
If Perry had won five more games than did Dana, and Dana won 11 games, then Perry won 11+5=<<11+5=16>>... |
John places a total of 15 red Easter eggs in several green baskets and a total of 30 blue Easter eggs in some yellow baskets. Each basket contains the same number of eggs, and there are at least 3 eggs in each basket. How many eggs did John put in each basket? | 15 |
Triangle $ABC$ lies in the cartesian plane and has an area of $70$. The coordinates of $B$ and $C$ are $(12,19)$ and $(23,20),$ respectively, and the coordinates of $A$ are $(p,q).$ The line containing the median to side $BC$ has slope $-5.$ Find the largest possible value of $p+q.$
[asy]defaultpen(fontsize(8)); size(1... | 47 |
Given $\sin \alpha = \frac{3}{5}$ and $\cos (\alpha - \beta) = \frac{12}{13}$, where $0 < \alpha < \beta < \frac{\pi}{2}$, determine the value of $\sin \beta$. | \frac{56}{65} |
Given that there are 5 advertisements (3 commercial advertisements and 2 Olympic promotional advertisements), the last advertisement is an Olympic promotional advertisement, and the two Olympic promotional advertisements cannot be broadcast consecutively, determine the number of different broadcasting methods. | 36 |
How many integers between $100$ and $999$, inclusive, have the property that some permutation of its digits is a multiple of $11$ between $100$ and $999?$ For example, both $121$ and $211$ have this property.
$\mathrm{\textbf{(A)} \ }226\qquad \mathrm{\textbf{(B)} \ } 243 \qquad \mathrm{\textbf{(C)} \ } 270 \qquad \mat... | 226 |
Peter brought a bag of candies to ten friends and distributed them so that each received the same amount. Later, he realized that the bag contained the smallest possible number of candies that could also be distributed in such a way that each friend received a different (but non-zero) number of candies.
Determine how ... | 60 |
Elsa started the day with 40 marbles. At breakfast, she lost 3 marbles while playing. At lunchtime, she gave her best friend Susie 5 marbles. In the afternoon, Elsa's mom bought her a new bag with 12 marbles. Susie came back and gave Elsa twice as many marbles as she received at lunch. How many marbles did Elsa en... | Elsa had 40 marbles - 3 - 5 = <<40-3-5=32>>32 marbles.
Elsa then gained 12 marbles + 32 = <<12+32=44>>44 marbles.
At dinner Elsa received 2 * 5 marbles from Susie = <<2*5=10>>10 marbles.
In total Elsa had 44 marbles + 10 = <<44+10=54>>54 marbles.
#### 54 |
Pyramid $OABCD$ has square base $ABCD,$ congruent edges $\overline{OA}, \overline{OB}, \overline{OC},$ and $\overline{OD},$ and $\angle AOB=45^\circ.$ Let $\theta$ be the measure of the dihedral angle formed by faces $OAB$ and $OBC.$ Given that $\cos \theta=m+\sqrt{n},$ where $m_{}$ and $n_{}$ are integers, find $m+n.$ | 5 |
Jessica has three marbles colored red, green, and blue. She randomly selects a non-empty subset of them (such that each subset is equally likely) and puts them in a bag. You then draw three marbles from the bag with replacement. The colors you see are red, blue, red. What is the probability that the only marbles in the... | \frac{27}{35} |
Flights are arranged between 13 countries. For $ k\ge 2$ , the sequence $ A_{1} ,A_{2} ,\ldots A_{k}$ is said to a cycle if there exist a flight from $ A_{1}$ to $ A_{2}$ , from $ A_{2}$ to $ A_{3}$ , $ \ldots$ , from $ A_{k \minus{} 1}$ to $ A_{k}$ , and from $ A_{k}$ to $ A_{1}$ . What is the smallest... | 79 |
Compute without using a calculator: $12!/11!$ | 12 |
Given three composite numbers \( A, B, C \) that are pairwise coprime and \( A \times B \times C = 11011 \times 28 \). What is the maximum value of \( A + B + C \)? | 1626 |
Given the function
$$
f(x)=
\begin{cases}
1 & (1 \leq x \leq 2) \\
\frac{1}{2}x^2 - 1 & (2 < x \leq 3)
\end{cases}
$$
define $h(a) = \max\{f(x) - ax \mid x \in [1, 3]\} - \min\{f(x) - ax \mid x \in [1, 3]\}$ for any real number $a$.
