problem stringlengths 10 5.15k | answer stringlengths 0 1.23k |
|---|---|
$\triangle ABC$ is similar to $\triangle DEF$ . What is the number of centimeters in the length of $\overline{EF}$ ? Express your answer as a decimal to the nearest tenth.
[asy]
draw((0,0)--(8,-2)--(5,4)--cycle);
label("8cm",(2.5,2),NW);
label("5cm",(6.1,1),NE);
draw((12,0)--(18,-1.5)--(15.7,2.5)--cycle);
label("$A$",... | 4.8 |
Dimitri eats 3 burgers per day. Each burger has a total of 20 calories. How many calories will he get after two days? | The total number of calories he gets per day is 20 x 3 = <<20*3=60>>60.
Therefore the total number of calories he will get after 2 days is 60 x 2 = <<60*2=120>>120.
#### 120 |
It is known that each side and diagonal of a regular polygon is colored in one of exactly 2018 different colors, and not all sides and diagonals are the same color. If a regular polygon contains no two-colored triangles (i.e., a triangle whose three sides are precisely colored with two colors), then the coloring of the... | 2017^2 |
Given that the polynomial \(x^2 - kx + 24\) has only positive integer roots, find the average of all distinct possibilities for \(k\). | 15 |
Cylinder $B$'s height is equal to the radius of cylinder $A$ and cylinder $B$'s radius is equal to the height $h$ of cylinder $A$. If the volume of cylinder $A$ is twice the volume of cylinder $B$, the volume of cylinder $A$ can be written as $N \pi h^3$ cubic units. What is the value of $N$?
[asy]
size(4cm,4cm);
path... | 4 |
An urn contains $4$ green balls and $6$ blue balls. A second urn contains $16$ green balls and $N$ blue balls. A single ball is drawn at random from each urn. The probability that both balls are of the same color is $0.58$. Find $N$. | 144 |
A lemming sits at a corner of a square with side length $10$ meters. The lemming runs $6.2$ meters along a diagonal toward the opposite corner. It stops, makes a $90^{\circ}$ right turn and runs $2$ more meters. A scientist measures the shortest distance between the lemming and each side of the square. What is the aver... | 5 |
Out of the 200 cookies that Javier baked from the recipe he learned online, his wife took 30%, and his daughter took 40 from the remaining cookies. If he ate half of the remaining cookies, how many cookies did they not eat? | From the 200 cookies, Javier's wife ate 30/100*200 = <<30/100*200=60>>60
After the wife ate 60 cookies, the number of cookies that remained was 200-60 = <<200-60=140>>140
The daughter also ate 40 cookies leaving 140-40 = <<140-40=100>>100 cookies.
If Javier ate half of the remaining cookies, he ate 1/2*100 = 50.
The nu... |
If two distinct numbers are selected at random from the first seven prime numbers, what is the probability that their sum is an even number? Express your answer as a common fraction. | \frac{5}{7} |
Let $p$, $q$, and $r$ be the roots of the equation $x^3 - 15x^2 + 25x - 10 = 0$. Find the value of $(1+p)(1+q)(1+r)$. | 51 |
What is the least common multiple of 12, 18 and 30? | 180 |
Mr. Green measures his rectangular garden by walking two of the sides and finds that it is $15$ steps by $20$ steps. Each of Mr. Green's steps is $2$ feet long. Mr. Green expects a half a pound of potatoes per square foot from his garden. How many pounds of potatoes does Mr. Green expect from his garden? | 600 |
Divide the product of the first five positive composite integers by the product of the next five composite integers. Express your answer as a common fraction. | \frac{1}{42} |
If the inequality $x^{2}+ax+1 \geqslant 0$ holds for all $x \in (0, \frac{1}{2}]$, find the minimum value of $a$. | -\frac{5}{2} |
The equation $y = -16t^2 + 34t + 25$ describes the height (in feet) of a ball thrown upwards at $34$ feet per second from $25$ feet above the ground. Determine the time (in seconds) when the ball will hit the ground. | \frac{25}{8} |
Brenda raises mice, and her adult mice recently had three litters of 8 each. She gave a sixth of the baby mice to her friend Robbie to keep as pets. She sold three times the number of babies she gave Robbie to a pet store. Half of the remaining mice were sold to snake owners as feeder mice. How many baby mice did Brend... | Brenda’s mice had 3 * 8 = <<3*8=24>>24 baby mice.
