problem stringlengths 10 5.15k | answer stringlengths 0 1.23k |
|---|---|
Given that Xiao Ming's elder brother was born in a year that is a multiple of 19, calculate his age in 2013. | 18 |
Given a right prism with all vertices on the same sphere, with a height of $4$ and a volume of $32$, the surface area of this sphere is ______. | 32\pi |
The sum of $n$ terms of an arithmetic progression is $153$, and the common difference is $2$. If the first term is an integer, and $n>1$, then the number of possible values for $n$ is: | 5 |
You want to paint some edges of a regular dodecahedron red so that each face has an even number of painted edges (which can be zero). Determine from How many ways this coloration can be done.
Note: A regular dodecahedron has twelve pentagonal faces and in each vertex concur three edges. The edges of the dodecahedron a... | 2048 |
To run his grocery store, Mr. Haj needs $4000 a day. This money is used to pay for orders done, delivery costs and employees' salaries. If he spends 2/5 of the total operation costs on employees' salary and 1/4 of the remaining amount on delivery costs, how much money does he pay for the orders done? | The total amount of money Mr. Haj used to pay for employee salary is 2/5*$4000 = $<<2/5*4000=1600>>1600
After paying the employee salaries, Mr. Haj remains with $4000-$1600 = $<<4000-1600=2400>>2400
He also uses 1/4*$2400= $<<2400/4=600>>600 on delivery costs.
The remaining amount of money that he uses to pay for order... |
Let $A = \left\{a_{1}, a_{2}, \cdots, a_{n}\right\}$ be a set of numbers, and let the arithmetic mean of all elements in $A$ be denoted by $P(A)\left(P(A)=\frac{a_{1}+a_{2}+\cdots+a_{n}}{n}\right)$. If $B$ is a non-empty subset of $A$ such that $P(B) = P(A)$, then $B$ is called a "balance subset" of $A$. Find the numbe... | 51 |
From the 2015 natural numbers between 1 and 2015, what is the maximum number of numbers that can be found such that their product multiplied by 240 is a perfect square? | 134 |
If $f(x) = 5x^2 - 2x - 1$, then $f(x + h) - f(x)$ equals: | h(10x+5h-2) |
John takes a 20-foot log and cuts it in half. If each linear foot of the log weighs 150 pounds how much does each cut piece weigh? | Each cut piece is 20/2=<<20/2=10>>10 feet long
So they each weigh 10*150=<<10*150=1500>>1500 pounds
#### 1500 |
Suppose \( a, b \), and \( c \) are real numbers with \( a < b < 0 < c \). Let \( f(x) \) be the quadratic function \( f(x) = (x-a)(x-c) \) and \( g(x) \) be the cubic function \( g(x) = (x-a)(x-b)(x-c) \). Both \( f(x) \) and \( g(x) \) have the same \( y \)-intercept of -8 and \( g(x) \) passes through the point \( (... | \frac{8}{3} |
What was the range of temperatures on Monday in Fermatville, given that the minimum temperature was $-11^{\circ} \mathrm{C}$ and the maximum temperature was $14^{\circ} \mathrm{C}$? | 25^{\circ} \mathrm{C} |
For a finite sequence $B = (b_1, b_2, \dots, b_n)$ of numbers, the Cesaro sum is defined as
\[\frac{T_1 + T_2 + \cdots + T_n}{n},\]
where $T_k = b_1 + b_2 + \cdots + b_k$ for $1 \leq k \leq n$.
