problem stringlengths 10 5.15k | answer stringlengths 0 1.23k |
|---|---|
David drives from his home to the airport to catch a flight. He drives $35$ miles in the first hour, but realizes that he will be $1$ hour late if he continues at this speed. He increases his speed by $15$ miles per hour for the rest of the way to the airport and arrives $30$ minutes early. How many miles is the airpor... | 210 |
Suppose that $\frac{2}{3}$ of $10$ bananas are worth as much as $8$ oranges. How many oranges are worth as much as $\frac{1}{2}$ of $5$ bananas? | 3 |
40 less than 10 times Diaz's age is 20 more than 10 times Sierra's age. If Sierra is currently 30 years old, how old will Diaz be 20 years from now? | If Sierra is currently 30 years old, 10 times her age is 30*10 = <<30*10=300>>300.
Twenty more than 10 times Sierra's age is 300+20 = 320
320 is 40 less than ten times Diaz's age so 10 times Diaz's age is 320+40 = 360
If ten times Diaz's age is 360, Diaz is 360/10 = <<360/10=36>>36 years old.
Twenty years from now, Dia... |
What is 25% of 60? | 15 |
In the rectangular coordinate system $XOY$, there is a line $l:\begin{cases} & x=t \\ & y=-\sqrt{3}t \\ \end{cases}(t$ is a parameter$)$, and a curve ${C_{1:}}\begin{cases} & x=\cos \theta \\ & y=1+\sin \theta \\ \end{cases}(\theta$ is a parameter$)$. Establish a polar coordinate system with the origin $O$ of this rect... | 4- \sqrt{3} |
Contractor Steve agreed to complete a job in 30 days. After 6 days he found that the 8 people assigned to the work had already done $\frac{1}{3}$ of the job. If everyone works at the same rate, what is the least number of people he must keep on the job to ensure that the job will be completed on time? | 4 |
Zara has a collection of $4$ marbles: an Aggie, a Bumblebee, a Steelie, and a Tiger. She wants to display them in a row on a shelf, but does not want to put the Steelie and the Tiger next to one another. In how many ways can she do this? | 12 |
Let $A$ and $B$ be two points on the parabola $y = x^2,$ such that when the tangents at $A$ and $B$ drawn, they are perpendicular. Then for any such pair of tangents, the $y$-coordinate of their point of intersection $P$ is always the same. Find this $y$-coordinate.
[asy]
unitsize(1.5 cm);
real parab (real x) {
r... | -\frac{1}{4} |
If $\lceil{\sqrt{x}}\rceil=12$, how many possible integer values of $x$ are there? | 23 |
Caleb, Andy and Billy went on a picnic with their father. Billy took 6 candies with him, Caleb took 11 and Andy left with 9. On the way, their father bought a packet of 36 candies. He gave 8 candies to Billy, 11 to Caleb and the rest to Andy. How many more candies does Andy now have than Caleb? | Billy took some candies and got more from his father for a total of 6+8 = <<6+8=14>>14 candies
Caleb has a total of 11+11 = <<11+11=22>>22 candies
Their father gave out 8+11 = <<8+11=19>>19 candies from a pack of 36
He gave the rest which is 36-19 = <<36-19=17>>17 candies to Andy
Andy has a total of 9+17 = <<9+17=26>>2... |
Arrange positive integers that are neither perfect squares nor perfect cubes (excluding 0) in ascending order as 2, 3, 5, 6, 7, 10, ..., and determine the 1000th number in this sequence. | 1039 |
How many irreducible fractions with a numerator of 2015 are there that are less than \( \frac{1}{2015} \) and greater than \( \frac{1}{2016} \)? | 1440 |
What is the total number of digits used when the first 2002 positive even integers are written? | 7456 |
A jar has $10$ red candies and $10$ blue candies. Terry picks two candies at random, then Mary picks two of the remaining candies at random. Given that the probability that they get the same color combination, irrespective of order, is $m/n,$ where $m$ and $n$ are relatively prime positive integers, find $m+n.$
| 441 |
There are 35 bottles of milk on the grocery store shelf. Jason buys 5 of the bottles and Harry buys 6 more. How many bottles of milk are left on the store shelf after Jason and Harry purchased milk? | Jason and Harry purchased a total of 5 + 6 = <<5+6=11>>11 bottles of milk.
