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If $x$ is a number satisfying the equation $\sqrt[3]{x+9}-\sqrt[3]{x-9}=3$, then $x^2$ is between:
|
75 \text{ and } 85
|
A portion of the graph of $f(x)=ax^3+bx^2+cx+d$ is shown below.
What is the value of $8a-4b+2c-d$?
[asy]
import graph; size(7cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-3.25,xmax=4.25,ymin=-9.25,ymax=4.25;
pen cqcqcq=rgb(0.75,0.75,0.75);
/*grid*/ pen gs=linewidth(0.7)+cqcqcq+linetype("2 2"); real gx=1,gy=1;
for(real i=ceil(xmin/gx)*gx;i<=floor(xmax/gx)*gx;i+=gx) draw((i,ymin)--(i,ymax),gs); for(real i=ceil(ymin/gy)*gy;i<=floor(ymax/gy)*gy;i+=gy) draw((xmin,i)--(xmax,i),gs);
Label laxis; laxis.p=fontsize(10);
xaxis("",xmin,xmax,Ticks(laxis,Step=1.0,Size=2,NoZero),Arrows(6),above=true); yaxis("",ymin,ymax,Ticks(laxis,Step=1.0,Size=2,NoZero),Arrows(6),above=true);
real f1(real x){return x*(x-1)*(x-2)/8;} draw(graph(f1,-3.25,4.25),linewidth(0.75));
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);
[/asy]
|
3
|
Two chess players, A and B, are in the midst of a match. Player A needs to win 2 more games to be the final winner, while player B needs to win 3 more games. If each player has a probability of $\frac{1}{2}$ to win any given game, then calculate the probability of player A becoming the final winner.
|
\frac{11}{16}
|
Find the values of $a$, $b$, and $c$ such that the equation $\sin^2 x + \sin^2 3x + \sin^2 4x + \sin^2 5x = 3$ can be reduced to an equivalent form involving $\cos ax \cos bx \cos cx = 0$ for some positive integers $a$, $b$, and $c$, and then find $a + b + c$.
|
12
|
A fair coin is to be tossed $10_{}^{}$ times. Let $\frac{i}{j}^{}_{}$, in lowest terms, be the probability that heads never occur on consecutive tosses. Find $i+j_{}^{}$.
|
73
|
Compute the sum:
\[ 2(1 + 2(1 + 2(1 + 2(1 + 2(1 + 2(1 + 2(1 + 2(1 + 2)))))))) \]
|
1022
|
If $x$ and $y$ are positive integers such that $5x+3y=100$, what is the greatest possible value of $xy$?
|
165
|
Find all positive real numbers $t$ with the following property: there exists an infinite set $X$ of real numbers such that the inequality \[ \max\{|x-(a-d)|,|y-a|,|z-(a+d)|\}>td\] holds for all (not necessarily distinct) $x,y,z\in X$, all real numbers $a$ and all positive real numbers $d$.
|
t < \frac{1}{2}
|
Calculate the monotonic intervals of $F(x)=\int_{0}^{x}{(t^{2}+2t-8)dt}$ for $x > 0$.
(1) Determine the monotonic intervals of $F(x)$.
(2) Find the maximum and minimum values of the function $F(x)$ on the interval $[1,2]$.
|
-\frac{28}{3}
|
It takes 3 ounces of wax to detail Kellan’s car and 4 ounces to detail his SUV. He bought an 11-ounce bottle of vehicle wax, but spilled 2 ounces before using it. How many ounces does he have left after waxing his car and SUV?
|
Kellan used 3 + 4 = <<3+4=7>>7 ounces of wax for his car and SUV.
Since he spilled 2 ounces, he has 11 - 2 - 7 = <<11-2-7=2>>2 ounces of wax left.
#### 2
|
Given vectors $\overset{→}{a}=(\cos x,-1+\sin x)$ and $\overset{→}{b}=(2\cos x,\sin x)$,
(1) Express $\overset{→}{a}·\overset{→}{b}$ in terms of $\sin x$.