1. Find the value of $h(0)$.
2. Find the expression for $h(a)$ and its minimum val... | \frac{5}{4} |
How many 3-letter words can we make from the letters A, B, C, and D, if we are allowed to repeat letters, and we must use the letter A at least once? (Here, a word is an arbitrary sequence of letters.) | 37 |
Let $A = (8,0,0),$ $B = (0,-4,0),$ $C = (0,0,6),$ and $D = (0,0,0).$ Find the point $P$ such that
\[AP = BP = CP = DP.\] | (4,-2,3) |
Let \( XYZ \) be a triangle with \( \angle X = 60^\circ \) and \( \angle Y = 45^\circ \). A circle with center \( P \) passes through points \( A \) and \( B \) on side \( XY \), \( C \) and \( D \) on side \( YZ \), and \( E \) and \( F \) on side \( ZX \). Suppose \( AB = CD = EF \). Find \( \angle XPY \) in degrees. | 120 |
Given that the sequence $\{a_n\}$ is a geometric sequence, and the sequence $\{b_n\}$ is an arithmetic sequence. If $a_1-a_6-a_{11}=-3\sqrt{3}$ and $b_1+b_6+b_{11}=7\pi$, then the value of $\tan \frac{b_3+b_9}{1-a_4-a_3}$ is ______. | -\sqrt{3} |
Find the number of positive integers $n \le 1000$ that can be expressed in the form
\[\lfloor x \rfloor + \lfloor 2x \rfloor + \lfloor 3x \rfloor = n\]for some real number $x.$ | 667 |
Factor $w^4-16$ as far as possible, where the factors are monic polynomials with real coefficients. | (w-2)(w+2)(w^2+4) |
John releases 3 videos on his channel a day. Two of them are short 2 minute videos and 1 of them is 6 times as long. Assuming a 7-day week, how many minutes of video does he release per week? | He releases 1 video that is 2*6=12 minutes long
So each day he releases 2+2+12=<<2+2+12=16>>16 minutes of videos
That means he releases 16*7=<<16*7=112>>112 minutes of videos
#### 112 |
The volume of a cube in cubic meters and its surface area in square meters is numerically equal to four-thirds of the sum of the lengths of its edges in meters. What is the total volume in cubic meters of twenty-seven such cubes? | 216 |
A list of five positive integers has all of the following properties:
$\bullet$ The only integer in the list that occurs more than once is $8,$
$\bullet$ its median is $9,$ and
$\bullet$ its average (mean) is $10.$
What is the largest possible integer that could appear in the list? | 15 |
Evaluate $\left\lceil3\left(6-\frac12\right)\right\rceil$. | 17 |
How many ways can the eight vertices of a three-dimensional cube be colored red and blue such that no two points connected by an edge are both red? Rotations and reflections of a given coloring are considered distinct. | 35 |
The minimum value of the function $y = \sin 2 \cos 2x$ is ______. | - \frac{1}{2} |
Solve
\[(x^3 + 3x^2 \sqrt{2} + 6x + 2 \sqrt{2}) + (x + \sqrt{2}) = 0.\]Enter all the solutions, separated by commas. | -\sqrt{2}, -\sqrt{2} + i, -\sqrt{2} - i |
The integers $1,2,4,5,6,9,10,11,13$ are to be placed in the circles and squares below with one number in each shape. Each integer must be used exactly once and the integer in each circle must be equal to the sum of the integers in the two neighbouring squares. If the integer $x$ is placed in the leftmost square and the... | 20 |
Coach Grunt is preparing the 5-person starting lineup for his basketball team, the Grunters. There are 12 players on the team. Two of them, Ace and Zeppo, are league All-Stars, so they'll definitely be in the starting lineup. How many different starting lineups are possible? (The order of the players in a basketbal... | 120 |
An acute isosceles triangle, $ABC$, is inscribed in a circle. Through $B$ and $C$, tangents to the circle are drawn, meeting at point $D$. If $\angle ABC = \angle ACB = 3 (\angle D$) and $\angle BAC = t \pi$ in radians, then find $t$.
