problem stringlengths 10 5.15k | answer stringlengths 0 1.23k |
|---|---|
Peter and Kristin are to read 20 fantasy novels each in a week. Peter can read three times as fast as Kristin. If Peter reads one book in 18 hours, how long will Kristin read half of her books? | Since Peter reads three times as fast as Kristin, Kristin will take 18*3 = <<18*3=54>>54 hours to read one book.
Kristin reads half of her books, a total of 20/2 = <<20/2=10>>10 books.
The time Kristin will take to read half of her books is 10*54 = <<10*54=540>>540
#### 540 |
What is the period of $y = \tan \frac{x}{2}$? | 2 \pi |
Yu Semo and Yu Sejmo have created sequences of symbols $\mathcal{U} = (\text{U}_1, \ldots, \text{U}_6)$ and $\mathcal{J} = (\text{J}_1, \ldots, \text{J}_6)$ . These sequences satisfy the following properties.
- Each of the twelve symbols must be $\Sigma$ , $\#$ , $\triangle$ , or $\mathbb{Z}$ .
- In each of th... | 24 |
Kat decides she wants to start a boxing career. She gets a gym membership and spends 1 hour in the gym 3 times a week doing strength training. She also trained at the boxing gym 4 times a week for 1.5 hours. How many hours a week does she train? | She strength trains 3*1=<<3*1=3>>3 hours a week
She does boxing training 4*1.5=<<4*1.5=6>>6 hours a week
So she trains a total of 3+6=<<3+6=9>>9 hours a week
#### 9 |
A board game spinner is divided into three regions labeled $A$, $B$ and $C$. The probability of the arrow stopping on region $A$ is $\frac{1}{3}$ and on region $B$ is $\frac{1}{2}$. What is the probability of the arrow stopping on region $C$? Express your answer as a common fraction. | \frac{1}{6} |
Find the minimum value of the following function $f(x) $ defined at $0<x<\frac{\pi}{2}$ .
\[f(x)=\int_0^x \frac{d\theta}{\cos \theta}+\int_x^{\frac{\pi}{2}} \frac{d\theta}{\sin \theta}\] | \ln(3 + 2\sqrt{2}) |
In a certain sequence, the first term is \(a_1 = 1010\) and the second term is \(a_2 = 1011\). The values of the remaining terms are chosen so that \(a_n + a_{n+1} + a_{n+2} = 2n\) for all \(n \geq 1\). Determine \(a_{1000}\). | 1676 |
Luke takes a 70-minute bus to work every day. His coworker Paula takes 3/5 of this time to arrive by bus at work. If Luke takes a bike ride back home every day, 5 times slower than the bus, and Paula takes the bus back home, calculate the total amount of time, in minutes, they take traveling from home to work and back ... | When Luke takes the 70-minute bus to work, his coworker Paula takes 3/5*70 = <<70*3/5=42>>42 minutes to travel from home to work.
From work to home, Paula spends the same amount of time she spent going to work, giving a total of 42+42 = <<42+42=84>>84 minutes in a day traveling.
From work to home, while riding the bike... |
What is the largest integer that must divide the product of any $4$ consecutive integers? | 24 |
The $120$ permutations of $AHSME$ are arranged in dictionary order as if each were an ordinary five-letter word.
