problem stringlengths 10 5.15k | answer stringlengths 0 1.23k |
|---|---|
Jean needs to buy 10 new pairs of pants. A store is running a sale for 20% off. If the pants normally retail for $45 each how much will she need to pay for all of them after paying a 10% tax? | The discount is 45*.2=$<<45*.2=9>>9 off each pair
So she pays 45-9= $<<45-9=36>>36 each
So it will cost 36*10=$<<36*10=360>>360 before tax
The tax will add 360*.1=$<<360*.1=36>>36
So she has to pay 360+36=$<<360+36=396>>396
#### 396 |
Given that point \(Z\) moves on \(|z| = 3\) in the complex plane, and \(w = \frac{1}{2}\left(z + \frac{1}{z}\right)\), where the trajectory of \(w\) is the curve \(\Gamma\). A line \(l\) passes through point \(P(1,0)\) and intersects the curve \(\Gamma\) at points \(A\) and \(B\), and intersects the imaginary axis at p... | -\frac{25}{8} |
Given $\sin\left(\theta - \frac{\pi}{6}\right) = \frac{1}{4}$ with $\theta \in \left( \frac{\pi}{6}, \frac{2\pi}{3}\right)$, calculate the value of $\cos\left(\frac{3\pi}{2} + \theta\right)$. | \frac{\sqrt{15} + \sqrt{3}}{8} |
Geoff and Trevor each roll a fair six-sided die. What is the probability that the product of the numbers they roll is even? | \dfrac34 |
Given that \( I \) is the incenter of \( \triangle ABC \) and \( 5 \overrightarrow{IA} = 4(\overrightarrow{BI} + \overrightarrow{CI}) \). Let \( R \) and \( r \) be the radii of the circumcircle and the incircle of \( \triangle ABC \) respectively. If \( r = 15 \), then find \( R \). | 32 |
Solve for $x$: $3^{2x} = \sqrt{27}$. Express your answer as a common fraction. | \frac{3}{4} |
Vincent’s washer broke so he had to go to the laundromat. On Wednesday he washed six loads of clothes. The next day he had time to wash double the number of loads he did the day before. On Friday he had a test and could only manage half of the loads he did on Thursday. On Saturday the laundromat closed at noon and he c... | On Thursday he washed 6*2=<<6*2=12>>12 loads of laundry
On Friday he washed 12/2=<<12/2=6>>6 loads of laundry
On Saturday he washed 6/3=<<6/3=2>>2 loads of laundry
In total he washed 6+12+6+2= <<6+12+6+2=26>>26 loads of laundry
#### 26 |
If the ratio of $b$ to $a$ is 3, then what is the value of $a$ when $b=12-5a$? | \frac{3}{2} |
Two lines are perpendicular and intersect at point $O$. Points $A$ and $B$ move along these two lines at a constant speed. When $A$ is at point $O$, $B$ is 500 yards away from point $O$. After 2 minutes, both points $A$ and $B$ are equidistant from $O$. After another 8 minutes, they are still equidistant from $O$. Find... | \frac{2}{3} |
John works a job that offers performance bonuses. He makes $80 a day and works for 8 hours. He has the option of working hard to earn the performance bonus of an extra $20 a day, but the extra effort results in a 2-hour longer workday. How much does John make per hour if he decides to earn the bonus? | First, we need to determine the length of John's workday if he decides to earn the bonus. We do this by performing 8+2= <<8+2=10>>10 hours for his workday.
Next, we need to determine his overall pay. We do this by performing 80+20=<<80+20=100>>100 dollars a day.
We then determine John's hourly rate by dividing his pay ... |
Find $\tan G$ in the right triangle shown below.