If the Cesaro sum of the 100-term sequence $(b_1, b_2, \dots, b_{100})$ is 1200, where $b_1 = 2$, calculate the Cesaro sum o... | 1191 |
For a natural number \( N \), if at least six of the nine natural numbers from 1 to 9 are factors of \( N \), then \( N \) is called a “six-match number.” Find the smallest "six-match number" greater than 2000. | 2016 |
Given the integers \( a, b, c \) that satisfy \( a + b + c = 2 \), and
\[
S = (2a + bc)(2b + ca)(2c + ab) > 200,
\]
find the minimum value of \( S \). | 256 |
Consider the addition problem: \begin{tabular}{ccccc} & C & A & S & H \\ + & & & M & E \\ \hline O & S & I & D & E \end{tabular} where each letter represents a base-ten digit, and $C, M, O \neq 0$. (Distinct letters are allowed to represent the same digit) How many ways are there to assign values to the letters so that... | 0 |
John believes that the amount of sleep he gets the night before a test and his score on that test are inversely related. On his first exam, he got eight hours of sleep and scored 70 on the exam. To the nearest tenth, how many hours does John believe he must sleep the night before his second exam so that the average o... | 6.2 |
Matt has a peanut plantation that is 500 feet by 500 feet. 1 square foot of peanuts can make 50 grams of peanuts. If it takes 20 grams of peanuts to make 5 grams of peanut butter and 1 kg of peanut butter sells for $10 how much does he make from his plantation? | His plantation is 500*500=<<500*500=250000>>250,000 square feet
That means he gets 250,000*50=<<250000*50=12500000>>12,500,000 grams of peanuts
So he makes 12,500,000*5/20=<<12500000*5/20=3125000>>3,125,000 grams of peanut butter
So he has 3,125,000/1000=<<3125000/1000=3125>>3125 kg of peanut butter
That means it sells... |
Find all real values of $x$ that satisfy $\frac{x(x+1)}{(x-4)^2} \ge 12.$ (Give your answer in interval notation.) | [3, 4) \cup \left(4, \frac{64}{11}\right] |
Let $n=2^{31}3^{19}.$ How many positive integer divisors of $n^2$ are less than $n_{}$ but do not divide $n_{}$? | 589 |
Evaluate $i^{11} + i^{111}$. | -2i |
An ordinary $6$-sided die has a number on each face from $1$ to $6$ (each number appears on one face). How many ways can I paint two faces of a die red, so that the numbers on the red faces don't add up to $7$? | 12 |
A list of $2018$ positive integers has a unique mode, which occurs exactly $10$ times. What is the least number of distinct values that can occur in the list? | 225 |
There were 133 people at a camp. There were 33 more boys than girls. How many girls were at the camp? | If there were 33 fewer boys at camp, there would be the same number of both boys and girls or 133 - 33 = <<133-33=100>>100 people.
That means that there were 100 / 2 = <<100/2=50>>50 girls at the camp.
#### 50 |
Given the set $A=\{(x,y) \,|\, |x| \leq 1, |y| \leq 1, x, y \in \mathbb{R}\}$, and $B=\{(x,y) \,|\, (x-a)^2+(y-b)^2 \leq 1, x, y \in \mathbb{R}, (a,b) \in A\}$, then the area represented by set $B$ is \_\_\_\_\_\_. | 12 + \pi |
There are 5 yellow pegs, 4 red pegs, 3 green pegs, 2 blue pegs, and 1 orange peg to be placed on a triangular peg board. In how many ways can the pegs be placed so that no (horizontal) row or (vertical) column contains two or more pegs of the same color? (Any two pegs of the same color are indistinguishable.)