So, there are 35 - 11 = <<35-11=24>>24 bottles of milk left on the store shelf.
#### 24 |
At Snowflake Plastics, each employee gets 10 sick days and 10 vacation days per year. If Mark uses half his allotment of both types of days in a year, how many hours' worth of days does he have left if each day covers an 8-hour long workday? | First, we add the two types of days together to find 10+10=<<10+10=20>>20 days in total between the two types.
We then divide this number in half to find Mark's remaining number of days, getting 20/2= <<10=10>>10 days remaining.
Since each day counts as 8 hours, we multiply 10*8= <<10*8=80>>80 hours' worth of days rema... |
An 18 inch by 24 inch painting is mounted in a wooden frame where the width of the wood at the top and bottom of the frame is twice the width of the wood at the sides. If the area of the frame is equal to the area of the painting, find the ratio of the shorter side to the longer side of this frame. | 2:3 |
Admiral Ackbar needs to send a 5-character message through hyperspace to the Rebels. Each character is a lowercase letter, and the same letter may appear more than once in a message. When the message is beamed through hyperspace, the characters come out in a random order. Ackbar chooses his message so that the Rebels h... | 26 |
A rectangle has positive integer side lengths and an area of 24. What perimeter of the rectangle cannot be? | 36 |
Let the function \( f(x) = x^3 + 3x^2 + 6x + 14 \), and \( f(a) = 1 \), \( f(b) = 19 \). Then \( a + b = \quad \). | -2 |
The function $f : \mathbb{R} \to \mathbb{R}$ satisfies
\[f(x) + 2f(1 - x) = 3x^2\]for all $x.$ Find $f(4).$ | 2 |
Xiaoming's family raises chickens and pigs in a ratio of 26:5, and sheep to horses in a ratio of 25:9, while the ratio of pigs to horses is 10:3. Find the ratio of chickens, pigs, horses, and sheep. | 156:30:9:25 |
Let $x, y, z$ be real numbers such that
\[
x + y + z = 5,
\]
\[
x^2 + y^2 + z^2 = 11.
\]
Find the smallest and largest possible values of $x$, and compute their sum. | \frac{10}{3} |
Find the largest natural number \( n \) for which the product of the numbers \( n, n+1, n+2, \ldots, n+20 \) is divisible by the square of one of these numbers. | 20 |
The café has enough chairs to seat $312_8$ people. If $3$ people are supposed to sit at one table, how many tables does the café have? | 67 |
Eighteen hours ago, Beth and I took 100 photographs of our project. Today, Beth and I will take 20% fewer photographs of the same project. If we were to take 300 photographs of the project, how many photographs would we take to reach the target? | If you took 100 photographs of the project 18 hours ago, and today 20% few photographs have been taken, then 20/100*100 = 20 fewer photographs of the project have been taken today.
The total number of photographs of the project that have been taken today is 100-20 = <<100-20=80>>80
So far, you've taken 80+100 = <<80+10... |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively, where $a=2$, $c=3$, and it satisfies $(2a-c)\cdot\cos B=b\cdot\cos C$. Find the value of $\overrightarrow{AB}\cdot\overrightarrow{BC}$. | -3 |
If the line $(m+2)x+3y+3=0$ is parallel to the line $x+(2m-1)y+m=0$, then the real number $m=$ \_\_\_\_\_\_. | -\frac{5}{2} |
Given that the batch of rice contains 1512 bushels and a sample of 216 grains contains 27 grains of wheat, calculate the approximate amount of wheat mixed in this batch of rice. | 189 |
In a right-angled triangle in which all side lengths are integers, one has a cathetus length $1994$ . Determine the length of the hypotenuse. | 994010 |
In the polygon shown, each side is perpendicular to its adjacent sides, and all 28 of the sides are congruent. The perimeter of the polygon is 56. Find the area of the polygon.
