problem stringlengths 10 5.15k | answer stringlengths 0 1.23k |
|---|---|
A park warden has issued 24 citations over the past three hours. He issued the same number for littering as he did for off-leash dogs, and he issued double the number of other citations for parking fines. How many littering citations did the warden issue? | Let L be the number of littering citations the warden issued.
He issued 2L citations for littering and off-leash dogs, and 2 * 2L = <<2*2=4>>4L citations for parking fines.
In all, he issued 2L + 4L = 6L = 24 citations.
Thus, he issued L = 24 / 6 = <<24/6=4>>4 citations for littering.
#### 4 |
How many 10-digit positive integers have all digits either 1 or 2, and have two consecutive 1's?
| 880 |
Marina solved the quadratic equation $9x^2-18x-720=0$ by completing the square. In the process, she came up with the equivalent equation $$(x+r)^2 = s,$$where $r$ and $s$ are constants.
What is $s$? | 81 |
Let $f(x)$ be a polynomial of degree 2006 with real coefficients, and let its roots be $r_1,$ $r_2,$ $\dots,$ $r_{2006}.$ There are exactly 1006 distinct values among
\[|r_1|, |r_2|, \dots, |r_{2006}|.\]What is the minimum number of real roots that $f(x)$ can have? | 6 |
Compute the multiplicative inverse of $201$ modulo $299$. Express your answer as an integer from $0$ to $298$. | 180 |
Given that vehicles are not allowed to turn back at a crossroads, calculate the total number of driving routes. | 12 |
Let $ABCD$ be a convex quadrilateral with $AB = CD = 10$, $BC = 14$, and $AD = 2\sqrt{65}$. Assume that the diagonals of $ABCD$ intersect at point $P$, and that the sum of the areas of triangles $APB$ and $CPD$ equals the sum of the areas of triangles $BPC$ and $APD$. Find the area of quadrilateral $ABCD$. | 70 |
Let $a$ and $b$ be positive real numbers with $a\ge b$. Let $\rho$ be the maximum possible value of $\frac {a}{b}$ for which the system of equations $$
a^2 + y^2 = b^2 + x^2 = (a - x)^2 + (b - y)^2
$$has a solution in $(x,y)$ satisfying $0\le x < a$ and $0\le y < b$. Find $\rho^2.$ | \frac{4}{3} |
In triangle $\triangle ABC$, a line passing through the midpoint $E$ of the median $AD$ intersects sides $AB$ and $AC$ at points $M$ and $N$ respectively. Let $\overrightarrow{AM} = x\overrightarrow{AB}$ and $\overrightarrow{AN} = y\overrightarrow{AC}$ ($x, y \neq 0$), then the minimum value of $4x+y$ is \_\_\_\_\_\_. | \frac{9}{4} |
Two real numbers $x$ and $y$ are such that $8 y^{4}+4 x^{2} y^{2}+4 x y^{2}+2 x^{3}+2 y^{2}+2 x=x^{2}+1$. Find all possible values of $x+2 y^{2}$. | \frac{1}{2} |
In the number \(2016 * * * * 02 *\), each of the 5 asterisks needs to be replaced with any of the digits \(0, 2, 4, 7, 8, 9\) (digits can be repeated) so that the resulting 11-digit number is divisible by 6. In how many ways can this be done? | 1728 |
Peter has $2022$ pieces of magnetic railroad cars, which are of two types: some have the front with north and the rear with south magnetic polarity, and some have the rear with north and the rear with south magnetic polarity (on these railroad cars the front and the rear can be distinguished). Peter wants to decide whe... | 2021 |
What is the greatest number of points of intersection that can occur when $2$ different circles and $2$ different straight lines are drawn on the same piece of paper? | 11 |
A teacher finds that when she offers candy to her class of 30 students, the mean number of pieces taken by each student is 5. If every student takes some candy, what is the greatest number of pieces one student could have taken? | 121 |
It is known that
$$
\sqrt{9-8 \sin 50^{\circ}}=a+b \sin c^{\circ}
$$
for exactly one set of positive integers \((a, b, c)\), where \(0 < c < 90\). Find the value of \(\frac{b+c}{a}\). | 14 |
Find the remainder when $6x^4-14x^3-4x^2+2x-26$ is divided by $2x - 6.$ | 52 |
Real numbers $X_1, X_2, \dots, X_{10}$ are chosen uniformly at random from the interval $[0,1]$ . If the expected value of $\min(X_1,X_2,\dots, X_{10})^4$ can be expressed as a rational number $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$ , what is $m+n$ ?