[asy]
d... | 1 |
Find the greatest common divisor of $7!$ and $(5!)^2.$ | 720 |
Back in 1930, Tillie had to memorize her multiplication facts from $0 \times 0$ to $12 \times 12$. The multiplication table she was given had rows and columns labeled with the factors, and the products formed the body of the table. To the nearest hundredth, what fraction of the numbers in the body of the table are odd? | 0.21 |
Let \( p \in \mathbb{R} \). In the complex plane, consider the equation
\[ x^2 - 2x + 2 = 0, \]
whose two complex roots are represented by points \( A \) and \( B \). Also consider the equation
\[ x^2 + 2px - 1 = 0, \]
whose two complex roots are represented by points \( C \) and \( D \). If the four points \( A \), \(... | -1 |
What are the mode and median of the set of ages $29$, $27$, $31$, $31$, $31$, $29$, $29$, and $31$? | 30 |
Use the angle addition formula for cosine to simplify the expression $\cos 54^{\circ}\cos 24^{\circ}+2\sin 12^{\circ}\cos 12^{\circ}\sin 126^{\circ}$. | \frac{\sqrt{3}}{2} |
Jimmy wants to order a pizza at a new place. The large pizza costs $10.00 and is cut into 8 slices. The first topping costs $2.00, the next 2 toppings cost $1.00 each and the rest of the toppings cost $0.50. If he orders a large pizza with pepperoni, sausage, ham, olives, mushrooms, bell peppers and pineapple. How ... | The pizza costs $10 and the first topping, pepperoni costs $2 so right now it costs 10=2 = $12.00
The next 2 toppings, sausage and ham, cost $1.00 each so 2*1 = $<<2*1=2.00>>2.00
The remaining toppings, olives, mushrooms, bell peppers and pineapple, cost $0.50 each so they cost 4*.50 = $<<4*.50=2.00>>2.00
All total, Ji... |
One side of a rectangle (the width) was increased by 10%, and the other side (the length) by 20%.
a) Could the perimeter increase by more than 20% in this case?
b) Find the ratio of the sides of the original rectangle if it is known that the perimeter of the new rectangle is 18% greater than the perimeter of the origi... | 1:4 |
Jaymee is 2 years older than twice the age of Shara. If Shara is 10 years old, how old is Jaymee? | Twice the age of Shara is 10 x 2 = <<10*2=20>>20.
So, Jaymee is 20 + 2 = <<20+2=22>>22 years old.
#### 22 |
Given that $x = \frac{3}{5}$ is a solution to the equation $30x^2 + 13 = 47x - 2$, find the other value of $x$ that will solve the equation. Express your answer as a common fraction. | \frac{5}{6} |
Given a four-digit positive integer $\overline{abcd}$, if $a+c=b+d=11$, then this number is called a "Shangmei number". Let $f(\overline{abcd})=\frac{{b-d}}{{a-c}}$ and $G(\overline{abcd})=\overline{ab}-\overline{cd}$. For example, for the four-digit positive integer $3586$, since $3+8=11$ and $5+6=11$, $3586$ is a "Sh... | -3 |
Let $\mathbf{a}$ and $\mathbf{b}$ be unit vectors such that $\mathbf{a} + 2 \mathbf{b}$ and $5 \mathbf{a} - 4 \mathbf{b}$ are orthogonal. Find the angle between $\mathbf{a}$ and $\mathbf{b},$ in degrees.
Note: A unit vector is a vector of magnitude 1. | 60^\circ |
Two sides of a regular $n$-gon are extended to meet at a $28^{\circ}$ angle. What is the smallest possible value for $n$? | 45 |
The numbers $1, 2, 3, 4, 5, 6, 7,$ and $8$ are randomly written on the faces of a regular octahedron so that each face contains a different number. The probability that no two consecutive numbers, where $8$ and $1$ are considered to be consecutive, are written on faces that share an edge is $m/n,$ where $m$ and $n$ are... | 85 |
Each of the squares of an $8 \times 8$ board can be colored white or black. Find the number of colorings of the board such that every $2 \times 2$ square contains exactly 2 black squares and 2 white squares. | 8448 |
Given point \( A(4,0) \) and \( B(2,2) \), while \( M \) is a moving point on the ellipse \(\frac{x^{2}}{25} + \frac{y^{2}}{9} = 1\), the maximum value of \( |MA| + |MB| \) is ______. | 10 + 2\sqrt{10} |
Find $10110_2\times10100_2\div10_2$. Express your answer in base 2. | 11011100_2 |
(1) Consider the function $f(x) = |x - \frac{5}{2}| + |x - a|$, where $x \in \mathbb{R}$. If the inequality $f(x) \geq a$ holds true for all $x \in \mathbb{R}$, find the maximum value of the real number $a$.
