problem stringlengths 10 5.15k | answer stringlengths 0 1.23k |
|---|---|
Given $$\frac{\cos\alpha + \sin\alpha}{\cos\alpha - \sin\alpha} = 2$$, find the value of $$\frac{1 + \sin4\alpha - \cos4\alpha}{1 + \sin4\alpha + \cos4\alpha}$$. | \frac{3}{4} |
Seven sticks with lengths 2, 3, 5, 7, 11, 13 and 17 inches are placed in a box. Three of the sticks are randomly selected. What is the probability that a triangle can be formed by joining the endpoints of the sticks? Express your answer as a common fraction. | \frac{9}{35} |
You have six blocks in a row, labeled 1 through 6, each with weight 1. Call two blocks $x \leq y$ connected when, for all $x \leq z \leq y$, block $z$ has not been removed. While there is still at least one block remaining, you choose a remaining block uniformly at random and remove it. The cost of this operation is th... | \frac{163}{10} |
If a rectangular prism has a length of $l$, a width of $w$, and a height of $h$, then the length of its diagonal is equal to $\sqrt{l^2 + w^2 + h^2}$. Suppose $l = 3$ and $h = 12$; if the length of the diagonal is $13$, what is the width? | 4 |
Grace started her own landscaping business. She charges $6 an hour for mowing lawns, $11 for pulling weeds and $9 for putting down mulch. In September she mowed lawns for 63 hours, pulled weeds for 9 hours and put down mulch for 10 hours. How much money did she earn in September? | Grace mowed lawns for 63 hours at $6/hour, so earned 63 * 6 = $<<63*6=378>>378.
Grace pulled weeds for 9 hours at $11/hour, so earned 9 * 11 = $<<9*11=99>>99.
Grace put down mulch for 10 hours at $9/hour, so earned 10 * 9 = $<<10*9=90>>90.
During September Grace earned $378 + $99 + $90 = $<<378+99+90=567>>567.
#### 567 |
Maria collects stamps and wants to enlarge her collection. She has collected 40 stamps so far and plans to have 20% more. How many stamps in total does Maria want to collect? | Maria plans on having 40 * 20/100 = <<40*20/100=8>>8 more stamps in her collection.
This means she wants to have 40 + 8 = <<40+8=48>>48 stamps.
#### 48 |
Find the minimum value of
\[x^2 + 8x + \frac{64}{x^3}\]for $x > 0.$ | 28 |
Determine the smallest natural number $n =>2$ with the property:
For every positive integers $a_1, a_2,. . . , a_n$ the product of all differences $a_j-a_i$ ,
$1 <=i <j <=n$ , is divisible by 2001. | 30 |
Three planes are going to the same place but each has a different number of passengers. The first plane has 50, the second had 60, and the third has 40. An empty plane can go 600 MPH, but each passenger makes it go 2 MPH slower. What is their average speed? | The first plane goes 100 MPH slower than empty because 50 x 2 = <<50*2=100>>100
The first plane goes 500 MPH because 600 - 100 = <<600-100=500>>500
The second plane goes 120 MPH slower because 60 x 2 = <<60*2=120>>120
The second plane goes 480 because 600 - 120 = <<600-120=480>>480
The third plane goes 80 MPH slower be... |
Xiaoming saw a tractor pulling a rope slowly on the road. Xiaoming decided to measure the length of the rope. If Xiaoming walks in the same direction as the tractor, it takes 140 steps to walk from one end of the rope to the other end; if Xiaoming walks in the opposite direction of the tractor, it takes 20 steps. The s... | 35 |
Suppose that \(A\) and \(B\) are digits such that:
\[ \begin{array}{r}
AAA \\
AAB \\
ABB \\
+\ BBB \\
\hline
1503 \\
\end{array} \]
What is the value of \(A^3 + B^2\)? | 57 |
$S=\{1,2, \ldots, 6\} .$ Then find out the number of unordered pairs of $(A, B)$ such that $A, B \subseteq S$ and $A \cap B=\phi$ | 365 |
Patrick is half the age of his elder brother Robert. If Robert will turn 30 after 2 years, how old is Patrick now? | The age of Robert now is 30 - 2 = <<30-2=28>>28 years old.
