problem stringlengths 10 5.15k | answer stringlengths 0 1.23k |
|---|---|
Find the sum of the digits of the number \( A \), if \( A=2^{63} \cdot 4^{25} \cdot 5^{106}-2^{22} \cdot 4^{44} \cdot 5^{105}-1 \). | 959 |
The product of three different positive integers is equal to $7^3$. What is the sum of the three integers? | 57 |
My brother and I have thirty minutes to go to school, without being late. It takes us 15 minutes to arrive at the school gate, and another 6 minutes to get to the school building. How much time do we have to get to our room, without being late? | The total time we spent going to school is 15 minutes + 6 minutes = <<15+6=21>>21 minutes.
We have 30 minutes - 21 minutes = <<30-21=9>>9 minutes left not to be late.
#### 9 |
Riley has 64 cubes with dimensions \(1 \times 1 \times 1\). Each cube has its six faces labeled with a 2 on two opposite faces and a 1 on each of its other four faces. The 64 cubes are arranged to build a \(4 \times 4 \times 4\) cube. Riley determines the total of the numbers on the outside of the \(4 \times 4 \times 4... | 49 |
What is the value of $1234 + 2341 + 3412 + 4123$ | 11110 |
12 balls numbered 1 through 12 are placed in a bin. In how many ways can 3 balls be drawn, in order, from the bin, if each ball remains outside the bin after it is drawn? | 1320 |
Li Qiang rented a piece of land from Uncle Zhang, for which he has to pay Uncle Zhang 800 yuan and a certain amount of wheat every year. One day, he did some calculations: at that time, the price of wheat was 1.2 yuan per kilogram, which amounted to 70 yuan per mu of land; but now the price of wheat has risen to 1.6 yu... | 20 |
Let $f : \mathbb{R} \to \mathbb{R}$ be a function such that
\[f(xf(y) + x) = xy + f(x)\]for all $x,$ $y.$
Let $n$ be the number of possible values of $f(2),$ and let $s$ be the sum of all possible values of $f(2).$ Find $n \times s.$ | 0 |
Given that the central angle of a sector is $\frac{3}{2}$ radians, and its radius is 6 cm, then the arc length of the sector is \_\_\_\_\_\_ cm, and the area of the sector is \_\_\_\_\_\_ cm<sup>2</sup>. | 27 |
What is the smallest odd number with four different prime factors? | 1155 |
John wants to find all the five-letter words that begin and end with the same letter. How many combinations of letters satisfy this property? | 456976 |
An integer has exactly 4 prime factors, and the sum of the squares of these factors is 476. Find this integer. | 1989 |
A single bench section at a school event can hold either $7$ adults or $11$ children. When $N$ bench sections are connected end to end, an equal number of adults and children seated together will occupy all the bench space. What is the least possible positive integer value of $N?$ | 77 |
Find all prime numbers $p,q,r$ , such that $\frac{p}{q}-\frac{4}{r+1}=1$ | \[
(7, 3, 2), (3, 2, 7), (5, 3, 5)
\] |
Given a random variable $X \sim B(10, 0.6)$, calculate the values of $E(X)$ and $D(X)$. | 2.4 |
Armand is playing a guessing game with his dad where he has to guess a number his dad is thinking of. His dad tells him that the number, when multiplied by 3, is three less than twice 51. What is the number? | 102 is twice 51 because 2 x 51 = <<2*51=102>>102
Three less than 102 is 99 because 102 -3 = <<102-3=99>>99
The number is 33 because 99 / 3 = <<99/3=33>>33
#### 33 |
Let the sequence \(b_1, b_2, b_3, \dots\) be defined such that \(b_1 = 24\), \(b_{12} = 150\), and for all \(n \geq 3\), \(b_n\) is the arithmetic mean of the first \(n - 1\) terms. Find \(b_2\). | 276 |
The numbers from 1 to 9 are placed in the cells of a \(3 \times 3\) grid such that the sum of the numbers on one diagonal is 7 and on the other diagonal is 21. What is the sum of the numbers in the five shaded cells? | 25 |
How many prime numbers are divisible by $39$ ? | 0 |
Which are more: three-digit numbers where all digits have the same parity (all even or all odd), or three-digit numbers where adjacent digits have different parity? | 225 |
Side $AB$ of triangle $ABC$ has length 8 inches. Line $DEF$ is drawn parallel to $AB$ so that $D$ is on segment $AC$, and $E$ is on segment $BC$. Line $AE$ extended bisects angle $FEC$. If $DE$ has length $5$ inches, then the length of $CE$, in inches, is: | \frac{40}{3} |
Given a sequence $a_1,$ $a_2,$ $a_3,$ $\dots,$ let $S_n$ denote the sum of the first $n$ terms of the sequence.