(2) Given positive numbers $x$, $y$, and $z$ satisfying $x + 2y + 3z = 1$, find the minimum value of $\frac{3}{x... | 16 + 8\sqrt{3} |
Let $f_{1}(x)=\sqrt{1-x}$, and for integers $n \geq 2$, let \[f_{n}(x)=f_{n-1}\left(\sqrt{n^2 - x}\right).\]Let $N$ be the largest value of $n$ for which the domain of $f_n$ is nonempty. For this value of $N,$ the domain of $f_N$ consists of a single point $\{c\}.$ Compute $c.$ | -231 |
Regular hexagon $ABCDEF$ is the base of right pyramid $\allowbreak PABCDEF$. If $PAD$ is an equilateral triangle with side length 8, then what is the volume of the pyramid? | 96 |
Two circles of radius 3 are centered at $(3,0)$ and at $(0,3)$. Determine the area of their overlapping interiors. Express your answer in expanded form in terms of $\pi$. | \frac{9}{2}\pi - 9 |
Let $x,$ $y,$ and $z$ be positive real numbers such that $x + y + z = 1.$ Find the maximum value of $x^3 y^2 z.$ | \frac{1}{432} |
A wooden cube, whose edges are one centimeter long, rests on a horizontal surface. Illuminated by a point source of light that is $x$ centimeters directly above an upper vertex, the cube casts a shadow on the horizontal surface. The area of the shadow, which does not include the area beneath the cube is 48 square centi... | 166 |
In a certain place, four people, $A$, $B$, $C$, and $D$, successively contracted the novel coronavirus, with only $A$ having visited an epidemic area.
1. If the probabilities of $B$, $C$, and $D$ being infected by $A$ are $\frac{1}{2}$ each, what is the probability that exactly one of $B$, $C$, and $D$ is infected with... | \frac{11}{6} |
An integer partition, is a way of writing n as a sum of positive integers. Two sums that differ only in the order of their summands are considered the same partition.
[quote]For example, 4 can be partitioned in five distinct ways:
4
3 + 1
2 + 2
2 + 1 + 1
1 + 1 + 1 + 1[/quote]
The number of partitions of n is given by t... | 1, 3, 5 |
The ferry "Yi Rong" travels at a speed of 40 kilometers per hour. On odd days, it travels downstream from point $A$ to point $B$, while on even days, it travels upstream from point $B$ to point $A$ (with the water current speed being 24 kilometers per hour). On one odd day, when the ferry reached the midpoint $C$, it ... | 192 |
What integer $n$ satisfies $0\le n<19$ and $$-200\equiv n\pmod{19}~?$$ | 9 |
Find the smallest real number \(\lambda\) such that the inequality
$$
5(ab + ac + ad + bc + bd + cd) \leq \lambda abcd + 12
$$
holds for all positive real numbers \(a, b, c, d\) that satisfy \(a + b + c + d = 4\). | 18 |
If two stagecoaches travel daily from Bratislava to Brașov, and likewise, two stagecoaches travel daily from Brașov to Bratislava, and considering that the journey takes ten days, how many stagecoaches will you encounter on your way when traveling by stagecoach from Bratislava to Brașov? | 20 |
Sarah intended to multiply a two-digit number and a three-digit number, but she left out the multiplication sign and simply placed the two-digit number to the left of the three-digit number, thereby forming a five-digit number. This number is exactly nine times the product Sarah should have obtained. What is the sum of... | 126 |
Suppose that $a$ and $b$ are positive integers such that $a-b=6$ and $\text{gcd}\left(\frac{a^3+b^3}{a+b}, ab\right) = 9$. Find the smallest possible value of $b$. | 3 |
If $a @ b$ is defined as $a @ b$ = $3a - 3b$, what is the value of $3 @ 5$? | -6 |
What is the $111$th digit after the decimal point when $\frac{33}{555}$ is expressed as a decimal? | 9 |
Let $P(x)$ be a polynomial such that when $P(x)$ is divided by $x - 19,$ the remainder is 99, and when $P(x)$ is divided by $x - 99,$ the remainder is 19. What is the remainder when $P(x)$ is divided by $(x - 19)(x - 99)$? | -x + 118 |
Given a right triangle \(ABC\), point \(D\) is located on the extension of hypotenuse \(BC\) such that line \(AD\) is tangent to the circumcircle \(\omega\) of triangle \(ABC\). Line \(AC\) intersects the circumcircle of triangle \(ABD\) at point \(E\). It turns out that the angle bisector of \(\angle ADE\) is tangent ... | 1:2 |
The sum of all the roots of $4x^3-8x^2-63x-9=0$ is: | 2 |
Compute the value of $\sqrt{105^{3}-104^{3}}$, given that it is a positive integer. | 181 |
Let the hyperbola $C: \frac{x^2}{a^2} - y^2 = 1$ ($a > 0$) intersect with the line $l: x + y = 1$ at two distinct points $A$ and $B$.
