problem stringlengths 10 5.15k | answer stringlengths 0 1.23k |
|---|---|
Yanna bought 60 apples. She gave eighteen apples to Zenny. She gave six more apples to Andrea and kept the rest. How many apples did she keep? | After giving 18 apples to Zenny, Yanna remained with 60 - 18 = <<60-18=42>>42 apples.
Since she also gave Andrea 6 apples, Yanna remained with 42 - 6 = <<36=36>>36 apples
Therefore, Yanna kept 36 apples.
#### 36 |
A point is selected at random from the portion of the number line shown here. What is the probability that the point is closer to 4 than to 0? Express your answer as a decimal to the nearest tenth.
[asy]unitsize(0.2inch);
draw((0,0)--(5,0));
draw((0,-0.1)--(0,0.1));
draw((1,-0.1)--(1,0.1));
draw((2,-0.1)--(2,0.1));
dr... | .6 |
The robotics club has 30 members: 12 boys and 18 girls. A 6-person committee is chosen at random. What is the probability that the committee has at least 1 boy and at least 1 girl? | \frac{574,287}{593,775} |
Sergey, while being a student, worked part-time at a student cafe throughout the year. Sergey's salary was 9000 rubles per month. In the same year, Sergey paid 100000 rubles for his treatment at a healthcare institution and purchased medication prescribed by a doctor for 20000 rubles (eligible for deduction). The foll... | 14040 |
In the rectangular coordinate system xOy, the parametric equation of line l is $$\begin{cases} \overset{x=t}{y=1+t}\end{cases}$$ (t is the parameter), line m is parallel to line l and passes through the coordinate origin, and the parametric equation of circle C is $$\begin{cases} \overset{x=1+cos\phi }{y=2+sin\phi }\en... | 2+ \sqrt {2} |
Let $f(x) = x^2 + 6x + c$ for all real numbers $x$, where $c$ is some real number. For what values of $c$ does $f(f(x))$ have exactly $3$ distinct real roots? | \frac{11 - \sqrt{13}}{2} |
A bug is on the edge of a ceiling of a circular room with a radius of 65 feet. The bug walks straight across the ceiling to the opposite edge, passing through the center of the circle. It next walks straight to another point on the edge of the circle but not back through the center. If the third part of its journey, ba... | 313 |
Given that $0\le x_3 \le x_2 \le x_1\le 1$ and $(1-x_1)^2+(x_1-x_2)^2+(x_2-x_3)^2+x_3^2=\frac{1}{4},$ find $x_1.$ | \frac{3}{4} |
The vertical drops of six roller coasters at Cantor Amusement Park are shown in the table.
\[
\begin{tabular}{|l|c|}
\hline
Cyclone & 180 feet \\ \hline
Gravity Rush & 120 feet \\ \hline
Screamer & 150 feet \\ \hline
Sky High & 310 feet \\ \hline
Twister & 210 feet \\ \hline
Loop de Loop & 190 feet \\ \hline
\end{tabu... | 8.\overline{3} |
If the price of a stamp is 33 cents, what is the maximum number of stamps that could be purchased with $\$32$? | 96 |
What is the sum of the prime numbers between 10 and 20? | 60 |
Pauline is buying school supplies. The total amount of all the items she wants to buy add up to $150 before sales tax. Sales tax is 8% of the total amount purchased. How much will Pauline spend on all the items, including sales tax? | Sales tax is 8/100 x $150= $<<8/100*150=12>>12.
Therefore, Pauline wil spend $150 + $12 = $<<150+12=162>>162, including sales tax.
