problem stringlengths 10 5.15k | answer stringlengths 0 1.23k |
|---|---|
A book of one hundred pages has its pages numbered from 1 to 100. How many pages in this book have the digit 5 in their numbering? (Note: one sheet has two pages.)
(a) 13
(b) 14
(c) 15
(d) 16
(e) 17 | 15 |
While on vacation in New York, Greg went out for a lunch that cost $100. If sales tax in New York is 4% and he left a 6% tip, how much did Greg pay? | The sales tax is worth $100 x 4/100 = $<<100*4/100=4>>4.
Greg left $100 x 6/100 = $<<100*6/100=6>>6 tip.
So, the total cost for the tax and the tip is $4 + $6 = $<<4+6=10>>10
Therefore, Greg paid $100 + $10 = $<<100+10=110>>110 in all.
#### 110 |
In rectangle $PQRS$, $PQ=7$ and $QR =4$. Points $X$ and $Y$ are on $\overline{RS}$ so that $RX = 2$ and $SY=3$. Lines $PX$ and $QY$ intersect at $Z$. Find the area of $\triangle PQZ$. | 19.6 |
Let $r(x)$ have a domain $\{0,1,2,3\}$ and a range $\{1,3,5,7\}$. Let $s(x)$ be defined on the domain $\{1,2,3,4,5,6\}$ with the function rule $s(x) = 2x + 1$. Determine the sum of all possible values of $s(r(x))$ where $r(x)$ outputs only odd numbers. | 21 |
A circle with radius $\frac{\sqrt{2}}{2}$ and a regular hexagon with side length 1 share the same center. Calculate the area inside the circle, but outside the hexagon. | \frac{\pi}{2} - \frac{3\sqrt{3}}{2} |
Given that \( a > 0 \), if \( f(g(h(a))) = 17 \), where \( f(x) = x^2 + 5 \), \( g(x) = x^2 - 3 \), and \( h(x) = 2x + 1 \), what is the value of \( a \)? | \frac{-1 + \sqrt{3 + 2\sqrt{3}}}{2} |
Darryl has a six-sided die with faces $1, 2, 3, 4, 5, 6$ . He knows the die is weighted so that one face
comes up with probability $1/2$ and the other five faces have equal probability of coming up. He
unfortunately does not know which side is weighted, but he knows each face is equally likely
to be the weighted one... | 3/26 |
The image depicts a top-down view of a three-layered pyramid made of 14 identical cubes. Each cube is assigned a natural number in such a way that the numbers corresponding to the cubes in the bottom layer are all different, and the number on any other cube is the sum of the numbers on the four adjacent cubes from the ... | 64 |
let $x,y,z$ be positive reals , such that $x+y+z=1399$ find the
$$ \max( [x]y + [y]z + [z]x ) $$
( $[a]$ is the biggest integer not exceeding $a$ ) | 652400 |
Evaluate $(x^x)^{(x^x)}$ at $x = 2$. | 256 |
Divide a circle with a circumference of 24 into 24 equal segments. Select 8 points from the 24 segment points such that the arc length between any two chosen points is not equal to 3 or 8. How many different ways are there to choose such a set of 8 points? Provide reasoning. | 258 |
In right triangle \( ABC \), a point \( D \) is on hypotenuse \( AC \) such that \( BD \perp AC \). Let \(\omega\) be a circle with center \( O \), passing through \( C \) and \( D \) and tangent to line \( AB \) at a point other than \( B \). Point \( X \) is chosen on \( BC \) such that \( AX \perp BO \). If \( AB = ... | 8041 |
Find the maximum value of the positive real number \( A \) such that for any real numbers \( x, y, z \), the inequality
$$
x^{4} + y^{4} + z^{4} + x^{2} y z + x y^{2} z + x y z^{2} - A(x y + y z + z x)^{2} \geq 0
$$
holds. | \frac{2}{3} |
We define a function $f(x)$ such that $f(11)=34$, and if there exists an integer $a$ such that $f(a)=b$, then $f(b)$ is defined and
$f(b)=3b+1$ if $b$ is odd
$f(b)=\frac{b}{2}$ if $b$ is even.
