problem stringlengths 10 5.15k | answer stringlengths 0 1.23k |
|---|---|
The players on a basketball team made some three-point shots, some two-point shots, and some one-point free throws. They scored as many points with two-point shots as with three-point shots. Their number of successful free throws was one more than their number of successful two-point shots. The team's total score was $... | 13 |
The ratio of measures of two complementary angles is 4 to 5. The smallest measure is increased by $10\%$. By what percent must the larger measure be decreased so that the two angles remain complementary? | 8\% |
A house and store were sold for $12,000 each. The house was sold at a loss of 20% of the cost, and the store at a gain of 20% of the cost. The entire transaction resulted in: | -$1000 |
Let $ABCD$ be a convex quadrilateral such that $AB + BC = 2021$ and $AD = CD$ . We are also given that $\angle ABC = \angle CDA = 90^o$ . Determine the length of the diagonal $BD$ . | \frac{2021 \sqrt{2}}{2} |
It takes 20 minutes for John to go to the bathroom 8 times. How long would it take to go 6 times? | He spends 20/8=<<20/8=2.5>>2.5 minutes each time he goes in
So it would take 2.5*6=<<2.5*6=15>>15 minutes to go 6 times
#### 15 |
Let $f$ be a function defined on the positive integers, such that
\[f(xy) = f(x) + f(y)\]for all positive integers $x$ and $y.$ Given $f(10) = 14$ and $f(40) = 20,$ find $f(500).$ | 39 |
Find the area of the triangle bounded by the $y$-axis and the lines $y-3x=-2$ and $3y+x=12$. | \frac{27}{5} |
Let $a\star b = \dfrac{\sqrt{a+b}}{\sqrt{a-b}}$. If $ x \star 24 = 7$, find $x$. | 25 |
What is $\frac{1}{3}$ of $\frac{1}{4}$ of $\frac{1}{5}$ of 60? | 1 |
Bret takes a 9 hour train ride to go to Boston. He spends 2 hours reading a book, 1 hour to eat his dinner, and 3 hours watching movies on his computer. How many hours does he have left to take a nap? | For all his activities on the train, Bret spends 2 + 1 + 3 = <<2+1+3=6>>6 hours.
For his nap, he has 9 - 6 = <<9-6=3>>3 hours left.
#### 3 |
Let $f(x) = x^{2}(1-x)^{2}$. What is the value of the sum
\[f \left(\frac{1}{2019} \right)-f \left(\frac{2}{2019} \right)+f \left(\frac{3}{2019} \right)-f \left(\frac{4}{2019} \right)+\cdots + f \left(\frac{2017}{2019} \right) - f \left(\frac{2018}{2019} \right)?\] | 0 |
In the diagram, \(PQ\) is a diameter of a larger circle, point \(R\) is on \(PQ\), and smaller semi-circles with diameters \(PR\) and \(QR\) are drawn. If \(PR = 6\) and \(QR = 4\), what is the ratio of the area of the shaded region to the area of the unshaded region? | 2: 3 |
Given $tan({θ+\frac{π}{{12}}})=2$, find $sin({\frac{π}{3}-2θ})$. | -\frac{3}{5} |
If there are exactly three points on the circle $x^{2}+y^{2}-2x-6y+1=0$ that are at a distance of $2$ from the line $y=kx$, determine the possible values of $k$. | \frac{4}{3} |
**The first term of a sequence is $2089$. Each succeeding term is the sum of the squares of the digits of the previous term. What is the $2089^{\text{th}}$ term of the sequence?** | 16 |
The set \( M = \left\{(x, y) \mid \log_{4} x + \log_{4} y \leq 1, x, y \in \mathbf{N}^{*}\right\} \) has how many subsets? | 256 |
The hospital has 11 doctors and 18 nurses. If 5 doctors and 2 nurses quit, how many doctors and nurses are left? | There are 11 doctors - 5 doctors = <<11-5=6>>6 doctors left.
There are 18 nurses - 2 nurses = <<18-2=16>>16 nurses left.
In total there are 6 doctors + 16 nurses = <<6+16=22>>22 doctors and nurses left.