[asy]
unitsize(0.5 cm);
draw((3,0)--(4,0)--(4,1)--(5,1)--(5,2)--(6,2)--(6,3)--(7,3)--(7,4)--(6,4)--(6,5)--(5,5)--(5,6)--(4,6)--(4,7)--(3,7)... | 100 |
Let positive integers $a$, $b$, $c$ satisfy $ab + bc = 518$ and $ab - ac = 360$. The maximum value of $abc$ is ____. | 1008 |
The measure of angle $ACB$ is 40 degrees. If ray $CA$ is rotated 480 degrees about point $C$ in a clockwise direction, what will be the positive measure of the new acute angle $ACB$, in degrees?
[asy]
draw((0,0)--dir(40),linewidth(1),Arrow);
draw((0,0)--dir(0),linewidth(1),Arrow);
dot(.8dir(40));
dot(.8dir(0));
dot((0... | 80 |
Simplify
\[\frac{\tan^3 75^\circ + \cot^3 75^\circ}{\tan 75^\circ + \cot 75^\circ}.\] | 13 |
Given a sequence $\{a_{n}\}$ that satisfies the equation: ${a_{n+1}}+{({-1})^n}{a_n}=3n-1$ ($n∈{N^*}$), calculate the sum of the first $60$ terms of the sequence $\{a_{n}\}$. | 2760 |
A "double-single" number is a three-digit number made up of two identical digits followed by a different digit. For example, 553 is a double-single number. How many double-single numbers are there between 100 and 1000? | 81 |
For what ratio of the bases of a trapezoid does there exist a line on which the six points of intersection with the diagonals, the lateral sides, and the extensions of the bases of the trapezoid form five equal segments? | 1:2 |
For what values of $j$ does the equation $(2x+7)(x-5) = -43 + jx$ have exactly one real solution? Express your answer as a list of numbers, separated by commas. | 5,\,-11 |
There are 3 consecutive odd integers that have a sum of -147. What is the largest number? | Let N = smallest number
N + 2 = next number
N + 4 = largest number
N + (N + 2) + (N + 4) = -147
3N + 6 = -147
3N = <<-153=-153>>-153
N = -51
The largest number is <<-47=-47>>-47.
#### -47 |
Calculate $[x]$, where $x = -3.7 + 1.5$. | -3 |
Nine points are evenly spaced at intervals of one unit around a $3 \times 3$ square grid, such that each side of the square has three equally spaced points. Two of the 9 points are chosen at random. What is the probability that the two points are one unit apart?
A) $\frac{1}{3}$
B) $\frac{1}{4}$
C) $\frac{1}{5}$
D) $\f... | \frac{1}{3} |
Sue and her sister buy a $2,100 car. They agree to split the cost based on the percentage of days use. Sue's sister will drive the car 4 days a week and Sue will get the rest of the days. How much does Sue have to pay? | Sue will drive 7-4=<<7-4=3>>3 days a week
So she will pay 2100*3/7=$<<2100*3/7=900>>900
#### 900 |
The difference in ages between Richard and Hurley is 20. If Hurley is 14 years old, what are their combined ages 40 years from now? | If Hurley is 14 years old, and the difference in ages between Richard and Hurley is 20, Richard is 20+14 = <<20+14=34>>34
Forty years from now, Hurley will be 14+40 = <<14+40=54>>54 years old.
Richard will be 40+34 = <<40+34=74>>74 years forty years from now.
