problem stringlengths 10 5.15k | answer stringlengths 0 1.23k |
|---|---|
If eight apples cost the same as four bananas, and two bananas cost the same as three cucumbers, how many cucumbers can Tyler buy for the price of 16 apples? | 12 |
Find the smallest natural decimal number \(n\) whose square starts with the digits 19 and ends with the digits 89. | 1383 |
In the plane rectangular coordinate system $xOy$, the parametric equations of the line $l$ are $\left\{\begin{array}{l}x=2t+1,\\ y=2t\end{array}\right.$ (where $t$ is a parameter). Taking the coordinate origin $O$ as the pole and the non-negative half-axis of the $x$-axis as the polar axis, the polar coordinate equatio... | \frac{\sqrt{14}}{2} |
What is the modular inverse of $11$, modulo $1000$?
Express your answer as an integer from $0$ to $999$, inclusive. | 91 |
$(1)$ When throwing two uniformly weighted dice, let $A=$"the first time an odd number appears," $B=$"the sum of the two dice is a multiple of $3$." Determine whether events $A$ and $B$ are independent, and explain the reason;<br/>$(2)$ Athletes A and B are participating in a shooting assessment test. Each person has t... | 0.2212 |
A sphere with center $O$ has radius $6$. A triangle with sides of length $15, 15,$ and $24$ is situated in space so that each of its sides is tangent to the sphere. What is the distance between $O$ and the plane determined by the triangle? | 2\sqrt{5} |
Gina has two bank accounts. Each account has a quarter of the balance in Betty's account. If Betty's account balance is $3,456, what is the combined balance of both Gina's accounts? | Each of Gina's account has $3456 / 4 accounts = $<<3456/4=864>>864/account
Her combined account balance is therefore $864 + $864 = $<<864+864=1728>>1728
#### 1728 |
The stem-and-leaf plot displays the lengths of songs on an album in minutes and seconds. There are 18 songs on the album. In the plot, $3\ 45$ represents $3$ minutes, $45$ seconds, which is equivalent to $225$ seconds. What is the median length of the songs? Express your answer in seconds.
\begin{tabular}{c|ccccc}
0&3... | 147.5 |
In triangle $ABC$, $BC = 8$. The length of median $AD$ is 5. Let $M$ be the largest possible value of $AB^2 + AC^2$, and let $m$ be the smallest possible value. Find $M - m$. | 0 |
Find the value of $x$ between 0 and 180 such that
\[\tan (120^\circ - x^\circ) = \frac{\sin 120^\circ - \sin x^\circ}{\cos 120^\circ - \cos x^\circ}.\] | 100 |
The perimeter of a particular square and the circumference of a particular circle are equal. What is the ratio of the area of the square to the area of the circle? Express your answer as a common fraction in terms of $\pi$. | \frac{\pi}{4} |
Define the Fibonacci numbers by $F_{0}=0, F_{1}=1, F_{n}=F_{n-1}+F_{n-2}$ for $n \geq 2$. For how many $n, 0 \leq n \leq 100$, is $F_{n}$ a multiple of 13? | 15 |
Suppose that $x_1+1 = x_2+2 = x_3+3 = \cdots = x_{1000}+1000 = x_1 + x_2 + x_3 + \cdots + x_{1000} + 1001$. Find the value of $\left\lfloor |S| \right\rfloor$, where $S = \sum_{n=1}^{1000} x_n$. | 501 |
Given that $f(x)$ is an odd function defined on $\mathbb{R}$, when $x > 0$, $f(x)=2^{x}+ \ln \frac{x}{4}$. Let $a_{n}=f(n-5)$, then the sum of the first $8$ terms of the sequence $\{a_{n}\}$ is $\_\_\_\_\_\_\_\_\_.$ | -16 |
Three 12 cm $\times$12 cm squares are each cut into two pieces $A$ and $B$, as shown in the first figure below, by joining the midpoints of two adjacent sides. These six pieces are then attached to a regular hexagon, as shown in the second figure, so as to fold into a polyhedron. What is the volume (in $\mathrm{cm}^3$)... | 864 |
Brianna reads two books a month. This year, she was given six new books as a gift, she bought eight new books, and she plans to borrow two fewer new books than she bought from the library. How many of her old books from her old book collection will she have to reread to have two books to read a month this year? | Brianna needs 12 * 2 = <<12*2=24>>24 books to get her through the year.