(2) Find the maximum value of $\overset{→}{a}·\overset{→}{b}$ and the corresponding value of $x$.
|
\frac{9}{4}
|
Xiao Zhang has three watches. The first watch runs 2 minutes fast every hour, the second watch runs 6 minutes fast, and the third watch runs 16 minutes fast. If the minute hands of the three watches are currently all pointing in the same direction, after how many hours will the three minute hands point in the same direction again?
|
30
|
Find the least integer value of $x$ for which $3|x| - 2 > 13$.
|
-6
|
Let's call any natural number "very prime" if any number of consecutive digits (in particular, a digit or number itself) is a prime number. For example, $23$ and $37$ are "very prime" numbers, but $237$ and $357$ are not. Find the largest "prime" number (with justification!).
|
373
|
Given the real numbers \( x \) and \( y \) satisfy the equation \( 2x^2 + 3xy + 2y^2 = 1 \), find the minimum value of \( x + y + xy \).
|
-\frac{9}{8}
|
24 gallons per second of water flows down a waterfall. If you have an empty basin that can hold 260 gallons of water. The basin leaks water at 4 gallons per second. how long would it take to fill the basin with water from the waterfall in seconds?
|
24 gallons in minus 4 gallons out is 24-4=<<24-4=20>>20 gallons per second
260 gallons divided by 20 gallons per second 260/20 = <<260/20=13>>13 seconds
#### 13
|
Find the area of triangle $DEF$ below, where $DF = 8$ and $\angle D = 45^\circ$.
[asy]
unitsize(1inch);
pair D,E,F;
D = (0,0);
E= (sqrt(2),0);
F = (0,sqrt(2));
draw (D--E--F--D,linewidth(0.9));
draw(rightanglemark(E,D,F,3));
label("$D$",D,S);
label("$E$",E,S);
label("$F$",F,N);
label("$8$",F/2,W);
label("$45^\circ$",(1.25,0),N);
[/asy]
|
32
|
In a factor tree, each value is the product of the two values below it, unless a value is a prime number already. What is the value of $A$ on the factor tree shown?
[asy]
draw((-1,-.3)--(0,0)--(1,-.3),linewidth(1));
draw((-2,-1.3)--(-1.5,-.8)--(-1,-1.3),linewidth(1));
draw((1,-1.3)--(1.5,-.8)--(2,-1.3),linewidth(1));
label("A",(0,0),N);
label("B",(-1.5,-.8),N);
label("3",(-2,-1.3),S);
label("C",(1.5,-.8),N);
label("D",(-1,-1.3),S);
label("5",(1,-1.3),S);
label("E",(2,-1.3),S);
draw((-1.5,-2.3)--(-1,-1.8)--(-.5,-2.3),linewidth(1));
draw((1.5,-2.3)--(2,-1.8)--(2.5,-2.3),linewidth(1));
label("3",(-1.5,-2.3),S);
label("2",(-.5,-2.3),S);
label("5",(1.5,-2.3),S);
label("2",(2.5,-2.3),S);
[/asy]
|
900
|
Determine the number of ordered pairs $(a,b)$ of integers such that $\log_a b + 6\log_b a=5, 2 \leq a \leq 2005,$ and $2 \leq b \leq 2005.$
|
54
|
Kyle can lift 60 more pounds this year, which is 3 times as much as he could lift last year. How many pounds can Kyle lift in all?
|
Since Kyle can lift 60 more pounds and that is 3 times what he could do before, that means that last year Kyle could lift 60 pounds / 3 = <<60/3=20>>20 pounds.
Kyle can now lift 60 pounds + 20 pounds he could lift last year = <<60+20=80>>80 pounds.
#### 80
|
Find a polynomial $ p\left(x\right)$ with real coefficients such that
$ \left(x\plus{}10\right)p\left(2x\right)\equal{}\left(8x\minus{}32\right)p\left(x\plus{}6\right)$
for all real $ x$ and $ p\left(1\right)\equal{}210$.
|
2(x + 4)(x - 4)(x - 8)
|
Pat wants to buy four donuts from an ample supply of three types of donuts: glazed, chocolate, and powdered. How many different selections are possible?
|
15
|
How many perfect cubes are between 100 and 900?
|
5
|
Find the modular inverse of \( 31 \), modulo \( 45 \).
Express your answer as an integer from \( 0 \) to \( 44 \), inclusive.
|
15
|
Jason goes to the library 4 times more often than William goes. If William goes 2 times per week to the library, how many times does Jason go to the library in 4 weeks?
|
The number of times Jason goes to the library per week is 4 * 2 = <<4*2=8>>8 times.
The number of times he goes to the library in 4 weeks is 8 * 4 = <<8*4=32>>32 times.