[asy]
pair H,F,G;
H = (0,0);
G = (15,0);
F = (0,8);
draw(F--G--H--F);
draw(rightanglemark(F,H,G,20));
label("$H$",H,SW);
label("$G$",G,SE);
label("$F$",F,N);
label("$17$",(F+G)/2,NE);
label("$15$",G/2,S);
[/asy] | \frac{8}{15} |
Let $P(z) = z^8 + \left(4\sqrt{3} + 6\right)z^4 - \left(4\sqrt{3} + 7\right)$. What is the minimum perimeter among all the $8$-sided polygons in the complex plane whose vertices are precisely the zeros of $P(z)$? | 8 \sqrt{2} |
Ivan Tsarevich is fighting the Dragon Gorynych on the Kalinov Bridge. The Dragon has 198 heads. With one swing of his sword, Ivan Tsarevich can cut off five heads. However, new heads immediately grow back, the number of which is equal to the remainder when the number of heads left after Ivan's swing is divided by 9. If... | 40 |
In the triangular pyramid $P-ABC$, $PA\bot $ plane $ABC$, $\triangle ABC$ is an isosceles triangle, where $AB=BC=2$, $\angle ABC=120{}^\circ $, and $PA=4$. The surface area of the circumscribed sphere of the triangular pyramid $P-ABC$ is __________. | 32\pi |
How many positive integer multiples of $210$ can be expressed in the form $6^{j} - 6^{i}$, where $i$ and $j$ are integers and $0 \leq i < j \leq 49$? | 600 |
Ahmed is 11 years old and Fouad is 26 years old. In how many years will Fouad's age be double Ahmed's current age? | Let X be the number of years before Fouad's age doubles Ahmed's age.
So (X+11)*2 = X+26.
So X*2 + 22 = X + 26.
So X = 26 - 22 = <<26-22=4>>4 years.
#### 4 |
In a shooting contest, 8 targets are arranged in two columns with 3 targets and one column with 2 targets. The rules are:
- The shooter can freely choose which column to shoot at.
- He must attempt the lowest target not yet hit.
a) If the shooter ignores the second rule, in how many ways can he choose only 3 positio... | 560 |
A sphere is inscribed in a right cone with base radius $15$ cm and height $30$ cm. The radius of the sphere can be expressed as $b\sqrt{d} - g$ cm, where $g = b + 6$. What is the value of $b + d$? | 12.5 |
$\Phi$ is the union of all triangles that are symmetric of the triangle $ABC$ wrt a point $O$ , as point $O$ moves along the triangle's sides. If the area of the triangle is $E$ , find the area of $\Phi$ . | 2E |
Calculate: $(243)^{\frac35}$ | 27 |
Patty decides that to convince her brother and sister to do her chores in exchange for cookies. Patty agrees to give them 3 cookies for every chore they do. Each kid normally has 4 chores to do per week. Patty has $15 to buy cookies. Each pack of cookies contains 24 cookies and costs $3. How many weeks can Patty go wit... | Patty can buy 5 packs of cookies because 15 / 3 = <<15/3=5>>5
Patty gets 120 cookies because 5 x 24 = <<5*24=120>>120
Patty would need to pay 12 cookies per week because 3 x 4 = <<3*4=12>>12
Patty can avoid 10 weeks of chores because 120 / 12 = <<120/12=10>>10
#### 10 |
Let $x$ and $y$ be positive real numbers such that $3x + 4y < 72.$ Find the maximum value of
\[xy (72 - 3x - 4y).\] | 1152 |
If $\frac{1}{4}$ of $2^{30}$ is $4^x$, then what is the value of $x$ ? | 14 |
If Jason eats three potatoes in 20 minutes, how long, in hours, would it take for him to eat all 27 potatoes cooked by his wife? | Since 1 hour is 60 minutes, 20 minutes is 20/60=1/3 of an hour.
Eating 3 potatoes per 20 minutes is 3/(1/3)=<<3/(1/3)=9>>9 potatoes per hour.
At 9 potatoes per hour, it will take Jason 27/9=3 hours to eat all of the potatoes.