So, Patrick is 28 / 2 = <<28/2=14>>14 years old.
#### 14 |
Find the distance between the foci of the hyperbola $x^2 - 6x - 4y^2 - 8y = 27.$ | 4 \sqrt{10} |
In a pocket, there are several balls of three different colors (enough in quantity), and each time 2 balls are drawn. To ensure that the result of drawing is the same 5 times, at least how many times must one draw? | 25 |
$ABCD$ is a square where each side measures 4 units. $P$ and $Q$ are the midpoints of $\overline{BC}$ and $\overline{CD},$ respectively. Find $\sin \phi$ where $\phi$ is the angle $\angle APQ$.
 | \frac{3}{5} |
What is the units digit of $\frac{20 \cdot 21 \cdot 22 \cdot 23 \cdot 24 \cdot 25}{1000}$? | 2 |
Eighty-five cans were collected. LaDonna picked up 25 cans. Prikya picked up twice as many times as many cans as LaDonna. Yoki picked up the rest of the cans. How many cans did Yoki pick up? | LaDonna = <<25=25>>25 cans
Prikya is twice as many as LaDonna = 2*25 = <<2*25=50>>50
Yoki picked up the rest 85 - 25 - 50 = <<85-25-50=10>>10
Yoki picked up 10 cans.
#### 10 |
Jack is making flower crowns out of red, pink and white daisies. There are nine times as many pink daisies as white daisies and three less than four times as many red daisies as pink daisies. If there are 6 white daisies, how many daisies are there total? | First find the total number of pink daisies: 9 * 6 daisies = <<9*6=54>>54 daisies
Then multiply that number by 4: 54 daisies * 4 = <<54*4=216>>216 daisies
Then subtract 3 to find the total number of red daisies: 216 daisies - 3 daisies = <<216-3=213>>213 daisies
Then add the number of each color of daisies to find the ... |
Xiao Ming observed a faucet that was continuously dripping water due to damage. In order to investigate the waste caused by the water leakage, Xiao Ming placed a graduated cylinder under the faucet to collect water and recorded the total amount of water in the cylinder every minute. However, due to a delay in starting ... | 144 |
The product of two positive integers plus their sum is 103. The integers are relatively prime, and each is less than 20. What is the sum of the two integers? | 19 |
Twelve standard 6-sided dice are rolled. What is the probability that exactly two of the dice show a 1? Express your answer as a decimal rounded to the nearest thousandth. | 0.303 |
Tom's age is $T$ years, which is also the sum of the ages of his three children. His age $N$ years ago was twice the sum of their ages then. What is $T/N$? | 5 |
Let the sides opposite to angles $A$, $B$, $C$ in $\triangle ABC$ be $a$, $b$, $c$ respectively, and $\cos B= \frac {3}{5}$, $b=2$
(Ⅰ) When $A=30^{\circ}$, find the value of $a$;
(Ⅱ) When the area of $\triangle ABC$ is $3$, find the value of $a+c$. | 2 \sqrt {7} |
What is the sum of all real numbers \(x\) for which \(|x^2 - 14x + 45| = 3?\)
A) 12
B) 14
C) 16
D) 18 | 14 |
How many numbers are in the list $ 4, 6, 8, \ldots, 128, 130 ?$ | 64 |
Lyla and Isabelle run on a circular track both starting at point \( P \). Lyla runs at a constant speed in the clockwise direction. Isabelle also runs in the clockwise direction at a constant speed 25% faster than Lyla. Lyla starts running first and Isabelle starts running when Lyla has completed one third of one lap. ... | 17 |
How many positive integers $n$ satisfy\[\dfrac{n+1000}{70} = \lfloor \sqrt{n} \rfloor?\](Recall that $\lfloor x\rfloor$ is the greatest integer not exceeding $x$.)
