problem stringlengths 10 5.15k | answer stringlengths 0 1.23k |
|---|---|
Four days ago, Carlos bought a bag of gummy worms. Every day he ate half of the remaining gummy worms. After eating half of the remainder on the fourth day, he now has 4 gummy worms left. How many gummy worms were in the bag when Carlos bought it? | Let G be the number of gummy worms in the bag when Carlos bought it.
On the first day, he ate half the gummy worms, so he had G / 2 gummy worms.
On the second day, he ate half of the remaining gummy worms, so he had G / 2 / 2 = G / 4 gummy worms.
On the third day, he ate another half, so he had G / 4 / 2 = G / 8 gummy ... |
Given the sequence $\{a\_n\}$ that satisfies $a\_n-(-1)^{n}a\_{n-1}=n$ $(n\geqslant 2)$, and $S\_n$ is the sum of the first $n$ terms of the sequence, find the value of $S\_{40}$. | 440 |
Solve the following equations:
a) $\log _{1 / 5} \frac{2+x}{10}=\log _{1 / 5} \frac{2}{x+1}$;
b) $\log _{3}\left(x^{2}-4 x+3\right)=\log _{3}(3 x+21)$;
c) $\log _{1 / 10} \frac{2 x^{2}-54}{x+3}=\log _{1 / 10}(x-4)$;
d) $\log _{(5+x) / 3} 3=\log _{-1 /(x+1)} 3$. | -4 |
When programming a computer to print the first 10,000 natural numbers greater than 0: $1,2,3, \cdots, 10000$, the printer unfortunately has a malfunction. Each time it prints the digits 7 or 9, it prints $x$ instead. How many numbers are printed incorrectly? | 5904 |
What is the sum of the tens digit and the ones digit of the integer form of $(2+3)^{23}$? | 7 |
If $2^{x-3}=4^2$, find $x$. | 7 |
How many zeros are in the expansion of $999,\!999,\!999,\!998^2$? | 11 |
Let \( S = \{1, 2, \ldots, 280\} \). Find the smallest natural number \( n \) such that every \( n \)-element subset of \( S \) contains 5 pairwise coprime numbers. | 217 |
Six identical rectangles are arranged to form a larger rectangle \( ABCD \). The area of \( ABCD \) is 6000 square units. What is the length \( z \), rounded off to the nearest integer? | 32 |
Find the smallest \( n \) such that whenever the elements of the set \(\{1, 2, \ldots, n\}\) are colored red or blue, there always exist \( x, y, z, w \) (not necessarily distinct) of the same color such that \( x + y + z = w \). | 11 |
Calculate the volume of the tetrahedron with vertices at points \( A_{1}, A_{2}, A_{3}, A_{4} \). Additionally, find its height dropped from vertex \( A_{4} \) onto the face \( A_{1} A_{2} A_{3} \).
Vertices:
- \( A_{1}(-1, 2, 4) \)
- \( A_{2}(-1, -2, -4) \)
- \( A_{3}(3, 0, -1) \)
- \( A_{4}(7, -3, 1) \) | 24 |
During the first year, ABC's stock price starts at $ \$100 $ and increases $ 100\% $. During the second year, its stock price goes down $ 25\% $ from its price at the end of the first year. What is the price of the stock, in dollars, at the end of the second year? | \$150 |
What is the 100th letter in the pattern ABCABCABC...? | A |
How many times should two dice be rolled so that the probability of getting two sixes at least once is greater than $1/2$? | 25 |
A teacher received letters on Monday to Friday with counts of $10$, $6$, $8$, $5$, $6$ respectively. Calculate the standard deviation of this data set. | \dfrac {4 \sqrt {5}}{5} |
Find the smallest positive integer that cannot be expressed in the form $\frac{2^{a}-2^{b}}{2^{c}-2^{d}}$, where $a$, $b$, $c$, and $d$ are all positive integers. | 11 |
Alpha and Beta both took part in a two-day problem-solving competition. At the end of the second day, each had attempted questions worth a total of 500 points. Alpha scored 160 points out of 300 points attempted on the first day, and scored 140 points out of 200 points attempted on the second day. Beta who did not atte... | 849 |
A tank with a capacity of 8000 gallons is 3/4 full. Daxton empty's the tank by 40% of the total volume of water in the tank to water his vegetable farm. He then fills the tank with 30% of the volume of water remaining in the tank. Calculate the final volume of water in the tank. | The volume of water that is in the tank initially is 3/4*8000 = <<3/4*8000=6000>>6000 gallons.