[asy]
import graph;
unitsize(2 cm);
pair O, A, B, C, D;
O = (0,0);
A = dir(90);
B... | \frac{5}{11} |
Find the product of all integer divisors of $105$ that also divide $14$. (Recall that the divisors of an integer may be positive or negative.) | 49 |
Given the function $f(x)=2\cos^2\frac{x}{2}+\sin x-1$. Find:
- $(Ⅰ)$ The minimum positive period, monotonic decreasing interval, and symmetry center of $f(x)$.
- $(Ⅱ)$ When $x\in \left[-\pi ,0\right]$, find the minimum value of $f(x)$ and the corresponding value of $x$. | -\frac{3\pi}{4} |
A store sells a batch of football souvenir books, with a cost price of $40$ yuan per book and a selling price of $44$ yuan per book. The store can sell 300 books per day. The store decides to increase the selling price, and after investigation, it is found that for every $1$ yuan increase in price, the daily sales decr... | 2640 |
Jarris the triangle is playing in the \((x, y)\) plane. Let his maximum \(y\) coordinate be \(k\). Given that he has side lengths 6, 8, and 10 and that no part of him is below the \(x\)-axis, find the minimum possible value of \(k\). | 24/5 |
An eagle can fly 15 miles per hour; a falcon can fly 46 miles per hour; a pelican can fly 33 miles per hour; a hummingbird can fly 30 miles per hour. If the eagle, the falcon, the pelican, and the hummingbird flew for 2 hours straight, how many miles in total did the birds fly? | An eagle can fly 15 x 2 = <<15*2=30>>30 miles for 2 hours.
A falcon can fly 46 x 2 = <<46*2=92>>92 miles for 2 hours.
A pelican can fly 33 x 2 = <<33*2=66>>66 miles for 2 hours.
A hummingbird can fly 30 x 2 = <<30*2=60>>60 miles for 2 hours.
Therefore, the birds flew a total of 30 + 92 + 66 + 60 = <<30+92+66+60=248>>24... |
Tom dances 4 times a week for 2 hours at a time and does this every year for 10 years. How many hours did he dance? | He dances 4*2=<<4*2=8>>8 hours per week
He dances for 52*10=<<52*10=520>>520 weeks
So in total he danced for 8*520=<<8*520=4160>>4160 hours
#### 4160 |
Let $p(x)$ be a polynomial of degree strictly less than $100$ and such that it does not have $(x^3-x)$ as a factor. If $$ \frac{d^{100}}{dx^{100}}\bigg(\frac{p(x)}{x^3-x}\bigg)=\frac{f(x)}{g(x)} $$ for some polynomials $f(x)$ and $g(x)$ then find the smallest possible degree of $f(x)$ . | 200 |
What is the measure of the smaller angle between the hands of a 12-hour clock at 12:25 pm, in degrees? Express your answer as a decimal to the nearest tenth. | 137.5\text{ degrees} |
A right rectangular prism has edge lengths $\log_{5}x, \log_{8}x,$ and $\log_{10}x.$ Given that the sum of its surface area and volume is twice its volume, find the value of $x$.
A) $1,000,000$
B) $10,000,000$
C) $100,000,000$
D) $1,000,000,000$
E) $10,000,000,000$ | 100,000,000 |
Let $f(x) = 4\cos(wx+\frac{\pi}{6})\sin(wx) - \cos(2wx) + 1$, where $0 < w < 2$.