$\textbf{(A) } 2 \qquad\textbf{(B) } 4 \qquad\textbf{(C) } 6 \qquad\textbf{(D) } 30 \qquad\textbf{(E) } 32$
| 6 |
Xiao Jun is playing a dice game. He starts at the starting square. If he rolls a 1 to 5, he moves forward by the number of spaces shown on the dice. If he rolls a 6 or moves beyond the final square at any time, he must immediately return to the starting square. How many possible ways are there for Xiao Jun to roll the ... | 19 |
In a trapezoid $ABCD$ with $AB$ parallel to $CD$, the diagonals $AC$ and $BD$ intersect at $E$. If the area of triangle $ABE$ is 50 square units, and the area of triangle $ADE$ is 20 square units, what is the area of trapezoid $ABCD$? | 98 |
Judy uses 10 pencils during her 5 day school week. A 30 pack of pencils costs $4. How much will she spend on pencils over 45 days? | She uses 2 pencils a day because 10 / 5 = <<10/5=2>>2
She will use 90 pencils in 45 days because 2 x 45 = <<2*45=90>>90
She will need 3 packs of pencils because 90 / 30 = <<90/30=3>>3
She will spend $12 because 3 x 4 = <<3*4=12>>12
#### 12 |
The set $S = \{1, 2, 3, \ldots , 49, 50\}$ contains the first $50$ positive integers. After the multiples of 2 and the multiples of 3 are removed, how many integers remain in the set $S$? | 17 |
Together 3 friends watched 411 short videos. Kelsey watched 43 more than Ekon. Ekon watched 17 less than Uma. How many videos did Kelsey watch? | Let U = the number of videos Uma watched
Ekon = U - 17
Kelsey = (U - 17) + 43 = U + <<(-17)+43=26>>26
U + U - 17 + U + 26 = 411
3U + 9 = 411
3U = 402
U = <<134=134>>134
Kelsey = 134 + 26 = <<134+26=160>>160 videos
Kelsey watched 160 videos.
#### 160 |
For how many integer values of $n$ between 1 and 474 inclusive does the decimal representation of $\frac{n}{475}$ terminate? | 24 |
\(1.25 \times 67.875 + 125 \times 6.7875 + 1250 \times 0.053375\). | 1000 |
For \(0 \leq x \leq 1\) and positive integer \(n\), let \(f_0(x) = |1 - 2x|\) and \(f_n(x) = f_0(f_{n-1}(x))\). How many solutions are there to the equation \(f_{10}(x) = x\) in the range \(0 \leq x \leq 1\)? | 2048 |
What is the value of $x$ in the plane figure shown?
[asy]
pair A;
draw(dir(40)--A); draw(dir(200)--A); draw(dir(300)--A);
label("$160^{\circ}$",A,dir(120)); label("$x^{\circ}$",A,dir(250)); label("$x^{\circ}$",A,dir(350));
[/asy] | 100 |
Calculate $5\cdot5! + 4\cdot4!+4!$. | 720 |
Weiming Real Estate Company sold a house to Mr. Qian at a 5% discount off the list price. Three years later, Mr. Qian sold the house to Mr. Jin at a price 60% higher than the original list price. Considering the total inflation rate of 40% over three years, Mr. Qian actually made a profit at a rate of % (rounded to... | 20.3 |
Don can paint 3 tiles a minute, Ken can paint 2 more tiles a minute than Don and Laura can paint twice as many tiles as Ken. Kim can paint 3 fewer tiles than Laura can in a minute. How many tiles can Don, Ken, Laura and Kim paint in 15 minutes? | Ken paints 3 tiles/minute + 2 tiles/minute = <<3+2=5>>5 tiles/minute.
Laura can paint 5 tiles/minute x 2 = <<5*2=10>>10 tiles/minute.
Kim can paint 10 tiles/minute - 3 tiles/minute = <<10-3=7>>7 tiles/minute.