She will borrow 8 - 2 = <<8-2=6>>6 new books from the library.
With the books she was given, the books she bought, and the library books, she will have 6 + 8 + 6 = <<6+8+6=20>>20 new books to read this year.
Thus, she will need to reread 24 - 20 = ... |
Dave arrives at an airport which has twelve gates arranged in a straight line with exactly $100$ feet between adjacent gates. His departure gate is assigned at random. After waiting at that gate, Dave is told the departure gate has been changed to a different gate, again at random. Let the probability that Dave walks $... | 52 |
The polynomial \( x^{103} + Cx + D \) is divisible by \( x^2 + 2x + 1 \) for some real numbers \( C \) and \( D \). Find \( C + D \). | -1 |
Andrew works in a company that provides a generous vacation allotment: for every 10 days worked, you get 1 vacation day. If last year Andrew worked 300 days and took 5 days off in March and twice as many in September, how many more vacation days can Andrew still take? | If for every 10 days worked Andrew earns 1 day of vacation, then over 300 days worked he would have earned 300/10=<<300/10=30>>30 days of vacation
We know he took 5 days off in March and that in September twice as much. That means in September he took 5*2=<<5*2=10>>10 days off
In total, Andrew has used 5 vacation days ... |
Given the function $f(x)=\cos (2x+ \frac {\pi}{4})$, if we shrink the x-coordinates of all points on the graph of $y=f(x)$ to half of their original values while keeping the y-coordinates unchanged; and then shift the resulting graph to the right by $|\varphi|$ units, and the resulting graph is symmetric about the orig... | \frac {3\pi}{16} |
In triangle $DEF$, $DE = 6$, $EF = 8$, and $FD = 10$. Point $Q$ is randomly selected inside triangle $DEF$. What is the probability that $Q$ is closer to $D$ than it is to either $E$ or $F$? | \frac{1}{2} |
GiGi took out a big bowl of mushrooms from the refrigerator. She cut each mushroom into 4 pieces. Her twins, Kenny and Karla sprinkled mushrooms on their pizzas and baked them in the oven. Kenny grabbed a handful and sprinkled 38 mushroom pieces on his pizza. Karla scooped them up with both hands and sprinkled 42 m... | Kenny used 38 / 4 = <<38/4=9.5>>9.5 mushrooms on his pizza.
Karla used 42 / 4 = <<42/4=10.5>>10.5 mushrooms on her pizza.
Together, the twins used 9.5 + 10.5 = <<9.5+10.5=20>>20 mushrooms.
There are 8 / 4 = <<8/4=2>>2 mushrooms left on the cutting board.
GiGi cut up 20 + 2 = <<20+2=22>>22 mushrooms at the beginning.
##... |
The chord \( AB \) divides the circle into two arcs, with the smaller arc being \( 130^{\circ} \). The larger arc is divided by chord \( AC \) in the ratio \( 31:15 \) from point \( A \). Find the angle \( BAC \). | 37.5 |
What is the least common multiple of 3, 4, 6 and 15? | 60 |
Points $A$, $B$, $C$, and $T$ are in space such that each of $\overline{TA}$, $\overline{TB}$, and $\overline{TC}$ is perpendicular to the other two. If $TA = TB = 12$ and $TC = 6$, then what is the distance from $T$ to face $ABC$? | 2\sqrt{6} |
Given the function $f(x)=|x-1|+|x+1|$.
(I) Solve the inequality $f(x) < 3$;
(II) If the minimum value of $f(x)$ is $m$, let $a > 0$, $b > 0$, and $a+b=m$, find the minimum value of $\frac{1}{a}+ \frac{2}{b}$. | \frac{3}{2}+ \sqrt{2} |
Fred spent half of his allowance going to the movies. He washed the family car and earned 6 dollars. What is his weekly allowance if he ended with 14 dollars? | Before washing the car Fred had $14 - $6 = $<<14-6=8>>8.
Fred spent half his allowance for the movies, so 2 * $8 = $<<2*8=16>>16 allowance.