She gave Robbie 24 / 6 = <<24/6=4>>4 mice.
Thus, she sold 3 * 4 = <<3*4=12>>12 mice to the pet store.
She had 24 - 12 - 4 = <<24-12-4=8>>8 mice remaining.
She sold 8 / 2 = <<8/2=4>>4 as feeder mice.
Thus, Brenda had 8 - 4 = <<8-4=4>>4 mice left.
#### 4 |
Let \( f(x) \) be the polynomial \( (x - a_1)(x - a_2)(x - a_3)(x - a_4)(x - a_5) \) where \( a_1, a_2, a_3, a_4, \) and \( a_5 \) are distinct integers. Given that \( f(104) = 2012 \), evaluate \( a_1 + a_2 + a_3 + a_4 + a_5 \). | 17 |
Ed and Sue bike at equal and constant rates. Similarly, they jog at equal and constant rates, and they swim at equal and constant rates. Ed covers $74$ kilometers after biking for $2$ hours, jogging for $3$ hours, and swimming for $4$ hours, while Sue covers $91$ kilometers after jogging for $2$ hours, swimming for $3$... | 314 |
Carla is dividing up insurance claims among 3 agents. Missy can handle 15 more claims than John, who can handle 30% more claims than Jan. If Jan can handle 20 claims, how many claims can Missy handle? | First find the additional number of claims John can handle: 30% * 20 claims = <<30*.01*20=6>>6 claims
Then add that amount to Jan's number of claims to find John's number of claims: 20 claims + 6 claims = <<20+6=26>>26 claims
Then add the 15 additional claims Missy can handle to find her number: 26 claims + 15 claims =... |
Susie's pet lizard Moe takes 10 seconds to eat 40 pieces of cuttlebone each day. How long would it take Moe to eat 800 pieces? | If it takes Moe 10 seconds to eat 40 cuttlebones, then in one second she eats 40 cuttlebones / 10 seconds = <<40/10=4>>4 cuttlebones/second
Since the cat eats 4 pieces of cuttlebone in 1 second, 800 pieces will be 800 cuttlebones / 4 cuttlebones/second = <<800/4=200>>200 seconds.
#### 200 |
If the points $(1,y_1)$ and $(-1,y_2)$ lie on the graph of $y=ax^2+bx+c$, and $y_1-y_2=-6$, then $b$ equals: | -3 |
Find the equation of the plane passing through $(-1,1,1)$ and $(1,-1,1),$ and which is perpendicular to the plane $x + 2y + 3z = 5.$ Enter your answer in the form
\[Ax + By + Cz + D = 0,\]where $A,$ $B,$ $C,$ $D$ are integers such that $A > 0$ and $\gcd(|A|,|B|,|C|,|D|) = 1.$ | x + y - z + 1 = 0 |
What percent of square $PQRS$ is shaded? All angles in the diagram are right angles. [asy]
import graph;
defaultpen(linewidth(0.8));
xaxis(0,7,Ticks(1.0,NoZero));
yaxis(0,7,Ticks(1.0,NoZero));
fill((0,0)--(2,0)--(2,2)--(0,2)--cycle);
fill((3,0)--(5,0)--(5,5)--(0,5)--(0,3)--(3,3)--cycle);
fill((6,0)--(7,0)--(7,7)--(0,... | 67.35\% |
Ten unit cubes are glued together as shown. How many square units are in the surface area of the resulting solid?