All 4 can paint 3 tiles/minute + 5 tiles/minute + 10 tiles/minute + 7 tiles/minute = 25 tiles/minute.
In 15 min... |
Find the area of triangle $QCD$ given that $Q$ is the intersection of the line through $B$ and the midpoint of $AC$ with the plane through $A, C, D$ and $N$ is the midpoint of $CD$. | \frac{3 \sqrt{3}}{20} |
An artist uses 3 ounces of paint for every large canvas they cover, and 2 ounces of paint for every small canvas they cover. They have completed 3 large paintings and 4 small paintings. How many ounces of paint have they used? | The large paintings have used 3 * 3 = <<3*3=9>>9 ounces.
The small paintings have used 2 * 4 = <<2*4=8>>8 ounces.
In total, all the paintings have used 9 + 8 = <<9+8=17>>17 ounces.
#### 17 |
The sky currently has 4 times as many cirrus clouds as cumulus clouds, and 12 times as many cumulus clouds as cumulonimbus clouds. If the sky currently has 3 cumulonimbus clouds, how many cirrus clouds are in the sky at this moment? | The sky has 3*12=<<3*12=36>>36 cumulus clouds.
The sky has 4*36=<<4*36=144>>144 cirrus clouds.
#### 144 |
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 |
Find the largest integer $n$ satisfying the following conditions:
(i) $n^2$ can be expressed as the difference of two consecutive cubes;
(ii) $2n + 79$ is a perfect square.
| 181 |
Define the operations:
\( a \bigcirc b = a^{\log _{7} b} \),
\( a \otimes b = a^{\frac{1}{\log ^{6} b}} \), where \( a, b \in \mathbb{R} \).
A sequence \(\{a_{n}\} (n \geqslant 4)\) is given such that:
\[ a_{3} = 3 \otimes 2, \quad a_{n} = (n \otimes (n-1)) \bigcirc a_{n-1}. \]
Then, the integer closest to \(\log _... | 11 |
Let \( A \) be a set with 225 elements, and \( A_{1}, A_{2}, \cdots, A_{11} \) be 11 subsets of \( A \) each containing 45 elements, such that for any \( 1 \leq i < j \leq 11 \), \(|A_{i} \cap A_{j}| = 9\). Find the minimum value of \(|A_{1} \cup A_{2} \cup \cdots \cup A_{11}|\). | 165 |
Compute
\[\sum_{1 \le a < b < c} \frac{1}{3^a 5^b 7^c}.\]
(The sum is taken over all triples \((a,b,c)\) of positive integers such that \(1 \le a < b < c\).) | \frac{1}{21216} |
I have a bag with only red, blue, and green marbles. The ratio of red marbles to blue marbles to green marbles is $1:5:3$. There are 27 green marbles in the bag. How many marbles are there in the bag? | 81 |
What is the 200th term of the increasing sequence of positive integers formed by omitting only the perfect squares? | 214 |
The ratio of the measures of two angles of a triangle is 2, and the difference in lengths of the sides opposite to these angles is 2 cm; the length of the third side of the triangle is 5 cm. Calculate the area of the triangle. | 3.75\sqrt{7} |
Given that $a-b=5$ and $a^2+b^2=35$, find $a^3-b^3$. | 200 |
What is the sum of the roots of $x^2 - 4x + 3 = 0$? | 4 |
The graph of $y = \frac{p(x)}{q(x)}$ is shown below, where $p(x)$ is linear and $q(x)$ is quadratic. (Assume that the grid lines are at integers.)