If $a_1 = 1$ and
\[a_n = \frac{2S_n^2}{2S_n - 1}\]for all $n \ge 2,$ then find $a_{100}.$ | -\frac{2}{39203} |
In the equilateral triangle \(ABC\), point \(T\) is its centroid, point \(R\) is the reflection of \(T\) across the line \(AB\), and point \(N\) is the reflection of \(T\) across the line \(BC\).
Determine the ratio of the areas of triangles \(ABC\) and \(TRN\). | 3:1 |
If 6 students want to sign up for 4 clubs, where students A and B do not join the same club, and every club must have at least one member with each student only joining one club, calculate the total number of different registration schemes. | 1320 |
Suppose
\[\frac{1}{x^3-x^2-21x+45}=\frac{A}{x+5}+\frac{B}{x-3} + \frac{C}{(x - 3)^2}\]where $A$, $B$, and $C$ are real constants. What is $A$? | \frac{1}{64} |
Brian writes down four integers $w > x > y > z$ whose sum is $44$. The pairwise positive differences of these numbers are $1, 3, 4, 5, 6,$ and $9$. What is the sum of the possible values for $w$? | 31 |
Jane picked 64 apples at the orchard. She sorted them into 4 different baskets to send to her friends. When Jane wasn't looking her sister took 3 apples from each of the baskets. How many apples are in each basket now? | Before her sister took some there were 64/4=<<64/4=16>>16 apples in each basket
After her sister took some each pile had 16-3=<<16-3=13>>13 apples in each basket
#### 13 |
Find the smallest four-digit number SEEM for which there is a solution to the puzzle MY + ROZH = SEEM. (The same letters correspond to the same digits, different letters - different.) | 2003 |
Find the distance from the point $(1,-1,2)$ to the line passing through $(-2,2,1)$ and $(-1,-1,3).$ | \sqrt{5} |
While Travis is having fun on cubes, Sherry is hopping in the same manner on an octahedron. An octahedron has six vertices and eight regular triangular faces. After five minutes, how likely is Sherry to be one edge away from where she started? | \frac{11}{16} |
The Weston Junior Football Club has 24 players on its roster, including 4 goalies. During a training session, a drill is conducted wherein each goalie takes turns defending the goal while the remaining players (including the other goalies) attempt to score against them with penalty kicks.