*2016 CCA Math Bonanza Lightning... | 1002 |
How many distinct arrangements of the letters in the word "balloon" are there? | 1260 |
There are 20 chairs arranged in a circle. There are \(n\) people sitting in \(n\) different chairs. These \(n\) people stand, move \(k\) chairs clockwise, and then sit again. After this happens, exactly the same set of chairs is occupied. For how many pairs \((n, k)\) with \(1 \leq n \leq 20\) and \(1 \leq k \leq 20\) ... | 72 |
Find the $\emph{positive}$ real number(s) $x$ such that $\frac{1}{2}\left( 3x^2-1\right) = \left( x^2-50x-10\right)\left( x^2+25x+5\right)$. | 25 + 2\sqrt{159} |
Sandro has six times as many daughters as sons. If he currently has three sons, how many children does he have? | If Sandro has three sons, the total number of daughters, six times the number of sons, is 3 sons * 6 daughters/son = <<3*6=18>>18 daughters.
The total number of children is 18 daughters + 3 sons = <<18+3=21>>21 children.
#### 21 |
John took a test with 80 questions. For the first 40 questions, she got 90% right. For the next 40 questions, she gets 95% right. How many total questions does she get right? | He gets 40*.9=<<40*.9=36>>36 questions right for the first part.
For the second part, he gets 40*.95=<<40*.95=38>>38 questions right.
So he gets a total of 36+38=<<36+38=74>>74 questions right.
#### 74 |
Let S$_{n}$ denote the sum of the first $n$ terms of the arithmetic sequence {a$_{n}$} with a common difference d=2. The terms a$_{1}$, a$_{3}$, and a$_{4}$ form a geometric sequence. Find the value of S$_{8}$. | -8 |
If the least common multiple of $A$ and $B$ is $120$, and the ratio of $A$ to $B$ is $3:4$, then what is their greatest common divisor? | 10 |
Sharik and Matroskin ski on a circular track, half of which is an uphill slope and the other half is a downhill slope. Their speeds are identical on the uphill slope and are four times less than their speeds on the downhill slope. The minimum distance Sharik falls behind Matroskin is 4 km, and the maximum distance is 1... | 24 |
The smallest sum one could get by adding three different numbers from the set $\{ 7,25,-1,12,-3 \}$ is | 3 |
The price of two kilograms of sugar and five kilograms of salt is $5.50. If a kilogram of sugar costs $1.50, then how much is the price of three kilograms of sugar and a kilogram of salt? | Two kilograms of sugar is priced at $1.50 x 2 = $<<2*1.5=3>>3.
Three kilograms of sugar is priced at $1.50 x 3 = $<<3*1.5=4.50>>4.50.
So five kilograms of salt is priced at $5.50 - $3 = $2.50.
Thus, a kilogram of salt is priced at $2.50/5 = $<<2.5/5=0.50>>0.50.
Hence, three kilograms of sugar and a kilogram of salt.s p... |
A child lines up $2020^2$ pieces of bricks in a row, and then remove bricks whose positions are square numbers (i.e. the 1st, 4th, 9th, 16th, ... bricks). Then he lines up the remaining bricks again and remove those that are in a 'square position'. This process is repeated until the number of bricks remaining drops b... | 240 |
A number is called *6-composite* if it has exactly 6 composite factors. What is the 6th smallest 6-composite number? (A number is *composite* if it has a factor not equal to 1 or itself. In particular, 1 is not composite.)
*Ray Li.* | 441 |
On a plane, 6 lines intersect pairwise, but only three pass through the same point. Find the number of non-overlapping line segments intercepted. | 21 |
Sir Alex plays the following game on a row of 9 cells. Initially, all cells are empty. In each move, Sir Alex is allowed to perform exactly one of the following two operations:
[list=1]
[*] Choose any number of the form $2^j$, where $j$ is a non-negative integer, and put it into an empty cell.
[*] Choose two (not neces... | 2 \sum_{i=0}^{8} \binom{n}{i} - 1 |
Calculate the volume of a tetrahedron with vertices at points $A_{1}, A_{2}, A_{3}, A_{4}$, and its height dropped from vertex $A_{4}$ to the face $A_{1} A_{2} A_{3}$.