When Daxton empties 40% of the water, he pumps out 40/100*6000 = <<2400=2400>>2400 gallons of water.
The total volume of water remaining in the tank is 6000-2400 = <<6000-2400=3600>>3600 gallons.
Daxton then fills the tank wi... |
Let $f(x) = 4x - 9$ and $g(f(x)) = 3x^2 + 4x - 2.$ Find $g(-10).$ | \frac{-45}{16} |
Given $a$ and $b$ are positive numbers such that $a^b = b^a$ and $b = 4a$, solve for the value of $a$. | \sqrt[3]{4} |
How many tetrahedrons can be formed using the vertices of a regular triangular prism? | 12 |
Jack needs to put his shoes on, then help both his toddlers tie their shoes. If it takes Jack 4 minutes to put his shoes on, and three minutes longer to help each toddler with their shoes, how long does it take them to get ready? | First figure out how long it takes to help one toddler: 4 minutes + 3 minutes = <<4+3=7>>7 minutes
Then multiply the time per toddler by the number of toddlers: 7 minutes/toddler * 2 toddlers = <<7*2=14>>14 minutes
Now add the time spent helping the toddlers to the time Jack spends on his own shoes: 14 minutes + 4 minu... |
Let $ABC$ be a triangle with incenter $I$, centroid $G$, and $|AC|>|AB|$. If $IG\parallel BC$, $|BC|=2$, and $\text{Area}(ABC)=3\sqrt{5}/8$, calculate $|AB|$. | \frac{9}{8} |
If \( \sqrt{\frac{3}{x} + 3} = \frac{5}{3} \), solve for \( x \). | -\frac{27}{2} |
What value of $k$ will make $x^2 - 16x + k$ the square of a binomial? | 64 |
How many obtuse interior angles are in an obtuse triangle? | 1 |
Karen has seven envelopes and seven letters of congratulations to various HMMT coaches. If she places the letters in the envelopes at random with each possible configuration having an equal probability, what is the probability that exactly six of the letters are in the correct envelopes? | 0 |
When $3z^3-4z^2-14z+3$ is divided by $3z+5$, the quotient is $z^2-3z+\frac{1}{3}$. What is the remainder? | \frac{4}{3} |
Find the largest positive integer $n$ such that
\[\sin^n x + \cos^n x \ge \frac{1}{n}\]for all real numbers $x.$ | 8 |
Marcos has to get across a 5 mile lake in his speedboat in 10 minutes so he can make it to work on time. How fast does he need to go in miles per hour to make it? | 10 minutes for 5 miles means 10 minutes / 5 miles = <<10/5=2>>2 minutes/mile
1 hour is 60 minutes so 60 minutes/hour / 2 minutes/mile = 30 miles/hour
#### 30 |
Johann and two friends are to deliver 180 pieces of certified mail. His friends each deliver 41 pieces of mail. How many pieces of mail does Johann need to deliver? | Friends = 41 * 2 = <<41*2=82>>82
Johann = 180 - 82 = <<180-82=98>>98
Johann must deliver 98 pieces of mail.
#### 98 |
In a factor tree, each value is the product of the two values below it, unless a value is a prime number already. What is the value of $A$ on the factor tree shown?