#### 162 |
Let $ ABCD$ be a unit square (that is, the labels $ A, B, C, D$ appear in that order around the square). Let $ X$ be a point outside of the square such that the distance from $ X$ to $ AC$ is equal to the distance from $ X$ to $ BD$ , and also that $ AX \equal{} \frac {\sqrt {2}}{2}$ . Determine the valu... | 5/2 |
Six consecutive prime numbers have sum \( p \). Given that \( p \) is also a prime, determine all possible values of \( p \). | 41 |
Ten points are marked on a circle. How many distinct convex polygons of three or more sides can be drawn using some (or all) of the ten points as vertices?
| 968 |
Jane bought pens for her three students. Each student received 62 red pens and 43 black pens. They pooled their pens together and took them from the pool whenever they needed them. After the first month, they had taken a total of 37 pens from the pool. After the second month, they had taken another 41 pens from the poo... | Each student got 62 + 43 = <<62+43=105>>105 pens to start with.
They pooled them to get 105 * 3 = <<105*3=315>>315 pens in the pool.
They took 37 pens in the first month, leaving 315 - 37 = 278 pens in the pool.
They took 41 pens in the second month, leaving 278 - 41 = 237 pens in the pool.
237 pens were split into thr... |
Over all real numbers $x$ and $y$ such that $$x^{3}=3 x+y \quad \text { and } \quad y^{3}=3 y+x$$ compute the sum of all possible values of $x^{2}+y^{2}$. | 15 |
The image shows a 3x3 grid where each cell contains one of the following characters: 华, 罗, 庚, 杯, 数, 学, 精, 英, and 赛. Each character represents a different number from 1 to 9, and these numbers satisfy the following conditions:
1. The sum of the four numbers in each "田" (four cells in a square) is equal.
2. 华 $\times$ 华 ... | 120 |
A housewife goes to the market. She spent 2/3 of her $150. How much does she have left? | The housewife has 3/3 - 2/3 = 1/3 of $150 left.
This means she has $150 * (1/3) = $<<150*(1/3)=50>>50 left.
#### 50 |
A certain TV station randomly selected $100$ viewers to evaluate a TV program in order to understand the evaluation of the same TV program by viewers of different genders. It is known that the ratio of the number of "male" to "female" viewers selected is $9:11$. The evaluation results are divided into "like" and "disli... | 212.5 |
Let $x$ be a real number such that
\[x^2 + 4 \left( \frac{x}{x - 2} \right)^2 = 45.\]Find all possible values of $y = \frac{(x - 2)^2 (x + 3)}{2x - 3}.$ Enter all possible values, separated by commas. | 2,16 |
Herman likes to feed the birds in December, January and February. He feeds them 1/2 cup in the morning and 1/2 cup in the afternoon. How many cups of food will he need for all three months? | December has 31 days, January has 31 days and February has 28 days for a total of 31+31+28 = <<31+31+28=90>>90 days
He feeds them 1/2 cup in the morning and 1/2 cup in the afternoon for a total of 1/2+1/2 = <<1/2+1/2=1>>1 cup per day
If he feeds them 1 cup per day for 90 days then he will need 1*90 = <<1*90=90>>90 cups... |
How many positive integers, including $1,$ are divisors of both $40$ and $72?$ | \mbox{four} |
Given the sequence \(\{a_{n}\}\) with the sum of the first \(n\) terms \(S_{n} = n^{2} - 1\) \((n \in \mathbf{N}_{+})\), find \(a_{1} + a_{3} + a_{5} + a_{7} + a_{9} = \). | 44 |
Let $x,$ $y,$ $z$ be real numbers such that $-1 < x,$ $y,$ $z < 1.$ Find the minimum value of
\[\frac{1}{(1 - x)(1 - y)(1 - z)} + \frac{1}{(1 + x)(1 + y)(1 + z)}.\] | 2 |