#### 32
|
Mark is filling a punch bowl that can hold 16 gallons of punch. He fills it part way, then his cousin comes along and drinks half the punch in the bowl. Mark starts to refill the bowl and adds 4 more gallons, but then his friend Sally comes in and drinks 2 more gallons of punch. After that, Mark has to add 12 gallons of punch to completely fill the bowl. How much punch did Mark initially add to the bowl?
|
First, figure out how much punch was left in the bowl before Mark refilled it by subtracting the 12 gallons he added from the bowl's total capacity: 16 - 12 = <<16-12=4>>4 gallons
Next, figure out how much punch was in the bowl before Sally came along: 4 + 2 = <<4+2=6>>6 gallons.
Next, figure out how much punch was in the bowl before Mark started to refill it: 6 - 4 = <<6-4=2>>2 gallons
Finally, multiply that amount by 2 to find out how much punch there was before the cousin drank half: 2 * 2 = <<4=4>>4 gallons
#### 4
|
There are six clearly distinguishable frogs sitting in a row. Two are green, three are red, and one is blue. Green frogs refuse to sit next to the red frogs, for they are highly poisonous. In how many ways can the frogs be arranged?
|
24
|
The slope angle of the tangent line to the curve $y=\frac{1}{2}x^2+2$ at the point $\left(-1, \frac{5}{2}\right)$ is what?
|
\frac{3\pi}{4}
|
Let $\omega$ be a circle with radius $1$ . Equilateral triangle $\vartriangle ABC$ is tangent to $\omega$ at the midpoint of side $BC$ and $\omega$ lies outside $\vartriangle ABC$ . If line $AB$ is tangent to $\omega$ , compute the side length of $\vartriangle ABC$ .
|
\frac{2 \sqrt{3}}{3}
|
The sum of the areas of all triangles whose vertices are also vertices of a $1$ by $1$ by $1$ cube is $m + \sqrt{n} + \sqrt{p},$ where $m, n,$ and $p$ are integers. Find $m + n + p.$
|
348
|
Every week, Lucas makes 4 pieces of chocolate candy for each of his students on Monday. He made 40 pieces of chocolate candy last Monday. This upcoming Monday, 3 of Lucas' students will not be coming to class. How many pieces of chocolate candy will Lucas make for his class on Monday?
|
Since Lucas makes 4 pieces of chocolate per student, he has 40 / 4 = <<40/4=10>>10 students.
This upcoming Monday, there will be 10 - 3 = <<10-3=7>>7 students.
Lucas will make 4 * 7 = <<4*7=28>>28 pieces of chocolate candy.
#### 28
|
For real numbers $a$ and $b$, define $a * b=(a-b)^2$. What is $(x-y)^2*(y-x)^2$?
|
0
|
Each cell of a $100 \times 100$ board is painted in either blue or white. We call a cell balanced if it has an equal number of blue and white neighboring cells. What is the maximum number of balanced cells that can be found on the board? (Cells are considered neighbors if they share a side.)
|
9608
|
At breakfast, lunch, and dinner, Joe randomly chooses with equal probabilities either an apple, an orange, or a banana to eat. On a given day, what is the probability that Joe will eat at least two different kinds of fruit?
|
\frac{8}{9}
|
The Screamers are coached by Coach Yellsalot. The Screamers have 12 players, but two of them, Bob and Yogi, refuse to play together. How many starting lineups (of 5 players) can Coach Yellsalot make, if the starting lineup can't contain both Bob and Yogi? (The order of the 5 players in the lineup does not matter; that is, two lineups are the same if they consist of the same 5 players.)
|
672
|
The concept of negative numbers first appeared in the ancient Chinese mathematical book "Nine Chapters on the Mathematical Art." If income of $5$ yuan is denoted as $+5$ yuan, then expenses of $5$ yuan are denoted as $-5$ yuan.
|
-5
|
Sally has six red cards numbered 1 through 6 and seven blue cards numbered 2 through 8. She attempts to create a stack where the colors alternate, and the number on each red card divides evenly into the number on each neighboring blue card. Can you determine the sum of the numbers on the middle three cards of this configuration?