#### 22 |
Find the maximum value of \( x + y \), given that \( x^2 + y^2 - 3y - 1 = 0 \). | \frac{\sqrt{26}+3}{2} |
The side \( AB \) of triangle \( ABC \) is longer than side \( AC \), and \(\angle A = 40^\circ\). Point \( D \) lies on side \( AB \) such that \( BD = AC \). Points \( M \) and \( N \) are the midpoints of segments \( BC \) and \( AD \) respectively. Find the angle \( \angle BNM \). | 20 |
Sun City has 1000 more than twice as many people as Roseville City. Roseville city has 500 less than thrice as many people as Willowdale city. If Willowdale city has 2000 people, how many people live in Sun City? | Roseville city has 3*2000-500=<<3*2000-500=5500>>5500 people.
Sun City has 2*5500+1000=<<2*5500+1000=12000>>12000 people.
#### 12000 |
Mr Julien's store has 400 marbles remaining after the previous day's sales. Twenty customers came into the store, and each bought 15 marbles. How many marbles remain in the store? | If each of the 20 customers bought 15 marbles, then they bought 20*15 = <<20*15=300>>300 marbles in total from Mr Julien's store.
The number of marbles remaining in the store is 400-300 = <<400-300=100>>100 marbles
#### 100 |
Find the greatest $a$ such that $\frac{7\sqrt{(2a)^2+(1)^2}-4a^2-1}{\sqrt{1+4a^2}+3}=2$. | \sqrt{2} |
Alexa has a lemonade stand where she sells lemonade for $2 for one cup. If she spent $20 on ingredients, how many cups of lemonade does she need to sell to make a profit of $80? | To make a profit of $80, she needs to make $80 + $20 = $<<80+20=100>>100
She needs to sell $100 / $2 = <<100/2=50>>50 cups of lemonade
#### 50 |
Simplify first, then evaluate: $\frac{{m^2-4m+4}}{{m-1}}÷(\frac{3}{{m-1}}-m-1)$, where $m=\sqrt{3}-2$. | \frac{-3+4\sqrt{3}}{3} |
Compute the number of tuples $\left(a_{0}, a_{1}, a_{2}, a_{3}, a_{4}, a_{5}\right)$ of (not necessarily positive) integers such that $a_{i} \leq i$ for all $0 \leq i \leq 5$ and $$a_{0}+a_{1}+\cdots+a_{5}=6$$ | 2002 |
Find the product of the divisors of $50$. | 125,\!000 |
The number of diagonals that can be drawn in a polygon of 100 sides is: | 4850 |
Given that points $A$ and $B$ lie on the curves $C_{1}: x^{2}-y+1=0$ and $C_{2}: y^{2}-x+1=0$ respectively, what is the minimum value of the distance $|AB|$? | \frac{3\sqrt{2}}{4} |
In triangle $ABC$, $\tan \angle CAB = 22/7$, and the altitude from $A$ divides $BC$ into segments of length 3 and 17. What is the area of triangle $ABC$? | 110 |
How many positive 3-digit numbers are divisible by 11? | 81 |
Altitudes $\overline{AX}$ and $\overline{BY}$ of acute triangle $ABC$ intersect at $H$. If $\angle BAC = 61^\circ$ and $\angle ABC = 73^\circ$, then what is $\angle CHX$? | 73^\circ |
Dr. Fu Manchu has a bank account that has an annual interest rate of 6 percent, but it compounds monthly. If this is equivalent to a bank account that compounds annually at a rate of $r$ percent, then what is $r$? (Give your answer to the nearest hundredth.) | 6.17 |
Let $f : \mathbb{R} \to \mathbb{R}$ be a function such that $f(1) = 1$ and
\[f(x^2 - y^2) = (x - y) (f(x) + f(y))\]for all real numbers $x$ and $y.$
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 $n \times s.$ | 2 |
Find all values of $k$ for which the positive difference between the solutions of
\[5x^2 + 4x + k = 0\]equals the sum of the squares of these solutions. Enter all possible values of $k,$ separated by commas. | \frac{3}{5}, -\frac{12}{5} |
If the sum of two numbers is $1$ and their product is $1$, then the sum of their cubes is: | -2 |
Andrea notices that the 40-foot tree next to her is casting a 10-foot shadow. How tall, in inches, is Andrea if she is casting a 15-inch shadow at the same time? | 60 |
Find $r$ such that $\log_{81} (2r-1) = -1/2$. | \frac{5}{9} |
Twelve tiles numbered $1$ through $12$ are turned face down. One tile is turned up at random, and an eight-sided die (numbered 1 to 8) is rolled. What is the probability that the product of the numbers on the tile and the die will be a square?