1. If $x = \frac{\pi}{4}$ is a symmetry axis of the function $f(x)$, find the period $T$ of the function.
2. If the function $f(x)$ is increasing on the interval $[-\frac{\pi}{6}, \frac{\pi}{3}]$, find the maximum value of $w$. | \frac{3}{4} |
Let $L$ be the intersection point of the diagonals $C E$ and $D F$ of a regular hexagon $A B C D E F$ with side length 4. The point $K$ is defined such that $\overrightarrow{L K}=3 \overrightarrow{F A}-\overrightarrow{F B}$. Determine whether $K$ lies inside, on the boundary, or outside of $A B C D E F$, and find the l... | \frac{4 \sqrt{3}}{3} |
Given real numbers $a$ and $b$ satisfying the equation $\sqrt{(a-1)^2} + \sqrt{(a-6)^2} = 10 - |b+3| - |b-2|$, find the maximum value of $a^2 + b^2$. | 45 |
Eighty percent of adults drink coffee and seventy percent drink tea. What is the smallest possible percent of adults who drink both coffee and tea? | 50\% |
Given an ellipse $C$: $\frac{{x}^{2}}{3}+{y}^{2}=1$ with left focus and right focus as $F_{1}$ and $F_{2}$ respectively, the line $y=x+m$ intersects $C$ at points $A$ and $B$. Determine the value of $m$ such that the area of $\triangle F_{1}AB$ is twice the area of $\triangle F_{2}AB$. | -\frac{\sqrt{2}}{3} |
Gina is figuring out how much she'll have to spend on college this year. She's taking 14 credits that cost $450 each, and she has to pay $120 for each of her 5 textbooks plus a $200 facilities fee. How much will Gina spend in total? | First, find out how much will Gina pay for tuition by multiplying her number of credits by the cost per credit: 14 * 450 = $<<14*450=6300>>6300.
Then find out how much she pays for textbooks by multiplying the number of books by the cost per book: 5 * 120 = $<<5*120=600>>600.
Finally, add up all Gina's costs to find th... |
Given that $\triangle ABC$ is an acute triangle, vector $\overrightarrow{m}=(\cos (A+ \frac{\pi}{3}),\sin (A+ \frac{\pi}{3}))$, $\overrightarrow{n}=(\cos B,\sin B)$, and $\overrightarrow{m} \perp \overrightarrow{n}$.
(I) Find the value of $A-B$;
(II) If $\cos B= \frac{3}{5}$, and $AC=8$, find the length of $BC$. | 4\sqrt{3}+3 |
Given a pair of concentric circles, chords $AB,BC,CD,\dots$ of the outer circle are drawn such that they all touch the inner circle. If $\angle ABC = 75^{\circ}$ , how many chords can be drawn before returning to the starting point ?
 | 24 |
Let $(x_1,y_1),$ $(x_2,y_2),$ $\dots,$ $(x_n,y_n)$ be the solutions to
\begin{align*}
|x - 5| &= |y - 12|, \\
|x - 12| &= 3|y - 5|.
\end{align*}
Find $x_1 + y_1 + x_2 + y_2 + \dots + x_n + y_n.$ | 70 |
If a podcast series that lasts for 837 minutes needs to be stored on CDs and each CD can hold up to 75 minutes of audio, determine the number of minutes of audio that each CD will contain. | 69.75 |
Find all ordered pairs of integers $(x, y)$ such that $3^{x} 4^{y}=2^{x+y}+2^{2(x+y)-1}$. | (0,1), (1,1), (2,2) |
What is the sum and product of all values of $x$ such that $x^2 = 18x - 16$? | 16 |
Given that the function $y=f(x)$ is an odd function defined on $R$, when $x\leqslant 0$, $f(x)=2x+x^{2}$. If there exist positive numbers $a$ and $b$ such that when $x\in[a,b]$, the range of $f(x)$ is $[\frac{1}{b}, \frac{1}{a}]$, find the value of $a+b$. | \frac{3+ \sqrt{5}}{2} |
Bonnie makes the frame of a cube out of 12 pieces of wire that are each six inches long. Meanwhile Roark uses 1-inch-long pieces of wire to make a collection of unit cube frames that are not connected to each other. The total volume of Roark's cubes is the same as the volume of Bonnie's cube. What is the ratio of the t... | \dfrac{1}{36} |
Square $ABCD$ has side length $1$ unit. Points $E$ and $F$ are on sides $AB$ and $CB$, respectively, with $AE = CF$. When the square is folded along the lines $DE$ and $DF$, sides $AD$ and $CD$ coincide and lie on diagonal $BD$. The length of segment $AE$ can be expressed in the form $\sqrt{k}-m$ units. What is the ... | 3 |
(1) When $x \in \left[\frac{\pi}{6}, \frac{7\pi}{6}\right]$, find the maximum value of the function $y = 3 - \sin x - 2\cos^2 x$.
