problem stringlengths 10 5.15k | answer stringlengths 0 1.23k |
|---|---|
Given that $\theta$ is an angle in the third quadrant, if $\sin^{4}{\theta} + \cos^{4}{\theta} = \frac{5}{9}$, find the value of $\sin{2\theta}$. | \frac{2\sqrt{2}}{3} |
A rectangular garden measuring 60 feet by 20 feet is enclosed by a fence. In a redesign to maximize the area using the same amount of fencing, its shape is changed to a circle. How many square feet larger or smaller is the new garden compared to the old one? | 837.62 |
A regular hexagon $ABCDEF$ has sides of length three. Find the area of $\bigtriangleup ACE$. Express your answer in simplest radical form. | \frac{9\sqrt{3}}{4} |
Given the power function $f(x) = kx^a$ whose graph passes through the point $\left( \frac{1}{3}, 81 \right)$, find the value of $k + a$. | -3 |
In the expansion of $(a + b)^n$ there are $n + 1$ dissimilar terms. The number of dissimilar terms in the expansion of $(a + b + c)^{10}$ is: | 66 |
There are 6 locked suitcases and 6 keys for them. However, it is unknown which key opens which suitcase. What is the minimum number of attempts needed to ensure that all suitcases are opened? How many attempts are needed if there are 10 suitcases and 10 keys? | 45 |
The sides of triangle $CAB$ are in the ratio of $2:3:4$. Segment $BD$ is the angle bisector drawn to the shortest side, dividing it into segments $AD$ and $DC$. What is the length, in inches, of the longer subsegment of side $AC$ if the length of side $AC$ is $10$ inches? Express your answer as a common fraction. | \frac {40}7 |
There exist positive integers $a,$ $b,$ and $c$ such that
\[3 \sqrt{\sqrt[3]{5} - \sqrt[3]{4}} = \sqrt[3]{a} + \sqrt[3]{b} - \sqrt[3]{c}.\]Find $a + b + c.$ | 47 |
Ten standard 6-sided dice are rolled. What is the probability that exactly one of the dice shows a 1? Express your answer as a decimal rounded to the nearest thousandth. | 0.323 |
In square $ABCD$, points $E$ and $H$ lie on $\overline{AB}$ and $\overline{DA}$, respectively, so that $AE=AH.$ Points $F$ and $G$ lie on $\overline{BC}$ and $\overline{CD}$, respectively, and points $I$ and $J$ lie on $\overline{EH}$ so that $\overline{FI} \perp \overline{EH}$ and $\overline{GJ} \perp \overline{EH}$. ... | 8-4\sqrt{2} |
Working 22 hours in the second week of June, Xenia was able to earn $\$$47.60 more than during the first week of June when she worked 15 hours. If her hourly wage was constant, how many dollars did she earn during the first two weeks of June? Express your answer to the nearest hundredth. | \$ 251.60 |
A company allocates 5 employees to 3 different departments, with each department being allocated at least one employee. Among them, employees A and B must be allocated to the same department. Calculate the number of different allocation methods. | 36 |
Find the smallest multiple of 9 that does not contain any odd digits. | 288 |
Noemi lost $400 on roulette and $500 on blackjack. How much money did Noemi begin with if she still had $800 in her purse? | The total amount of money that she lost is $400+$500 = $<<400+500=900>>900
If she remained with $900, she initially had $900+$800 = $<<900+800=1700>>1700
#### 1700 |
18.14 People are participating in a round-robin Japanese chess tournament. Each person plays against 13 others, with no draws in the matches. Find the maximum number of "circular triples" (where each of the three participants wins against one and loses to another) in the tournament. | 112 |
Let \(a, b, c\) be the side lengths of a right triangle, with \(a \leqslant b < c\). Determine the maximum constant \(k\) such that the inequality \(a^{2}(b+c) + b^{2}(c+a) + c^{2}(a+b) \geqslant k a b c\) holds for all right triangles, and specify when equality occurs. | 2 + 3\sqrt{2} |
A token starts at the point $(0,0)$ of an $xy$-coordinate grid and then makes a sequence of six moves. Each move is 1 unit in a direction parallel to one of the coordinate axes. Each move is selected randomly from the four possible directions and independently of the other moves. The probability the token ends at a poi... | 391 |
An equilateral triangle $ABC$ is divided by nine lines parallel to $BC$ into ten bands that are equally wide. We colour the bands alternately red and blue, with the smallest band coloured red. The difference between the total area in red and the total area in blue is $20$ $\text{cm}^2$ .