A) 16
B) 17
C) 18
D) 19
E) 20
|
17
|
If $f(x) = (x-1)^3 + 1$, calculate the value of $f(-5) + f(-4) + \ldots + f(0) + \ldots + f(7)$.
|
13
|
Call an integer \( n > 1 \) radical if \( 2^n - 1 \) is prime. What is the 20th smallest radical number?
|
4423
|
Consider a string of $n$ $8$'s, $8888\cdots88$, into which $+$ signs are inserted to produce an arithmetic expression. For how many values of $n$ is it possible to insert $+$ signs so that the resulting expression has value $8000$?
|
1000
|
The values of a function $f(x)$ are given below:
\begin{tabular}{|c||c|c|c|c|c|} \hline $x$ & 3 & 4 & 5 & 6 & 7 \\ \hline $f(x)$ & 10 & 17 & 26 & 37 & 50 \\ \hline \end{tabular}Evaluate $f^{-1}\left(f^{-1}(50)\times f^{-1}(10)+f^{-1}(26)\right)$.
|
5
|
What is $6 \div 0.\overline{6}$?
|
9
|
When ethane is mixed with chlorine gas under lighting conditions, determine the maximum number of substances that can be generated.
|
10
|
If the function \( f(x) = (x^2 - 1)(x^2 + ax + b) \) satisfies \( f(x) = f(4 - x) \) for any \( x \in \mathbb{R} \), what is the minimum value of \( f(x) \)?
|
-16
|
Annie has $120. The restaurant next door sells hamburgers for $4 each. The restaurant across the street sells milkshakes for $3 each. Annie buys 8 hamburgers and 6 milkshakes. How much money, in dollars, does she have left?
|
Annie spends 4*8=<<4*8=32>>32 dollars on hamburgers.
Annie spends 3*6=<<3*6=18>>18 dollars on milkshakes.
Annie spends 32+18=<<32+18=50>>50 dollars on hamburgers and milkshakes.
Annie has 120-50=<<120-50=70>>70 dollars left.
#### 70
|
In triangle ABC, the sides opposite to angles A, B, and C are a, b, and c, respectively. Given that $$\frac {sin2B}{ \sqrt {3}cos(B+C)-cosCsinB}= \frac {2b}{c}$$.
(I) Find the measure of angle A.
(II) If $$a= \sqrt {3}$$, find the maximum area of triangle ABC.
|
\frac { \sqrt {3}}{4}
|
Leila bought a living room set consisting of a sofa worth $1,250, 2 armchairs costing $425 each and a coffee table. The total amount of the invoice is $2,430. What is the price of the coffee table?
|
The price of 2 armchairs is $425 x 2 = $<<425*2=850>>850. Mrs. Dubois will pay $850 for the 2 seats.
Thus the Coffee table price is $2,430 – ($1,250 + $850) = $2,430 – $2,100 = $<<2430-2100=330>>330
#### 330
|
The squares of a chessboard are labelled with numbers, as shown below.
[asy]
unitsize(0.8 cm);
int i, j;
for (i = 0; i <= 8; ++i) {
draw((i,0)--(i,8));
draw((0,i)--(8,i));
}
for (i = 0; i <= 7; ++i) {
for (j = 0; j <= 7; ++j) {
label("$\frac{1}{" + string(i + 8 - j) + "}$", (i + 0.5, j + 0.5));
}}
[/asy]
Eight of the squares are chosen, so that there is exactly one chosen square in each row and each column. Find the minimum sum of the labels of the eight chosen squares.
|
1
|
The sum of the three sides of a triangle is 50. The right side of the triangle is 2 cm longer than the left side. Find the value of the triangle base if the left side has a value of 12 cm.
|
If the left side of the triangle is 12, and the right side is 2 cm longer, the right side is 12+2 = <<12+2=14>>14 cm long.
The sum of the left and right sides of the triangle is 14+12 = <<14+12=26>>26 cm.
If the sum of all sides of the is 50, and the left and right sides of the triangle are 26, the triangle base is 50-26 = <<50-26=24>>24 cm.