(2) Given that $5\sin\beta = \sin(2\alpha + \beta)$ and $\tan(\alpha + \beta) = \frac{9}{4}$, find $\tan \alpha$. | \frac{3}{2} |
At the beginning of a trip, the mileage odometer read $56,200$ miles. The driver filled the gas tank with $6$ gallons of gasoline. During the trip, the driver filled his tank again with $12$ gallons of gasoline when the odometer read $56,560$. At the end of the trip, the driver filled his tank again with $20$ gallons o... | 26.9 |
Find the number of ways to distribute 4 pieces of candy to 12 children such that no two consecutive children receive candy. | 105 |
A right pyramid has a square base where each side measures 15 cm. The height of the pyramid, measured from the center of the base to the peak, is 15 cm. Calculate the total length of all edges of the pyramid. | 60 + 4\sqrt{337.5} |
A wire is cut into two pieces, one of length $a$ and the other of length $b$. The piece of length $a$ is bent to form an equilateral triangle, and the piece of length $b$ is bent to form a regular hexagon. The triangle and the hexagon have equal area. What is $\frac{a}{b}$? | \frac{\sqrt6}2 |
Let \[f(x) =
\begin{cases}
2x + 9 &\text{if }x<-2, \\
5-2x&\text{if }x\ge -2.
\end{cases}
\]Find $f(3).$ | -1 |
Given the random variable $\eta\sim B(n,p)$, and $E(2\eta)=8$, $D(4\eta)=32$, find the respective values of $n$ and $p$. | 0.5 |
Calculate the sum of the repeating decimals $0.\overline{2}$, $0.\overline{02}$, and $0.\overline{0002}$ as a common fraction. | \frac{224422}{9999} |
Calculate the value of \[\cot(\cot^{-1}5 + \cot^{-1}11 + \cot^{-1}17 + \cot^{-1}23).\] | \frac{97}{40} |
Solve for $x$: $\frac{5x}{(x+3)} - \frac3{(x+3)} = \frac1{(x+3)}$ | \frac{4}{5} |
What is the slope of the line passing through $(-3,5)$ and $(2,-5)$? | -2 |
Find the number of square units in the area of the triangle.
[asy]size(125);
draw( (-10,-2) -- (2,10), Arrows);
draw( (0,-2)-- (0,10) ,Arrows);
draw( (5,0) -- (-10,0),Arrows);
label("$l$",(2,10), NE);
label("$x$", (5,0) , E);
label("$y$", (0,-2) , S);
filldraw( (-8,0) -- (0,8) -- (0,0) -- cycle, lightgray);
dot( (-2,... | 32 |
Twenty points are equally spaced around the circumference of a circle. Kevin draws all the possible chords that connect pairs of these points. How many of these chords are longer than the radius of the circle but shorter than its diameter? | 120 |
What is the value of \(1.90 \frac{1}{1-\sqrt[4]{3}}+\frac{1}{1+\sqrt[4]{3}}+\frac{2}{1+\sqrt{3}}\)? | -2 |
If a regular polygon has a total of nine diagonals, how many sides does it have? | 6 |
Consider two lines: line $l$ parametrized as
\begin{align*}
x &= 1 + 4t,\\
y &= 4 + 3t
\end{align*}and the line $m$ parametrized as
\begin{align*}
x &=-5 + 4s\\
y &= 6 + 3s.
\end{align*}Let $A$ be a point on line $l$, $B$ be a point on line $m$, and let $P$ be the foot of the perpendicular from $A$ to line $m$.