Their combined ages forty years from now will be 74+54 = <<7... |
There are 20 rooms, with some lights on and some lights off. The people in these rooms want to have their lights in the same state as the majority of the other rooms. Starting with the first room, if the majority of the remaining 19 rooms have their lights on, the person will turn their light on; otherwise, they will t... | 20 |
In square \(ABCD\), \(P\) is the midpoint of \(DC\) and \(Q\) is the midpoint of \(AD\). If the area of the quadrilateral \(QBCP\) is 15, what is the area of square \(ABCD\)? | 24 |
Find all rational roots of
\[4x^4 - 3x^3 - 13x^2 + 5x + 2 = 0\]Enter all the rational roots, separated by commas. | 2,-\frac{1}{4} |
There are $2018$ players sitting around a round table. At the beginning of the game we arbitrarily deal all the cards from a deck of $K$ cards to the players (some players may receive no cards). In each turn we choose a player who draws one card from each of the two neighbors. It is only allowed to choose a player whos... | 2017 |
When studying the traffic situation in a certain city, road density refers to the number of vehicles passing through a certain section of road divided by time, and vehicle density is the number of vehicles passing through a certain section of road divided by the length of that section. The traffic flow is defined as $v... | \frac{28800}{7} |
At the end of a circus act, there are 12 dogs on stage. Half of the dogs are standing on their back legs and the other half are standing on all 4 legs. How many dog paws are on the ground? | There are 12 dogs and half are standing on their back legs so that means 12/2 = <<12/2=6>>6 are standing on their back legs
A dog has 2 back legs and only 6 are standing on their back legs meaning that there are 2*6 = <<2*6=12>>12 paws on the ground
The other 6 dogs are standing on all 4 legs which means these 6 dogs h... |
A professor is assigning grades to a class of 10 students. As a very kind professor, he only gives out A's, B's, and C's. How many ways can the professor assign grades to all his students? | 59049 |
There is a set of points \( M \) on a plane and seven different circles \( C_{1}, C_{2}, \dots, C_{7} \). Circle \( C_{7} \) passes through exactly 7 points in \( M \); circle \( C_{6} \) passes through exactly 6 points in \( M \); ..., circle \( C_{1} \) passes through exactly 1 point in \( M \). Determine the minimum... | 12 |
Let \( x \) and \( y \) be positive numbers, and let \( s \) be the smallest of the numbers \( x \), \( y + \frac{1}{x} \), and \( \frac{1}{y} \). Find the maximum possible value of \( s \). For which values of \( x \) and \( y \) is it achieved? | \sqrt{2} |
Find the smallest positive integer $k$ such that $1^2 + 2^2 + 3^2 + \ldots + k^2$ is a multiple of $360$. | 72 |
For every integer $n\ge2$, let $\text{pow}(n)$ be the largest power of the largest prime that divides $n$. For example $\text{pow}(144)=\text{pow}(2^4\cdot3^2)=3^2$. What is the largest integer $m$ such that $2010^m$ divides
$\prod_{n=2}^{5300}\text{pow}(n)$? | 77 |
How many numbers are in the list $250, 243, 236, \ldots, 29, 22?$ | 34 |
The polynomial $f(x) = x^3 + x^2 + 2x + 3$ has three distinct roots. Let $g(x) = x^3+bx^2+cx+d$ be a cubic polynomial with leading coefficient $1$ such that the roots of $g(x)$ are the squares of the roots of $f(x)$. Find the ordered triple $(b,c,d)$. | (3,-2,-9) |
The number \( N \) has the smallest positive divisor 1, the second largest positive divisor \( k \), and the third largest positive divisor \( m \). Moreover, \( k^k + m^m = N \). What is \( N \)? | 260 |
Compute
\[e^{2 \pi i/13} + e^{4 \pi i/13} + e^{6 \pi i/13} + \dots + e^{24 \pi i/13}.\] | -1 |
If for any positive integer \( m \), the set
$$
\{m, m+1, m+2, \cdots, m+99\}
$$
in any \( n \)-element subset with \( n \geq 3 \), there are always three elements that are pairwise coprime, find the smallest value of \( n \). | 68 |