#### 24
|
Let \( a_{n} \) represent the closest positive integer to \( \sqrt{n} \) for \( n \in \mathbf{N}^{*} \). Suppose \( S=\frac{1}{a_{1}}+\frac{1}{a_{2}}+\cdots+\frac{1}{a_{2000}} \). Determine the value of \( [S] \).
|
88
|
Evaluate
\[\begin{vmatrix} 0 & \sin \alpha & -\cos \alpha \\ -\sin \alpha & 0 & \sin \beta \\ \cos \alpha & -\sin \beta & 0 \end{vmatrix}.\]
|
0
|
Given a triangle \( \triangle ABC \) with \(\angle B = 90^\circ\). The incircle touches sides \(BC\), \(CA\), and \(AB\) at points \(D\), \(E\), and \(F\) respectively. Line \(AD\) intersects the incircle at another point \(P\), and \(PF \perp PC\). Find the ratio of the side lengths of \(\triangle ABC\).
|
3:4:5
|
Cindy can run at 3 miles per hour and walk at 1 mile per hour. If she runs for half a mile and then walks for half a mile, how many minutes will it take her to travel the full mile?
|
It will take her 0.5 miles / 3 mph * (60 minutes / 1 hour) = <<0.5/3*60=10>>10 minutes to run half a mile.
It will take her 0.5 miles / 1 mph * (60 minutes / 1 hour) = <<0.5/1*60=30>>30 minutes to walk half a mile.
Her total time is 10 + 30 = <<10+30=40>>40 minutes.
#### 40
|
Using arithmetic operation signs, write the largest natural number using two twos.
|
22
|
One of the clubs at Georgia's school was selling carnations for a fundraising event. A carnation costs $0.50. They also had a special where you could buy one dozen carnations for $4.00. If Georgia sent a dozen carnations to each of her 5 teachers and bought one carnation for each of her 14 friends, how much money would she spend?
|
She wants to send a dozen carnations that cost $4/dz to each of her 5 teachers, so 4*5 = $<<4*5=20>>20
She has 14 friends and she knows a dozen carnations is cheaper than buying 12 single carnations. So 14-12 = <<14-12=2>>2 single carnations are needed
A single carnation cost $0.50 so 2*.50 = $<<2*.50=1.00>>1.00
All total she will spend 20+4+1 = $<<20+4+1=25.00>>25.00 on carnations
#### 25
|
A peacock is a ten-digit positive integer that uses each digit exactly once. Compute the number of peacocks that are exactly twice another peacock.
|
184320
|
Point \(C\) divides diameter \(AB\) in the ratio \(AC:BC = 2:1\). A point \(P\) is selected on the circle. Determine the possible values that the ratio \(\tan \angle PAC: \tan \angle APC\) can take. Specify the smallest such value.
|
1/2
|
For how many two-digit natural numbers \( n \) are exactly two of the following three statements true: (A) \( n \) is odd; (B) \( n \) is not divisible by 3; (C) \( n \) is divisible by 5?
|
33
|
Solve the following system of equations: \begin{align*}
3x-5y&=-11,\\
7x+2y&=-12.
\end{align*}Express your answer as an ordered pair $(x,y).$
|
(-2,1)
|
Let $ABCD$ be an isosceles trapezoid with $AD=BC$ and $AB<CD.$ Suppose that the distances from $A$ to the lines $BC,CD,$ and $BD$ are $15,18,$ and $10,$ respectively. Let $K$ be the area of $ABCD.$ Find $\sqrt2 \cdot K.$
|
270
|
Cara is sitting at a circular table with her five friends as shown below. How many different possible pairs of people could Cara be sitting between?
[asy]
draw(circle((0,0),1));
label("$\_$",1.5dir(0));
label("$\_$",1.5dir(60));
label("Cara",1.5dir(120));
label("$\_$",1.5dir(180));
label("$\_$",1.5dir(240));
label("$\_$",1.5dir(300));
[/asy]
|
10
|
Conduct an experiment by throwing 2 dice, and denote the point P with the coordinates $(x, y)$, where $x$ represents the number shown on the first die, and $y$ represents the number shown on the second die.
(I) Find the probability that point P lies on the line $y = x$.
(II) Find the probability that point P does not lie on the line $y = x + 1$.
(III) Find the probability that the coordinates of point P $(x, y)$ satisfy $16 < x^2 + y^2 \leq 25$.
|
\frac{7}{36}
|
If $2x+1=8$, then $4x+1=$
|
15
|
There are two pairs $(x,y)$ of real numbers that satisfy the equation $x+y = 3xy = 4$. Given that the solutions $x$ are in the form $x = \frac{a \pm b\sqrt{c}}{d}$ where $a$, $b$, $c$, and $d$ are positive integers and the expression is completely simplified, what is the value of $a + b + c + d$?
|
17
|
Compute $\dbinom{50}{2}$.
|
1225
|
A squirrel travels at a constant 4 miles per hour. How long does it take for this squirrel to travel 1 mile? Express your answer in minutes.
|
15
|
The half-hour newscast includes 12 minutes of national news, 5 minutes of international news, 5 minutes of sports, and 2 minutes of weather forecasts. The rest is advertisements. How many minutes of advertising are in the newscast?
|
12+5+5+2=<<12+5+5+2=24>>24 minutes are accounted for.