How many penalty kicks are n... | 92 |
For some integer $m$, the polynomial $x^3 - 2011x + m$ has the three integer roots $a$, $b$, and $c$. Find $|a| + |b| + |c|$. | 98 |
Compute the value of the expression:
\[ 2(1 + 2(1 + 2(1 + 2(1 + 2(1 + 2(1 + 2(1 + 2(1 + 2)))))))) \] | 1022 |
A four-digit palindrome between $1000$ and $10,000$ is chosen at random. What is the probability that it is divisible by $9$? | \frac{1}{10} |
The value of the expression $\frac{\sin 10^{\circ}}{1-\sqrt{3}\tan 10^{\circ}}$ can be simplified using trigonometric identities and calculated exactly. | \frac{1}{4} |
The imaginary part of the complex number $z = -4i + 3$ is $-4i$. | -4 |
Jon's laundry machine can do 5 pounds of laundry at a time. 4 shirts weigh 1 pound and 2 pairs of pants weigh 1 pound. If he needs to wash 20 shirts and 20 pants, how many loads of laundry does he have to do? | The shirts weigh 20/4=<<20/4=5>>5 pounds
The pants weigh 20/2=<<20/2=10>>10 pounds
So he needs to do 10+5=<<10+5=15>>15 pounds of laundry
So that would take 15/5=<<15/5=3>>3 loads of laundry
#### 3 |
What is the smallest square number whose first five digits are 4 and the sixth digit is 5? | 666667 |
In a rectangle $P Q R S$ with $P Q=5$ and $Q R=3$, $P R$ is divided into three segments of equal length by points $T$ and $U$. What is the area of quadrilateral $S T Q U$? | 5 |
Hot dog buns come in packages of 8. For the school picnic, Mr. Gates bought 30 packages of hot dog buns. He has four classes with 30 students in each class. How many hot dog buns can each of Mr. Gates' students get? | Mr. Gates bought 30 x 8 = <<30*8=240>>240 hot dog buns.
In his four classes, he has a total of 4 x 30 = <<4*30=120>>120 students.
Therefore, each of his students can get 240/120 = <<240/120=2>>2 hot dog buns.
#### 2 |
In triangle $ABC$, the angle bisectors $AA_{1}$ and $BB_{1}$ intersect at point $O$. Find the ratio $AA_{1} : OA_{1}$ given $AB=6, BC=5$, and $CA=4$. | 3 : 1 |
Square $ABCD$ is inscribed in the region bound by the parabola $y = x^2 - 8x + 12$ and the $x$-axis, as shown below. Find the area of square $ABCD.$
[asy]
unitsize(0.8 cm);
real parab (real x) {
return(x^2 - 8*x + 12);
}
pair A, B, C, D;
real x = -1 + sqrt(5);
A = (4 - x,0);
B = (4 + x,0);
C = (4 + x,-2*x);
D = ... | 24 - 8 \sqrt{5} |
Given six senior students (including 4 boys and 2 girls) are arranged to intern at three schools, A, B, and C, with two students at each school, and the two girls cannot be at the same school or at school C, and boy A cannot go to school A, calculate the total number of different arrangements. | 18 |
Two distinct positive integers $a$ and $b$ are factors of 48. If $a\cdot b$ is not a factor of 48, what is the smallest possible value of $a\cdot b$? | 32 |
Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ have an angle of $30^{\circ}$ between them, and $|\overrightarrow{a}|=\sqrt{3}$, $|\overrightarrow{b}|=1$,
$(1)$ Find the value of $|\overrightarrow{a}-2\overrightarrow{b}|$
$(2)$ Let vector $\overrightarrow{p}=\overrightarrow{a}+2\overrightarrow{b}$, $\over... | -1 |
Let $S$ be a finite set of points in the plane. A linear partition of $S$ is an unordered pair $\{A,B\}$ of subsets of $S$ such that $A \cup B = S$, $A \cap B = \emptyset$, and $A$ and $B$ lie on opposite sides of some straight line disjoint from $S$ ($A$ or $B$ may be empty). Let $L_S$ be the number of linear partitio... | \binom{n}{2} + 1 |
Calculate: (1) $( \frac {3}{2})^{-2}-(-4.5)^{0}-( \frac {8}{27})^{ \frac {2}{3}}$;
(2) $\frac {2}{3}\text{lg}8+\text{lg}25-3^{2\log_{3}5}+16^{ \frac {3}{4}}$. | -15 |
Note that $9^2 = 81$, which contains no zeros; $99^2 = 9801$, which contains 1 zero; and $999^2 = 998,\!001$, which contains 2 zeros. Assuming this pattern continues, how many zeros are in the expansion of $99,\!999,\!999^2$? | 7 |