Six orange candies and four purple candies are available to create different flavors. A flavor is considered different if the percentage of orange candies is different. Combine some or all of these ten candies to determine how many unique flavors can be created based on their ratios. | 14 |
Let $f(n)$ be the base-10 logarithm of the sum of the elements of the $n$th row in Pascal's triangle. Express $\frac{f(n)}{\log_{10} 2}$ in terms of $n$. Recall that Pascal's triangle begins
\begin{tabular}{rccccccccc}
$n=0$:& & & & & 1\\\noalign{\smallskip\smallskip}
$n=1$:& & & & 1 & & 1\\\noalign{\smallskip\smallsk... | n |
Represent the number 1000 as a sum of the maximum possible number of natural numbers, the sums of the digits of which are pairwise distinct. | 19 |
If \( a \) is the smallest cubic number divisible by 810, find the value of \( a \). | 729000 |
There are 20 sandcastles on Mark's beach, each with 10 towers. On Jeff's beach, there are three times as many castles as on Mark's beach, each with 5 towers. What is the combined total number of sandcastles and towers on Mark's and Jeff's beaches? | The total number of towers at Mark's beach is 20*10 = <<20*10=200>>200 towers.
There are 200+20 = <<200+20=220>>220 sandcastles and towers on Mark's beach.
Jeff's beach has 3*20 = <<3*20=60>>60 sandcastles, three times as many as the number of sandcastles in Mark's beach.
The number of towers in each of the sandcastles... |
Given that all three vertices of \(\triangle ABC\) lie on the parabola defined by \(y = 4x^2\), with \(A\) at the origin and \(\overline{BC}\) parallel to the \(x\)-axis, calculate the length of \(BC\), given that the area of the triangle is 128. | 4\sqrt[3]{4} |
A tetrahedron \(ABCD\) has six edges with lengths \(7, 13, 18, 27, 36, 41\) units. If the length of \(AB\) is 41 units, then the length of \(CD\) is | 27 |
There are chickens roaming the chicken farm. The roosters outnumber the hens 2 to 1. If there are 9,000 chickens on the chicken farm, how many roosters are there? | There are 9000/(2+1)=<<9000/(2+1)=3000>>3000 hens on the farm.
There are 3000*2=<<3000*2=6000>>6000 roosters on the farm.
#### 6000 |
In a graveyard, there are 20 skeletons. Half of these skeletons are adult women, and the remaining number are split evenly between adult men and children. If an adult woman has 20 bones in their body, and a male has 5 more than this, and a child has half as many as an adult woman, how many bones are in the graveyard? | We first need to figure out how many of each type of skeleton there are. Since half the 20 skeletons are adult women, that means there are 20/2=<<20/2=10>>10 adult women's skeletons.
The remaining half, 10, is split between adult men and children, meaning there are 10/2= <<10/2=5>>5 of each.
Since an adult woman has 20... |
A collection of seven positive integers has a mean of 6, a unique mode of 4, and a median of 6. If a 12 is added to this collection, what is the new median? | 6.5 |
Let $n$ be a positive integer such that $1 \leq n \leq 1000$ . Let $M_n$ be the number of integers in the set $X_n=\{\sqrt{4 n+1}, \sqrt{4 n+2}, \ldots, \sqrt{4 n+1000}\}$ . Let $$ a=\max \left\{M_n: 1 \leq n \leq 1000\right\} \text {, and } b=\min \left\{M_n: 1 \leq n \leq 1000\right\} \text {. } $$ Find $a-b... | 22 |
Chinese mathematician Hua Luogeng saw a brain teaser in a magazine that the passenger next to him was reading while on a trip abroad: find the cube root of $59319$. Hua Luogeng blurted out the answer, astonishing everyone. They quickly asked about the calculation's mystery. Do you know how he calculated the result quic... | 58 |
For a given list of three numbers, the operation "changesum" replaces each number in the list with the sum of the other two. For example, applying "changesum" to \(3,11,7\) gives \(18,10,14\). Arav starts with the list \(20,2,3\) and applies the operation "changesum" 2023 times. What is the largest difference between t... | 18 |
**
Consider a set of numbers $\{1, 2, 3, 4, 5, 6, 7\}$. Two different natural numbers are selected at random from this set. What is the probability that the greatest common divisor (gcd) of these two numbers is one? Express your answer as a common fraction.