[asy]
draw((-1,-.3)--(0,0)--(1,-.3),linewidth(1));
draw((-2,-1.3)--(-1.5,-.8)--(-1,-1.3),linewidth(1));
draw((1,-1.3)--(1.5,-.8)--(2,-1.3),linewidth(1));
... | 900 |
The values of $x$ and $y$ are always positive, and $x^2$ and $y$ vary inversely. If $y$ is 10 when $x$ is 2, then find $x$ when $y$ is 4000. | \frac{1}{10} |
Compute the sum of the geometric series $-1 -3-9-27 -81-243-729$. | -1093 |
\[
1.047. \left(\frac{\sqrt{561^{2} - 459^{2}}}{4 \frac{2}{7} \cdot 0.15 + 4 \frac{2}{7} : \frac{20}{3}} + 4 \sqrt{10}\right) : \frac{1}{3} \sqrt{40}
\] | 125 |
$A B C D E$ is a cyclic convex pentagon, and $A C=B D=C E . A C$ and $B D$ intersect at $X$, and $B D$ and $C E$ intersect at $Y$. If $A X=6, X Y=4$, and $Y E=7$, then the area of pentagon $A B C D E$ can be written as $\frac{a \sqrt{b}}{c}$, where $a, b, c$ are integers, $c$ is positive, $b$ is square-free, and $\oper... | 2852 |
Three people, A, B, and C, start from point $A$ to point $B$. A starts at 8:00, B starts at 8:20, and C starts at 8:30. They all travel at the same speed. Ten minutes after C starts, the distance from A to point $B$ is exactly half the distance from B to point $B$. At this time, C is 2015 meters away from point $B$. Ho... | 2418 |
A unit cube has vertices $P_1,P_2,P_3,P_4,P_1',P_2',P_3',$ and $P_4'$. Vertices $P_2$, $P_3$, and $P_4$ are adjacent to $P_1$, and for $1\le i\le 4,$ vertices $P_i$ and $P_i'$ are opposite to each other. A regular octahedron has one vertex in each of the segments $\overline{P_1P_2}$, $\overline{P_1P_3}$, $\overline{P_1... | \frac{3 \sqrt{2}}{4} |
The difference \(\sqrt{|40 \sqrt{2}-57|}-\sqrt{40 \sqrt{2}+57}\) is an integer. Find this number. | -10 |
Given a six-digit phone number, how many different seven-digit phone numbers exist such that, by crossing out one digit, you obtain the given six-digit number? | 70 |
Right triangle $ACD$ with right angle at $C$ is constructed outwards on the hypotenuse $\overline{AC}$ of isosceles right triangle $ABC$ with leg length $1$, as shown, so that the two triangles have equal perimeters. What is $\sin(2\angle BAD)$? | \frac{7}{9} |
Sixteen 6-inch wide square posts are evenly spaced with 4 feet between them to enclose a square field. What is the outer perimeter, in feet, of the fence? | 56 |
What is the sum of three consecutive even integers if the sum of the first and third integers is $128$? | 192 |
An old car can drive 8 miles in one hour. After 5 hours of constant driving, the car needs to get cooled down which takes 1 hour. How many miles can this car drive in 13 hours? | In 5 hours the car can drive 8 * 5 = <<8*5=40>>40 miles.
One cycle of driving and pause takes 5 + 1 = <<5+1=6>>6 hours.
In 13 hours this cycle can fit 13 / 6 = 2 full cycles with a remainder of 1 hour
Two cycles will last for 2 * 6 = <<2*6=12>>12 hours.
During two full cycles, the car will drive 40 * 2 = <<40*2=80>>80 ... |
Find the absolute value of the difference of single-digit integers \( C \) and \( D \) such that in base \( 5 \):
$$ \begin{array}{c@{}c@{\;}c@{}c@{}c@{}c}
& & & D & D & C_5 \\
& & & \mathbf{3} & \mathbf{2} & D_5 \\
& & + & C & \mathbf{2} & \mathbf{4_5} \\
\cline{2-6}
& & C & \mathbf{2} & \mathbf{3} & \mathbf{1_5} \\
\... | 1_5 |
Let $ABC$ be a triangle with $|AB|=18$ , $|AC|=24$ , and $m(\widehat{BAC}) = 150^\circ$ . Let $D$ , $E$ , $F$ be points on sides $[AB]$ , $[AC]$ , $[BC]$ , respectively, such that $|BD|=6$ , $|CE|=8$ , and $|CF|=2|BF|$ . Let $H_1$ , $H_2$ , $H_3$ be the reflections of the orthocenter of triangle $AB... | 96 |
Let $a^2=\frac{16}{44}$ and $b^2=\frac{(2+\sqrt{5})^2}{11}$, where $a$ is a negative real number and $b$ is a positive real number. If $(a+b)^3$ can be expressed in the simplified form $\frac{x\sqrt{y}}{z}$ where $x$, $y$, and $z$ are positive integers, what is the value of the sum $x+y+z$? | 181 |
How many multiples of 5 are there between 5 and 205? | 41 |