[asy]
draw((0,0)--(30,0)--(30,10)--(0,10)--cycle);
draw((10,0)--(10,10));
draw((20,0)--(20,10));
draw((5,15)--(35,15));
draw((0,10)--(5,15));
draw((10,10)--(15,15));
draw((20,10)--(25,15));
draw((35,15)-... | 34\text{ sq. units} |
Evaluate $\log_2 (4^2)$. | 4 |
How many total days were there in the years 2001 through 2004? | 1461 |
Tom rents a helicopter for 2 hours a day for 3 days. It cost $75 an hour to rent. How much did he pay? | He rented the helicopter for 2*3=<<2*3=6>>6 hours
So he paid 6*75=$<<6*75=450>>450
#### 450 |
On September 10, 2005, the following numbers were drawn in the five-number lottery: 4, 16, 22, 48, 88. All five numbers are even, exactly four of them are divisible by 4, three by 8, and two by 16. In how many ways can five different numbers with these properties be selected from the integers ranging from 1 to 90? | 15180 |
23. Two friends, Marco and Ian, are talking about their ages. Ian says, "My age is a zero of a polynomial with integer coefficients."
Having seen the polynomial \( p(x) \) Ian was talking about, Marco exclaims, "You mean, you are seven years old? Oops, sorry I miscalculated! \( p(7) = 77 \) and not zero."
"Yes, I am o... | 14 |
From the $7$ integers from $2$ to $8$, randomly select $2$ different numbers. The probability that these $2$ numbers are coprime is ______. | \frac{2}{3} |
Johnny spent 3 hours working on a job that paid $7 per hour, 2 hours working on a job that paid $10 an hour, and 4 hours working on a job that paid $12 an hour. Assuming he repeats this process 5 days in a row, how much does Johnny make? | First, we need to determine how much Johnny makes in one day. To start, we perform 3*7=<<3*7=21>>21 dollars for the first job.
Second, we perform 2*10=<<20=20>>20 dollars for the second job.
Third, we perform 4*12= <<4*12=48>>48 dollars for the third job.
Altogether, Johnny makes 21+20+48= <<21+20+48=89>>89 dollars in ... |
In order to purchase new headphones costing 275 rubles, Katya decided to save money by spending less on sports activities. Until now, she had bought a single-visit pass to the swimming pool, including a trip to the sauna, for 250 rubles to warm up. However, now that summer has arrived, there is no longer a need to visi... | 11 |
Is
\[f(x) = \frac{5^x - 1}{5^x + 1}\]an even function, odd function, or neither?
Enter "odd", "even", or "neither". | \text{odd} |
Points $B$ and $C$ lie on $\overline{AD}$. The length of $\overline{AB}$ is $4$ times the length of $\overline{BD}$, and the length of $\overline{AC}$ is $9$ times the length of $\overline{CD}$. The length of $\overline{BC}$ is what fraction of the length of $\overline{AD}$? | \frac{1}{10} |
Given the polynomial $f(x) = 4x^5 + 2x^4 + 3.5x^3 - 2.6x^2 + 1.7x - 0.8$, find the value of $V_1$ when calculating $f(5)$ using the Horner's Method. | 22 |
Claire begins with 40 sweets. Amy gives one third of her sweets to Beth, Beth gives one third of all the sweets she now has to Claire, and then Claire gives one third of all the sweets she now has to Amy. Given that all the girls end up having the same number of sweets, determine the number of sweets Beth had originall... | 50 |
Given that $P$ is a moving point on the parabola $y^{2}=4x$, and $Q$ is a moving point on the circle $x^{2}+(y-4)^{2}=1$, the minimum value of the sum of the distance from point $P$ to point $Q$ and the distance from point $P$ to the directrix of the parabola is ______. | \sqrt{17}-1 |
If the function $G$ has a maximum value of $M$ and a minimum value of $N$ on $m\leqslant x\leqslant n\left(m \lt n\right)$, and satisfies $M-N=2$, then the function is called the "range function" on $m\leqslant x\leqslant n$. <br/>$(1)$ Functions ① $y=2x-1$; ② $y=x^{2}$, of which function ______ is the "range function"... | \frac{1}{8} |
Let \((a,b,c,d)\) be an ordered quadruple of integers, each in the set \(\{-2, -1, 0, 1, 2\}\). Determine the count of such quadruples for which \(a\cdot d - b\cdot c\) is divisible by 4. | 81 |
Given circle $C: (x-2)^{2} + (y-2)^{2} = 8-m$, if circle $C$ has three common tangents with circle $D: (x+1)^{2} + (y+2)^{2} = 1$, then the value of $m$ is ______. | -8 |