T... | \begin{pmatrix}-6 \\ 8 \end{pmatrix} |
James replaces the coffee for the household. There are 3 other people in the house and everyone drinks 2 cups of coffee a day. It takes .5 ounces of coffee per cup of coffee. If coffee costs $1.25 an ounce, how much does he spend on coffee a week? | There are 3+1=<<3+1=4>>4 people in the house
So they drink 4*2=<<4*2=8>>8 cups of coffee a day
That means they use 8*.5=<<8*.5=4>>4 ounces of coffee a day
So they use 4*7=<<4*7=28>>28 ounces of coffee a week
So he spends 28*1.25=$<<28*1.25=35>>35
#### 35 |
The number of solutions in positive integers of $2x+3y=763$ is: | 127 |
Define a function $g :\mathbb{N} \rightarrow \mathbb{R}$ Such that $g(x)=\sqrt{4^x+\sqrt {4^{x+1}+\sqrt{4^{x+2}+...}}}$ .
Find the last 2 digits in the decimal representation of $g(2021)$ . | 53 |
Find the focus of the parabola $y = -3x^2 - 6x.$ | \left( -1, \frac{35}{12} \right) |
A school is adding 5 rows of seats to the auditorium. Each row has 8 seats and each seat costs $30. A parent, being a seat manufacturer, offered a 10% discount on each group of 10 seats purchased. How much will the school pay for the new seats? | Ten seats amount to $30 x 10 = $<<30*10=300>>300.
So there is $300 x 10/100 = $<<300*10/100=30>>30 discount for each 10 seats purchased.
Thus, the total cost for every 10 seats is $300 - $30 = $<<300-30=270>>270.
The school is going to buy 5 x 8 = <<5*8=40>>40 seats.
This means that the school is going to buy 40/10 = <... |
The sum of the first n terms of the sequence {a_n} is S_n = n^2 + n + 1, and b_n = (-1)^n a_n (n ∈ N^*). Determine the sum of the first 50 terms of the sequence {b_n}. | 49 |
Let \(r(x)\) have a domain of \(\{-2,-1,0,1\}\) and a range of \(\{-1,0,2,3\}\). Let \(t(x)\) have a domain of \(\{-1,0,1,2,3\}\) and be defined as \(t(x) = 2x + 1\). Furthermore, \(s(x)\) is defined on the domain \(\{1, 2, 3, 4, 5, 6\}\) by \(s(x) = x + 2\). What is the sum of all possible values of \(s(t(r(x)))\)? | 10 |
A right square pyramid with base edges of length $8\sqrt{2}$ units each and slant edges of length 10 units each is cut by a plane that is parallel to its base and 3 units above its base. What is the volume, in cubic units, of the new pyramid that is cut off by this plane? [asy]
import three;
size(2.5inch);
currentproje... | 32 |
Define $H_n = 1+\frac{1}{2}+\cdots+\frac{1}{n}$ . Let the sum of all $H_n$ that are terminating in base 10 be $S$ . If $S = m/n$ where m and n are relatively prime positive integers, find $100m+n$ .
*Proposed by Lewis Chen* | 9920 |
Determine the least positive period $q$ of the functions $g$ such that $g(x+2) + g(x-2) = g(x)$ for all real $x$. | 12 |
Write the process of using the Horner's algorithm to find the value of the function $\_(f)\_()=1+\_x+0.5x^2+0.16667x^3+0.04167x^4+0.00833x^5$ at $x=-0.2$. | 0.81873 |
For $n$ a positive integer, let $R(n)$ be the sum of the remainders when $n$ is divided by $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, and $10$. For example, $R(15) = 1+0+3+0+3+1+7+6+5=26$. How many two-digit positive integers $n$ satisfy $R(n) = R(n+1)\,?$ | 2 |
Two children at a time can play pairball. For $90$ minutes, with only two children playing at time, five children take turns so that each one plays the same amount of time. The number of minutes each child plays is | 36 |
Find the largest positive number \( c \) such that for every natural number \( n \), the inequality \( \{n \sqrt{2}\} \geqslant \frac{c}{n} \) holds, where \( \{n \sqrt{2}\} = n \sqrt{2} - \lfloor n \sqrt{2} \rfloor \) and \( \lfloor x \rfloor \) denotes the integer part of \( x \). Determine the natural number \( n \)... | \frac{1}{2\sqrt{2}} |
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