#### 3 |
Tyler has 21 CDs. He gives away a third of his CDs to his friend. Then he goes to the music store and buys 8 brand new CDs. How many CDs does Tyler have now? | Tyler gives away 21 / 3 = <<21/3=7>>7 CDs
After giving away a third of his CDs, Tyler has 21 - 7 = <<21-7=14>>14 CDs
After buying new CDs, Tyler has 14 + 8 = <<14+8=22>>22 CDs
#### 22 |
A standard deck of 52 cards has 13 ranks (Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, King) and 4 suits ($\spadesuit$, $\heartsuit$, $\diamondsuit$, and $\clubsuit$), such that there is exactly one card for any given rank and suit. Two of the suits ($\spadesuit$ and $\clubsuit$) are black and the other two suits ($\... | \dfrac{1}{13} |
A cylinder with a volume of 21 is inscribed in a cone. The plane of the upper base of this cylinder cuts off a truncated cone with a volume of 91 from the original cone. Find the volume of the original cone. | 94.5 |
In a square \(ABCD\), let \(P\) be a point on the side \(BC\) such that \(BP = 3PC\) and \(Q\) be the midpoint of \(CD\). If the area of the triangle \(PCQ\) is 5, what is the area of triangle \(QDA\)? | 20 |
In triangle $\triangle ABC$, the length of the side opposite angle $A$ is equal to 2, the vector $\overrightarrow {m} = (2, 2\cos^2 \frac {B+C}{2}-1)$, and the vector $\overrightarrow {n} = (\sin \frac {A}{2}, -1)$.
(1) Find the size of angle $A$ when the dot product $\overrightarrow {m} \cdot \overrightarrow {n}$ re... | \sqrt{3} |
Place the numbers 1, 2, 3, 4, 5, 6, 7, and 8 on the eight vertices of a cube such that the sum of any three numbers on a face is at least 10. Find the minimum sum of the four numbers on any face. | 16 |
What is the sum of all positive integers less than 100 that are squares of perfect squares? | 98 |
$A, B, C$ and $D$ are distinct positive integers such that the product $AB = 60$, the product $CD = 60$ and $A - B = C + D$ . What is the value of $A$? | 20 |
Factor completely over the set of polynomials with integer coefficients:
\[4(x + 5)(x + 6)(x + 10)(x + 12) - 3x^2.\] | (2x^2 + 35x + 120)(x + 8)(2x + 15) |
Tim has 30 less apples than Martha, and Harry has half as many apples as Tim. If Martha has 68 apples, how many apples does Harry have? | Tim has 68-30 = <<68-30=38>>38 apples.
Harry has 38/2 = <<38/2=19>>19 apples.
#### 19 |
Let $x$ and $y$ be real numbers such that
\[xy - \frac{x}{y^2} - \frac{y}{x^2} = 3.\]Find the sum of all possible values of $(x - 1)(y - 1).$ | 5 |
The clock shows 00:00, and the hour and minute hands coincide. Considering this coincidence to be number 0, determine after what time interval (in minutes) they will coincide the 19th time. If necessary, round the answer to two decimal places following the rounding rules. | 1243.64 |
Bob rolls a fair six-sided die each morning. If Bob rolls a composite number, he eats sweetened cereal. If he rolls a prime number, he eats unsweetened cereal. If he rolls a 1, then he rolls again. In a non-leap year, what is the expected value of the difference between the number of days Bob eats unsweetened cereal an... | 73 |
A point $Q$ lies inside the triangle $\triangle DEF$ such that lines drawn through $Q$ parallel to the sides of $\triangle DEF$ divide it into three smaller triangles $u_1$, $u_2$, and $u_3$ with areas $16$, $25$, and $36$ respectively. Determine the area of $\triangle DEF$. | 77 |
Two more than three times $B$ is equal to 20. What is the value of $B$? | 6 |
A plane intersects a right circular cylinder of radius $3$ forming an ellipse. If the major axis of the ellipse is $75\%$ longer than the minor axis, the length of the major axis is: | 10.5 |
In a square, points A and B are midpoints of two adjacent sides. A line segment is drawn from point A to the opposite vertex of the side that does not contain B, forming a triangle. What fraction of the interior of the square is shaded, if the triangle is left unshaded?