What is the area of tri... | 200 |
What is the value of $x$ for which $(8-x)^2=x^2$? | 4 |
Let $F_n$ be the $n$th Fibonacci number, where as usual $F_1 = F_2 = 1$ and $F_{n + 1} = F_n + F_{n - 1}.$ Then
\[\prod_{k = 2}^{100} \left( \frac{F_k}{F_{k - 1}} - \frac{F_k}{F_{k + 1}} \right) = \frac{F_a}{F_b}\]for some positive integers $a$ and $b.$ Enter the ordered pair $(a,b).$ | (100,101) |
In our number system the base is ten. If the base were changed to four you would count as follows:
$1,2,3,10,11,12,13,20,21,22,23,30,\ldots$ The twentieth number would be: | 110 |
For any positive integer \( n \), let \( f(n) = 70 + n^2 \) and let \( g(n) \) be the greatest common divisor (GCD) of \( f(n) \) and \( f(n+1) \). Find the greatest possible value of \( g(n) \). | 281 |
There are two values of $a$ for which the equation $4x^2+ax+8x+9=0$ has only one solution for $x$. What is the sum of those values of $a$? | -16 |
Let $S_n$ be the sum of the reciprocals of the non-zero digits of the integers from $1$ to $10^n$ inclusive. Find the smallest positive integer $n$ for which $S_n$ is an integer.
| 63 |
Let $ABCD$ be an isosceles trapezoid with $\overline{BC} \parallel \overline{AD}$ and $AB=CD$. Points $X$ and $Y$ lie on diagonal $\overline{AC}$ with $X$ between $A$ and $Y$. Suppose $\angle AXD = \angle BYC = 90^\circ$, $AX = 3$, $XY = 1$, and $YC = 2$. What is the area of $ABCD?$ | $3\sqrt{35}$ |
If $a$ and $b$ are the roots of the equation $x^2-2cx-5d = 0$ , $c$ and $d$ are the roots of the equation $x^2-2ax-5b=0$ , where $a,b,c,d$ are distinct real numbers, what is $a+b+c+d$ ? | 30 |
Suppose that $a$ and $b$ are nonzero real numbers, and that the equation $x^2+ax+b=0$ has solutions $a$ and $b$. Find the ordered pair $(a,b).$ | (1,-2) |
For a New Year’s Eve appetizer, Alex is serving caviar with potato chips and creme fraiche. He buys 3 individual bags of potato chips for $1.00 each and dollops each chip with some creme fraiche that costs $5.00 and then the $73.00 caviar. This appetizer will serve 3 people. How much will this appetizer cost per per... | He buys 3 bags of potato chips for $1.00 each so that’s 3*1 = $<<3*1=3.00>>3.00
The chips are $3.00, the creme fraiche is $5.00 and the caviar is $73.00 for a total of 3+5+73 = $<<3+5+73=81.00>>81.00
The appetizer costs $81.00 and will serve 3 people so it will cost 81/3 = $<<81/3=27.00>>27.00 per person
#### 27 |
For the graph $y = mx + 3$, determine the maximum value of $a$ such that the line does not pass through any lattice points for $0 < x \leq 50$ when $\frac{1}{3} < m < a$. | \frac{17}{51} |
How many positive integers less than $200$ are multiples of $5$, but not multiples of either $10$ or $6$? | 20 |
In the polynomial $x^4 - 18x^3 + kx^2 + 200x - 1984 = 0$ , the product of $2$ of its roots is $- 32$ . Find $k$ . | \[ k = 86 \] |
Calculate the definite integral:
$$
\int_{0}^{\pi} 2^{4} \sin ^{4}\left(\frac{x}{2}\right) \cos ^{4}\left(\frac{x}{2}\right) d x
$$ | \frac{3\pi}{8} |
The sequence $(x_n)$ is defined by $x_1 = 25$ and $x_k = x_{k - 1}^2 + x_{k - 1}$ for all $k \ge 2.$ Compute
\[\frac{1}{x_1 + 1} + \frac{1}{x_2 + 1} + \frac{1}{x_3 + 1} + \dotsb.\] | \frac{1}{25} |
Marcy's grade is electing their class president. Marcy got three times as many votes as Barry, who got twice as many as 3 more than the number of votes Joey got. If Joey got 8 votes, how many votes did Marcy get? | First, add three to Joey's vote total: 8 votes + 3 votes = <<8+3=11>>11 votes
Then double that number to find the number of votes Barry got: 11 votes * 2 = <<11*2=22>>22 votes