Let $z_1$, $z_2$, $z_3$, $\dots$, $z_{12}$ be the 12 zeroes of the polynomial $z^{12} - 2^{36}$. For each $j$, let $w_j$ be one of $z_j$ or $iz_j$. Find the maximum possible value of the real part of
\[\sum_{j = 1}^{12} w_j.\] | 16 + 16 \sqrt{3} |
Find the largest real number $c$ such that \[x_1^2 + x_2^2 + \dots + x_{101}^2 \geq cM^2\]whenever $x_1,x_2,\ldots,x_{101}$ are real numbers such that $x_1+x_2+\cdots+x_{101}=0$ and $M$ is the median of $x_1,x_2,\ldots,x_{101}.$ | \tfrac{5151}{50} |
Several cuboids with edge lengths of $2, 7, 13$ are arranged in the same direction to form a cube with an edge length of 2002. How many small cuboids does a diagonal of the cube pass through? | 1210 |
A group of children riding on bicycles and tricycles rode past Billy Bob's house. Billy Bob counted $7$ children and $19$ wheels. How many tricycles were there? | 5 |
The cost of purchasing a car is 150,000 yuan, and the annual expenses for insurance, tolls, and gasoline are about 15,000 yuan. The maintenance cost for the first year is 3,000 yuan, which increases by 3,000 yuan each year thereafter. Determine the best scrap year limit for this car. | 10 |
On a party with 99 guests, hosts Ann and Bob play a game (the hosts are not regarded as guests). There are 99 chairs arranged in a circle; initially, all guests hang around those chairs. The hosts take turns alternately. By a turn, a host orders any standing guest to sit on an unoccupied chair $c$. If some chair adjace... | 34 |
Susan is playing a board game with 48 spaces from the starting space to the winning end space of the game. On the first turn, she moves forward eight spaces. On the second turn, she moves two spaces, but lands on a space that sends her back five spaces. On the third turn, she moves forward six more spaces. How many spa... | In the first three turns, Susan has moved 8 + 2 - 5 + 6 = <<8+2-5+6=11>>11 spaces.
Susan has 48 - 11 = <<48-11=37>>37 spaces to move to win the game.
#### 37 |
John buys 500 newspapers. Each newspaper sells for $2. He sells 80% of them. He buys them all for 75% less than the price at which he sells them. How much profit does he make? | He sells 500*.8=<<500*.8=400>>400 newspapers
He sells them for 400*2=$<<400*2=800>>800
He gets a discount of 2*.75=$<<2*.75=1.5>>1.5 on the newspapers
So he buys them for 2-1.5=$<<2-1.5=.5>>.5
So he spent 500*.5=$<<500*.5=250>>250 buying them
That means he made a profit of 800-250=$<<800-250=550>>550
#### 550 |
James listens to super-fast music. It is 200 beats per minute. He listens to 2 hours of music a day. How many beats does he hear per week? | He listens to 2*60=<<2*60=120>>120 minutes of music a day
So he hears 200*120=<<200*120=24000>>24000 beats per day
That means he hears 24,000*7=<<24000*7=168000>>168,000 beats per week
#### 168000 |
Given points A (-3, 5) and B (2, 15), find a point P on the line $l: 3x - 4y + 4 = 0$ such that $|PA| + |PB|$ is minimized. The minimum value is \_\_\_\_\_\_. | 5\sqrt{13} |
In an $8 \times 8$ table, 23 cells are black, and the rest are white. In each white cell, the sum of the black cells located in the same row and the black cells located in the same column is written. Nothing is written in the black cells. What is the maximum value that the sum of the numbers in the entire table can tak... | 234 |
A rectangular tile measures 3 inches by 4 inches. What is the fewest number of these tiles that are needed to completely cover a rectangular region that is 2 feet by 5 feet? | 120 |