There are 30-24=<<30-24=6>>6 minutes of ads.
#### 6
|
For which integer $a$ does $x^2 - x + a$ divide $x^{13} + x + 90$?
|
2
|
What is the area of the circle defined by $x^2-6x +y^2-14y +33=0$ that lies beneath the line $y=7$?
|
\frac{25\pi}{2}
|
Given the function $f(x)=|\log_{x}|$, $a > b > 0$, $f(a)=f(b)$, calculate the minimum value of $\frac{a^2+b^2}{a-b}$.
|
2\sqrt{2}
|
Let $\mathcal{S}$ be the set $\lbrace1,2,3,\ldots,10\rbrace$ Let $n$ be the number of sets of two non-empty disjoint subsets of $\mathcal{S}$. (Disjoint sets are defined as sets that have no common elements.) Find the remainder obtained when $n$ is divided by $1000$.
|
501
|
Given an equilateral triangle \( \triangle ABC \) with a side length of 1,
\[
\overrightarrow{AP} = \frac{1}{3}(\overrightarrow{AB} + \overrightarrow{AC}), \quad \overrightarrow{AQ} = \overrightarrow{AP} + \frac{1}{2}\overrightarrow{BC}.
\]
Find the area of \( \triangle APQ \).
|
\frac{\sqrt{3}}{12}
|
For each integer $n$ greater than 1, let $G(n)$ be the number of solutions of the equation $\sin x = \sin (n^2 x)$ on the interval $[0, 2\pi]$. What is $\sum_{n=2}^{100} G(n)$?
|
676797
|
Let $x = (2 + \sqrt{3})^{1000},$ let $n = \lfloor x \rfloor,$ and let $f = x - n.$ Find
\[x(1 - f).\]
|
1
|
A bag contains 10 red marbles and 6 blue marbles. Three marbles are selected at random and without replacement. What is the probability that one marble is red and two are blue? Express your answer as a common fraction.
|
\frac{15}{56}
|
Together Felipe and Emilio needed a combined time of 7.5 years to build their homes. Felipe finished in half the time of Emilio. How many months did it take Felipe to build his house?
|
Let F = the number of years Felipe needed to build his house
Emilio = <<2=2>>2F
3F = 7.5 years
F = <<2.5=2.5>>2.5 years
2.5 years = 30 months
It took Felipe 30 months to build his house.
#### 30
|
Given a $4 \times 4$ square grid partitioned into $16$ unit squares, each of which is painted white or black with a probability of $\frac{1}{2}$, determine the probability that the grid is entirely black after a $90^{\circ}$ clockwise rotation and any white square landing in a position previously occupied by a black square is repainted black.
|
\frac{1}{65536}
|
Given an ellipse $C$: $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1 (a > b > 0)$ with its upper vertex at $(0,2)$ and an eccentricity of $\frac{\sqrt{5}}{3}$.
(1) Find the equation of ellipse $C$;
(2) From a point $P$ on the ellipse $C$, draw two tangent lines to the circle $x^{2}+y^{2}=1$, with the tangent points being $A$ and $B$. When the line $AB$ intersects the $x$-axis and $y$-axis at points $N$ and $M$, respectively, find the minimum value of $|MN|$.
|
\frac{5}{6}
|
What is the average student headcount for the spring terms of the `02-`03, `03-`04 and `04-`05 academic years? Express your answer to the nearest whole number.