#### 16 |
On a digital clock, the date is always displayed as an eight-digit number, such as January 1, 2011, which is displayed as 20110101. What is the last date in 2011 that is divisible by 101? This date is represented as $\overline{2011 \mathrm{ABCD}}$. What is $\overline{\mathrm{ABCD}}$? | 1221 |
The Big Sixteen Basketball League consists of two divisions, each with eight teams. Each team plays each of the other teams in its own division three times and every team in the other division twice. How many league games are scheduled? | 296 |
The running time of Beast of War: Armoured Command is 10 minutes longer than that of Alpha Epsilon, which is 30 minutes shorter than that of Millennium. If Millennium runs for 2 hours, what is the running time of Beast of War: Armoured Command in minutes? | One hour contains 60 minutes so 2 hours contain 60*2 = <<2*60=120>>120 minutes
Alpha Epsilon's running time is 30 minutes less than that of Millennium (which runs for 120 minutes) hence 120-30 = <<120-30=90>>90 minutes
Beast of War: Armoured Command runs for 10 minutes more than Alpha Epsilon hence 90+10 = <<10+90=100>... |
The $\textit{arithmetic derivative}$ $D(n)$ of a positive integer $n$ is defined via the following rules:
- $D(1) = 0$ ;
- $D(p)=1$ for all primes $p$ ;
- $D(ab)=D(a)b+aD(b)$ for all positive integers $a$ and $b$ .
Find the sum of all positive integers $n$ below $1000$ satisfying $D(n)=n$ . | 31 |
In right triangle $DEF$, where $DE = 15$, $DF = 9$, and $EF = 12$, calculate the distance from point $F$ to the midpoint of segment $DE$. | 7.5 |
Cynthia wants floor-to-ceiling curtains made with an additional 5" of material so it will pool at the bottom. If her room is 8 feet tall, how long will the curtains need to be? | There are 12 inches in 1 foot and her room height is 8 feet so that's 12*8 = <<12*8=96>>96 inches tall
She wants an additional 5 inches of material added so the 96 inch long curtains will pool so she needs 5+96 = <<5+96=101>>101 inch long curtains
#### 101 |
In a table tennis team of 5 players, which includes 2 veteran players and 3 new players, we need to select 3 players to be ranked as No. 1, No. 2, and No. 3 for a team competition. The selection must ensure that among the 3 chosen players, there is at least 1 veteran player and among players No. 1 and No. 2, there is a... | 48 |
In triangle $ABC$, point $D$ is on side $BC$ such that $BD:DC = 1:2$. A line through $A$ and $D$ intersects $BC$ at $E$. If the area of triangle $ABE$ is $30$, find the total area of triangle $ABC$. | 90 |
The solid $S$ consists of the set of all points $(x,y,z)$ such that $|x| + |y| \le 1,$ $|x| + |z| \le 1,$ and $|y| + |z| \le 1.$ Find the volume of $S.$ | 2 |
How many integers between $\frac{23}{3}$ and $\frac{65}{2}$ are multiples of $5$ or $3$? | 11 |
How many (possibly empty) sets of lattice points $\{P_1, P_2, ... , P_M\}$ , where each point $P_i =(x_i, y_i)$ for $x_i
, y_i \in \{0, 1, 2, 3, 4, 5, 6\}$ , satisfy that the slope of the line $P_iP_j$ is positive for each $1 \le i < j \le M$ ? An infinite slope, e.g. $P_i$ is vertically above $P_j$ , does... | 3432 |
In a pile of apples, the ratio of large apples to small apples is $9:1$. Now, a fruit sorting machine is used for screening, with a probability of $5\%$ that a large apple is sorted as a small apple and a probability of $2\%$ that a small apple is sorted as a large apple. Calculate the probability that a "large apple" ... | \frac{855}{857} |
Divide the sides of a unit square \(ABCD\) into 5 equal parts. Let \(D'\) denote the second division point from \(A\) on side \(AB\), and similarly, let the second division points from \(B\) on side \(BC\), from \(C\) on side \(CD\), and from \(D\) on side \(DA\) be \(A'\), \(B'\), and \(C'\) respectively. The lines \(... | \frac{9}{29} |
Find the maximum number of different integers that can be selected from the set $ \{1,2,...,2013\}$ so that no two exist that their difference equals to $17$ . | 1010 |
Complete the following questions:
$(1)$ Calculate: $(\sqrt{8}-\sqrt{\frac{1}{2}})\div \sqrt{2}$.