What is the smallest possible number of integers in the domain of $f$? | 15 |
What is the largest four-digit number that is divisible by 4? | 9996 |
In triangle $\triangle XYZ$, the medians $\overline{XM}$ and $\overline{YN}$ are perpendicular. If $XM=12$ and $YN=18$, then what is the area of $\triangle XYZ$? | 144 |
What is the smallest prime number dividing the sum $3^{11}+5^{13}$? | 2 |
A tangent line is drawn to the moving circle $C: x^2 + y^2 - 2ay + a^2 - 2 = 0$ passing through the fixed point $P(2, -1)$. If the point of tangency is $T$, then the minimum length of the line segment $PT$ is \_\_\_\_\_\_. | \sqrt {2} |
Given a hyperbola \( x^{2} - y^{2} = t \) (where \( t > 0 \)), the right focus is \( F \). Any line passing through \( F \) intersects the right branch of the hyperbola at points \( M \) and \( N \). The perpendicular bisector of \( M N \) intersects the \( x \)-axis at point \( P \). When \( t \) is a positive real nu... | \frac{\sqrt{2}}{2} |
Find the minimum value of $\sqrt{x^2+y^2}$ if $5x+12y=60$. | \frac{60}{13} |
How many three-digit whole numbers contain at least one digit 6 or at least one digit 8? | 452 |
When $n$ is divided by 6, a remainder of 1 is given. What is the remainder when $n+2010$ is divided by 6? | 1 |
All positive integers whose digits add up to 12 are listed in increasing order. What is the eleventh number in that list? | 156 |
Compute \[\lfloor \sqrt{1} \rfloor + \lfloor \sqrt{2} \rfloor + \lfloor \sqrt{3} \rfloor + \cdots + \lfloor \sqrt{25} \rfloor.\] | 75 |
The function $f(x)$ satisfies
\[b^2 f(a) = a^2 f(b)\]for all real numbers $a$ and $b.$ If $f(2) \neq 0,$ find
\[\frac{f(5) - f(1)}{f(2)}.\] | 6 |
Find the set of $x$-values satisfying the inequality $|\frac{5-x}{3}|<2$. [The symbol $|a|$ means $+a$ if $a$ is positive,
$-a$ if $a$ is negative,$0$ if $a$ is zero. The notation $1<a<2$ means that a can have any value between $1$ and $2$, excluding $1$ and $2$. ] | -1 < x < 11 |
The number $5\,41G\,507\,2H6$ is divisible by $40.$ Determine the sum of all distinct possible values of the product $GH.$ | 225 |
Given a bag contains 28 red balls, 20 green balls, 12 yellow balls, 20 blue balls, 10 white balls, and 10 black balls, determine the minimum number of balls that must be drawn to ensure that at least 15 balls of the same color are selected. | 75 |
Given a circle $C$ passes through points $A(-1,0)$ and $B(3,0)$, and the center of the circle is on the line $x-y=0$.
$(1)$ Find the equation of circle $C$;
$(2)$ If point $P(x,y)$ is any point on circle $C$, find the maximum and minimum distance from point $P$ to the line $x+2y+4=0$. | \frac{2}{5}\sqrt{5} |
Find the least positive integer $n$ such that when $3^n$ is written in base $143$, its two right-most digits in base $143$ are $01$. | 195 |
Find the largest integer $k$ such that for all integers $x$ and $y$, if $xy + 1$ is divisible by $k$, then $x + y$ is also divisible by $k$. | 24 |
For what value of $k$ does the equation $\frac{x-1}{x-2} = \frac{x-k}{x-6}$ have no solution for $x$? | 5 |
Insert two numbers between 1 and 2 to form an arithmetic sequence. What is the common difference? | \frac{1}{3} |
Given that the area of a rectangle is 192 and its length is 24, what is the perimeter of the rectangle? | 64 |
A movie theatre has 250 seats. The cost of a ticket is $6 for an adult and $4 for a child. The theatre is full and contains 188 children. What is the total ticket revenue for this movie session? | The number of adults present is 250 – 188 = <<250-188=62>>62 adults
The revenue generated by adults is 62 * 6 = $<<62*6=372>>372
The revenue generated by children is 188 * 4 = $<<188*4=752>>752
So the total ticket revenue is 372 + 752= $<<372+752=1124>>1124
#### 1124 |
Joel selected an acute angle $x$ (strictly between 0 and 90 degrees) and wrote the values of $\sin x$, $\cos x$, and $\tan x$ on three different cards. Then he gave those cards to three students, Malvina, Paulina, and Georgina, one card to each, and asked them to figure out which trigonometric function (sin, cos, or t... | \frac{1 + \sqrt{5}}{2} |
On Sunday Trey is going to do some chores at home. First, he wants to make a To Do list and count up how many things he has to do that day and how long it should take. He has several items under 'clean the house,' 'take a shower' and then 'make dinner.' In total there are 7 things to do to clean the house; 1 thing to d... | On Trey's list there are 7 + 1 + 4 things to do = <<7+1+4=12>>12 things to do in total.