[asy]
unitsize(0.6 cm);
real func (real x) {
return (2*x/((x - 2)*(x + 3)));
}
int i;
for (i = -5; i <= 5; ++i) {
draw((i,-5)--(i,5),gray(0.7));
draw((-5,i)--(5,i... | \frac{1}{3} |
Simplify
\[\frac{\cos x}{1 + \sin x} + \frac{1 + \sin x}{\cos x}.\] | 2 \sec x |
An ant moves on the following lattice, beginning at the dot labeled $A$. Each minute he moves to one of the dots neighboring the dot he was at, choosing from among its neighbors at random. What is the probability that after 5 minutes he is at the dot labeled $B$? [asy]
draw((-2,0)--(2,0));
draw((0,-2)--(0,2));
draw((1,... | \frac{1}{4} |
Given an ellipse $C$: $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b>0)$ with eccentricity $\frac{2\sqrt{2}}{3}$, the line $y=\frac{1}{2}$ intersects $C$ at points $A$ and $B$, where $|AB|=3\sqrt{3}$.
$(1)$ Find the equation of $C$;
$(2)$ Let the left and right foci of $C$ be $F_{1}$ and $F_{2}$ respectively. The line pass... | 12 |
Six friends earn $25, $30, $35, $45, $50, and $60. Calculate the amount the friend who earned $60 needs to distribute to the others when the total earnings are equally shared among them. | 19.17 |
Let the set $\mathcal{S} = \{8, 5, 1, 13, 34, 3, 21, 2\}.$ Susan makes a list as follows: for each two-element subset of $\mathcal{S},$ she writes on her list the greater of the set's two elements. Find the sum of the numbers on the list. | 484 |
Arnold and Kevin are playing a game in which Kevin picks an integer \(1 \leq m \leq 1001\), and Arnold is trying to guess it. On each turn, Arnold first pays Kevin 1 dollar in order to guess a number \(k\) of Arnold's choice. If \(m \geq k\), the game ends and he pays Kevin an additional \(m-k\) dollars (possibly zero)... | 859 |
What is the sum of all two-digit positive integers whose squares end with the digits 25? | 495 |
What is the largest prime factor of 2323? | 101 |
Given that the mean score of the students in the first section is 92 and the mean score of the students in the second section is 78, and the ratio of the number of students in the first section to the number of students in the second section is 5:7, calculate the combined mean score of all the students in both sections... | \frac{1006}{12} |
Compute the number of positive integers less than or equal to $10000$ which are relatively prime to $2014$ . | 4648 |
The function \( f: \mathbf{N}^{\star} \rightarrow \mathbf{R} \) satisfies \( f(1)=1003 \), and for any positive integer \( n \), it holds that
\[ f(1) + f(2) + \cdots + f(n) = n^2 f(n). \]
Find the value of \( f(2006) \). | \frac{1}{2007} |
Find all values of the real number $a$ so that the four complex roots of
\[z^4 - 6z^3 + 11az^2 - 3(2a^2 + 3a - 3) z + 1 = 0\]form the vertices of a parallelogram in the complex plane. Enter all the values, separated by commas. | 3 |
Let $S$ be a subset of $\{1, 2, . . . , 500\}$ such that no two distinct elements of S have a
product that is a perfect square. Find, with proof, the maximum possible number of elements
in $S$ . | 306 |
Selene has 120 cards numbered from 1 to 120, inclusive, and she places them in a box. Selene then chooses a card from the box at random. What is the probability that the number on the card she chooses is a multiple of 2, 4, or 5? Express your answer as a common fraction. | \frac{11}{20} |
A spiral staircase turns $270^\circ$ as it rises 10 feet. The radius of the staircase is 3 feet. What is the number of feet in the length of the handrail? Express your answer as a decimal to the nearest tenth. | 17.3 |
Cyclic pentagon $ABCDE$ has a right angle $\angle ABC=90^{\circ}$ and side lengths $AB=15$ and $BC=20$. Supposing that $AB=DE=EA$, find $CD$. | 7 |
For how many ordered pairs of positive integers $(x,y),$ with $y<x\le 100,$ are both $\frac xy$ and $\frac{x+1}{y+1}$ integers? | 85 |
Evaluate
\[\sum_{m = 1}^\infty \sum_{n = 1}^\infty \frac{1}{mn(m + n + 1)}.\] | 2 |
Chuck breeds dogs. He has 3 pregnant dogs. They each give birth to 4 puppies. Each puppy needs 2 shots and each shot costs $5. How much did the shots cost? | He has 3*4=<<3*4=12>>12 puppies
So they need 12*2=<<12*2=24>>24 shots
That means the vaccines cost 24*5=$<<24*5=120>>120
#### 120 |
Let $a$ and $b$ be real numbers such that $a^5b^8=12$ and $a^8b^{13}=18$ . Find $ab$ . | \frac{128}{3} |
Given the integers from 1 to 25, Ajibola wants to remove the smallest possible number of integers so that the remaining integers can be split into two groups with equal products. What is the sum of the numbers which Ajibola removes? | 79 |
A group of 4 fruit baskets contains 9 apples, 15 oranges, and 14 bananas in the first three baskets and 2 less of each fruit in the fourth basket. How many fruits are there? | For the first three baskets, the number of apples and oranges in one basket is 9+15=<<9+15=24>>24
In total, together with bananas, the number of fruits in one basket is 24+14=<<24+14=38>>38 for the first three baskets.