The last letter of the $86$th word in this list is: | E |
If $8^x = 32$, then $x$ equals: | \frac{5}{3} |
Find the coefficient of the $x^3$ term in the expansion of the product $$(3x^3 + 2x^2 + 4x + 5)(4x^2 + 5x + 6).$$ | 44 |
Which of the following fractions has the greatest value: $\frac{3}{10}$, $\frac{4}{7}$, $\frac{5}{23}$, $\frac{2}{3}$, $\frac{1}{2}$? | \frac{2}{3} |
What is the product of the numerator and the denominator when $0.\overline{018}$ is expressed as a fraction in lowest terms? | 222 |
Let $f(x)$ be the function defined on $-1\le x\le 1$ by the formula $$f(x)=1-\sqrt{1-x^2}.$$This is a graph of $y=f(x)$: [asy]
import graph; size(4cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-1.5,xmax=1.5,ymin=-1.5,ymax=1.5;
pen cqcqcq=rgb(0.75,0.75,0.75);
/*grid*/... | 0.57 |
Given a function defined on the set of positive integers as follows:
\[ f(n) = \begin{cases}
n - 3, & \text{if } n \geq 1000 \\
f[f(n + 7)], & \text{if } n < 1000
\end{cases} \]
Find the value of \( f(90) \). | 999 |
In the Cartesian coordinate plane $(xOy)$, a point $A(2,0)$, a moving point $B$ on the curve $y= \sqrt {1-x^{2}}$, and a point $C$ in the first quadrant form an isosceles right triangle $ABC$ with $\angle A=90^{\circ}$. The maximum length of the line segment $OC$ is _______. | 1+2 \sqrt {2} |
Let $p(x)$ and $q(x)$ be two cubic polynomials such that $p(0)=-24, q(0)=30$, and $p(q(x))=q(p(x))$ for all real numbers $x$. Find the ordered pair $(p(3), q(6))$. | (3,-24) |
We draw the diagonals of the convex quadrilateral $ABCD$, then find the centroids of the 4 triangles formed. What fraction of the area of quadrilateral $ABCD$ is the area of the quadrilateral determined by the 4 centroids? | \frac{2}{9} |
There are $2n$ complex numbers that satisfy both $z^{28} - z^{8} - 1 = 0$ and $\mid z \mid = 1$. These numbers have the form $z_{m} = \cos\theta_{m} + i\sin\theta_{m}$, where $0\leq\theta_{1} < \theta_{2} < \ldots < \theta_{2n} < 360$ and angles are measured in degrees. Find the value of $\theta_{2} + \theta_{4} + \ldo... | 840 |
The product of the digits of 4321 is 24. How many distinct four-digit positive integers are such that the product of their digits equals 18? | 24 |
What is the smallest positive integer $n$ for which $n^2$ is divisible by 18 and $n^3$ is divisible by 640? | 120 |
Given $m$ and $n∈\{lg2+lg5,lo{g}_{4}3,{(\frac{1}{3})}^{-\frac{3}{5}},tan1\}$, the probability that the function $f\left(x\right)=x^{2}+2mx+n^{2}$ has two distinct zeros is ______. | \frac{3}{8} |
A right triangle when rotating around a large leg forms a cone with a volume of $100\pi$ . Calculate the length of the path that passes through each vertex of the triangle at rotation of $180^o$ around the point of intersection of its bisectors, if the sum of the diameters of the circles, inscribed in the triangle a... | 30 |
Find the smallest positive integer divisible by $10$, $11$, and $12$. | 660 |
A firecracker was thrown vertically upwards with a speed of 20 m/s. Three seconds after the start of its flight, it exploded into two unequal parts, the mass ratio of which is $1: 2$. The smaller fragment immediately after the explosion flew horizontally at a speed of $16 \mathrm{~m}/\mathrm{s}$. Find the magnitude of... | 17 |
Let $S$ be a list of positive integers--not necessarily distinct--in which the number $68$ appears. The average (arithmetic mean) of the numbers in $S$ is $56$. However, if $68$ is removed, the average of the remaining numbers drops to $55$. What is the largest number that can appear in $S$? | 649 |
Rectangle ABCD has dimensions AB=CD=4 and BC=AD=8. The rectangle is rotated 90° clockwise about corner D, then rotated 90° clockwise about the corner C's new position after the first rotation. What is the length of the path traveled by point A?