$(2)$ Calculate: $2\sqrt{3}\times (\sqrt{12}-3\sqrt{75}+\frac{1}{3}\sqrt{108})$.
$(3)$ Given $a=3+2\sqrt{2}$ and $b=3-2\sqrt{2}$, find the value of the algebraic expression $a^{2}-3ab+b^{2}$.
$(4)$ Solve the equatio... | -3 |
Given the function $$f(x)= \begin{cases} \sqrt {x}+3, & x\geq0 \\ ax+b, & x<0\end{cases}$$ satisfies the condition: $y=f(x)$ is a monotonic function on $\mathbb{R}$ and $f(a)=-f(b)=4$, then the value of $f(-1)$ is \_\_\_\_\_\_. | -3 |
Rectangle $ABCD$ is divided into four parts of equal area by five segments as shown in the figure, where $XY = YB + BC + CZ = ZW = WD + DA + AX$, and $PQ$ is parallel to $AB$. Find the length of $AB$ (in cm) if $BC = 19$ cm and $PQ = 87$ cm. | 193 |
Give an example of a function that, when \( x \) is equal to a known number, takes the form \( \frac{0}{0} \), but as \( x \) approaches this number, tends to a certain limit. | \frac{8}{7} |
Darma can eat 20 peanuts in 15 seconds. At this same rate, how many peanuts could she eat in 6 minutes? | There are 60 * 6 = <<60*6=360>>360 seconds in 6 minutes.
Then there are 360/15 = <<360/15=24>>24 sets of 15 seconds.
So, Darma could eat 24 x 20 = <<24*20=480>>480 peanuts.
#### 480 |
A triangle has three sides of the following side lengths: $7$, $10$, and $x^2$. What are all of the positive integer values of $x$ such that the triangle exists? Separate your answers using commas and express them in increasing order. | 2, 3, \text{ and } 4 |
Evaluate \( ((a^b)^a + c) - ((b^a)^b + c) \) for \( a = 3 \), \( b = 4 \), and \( c = 5 \). | -16245775 |
The focus of the parabola $y^{2}=4x$ is $F$, and the equation of the line $l$ is $x=ty+7$. Line $l$ intersects the parabola at points $M$ and $N$, and $\overrightarrow{MF}⋅\overrightarrow{NF}=0$. The tangents to the parabola at points $M$ and $N$ intersect at point $P$. Find the area of $\triangle PMN$. | 108 |
The operation $\star$ is defined as $a \star b = a + \frac{a}{b}$. What is the value of $12 \star 3$? | 16 |
Points $A, B, C$, and $D$ are on a line in that order. The distance from $A$ to $D$ is 24. The distance from $B$ to $D$ is 3 times the distance from $A$ to $B$. Point $C$ is halfway between $B$ and $D$. What is the distance from $A$ to $C$? | 15 |
For what base is the representation of $285_{10}$ a four digit number whose final digit is odd? | 6 |
Let \( a_{1}, a_{2}, \ldots, a_{2000} \) be real numbers in the interval \([0,1]\). Find the maximum possible value of
\[
\sum_{1 \leq i < j \leq 2000}(j - i) \left| a_{j} - a_{i} \right|
\] | 1000000000 |
Jarris is a weighted tetrahedral die with faces $F_{1}, F_{2}, F_{3}, F_{4}$. He tosses himself onto a table, so that the probability he lands on a given face is proportional to the area of that face. Let $k$ be the maximum distance any part of Jarris is from the table after he rolls himself. Given that Jarris has an i... | 12 |
Determine the coefficient of \(x^{29}\) in the expansion of \(\left(1 + x^{5} + x^{7} + x^{9}\right)^{16}\). | 65520 |
Two circles of radius 3 are centered at $(3,0)$ and at $(0,3)$. What is the area of the intersection of the interiors of the two circles? Express your answer in fully expanded form in terms of $\pi$. | \frac{9\pi - 18}{2} |
Nina has four times more math homework and eight times more reading homework than Ruby. If Ruby has six math homework and two reading homework, how much homework is Nina having altogether? | Nina has four times more math homework than Ruby, meaning she has 6*4 = <<4*6=24>>24 more math homework than Ruby.