Then triple that number to find Marcy's vote total: 22 votes * 3 = <<22*3=66>>66 votes
#### 66 |
Out of 100 externally identical marbles, one is radioactive, but I don't know which one it is. A friend of mine would buy only non-radioactive marbles from me, at a price of 1 forint each. Another friend of mine has an instrument that can determine whether or not there is a radioactive marble among any number of marbl... | 92 |
Leonard and his brother Michael bought presents for their father. Leonard bought a wallet at $50 and two pairs of sneakers at $100 each pair, while Michael bought a backpack at $100 and two pairs of jeans at $50 each pair. How much did they spend in all? | Two pairs of sneakers cost 2 x $100 = $<<2*100=200>>200.
So, Leonard spent $50 + $200 = $<<50+200=250>>250.
Two pairs of jeans cost 2 x $50 = $<<2*50=100>>100.
Thus, Michael spent $100 + $100 = $<<100+100=200>>200.
Therefore, they spent a total of $250 + $200 = $<<250+200=450>>450 in all.
#### 450 |
One ticket to the underground costs $3. In one minute, the metro sells an average of 5 such tickets. What will be the earnings from the tickets in 6 minutes? | During one minute the metro sells 5 tickets, so they earn 5 * 3 = $<<5*3=15>>15.
So in 6 minutes they would earn 15 * 6 = $<<15*6=90>>90.
#### 90 |
Arnaldo claimed that one billion is the same as one million millions. Professor Piraldo corrected him and said, correctly, that one billion is the same as one thousand millions. What is the difference between the correct value of one billion and Arnaldo's assertion? | 999000000000 |
A triangle has one side of length 5 cm, another side of length 12 cm, and includes a right angle. What is the shortest possible length of the third side of the triangle? Express your answer in centimeters as a decimal to the nearest hundredth. | 10.91 |
Given the function \( f(x)=\sin \omega x+\cos \omega x \) where \( \omega > 0 \) and \( x \in \mathbb{R} \), if the function \( f(x) \) is monotonically increasing on the interval \( (-\omega, \omega) \) and the graph of the function \( y=f(x) \) is symmetric with respect to the line \( x=\omega \), determine the value... | \frac{\sqrt{\pi}}{2} |
Professor Severus Snape brewed three potions, each in a volume of 600 ml. The first potion makes the drinker intelligent, the second makes them beautiful, and the third makes them strong. To have the effect of the potion, it is sufficient to drink at least 30 ml of each potion. Severus Snape intended to drink his poti... | 40 |
If \( 50\% \) of \( N \) is 16, what is \( 75\% \) of \( N \)? | 24 |
In 1980, the per capita income in our country was $255; by 2000, the standard of living had reached a moderately prosperous level, meaning the per capita income had reached $817. What was the annual average growth rate? | 6\% |
Let \( x = \sqrt{1 + \frac{1}{1^{2}} + \frac{1}{2^{2}}} + \sqrt{1 + \frac{1}{2^{2}} + \frac{1}{3^{2}}} + \cdots + \sqrt{1 + \frac{1}{2012^{2}} + \frac{1}{2013^{2}}} \). Find the value of \( x - [x] \), where \( [x] \) denotes the greatest integer not exceeding \( x \). | \frac{2012}{2013} |
Let $A B C D$ be a rectangle such that $A B=20$ and $A D=24$. Point $P$ lies inside $A B C D$ such that triangles $P A C$ and $P B D$ have areas 20 and 24, respectively. Compute all possible areas of triangle $P A B$. | 98, 118, 122, 142 |
Determine the number of positive integers $n$ satisfying:
- $n<10^6$
- $n$ is divisible by 7
- $n$ does not contain any of the digits 2,3,4,5,6,7,8.