In a regional frisbee league, teams have 7 members and each of the 5 teams takes turns hosting matches. At each match, each team selects three members of that team to be on the match committee, except the host team, which selects four members. How many possible 13-member match committees are there? | 262,609,375 |
Given the set \( M = \{1, 2, \cdots, 2017\} \) which consists of the first 2017 positive integers, if one element is removed from \( M \) such that the sum of the remaining elements is a perfect square, what is the removed element? | 1677 |
A bag of grapes is to be distributed evenly to 5 kids in a class, and the grapes that are left over will be thrown out. If each student receives the greatest possible number of grapes, what is the greatest possible number of grapes that could be thrown out? | 4 |
A circle made of wire and a rectangle are arranged in such a way that the circle passes through two vertices $A$ and $B$ and touches the side $CD$. The length of side $CD$ is 32.1. Find the ratio of the sides of the rectangle, given that its perimeter is 4 times the radius of the circle. | 4:1 |
Mrs. Thomson received an incentive worth $240. She spent 1/3 of the money on food and 1/5 of it on clothes. Then, she put in her savings account 3/4 of the remaining money. How much money did Mrs. Thomson save? | Mrs. Thomson spent $240 x 1/3 = $<<240*1/3=80>>80 on food.
She spent $240 x 1/5 = $<<240*1/5=48>>48 on clothes.
So she spent a total of $80 + $48 = $<<80+48=128>>128 on food and clothes.
Hence, she had $240 - $128 = $<<240-128=112>>112 left after buying food and clothes.
Then, Mrs. Thomson saved $112 x 3/4 = $<<112*3/4... |
The base of a pyramid is an equilateral triangle with a side length of 1. Out of the three vertex angles at the apex of the pyramid, two are right angles.
Find the maximum volume of the pyramid. | \frac{1}{16} |
Let $ABC$ be a triangle such that $AB=6,BC=5,AC=7.$ Let the tangents to the circumcircle of $ABC$ at $B$ and $C$ meet at $X.$ Let $Z$ be a point on the circumcircle of $ABC.$ Let $Y$ be the foot of the perpendicular from $X$ to $CZ.$ Let $K$ be the intersection of the circumcircle of $BCY$ wit... | 147/10 |
How many distinct pairs of integers \(x, y\) are there between 1 and 1000 such that \(x^{2} + y^{2}\) is divisible by 49? | 10153 |
What is $1010101_2 + 111000_2$? Write your answer in base $10$. | 141 |
Twenty seven unit cubes are painted orange on a set of four faces so that two non-painted faces share an edge. The 27 cubes are randomly arranged to form a $3\times 3 \times 3$ cube. Given the probability of the entire surface area of the larger cube is orange is $\frac{p^a}{q^br^c},$ where $p,q,$ and $r$ are distinct ... | 74 |
Consider the paper triangle whose vertices are $(0,0), (34,0),$ and $(16,24).$ The vertices of its midpoint triangle are the midpoints of its sides. A triangular pyramid is formed by folding the triangle along the sides of its midpoint triangle. What is the volume of this pyramid?
| 408 |
So, Xiao Ming's elder brother was born in a year that is a multiple of 19. In 2013, determine the possible ages of Xiao Ming's elder brother. | 18 |
Two \(1 \times 1\) squares are removed from a \(5 \times 5\) grid as shown. Determine the total number of squares of various sizes on the grid. | 55 |
Given that $θ$ is a real number, if the complex number $z=\sin 2θ-1+i( \sqrt {2}\cos θ-1)$ is a purely imaginary number, then the imaginary part of $z$ is _______. | -2 |
The average of the data $x_1, x_2, \ldots, x_8$ is 6, and the standard deviation is 2. Then, the average and the variance of the data $2x_1-6, 2x_2-6, \ldots, 2x_8-6$ are | 16 |
There are 8 Olympic volunteers, among them volunteers $A_{1}$, $A_{2}$, $A_{3}$ are proficient in Japanese, $B_{1}$, $B_{2}$, $B_{3}$ are proficient in Russian, and $C_{1}$, $C_{2}$ are proficient in Korean. One volunteer proficient in Japanese, Russian, and Korean is to be selected from them to form a group.