The total number of math homework that Nina has is 24+6= <<24+6=30>>30
Nina also has eight times more reading homework than Ruby, a total of 8*2 = <<8*2=16>>16 more reading homework.
In total, Nina has 16+... |
Calculate the distance between the foci of the ellipse defined by the equation
\[\frac{x^2}{36} + \frac{y^2}{9} = 9.\] | 2\sqrt{3} |
Let $x,$ $y,$ $z$ be positive real numbers. Find the set of all possible values of
\[f(x,y,z) = \frac{x}{x + y} + \frac{y}{y + z} + \frac{z}{z + x}.\] | (1,2) |
Rylee is bored and decides to count the number of leaves falling off the tree in her backyard. 7 leaves fall in the first hour. For the second and third hour, the leaves fall at a rate of 4 per hour. What is the average number of leaves which fell per hour? | 7 leaves fell in the first hour.
4 leaves fell in the second hour.
4 leaves fell in the third hour.
The total number of leaves that fell during the 3 hours is 7 + 4 + 4 = <<7+4+4=15>>15 leaves.
The average number of leaves that fell per hour is 15 leaves / 3 hours = <<15/3=5>>5 leaves per hour.
#### 5 |
A whole number is said to be ''9-heavy'' if the remainder when the number is divided by 9 is greater than 5. What is the least three-digit 9-heavy whole number? | 105 |
Let $N = 99999$. Then $N^3 = \ $ | 999970000299999 |
A band's members each earn $20 per gig. If there are 4 members and they've earned $400, how many gigs have they played? | The band earns $80 per gig because 4 x 20 = <<4*20=80>>80
They played 5 gigs because 400 / 80 = <<400/80=5>>5
#### 5 |
Stuart has drawn a pair of concentric circles, as shown. He draws chords $\overline{AB}$, $\overline{BC}, \ldots$ of the large circle, each tangent to the small one. If $m\angle ABC=75^\circ$, then how many segments will he draw before returning to his starting point at $A$? [asy]
size(100); defaultpen(linewidth(0.8)... | 24 |
The front wheel of Georgina's bicycle has a diameter of 0.75 metres. She cycled for 6 minutes at a speed of 24 kilometres per hour. The number of complete rotations that the wheel made during this time is closest to: | 1020 |
The median of the set of numbers $\{$12, 38, 45, $x$, 14$\}$ is five less than the mean. If $x$ is a negative integer, what is the value of $x$? | -14 |
Find the sum\[1+11+111+\cdots+\underbrace{111\ldots111}_{n\text{ digits}}.\] | \frac{10^{n+1} - 10 - 9n}{81} |
On a four-day trip, Carrie drove 135 miles the first day, 124 miles more the second day, 159 miles the third day, and 189 miles the fourth day. If she had to charge her phone every 106 miles, how many times did she charge her phone for the whole trip? | Carrie drove 135 + 124 = <<135+124=259>>259 miles on the second day
Carrie drove 135 + 259 + 159 + 189 = <<135+259+159+189=742>>742 miles for the whole trip.
Carries charged her phone 742 / 106 = <<742/106=7>>7 times.
#### 7 |
During her birthday, her parents have decided to give Laura and her 2 younger brothers new cellphones. However, they are confused between the innumerable service providers. Assuming no child wants a provider that another sibling has, and that there are 20 service providers, in how many ways can the parents grant the c... | 6840 |
Ryan works in an office that has an even number of men and women working there. Ryan participates in a meeting composed of 4 men and 6 women who are pulled from the office floor. This reduces the number of women working on the office floor by 20%. How many people work at Ryan's office? | Since 6 women are 20% of the total number of women working there, that means there are 6*5= <<6*5=30>>30 women working there in total.
Since there are an even number of women and men working in the office, that means there are 30*2= <<30*2=60>>60 people working there in total, since there are 2 sexes.