| 104 |
Given two fixed points $A(-1,0)$ and $B(1,0)$, and a moving point $P(x,y)$ on the line $l$: $y=x+3$, an ellipse $C$ has foci at $A$ and $B$ and passes through point $P$. Find the maximum value of the eccentricity of ellipse $C$. | \dfrac{\sqrt{5}}{5} |
In $\triangle ABC$, the side lengths are: $AB = 17, BC = 20$ and $CA = 21$. $M$ is the midpoint of side $AB$. The incircle of $\triangle ABC$ touches $BC$ at point $D$. Calculate the length of segment $MD$.
A) $7.5$
B) $8.5$
C) $\sqrt{8.75}$
D) $9.5$ | \sqrt{8.75} |
Sally teaches elementary school and is given $320 to spend on books for her students. A reading book costs $12 and there are 30 students in her class. Unfortunately, if the money she is given by the school to pay for books is not sufficient, she will need to pay the rest out of pocket. How much money does Sally need to... | To purchase a book for each student, Sally needs: 30 x $12 = $<<30*12=360>>360.
She thus needs to pay $360 - $320 = $<<360-320=40>>40 out of pocket.
#### 40 |
Simplify $\sqrt{28x} \cdot \sqrt{15x} \cdot \sqrt{21x}$. Express your answer in simplest radical form in terms of $x$.
Note: When entering a square root with more than one character, you must use parentheses or brackets. For example, you should enter $\sqrt{14}$ as "sqrt(14)" or "sqrt{14}". | 42x\sqrt{5x} |
A game show offers a contestant three prizes A, B and C, each of which is worth a whole number of dollars from $$ 1$ to $$ 9999$ inclusive. The contestant wins the prizes by correctly guessing the price of each prize in the order A, B, C. As a hint, the digits of the three prices are given. On a particular day, the dig... | 420 |
What is the remainder when 5462 is divided by 9? | 8 |
A flag consists of three horizontal strips of fabric, each of a solid color, from the choices of red, white, blue, green, or yellow. If no two adjacent strips can be the same color, and an additional rule that no color can be used more than twice, how many distinct flags are possible? | 80 |
Given $ab+bc+cd+da = 30$ and $b+d = 5$, find $a+c$. | 6 |
Chris wants to hold his breath underwater for 90 straight seconds so he starts training each day. On the first day, he holds it for 10 seconds. On the second day, he holds it for 20 seconds. On the third day, he holds it for 30 seconds. After realizing that he can hold it for ten extra seconds each day, he realizes he ... | He still has to improve by 60 more seconds because 90 - 30 = <<90-30=60>>60
This will take 6 days because 60 / 10 = <<60/10=6>>6
#### 6 |
What percent of the prime numbers less than 12 are divisible by 2? | 20\% |
Let \( c_{n}=11 \ldots 1 \) be a number in which the decimal representation contains \( n \) ones. Then \( c_{n+1}=10 \cdot c_{n}+1 \). Therefore:
\[ c_{n+1}^{2}=100 \cdot c_{n}^{2} + 22 \ldots 2 \cdot 10 + 1 \]
For example,
\( c_{2}^{2}=11^{2}=(10 \cdot 1+1)^{2}=100+2 \cdot 10+1=121 \),
\( c_{3}^{2} = 111^{2} = 1... | 11111111 |
Diane bakes four trays with 25 gingerbreads in each tray and three trays with 20 gingerbreads in each tray. How many gingerbreads does Diane bake? | Four trays have 4 x 25 = <<4*25=100>>100 gingerbreads.
And three trays have 3 x 20 = <<3*20=60>>60 gingerbreads.
Therefore, Diane bakes 100 + 60 = <<100+60=160>>160 gingerbreads.