A) $\frac{1}{12}$
B) $\frac{1}{8}$
C) $\frac{1}{6}$
D) $\frac{1}{4}$
E) $\fr... | \frac{1}{6} |
How many three-digit numbers are composed of three distinct digits such that one digit is the average of the other two? | 112 |
Phil and Andre decide to order some pizza. They get a small cheese and a large pepperoni. The small pizza has 8 slices, The large has 14 slices. They have both eaten 9 slices already. How many pieces are left per person? | There are 22 pieces of pizza because 14 plus 8 equals <<14+8=22>>22.
They have eaten 18 pieces of pizza because 9 plus 9 equals <<9+9=18>>18.
There are 4 pieces of pizza left because 22 minus 18 equals <<22-18=4>>4.
There are 2 pieces per person because 4 divided by 2 equals <<4/2=2>>2.
#### 2 |
At a circular table, there are 5 people seated: Arnaldo, Bernaldo, Cernaldo, Dernaldo, and Ernaldo, each in a chair. Analyzing clockwise, we have:
I. There is 1 empty chair between Arnaldo and Bernaldo;
II. There are 5 chairs between Bernaldo and Cernaldo;
III. There are 4 chairs between Dernaldo and Ernaldo, almost... | 12 |
Sophia chooses a real number uniformly at random from the interval $[0, 3000]$. Independently, Oliver chooses a real number uniformly at random from the interval $[0, 4500]$. What is the probability that Oliver's number is at least twice as large as Sophia's number?
**A) $\frac{1}{8}$**
**B) $\frac{3}{8}$**
**C) $\frac... | \frac{5}{8} |
Given $f(x)= \frac{1}{2^{x}+ \sqrt {2}}$, use the method for deriving the sum of the first $n$ terms of an arithmetic sequence to find the value of $f(-5)+f(-4)+…+f(0)+…+f(5)+f(6)$. | 3 \sqrt {2} |
Place 5 balls, numbered 1, 2, 3, 4, 5, into three different boxes, with two boxes each containing 2 balls and the other box containing 1 ball. How many different arrangements are there? (Answer with a number). | 90 |
Wilfred eats 4 carrots on Tuesday and 6 carrots on Wednesday. If Wilfred wants to eat a total of 15 carrots from Tuesday to Thursday, how many carrots does Wilfred need to eat on Thursday? | Across Tuesday and Wednesday, Wilfred eats 4 + 6 = <<4+6=10>>10 carrots
On Thursday, Wilfred needs to eat 15 - 10 = <<15-10=5>>5 carrots
#### 5 |
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 |
A and B are playing a guessing game. First, A thinks of a number, denoted as $a$, then B guesses the number A was thinking of, denoting B's guess as $b$. Both $a$ and $b$ belong to the set $\{0,1,2,…,9\}$. The probability that $|a-b|\leqslant 1$ is __________. | \dfrac{7}{25} |
A vampire drains three people a week. His best friend is a werewolf who eats five people a week, but only fresh ones, never drained ones. How many weeks will a village of 72 people last them both? | The vampire and werewolf need 3 + 5 = <<3+5=8>>8 people a week.
A village of 72 people will last them both 72 / 8 = <<72/8=9>>9 weeks.
#### 9 |
Two regular polygons have the same perimeter. If the first has 38 sides and a side length twice as long as the second, how many sides does the second have? | 76 |
Sam and Drew have a combined age of 54. Sam is half of Drew's age. How old is Sam? | Let x be Drew's age.
Sam is (1/2)x
x+(1/2)x=54
(3/2)x=54
3x=108
x=<<36=36>>36
Sam is 36/2=<<36/2=18>>18 years old
#### 18 |
It is known that there are four different venues $A$, $B$, $C$, $D$ at the Flower Expo. Person A and person B each choose 2 venues to visit. The probability that exactly one venue is the same in their choices is ____. | \frac{2}{3} |
Three people, Jia, Yi, and Bing, participated in a competition and they took the top 3 places (with no ties). Jia said: "I am first", Yi said: "I am not first", and Bing said: "I am not third". Only one of them is telling the truth. If the rankings of Jia, Yi, and Bing are respectively $A, B, C$, then the three-digit n... | 312 |
Jen has 10 more ducks than four times the number of chickens. If Jen has 150 ducks, how many total birds does she have? | Let the number of chickens be c.
If there are 150 ducks, we can describe the number of chickens as 150 =10+4c.
So, 4c = 150.
And, there are c = <<35=35>>35 chickens.
In total, Jen has 35 chickens+150 ducks =<<35+150=185>>185 birds.