(Ⅰ) Cal... | \dfrac {5}{6} |
Our school's girls volleyball team has 14 players, including a set of 3 triplets: Alicia, Amanda, and Anna. In how many ways can we choose 6 starters if exactly one of the triplets is in the starting lineup? | 1386 |
Let P_{1}, P_{2}, \ldots, P_{6} be points in the complex plane, which are also roots of the equation x^{6}+6 x^{3}-216=0. Given that P_{1} P_{2} P_{3} P_{4} P_{5} P_{6} is a convex hexagon, determine the area of this hexagon. | 9 \sqrt{3} |
An equilateral triangle $ABC$ shares a common side $BC$ with a square $BCDE,$ as pictured. What is the number of degrees in $\angle DAE$ (not pictured)? [asy]
pair pA, pB, pC, pD, pE;
pA = (0, 0);
pB = pA + dir(300);
pC = pA + dir(240);
pD = pC + dir(270);
pE = pB + dir(270);
draw(pA--pB--pC--pA);
draw(pB--pC--pD--pE--... | 30^\circ. |
Rural School USA has 105 students enrolled. There are 60 boys and 45 girls. If $\frac{1}{10}$ of the boys and $\frac{1}{3}$ of the girls are absent on one day, what percent of the total student population is absent? | 20 \% |
John assembles computers and sells prebuilt computers as a business. The parts for the computer cost $800. He sells the computers for 1.4 times the value of the components. He manages to build 60 computers a month. He has to pay $5000 a month in rent and another $3000 in non-rent extra expenses a month. How much pr... | He sells each computer for 1.4*800=$<<1.4*800=1120>>1120
That means he makes a profit of 1120-800=$<<1120-800=320>>320 per computer
So he makes 320*60=$<<320*60=19200>>19200 from selling computers
He has extra expenses of 5000+3000=$<<5000+3000=8000>>8000 a month
So his profit is 19200-8000=$<<19200-8000=11200>>11,200
... |
Emmalyn decided to paint fences in her neighborhood for twenty cents per meter. If there were 50 fences in the neighborhood that she had to paint and each fence was 500 meters long, calculate the total amount she earned from painting the fences. | The total length for the fifty fences is 50*500 = <<50*500=25000>>25000 meters.
If Emmalyn charged twenty cents to paint a meter of a fence, the total income she got from painting the fences is $0.20*25000 =$5000
#### 5000 |
In the vertices of a convex 2020-gon, numbers are placed such that among any three consecutive vertices, there is both a vertex with the number 7 and a vertex with the number 6. On each segment connecting two vertices, the product of the numbers at these two vertices is written. Andrey calculated the sum of the numbers... | 1010 |
Through two vertices of an equilateral triangle \(ABC\) with an area of \(21 \sqrt{3} \ \text{cm}^2\), a circle is drawn such that two sides of the triangle are tangent to the circle. Find the radius of this circle. | 2\sqrt{7} |
The function \( f \) satisfies the equation \((x-1) f(x) + f\left(\frac{1}{x}\right) = \frac{1}{x-1}\) for each value of \( x \) not equal to 0 and 1. Find \( f\left(\frac{2016}{2017}\right) \). | 2017 |
Given a unit square region $R$ and an integer $n \geq 4$, determine how many points are $80$-ray partitional but not $50$-ray partitional. | 7062 |
In the tetrahedron \( P-ABC \), \( \triangle ABC \) is an equilateral triangle with a side length of \( 2\sqrt{3} \), \( PB = PC = \sqrt{5} \), and the dihedral angle between \( P-BC \) and \( BC-A \) is \( 45^\circ \). Find the surface area of the circumscribed sphere around the tetrahedron \( P-ABC \). | 25\pi |
Misty's favorite number is 3 times smaller than Glory's favorite number. If Glory's favorite number is 450, what's the sum of their favorite numbers? | Misty's favorite number is 3 times smaller than Glory's, meaning her favorite number is 450 / 3 = <<450/3=150>>150
The sum of their favorite numbers is 150 + 450 = <<150+450=600>>600
#### 600 |
How many digits does the smallest repeating block in the decimal expansion of $\frac{5}{7}$ contain? | 6 |
A game board consists of $64$ squares that alternate in color between black and white. The figure below shows square $P$ in the bottom row and square $Q$ in the top row. A marker is placed at $P.$ A step consists of moving the marker onto one of the adjoining white squares in the row above. How many $7$-step paths are ... | 28 |
Two brick walls are being built. Each wall has 30 bricks in a single row and there are 50 rows in each wall. How many bricks will be used to make both walls? | Each brick wall needs 30 x 50 = <<30*50=1500>>1500 bricks.