#### 160 |
A number $p$ is $perfect$ if the sum of its divisors, except $p$ is $p$. Let $f$ be a function such that:
$f(n)=0$, if n is perfect
$f(n)=0$, if the last digit of n is 4
$f(a.b)=f(a)+f(b)$
Find $f(1998)$ | 0 |
Each of two boxes contains four chips numbered $1$, $2$, $3$, and $4$. Calculate the probability that the product of the numbers on the two chips is a multiple of $4$. | \frac{1}{2} |
Evaluate $\left|\frac12 - \frac38i\right|$. | \frac58 |
A line has a slope of $-7$ and contains the point $(3,0)$. The equation of this line can be written in the form $y = mx+b$. What is the value of $m+b$? | 14 |
Compute the number of complex numbers $z$ with $|z|=1$ that satisfy $$1+z^{5}+z^{10}+z^{15}+z^{18}+z^{21}+z^{24}+z^{27}=0$$ | 11 |
Stan weighs 5 more pounds than Steve. Steve is eight pounds lighter than Jim. If Jim weighs 110 pounds and the three of them crowd onto a scale at the same time, what is their total weight? | Steve weighs 110 - 8 = <<110-8=102>>102 pounds.
Stan weighs 102 + 5 = <<102+5=107>>107 pounds.
Their total weight is 102 + 107 + 110 = <<102+107+110=319>>319 pounds.
#### 319 |
Eight friends ate at a restaurant and agreed to share the bill equally. Because Judi forgot her money, each of her seven friends paid an extra \$2.50 to cover her portion of the total bill. What was the total bill? | $120 |
A trapezoid is divided into seven strips of equal width. What fraction of the trapezoid's area is shaded? Explain why your answer is correct. | 4/7 |
During the 2011 Universiade in Shenzhen, a 12-person tour group initially stood in two rows with 4 people in the front row and 8 people in the back row. The photographer plans to keep the order of the front row unchanged, and move 2 people from the back row to the front row, ensuring that these two people are not adjac... | 560 |
Julio makes a mocktail every evening. He uses 1 tablespoon of lime juice and tops with 1 cup of sparkling water. He can usually squeeze 2 tablespoons of lime juice per lime. After 30 days, if limes are 3 for $1.00, how much will he have spent on limes? | He uses 1 tablespoon of lime juice per drink so over 30 days that's 1*30 = <<1*30=30>>30 tablespoons of lime juice
He can get 2 tablespoons of lime juice per lime and he will use 30 tablespoons of lime juice so he will need 30/2 = 15 limes
The limes are 3 for $1.00 and he needs 15 limes so that's 15/3 = $5.00
#### 5 |
In isosceles trapezoid $ABCD$ where $AB$ (shorter base) is 10 and $CD$ (longer base) is 20. The non-parallel sides $AD$ and $BC$ are extended to meet at point $E$. What is the ratio of the area of triangle $EAB$ to the area of trapezoid $ABCD$? | \frac{1}{3} |
Joe has exactly enough paint to paint the surface of a cube whose side length is 2. It turns out that this is also exactly enough paint to paint the surface of a sphere. If the volume of this sphere is $\frac{K \sqrt{6}}{\sqrt{\pi}}$, then what is $K$? | K=8 |
Ellen is baking bread. It takes 3 hours to rise 1 ball of dough, and then another 2 hours to bake it. If she makes 4 balls of dough one after another and then bakes them one after another when they're done rising, how many hours will it take? | Ellen takes 4*3=<<4*3=12>>12 hours in total to rise all balls of dough.
It will take 4*2=<<4*2=8>>8 more hours to bake each one.
It will take her 12+8=<<12+8=20>>20 hours to make all of the bread.