[asy]
filldraw((0,0)--(2,0)--(2,2)--(0,2)--cycle... | \frac{3}{4} |
How many distinct arrangements of the letters in the word "example" are there? | 5040 |
Each of $6$ balls is randomly and independently painted either black or white with equal probability. What is the probability that every ball is different in color from more than half of the other $5$ balls? | \frac{5}{16} |
In a tournament, any two players play against each other. Each player earns one point for a victory, 1/2 for a draw, and 0 points for a loss. Let \( S \) be the set of the 10 lowest scores. We know that each player obtained half of their score by playing against players in \( S \).
a) What is the sum of the scores of ... | 25 |
What is the sum of the positive integer divisors of 23? | 24 |
Al gets the disease algebritis and must take one green pill and one pink pill each day for two weeks. A green pill costs $1 more than a pink pill, and Al's pills cost a total of $546 for the two weeks. How much does one green pill cost? | $19 |
In response to the call for rural revitalization, Xiao Jiao, a college graduate who has successfully started a business in another place, resolutely returned to her hometown to become a new farmer and established a fruit and vegetable ecological planting base. Recently, in order to fertilize the vegetables in the base,... | 500 |
Tony drives a car that gets 25 miles to the gallon. He drives 50 miles round trip to work 5 days a week. His tank holds 10 gallons. He begins the week with a full tank and when he runs out he fills up at the local gas station for $2 a gallon. How much money does Tony spend on gas in 4 weeks? | First, we need to determine how many miles Tony drives in 4 weeks. We do this by performing 50*5=<<50*5=250>>250 miles per week.
Next, we multiply the weekly total by 4 by performing 4*250= <<4*250=1000>>1000 miles in 4 weeks.
Now, we need to find out how many times Tony needs to fill up his gas tank. His gas tank hold... |
A high school's 11th-grade class 1 has 45 male students and 15 female students. The teacher uses stratified sampling to form a 4-person extracurricular interest group. Calculate the probability of a student being selected for this group and the number of male and female students in the extracurricular interest group. ... | 0.5 |
Let $\mathbf{a},$ $\mathbf{b},$ and $\mathbf{c}$ be nonzero vectors, no two of which are parallel, such that
\[(\mathbf{a} \times \mathbf{b}) \times \mathbf{c} = \frac{1}{3} \|\mathbf{b}\| \|\mathbf{c}\| \mathbf{a}.\]Let $\theta$ be the angle between $\mathbf{b}$ and $\mathbf{c}.$ Find $\sin \theta.$ | \frac{2 \sqrt{2}}{3} |
Find the number of real solutions to the equation
\[\frac{1}{x - 1} + \frac{2}{x - 2} + \frac{3}{x - 3} + \dots + \frac{120}{x - 120} = x.\] | 121 |
The largest three-digit number divided by an integer, with the quotient rounded to one decimal place being 2.5, will have the smallest divisor as: | 392 |
The expression \(\frac{3}{10}+\frac{3}{100}+\frac{3}{1000}\) is equal to: | 0.333 |
On the AMC 8 contest Billy answers 13 questions correctly, answers 7 questions incorrectly and doesn't answer the last 5. What is his score? | 13 |
The second and fifth terms of an arithmetic sequence are 17 and 19, respectively. What is the eighth term? | 21 |
Alicia has $n$ candies, where $n$ is a positive integer with three digits. If she buys $5$ more, she will have a multiple of $8$. If she loses $8$, she will have a multiple of $5$. What is the smallest possible value of $n$? | 123 |
Given the function $f\left(x\right)=\cos x+\left(x+1\right)\sin x+1$ on the interval $\left[0,2\pi \right]$, find the minimum and maximum values of $f(x)$. | \frac{\pi}{2}+2 |
Compute the number of ways to tile a $3 \times 5$ rectangle with one $1 \times 1$ tile, one $1 \times 2$ tile, one $1 \times 3$ tile, one $1 \times 4$ tile, and one $1 \times 5$ tile. (The tiles can be rotated, and tilings that differ by rotation or reflection are considered distinct.) | 40 |
All the numbers 2, 3, 4, 5, 6, 7 are assigned to the six faces of a cube, one number to each face. For each of the eight vertices of the cube, a product of three numbers is computed, where the three numbers are the numbers assigned to the three faces that include that vertex. What is the greatest possible value of the ... | 729 |
Queen High School has $1500$ students, and each student takes $6$ classes per day. Each teacher teaches $5$ classes, with each class having $25$ students and $1$ teacher. How many teachers are there at Queen High School? | 72 |
Monroe made 200 granola bars for her family. She and her husband ate 80, and the rest was divided equally among her children. If each child received 20 granola bars, how many children are in the family? | If Monroe and her husband ate 80 granola bars, then the number of granola bars the children shared is 200-80 = 120
Each child received 20 bars, and there were 120 granola bars, meaning there were 120/20 = <<120/20=6>>6 children in the family.