#### 185 |
A class of 30 students was asked what they did on their winter holiday. 20 students said that they went skating. 9 students said that they went skiing. Exactly 5 students said that they went skating and went skiing. How many students did not go skating and did not go skiing? | 6 |
A basketball team played 40 games and won 70% of the games. It still had 10 games to play. How many games can they lose to win 60% of their games? | The team has won 40 x 70/100 = <<40*70/100=28>>28 games out of 40.
There a total of 40 + 10 = <<40+10=50>>50 games to be played.
The team has to win a total of is 50 x 60/100 = <<50*60/100=30>>30 games.
So, it still needs to win 30 - 28 = <<30-28=2>>2 more games.
Hence, they can still lose 10 - 2 = <<10-2=8>>8 games.
#... |
The sum of three numbers is $20$. The first is four times the sum of the other two. The second is seven times the third. What is the product of all three? | 28 |
$A$,$B$,$C$,$D$,$E$,$F$ are 6 students standing in a row to participate in a literary performance. If $A$ does not stand at either end, and $B$ and $C$ must be adjacent, then the total number of different arrangements is ____. | 144 |
In the Cartesian coordinate system $xOy$, a line segment of length $\sqrt{2}+1$ has its endpoints $C$ and $D$ sliding on the $x$-axis and $y$-axis, respectively. It is given that $\overrightarrow{CP} = \sqrt{2} \overrightarrow{PD}$. Let the trajectory of point $P$ be curve $E$.
(I) Find the equation of curve $E$;
(II... | \frac{\sqrt{6}}{2} |
Given functions $y_1=\frac{k_1}{x}$ and $y_{2}=k_{2}x+b$ ($k_{1}$, $k_{2}$, $b$ are constants, $k_{1}k_{2}\neq 0$).<br/>$(1)$ If the graphs of the two functions intersect at points $A(1,4)$ and $B(a,1)$, find the expressions of functions $y_{1}$ and $y_{2}$.<br/>$(2)$ If point $C(-1,n)$ is translated $6$ units upwards ... | -6 |
Given the array: $(1,1,1)$, $(2,2,4)$, $(3,4,12)$, $(4,8,32)$, $\ldots$, $(a_{n}, b_{n}, c_{n})$, find the value of $c_{7}$. | 448 |
In right triangle $ABC$ with $\angle A = 90^\circ$, we have $AB = 6$ and $BC = 10$. Find $\cos C$. | \frac45 |
Calculate the sum of $5.46$, $2.793$, and $3.1$ as a decimal. | 11.353 |
Let $M$ be the second smallest positive integer that is divisible by every positive integer less than 10 and includes at least one prime number greater than 10. Find the sum of the digits of $M$. | 18 |
Given a finite sequence $S=(a_1,a_2,\ldots ,a_n)$ of $n$ real numbers, let $A(S)$ be the sequence
$\left(\frac{a_1+a_2}{2},\frac{a_2+a_3}{2},\ldots ,\frac{a_{n-1}+a_n}{2}\right)$
of $n-1$ real numbers. Define $A^1(S)=A(S)$ and, for each integer $m$, $2\le m\le n-1$, define $A^m(S)=A(A^{m-1}(S))$. Suppose $x>0$, and le... | \sqrt{2}-1 |
A set \( \mathcal{S} \) of distinct positive integers has the property that for every integer \( x \) in \( \mathcal{S}, \) the arithmetic mean of the set of values obtained by deleting \( x \) from \( \mathcal{S} \) is an integer. Given that 1 belongs to \( \mathcal{S} \) and that 2310 is the largest element of \( \ma... | 20 |
The line $y = \frac{5}{3} x - \frac{17}{3}$ is to be parameterized using vectors. Which of the following options are valid parameterizations?