#### 20 |
Given that \(\frac{x+y}{x-y}+\frac{x-y}{x+y}=3\). Find the value of the expression \(\frac{x^{2}+y^{2}}{x^{2}-y^{2}}+\frac{x^{2}-y^{2}}{x^{2}+y^{2}}\). | 13/6 |
Convert $427_8$ to base 5. | 2104_5 |
Consider an equilateral triangle and a square both inscribed in a unit circle such that one side of the square is parallel to one side of the triangle. Compute the area of the convex heptagon formed by the vertices of both the triangle and the square. | \frac{3+\sqrt{3}}{2} |
A sequence $(a_n)$ of real numbers is defined by $a_0=1$, $a_1=2015$ and for all $n\geq1$, we have
$$a_{n+1}=\frac{n-1}{n+1}a_n-\frac{n-2}{n^2+n}a_{n-1}.$$
Calculate the value of $\frac{a_1}{a_2}-\frac{a_2}{a_3}+\frac{a_3}{a_4}-\frac{a_4}{a_5}+\ldots+\frac{a_{2013}}{a_{2014}}-\frac{a_{2014}}{a_{2015}}$. | 3021 |
In triangle $ABC,$ $\sin A = \frac{3}{5}$ and $\cos B = \frac{5}{13}.$ Find $\cos C.$ | \frac{16}{65} |
How many different rectangles with sides parallel to the grid can be formed by connecting four of the dots in a $5 \times 5$ square array of dots? | 100 |
There are $10000$ trees in a park, arranged in a square grid with $100$ rows and $100$ columns. Find the largest number of trees that can be cut down, so that sitting on any of the tree stumps one cannot see any other tree stump.
| 2500 |
Naomi has three colors of paint which she uses to paint the pattern below. She paints each region a solid color, and each of the three colors is used at least once. If Naomi is willing to paint two adjacent regions with the same color, how many color patterns could Naomi paint?
[asy]
size(150);
defaultpen(linewidth(... | 540 |
If the price of a stamp is 45 cents, what is the maximum number of stamps that could be purchased with $50? | 111 |
Let acute triangle $ABC$ have circumcenter $O$, and let $M$ be the midpoint of $BC$. Let $P$ be the unique point such that $\angle BAP=\angle CAM, \angle CAP=\angle BAM$, and $\angle APO=90^{\circ}$. If $AO=53, OM=28$, and $AM=75$, compute the perimeter of $\triangle BPC$. | 192 |
Joe's quiz scores were 88, 92, 95, 81, and 90, and then he took one more quiz and scored 87. What was his mean score after all six quizzes? | 88.83 |
Given that $f(\alpha) = \cos\alpha \sqrt{\frac{\cot\alpha - \cos\alpha}{\cot\alpha + \cos\alpha}} + \sin\alpha \sqrt{\frac{\tan\alpha - \sin\alpha}{\tan\alpha + \sin\alpha}}$, and $\alpha$ is an angle in the second quadrant.
(1) Simplify $f(\alpha)$.
(2) If $f(-\alpha) = \frac{1}{5}$, find the value of $\frac{1}{\t... | -\frac{7}{12} |
Given plane vectors $\overrightarrow{a}$, $\overrightarrow{b}$, $\overrightarrow{c}$ satisfy $\langle \overrightarrow{a}, \overrightarrow{b} \rangle = 60^\circ$, and $\{|\overrightarrow{a}|, |\overrightarrow{b}|, |\overrightarrow{c}|\} = \{1, 2, 3\}$, calculate the maximum value of $|\overrightarrow{a}+ \overrightarrow... | \sqrt{7}+3 |
Given an ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ($a > b > 0$), a line passing through points A($-a, 0$) and B($0, b$) has an inclination angle of $\frac{\pi}{6}$, and the distance from origin to this line is $\frac{\sqrt{3}}{2}$.
(1) Find the equation of the ellipse.
(2) Suppose a line with a positive slope p... | k = \frac{7}{6} |
Given non-zero vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfy $|\overrightarrow{a}|=2|\overrightarrow{b}|$, and $(\overrightarrow{a}-\overrightarrow{b})\bot \overrightarrow{b}$, then the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is ______. | \frac{\pi}{3} |
The vertex of the parabola described by the equation $y=-3x^2-30x-81$ is $(m,n)$. What is $n$? | -6 |
Sarah stands at $(0,0)$ and Rachel stands at $(6,8)$ in the Euclidean plane. Sarah can only move 1 unit in the positive $x$ or $y$ direction, and Rachel can only move 1 unit in the negative $x$ or $y$ direction. Each second, Sarah and Rachel see each other, independently pick a direction to move at the same time, and m... | \[
\frac{63}{64}
\] |
In triangle \( \triangle ABC \), \( M \) is the midpoint of side \( BC \), and \( N \) is the midpoint of segment \( BM \). Given \( \angle A = \frac{\pi}{3} \) and the area of \( \triangle ABC \) is \( \sqrt{3} \), find the minimum value of \( \overrightarrow{AM} \cdot \overrightarrow{AN} \). | \sqrt{3} + 1 |
Three points are chosen uniformly at random on a circle. What is the probability that no two of these points form an obtuse triangle with the circle's center? | \frac{3}{16} |
Rick held a fundraiser and is figuring out who still owes money. He marked on a sheet how much everyone has promised to contribute, but he forgot to note how much some people had actually paid. Altogether, everyone has promised to contribute $400, and Rick has actually received $285. Sally, Amy, Derek, and Carl all st... | In total, Rick is still owed 400 - 285 = $<<400-285=115>>115.