In a parking lot, there are seven parking spaces numbered from 1 to 7. Now, two different trucks and two different buses are to be parked at the same time, with each parking space accommodating at most one vehicle. If vehicles of the same type are not parked in adjacent spaces, there are a total of ▲ different parking ... | 840 |
A Martian traffic light consists of six identical bulbs arranged in two horizontal rows (one below the other) with three bulbs in each row. A rover driver in foggy conditions can distinguish the number and relative positions of the lit bulbs on the traffic light (for example, if two bulbs are lit, whether they are in t... | 44 |
I ponder some numbers in bed, all products of three primes I've said, apply $\phi$ they're still fun: $$n=37^{2} \cdot 3 \ldots \phi(n)= 11^{3}+1 ?$$ now Elev'n cubed plus one. What numbers could be in my head? | 2007, 2738, 3122 |
Given that $\cos α=-\dfrac{4}{5}\left(\dfrac{π}{2}<α<π\right)$, find $\cos\left(\dfrac{π}{6}-α\right)$ and $\cos\left(\dfrac{π}{6}+α\right)$. | -\dfrac{3+4\sqrt{3}}{10} |
Let $x$, $y$, $z$, $u$, and $v$ be positive integers with $x+y+z+u+v=2505$. Let $N$ be the largest of the sums $x+y$, $y+z$, $z+u$, and $u+v$. Determine the smallest possible value of $N$. | 1253 |
Determine the number of four-digit integers $n$ such that $n$ and $2n$ are both palindromes. | 20 |
Solve for $x$: $5 - x = 8$. | -3 |
Janet filmed a new movie that is 60% longer than her previous 2-hour long movie. Her previous movie cost $50 per minute to film, and the newest movie cost twice as much per minute to film as the previous movie. What was the total amount of money required to film Janet's entire newest film? | The first movie was 2*60=<<2*60=120>>120 minutes
So this movie is 120*.6=<<120*.6=72>>72 minutes longer
So this movie is 192 minutes
It also cost 50*2=$<<50*2=100>>100 per minute to film
So it cost 192*100=$1920
#### 1920 |
Suppose that $f$ is a function and $f^{-1}$ is the inverse of $f$. If $f(3)=4$, $f(5)=1$, and $f(2)=5$, evaluate $f^{-1}\left(f^{-1}(5)+f^{-1}(4)\right)$. | 2 |
The teacher told the class that if they averaged at least 75% on their final exam, they could have a pizza party. Everyone took the exam on Monday, except for William, who was allowed to take it on Tuesday. If there are 30 people in the class, and the average before he took the test was a 74%, what score does he have t... | Let x be the score that William needs to achieve.
Since there are 30 people in the class, there are 30 - 1 = <<30-1=29>>29 people without him.
Since their average was a 74%, this means that (29 * 74 + x) / 30 >= 75.
Thus, 29 * 74 + x >= 30 * 75
Thus, 29 * 74 + x >= 2250
Thus, 2146 + x >= 2250
Thus, x >= 2250 - 2146
Thu... |
Let $s(n)$ denote the number of 1's in the binary representation of $n$. Compute $$\frac{1}{255} \sum_{0 \leq n<16} 2^{n}(-1)^{s(n)}$$ | 45 |
The graph of the rational function $\frac{p(x)}{q(x)}$ is shown below, with a horizontal asymptote of $y = 0$ and a vertical asymptote of $ x=-1 $. If $q(x)$ is quadratic, $p(2)=1$, and $q(2) = 3$, find $p(x) + q(x).$
[asy]
size(8cm);
import graph;
Label f;
f.p=fontsize(6);
real f(real x) {return (x-1)/((x-1)*(x+1)... | x^2 + x - 2 |
Three of the four vertices of a rectangle are $(3, 7)$, $(12, 7)$, and $(12, -4)$. What is the area of the intersection of this rectangular region and the region inside the graph of the equation $(x - 3)^2 + (y + 4)^2 = 16$? | 4\pi |