Each thing takes 10 minutes to do, so 10 minutes x 12 things = <<10*12=120>>120 minutes.
An hour has 60 minutes in it, so 120 minutes / 60 minutes in an hour = <<120/60=2>>2 total hours it will take Trey to complete his list.
#### 2 |
Let $\pi$ be a uniformly random permutation of the set $\{1,2, \ldots, 100\}$. The probability that $\pi^{20}(20)=$ 20 and $\pi^{21}(21)=21$ can be expressed as $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. Compute $100 a+b$. (Here, $\pi^{k}$ means $\pi$ iterated $k$ times.) | 1025 |
If the matrix $\mathbf{A}$ has an inverse and $(\mathbf{A} - 2 \mathbf{I})(\mathbf{A} - 4 \mathbf{I}) = \mathbf{0},$ then find
\[\mathbf{A} + 8 \mathbf{A}^{-1}.\] | \begin{pmatrix} 6 & 0 \\ 0 & 6 \end{pmatrix} |
The number of games won by five basketball teams is shown in a bar chart. The teams' names are not displayed. The following clues provide information about the teams:
1. The Hawks won more games than the Falcons.
2. The Warriors won more games than the Knights, but fewer games than the Royals.
3. The Knights won more ... | 33 |
Two different integers from 1 through 20 inclusive are chosen at random. What is the probability that both numbers are prime? Express your answer as a common fraction. | \dfrac{14}{95} |
Let $ABCD$ be a rectangle with sides $AB,BC,CD$ and $DA$ . Let $K,L$ be the midpoints of the sides $BC,DA$ respectivily. The perpendicular from $B$ to $AK$ hits $CL$ at $M$ . Find $$ \frac{[ABKM]}{[ABCL]} $$ | 2/3 |
A majority of the $30$ students in Ms. Demeanor's class bought pencils at the school bookstore. Each of these students bought the same number of pencils, and this number was greater than $1$. The cost of a pencil in cents was greater than the number of pencils each student bought, and the total cost of all the pencils ... | 11 |
Suppose $a$, $b$ and $c$ are integers such that the greatest common divisor of $x^2+ax+b$ and $x^2+bx+c$ is $x+1$ (in the set of polynomials in $x$ with integer coefficients), and the least common multiple of $x^2+ax+b$ and $x^2+bx+c$ is $x^3-4x^2+x+6$. Find $a+b+c$. | -6 |
A hyperbola with its center shifted to $(1,1)$ passes through point $(4, 2)$. The hyperbola opens horizontally, with one of its vertices at $(3, 1)$. Determine $t^2$ if the hyperbola also passes through point $(t, 4)$. | 36 |
Given set \( A = \{0, 1, 2, 3, 4, 5, 9\} \), and \( a, b \in A \) where \( a \neq b \). The number of functions of the form \( y = -a x^2 + (4 - b)x \) whose vertex lies in the first quadrant is ___.