#### 6 |
What is $(5^{-2})^0 + (5^0)^3$? | 2 |
Given positive integers \( x, y, z \) that satisfy the condition \( x y z = (14 - x)(14 - y)(14 - z) \), and \( x + y + z < 28 \), what is the maximum value of \( x^2 + y^2 + z^2 \)? | 219 |
Determine the sum of the fifth and sixth elements in Row 20 of Pascal's triangle. | 20349 |
Auston is 60 inches tall. Using the conversion 1 inch = 2.54 cm, how tall is Auston in centimeters? Express your answer as a decimal to the nearest tenth. | 152.4 |
Brian is taping up some boxes. Each box needs three pieces of tape, one as long as the long side and two as long as the short side. If Brian tapes up 5 boxes that measure 15 inches by 30 inches and 2 boxes that measure 40 inches square, how much tape does he need? | First find the amount of tape each 15 in x 30 in box needs on the short sides: 2 sides * 15 inches/side = <<2*15=30>>30 inches
Then add that to the amount of tape each 15 in x 30 in box needs on the long side to find the total amount needed per box: 30 inches + 30 inches = <<30+30=60>>60 inches
Then multiply that amoun... |
James buys steaks for buy one get one free. The price is $15 per pound and he buys 20 pounds. How much did he pay for the steaks? | He pays for 20/2=<<20/2=10>>10 pounds of steaks
That cost 10*15=$<<10*15=150>>150
#### 150 |
Find $a$ if $a$ and $b$ are integers such that $x^2 - x - 1$ is a factor of $ax^{17} + bx^{16} + 1$. | 987 |
Find all non-negative solutions to the equation $2013^x+2014^y=2015^z$ | (0,1,1) |
John has 2 umbrellas in his house and 1 in the car. If they cost $8 each, how much did he pay in total? | He has 2+1=<<2+1=3>>3 umbrellas
That means he paid 3*8=$<<3*8=24>>24
#### 24 |
Bridgette has 2 dogs, 3 cats, and 4 birds. She gives the dogs a bath twice a month. She gives the cats a bath once a month. She gives the birds a bath once every 4 months. In a year, how many baths does she give? | Each dog gets 24 baths a year because 2 x 12 = <<2*12=24>>24
Each cat gets 12 baths a year because 1 x 12 = <<1*12=12>>12
Each bird averages .25 baths per month because 1 / 4 = <<1/4=.25>>.25
Each bird gets 3 baths a year because .25 x 12 = <<.25*12=3>>3
She gives 48 dog baths because 2 x 24 = <<2*24=48>>48
She gives 7... |
Given a convex quadrilateral \( ABCD \) with an interior point \( P \) such that \( P \) divides \( ABCD \) into four triangles \( ABP, BCP, CDP, \) and \( DAP \). Let \( G_1, G_2, G_3, \) and \( G_4 \) denote the centroids of these triangles, respectively. Determine the ratio of the area of quadrilateral \( G_1G_2G_3G... | \frac{1}{9} |
Let $S$ be a subset of $\{1,2,3,...,50\}$ such that no pair of distinct elements in $S$ has a sum divisible by $7$. What is the maximum number of elements in $S$?