John goes to the store and pays with a 20 dollar bill. He buys 3 sodas and gets $14 in change. How much did each soda cost? | He paid 20-14=$<<20-14=6>>6
So each soda cost 6/3=$<<6/3=2>>2
#### 2 |
Given the sequence $\{a_n\}$ is a non-zero arithmetic sequence, $S_n$ denotes the sum of the first $n$ terms, and $S_{2n-1} = a_n^2$ for any $n \in \mathbb{N^*}$. If the inequality $\dfrac{1}{a_1a_2} + \dfrac{1}{a_2a_3} + \ldots + \dfrac{1}{a_na_{n+1}} \leqslant n\log_{\frac{1}{8}}\lambda$ holds for any $n \in \mathbb{... | \frac{1}{2} |
Given the function $g(x) = \frac{6x^2 + 11x + 17}{7(2 + x)}$, find the minimum value of $g(x)$ for $x \ge 0$. | \frac{127}{24} |
Eight women of different heights are at a party. Each woman decides to only shake hands with women shorter than herself. How many handshakes take place? | 0 |
A grocery store manager decides to design a more compact pyramid-like stack of apples with a rectangular base of 4 apples by 6 apples. Each apple above the first level still rests in a pocket formed by four apples below, and the stack is completed with a double row of apples on top. Determine the total number of apples... | 53 |
Let point P be the intersection point in the first quadrant of the hyperbola $\frac{x^{2}}{a^{2}}- \frac{y^{2}}{b^{2}}=1 (a > 0, b > 0)$ and the circle $x^{2}+y^{2}=a^{2}+b^{2}$. F\1 and F\2 are the left and right foci of the hyperbola, respectively, and $|PF_1|=3|PF_2|$. Find the eccentricity of the hyperbola. | \frac{\sqrt{10}}{2} |
Randomly select a number $x$ in the interval $[-1, 1]$. The probability that the value of $\cos \frac{\pi x}{2}$ falls between 0 and $\frac{1}{2}$ is ______. | \frac{1}{3} |
Given that Xiao Ming ran a lap on a 360-meter circular track at a speed of 5 meters per second in the first half of the time and 4 meters per second in the second half of the time, determine the time taken to run in the second half of the distance. | 44 |
On the base \(AC\) of an isosceles triangle \(ABC\), a point \(E\) is taken, and on the sides \(AB\) and \(BC\), points \(K\) and \(M\) are taken such that \(KE \parallel BC\) and \(EM \parallel AB\). What fraction of the area of triangle \(\mathrm{ABC}\) is occupied by the area of triangle \(KEM\) if \(BM:EM = 2:3\)? | 6/25 |
Point \( M \) belongs to the edge \( CD \) of the parallelepiped \( ABCDA_1B_1C_1D_1 \), where \( CM: MD = 1:2 \). Construct the section of the parallelepiped with a plane passing through point \( M \) parallel to the lines \( DB \) and \( AC_1 \). In what ratio does this plane divide the diagonal \( A_1C \) of the par... | 1 : 11 |
Find the smallest positive number \( c \) with the following property: For any integer \( n \geqslant 4 \) and any set \( A \subseteq \{1, 2, \ldots, n\} \), if \( |A| > c n \), then there exists a function \( f: A \rightarrow \{1, -1\} \) such that \( \left|\sum_{a \in A} f(a) \cdot a\right| \leq 1 \). | 2/3 |
An eight-sided die numbered from 1 to 8 is rolled, and $Q$ is the product of the seven numbers that are visible. What is the largest number that is certain to divide $Q$? | 192 |
Find all the values of $m$ for which the zeros of $2 x^{2}-m x-8$ differ by $m-1$. | 6,-\frac{10}{3} |
There are $4$ cards, marked with $0$, $1$, $2$, $3$ respectively. If two cards are randomly drawn from these $4$ cards to form a two-digit number, what is the probability that this number is even? | \frac{5}{9} |
Let $a,$ $b,$ and $c$ be nonzero real numbers such that $a + b + c = 3$. Simplify:
\[
\frac{1}{b^2 + c^2 - 3a^2} + \frac{1}{a^2 + c^2 - 3b^2} + \frac{1}{a^2 + b^2 - 3c^2}.