Grisha wrote 100 numbers on the board. Then he increased each number by 1 and noticed that the product of all 100 numbers did not change. He increased each number by 1 again, and again the product of all the numbers did not change, and so on. Grisha repeated this procedure $k$ times, and each of the $k$ times the produ... | 99 |
0.8 + 0.02 | 0.82 |
A rectangle has one side of length 5 and the other side less than 4. When the rectangle is folded so that two opposite corners coincide, the length of the crease is \(\sqrt{6}\). Calculate the length of the other side. | \sqrt{5} |
In $\triangle ABC$, points $E$ and $F$ are on $AB$ and $BC$, respectively, such that $AE = BF$ and $BE = CF$. If $\angle BAC = 70^{\circ}$, what is the measure of $\angle ABC$? | 40^{\circ} |
Simplify $\sqrt[3]{1+8} \cdot \sqrt[3]{1+\sqrt[3]{8}}$. | 3 |
How much greater, in square inches, is the area of a circle of radius 20 inches than a circle of diameter 20 inches? Express your answer in terms of $\pi$. | 300\pi |
A rectangle in the coordinate plane has vertices at $(0, 0), (1000, 0), (1000, 1000),$ and $(0, 1000)$. Compute the radius $d$ to the nearest tenth such that the probability the point is within $d$ units from any lattice point is $\tfrac{1}{4}$. | 0.3 |
$\triangle ABC$ is inscribed in a semicircle of radius $r$ so that its base $AB$ coincides with diameter $AB$. Point $C$ does not coincide with either $A$ or $B$. Let $s=AC+BC$. Then, for all permissible positions of $C$:
$\textbf{(A)}\ s^2\le8r^2\qquad \textbf{(B)}\ s^2=8r^2 \qquad \textbf{(C)}\ s^2 \ge 8r^2 \qquad\\ ... | s^2 \le 8r^2 |
The batting cage sells golf balls by the dozen. They charge $30 for 3 dozen. Dan buys 5 dozen, Gus buys 2 dozen, and Chris buys 48 golf balls. How many golf balls do they purchase in total, assuming 12 golf balls are 1 dozen? | Dan gets 5*12 = <<5*12=60>>60 golf balls.
Gus gets 2*12 = <<2*12=24>>24 golf balls.
In total they purchase 60+24+48 = <<60+24+48=132>>132 golf balls
#### 132 |
Josie received $50 as a gift. She plans to buy two cassette tapes that cost $9 each and a headphone set that costs $25. How much money will she have left? | The cassette tapes will cost 2 × 9 = $<<2*9=18>>18.
She spends 18 + 25 = $<<18+25=43>>43 in total.
Josie will have 50 - 43 = $<<50-43=7>>7 left.
#### 7 |
Let \( f(x) = \frac{x + a}{x^2 + \frac{1}{2}} \), where \( x \) is a real number and the maximum value of \( f(x) \) is \( \frac{1}{2} \) and the minimum value of \( f(x) \) is \( -1 \). If \( t = f(0) \), find the value of \( t \). | -\frac{1}{2} |
Cynthia and Lynnelle are collaborating on a problem set. Over a $24$ -hour period, Cynthia and Lynnelle each independently pick a random, contiguous $6$ -hour interval to work on the problem set. Compute the probability that Cynthia and Lynnelle work on the problem set during completely disjoint intervals of time. | 4/9 |
Find the minimum of the function
\[\frac{xy}{x^2 + y^2}\]in the domain $\frac{2}{5} \le x \le \frac{1}{2}$ and $\frac{1}{3} \le y \le \frac{3}{8}.$ | \frac{6}{13} |
Heidi's apartment has 3 times as many rooms as Danielle's apartment. Grant's apartment has 1/9 as many rooms as Heidi's apartment. If Grant's apartment has 2 rooms, how many rooms does Danielle's apartment have? | Heidi's apartment has 2*9=<<2*9=18>>18 rooms.
Danielle's apartment has 18/3=<<18/3=6>>6 rooms.
#### 6 |
Consider the sequence of six real numbers 60, 10, 100, 150, 30, and $x$ . The average (arithmetic mean) of this sequence is equal to the median of the sequence. What is the sum of all the possible values of $x$ ? (The median of a sequence of six real numbers is the average of the two middle numbers after all the n... | 135 |
By what common fraction does $0.\overline{81}$ exceed $0.81$? | \frac{9}{1100} |
Three natural numbers are written on the board: two ten-digit numbers \( a \) and \( b \), and their sum \( a + b \). What is the maximum number of odd digits that could be written on the board? | 30 |
Consider the function $g(x)$ satisfying
\[ g(xy) = 2g(x)g(y) \]
for all real numbers $x$ and $y$ and $g(0) = 2.$ Find $g(10)$. | \frac{1}{2} |
Given the function $f(x) = x^3 - 6x + 5, x \in \mathbb{R}$.