A) $8\sqrt{5}\pi$
B) $4\sqrt{5}\pi$
C) $2\sqrt{2}\pi$
D) $4\sqrt{2}\p... | 4\sqrt{5}\pi |
Let $x$ , $y$ , $z$ be positive integers satisfying $x<y<z$ and $x+xy+xyz=37$ . Find the greatest possible value of $x+y+z$ . | 20 |
In the number $56439.2071$, the value of the place occupied by the digit 6 is how many times as great as the value of the place occupied by the digit 2? | 10,000 |
The perpendicular bisectors of the sides of triangle $ABC$ meet its circumcircle at points $A',$ $B',$ and $C',$ as shown. If the perimeter of triangle $ABC$ is 35 and the radius of the circumcircle is 8, then find the area of hexagon $AB'CA'BC'.$
[asy]
unitsize(2 cm);
pair A, B, C, Ap, Bp, Cp, O;
O = (0,0);
A = di... | 140 |
On a certain math exam, $10\%$ of the students got $70$ points, $25\%$ got $80$ points, $20\%$ got $85$ points, $15\%$ got $90$ points, and the rest got $95$ points. What is the difference between the mean and the median score on this exam? | 1 |
An equilateral triangle and a square both have perimeters of 48 inches. What is the ratio of the length of the side of the triangle to the length of the side of the square? Express your answer as a common fraction. | \frac43 |
Find $\cot (-60^\circ).$ | -\frac{\sqrt{3}}{3} |
Marcus has three times as many cheese crackers as Mona. Nicholas has 6 more crackers than Mona. If Marcus has 27 crackers, how many crackers does Nicholas have? | Mona has 27 / 3 = <<27/3=9>>9 cheese crackers.
Nicholas has 9 + 6 = <<9+6=15>>15 cheese crackers.
#### 15 |
Determine how many solutions the following equation has:
\[
\frac{(x-1)(x-2)(x-3)\dotsm(x-50)}{(x-2^2)(x-4^2)(x-6^2)\dotsm(x-24^2)} = 0
\] | 47 |
When three different numbers from the set $\{ -3, -2, -1, 4, 5 \}$ are multiplied, the largest possible product is | 30 |
A fair die is rolled six times. The probability of rolling at least a five at least five times is | \frac{13}{729} |
The longest seminar session and the closing event lasted a total of $4$ hours and $45$ minutes plus $135$ minutes, plus $500$ seconds. Convert this duration to minutes and determine the total number of minutes. | 428 |
James was 2/3s as tall as his uncle who is 72 inches. He gets a growth spurt that makes him gain 10 inches. How much taller is his uncle than James now? | He was 72*2/3=<<72*2/3=48>>48 inches tall before.
He is now 48+10=<<48+10=58>>58 inches.
So his uncle is 72-58=<<72-58=14>>14 inches taller.
#### 14 |
Consider a square-based pyramid (with base vertices $A, B, C, D$) with equal side edges, and let the apex be $E$. Let $P$ be the point that divides the side edge $A E$ in a ratio of 3:1, such that $E P : P A = 3$, and let $Q$ be the midpoint of the side edge $C E$. In what ratio does the plane passing through points $D... | 4/3 |
In the triangle shown, $n$ is a positive integer, and $\angle A > \angle B > \angle C$. How many possible values of $n$ are there? [asy]
draw((0,0)--(1,0)--(.4,.5)--cycle);
label("$A$",(.4,.5),N); label("$B$",(1,0),SE); label("$C$",(0,0),SW);
label("$2n + 12$",(.5,0),S); label("$3n - 3$",(.7,.25),NE); label("$2n + 7$"... | 7 |
Find the minimum value of the function \( f(x)=\cos 4x + 6\cos 3x + 17\cos 2x + 30\cos x \) for \( x \in \mathbb{R} \). | -18 |
If $\tan (\alpha+\beta)= \frac {3}{4}$ and $\tan (\alpha- \frac {\pi}{4})= \frac {1}{2}$, find the value of $\tan (\beta+ \frac {\pi}{4})$. | \frac {2}{11} |
Find constants $A,$ $B,$ and $C$ so that
\[\frac{4x}{(x - 5)(x - 3)^2} = \frac{A}{x - 5} + \frac{B}{x - 3} + \frac{C}{(x - 3)^2}.\]Enter the ordered triple $(A,B,C).$ | (5,-5,-6) |
In triangle $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. It is known that $4a = \sqrt{5}c$ and $\cos C = \frac{3}{5}$.