** | \frac{17}{21} |
Find the largest constant $c>0$ such that for every positive integer $n\ge 2$ , there always exist a positive divisor $d$ of $n$ such that $$ d\le \sqrt{n}\hspace{0.5cm} \text{and} \hspace{0.5cm} \tau(d)\ge c\sqrt{\tau(n)} $$ where $\tau(n)$ is the number of divisors of $n$ .
*Proposed by Mohd. Suhaimi ... | \frac{1}{\sqrt{2}} |
In the plane rectangular coordinate system $xOy$, the parametric equations of the line $l$ are $\left\{{\begin{array}{l}{x=t}\\{y=-1+\sqrt{3}t}\end{array}}\right.$ (where $t$ is a parameter). Establish a polar coordinate system with the coordinate origin $O$ as the pole and the positive half-axis of the $x$-axis as the... | \frac{2\sqrt{3}+1}{3} |
The diagram shows a square \(PQRS\). The arc \(QS\) is a quarter circle. The point \(U\) is the midpoint of \(QR\) and the point \(T\) lies on \(SR\). The line \(TU\) is a tangent to the arc \(QS\). What is the ratio of the length of \(TR\) to the length of \(UR\)? | 4:3 |
Let $Q$ be a point outside of circle $C$. A segment is drawn from $Q$ such that it is tangent to circle $C$ at point $R$. Meanwhile, a secant from $Q$ intersects $C$ at points $D$ and $E$, such that $QD < QE$. If $QD = 4$ and $QR = ED - QD$, then what is $QE$? | 16 |
The expression $\log_{y^6}{x}\cdot\log_{x^5}{y^2}\cdot\log_{y^4}{x^3}\cdot\log_{x^3}{y^4}\cdot\log_{y^2}{x^5}$ can be written as $a\log_y{x}$ for what constant $a$? | \frac16 |
How many real numbers $x$ are solutions to the following equation? \[ |x-1| = |x-2| + |x-3| \] | 2 |
Find the volume of the three-dimensional solid given by the inequality $\sqrt{x^{2}+y^{2}}+$ $|z| \leq 1$. | 2 \pi / 3 |
A zoo has a menagerie containing four pairs of different animals, one male and one female for each. The zookeeper wishes to feed the animals in a specific pattern: each time he feeds a single animal, the next one he feeds must be a different gender. If he starts by feeding the male giraffe, how many ways can he feed al... | 144 |
Given that $x$ is the median of the data set $1$, $2$, $3$, $x$, $5$, $6$, $7$, and the average of the data set $1$, $2$, $x^{2}$, $-y$ is $1$, find the minimum value of $y- \frac {1}{x}$. | \frac {23}{3} |
Let's modify the problem slightly. Sara writes down four integers $a > b > c > d$ whose sum is $52$. The pairwise positive differences of these numbers are $2, 3, 5, 6, 8,$ and $11$. What is the sum of the possible values for $a$? | 19 |
Ten adults enter a room, remove their shoes, and toss their shoes into a pile. Later, a child randomly pairs each left shoe with a right shoe without regard to which shoes belong together. The probability that for every positive integer $k<5$, no collection of $k$ pairs made by the child contains the shoes from exactly... | 28 |
On a given circle, six points $A$ , $B$ , $C$ , $D$ , $E$ , and $F$ are chosen at random, independently and uniformly with respect to arc length. Determine the probability that the two triangles $ABC$ and $DEF$ are disjoint, i.e., have no common points. | \[
\frac{3}{10}
\] |
Jim borrows $1500$ dollars from Sarah, who charges an interest rate of $6\%$ per month (which compounds monthly). What is the least integer number of months after which Jim will owe more than twice as much as he borrowed? | 12 |
Two days ago, the temperature in the morning went up 1.5 degrees every 2 hours. If the temperature was 50 degrees at 3 A.M., what was the temperature at 11 A.M.? | At 5 A.M., the temperature was 50 degrees + 1.5 degrees = <<50+1.5=51.5>>51.5 degrees.