\] | -3 |
Given the equation $x^{2}+4ax+3a+1=0 (a > 1)$, whose two roots are $\tan \alpha$ and $\tan \beta$, with $\alpha, \beta \in (-\frac{\pi}{2}, \frac{\pi}{2})$, find $\tan \frac{\alpha + \beta}{2}$. | -2 |
The volume of the solid of revolution generated by rotating the region bounded by the curve $y= \sqrt{2x}$, the line $y=x-4$, and the x-axis around the x-axis is \_\_\_\_\_\_. | \frac{128\pi}{3} |
In right triangle $ABC$, $\sin A = \frac{8}{17}$ and $\sin B = 1$. Find $\sin C$. | \frac{15}{17} |
Let $ABCDEFGH$ be a cube with each edge of length $s$. A right square pyramid is placed on top of the cube such that its base aligns perfectly with the top face $EFGH$ of the cube, and its apex $P$ is directly above $E$ at a height $s$. Calculate $\sin \angle FAP$. | \frac{\sqrt{2}}{2} |
The matrices
\[\begin{pmatrix} a & 1 & b \\ 2 & 2 & 3 \\ c & 5 & d \end{pmatrix} \quad \text{and} \quad \begin{pmatrix} -5 & e & -11 \\ f & -13 & g \\ 2 & h & 4 \end{pmatrix}\]are inverses. Find $a + b + c + d + e + f + g + h.$ | 45 |
Find the number of ordered pairs $(x,y)$ of positive integers that satisfy $x \le 2y \le 60$ and $y \le 2x \le 60$. | 480 |
The product of $7d^2-3d+g$ and $3d^2+hd-8$ is $21d^4-44d^3-35d^2+14d-16$. What is $g+h$? | -3 |
Call a positive integer an uphill integer if every digit is strictly greater than the previous digit. For example, $1357$, $89$, and $5$ are all uphill integers, but $32$, $1240$, and $466$ are not. How many uphill integers are divisible by $15$? | 6 |
Each day, Jenny ate $20\%$ of the jellybeans that were in her jar at the beginning of that day. At the end of second day, 32 remained. How many jellybeans were in the jar originally? | 50 |
A flea is jumping on the vertices of square \(ABCD\), starting from vertex \(A\). With each jump, it moves to an adjacent vertex with a probability of \(\frac{1}{2}\). The flea stops when it reaches the last vertex it has not yet visited. Determine the probability that each vertex will be the last one visited. | \frac{1}{3} |
Stacy has 32 berries. Steve takes 4 of Stacy's berries, and still has 7 less berries than Stacy started with. How many berries did Steve start with? | Let x be the number of berries Steve started with
x+4=32-7
x+4=25
x=<<21=21>>21 berries
#### 21 |
The restaurant has two types of tables: square tables that can seat 4 people, and round tables that can seat 9 people. If the number of diners exactly fills several tables, the restaurant manager calls this number a "wealth number." Among the numbers from 1 to 100, how many "wealth numbers" are there? | 88 |
A particle moves in the Cartesian plane according to the following rules:
From any lattice point $(a,b),$ the particle may only move to $(a+1,b), (a,b+1),$ or $(a+1,b+1).$
There are no right angle turns in the particle's path.
How many different paths can the particle take from $(0,0)$ to $(5,5)$?