$(Ⅰ)$ Find the value of $\sin A$.
$(Ⅱ)$ If $b = 11$, find the area of $\triangle ABC$. | 22 |
A rectangular box has a total surface area of 94 square inches. The sum of the lengths of all its edges is 48 inches. What is the sum of the lengths in inches of all of its interior diagonals? | 20\sqrt{2} |
Sienna gave Bailey half of her suckers. Jen ate half and gave the rest to Molly. Molly ate 2 and gave the rest to Harmony. Harmony kept 3 and passed the remainder to Taylor. Taylor ate one and gave the last 5 to Callie. How many suckers did Jen eat? | Taylor received 5 + 1 = <<5+1=6>>6 suckers.
Harmony was given 3 + 6 = <<3+6=9>>9 suckers.
Molly received 9 + 2 = <<9+2=11>>11 suckers from Jen.
Jen originally had 11 * 2 = <<11*2=22>>22 suckers.
Jen ate 22 – 11 = <<22-11=11>>11 suckers.
#### 11 |
For what values of the constant $c$ does the graph of $f(x) = \frac{x^2-x+c}{x^2+x-20}$ have exactly one vertical asymptote?
Enter all possible values, separated by commas. | -12 \text{ or } -30 |
A basketball player made 5 baskets during a game. Each basket was worth either 2 or 3 points. How many different numbers could represent the total points scored by the player? | 6 |
Jennifer wants to do origami, and she has a square of side length $ 1$ . However, she would prefer to use a regular octagon for her origami, so she decides to cut the four corners of the square to get a regular octagon. Once she does so, what will be the side length of the octagon Jennifer obtains? | 1 - \frac{\sqrt{2}}{2} |
A triangle in a Cartesian coordinate plane has vertices (5, -2), (10, 5) and (5, 5). How many square units are in the area of the triangle? Express your answer as a decimal to the nearest tenth. | 17.5 |
Given that the scores X of 10000 people approximately follow a normal distribution N(100,13^2), it is given that P(61 < X < 139)=0.997, find the number of people scoring no less than 139 points in this exam. | 15 |
If $V = gt + V_0$ and $S = \frac{1}{2}gt^2 + V_0t$, then $t$ equals: | \frac{2S}{V+V_0} |
A sequence $(c_n)$ is defined as follows: $c_1 = 1$, $c_2 = \frac{1}{3}$, and
\[c_n = \frac{2 - c_{n-1}}{3c_{n-2}}\] for all $n \ge 3$. Find $c_{100}$. | \frac{1}{3} |
Given a square $ABCD$ . Let $P\in{AB},\ Q\in{BC},\ R\in{CD}\ S\in{DA}$ and $PR\Vert BC,\ SQ\Vert AB$ and let $Z=PR\cap SQ$ . If $BP=7,\ BQ=6,\ DZ=5$ , then find the side length of the square. | 10 |
Jonas is trying to expand his wardrobe. He has 20 pairs of socks, 5 pairs of shoes, 10 pairs of pants and 10 t-shirts. How many pairs of socks does he need to buy to double the number of individual items in his wardrobe? | First, we must recognize that all pairs of shoes and socks are actually two items each, whereas the pants and the t-shirts are individual items. Therefore there are 20*2= <<20*2=40>>40 individual socks in 20 pairs of socks.
Similarly, there are 5*2=<<5*2=10>>10 individual shoes in 5 pairs of shoes.