$A_{1}(-2, 0, -4)$
$A_{2}(-1, 7, 1)$
$A_{3}(4, -8, -4)$
$A_{4}(1, -4, 6)$ | 5\sqrt{2} |
Leo has to write a 400-word story for his literature class. 10 words fit on each line of his notebook and 20 lines fit on each page. Lucas has filled one and a half pages. How many words does he have left to write? | Half a page is 20 / 2 =<<20/2=10>>10 lines.
In total, he has written 20 + 10 = <<20+10=30>>30 lines.
The 30 lines written are 30 * 10 = <<30*10=300>>300 words.
Lucas has 400 - 300 = <<400-300=100>>100 words left to write.
#### 100 |
Studying for her test, Mitchell had read ten chapters of a book before 4 o'clock. When it clocked 4, Mitchell had read 20 pages of the 11th chapter of the book she was studying from. After 4 o'clock, she didn't read the remaining pages of chapter eleven but proceeded and read 2 more chapters of the book. If each chapte... | Since each chapter of the book has 40 pages, Mitchell had read 10*40 = <<10*40=400>>400 pages from the first ten chapters.
After reading 20 pages of the eleventh chapter, the total number of pages that Mitchell had read is 400+20 = <<400+20=420>>420
The next two chapters that she read had 2*40 = <<2*40=80>>80 pages.
In... |
Consider a $10 \times 10$ grid of squares. One day, Daniel drops a burrito in the top left square, where a wingless pigeon happens to be looking for food. Every minute, if the pigeon and the burrito are in the same square, the pigeon will eat $10 \%$ of the burrito's original size and accidentally throw it into a rando... | 71.8 |
The graph of $y=ax^2+bx+c$ is given below, where $a$, $b$, and $c$ are integers. Find $a-b+c$.
[asy]
size(150);
Label f;
f.p=fontsize(4);
xaxis(-3,3,Ticks(f, 1.0));
yaxis(-4,4,Ticks(f, 1.0));
real f(real x)
{
return x^2+2x-1;
}
draw(graph(f,-2.7,.7),linewidth(1),Arrows(6));
[/asy] | -2 |
George collected 50 marbles in white, yellow, green, and red. Half of them are white, and 12 are yellow. There are 50% fewer green balls than yellow balls. How many marbles are red? | Half of the marbles are white, so there are 50 / 2 = <<50/2=25>>25 white marbles.
Green marbles are only 50% as many as yellow marbles, so there are 50/100 * 12 = <<50/100*12=6>>6 green marbles.
So currently we know that we have 25 + 12 + 6 = <<25+12+6=43>>43 marbles.
The rest are red, so there are 50 - 43 = <<50-43=7>... |
Calculate $\sin 9^\circ \sin 45^\circ \sin 69^\circ \sin 81^\circ.$ | \frac{0.6293 \sqrt{2}}{4} |
How many different routes are there from point $A$ to point $B$ in a 3x3 grid (where you can only move to the right or down along the drawn segments)?
[asy]
unitsize(0.09inch);
draw((0,0)--(15,0)--(15,15)--(0,15)--cycle);
draw((5,0)--(5,15));
draw((10,0)--(10,15));
draw((0,5)--(15,5));
draw((0,10)--(15,10));
dot((0,15... | 20 |
Given that three roots of $f(x) = x^{4} + ax^{2} + bx + c$ are $2, -3$, and $5$, what is the value of $a + b + c$? | 79 |
Find the angle $D A C$ given that $A B = B C$ and $A C = C D$, and the lines on which points $A, B, C, D$ lie are parallel with equal distances between adjacent lines. Point $A$ is to the left of $B$, $C$ is to the left of $B$, and $D$ is to the right of $C$. | 30 |
Given Farmer Euclid has a field in the shape of a right triangle with legs of lengths $5$ units and $12$ units, and he leaves a small unplanted square of side length S in the corner where the legs meet at a right angle, where $S$ is $3$ units from the hypotenuse, calculate the fraction of the field that is planted. | \frac{431}{480} |
To take quizzes, each of 30 students in a class is paired with another student. If the pairing is done randomly, what is the probability that Margo is paired with her best friend, Irma? Express your answer as a common fraction. | \frac{1}{29} |
Let $ABC$ be a triangle with side lengths $AB=6, AC=7,$ and $BC=8.$ Let $H$ be the orthocenter of $\triangle ABC$ and $H'$ be the reflection of $H$ across the midpoint $M$ of $BC.$ $\tfrac{[ABH']}{[ACH']}$ can be expressed as $\frac{p}{q}$ . Find $p+q$ .