Determine if there exists a positive integer \( m \) such that the equation
\[
\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{abc}=\frac{m}{a+b+c}
\]
has infinitely many solutions in positive integers \( (a, b, c) \). | 12 |
Let $(a_1,a_2,\ldots, a_{13})$ be a permutation of $(1, 2, \ldots, 13)$ . Ayvak takes this permutation and makes a series of *moves*, each of which consists of choosing an integer $i$ from $1$ to $12$ , inclusive, and swapping the positions of $a_i$ and $a_{i+1}$ . Define the *weight* of a permutation to be ... | 13703 |
What is the length of the segment of the number line whose endpoints satisfy $|x-\sqrt[5]{16}|=3$? | 6 |
In the expansion of $(x^{4}+y^{2}+\frac{1}{2xy})^{7}$, the constant term is ______. | \frac{105}{16} |
In triangle \(ABC\), the angle bisector \(BL\) is drawn. Find the area of the triangle, given that \(AL = 2\), \(BL = \sqrt{30}\), and \(CL = 5\). | \frac{7\sqrt{39}}{4} |
A sample has a capacity of $80$. After grouping, the frequency of the second group is $0.15$. Then, the frequency of the second group is ______. | 12 |
Max is planning a vacation for 8 people. The Airbnb rental costs $3200. They also plan on renting a car that will cost $800. If they plan on splitting the costs equally, how much will each person’s share be? | The total cost for the vacation rentals will be $3200 + $800 = $<<3200+800=4000>>4000.
The cost per person will be $4000 / 8 = $<<4000/8=500>>500.
#### 500 |
Eugene, Brianna, and Katie are going on a run. Eugene runs at a rate of 4 miles per hour. If Brianna runs $\frac{2}{3}$ as fast as Eugene, and Katie runs $\frac{7}{5}$ as fast as Brianna, how fast does Katie run? | \frac{56}{15} |
Annie walked 5 blocks from her house to the bus stop. She rode the bus 7 blocks to the coffee shop. Later, she came home the same way. How many blocks did Annie travel in all? | Annie traveled 5 + 7 = <<5+7=12>>12 blocks to the coffee shop.
Her round trip would take 12 x 2 = <<12*2=24>>24 blocks.
#### 24 |
Each day Maria must work $8$ hours. This does not include the $45$ minutes she takes for lunch. If she begins working at $\text{7:25 A.M.}$ and takes her lunch break at noon, then her working day will end at | \text{4:10 P.M.} |
Let $ABC$ be equilateral, and $D, E,$ and $F$ be the midpoints of $\overline{BC}, \overline{CA},$ and $\overline{AB},$ respectively. There exist points $P, Q,$ and $R$ on $\overline{DE}, \overline{EF},$ and $\overline{FD},$ respectively, with the property that $P$ is on $\overline{CQ}, Q$ is on $\overline{AR},$ and $R$... | 83 |
What is the least positive multiple of 21 that is greater than 380? | 399 |
Two isosceles triangles are given with equal perimeters. The base of the second triangle is 15% larger than the base of the first, and the leg of the second triangle is 5% smaller than the leg of the first triangle. Find the ratio of the sides of the first triangle. | \frac{2}{3} |
Alice and Bob play a game with a baseball. On each turn, if Alice has the ball, there is a 1/2 chance that she will toss it to Bob and a 1/2 chance that she will keep the ball. If Bob has the ball, there is a 2/5 chance that he will toss it to Alice, and if he doesn't toss it to Alice, he keeps it. Alice starts with th... | \frac{9}{20} |
Let $P$ be the plane passing through the origin with normal vector $\begin{pmatrix} 1 \\ 1 \\ -1 \end{pmatrix}.$ Find the matrix $\mathbf{R}$ such that for any vector $\mathbf{v},$ $\mathbf{R} \mathbf{v}$ is the reflection of $\mathbf{v}$ through plane $P.$ | \begin{pmatrix} \frac{1}{3} & -\frac{2}{3} & \frac{2}{3} \\ -\frac{2}{3} & \frac{1}{3} & \frac{2}{3} \\ \frac{2}{3} & \frac{2}{3} & \frac{1}{3} \end{pmatrix} |
A function $f$ has domain $[0,2]$ and range $[0,1]$. (The notation $[a,b]$ denotes $\{x:a \le x \le b \}$.) Let
\[g(x) = 1 - f(x + 1).\]Then the domain of $g(x)$ is $[a,b],$ and the range of $g(x)$ is $[c,d].$ Enter the ordered quadruple $(a,b,c,d).$ | (-1,1,0,1) |
In the diagram, the smaller circles touch the larger circle and touch each other at the center of the larger circle. The radius of the larger circle is $6.$ What is the area of the shaded region?