Derek owes half as much as Amy, so he owes $30 / 2 = $<<30/2=15>>15.
This means that Sally and Carl owe a combined total of $115 - $30 – $15 = $<<115-30-15=70>>70.
As they owe equal amounts, they therefore owe $70 / 2 = $<<70/2=35>>35 each.
#### 35 |
The shape shown is made up of three similar right-angled triangles. The smallest triangle has two sides of side-length 2, as shown. What is the area of the shape? | 14 |
Jason has a carriage house that he rents out. He’s charging $50.00 per day or $500.00 for 14 days. Eric wants to rent the house for 20 days. How much will it cost him? | He wants to rent for 20 days and there is a deal if you rent for 14 days so that leaves 20-14 = <<20-14=6>>6 individual days
Each individual day is $50.00 and he will have 6 individual days for a total of 50*6 = $<<50*6=300.00>>300.00
14 days costs $500.00 and 6 days costs $300.00 for a total of 500+300 = $800.00
#### ... |
An alarm clock gains 9 minutes each day. When going to bed at 22:00, the precise current time is set on the clock. At what time should the alarm be set so that it rings exactly at 6:00? Explain your answer. | 6:03 |
Movie tickets cost $5 each on a Monday, twice as much on a Wednesday, and five times as much as Monday on a Saturday. If Glenn goes to the movie theater on Wednesday and Saturday, how much does he spend? | He spends 5*2=$<<5*2=10>>10 on Wednesday to pay for a movie ticket.
He spends 5*5=$<<5*5=25>>25 on Saturday to pay for a movie ticket.
He spends a total of 10+25=$<<10+25=35>>35.
#### 35 |
There are 456 natives on an island, each of whom is either a knight who always tells the truth or a liar who always lies. All residents have different heights. Once, each native said, "All other residents are shorter than me!" What is the maximum number of natives who could have then said one minute later, "All other r... | 454 |
Terese thinks that running various distances throughout the week can make one healthy. On Monday, she runs 4.2 miles; Tuesday, 3.8 miles; Wednesday, 3.6 miles; and on Thursday, 4.4 miles. Determine the average distance Terese runs on each of the days she runs. | The total distance Terese runs is 4.2 + 3.8+4.4 +3.6 = <<4.2+3.8+4.4+3.6=16>>16 miles.
Therefore, she runs with an average distance of 16 /4 = <<16/4=4>>4 miles.
#### 4 |
Five people of heights $65,66,67,68$, and 69 inches stand facing forwards in a line. How many orders are there for them to line up, if no person can stand immediately before or after someone who is exactly 1 inch taller or exactly 1 inch shorter than himself? | 14 |
Doug constructs a square window using $8$ equal-size panes of glass. The ratio of the height to width for each pane is $5 : 2$, and the borders around and between the panes are $2$ inches wide. In inches, what is the side length of the square window? | 26 |
Julia collects old watches. She owns 20 silver watches, and three times as many bronze watches. She decided to buy gold watches to add to her collection, a number that represents 10% of all the watches she owns. How many watches does Julia own after this purchase? | Julia owns 3 * 20 = <<3*20=60>>60 bronze watches.
60+20 = <<60+20=80>>80 total silver and bronze watches
10% of all Julia's watches is 80 * 10/100 = <<80*10/100=8>>8 watches, and she decided to buy so many gold watches.
So in total Julia is in possession of 20 + 60 + 8 = <<20+60+8=88>>88 watches.
#### 88 |
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