*2022 CCA Math Bonanza Individual Round #1... | 251 |
For any real number \(x\), \(\lfloor x \rfloor\) denotes the largest integer less than or equal to \(x\). For example, \(\lfloor 4.2 \rfloor = 4\) and \(\lfloor 0.9 \rfloor = 0\).
If \(S\) is the sum of all integers \(k\) with \(1 \leq k \leq 999999\) and for which \(k\) is divisible by \(\lfloor \sqrt{k} \rfloor\), t... | 999999000 |
Martin has 18 goldfish. Each week 5 goldfish die. Martin purchases 3 new goldfish every week. How many goldfish will Martin have in 7 weeks? | In 7 weeks 5*7 = <<35=35>>35 goldfish die.
In 7 weeks, Martin purchases 3*7 = <<3*7=21>>21 goldfish.
In 7 weeks, Martin will have 18-35+21 = <<18-35+21=4>>4 goldfish.
#### 4 |
A casino table pays 3:2 if you get a blackjack with your first 2 cards. If you scored a blackjack and were paid $60, what was your original bet? | The payout for blackjack is 3:2 or 3/2=1.5 times my bet
If my original bet was X then I won 1.5*X my bet
Given that I was paid $60 we know $60 = 1.5X
X=$60/1.5
X=$<<40=40>>40 was my original bet
#### 40 |
For each positive integer $n$, let $S(n)$ denote the sum of the digits of $n$. For how many values of $n$ is $n+S(n)+S(S(n))=2007$? | 4 |
For what values of $x$ is \[\frac{x-10x^2+25x^3}{8-x^3}\]nonnegative? Answer as an interval. | [0,2) |
Three friends are in the park. Bob and Clarise are standing at the same spot and Abe is standing 10 m away. Bob chooses a random direction and walks in this direction until he is 10 m from Clarise. What is the probability that Bob is closer to Abe than Clarise is to Abe? | \frac{1}{3} |
Compute $\tan 3825^\circ$. | 1 |
Find all functions $f$ from the reals to the reals such that \[ \left(f(x)+f(z)\right)\left(f(y)+f(t)\right)=f(xy-zt)+f(xt+yz) \] for all real $x,y,z,t$. | f(x) = 0, \quad f(x) = \frac{1}{2}, \quad f(x) = x^2. |
The TV station continuously plays 5 advertisements, consisting of 3 different commercial advertisements and 2 different Olympic promotional advertisements. The requirements are that the last advertisement must be an Olympic promotional advertisement, and the 2 Olympic promotional advertisements can be played consecutiv... | 36 |
What integer $n$ satisfies $0\le n<9$ and $$-1111\equiv n\pmod 9~?$$ | 5 |
The matrix
\[\begin{pmatrix} \frac{4}{29} & -\frac{10}{29} \\ -\frac{10}{29} & \frac{25}{29} \end{pmatrix}\]corresponds to projecting onto a certain vector $\begin{pmatrix} x \\ y \end{pmatrix}.$ Find $\frac{y}{x}.$ | -\frac{5}{2} |
In rectangle \(ABCD\), points \(E\) and \(F\) lie on sides \(AB\) and \(CD\) respectively such that both \(AF\) and \(CE\) are perpendicular to diagonal \(BD\). Given that \(BF\) and \(DE\) separate \(ABCD\) into three polygons with equal area, and that \(EF = 1\), find the length of \(BD\). | \sqrt{3} |
The perpendicular bisectors of the sides of triangle $PQR$ meet its circumcircle at points $P',$ $Q',$ and $R',$ respectively. If the perimeter of triangle $PQR$ is 30 and the radius of the circumcircle is 7, then find the area of hexagon $PQ'RP'QR'.$ | 105 |
Given four real numbers $-9$, $a\_1$, $a\_2$, and $-1$ that form an arithmetic sequence, and five real numbers $-9$, $b\_1$, $b\_2$, $b\_3$, and $-1$ that form a geometric sequence, find the value of $b\_2(a\_2-a\_1)=$\_\_\_\_\_\_. | -8 |
The four-corner codes for the characters "ε", "ζ―", and "θ΅" are $2440$, $4199$, and $3088$, respectively. By concatenating these, the encoded value for "εζ―θ΅" is $244041993088$. If the digits in the odd positions remain unchanged and the digits in the even positions are replaced with their complements with respect to 9 (... | 254948903981 |
In the Cartesian coordinate system $xOy$, there is a circle $C_{1}$: $(x-2)^{2}+(y-4)^{2}=20$. With the origin $O$ as the pole and the positive half-axis of $x$ as the polar axis, a polar coordinate system is established. For $C_{2}$: $\theta= \frac {\pi}{3}(\rho\in\mathbb{R})$.