[asy]
unitsize(0.35 cm);
fill((1,0)--(1,11.7)--(4,11.7)--(4,0)--cycle,gray(.5));
fill((4,0)--(4,10.9)--(7,10.9)--(7,0)--cycle,gray(.7));
fill((8,0)--(8,11.5)--(11,11.5)--(11,0)--cycle,gray(.5));
fill((11,0)--(11,10.5)--(14,10.5)--(14,0)--cycle,gray(.7));
fill((15,0)--(15,11.6)--(18,11.6)--(18,0)--cycle,gray(.5));
fill((18,0)--(18,10.7)--(21,10.7)--(21,0)--cycle,gray(.7));
fill((22,0)--(22,11.3)--(25,11.3)--(25,0)--cycle,gray(.5));
draw((0,0)--(0,15)--(30,15)--(30,0)--cycle,linewidth(1));
label("11,700",(2.5,12.5), fontsize(10));
label("10,900",(5.5,12), fontsize(10));
label("11,500",(9.5,12.5), fontsize(10));
label("10,500",(12.5,11.5), fontsize(10));
label("11,600",(16.5,12.5), fontsize(10));
label("10,700",(19.5,11.5), fontsize(10));
label("11,300",(23.5,12), fontsize(10));
label("Student Headcount (2002-2003 to 2005-2006)",(15,17));
label("'02-'03",(4,0),S);
label("'03-'04",(11,0),S);
label("'04-'05",(18,0),S);
label("'05-'06",(25,0),S);
label("Academic Year",(15,-2),S);
fill((32,11)--(32,9)--(34,9)--(34,11)--cycle,gray(.5));
fill((32,8)--(32,6)--(34,6)--(34,8)--cycle,gray(.7));
label("Fall ",(34,10),E, fontsize(10));
label("Spring ",(34,7),E, fontsize(10));
[/asy]
|
10700
|
Brayden and Gavin were playing touch football against Cole and Freddy. Touchdowns were worth 7 points. Brayden and Gavin scored 7 touchdowns. Cole and Freddy's team scored 9 touchdowns. How many more points did Cole and Freddy have than Brayden and Gavin?
|
Brayden and Gavin scored 7 touchdowns * 7 points each = <<7*7=49>>49 points.
Cole and Freddy scored 9 touchdowns * 7 points each = <<9*7=63>>63 points.
Cole and Freddy's team had 63 - 49 points = <<63-49=14>>14 points more than Brayden and Gavin's team.
#### 14
|
A printing press is printing brochures. The press prints 20 single-page spreads, and twice as many double-page spreads. For each 4 pages printed for the spreads, the press prints a block of 4 ads, each of which take up a quarter of a page. The brochures can be arranged in any order as long as they are made up of 5 pages each. How many brochures is the printing press creating?
|
There are a total of 20 single-page spreads * 2 = <<20*2=40>>40 double-page spreads.
As these are made up of 2 pages each, there are 40 double-page spreads * 2 pages = <<40*2=80>>80 pages in the double-page spreads.
In total, there are 20 single-pages + 80 double-pages = <<20+80=100>>100 pages in the spreads.
A block of ads is printed every 4 pages, so there will be 100 pages / 4 pages/block = <<100/4=25>>25 blocks of ads.
This is a total of 25 blocks * 4 ads = <<25*4=100>>100 ads.
As each ad takes up a quarter of a page, this must create an additional 100 ads * 0.25 = <<100*0.25=25>>25 pages.
Therefore, the press has printed 100 pages from the spreads + 25 pages of ads = <<100+25=125>>125 pages.
Since brochures are made up of 5 pages each, this creates a total of 125 pages / 5 = <<125/5=25>>25 brochures.
#### 25
|
Given a population of $100$ individuals randomly numbered from $0$ to $99$, and a sample of size $10$ is drawn, with the units digit of the number drawn from the $k$-th group being the same as the units digit of $m + k$, where $m = 6$, find the number drawn from the 7-th group.
|
63
|
The game of Penta is played with teams of five players each, and there are five roles the players can play. Each of the five players chooses two of five roles they wish to play. If each player chooses their roles randomly, what is the probability that each role will have exactly two players?
|
\frac{51}{2500}
|
Angie bought 3 lbs. of coffee at the store today. Each lb. of coffee will brew about 40 cups of coffee. Angie drinks 3 cups of coffee every day. How many days will this coffee last her?
|
A pound of coffee makes 40 cups of coffee, so 3 lbs. coffee * 40 cups of coffee per lb. = <<3*40=120>>120 cups of coffee.
Angie drinks 3 cups of coffee a day, so she has enough coffee to last her 120 cups of coffee / 3 cups of coffee per day = <<120/3=40>>40 days
#### 40
|
We choose 100 points in the coordinate plane. Let $N$ be the number of triples $(A,B,C)$ of distinct chosen points such that $A$ and $B$ have the same $y$ -coordinate, and $B$ and $C$ have the same $x$ -coordinate. Find the greatest value that $N$ can attain considering all possible ways to choose the points.
|
8100
|
Let \(p\), \(q\), \(r\), and \(s\) be the roots of the polynomial \[x^4 + 10x^3 + 20x^2 + 15x + 6 = 0.\] Find the value of \[\frac{1}{pq} + \frac{1}{pr} + \frac{1}{ps} + \frac{1}{qr} + \frac{1}{qs} + \frac{1}{rs}.\]
|
\frac{10}{3}
|
Evaluate the sum
\[
\sum_{k=1}^{50} (-1)^k \cdot \frac{k^3 + k^2 + 1}{(k+1)!}.