#### 60 |
A school is arranging chairs in rows for an assembly. $11$ chairs make a complete row, and right now there are $110$ chairs total. The school wants to have as few empty seats as possible, but all rows of chairs must be complete. If $70$ students will attend the assembly, how many chairs should be removed? | 33 |
Given an ellipse and a hyperbola,\[\frac{x^2}{16} - \frac{y^2}{25} = 1\]and \[\frac{x^2}{K} + \frac{y^2}{25} = 1\], have the same asymptotes. Find the value of $K$. | 16 |
For positive integers $n$, denote $D(n)$ by the number of pairs of different adjacent digits in the binary (base two) representation of $n$. For example, $D(3) = D(11_{2}) = 0$, $D(21) = D(10101_{2}) = 4$, and $D(97) = D(1100001_{2}) = 2$. For how many positive integers less than or equal to $97$ does $D(n) = 2$? | 26 |
Let $\overline{CH}$ be an altitude of $\triangle ABC$. Let $R\,$ and $S\,$ be the points where the circles inscribed in the triangles $ACH\,$ and $BCH^{}_{}$ are tangent to $\overline{CH}$. If $AB = 1995\,$, $AC = 1994\,$, and $BC = 1993\,$, then $RS\,$ can be expressed as $m/n\,$, where $m\,$ and $n\,$ are relatively ... | 997 |
Seven cards numbered $1$ through $7$ are to be lined up in a row. Find the number of arrangements of these seven cards where one of the cards can be removed leaving the remaining six cards in either ascending or descending order. | 10 |
Define $\operatorname{gcd}(a, b)$ as the greatest common divisor of integers $a$ and $b$. Given that $n$ is the smallest positive integer greater than 1000 that satisfies:
$$
\begin{array}{l}
\operatorname{gcd}(63, n+120) = 21, \\
\operatorname{gcd}(n+63, 120) = 60
\end{array}
$$
Then the sum of the digits of $n$ is (... | 18 |
Triangle PQR is a right triangle with PQ = 6, QR = 8, and PR = 10. Point S is on PR, and QS bisects the right angle at Q. The inscribed circles of triangles PQS and QRS have radii rp and rq, respectively. Find rp/rq. | \frac{3}{28}\left(10-\sqrt{2}\right) |
Find the largest solution to \[\lfloor x \rfloor = 5 + 100 \{ x \},\]where $\{x\} = x - \lfloor x \rfloor.$ | 104.99 |
The hyperbola $C: x^{2}-y^{2}=2$ has its right focus at $F$. Let $P$ be any point on the left branch of the hyperbola, and point $A$ has coordinates $(-1,1)$. Find the minimum perimeter of $\triangle A P F$. | 3\sqrt{2} + \sqrt{10} |
The *equatorial algebra* is defined as the real numbers equipped with the three binary operations $\natural$ , $\sharp$ , $\flat$ such that for all $x, y\in \mathbb{R}$ , we have \[x\mathbin\natural y = x + y,\quad x\mathbin\sharp y = \max\{x, y\},\quad x\mathbin\flat y = \min\{x, y\}.\]
An *equatorial expression*... | 419 |
Fill in the blanks with unique digits in the following equation:
\[ \square \times(\square+\square \square) \times(\square+\square+\square+\square \square) = 2014 \]
The maximum sum of the five one-digit numbers among the choices is: | 35 |
Mindy is attempting to solve the quadratic equation by completing the square: $$100x^2+80x-144 = 0.$$ She rewrites the given quadratic equation in the form $$(dx + e)^2 = f,$$ where \(d\), \(e\), and \(f\) are integers and \(d > 0\). What are the values of \(d + e + f\)? | 174 |
What is the smallest result that can be obtained from the following process?
Choose three different numbers from the set $\{3,5,7,11,13,17\}$.
Add two of these numbers.