| 83 |
John buys 3 puzzles. The first puzzle has 1000 pieces. The second and third puzzles have the same number of pieces and each has 50% more pieces. How many total pieces are all the puzzles? | The second puzzle has 1000*.5=<<1000*.5=500>>500 more pieces than the first
So it has 1000+500=<<1000+500=1500>>1500 total pieces
That means those two puzzles have 1500*2=<<1500*2=3000>>3000 pieces
So in total there were 3000+1000=<<3000+1000=4000>>4000 pieces
#### 4000 |
The roots of the equation $2\sqrt{x} + 2x^{-\frac{1}{2}} = 5$ can be found by solving: | 4x^2-17x+4 = 0 |
Find the sum of all real numbers $x$ that are not in the domain of the function $$g(x) = \frac{1}{2 + \frac{1}{2 + \frac{1}{x}}}.$$ | -\frac{9}{10} |
Consider the function $f(x) = x^2 +2\sqrt{x}$. Evaluate $2f(2) - f(8)$. | -56 |
A list of five positive integers has a median of 3 and a mean of 11. What is the maximum possible value of the list's largest element? | 47 |
How many of the numbers from the set $\{1,\ 2,\ 3,\ldots,\ 50\}$ have a perfect square factor other than one? | 19 |
Convert $3206_7$ to a base 10 integer. | 1133 |
A hiker is exploring a trail. The trail has three sections: the first $25 \%$ of the trail is along a river, the next $\frac{5}{8}$ of the trail is through a forest, and the remaining 3 km of the trail is up a hill. How long is the trail? | 24 \text{ km} |
Sara conducted a survey among a group to find out how many people were aware that bats can transmit diseases. She found out that $75.3\%$ believed bats could transmit diseases. Among those who believed this, $60.2\%$ incorrectly thought that all bats transmit Zika virus, which amounted to 37 people. Determine how many ... | 81 |
The length of the median to the hypotenuse of an isosceles, right triangle is $10$ units. What is the length of a leg of the triangle, in units? Express your answer in simplest radical form. | 10\sqrt{2} |
Cagney can frost a cupcake every 18 seconds and Lacey can frost a cupcake every 40 seconds. Lacey starts working 1 minute after Cagney starts. Calculate the number of cupcakes that they can frost together in 6 minutes. | 27 |
The hypotenuse of a right triangle whose legs are consecutive even numbers is 34 units. What is the sum of the lengths of the two legs? | 46 |
Christi saw twice as many black bears as white bears and 40 more brown bears than black bears in a national park. If the number of black bears in the park is 60, calculate the population of bears in the park. | If there are 60 black bears in the park and 40 more brown bears than black bears, there are 60+40 = <<60+40=100>>100 brown bears.
The number of black and brown bears in the park is 100+60 = <<100+60=160>>160
Twice the number of black bears as white bears means 60/2 = <<60/2=30>>30 white bears.
The population of bears i... |
In a field of 500 clovers, 20% have four leaves and one quarter of these are purple clovers. Assuming these proportions are exactly correct, how many clovers in the field are both purple and four-leaved? | There are 500/5= <<500/5=100>>100 four leaf clovers
There are 100/4= <<100/4=25>>25 purple four leaf clovers
#### 25 |
How can we connect 50 cities with the minimum number of flight routes so that it's possible to travel from any city to any other city with no more than two layovers? | 49 |
Given the function $f(x) = \log_{m}(m - x)$, if the maximum value in the interval $[3, 5]$ is 1 greater than the minimum value, determine the real number $m$. | 3 + \sqrt{6} |
My co-worker Larry only likes numbers that are divisible by 4, such as 20, or 4,004. How many different ones digits are possible in numbers that Larry likes? | 5 |
Find $q(x)$ if the graph of $\frac{x^3-2x^2-5x+3}{q(x)}$ has vertical asymptotes at $2$ and $-2$, no horizontal asymptote, and $q(3) = 15$. | 3x^2 - 12 |
Let $x$ and $y$ be positive real numbers such that $x + y = 10.$ Find the minimum value of $\frac{1}{x} + \frac{1}{y}.$ | \frac{2}{5} |
A school canteen sells a sandwich at $2, a hamburger at $2, one stick of hotdog at $1, and a can of fruit juice at $2 each can. Selene buys three sandwiches and a can of fruit juice. Tanya buys two hamburgers and two cans of fruit juice. How much do Selene and Tanya spend together? | Three sandwiches cost 3 x $2 = $<<3*2=6>>6.
So, Selene spends $6 + $2 = $<<6+2=8>>8.
Two hamburgers cost 2 x $2 = $<<2*2=4>>4.
Two cans of fruit juice cost 2 x $2 = $<<2*2=4>>4.
Thus, Tanyah spends $4 +$4 = $<<4+4=8>>8.
Therefore, Selene and Tanya spend $8 + $8 = $<<8+8=16>>16.
#### 16 |
How many positive integers, not exceeding 100, are multiples of 2 or 3 but not 4? | 42 |
Calculate \(\int_{0}^{1} e^{-x^{2}} \, dx\) to an accuracy of 0.001. | 0.747 |
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