Thus, the two walls need 1500 x 2 = <<1500*2=3000>>3000 bricks.
#### 3000 |
Given the function $f(x)=4\sin({8x-\frac{π}{9}})$, $x\in \left[0,+\infty \right)$, determine the initial phase of this harmonic motion. | -\frac{\pi}{9} |
Compute $\begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix}^{2018}.$ | \begin{pmatrix} 1 & 0 \\ 2018 & 1 \end{pmatrix} |
There is a stack of 200 cards, numbered from 1 to 200 from top to bottom. Starting with the top card, the following operations are performed in sequence: remove the top card, then place the next card at the bottom of the stack; remove the new top card, then place the next card at the bottom of the stack… This process i... | 145 |
Tetrahedron $A B C D$ with volume 1 is inscribed in circumsphere $\omega$ such that $A B=A C=A D=2$ and $B C \cdot C D \cdot D B=16$. Find the radius of $\omega$. | \frac{5}{3} |
Consider the given functions: $$\begin{array}{ccc}
f(x) & = & 5x^2 - \frac{1}{x}+ 3\\
g(x) & = & x^2-k
\end{array}$$If $f(2) - g(2) = 2$, what is the value of $k$? | k = \frac{-33}{2} |
Let $C$ be the circle with equation $x^2+12y+57=-y^2-10x$. If $(a,b)$ is the center of $C$ and $r$ is its radius, what is the value of $a+b+r$? | -9 |
"Modulo $m$ graph paper" consists of a grid of $13^2$ points, representing all pairs of integer residues $(x,y)$ where $0\le x, y <13$. To graph a congruence on modulo $13$ graph paper, we mark every point $(x,y)$ that satisfies the congruence. Consider the graph of $$4x \equiv 3y + 1 \pmod{13}.$$ Find the sum of the $... | 14 |
The projection of $\begin{pmatrix} -8 \\ b \end{pmatrix}$ onto $\begin{pmatrix} 2 \\ 1 \end{pmatrix}$ is
\[-\frac{13}{5} \begin{pmatrix} 2 \\ 1 \end{pmatrix}.\]Find $b.$ | 3 |
Donny has $78 in his piggy bank. If Donny buys a kite for $8 and a frisbee for $9. How much money does Donny have left? | Donny has 78 - 8 = <<78-8=70>>70 after paying for the kite.
He has 70 - 9 = $<<70-9=61>>61 left after buying the frisbee as well as the kite.