Multiply their sum by the third number. | 36 |
Paula the painter and her two helpers each paint at constant, but different, rates. They always start at 8:00 AM, and all three always take the same amount of time to eat lunch. On Monday the three of them painted 50% of a house, quitting at 4:00 PM. On Tuesday, when Paula wasn't there, the two helpers painted only 24%... | 48 |
Given $f(x) = 2\sqrt{3}\sin x \cos x + 2\cos^2x - 1$,
(1) Find the maximum value of $f(x)$, as well as the set of values of $x$ for which $f(x)$ attains its maximum value;
(2) In $\triangle ABC$, if $a$, $b$, and $c$ are the lengths of sides opposite the angles $A$, $B$, and $C$ respectively, with $a=1$, $b=\sqrt{3}$... | \frac{\pi}{2} |
Person A and Person B decided to go to a restaurant. Due to high demand, Person A arrived first and took a waiting number, while waiting for Person B. After a while, Person B arrived but did not see Person A, so he also took a waiting number. While waiting, Person B saw Person A, and they compared their waiting numbers... | 35 |
Given $a, b, c > 0$ and $(a+b)bc = 5$, find the minimum value of $2a+b+c$. | 2\sqrt{5} |
How many ways can 8 teaching positions be allocated to three schools, given that each school receives at least one position, and School A receives at least two positions? | 10 |
What is the smallest positive integer that can be expressed as the sum of nine consecutive integers, the sum of ten consecutive integers, and the sum of eleven consecutive integers? | 495 |
Let $\bigtriangleup PQR$ be a triangle with $\angle P = 75^\circ$ and $\angle Q = 60^\circ$. A regular hexagon $ABCDEF$ with side length 1 is drawn inside $\triangle PQR$ so that side $\overline{AB}$ lies on $\overline{PQ}$, side $\overline{CD}$ lies on $\overline{QR}$, and one of the remaining vertices lies on $\overl... | 21 |
Given the hyperbola $C$: $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 (a > 0, b > 0)$ shares a common vertex with the hyperbola $\frac{x^{2}}{16} - \frac{y^{2}}{9} = 1$ and passes through the point $A(6, \sqrt{5})$.
1. Find the equation of the hyperbola $C$ and write out the equations of its asymptotes.
2. If point $... | \frac{28\sqrt{5}}{5} |
In triangle $ABC, AB=13, BC=15,$ and $CA = 14.$ Point $D$ is on $\overline{BC}$ with $CD=6.$ Point $E$ is on $\overline{BC}$ such that $\angle BAE\cong \angle CAD.$ Given that $BE=\frac pq$ where $p$ and $q$ are relatively prime positive integers, find $q.$ | 463 |
The curve $y = \sin x$ cuts the line whose equation is $y = \sin 70^\circ$ into segments having the successive ratios
\[\dots p : q : p : q \dots\]with $p < q.$ Compute the ordered pair of relatively prime positive integers $(p,q).$ | (1,8) |
Each letter of the alphabet is assigned a value $(A=1, B=2, C=3, ..., Z=26)$. The product of a four-letter list is the product of the values of its four letters. The product of the list $ADGI$ is $(1)(4)(7)(9) = 252$. What is the only other four-letter list with a product equal to the product of the list $PQRS$? Write ... | LQSX |
Evaluate the expression $\left\lceil{\frac54}\right\rceil+\left\lfloor{-\frac54}\right\rfloor$. | 0 |
The grading scale shown is used at Jones Junior High. The fifteen scores in Mr. Freeman's class were: \(\begin{tabular}[t]{lllllllll} 89, & 72, & 54, & 97, & 77, & 92, & 85, & 74, & 75, \\ 63, & 84, & 78, & 71, & 80, & 90. & & & \\ \end{tabular}\)
In Mr. Freeman's class, what percent of the students received a grade o... | 33\frac{1}{3}\% |
When numbers are represented in "base fourteen", the digit before it becomes full is fourteen. If in "base fourteen", the fourteen digits are sequentially noted as 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, ten, J, Q, K, then convert the three-digit number JQK in "base fourteen" into a "binary" number and determine the number of di... | 11 |
Alice and Bob play a similar game with a basketball. On each turn, if Alice has the ball, there is a 2/3 chance that she will toss it to Bob and a 1/3 chance that she will keep the ball. If Bob has the ball, there is a 1/4 chance that he will toss it to Alice, and a 3/4 chance that he keeps it. Alice starts with the ba... | \frac{5}{18} |
Among a group of 120 people, some pairs are friends. A [i]weak quartet[/i] is a set of four people containing exactly one pair of friends. What is the maximum possible number of weak quartets ? | 4769280 |
Adjacent sides of Figure 1 are perpendicular. Four sides of Figure 1 are removed to form Figure 2. What is the total length, in units, of the segments in Figure 2?
[asy]
draw((0,0)--(4,0)--(4,6)--(3,6)--(3,3)--(1,3)--(1,8)--(0,8)--cycle);
draw((7,8)--(7,0)--(11,0)--(11,6)--(10,6));
label("Figure 1",(2,0),S);
label("Fi... | 19 |
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