[asy]
size(100);
import graph;
filldraw(Circle((0,0),2),mediumgray);
filldraw(Circle((-1,0),1),white);
filldraw(Circle((1,0... | 18\pi |
Inside a cylinder with a base radius of 6, there are two spheres each with a radius of 6. The distance between the centers of the spheres is 13. If a plane is tangent to these two spheres and intersects the surface of the cylinder forming an ellipse, then the sum of the lengths of the major axis and the minor axis of t... | 25 |
A ball is made of white hexagons and black pentagons. There are 12 pentagons in total. How many hexagons are there?
A) 12
B) 15
C) 18
D) 20
E) 24 | 20 |
2500 chess kings have to be placed on a $100 \times 100$ chessboard so that
[b](i)[/b] no king can capture any other one (i.e. no two kings are placed in two squares sharing a common vertex);
[b](ii)[/b] each row and each column contains exactly 25 kings.
Find the number of such arrangements. (Two arrangements differ... | 2 |
Calculate:
$$
202.2 \times 89.8 - 20.22 \times 186 + 2.022 \times 3570 - 0.2022 \times 16900
$$ | 18198 |
During the "Cool Summer Happy Shopping" promotion held in a certain shopping mall, Xiao Yang bought $m$ items of type A goods priced at $5$ yuan each, and $n$ items of type B goods priced at $17 yuan each, spending a total of $203$ yuan. Then the maximum value of $m+n$ is ______. | 31 |
Determine the smallest positive integer $n$ such that $5^n\equiv n^5\pmod 3$. | 4 |
Let $ a, b \in \mathbb{N}$ with $ 1 \leq a \leq b,$ and $ M \equal{} \left[\frac {a \plus{} b}{2} \right].$ Define a function $ f: \mathbb{Z} \mapsto \mathbb{Z}$ by
\[ f(n) \equal{} \begin{cases} n \plus{} a, & \text{if } n \leq M, \\
n \minus{} b, & \text{if } n >M. \end{cases}
\]
Let $ f^1(n) \equal{} f(n),$ $ f_{i ... | \frac {a + b}{\gcd(a,b)} |
An author of a book got 6% of the total sales of the paper cover version of his books and 12% of the total sales of the hardcover version. If 32,000 copies of the paper cover version were sold at $0.20 each and 15,000 copies of the hardcover version were sold at $0.40 each, how much did the author earn? | The total earnings for the paper cover version were $0.20 x 32 000 = $<<0.20*32000=6400>>6,400.
So, the author earned $6,400 x 6/100 = $384 from the paper cover version.
The total earnings for the hardcover version were $0.40 x 15,000 = $<<0.40*15000=6000>>6,000.