#### 61 |
Either increasing the radius or the height of a cylinder by six inches will result in the same volume. The original height of the cylinder is two inches. What is the original radius in inches? | 6 |
A stalker throws a small nut from the Earth's surface at an angle of $\alpha=30^{\circ}$ to the horizontal with an initial speed $v_{0}=10 \, \mathrm{m}/\mathrm{s}$. The normal acceleration due to gravity is $g=10 \, \mathrm{m}/\mathrm{s}^{2}$. At the highest point of its trajectory, the nut enters a zone of gravitatio... | 250 |
Parallelogram $ABCD$ has area $1,\!000,\!000$. Vertex $A$ is at $(0,0)$ and all other vertices are in the first quadrant. Vertices $B$ and $D$ are lattice points on the lines $y = x$ and $y = kx$ for some integer $k > 1$, respectively. How many such parallelograms are there? (A lattice point is any point whose coordina... | 784 |
Given the curve $x^{2}-y-2\ln \sqrt{x}=0$ and the line $4x+4y+1=0$, find the shortest distance from any point $P$ on the curve to the line. | \dfrac{\sqrt{2}(1+\ln2)}{2} |
A public bus departs on schedule at 6:30, 7:00, and 7:30. Student Xiao Ming arrives at the station between 6:50 and 7:30 to catch the bus, and his time of arrival is random. The probability that his waiting time is no more than 10 minutes is ______. | \frac{1}{2} |
In the Cartesian coordinate system $xOy$, the equation of curve $C_{1}$ is $x^{2}+y^{2}-4x=0$. The parameter equation of curve $C_{2}$ is $\left\{\begin{array}{l}x=\cos\beta\\ y=1+\sin\beta\end{array}\right.$ ($\beta$ is the parameter). Establish a polar coordinate system with the coordinate origin as the pole and the ... | 2\sqrt{5} + 2 |
Convert the point $\left( 8, \frac{\pi}{4}, \sqrt{3} \right)$ in cylindrical coordinates to rectangular coordinates. | (4 \sqrt{2}, 4 \sqrt{2}, \sqrt{3}) |
Evaluate $x^2y^3z$ if $x = \frac13$, $y = \frac23$, and $z = -9$. | -\frac{8}{27} |
Given that $M(2,5)$ is the midpoint of $\overline{AB}$ and $A(3,1)$ is one endpoint, what is the product of the coordinates of point $B$? | 9 |
Bob, Tom, Sally, and Jerry had dinner at their favorite pizzeria. They decide to share 2 pizzas. Bob ate half of a pizza on his own. Tom ate one-third of a pizza. Sally wasn't very hungry and only ate one-sixth of a pizza, and Jerry ate a quarter of a pizza. If each pizza is cut into 12 slices, how many slices were lef... | Each pizza is 12 slices. Bob at 12 slices / 2 = <<12/2=6>>6 slices.
Tom ate one third, or 12 slices / 3 = <<12/3=4>>4 slices.
Sally ate one sixth, or 12 slices / 6 = <<12/6=2>>2 slices.
Jerry ate one quarter, or 12 slices / 4 = <<12/4=3>>3 slices.
In total, they ate 6 slices + 4 slices + 2 slices + 3 slices = <<6+4+2+3... |
A $44$-gon $Q_1$ is constructed in the Cartesian plane, and the sum of the squares of the $x$-coordinates of the vertices equals $176$. The midpoints of the sides of $Q_1$ form another $44$-gon, $Q_2$. Finally, the midpoints of the sides of $Q_2$ form a third $44$-gon, $Q_3$. Find the sum of the squares of the $x$-coor... | 44 |
Jenny makes and freezes pans of lasagna all week so she can sell them at the market on the weekend. It costs Jenny $10.00 in ingredients to make 1 pan of lasagna. If she makes and sells 20 pans over the weekend at $25.00 apiece, how much does she make after factoring in expenses? | Each pan costs $10.00 to make so for 20 pans, it will cost 10*20 = $<<10*20=200.00>>200.00
She sells each of the 20 pans for $25.00 each so she will make 20*25 = $<<20*25=500.00>>500.00
She makes $500.00 and spent $200.00 on ingredients so she makes 500-200 = $<<500-200=300.00>>300.00
#### 300 |
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