At 7 A.M., the temperature was 51.5 degrees + 1.5 degrees = <<51.5+1.5=53>>53 degrees.
At 9 A.M., the temperature was 53 degrees + 1.5 degrees = <<53+1.5=54.5>>54.5 degrees.
At 11 A.M., the temperature was 54.5 degrees + 1.5 degrees ... |
If $x+\frac1x=-5$, what is $x^3+\frac1{x^3}$? | -110 |
Given a set of data: 10, 10, x, 8, where the median is equal to the mean, find the median of this data set. | 10 |
What is the range of the function $$r(x) = \frac{1}{(1-x)^2}~?$$ Express your answer in interval notation. | (0,\infty) |
Define $p(n)$ to be th product of all non-zero digits of $n$ . For instance $p(5)=5$ , $p(27)=14$ , $p(101)=1$ and so on. Find the greatest prime divisor of the following expression:
\[p(1)+p(2)+p(3)+...+p(999).\] | 103 |
In the drawing, there is a grid consisting of 25 small equilateral triangles.
How many rhombuses can be formed from two adjacent small triangles? | 30 |
Consider a sequence $\{a_n\}$ of integers, satisfying $a_1=1, a_2=2$ and $a_{n+1}$ is the largest prime divisor of $a_1+a_2+\ldots+a_n$. Find $a_{100}$. | 53 |
What is the largest possible value of $8x^2+9xy+18y^2+2x+3y$ such that $4x^2 + 9y^2 = 8$ where $x,y$ are real numbers? | 26 |
If \( x \) and \( y \) are positive integers with \( x>y \) and \( x+x y=391 \), what is the value of \( x+y \)? | 39 |
Given that $x$ is a multiple of $12600$, what is the greatest common divisor of $g(x) = (5x + 7)(11x + 3)(17x + 8)(4x + 5)$ and $x$? | 840 |
A bag of caramel cookies has 20 cookies inside and a box of cookies has 4 bags in total. How many calories are inside the box if each cookie is 20 calories? | If 1 bag has 20 cookies and a box has 4 bags, then in total there are 20*4=<<20*4=80>>80 cookies inside the box
If 1 cookie has 20 calories and there are 80 cookies in the box, then the box has in total 20*80=<<20*80=1600>>1600 calories
#### 1600 |
Let vectors $\overrightarrow{a_{1}}=(1,5)$, $\overrightarrow{a_{2}}=(4,-1)$, $\overrightarrow{a_{3}}=(2,1)$, and let $\lambda_{1}, \lambda_{2}, \lambda_{3}$ be non-negative real numbers such that $\lambda_{1}+\frac{\lambda_{2}}{2}+\frac{\lambda_{3}}{3}=1$. Find the minimum value of $\left|\lambda_{1} \overrightarrow{a_... | 3\sqrt{2} |
The formula which expresses the relationship between $x$ and $y$ as shown in the accompanying table is:
\[\begin{tabular}[t]{|c|c|c|c|c|c|}\hline x&0&1&2&3&4\\\hline y&100&90&70&40&0\\\hline\end{tabular}\] | $y=100-5x-5x^{2}$ |
Two parabolas are the graphs of the equations $y=2x^2-10x-10$ and $y=x^2-4x+6$. Find all points where they intersect. List the points in order of increasing $x$-coordinate, separated by semicolons. | (8,38) |
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