(Ⅰ) Find the range of values for the eccentricity $e$ of the hyperbola $C$.
(Ⅱ) Let the intersection of line $l$ with the y-axis be $P$, and $\overrightarrow{PA} = \frac{5}{12} \overri... | \frac{17}{13} |
Given the real numbers \(a\) and \(b\) satisfying \(\left(a - \frac{b}{2}\right)^2 = 1 - \frac{7}{4} b^2\), let \(t_{\max}\) and \(t_{\min}\) denote the maximum and minimum values of \(t = a^2 + 2b^2\), respectively. Find the value of \(t_{\text{max}} + t_{\text{min}}\). | \frac{16}{7} |
Convert the point $(1,-\sqrt{3})$ in rectangular coordinates to polar coordinates. Enter your answer in the form $(r,\theta),$ where $r > 0$ and $0 \le \theta < 2 \pi.$ | \left( 2, \frac{5 \pi}{3} \right) |
Factor the expression $2x(x-3) + 3(x-3)$. | (2x+3)(x-3) |
3/5 of the mangoes on a mango tree are ripe. If Lindsay eats 60% of the ripe mangoes, calculate the number of ripe mangoes remaining if there were 400 mangoes on the tree to start with. | The number of ripe mangoes from the tree is 3/5*400 = <<3/5*400=240>>240
If Lindsay eats 60% of the ripe mangoes, she eats 60/100*240 = <<60/100*240=144>>144 mangoes.
The total number of ripe mangoes remaining is 240-144 =<<240-144=96>>96
#### 96 |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively. Given $\overrightarrow{m}=(\sin C,\sin B\cos A)$ and $\overrightarrow{n}=(b,2c)$ with $\overrightarrow{m}\cdot \overrightarrow{n}=0$.
(1) Find angle $A$;
(2) If $a=2 \sqrt {3}$ and $c=2$, find the area of $\triangle... | \sqrt {3} |
Find the minimum value of
\[2 \cos \theta + \frac{1}{\sin \theta} + \sqrt{2} \tan \theta\]for $0 < \theta < \frac{\pi}{2}.$ | 3 \sqrt{2} |
Find all integer values of the parameter \(a\) for which the system
\[
\begin{cases}
x - 2y = y^2 + 2, \\
ax - 2y = y^2 + x^2 + 0.25a^2
\end{cases}
\]
has at least one solution. In the answer, indicate the sum of the found values of the parameter \(a\). | 10 |
The sum of all real roots of the equation $|x^2-3x+2|+|x^2+2x-3|=11$ is . | \frac{5\sqrt{97}-19}{20} |
Let $f\left(x\right)=ax^{2}-1$ and $g\left(x\right)=\ln \left(ax\right)$ have an "$S$ point", then find the value of $a$. | \frac{2}{e} |
Rachel has the number 1000 in her hands. When she puts the number $x$ in her left pocket, the number changes to $x+1$. When she puts the number $x$ in her right pocket, the number changes to $x^{-1}$. Each minute, she flips a fair coin. If it lands heads, she puts the number into her left pocket, and if it lands tails,... | 13 |
Find the degree measure of the least positive angle $\theta$ for which
\[\cos 5^\circ = \sin 25^\circ + \sin \theta.\] | 35^\circ |
Bert's golden retriever has grown tremendously since it was a puppy. At 7 weeks old, the puppy weighed 6 pounds, but doubled in weight by week 9. It doubled in weight again at 3 months old, and doubled again at 5 months old. Finally, the dog reached its final adult weight by adding another 30 pounds by the time it w... | At 7 weeks old, the puppy weighed 6 pounds, but doubled in weight by week 9 to reach a weight of 6*2=<<6+6=12>>12 pounds.