(1) Find the equation of the tangent line to the function $f(x)$ at $x = 1$;
(2) Find the extreme values of $f(x)$ in the interval $[-2, 2]$. | 5 - 4\sqrt{2} |
If $1+2x+3x^2 + \dotsb=9$, find $x$. | \frac{2}{3} |
Multiply $2$ by $54$. For each proper divisor of $1,000,000$, take its logarithm base $10$. Sum these logarithms to get $S$, and find the integer closest to $S$. | 141 |
A quadrilateral $ABCD$ has a right angle at $\angle ABC$ and satisfies $AB = 12$ , $BC = 9$ , $CD = 20$ , and $DA = 25$ . Determine $BD^2$ .
.
| 769 |
Find the sum of the digits in the answer to
$\underbrace{9999\cdots 99}_{94\text{ nines}} \times \underbrace{4444\cdots 44}_{94\text{ fours}}$
where a string of $94$ nines is multiplied by a string of $94$ fours. | 846 |
Convert the binary number $110101_{(2)}$ to decimal. | 53 |
The polynomial $P(x) = x^3 + ax^2 + bx +c$ has the property that the mean of its zeros, the product of its zeros, and the sum of its coefficients are all equal. If the $y$-intercept of the graph of $y=
P(x)$ is 2, what is $b$? | -11 |
Define the sequence \(\{a_n\}\) where \(a_n = n^3 + 4\) for \(n \in \mathbf{N}_+\). Let \(d_n = \gcd(a_n, a_{n+1})\), which is the greatest common divisor of \(a_n\) and \(a_{n+1}\). Find the maximum value of \(d_n\). | 433 |
You are given that $x$ is directly proportional to $y^3$, and $y$ is inversely proportional to $\sqrt{z}$. If the value of $x$ is 3 when $z$ is $12$, what is the value of $x$ when $z$ is equal to $75$? Express your answer as a common fraction. | \frac{24}{125} |
Three students, A, B, and C, are playing badminton with the following rules:<br/>The player who loses two games in a row will be eliminated. Before the game, two players are randomly selected to play against each other, while the third player has a bye. The winner of each game will play against the player with the bye ... | \frac{7}{16} |
An ellipse is drawn with major and minor axes of lengths 10 and 8 respectively. Using one focus as a center, a circle is drawn that is tangent to the ellipse, with no part of the circle being outside the ellipse. Compute the radius of the circle. | 2 |
Ana's monthly salary was $2000$ in May. In June she received a 20% raise. In July she received a 20% pay cut. After the two changes in June and July, Ana's monthly salary was | 1920 |
The total in-store price for a blender is $\textdollar 129.95$. A television commercial advertises the same blender for four easy payments of $\textdollar 29.99$ and a one-time shipping and handling charge of $\textdollar 14.95$. Calculate the number of cents saved by purchasing the blender through the television adver... | 496 |
Triangle $ABC$ is isosceles with $AC = BC$ and $\angle ACB = 106^\circ.$ Point $M$ is in the interior of the triangle so that $\angle MAC = 7^\circ$ and $\angle MCA = 23^\circ.$ Find the number of degrees in $\angle CMB.$
[asy] pointpen = black; pathpen = black+linewidth(0.7); size(220); /* We will WLOG AB = 2 to draw... | 83^\circ |
Let $ y_0$ be chosen randomly from $ \{0, 50\}$ , let $ y_1$ be chosen randomly from $ \{40, 60, 80\}$ , let $ y_2$ be chosen randomly from $ \{10, 40, 70, 80\}$ , and let $ y_3$ be chosen randomly from $ \{10, 30, 40, 70, 90\}$ . (In each choice, the possible outcomes are equally likely to occur.) Let $ P... | 107 |
Given $$|\vec{a}|=3, |\vec{b}|=2$$. If $$\vec{a} \cdot \vec{b} = -3$$, then the angle between $$\vec{a}$$ and $$\vec{b}$$ is \_\_\_\_\_\_. | \frac{2}{3}\pi |
A facility has 7 consecutive parking spaces, and there are 3 different models of cars to be parked. If it is required that among the remaining 4 parking spaces, exactly 3 are consecutive, then the number of different parking methods is \_\_\_\_\_\_. | 72 |
On the extension of side $AD$ of rhombus $ABCD$, point $K$ is taken beyond point $D$. The lines $AC$ and $BK$ intersect at point $Q$. It is known that $AK=14$ and that points $A$, $B$, and $Q$ lie on a circle with a radius of 6, the center of which belongs to segment $AA$. Find $BK$. | 20 |
Today is my birthday and I'm three times older than I was six years ago. What is my age? | We know that my age divided by three is equal to my age minus six therefore X/3 = X-6 where X = My age
This means that X = 3X - 18 because we can multiply both sides by 3
This also means that -2X=-18 because we can subtract 3X from the right side.