$(1)$ Find the polar equation of $C_{1}... | 8+5\sqrt {3} |
Given Angie and Carlos are seated at a round table with three other people, determine the probability that Angie and Carlos are seated directly across from each other. | \frac{1}{2} |
Li Fang has 4 shirts of different colors, 3 skirts of different patterns, and 2 dresses of different styles. Calculate the total number of different choices she has for the May Day celebration. | 14 |
The Wolf and the three little pigs wrote a detective story "The Three Little Pigs-2", and then, together with Little Red Riding Hood and her grandmother, a cookbook "Little Red Riding Hood-2". The publisher gave the fee for both books to the pig Naf-Naf. He took his share and handed the remaining 2100 gold coins to the... | 700 |
The diagonals of a convex quadrilateral \(ABCD\) intersect at point \(E\). Given that \(AB = AD\) and \(CA\) is the bisector of angle \(C\), with \(\angle BAD = 140^\circ\) and \(\angle BEA = 110^\circ\), find angle \(CDB\). | 50 |
Given $$x \in \left(- \frac{\pi}{2}, \frac{\pi}{2}\right)$$ and $$\sin x + \cos x = \frac{1}{5}$$, calculate the value of $\tan 2x$. | -\frac{24}{7} |
Given that the asymptote equation of the hyperbola $y^{2}+\frac{x^2}{m}=1$ is $y=\pm \frac{\sqrt{3}}{3}x$, find the value of $m$. | -3 |
Let $m$ and $n$ be positive integers satisfying the conditions
$\quad\bullet\ \gcd(m+n,210)=1,$
$\quad\bullet\ m^m$ is a multiple of $n^n,$ and
$\quad\bullet\ m$ is not a multiple of $n.$
Find the least possible value of $m+n.$
| 407 |
A point \( M \) is chosen on the diameter \( AB \). Points \( C \) and \( D \), lying on the circumference on one side of \( AB \), are chosen such that \(\angle AMC=\angle BMD=30^{\circ}\). Find the diameter of the circle given that \( CD=12 \). | 8\sqrt{3} |
Consider a $5 \times 5$ grid of squares. Vladimir colors some of these squares red, such that the centers of any four red squares do not form an axis-parallel rectangle (i.e. a rectangle whose sides are parallel to those of the squares). What is the maximum number of squares he could have colored red? | 12 |
Rodney is a door-to-door salesman trying to sell home security systems. He gets a commission of $25 for each system he sells. He is canvassing a neighborhood of four streets with eight houses each. The first street gave him half the sales that the second street did, while every house on the third street turned him away... | Let S be the number of systems Rodney sold on the first street.
He sold 2S systems on the second street.
Based on his commission, he sold 175 / 25 = <<175/25=7>>7 systems.
In all, he sold S + 2S + 0 + 1 = 3S + 1 = 7 systems.
On the first and second streets, he sold 3S = 7 - 1 = 6 systems.
Thus, on the first street he s... |
In $\triangle ABC$, $AB= 425$, $BC=450$, and $AC=510$. An interior point $P$ is then drawn, and segments are drawn through $P$ parallel to the sides of the triangle. If these three segments are of an equal length $d$, find $d$.
| 306 |
How many numbers are in the list $ -48, -41, -34, \ldots, 65, 72?$ | 18 |
Consider a decreasing arithmetic sequence $\{a\_n\}$ with the sum of its first $n$ terms denoted as $S\_n$. If $a\_3a\_5=63$ and $a\_2+{a}\_{6} =16$,
(1) Find the general term formula of the sequence.