So, the author earned $6,000 x 12/100 = $<<6000*12/100=7... |
What is the greatest prime factor of $12! + 14!$? (Reminder: If $n$ is a positive integer, then $n!$ stands for the product $1\cdot 2\cdot 3\cdot \cdots \cdot (n-1)\cdot n$.) | 61 |
When three standard dice are tossed, the numbers $x, y, z$ are obtained. Find the probability that $xyz = 72$. | \frac{1}{36} |
Alice is thinking of a positive real number $x$, and Bob is thinking of a positive real number $y$. Given that $x^{\sqrt{y}}=27$ and $(\sqrt{x})^{y}=9$, compute $x y$. | 16 \sqrt[4]{3} |
Given that $a+b=3$ and $a^3+b^3=81$, find $ab$. | -6 |
The equation $\sin^2 x + \sin^2 2x + \sin^2 3x + \sin^2 4x = 2$ can be reduced to the equivalent equation
\[\cos ax \cos bx \cos cx = 0,\]for some positive integers $a,$ $b,$ and $c.$ Find $a + b + c.$ | 8 |
When a number is tripled and then decreased by 5, the result is 16. What is the original number? | 7 |
Square A has a perimeter of $24$ cm. Square B has an area equal to one-fourth the area of square A. What is the perimeter of square B?
[asy]
draw((0,0)--(7,0));
draw((7,0)--(7,7));
draw((7,7)--(0,7));
draw((0,7)--(0,0));
draw((11,2)--(11,5));
draw((11,5)--(14,5));
draw((14,5)--(14,2));
draw((14,2)--(11,2));
label("A... | 12 |
Out of two hundred ninth-grade students, $80\%$ received excellent grades on the first exam, $70\%$ on the second exam, and $59\%$ on the third exam. What is the minimum number of students who could have received excellent grades on all three exams?
| 18 |
Determine the number of ordered pairs $(m, n)$ that satisfy $m$ and $n \in \{-1,0,1,2,3\}$, and the equation $mx^2 + 2x + n = 0$ has real solutions. | 17 |
Petya has seven cards with the digits 2, 2, 3, 4, 5, 6, 8. He wants to use all the cards to form the largest natural number that is divisible by 12. What number should he get? | 8654232 |
Find a monic polynomial of degree $4,$ in $x,$ with rational coefficients such that $\sqrt{2} +\sqrt{3}$ is a root of the polynomial. | x^4-10x^2+1 |
In the plane rectangular coordinate system $xOy$, the parameter equation of the line $l$ is $\left\{{\begin{array}{l}{x=3-\frac{{\sqrt{3}}}{2}t,}\\{y=\sqrt{3}-\frac{1}{2}t}\end{array}}\right.$ (where $t$ is the parameter). Establish a polar coordinate system with the origin $O$ as the pole and the positive half-axis of... | \frac{\sqrt{3}}{2} |
In the rectangular coordinate system $(xOy)$, the parametric equation of line $l$ is given by $ \begin{cases} x=-\frac{1}{2}t \\ y=2+\frac{\sqrt{3}}{2}t \end{cases} (t\text{ is the parameter})$, and a circle $C$ with polar coordinate equation $\rho=4\cos\theta$ is established with the origin $O$ as the pole and the pos... | 4+2\sqrt{3} |
Find the number of sets $A$ that satisfy the three conditions: $\star$ $A$ is a set of two positive integers $\star$ each of the numbers in $A$ is at least $22$ percent the size of the other number $\star$ $A$ contains the number $30.$ | 129 |
A necklace is strung with gems in the order of A, B, C, D, E, F, G, H. Now, we want to select 8 gems from it in two rounds, with the requirement that only 4 gems can be taken each time, and at most two adjacent gems can be taken (such as A, B, E, F). How many ways are there to do this (answer with a number)? | 30 |
Two 8-sided dice, one blue and one yellow, are rolled. What is the probability that the blue die shows a prime number and the yellow die shows a number that is a power of 2? | \frac{1}{4} |
Eighty percent of dissatisfied customers leave angry reviews about a certain online store. Among satisfied customers, only fifteen percent leave positive reviews. This store has earned 60 angry reviews and 20 positive reviews. Using this data, estimate the probability that the next customer will be satisfied with the s... | 0.64 |
Consider the geometric sequence $3$, $\dfrac{9}{2}$, $\dfrac{27}{4}$, $\dfrac{81}{8}$, $\ldots$. Find the eighth term of the sequence. Express your answer as a common fraction. | \frac{6561}{128} |
A numerical sequence is defined by the conditions: \( a_{1}=1 \), \( a_{n+1}=a_{n}+\left \lfloor \sqrt{a_{n}} \right \rfloor \).