Therefore, Jonas has... |
Given the parametric equations of curve C as $$\begin{cases} x=2\cos\theta \\ y= \sqrt {3}\sin\theta \end{cases}(\theta\text{ is the parameter})$$, in the same Cartesian coordinate system, the points on curve C are transformed by the coordinate transformation $$\begin{cases} x'= \frac {1}{2}x \\ y'= \frac {1}{ \sqrt {3... | \frac {3 \sqrt {3}}{5} |
In a geometric sequence with positive terms $\{a_n\}$, where $a_5= \frac {1}{2}$ and $a_6 + a_7 = 3$, find the maximum positive integer value of $n$ such that $a_1 + a_2 + \ldots + a_n > a_1 a_2 \ldots a_n$. | 12 |
The set of $x$-values satisfying the inequality $2 \leq |x-1| \leq 5$ is: | -4\leq x\leq-1\text{ or }3\leq x\leq 6 |
If 15 bahs are equal to 24 rahs, and 9 rahs are equal in value to 15 yahs, how many bahs are equal in value to 1000 yahs? | 375 |
Express the sum as a common fraction: $.2 + .04 + .006 + .0008 + .00001.$ | \frac{24681}{100000} |
The bottoms of two vertical poles are 20 feet apart on a flat ground. One pole is 8 feet tall and the other is 18 feet tall. Simultaneously, the ground between the poles is sloped, with the base of the taller pole being 2 feet higher than the base of the shorter pole due to the slope. Calculate the length in feet of a ... | \sqrt{544} |
There were 100 people in attendance at the school dance. Ten percent of the attendees were school faculty and staff. Of the remaining attendees, two-thirds were girls. How many boys attended the school dance? | Of 100 people in attendance at the school dance, 10% were school faculty and staff, or a total of 0.1*100=<<100*0.1=10>>10 people.
This leaves 100-10=<<100-10=90>>90 students in attendance.
Of the remaining attendees, two-thirds were girls, or (2/3)*90=<<(2/3)*90=60>>60 attendees were girls.
Thus, the remaining 90-60=<... |
In a football tournament, 15 teams participated, each playing exactly once against every other team. A win awarded 3 points, a draw 1 point, and a loss 0 points.
After the tournament ended, it was found that some 6 teams each scored at least $N$ points. What is the maximum possible integer value of $N$? | 34 |
A 0-1 sequence of length $2^k$ is given. Alice can pick a member from the sequence, and reveal it (its place and its value) to Bob. Find the largest number $s$ for which Bob can always pick $s$ members of the sequence, and guess all their values correctly.
Alice and Bob can discuss a strategy before the game with the ... | k+1 |
Giselle will combine blue paint, green paint, and white paint in the ratio $3:2:4$, respectively. If she uses $12$ quarts of white paint, how many quarts of green paint should she use? | 6 |
Given $\sin\left(\frac{\pi}{3} - \alpha\right) = \frac{1}{3}$, calculate $\sin\left(\frac{\pi}{6} - 2\alpha\right)$. | - \frac{7}{9} |
1. Given that the line $l$ passing through the point $M(-3,0)$ is intercepted by the circle $x^{2}+(y+2)^{2}=25$ to form a chord of length $8$, what is the equation of line $l$?
2. Circles $C_{1}$: $x^{2}+y^{2}+2x+8y-8=0$ and $C_{2}$: $x^{2}+y^{2}-4x-4y-2=0$ intersect. What is the length of their common chord?
3. What ... | 16 |
Count how many 8-digit numbers there are that contain exactly four nines as digits. | 433755 |
In preparation for his mountain climbing, Arvin wants to run a total of 20 kilometers in a week. On the first day, he ran 2 kilometers. On each subsequent day, he increased his running distance by 1 kilometer over the previous day. If he runs for 5 days a week, how many kilometers did he run on the 5th day? | On the second day, Arvin ran 2 + 1 = <<2+1=3>>3 kilometers.