(A) $\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 4 \\ 1 \end{pmatrix} + t \begin{pmatrix} -3 \\ -5 \end{pmatrix}$
(B) $\begin{pmatrix} x \\ y \end{pmatrix} = \begin... | \text{A,C} |
In triangle $XYZ$, $XY = 5$, $YZ = 12$, $XZ = 13$, and $YM$ is the angle bisector from vertex $Y$. If $YM = l \sqrt{2}$, find $l$. | \frac{60}{17} |
Arrange the positive integers whose digits sum to 4 in ascending order. Which position does the number 2020 occupy in this sequence? | 28 |
Find the number of ordered triples of sets $(T_1, T_2, T_3)$ such that
1. each of $T_1, T_2$ , and $T_3$ is a subset of $\{1, 2, 3, 4\}$ ,
2. $T_1 \subseteq T_2 \cup T_3$ ,
3. $T_2 \subseteq T_1 \cup T_3$ , and
4. $T_3\subseteq T_1 \cup T_2$ . | 625 |
In the triangle $\triangle ABC$, let $G$ be the centroid, and let $I$ be the center of the inscribed circle. Let $\alpha$ and $\beta$ be the angles at the vertices $A$ and $B$, respectively. Suppose that the segment $IG$ is parallel to $AB$ and that $\beta = 2 \tan^{-1} (1/3)$. Find $\alpha$. | \frac{\pi}{2} |
A housewife saved $2.50 in buying a dress on sale. If she spent $25 for the dress, she saved about: | 9 \% |
Points $A_{1}$ and $C_{1}$ are located on the sides $BC$ and $AB$ of triangle $ABC$. Segments $AA_{1}$ and $CC_{1}$ intersect at point $M$.
In what ratio does line $BM$ divide side $AC$, if $AC_{1}: C_{1}B = 2: 3$ and $BA_{1}: A_{1}C = 1: 2$? | 1:3 |
Let $f(x)$ and $g(x)$ be nonzero polynomials such that
\[f(g(x)) = f(x) g(x).\]If $g(2) = 37,$ find $g(x).$ | x^2 + 33x - 33 |
For $1 \le n \le 100$, how many integers are there such that $\frac{n}{n+1}$ is a repeating decimal? | 86 |
If a four-digit number $\overline{a b c d}$ meets the condition $a + b = c + d$, it is called a "good number." For instance, 2011 is a "good number." How many "good numbers" are there? | 615 |
A conference center is setting up chairs in rows for a seminar. Each row can seat $13$ chairs, and currently, there are $169$ chairs set up. They want as few empty seats as possible but need to maintain complete rows. If $95$ attendees are expected, how many chairs should be removed? | 65 |
Given the function $f(x) = \frac {1}{2}x^{2} + x - 2\ln{x}$ ($x > 0$):
(1) Find the intervals of monotonicity for $f(x)$.
(2) Find the extreme values of the function $f(x)$. | \frac {3}{2} |
How many ways can 1995 be factored as a product of two two-digit numbers? (Two factorizations of the form $a\cdot b$ and $b\cdot a$ are considered the same). | 2 |
Let \( a, b, c \) be pairwise distinct positive integers such that \( a+b, b+c \) and \( c+a \) are all square numbers. Find the smallest possible value of \( a+b+c \). | 55 |
There are eight envelopes numbered 1 to 8. Find the number of ways in which 4 identical red buttons and 4 identical blue buttons can be put in the envelopes such that each envelope contains exactly one button, and the sum of the numbers on the envelopes containing the red buttons is more than the sum of the numbers on ... | 31 |
Given that α is an acute angle, cos(α+π/6) = 2/3, find the value of sinα. | \dfrac{\sqrt{15} - 2}{6} |
Compute the number of ordered pairs of integers $(x,y)$ with $1\le x<y\le 100$ such that $i^x+i^y$ is a real number. | 1850 |
How many positive multiples of nine are two-digit numbers? | 10 |
Susan is making jewelry with a repeating pattern that has 3 green beads, 5 purple beads, and twice as many red beads as green beads. If the pattern repeats three times per bracelet and 5 times per necklace, how many beads does she need to make 1 bracelets and 10 necklaces? | First find the number of red beads per repeat: 3 green * 2 red/green = <<3*2=6>>6 red
Then add the number of beads of each color to find the total number of bead per repeat: 6 beads + 3 beads + 5 beads = <<6+3+5=14>>14 beads
Then multiply the number of beads per repeat by the number of repeats per bracelet to find the ... |
The noon temperatures for seven consecutive days were $80^{\circ}$, $79^{\circ}$, $81^{\circ}$, $85^{\circ}$, $87^{\circ}$, $89^{\circ}$, and $87^{\circ}$ Fahrenheit. What is the mean noon temperature, in degrees Fahrenheit, for the week? | 84 |
Given the function $f(x)=x^{3}-x^{2}+1$.