\]
Determine the form of the answer as a difference between two terms, where each term is a fraction involving factorials.
|
\frac{126001}{51!}
|
Let $x,$ $y,$ $z$ be real numbers such that $x + 2y + z = 4.$ Find the maximum value of
\[xy + xz + yz.\]
|
4
|
A chessboard’s squares are labeled with numbers as follows:
[asy]
unitsize(0.8 cm);
int i, j;
for (i = 0; i <= 8; ++i) {
draw((i,0)--(i,8));
draw((0,i)--(8,i));
}
for (i = 0; i <= 7; ++i) {
for (j = 0; j <= 7; ++j) {
label("$\frac{1}{" + string(9 - i + j) + "}$", (i + 0.5, j + 0.5));
}}
[/asy]
Eight of the squares are chosen such that each row and each column has exactly one selected square. Find the maximum sum of the labels of these eight chosen squares.
|
\frac{8}{9}
|
What is the smallest positive integer $x$ that, when multiplied by $450$, results in a product that is a multiple of $800$?
|
32
|
From the set $\{1, 2, 3, \ldots, 10\}$, select 3 different elements such that the sum of these three numbers is a multiple of 3, and the three numbers cannot form an arithmetic sequence. Calculate the number of ways to do this.
|
22
|
Define a sequence of complex numbers by $z_1 = 0$ and
\[z_{n + 1} = z_n^2 + i\]for all $n \ge 1.$ In the complex plane, how far from the origin is $z_{111}$?
|
\sqrt{2}
|
Four years ago you invested some money at $10\%$ interest. You now have $\$439.23$ in the account. If the interest was compounded yearly, how much did you invest 4 years ago?
|
300
|
Find the largest possible value of $x$ in the simplified form $x=\frac{a+b\sqrt{c}}{d}$ if $\frac{5x}{6}+1=\frac{3}{x}$, where $a,b,c,$ and $d$ are integers. What is $\frac{acd}{b}$?
|
-55
|
The rails on a railroad are $30$ feet long. As the train passes over the point where the rails are joined, there is an audible click.
The speed of the train in miles per hour is approximately the number of clicks heard in:
|
20 seconds
|
Given vectors $\overrightarrow{m}=(\sin x,-1)$ and $\overrightarrow{n}=(\sqrt{3}\cos x,-\frac{1}{2})$, and the function $f(x)=(\overrightarrow{m}+\overrightarrow{n})\cdot\overrightarrow{m}$.
- (I) Find the interval where $f(x)$ is monotonically decreasing;
- (II) Given $a$, $b$, and $c$ are respectively the sides opposite to angles $A$, $B$, and $C$ in $\triangle ABC$, with $A$ being an acute angle, $a=2\sqrt{3}$, $c=4$, and $f(A)$ is exactly the maximum value of $f(x)$ on the interval $\left[0, \frac{\pi}{2}\right]$, find $A$, $b$, and the area $S$ of $\triangle ABC$.
|
2\sqrt{3}
|
Given the function $f(2x+1)=x^{2}-2x$, determine the value of $f(\sqrt{2})$.
|
\frac{5-4\sqrt{2}}{4}
|
There are real numbers $a, b, c,$ and $d$ such that $-20$ is a root of $x^3 + ax + b$ and $-21$ is a root of $x^3 + cx^2 + d.$ These two polynomials share a complex root $m + \sqrt{n} \cdot i,$ where $m$ and $n$ are positive integers and $i = \sqrt{-1}.$ Find $m+n.$
|
330
|
Choose any four distinct digits $w, x, y, z$ and form the four-digit numbers $wxyz$ and $zyxw$. What is the greatest common divisor of the numbers of the form $wxyz + zyxw + wxyz \cdot zyxw$?
|
11
|
For which natural number \( K \) does the expression \(\frac{n^{2}}{1.001^{n}}\) reach its maximum value?
|
2001
|
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