Therefore X = 9 because - 18/-2 = <<-18/-2=9>>9
#### 9 |
Let $L(m)$ be the $x$ coordinate of the left end point of the intersection of the graphs of $y=x^2-6$ and $y=m$, where $-6<m<6$. Let $r=[L(-m)-L(m)]/m$. Then, as $m$ is made arbitrarily close to zero, the value of $r$ is: | \frac{1}{\sqrt{6}} |
In triangle $ABC$, $AB=13$, $BC=14$, and $CA=15$. Distinct points $D$, $E$, and $F$ lie on segments $\overline{BC}$, $\overline{CA}$, and $\overline{DE}$, respectively, such that $\overline{AD}\perp\overline{BC}$, $\overline{DE}\perp\overline{AC}$, and $\overline{AF}\perp\overline{BF}$. The length of segment $\overline... | 21 |
There are 10 6-ounces of glasses that are only 4/5 full of water. How many ounces of water are needed to fill to the brim all those 10 glasses? | A glass is filled with 6 x 4/5 = 24/5 ounces of water.
So, 10 6-ounces of glasses contains a total of 24/5 x 10 = <<24/5*10=48>>48 ounces of water.
To fill the 10 glasses to the brim, 10 x 6 = <<10*6=60>>60 ounces of water are needed.
Hence, 60 - 48 = <<60-48=12>>12 ounces of water are still needed.
#### 12 |
The cost of 60 copies of the first volume and 75 copies of the second volume is 2700 rubles. In reality, the total payment for all these books was only 2370 rubles because a discount was applied: 15% off the first volume and 10% off the second volume. Find the original price of these books. | 20 |
In five years, Rehana will be three times as old as Phoebe. If Rehana is currently 25 years old, and Jacob, Rehana's brother, is 3/5 of Phoebe's current age, how old is Jacob now? | If Rehana is currently 25 years old, she will be 25+5=<<25+5=30>>30 in five years.
Rehana will be three times as old as Phoebe in five years, meaning Phoebe will be 30/3=10 years old in five years.
Currently, Phoebe is 10-5=<<10-5=5>>5 years old.
3/5 of Phoebe's age now is 3/5*5=<<3/5*5=3>>3 years, which is Jacob's age... |
How many distinct five-digit positive integers are there such that the product of their digits equals 16? | 15 |
Let the hyperbola $C:\frac{x^2}{a^2}-y^2=1\;(a>0)$ intersect the line $l:x+y=1$ at two distinct points $A$ and $B$.
$(1)$ Find the range of real numbers for $a$.
$(2)$ If the intersection point of the line $l$ and the $y$-axis is $P$, and $\overrightarrow{PA}=\frac{5}{12}\overrightarrow{PB}$, find the value of the ... | a = \frac{17}{13} |
Let $P$ be an interior point of triangle $ABC$ . Let $a,b,c$ be the sidelengths of triangle $ABC$ and let $p$ be it's semiperimeter. Find the maximum possible value of $$ \min\left(\frac{PA}{p-a},\frac{PB}{p-b},\frac{PC}{p-c}\right) $$ taking into consideration all possible choices of triangle $ABC$ and o... | \frac{2}{\sqrt{3}} |
Jackie and Phil have two fair coins and a third coin that comes up heads with probability $\frac47$. Jackie flips the three coins, and then Phil flips the three coins. Let $\frac {m}{n}$ be the probability that Jackie gets the same number of heads as Phil, where $m$ and $n$ are relatively prime positive integers. Find ... | 515 |
Solve for $x$: $(x-4)^3=\left(\frac18\right)^{-1}$ | 6 |
In a square $ABCD$ with side length $4$, find the probability that $\angle AMB$ is an acute angle. | 1-\dfrac{\pi}{8} |
A point $(x,y)$ is randomly picked from inside the rectangle with vertices $(0,0)$, $(3,0)$, $(3,2)$, and $(0,2)$. What is the probability that $x < y$? | \dfrac{1}{3} |
Seven cards numbered $1$ through $7$ are to be lined up in a row. Find the number of arrangements of these seven cards where one of the cards can be removed, leaving the remaining six cards in either ascending or descending order. | 74 |
To the eight-digit number 20222023, append one digit to the left and one digit to the right so that the resulting ten-digit number is divisible by 72. Determine all possible solutions. | 3202220232 |
What is the probability, expressed as a decimal, of drawing one marble which is either green or white from a bag containing 4 green, 3 white, and 8 black marbles? | 0.4667 |
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