On the third day, he ran 3 + 1 = <<3+1=4>>4 kilometers.
On the fourth day, he ran 4 + 1 = <<4+1=5>>5 kilometers.
So on the fifth day, he ran 5 + 1 = <<5+1=6>>6 kilometers.
#### 6 |
Arrange the 7 numbers $39, 41, 44, 45, 47, 52, 55$ in a sequence such that the sum of any three consecutive numbers is a multiple of 3. What is the maximum value of the fourth number in all such arrangements? | 47 |
Find the minimum value of $9^x - 3^x + 1$ over all real numbers $x.$ | \frac{3}{4} |
Let $P$ units be the increase in circumference of a circle resulting from an increase in $\pi$ units in the diameter. Then $P$ equals: | \pi^2 |
What is the sum of all positive integers less than 100 that are squares of perfect squares? | 98 |
For $a>0$ , denote by $S(a)$ the area of the part bounded by the parabolas $y=\frac 12x^2-3a$ and $y=-\frac 12x^2+2ax-a^3-a^2$ .
Find the maximum area of $S(a)$ . | \frac{8\sqrt{2}}{3} |
The teacher gave each of her $37$ students $36$ pencils in different colors. It turned out that each pair of students received exactly one pencil of the same color. Determine the smallest possible number of different colors of pencils distributed. | 666 |
The product of two positive integers plus their sum is 95. The integers are relatively prime, and each is less than 20. What is the sum of the two integers? | 18 |
In the land of Draconia, there are red, green, and blue dragons. Each dragon has three heads, and each head always tells the truth or always lies. Additionally, each dragon has at least one head that tells the truth. One day, 530 dragons sat around a round table, and each of them said:
- 1st head: "To my left is a gre... | 176 |
The cards in a stack are numbered consecutively from 1 to $2n$ from top to bottom. The top $n$ cards are removed to form pile $A$ and the remaining cards form pile $B$. The cards are restacked by alternating cards from pile $B$ and $A$, starting with a card from $B$. Given this process, find the total number of cards (... | 402 |
Chris’s internet bill is $45 per month for 100 GB and $0.25 for every 1 GB over. His bill for this month is $65. How many GB over was Chris charged for on this bill? | Let G stand for the number of GB billed, so the total bill is $0.25G + $45 flat rate charge = $65 total bill.
Now we will solve for G. $0.25G + $45 flat rate - $45 = $65 -$45. This simplifies to $0.25G = $20
$0.25G/$0.25 = $20/$0.25
G = 80 GB were charged on this bill.
#### 80 |
Determine $S$, the sum of all the real coefficients of the expansion of $(1+ix)^{2020}$, and find $\log_2(S)$. | 1010 |
A sequence consists of 2010 terms. Each term after the first is 1 larger than the previous term. The sum of the 2010 terms is 5307. What is the sum when every second term is added up, starting with the first term and ending with the second last term? | 2151 |
Given that $\{a_n\}$ is a geometric sequence with a common ratio of $q$, and $a_m$, $a_{m+2}$, $a_{m+1}$ form an arithmetic sequence.
(Ⅰ) Find the value of $q$;
(Ⅱ) Let the sum of the first $n$ terms of the sequence $\{a_n\}$ be $S_n$. Determine whether $S_m$, $S_{m+2}$, $S_{m+1}$ form an arithmetic sequence and exp... | -\frac{1}{2} |
A wire of length $80$cm is randomly cut into three segments. The probability that each segment is no less than $20$cm is $\_\_\_\_\_\_\_.$ | \frac{1}{16} |
Mike has two containers. One container is a rectangular prism with width 2 cm, length 4 cm, and height 10 cm. The other is a right cylinder with radius 1 cm and height 10 cm. Both containers sit on a flat surface. Water has been poured into the two containers so that the height of the water in both containers is the sa... | 7.2 |
In triangle $\triangle ABC$, $a=7$, $b=8$, $A=\frac{\pi}{3}$.