It doubled in weight again at 3 months old to reach a weight of 12*2=<<12*2=24>>24 pounds.
It doubled in weight again at 5 months old to reach a weight of 24*2=<<24*2=48>>48 pounds.
And it reached i... |
A dice is repeatedly rolled, and the upward-facing number is recorded for each roll. The rolling stops once three different numbers are recorded. If the sequence stops exactly after five rolls, calculate the total number of distinct recording sequences for these five numbers. | 840 |
You draw a rectangle that is 7 inches wide. It is 4 times as long as it is wide. What is the area of the rectangle? | The length of the rectangle is 4 * 7 inches = <<4*7=28>>28 inches.
The area of the rectangle is 7 inches * 28 inches = <<7*28=196>>196 square inches.
#### 196 |
Find a factor of 100140001 which lies between 8000 and 9000. | 8221 |
Points $A$ and $B$ have the same $y$-coordinate of 13, but different $x$-coordinates. What is the sum of the slope and the $y$-intercept of the line containing both points? | 13 |
199 people signed up to participate in a tennis tournament. In the first round, pairs of opponents are selected by drawing lots. The same method is used to pair opponents in the second, third, and all subsequent rounds. After each match, one of the two opponents is eliminated, and whenever the number of participants in... | 198 |
What is the base-10 integer 789 when expressed in base 7? | 2205_7 |
8 people are sitting around a circular table for a meeting, including one leader, one vice leader, and one recorder. If the recorder is seated between the leader and vice leader, how many different seating arrangements are possible (considering that arrangements that can be obtained by rotation are identical)? | 240 |
Point $P$ is inside $\triangle ABC$. Line segments $APD$, $BPE$, and $CPF$ are drawn with $D$ on $BC$, $E$ on $AC$, and $F$ on $AB$ (see the figure below). Given that $AP=6$, $BP=9$, $PD=6$, $PE=3$, and $CF=20$, find the area of $\triangle ABC$. | 108 |
If the graph of the function $f(x)=(x^{2}-4)(x^{2}+ax+b)$ is symmetric about the line $x=-1$, find the values of $a$ and $b$, and the minimum value of $f(x)$. | -16 |
The angle between the slant height of a cone and the base plane is $30^{\circ}$. The lateral surface area of the cone is $3 \pi \sqrt{3}$ square units. Determine the volume of a regular hexagonal pyramid inscribed in the cone. | \frac{27 \sqrt{2}}{8} |
\(ABCD\) is a convex quadrilateral where \(AB = 7\), \(BC = 4\), and \(AD = DC\). Also, \(\angle ABD = \angle DBC\). Point \(E\) is on segment \(AB\) such that \(\angle DEB = 90^\circ\). Find the length of segment \(AE\). | 1.5 |
If the graph of the power function $y=f(x)$ passes through the point $\left( -2,-\frac{1}{8} \right)$, find the value(s) of $x$ that satisfy $f(x)=27$. | \frac{1}{3} |
Camila has only gone hiking 7 times in her life. Amanda has gone on 8 times as many hikes as Camila, and Steven has gone on 15 more hikes than Amanda. If Camila wants to say that she has hiked as many times as Steven and plans to go on 4 hikes a week, how many weeks would it take Camila to achieve her goal? | Amanda has gone on 7 hikes x 8 = <<7*8=56>>56 hikes.
Steven has gone on 56 hikes + 15 hikes = <<56+15=71>>71 hikes.
Camila needs to go on another 71 hikes - 7 hikes = <<71-7=64>>64 hikes.