$(1)$ Find the equation of the tangent line to the function $f(x)$ at the point $(1,f(1))$;
$(2)$ Find the extreme values of the function $f(x)$. | \dfrac {23}{27} |
Find the number of real solutions to the equation
\[\frac{1}{x - 1} + \frac{2}{x - 2} + \frac{3}{x - 3} + \dots + \frac{100}{x - 100} = x.\] | 101 |
Caleb is baking a birthday cake for his grandfather. His grandfather is turning 79 years old. Caleb puts three colors of candles on the cake. He puts one candle for each year for his grandfather. He puts 27 yellow candles, 14 red candles and the rest are blue candles. How many blue candles did he use? | Caleb has put 27 + 14 = <<27+14=41>>41 yellow and red candles.
So, he used 79 - 41 = <<79-41=38>>38 blue candles.
#### 38 |
Given that $x, y,$ and $z$ are real numbers that satisfy: \begin{align*} x &= \sqrt{y^2-\frac{1}{16}}+\sqrt{z^2-\frac{1}{16}}, \\ y &= \sqrt{z^2-\frac{1}{25}}+\sqrt{x^2-\frac{1}{25}}, \\ z &= \sqrt{x^2 - \frac 1{36}}+\sqrt{y^2-\frac 1{36}}, \end{align*} and that $x+y+z = \frac{m}{\sqrt{n}},$ where $m$ and $n$ are posit... | 9 |
Sean buys 3 cans of soda, 2 soups, and 1 sandwich. Each soup cost as much as the 3 combined sodas. The sandwich cost 3 times as much as the soup. If the soda cost $1 how much did everything cost together? | The soda cost 3*1=$<<3*1=3>>3
So the soups cost 2*3=$<<2*3=6>>6
The sandwich cost 3*3=$<<3*3=9>>9
So the total cost of everything was 3+6+9=$<<3+6+9=18>>18
#### 18 |
On a circular track with a perimeter of 360 meters, three individuals A, B, and C start from the same point: A starts first, running counterclockwise. Before A completes one lap, B and C start simultaneously, running clockwise. When A and B meet for the first time, C is exactly halfway between them. After some time, w... | 90 |
Every positive integer $k$ has a unique factorial base expansion $(f_1,f_2,f_3,\ldots,f_m)$, meaning that $k=1!\cdot f_1+2!\cdot f_2+3!\cdot f_3+\cdots+m!\cdot f_m$, where each $f_i$ is an integer, $0\le f_i\le i$, and $0<f_m$. Given that $(f_1,f_2,f_3,\ldots,f_j)$ is the factorial base expansion of $16!-32!+48!-64!+\c... | 495 |
For each value of $x,$ $g(x)$ is defined to be the minimum value of the three numbers $3x + 3,$ $\frac{1}{3} x + 2,$ and $-\frac{1}{2} x + 8.$ Find the maximum value of $g(x).$ | \frac{22}{5} |
What is the value of $2-(-2)^{-2}$? | \frac{7}{4} |
The unit's digit (one's digit) of the product of any six consecutive positive whole numbers is | 0 |
It was Trevor's job to collect fresh eggs from the family's 4 chickens every morning. He got 4 eggs from Gertrude and 3 eggs from Blanche. Nancy laid 2 eggs as did Martha. On the way, he dropped 2 eggs. How many eggs did Trevor have left? | He collected 4+3+2+2 eggs for a total of <<4+3+2+2=11>>11 eggs.
He dropped 2 eggs so 11-2 = <<11-2=9>>9 eggs left
#### 9 |
Harry Potter is creating an enhanced magical potion called "Elixir of Life" (this is a very potent sleeping potion composed of powdered daffodil root and wormwood infusion. The concentration of the "Elixir of Life" is the percentage of daffodil root powder in the entire potion). He first adds a certain amount of wormwo... | 11 |
The area of an equilateral triangle ABC is 36. Points P, Q, R are located on BC, AB, and CA respectively, such that BP = 1/3 BC, AQ = QB, and PR is perpendicular to AC. Find the area of triangle PQR. | 10 |
Given $x > 0$, $y > 0$, and $2x+8y-xy=0$, find the minimum value of $x+y$. | 18 |
How many degrees are in the sum of the measures of the six numbered angles pictured? [asy]
draw((3,8)--(10,4)--(1,0)--cycle,linewidth(1));
draw((7,8)--(9,0)--(0,4)--cycle,linewidth(1));
label("1",(3,8),SSE);
label("2",(7,8),SSW);
label("3",(10,4),2W);
label("4",(9,0),NW+NNW);
label("5",(1,0),NE+NNE);
label("6",(0,4),2E... | 360^\circ |
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