(2) For what value of $n$ does $S\_n$ reach its maximum value? Also, find the maximum value.
(3) Calculate $|a\_1|+|a\... | 66 |
Among the natural numbers not exceeding 10,000, calculate the number of odd numbers with distinct digits. | 2605 |
A bowl contains 10 jellybeans (four red, one blue and five white). If you pick three jellybeans from the bowl at random and without replacement, what is the probability that exactly two will be red? Express your answer as a common fraction. | \frac{3}{10} |
Tonya spent $90.00 on art supplies. 4 canvases cost $40.00 and a mixed set of paints cost 1/2 that much. She also spent $15.00 on an easel, and the rest of the money on paintbrushes. How much money did she spend on paintbrushes? | The canvases cost $40.00 and she spent half that amount on paint so she spent 40/2 = $<<40/2=20.00>>20.00 on paint
She spent $40.00 on canvases, $20.00 on paint and $15.00 on an easel so she spent 40+20+15 = $<<40+20+15=75.00>>75.00
She spent $90.00 total and her other purchases totaled $75.00 so she spent 90-75 = $<<9... |
Find the value of $b$ such that the following equation in base $b$ is true:
$$\begin{array}{c@{}c@{}c@{}c@{}c@{}c@{}c}
&&8&7&3&6&4_b\\
&+&9&2&4&1&7_b\\
\cline{2-7}
&1&8&5&8&7&1_b.
\end{array}$$ | 10 |
If \( e^{i \theta} = \frac{3 + i \sqrt{2}}{4}, \) then find \( \cos 3\theta. \) | \frac{9}{64} |
Circle $A$ has radius $100$. Circle $B$ has an integer radius $r<100$ and remains internally tangent to circle $A$ as it rolls once around the circumference of circle $A$. The two circles have the same points of tangency at the beginning and end of circle $B$'s trip. How many possible values can $r$ have? | 8 |
According to the graph, what is the average monthly balance, in dollars, of David's savings account during the four-month period shown? [asy]
draw((0,0)--(13,0)--(13,8)--(0,8)--cycle,linewidth(1));
draw((0,2)--(13,2),linewidth(1));
draw((0,4)--(13,4),linewidth(1));
draw((0,6)--(13,6),linewidth(1));
draw((1,0)--(1,2)--(... | \$150 |
Suppose that $a,b,c,d$ are positive real numbers satisfying $(a+c)(b+d)=ac+bd$. Find the smallest possible value of
$$\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}.$$
[i]Israel[/i] | 8 |
Let \( n \) be the product of 3659893456789325678 and 342973489379256. Determine the number of digits of \( n \). | 34 |
Each of a group of $50$ girls is blonde or brunette and is blue eyed of brown eyed. If $14$ are blue-eyed blondes, $31$ are brunettes, and $18$ are brown-eyed, then the number of brown-eyed brunettes is | 13 |
Given positive integers \(a\), \(b\) \((a \leq b)\), a sequence \(\{ f_{n} \}\) satisfies:
\[ f_{1} = a, \, f_{2} = b, \, f_{n+2} = f_{n+1} + f_{n} \text{ for } n = 1, 2, \ldots \]
If for any positive integer \(n\), it holds that
\[ \left( \sum_{k=1}^{n} f_{k} \right)^2 \leq \lambda \cdot f_{n} f_{n+1}, \]
find the m... | 2 + \sqrt{5} |
An amusement park has a collection of scale models, with a ratio of $1: 20$, of buildings and other sights from around the country. The height of the United States Capitol is $289$ feet. What is the height in feet of its duplicate to the nearest whole number? | 14 |
Given a sequence $\{a\_n\}$ with its sum of the first $n$ terms denoted as $S\_n$. For any $n β β^β$, it is known that $S\_n = 2(a\_n - 1)$.
1. Find the general term formula for the sequence $\{a\_n\}$.
2. Insert $k$ numbers between $a\_k$ and $a_{k+1}$ to form an arithmetic sequence with a common difference $d$ such ... | 144 |
In the rectangle PQRS, side PQ is of length 2 and side QR is of length 4. Points T and U lie inside the rectangle such that βPQT and βRSU are equilateral triangles. Calculate the area of quadrilateral QRUT. | 4 - \sqrt{3} |
There are 18 green leaves on each of the 3 tea leaf plants. One-third of them turn yellow and fall off on each of the tea leaf plants. How many green leaves are left on the tea leaf plants? | In total, the 3 tea leaf plants have 18 x 3 = <<18*3=54>>54 green leaves.