Since there are three baskets each having 38 fruits, there are 3*38=<<3*38=114>>114 fruits in the fir... |
Let $x,y,$ and $z$ be real numbers satisfying the system \begin{align*} \log_2(xyz-3+\log_5 x)&=5,\\ \log_3(xyz-3+\log_5 y)&=4,\\ \log_4(xyz-3+\log_5 z)&=4.\\ \end{align*} Find the value of $|\log_5 x|+|\log_5 y|+|\log_5 z|$. | 265 |
In the Cartesian coordinate plane $xOy$, an ellipse $(E)$ has its center at the origin, passes through the point $A(0,1)$, and its left and right foci are $F_{1}$ and $F_{2}$, respectively, with $\overrightarrow{AF_{1}} \cdot \overrightarrow{AF_{2}} = 0$.
(I) Find the equation of the ellipse $(E)$;
(II) A line $l$ pa... | \frac{1}{4} |
If for any $x\in R$, $2x+2\leqslant ax^{2}+bx+c\leqslant 2x^{2}-2x+4$ always holds, then the maximum value of $ab$ is ______. | \frac{1}{2} |
Julie is preparing a speech for her class. Her speech must last between one-half hour and three-quarters of an hour. The ideal rate of speech is 150 words per minute. If Julie speaks at the ideal rate, what number of words would be an appropriate length for her speech? | 5650 |
Carla went to the zoo and counted 12 zebras in their enclosure. She then visited the camels and learned that there were only half as many camels as there were zebras. Next, she visited the monkeys and counted 4 times the number of monkeys as camels. Finally, she saw some giraffes and counted only 2 giraffes. How many m... | There were 12 zebras / 2 zebras/camel = 6 camels at the zoo.
There were 4 monkeys/camel x 6 camels = <<4*6=24>>24 monkeys at the zoo.
Therefore, Carla saw 24 monkeys - 2 giraffes = <<24-2=22>>22 more monkeys than giraffes.
#### 22 |
A mailman has to deliver 48 pieces of junk mail. There are 8 houses on the block. 2 of the houses have white mailboxes and 3 have red mailboxes. How many pieces of junk mail will each of those houses get? | The mailman has 48 pieces of junk mail / 8 houses = <<48/8=6>>6 pieces per house.
The houses with white mailboxes get 2 * 6 pieces = <<2*6=12>>12 pieces of junk mail.
The houses with red mailboxes get 3 * 6 pieces = <<3*6=18>>18 pieces of junk mail.