Camila will achieve her goal in 64 hikes / 4 hikes/week = <<64/4=16>>16 weeks.
#### 16 |
The regular hexagon \(ABCDEF\) has diagonals \(AC\) and \(CE\). The internal points \(M\) and \(N\) divide these diagonals such that \(AM: AC = CN: CE = r\). Determine \(r\) if it is known that points \(B\), \(M\), and \(N\) are collinear. | \frac{1}{\sqrt{3}} |
A triangle has sides of lengths 30, 70, and 80. When an altitude is drawn to the side of length 80, the longer segment of this side that is intercepted by the altitude is: | 65 |
Find the smallest two-digit prime number such that reversing the digits of the number produces an even number. | 23 |
What is the least natural number that can be added to 250,000 to create a palindrome? | 52 |
Janice bought 30 items each priced at 30 cents, 2 dollars, or 3 dollars. If her total purchase price was $\$$30.00, how many 30-cent items did she purchase? | 20 |
Suppose $x,y$ and $z$ are integers that satisfy the system of equations \[x^2y+y^2z+z^2x=2186\] \[xy^2+yz^2+zx^2=2188.\] Evaluate $x^2+y^2+z^2.$ | 245 |
Suppose $a$ and $b$ are single-digit positive integers chosen independently and at random. What is the probability that the point $(a,b)$ lies above the parabola $y=ax^2-bx$? | \frac{19}{81} |
Quadrilateral $ABCD$ has mid-segments $EF$ and $GH$ such that $EF$ goes from midpoint of $AB$ to midpoint of $CD$, and $GH$ from midpoint of $BC$ to midpoint of $AD$. Given that $EF$ and $GH$ are perpendicular, and the lengths are $EF = 18$ and $GH = 24$, find the area of $ABCD$. | 864 |
Jordan picked 54 mangoes from his tree. One-third of the mangoes were ripe while the other two-thirds were not yet ripe. Jordan kept 16 unripe mangoes and gave the remainder to his sister who pickled them in glass jars. If it takes 4 mangoes to fill a jar, how many jars of pickled mangoes can Jordan's sister make? | Jordan picked 54/3 = <<54/3=18>>18 ripe mangoes.
He picked 54 - 18 = <<54-18=36>>36 unripe mangoes.
Jordan gave 36 - 16 = <<36-16=20>>20 unripe mangoes to his sister.
Jordan's sister can make 20/4 = <<20/4=5>>5 jars of pickled mangoes.
#### 5 |
Given a regular triangular pyramid $P-ABC$ (the base triangle is an equilateral triangle, and the vertex $P$ is the center of the base) with all vertices lying on the same sphere, and $PA$, $PB$, $PC$ are mutually perpendicular, and the side length of the base equilateral triangle is $\sqrt{2}$, calculate the volume of... | \frac{\sqrt{3}\pi}{2} |
Carrie sends five text messages to her brother each Saturday and Sunday, and two messages on other days. Over four weeks, how many text messages does Carrie send? | 80 |
Let $\triangle X Y Z$ be a right triangle with $\angle X Y Z=90^{\circ}$. Suppose there exists an infinite sequence of equilateral triangles $X_{0} Y_{0} T_{0}, X_{1} Y_{1} T_{1}, \ldots$ such that $X_{0}=X, Y_{0}=Y, X_{i}$ lies on the segment $X Z$ for all $i \geq 0, Y_{i}$ lies on the segment $Y Z$ for all $i \geq 0,... | 1 |
Find the absolute value of the difference of single-digit integers $A$ and $B$ such that $$ \begin{array}{c@{}c@{\;}c@{}c@{}c@{}c} & & & B& B & A_6\\ & & & \mathbf{4} & \mathbf{1} & B_6\\& & + & A & \mathbf{1} & \mathbf{5_6}\\ \cline{2-6} & & A & \mathbf{1} & \mathbf{5} & \mathbf{2_6} \\ \end{array} $$Express your ans... | 1_6 |
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