1. Find the value of $\sin B$.
2. If $\triangle ABC$ is an obtuse triangle, find the height on side $BC$. | \frac{12\sqrt{3}}{7} |
Suppose that $f(x)$ is a linear function satisfying the equation $f(x) = 4f^{-1}(x) + 6$. Given that $f(1) = 4$, find $f(2)$. | 6 |
Solve for $x$: $0.05x + 0.07(30 + x) = 15.4$. | 110.8333 |
A $\textit{palindrome}$ is a number which reads the same forward as backward, for example 313 or 1001. Ignoring the colon, how many different palindromes are possible on a 12-hour digital clock displaying only the hours and minutes? (Notice a zero may not be inserted before a time with a single-digit hour value. Theref... | 57 |
On the game show $\text{\emph{Wheel of Fortune II}}$, you observe a spinner with the labels ["Bankrupt", "$\$700$", "$\$900$", "$\$200$", "$\$3000$", "$\$800$"]. Given that each region has equal area, determine the probability of earning exactly $\$2400$ in your first three spins. | \frac{1}{36} |
Freddy the frog is jumping around the coordinate plane searching for a river, which lies on the horizontal line $y = 24$. A fence is located at the horizontal line $y = 0$. On each jump Freddy randomly chooses a direction parallel to one of the coordinate axes and moves one unit in that direction. When he is at a point... | 273 |
The vertices of a $3 - 4 - 5$ right triangle are the centers of three mutually externally tangent circles, as shown. What is the sum of the areas of these circles?
[asy]unitsize(1cm);
draw(Circle((1.8,2.4),1),linewidth(0.7));
draw(Circle((0,0),2),linewidth(0.7));
draw(Circle((5,0),3),linewidth(0.7));
draw((0,0)--(5,... | 14\pi |
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 |
Prudence was starting a cupcake business. She figured that each cupcake cost $0.75 to make. The first 2 dozen that she made burnt and she had to throw them out. The next 2 came out perfectly and she ended up eating 5 cupcakes right away. Later that day she made 2 more dozen cupcakes and decided to eat 4 more. If sh... | She burnt 2 dozen, cooked 2 dozen and then another 2 dozen that evening. So she made 2+2+2 =<<2+2+2=6>>6 dozen cupcakes
Each dozen makes 12 cupcakes, so she made 6*12= <<6*12=72>>72 cupcakes
She threw away 2 dozen cupcakes, so 2*12 = <<2*12=24>>24 cupcakes
She made 72, threw away 24, ate 5 and then ate another 4 so she... |
Given a square pyramid \(M-ABCD\) with a square base such that \(MA = MD\), \(MA \perp AB\), and the area of \(\triangle AMD\) is 1, find the radius of the largest sphere that can fit into this square pyramid. | \sqrt{2} - 1 |
Two circles of radii 4 and 5 are externally tangent to each other and are circumscribed by a third circle. Find the area of the shaded region created in this way. Express your answer in terms of $\pi$. | 40\pi |
Convex quadrilateral $ABCD$ has $AB = 9$ and $CD = 12$. Diagonals $AC$ and $BD$ intersect at $E$, $AC = 14$, and $\triangle AED$ and $\triangle BEC$ have equal areas. What is $AE$? | 6 |
In how many distinct ways can I arrange my five keys on a keychain, if I want to put my house key next to my car key? Two arrangements are not considered different if the keys are in the same order (or can be made to be in the same order without taking the keys off the chain--that is, by reflection or rotation). | 6 |
Petya and Vasya are playing the following game. Petya thinks of a natural number \( x \) with a digit sum of 2012. On each turn, Vasya chooses any natural number \( a \) and finds out the digit sum of the number \( |x-a| \) from Petya. What is the minimum number of turns Vasya needs to determine \( x \) with certainty? | 2012 |
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