The houses with red and white mailboxes will get a total of 12 + 18 = ... |
Given an arithmetic sequence $\{a_n\}$ with the first term being a positive number, and the sum of the first $n$ terms is $S_n$. If $a_{1006}$ and $a_{1007}$ are the two roots of the equation $x^2 - 2012x - 2011 = 0$, then the maximum value of the positive integer $n$ for which $S_n > 0$ holds is ______. | 2011 |
Using the systematic sampling method to select 32 people for a questionnaire survey from 960 people, determine the number of people among the 32 whose numbers fall within the interval [200, 480]. | 10 |
A rectangular piece of paper with vertices $A B C D$ is being cut by a pair of scissors. The pair of scissors starts at vertex $A$, and then cuts along the angle bisector of $D A B$ until it reaches another edge of the paper. One of the two resulting pieces of paper has 4 times the area of the other piece. What is the ... | \frac{5}{2} |
Six students sign up for three different intellectual competition events. How many different registration methods are there under the following conditions? (Not all six students must participate)
(1) Each person participates in exactly one event, with no limit on the number of people per event;
(2) Each event is limi... | 216 |
Compute: $8 + 6(3-8)^2$. | 158 |
Find the product of $0.5$ and $0.8$. | 0.4 |
For the equation $6 x^{2}=(2 m-1) x+m+1$ with respect to $x$, there is a root $\alpha$ satisfying the inequality $-1988 \leqslant \alpha \leqslant 1988$, and making $\frac{3}{5} \alpha$ an integer. How many possible values are there for $m$? | 2385 |
A biased coin with the probability of landing heads as 1/3 is flipped 12 times. What is the probability of getting exactly 9 heads in the 12 flips? | \frac{1760}{531441} |
Determine the number of ways to select 4 representatives from a group of 5 male students and 4 female students to participate in an activity, ensuring that there are at least two males and at least one female among the representatives. | 100 |
Given $m \gt 0$, $n \gt 0$, and $m+2n=1$, find the minimum value of $\frac{(m+1)(n+1)}{mn}$. | 8+4\sqrt{3} |
In the geometric sequence $\{a_n\}$, if $a_2a_5= -\frac{3}{4}$ and $a_2+a_3+a_4+a_5= \frac{5}{4}$, calculate $\frac{1}{a_2}+ \frac{1}{a_3}+ \frac{1}{a_4}+ \frac{1}{a_5}$. | -\frac{5}{3} |
Tamara is 3 times Kim's height less 4 inches. Tamara and Kim have a combined height of 92 inches. How many inches tall is Tamara? | Let K = Kim's height
Tamara = 3K - 4
K + 3K - 4 = 92
4K - 4 = 92
4K = 96
Kim = <<24=24>>24 inches
Tamara = (3 * 24) - 4 = <<(3*24)-4=68>>68 inches
Tamara is 68 inches tall.
#### 68 |
The Ivanov family consists of three people: a father, a mother, and a daughter. Today, on the daughter's birthday, the mother calculated the sum of the ages of all family members and got 74 years. It is known that 10 years ago, the total age of the Ivanov family members was 47 years. How old is the mother now if she ga... | 33 |
The ticket price for a cinema is: 6 yuan per individual ticket, 40 yuan for a group ticket for every 10 people, and students enjoy a 10% discount. A school with 1258 students plans to watch a movie (teachers get in for free). The school should pay the cinema at least ____ yuan. | 4536 |
What is that largest positive integer $n$ for which $n^3+100$ is divisible by $n+10$?
| 890 |
A Senate committee has 5 Democrats and 5 Republicans. Assuming all politicians are distinguishable, in how many ways can they sit around a circular table without restrictions? (Two seatings are considered the same if one is a rotation of the other.) | 362,\!880 |
Consider a circular cone with vertex $V$, and let $ABC$ be a triangle inscribed in the base of the cone, such that $AB$ is a diameter and $AC=BC$. Let $L$ be a point on $BV$ such that the volume of the cone is 4 times the volume of the tetrahedron $ABCL$. Find the value of $BL/LV$. | \frac{\pi}{4-\pi} |
Let the set \(M = \{1,2,\cdots, 1000\}\). For any non-empty subset \(X\) of \(M\), let \(\alpha_X\) denote the sum of the largest and smallest numbers in \(X\). Find the arithmetic mean of all such \(\alpha_X\). | 1001 |
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