$\text{(A) } 6\quad \text{(B) } 7\quad \text{(C) } 14\quad \text{(D) } 22\quad \text{(E) } 23$
| 23 |
Add $101_2 + 11_2 + 1100_2 + 11101_2.$ Express your answer in base $2.$ | 110001_2 |
(1) Among the following 4 propositions:
① The converse of "If $a$, $G$, $b$ form a geometric sequence, then $G^2=ab$";
② The negation of "If $x^2+x-6\geqslant 0$, then $x > 2$";
③ In $\triangle ABC$, the contrapositive of "If $A > B$, then $\sin A > \sin B$";
④ When $0\leqslant \alpha \leqslant \pi$, if $8x^2-(8\si... | \frac{\sqrt{17}}{2} |
Angela and Barry share a piece of land. The ratio of the area of Angela's portion to the area of Barry's portion is $3: 2$. They each grow corn and peas on their piece of land. The entire piece of land is covered by corn and peas in the ratio $7: 3$. On Angela's portion of the land, the ratio of corn to peas is $4: 1$.... | $11: 9$ |
For real numbers $w$ and $z$, $\frac{\frac{1}{w} + \frac{1}{z}}{\frac{1}{w} - \frac{1}{z}} = 2014.$ What is $\frac{w+z}{w-z}$? | -2014 |
Determine the sum of all single-digit replacements for $z$ such that the number ${24{,}z38}$ is divisible by 6. | 12 |
A hurricane is approaching the southern coast of Texas, and a rancher is planning to move 400 head of cattle 60 miles to higher ground to protect them from possible inland flooding that might occur. His animal transport truck holds 20 head of cattle. Traveling at 60 miles per hour, what is the total driving time, in ... | Given the limited capacity of his transport vehicle (20 head of cattle), the 400 head of cattle will require 400/20=<<400/20=20>>20 trips using his transport vehicle.
Traveling to the site at 60 mph for 60 miles it will take 60/60=<<60/60=1>>1 hour to travel one-way.
Since each trip requires driving to and returning fr... |
Below is an arithmetic expression where 9 Chinese characters represent the digits 1 to 9, and different characters represent different digits. What is the maximum possible value for the expression?
草 $\times$ 绿 + 花儿 $\times$ 红 + 春光明 $\times$ 媚 | 6242 |
The absolute value of a number \( x \) is equal to the distance from 0 to \( x \) along a number line and is written as \( |x| \). For example, \( |8|=8, |-3|=3 \), and \( |0|=0 \). For how many pairs \( (a, b) \) of integers is \( |a|+|b| \leq 10 \)? | 221 |
Xanthia can read 100 pages per hour and Molly can read 50 pages per hour. If they each read the same book, and the book has 225 pages, how many more minutes than Xanthia would it take for Molly to finish reading the book? | 135 |
Compute
\[\sum_{n = 1}^\infty \frac{2n - 1}{n(n + 1)(n + 2)}.\] | \frac{3}{4} |
What is the sum of the integers from $-30$ to $50$, inclusive? | 810 |
The sum of the absolute values of the terms of a finite arithmetic progression is 100. If all its terms are increased by 1 or all its terms are increased by 2, the sum of the absolute values of the terms of the resulting progression will also be 100. What values can the quantity $n^{2} d$ take under these conditions, w... | 400 |
Find the least positive integer such that when its leftmost digit is deleted, the resulting integer is 1/19 of the original integer. | 95 |
In $\triangle XYZ$, $\angle X = 90^\circ$ and $\tan Z = \sqrt{3}$. If $YZ = 150$, what is $XY$? | 75\sqrt{3} |
Find the real solution(s) to the equation $(x+y)^{2}=(x+1)(y-1)$. | (-1,1) |
Given the real-coefficient polynomial \( f(x) = x^4 + a x^3 + b x^2 + c x + d \) that satisfies \( f(1) = 2 \), \( f(2) = 4 \), and \( f(3) = 6 \), find the set of all possible values of \( f(0) + f(4) \). | 32 |