How many perfect squares are among the first terms of this sequence that do not exceed 1,000,000? | 10 |
Trapezoid $A B C D$, with bases $A B$ and $C D$, has side lengths $A B=28, B C=13, C D=14$, and $D A=15$. Let diagonals $A C$ and $B D$ intersect at $P$, and let $E$ and $F$ be the midpoints of $A P$ and $B P$, respectively. Find the area of quadrilateral $C D E F$. | 112 |
A positive integer \( N \) and \( N^2 \) end with the same sequence of digits \(\overline{abcd}\), where \( a \) is a non-zero digit. Find \(\overline{abc}\). | 937 |
Tara has a shoebox that is 4 inches tall and 6 inches wide. She puts a square block inside that is 4 inches per side. How many square inches of the box are left uncovered? | The shoebox is 24 square inches because 4 x 6 = <<4*6=24>>24
The block is 16 square inches because 4 x 4 = <<4*4=16>>16
There are 8 square inches uncovered because 24 - 16 = <<24-16=8>>8
#### 8 |
How many different positive three-digit integers can be formed using only the digits in the set $\{2, 3, 5, 5, 7, 7, 7\}$ if no digit may be used more times than it appears in the given set of available digits? | 43 |
Find the maximum number of real roots to a polynomial of the form
\[x^n + x^{n - 1} + \dots + x + 1 = 0,\]where $n$ is a positive integer. | 1 |
An airplane has three sections: First Class (24 seats), Business Class ($25\%$ of the total number of seats) and Economy ($\frac{2}{3}$ of the total number of seats). How many seats does the plane have? | 288 |
A positive integer $n$ has $60$ divisors and $7n$ has $80$ divisors. What is the greatest integer $k$ such that $7^k$ divides $n$? | 2 |
Find the largest prime divisor of $25^2+72^2$.
| 157 |
Six students stand in a row for a photo. Among them, student A and student B are next to each other, student C is not next to either student A or student B. The number of different ways the students can stand is ______ (express the result in numbers). | 144 |
Determine the number of digits in the value of $2^{12} \times 5^8 $. | 10 |
Sadie, Ariana and Sarah are running a relay race. Each part of the race is a different length and covers different terrain. It takes Sadie 2 hours to run through the forest at an average speed of 3 miles per hour. Ariana sprints across the open field on her section at 6 miles per hour for half an hour. If Sarah runs al... | Sadie ran for 3 miles/hour * 2 hour = <<3*2=6>>6 miles.
Ariana covered a distance of 6 miles/hour * (1/2) hour = 3 miles.
Sarah ran for 4.5 hours – 2 hours – 0.5 hours = <<4.5-2-0.5=2>>2 hours.
In this time, Sarah traveled a distance of 4 miles/hour * 2 hours = <<4*2=8>>8 miles.
The total distance for the race was 6 mi... |
Given that \( x \) is a real number and \( y = \sqrt{x^2 - 2x + 2} + \sqrt{x^2 - 10x + 34} \). Find the minimum value of \( y \). | 4\sqrt{2} |
Determine the number of digits in the product of $84,123,457,789,321,005$ and $56,789,234,567,891$. | 32 |
Find $1^{234} + 4^6 \div 4^4$. | 17 |
Given $a= \int_{ 0 }^{ \pi }(\sin x-1+2\cos ^{2} \frac {x}{2})dx$, find the constant term in the expansion of $(a \sqrt {x}- \frac {1}{ \sqrt {x}})^{6}\cdot(x^{2}+2)$. | -332 |
Jack rewrites the quadratic $9x^2 - 30x - 42$ in the form of $(ax + b)^2 + c,$ where $a,$ $b,$ and $c$ are all integers. What is $ab$? | -15 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.