One-third of the yellow leaves falling off of all plants is 54 / 3 = <<54/3=18>>18.
The number of remaining green leaves on the plants is 54 - 18 = <<54-18=36>>36.
#### 36 |
Given the constraints \(x + 2y \leq 5\), \(2x + y \leq 4\), \(x \geq 0\), and \(y \geq 0\), find the coordinates \((x, y)\) where \(3x + 4y\) achieves its maximum value, and determine that maximum value. | 11 |
Each piece of candy in a store costs a whole number of cents. Casper has exactly enough money to buy either 12 pieces of red candy, 14 pieces of green candy, 15 pieces of blue candy, or $n$ pieces of purple candy. A piece of purple candy costs 20 cents. What is the smallest possible value of $n$? | 21 |
Hannah ran 9 kilometers on Monday. She ran 4816 meters on Wednesday and 2095 meters on Friday. How many meters farther did she run on Monday than Wednesday and Friday combined? | Wednesday + Friday = 4816 + 2095 = <<4816+2095=6911>>6911 meters
9 km = <<9*1000=9000>>9000 meters
9000 - 6911 = <<9000-6911=2089>>2089 meters
Hannah ran 2089 meters farther on Monday than Wednesday and Friday.
#### 2089 |
Given that $\sin \alpha + \cos \alpha = \frac{\sqrt{2}}{3}$, and $0 < \alpha < \pi$, find $\tan\left(\alpha - \frac{\pi}{4}\right) = \_\_\_\_\_\_\_\_\_\_.$ | 2\sqrt{2} |
Find $\frac{9}{10}+\frac{5}{6}$. Express your answer as a fraction in simplest form. | \frac{26}{15} |
In the equation $\frac {x(x - 1) - (m + 1)}{(x - 1)(m - 1)} = \frac {x}{m}$ the roots are equal when | -\frac{1}{2} |
Define the β operation: observe the following expressions:
$1$β$2=1\times 3+2=5$;
$4$β$(-1)=4\times 3-1=11$;
$(-5)$β$3=(-5)\times 3+3=-12$;
$(-6)$β$(-3)=(-6)\times 3-3=-21\ldots$
$(1)-1$β$2=$______, $a$β$b=$______;
$(2)$ If $a \lt b$, then $a$β$b-b$β$a$ ______ 0 (use "$ \gt $", "$ \lt $", or "$=$ to connect... | 16 |
In the geometric sequence $\{a_n\}$, if $a_5 + a_6 + a_7 + a_8 = \frac{15}{8}$ and $a_6a_7 = -\frac{9}{8}$, then find the value of $\frac{1}{a_5} + \frac{1}{a_6} + \frac{1}{a_7} + \frac{1}{a_8}$. | -\frac{5}{3} |
Audrey's key lime pie calls for 1/4 cup of key lime juice but she likes to double this amount to make it extra tart. Each key lime yields 1 tablespoon of juice. There are 16 tablespoons in 1 cup. How many key limes does Audrey need? | Her recipe calls for 1/4 cup of juice but she wants to double that so she needs 1/4+1/4 = 1/2 cup of lime juice
16 tablespoons are in 1 cup and she only needs 1/2 cup so she needs 16*.5 = <<16*.5=8>>8 tablespoons of juice
1 key lime yields 1 tablespoon of juice and she needs 8 tablespoons so she needs 1*8 = <<1*8=8>>8 ... |
How many different patterns can be made by shading exactly two of the nine squares? Patterns that can be matched by flips and/or turns are not considered different. For example, the patterns shown below are not considered different. | 8 |
The area of the base of a hemisphere is $144\pi$. The hemisphere is mounted on top of a cylinder that has the same radius as the hemisphere and a height of 10. What is the total surface area of the combined solid? Express your answer in terms of $\pi$. | 672\pi |
For distinct real numbers $x$ and $y$, let $M(x,y)$ be the larger of $x$ and $y$ and let $m(x,y)$ be the smaller of $x$ and $y$. If $a<b<c<d<e$, then
$M(M(a,m(b,c)),m(d,m(a,e)))=$ | b |
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