Suppose two distinct integers are chosen from between 5 and 17, inclusive. What is the probability that their product is odd? | \dfrac{7}{26} |
The arithmetic mean (average) of four numbers is $85$. If the largest of these numbers is $97$, then the mean of the remaining three numbers is | 81.0 |
Person A departs from location A to location B, while persons B and C depart from location B to location A. After A has traveled 50 kilometers, B and C start simultaneously from location B. A and B meet at location C, and A and C meet at location D. It is known that A's speed is three times that of C and 1.5 times that... | 130 |
If $x$ is a real number and $k$ is a nonnegative integer, recall that the binomial coefficient $\binom{x}{k}$ is defined by the formula
\[
\binom{x}{k} = \frac{x(x - 1)(x - 2) \dots (x - k + 1)}{k!} \, .
\]Compute the value of
\[
\frac{\binom{1/2}{2014} \cdot 4^{2014}}{\binom{4028}{2014}} \, .
\] | -\frac{1} { 4027} |
A school band found they could arrange themselves in rows of 6, 7, or 8 with no one left over. What is the minimum number of students in the band? | 168 |
Suppose that 7 boys and 13 girls line up in a row. Let $S$ be the number of places in the row where a boy and a girl are standing next to each other. For example, for the row $\text{GBBGGGBGBGGGBGBGGBGG}$ we have that $S=12$. The average value of $S$ (if all possible orders of these 20 people are considered) is closest... | 9 |
Given the function $$f(x)= \begin{cases} a^{x}, x<0 \\ ( \frac {1}{4}-a)x+2a, x\geq0\end{cases}$$ such that for any $x\_1 \neq x\_2$, the inequality $$\frac {f(x_{1})-f(x_{2})}{x_{1}-x_{2}}<0$$ holds true. Determine the range of values for the real number $a$. | \frac{1}{2} |
John gets lost on his way home. His normal trip is 150 miles and would take him 3 hours. He ends up driving 50 miles out of the way and has to get back on track. How long did the trip take if he kept the same speed? | His speed is 150/3=<<150/3=50>>50 mph
He was detoured 50*2=<<50*2=100>>100 miles
So that adds 100/50=<<100/50=2>>2 hours
So the total trip was 3+2=<<3+2=5>>5 hours
#### 5 |
Kristy, a sales representative earns a basic salary of $7.50 per hour plus a 16% commission on everything she sells. This month, she worked for 160 hours and sold $25000 worth of items. Her monthly budget for food, clothing, rent, transportation, bills and savings is 95% of her total monthly earnings and the rest will ... | Kristy earns a basic salary of $7.50 x 160 = $<<7.5*160=1200>>1200.
Her total commission is $25000 x 16/100 = $<<25000*16/100=4000>>4000.
Thus, her total earning is $1200 + $4000 = $<<1200+4000=5200>>5200.
Her monthly budget aside from the insurance is $5200 x 95/100 = $<<5200*95/100=4940>>4940.
Therefore, she allocate... |
Three circles have the same center O. Point X divides segment OP in the ratio 1:3, with X being closer to O. Calculate the ratio of the area of the circle with radius OX to the area of the circle with radius OP. Next, find the area ratio of a third circle with radius 2*OX to the circle with radius OP. | \frac{1}{4} |
During breaks, schoolchildren played table tennis. Any two schoolchildren played no more than one game against each other. At the end of the week, it turned out that Petya played half, Kolya - a third, and Vasya - one fifth of the total number of games played during the week. What could be the total number of games pla... | 30 |
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