| 21 |
A shopping mall's main staircase from the 1st floor to the 2nd floor consists of 15 steps. Each step has a height of 16 centimeters and a depth of 26 centimeters. The width of the staircase is 3 meters. If the cost of carpeting is 80 yuan per square meter, how much will it cost to buy the carpet needed for the staircas... | 1512 |
Regular hexagon $ABCDEF$ has vertices $A$ and $C$ at $(0,0)$ and $(7,1)$, respectively. What is its area? | 25\sqrt{3} |
Find \(n\) such that \(2^6 \cdot 3^3 \cdot n = 10!\). | 2100 |
Write $0.\overline{43}$ as a simplified fraction. | \frac{43}{99} |
A manufacturer built a machine which will address $500$ envelopes in $8$ minutes. He wishes to build another machine so that when both are operating together they will address $500$ envelopes in $2$ minutes. The equation used to find how many minutes $x$ it would require the second machine to address $500$ envelopes al... | $\frac{1}{8}+\frac{1}{x}=\frac{1}{2}$ |
John pays for half the cost of raising a child. It cost $10,000 a year for the first 8 years and then twice that much per year until the child is 18. University tuition then costs $250,000. How much did it cost? | The first 8 years cost 8*10,000=$<<8*10000=80000>>80,000
The next 18-8=<<18-8=10>>10 years
They cost 10000*2=$<<10000*2=20000>>20,000 per year
So they cost 20,000*10=$<<20000*10=200000>>200,000
So the cost was 200,000+80,000=$<<200000+80000=280000>>280,000
Adding in the cost of tuition brings the cost to 250,000+280,00... |
Jerome is taking a 150-mile bicycle trip. He wants to ride 12 miles for 12 days. How long will he ride on the 13th day to finish his goal? | Jerome rides a total of 12 x 12 = <<12*12=144>>144 miles in 12 days.
So, he will ride 150 - 144 = <<150-144=6>>6 miles on the 13th day to finish his goal.
#### 6 |
Given $f(x)=\sin (\omega x+\dfrac{\pi }{3})$ ($\omega > 0$), $f(\dfrac{\pi }{6})=f(\dfrac{\pi }{3})$, and $f(x)$ has a minimum value but no maximum value in the interval $(\dfrac{\pi }{6},\dfrac{\pi }{3})$, find the value of $\omega$. | \dfrac{14}{3} |
Consider the graph in 3-space of $0=xyz(x+y)(y+z)(z+x)(x-y)(y-z)(z-x)$. This graph divides 3-space into $N$ connected regions. What is $N$? | 48 |
What is the area of the polygon whose vertices are the points of intersection of the curves $x^2 + y^2 = 25$ and $(x-4)^2 + 9y^2 = 81?$ | 27 |
The cheerleading coach is ordering new uniforms. There are 4 cheerleaders who need a size 2, a certain number who need a size 6, and half that number who need a size 12. If there are 19 cheerleaders total, how many cheerleaders need a size 6? | Let s be the number of cheerleaders who need a size 6 and t be the number who need a size 12. We know that s = 2t and s + t + 4 = 19.
Substituting the first equation into the second equation, we get 2t + t + 4 = 19
Combining like terms, we get 3t + 4 = 19
Subtracting 4 from both sides, we get 3t = 15
Dividing both side... |
There are 5 integers written on a board. By summing them pairwise, the following set of 10 numbers is obtained: \( 5, 9, 10, 11, 12, 16, 16, 17, 21, 23 \). Determine which numbers are written on the board. Provide the product of these numbers in your answer. | 5292 |
The bar graph shows the results of a survey on color preferences. What percent preferred blue? | 24\% |
Define $a \Delta b = a^2 -b $. What is the value of $ (2^{4 \Delta13})\Delta(3^{3\Delta5})$ | -17 |
John has a 1 liter bottle of pure orange juice. He pours half of the contents of the bottle into a vat, fills the bottle with water, and mixes thoroughly. He then repeats this process 9 more times. Afterwards, he pours the remaining contents of the bottle into the vat. What fraction of the liquid in the vat is now wate... | \frac{5}{6} |
Let $b$ and $c$ be real numbers. If the polynomial $x^3 + bx^2 + cx + d$ has exactly one real root and $d = c + b + 1$, find the value of the product of all possible values of $c$. | -1 |
Let \( A = (2,0) \) and \( B = (8,6) \). Let \( P \) be a point on the parabola \( y^2 = 8x \). Find the smallest possible value of \( AP + BP \). | 10 |
A coin that comes up heads with probability $p > 0$ and tails with probability $1 - p > 0$ independently on each flip is flipped $8$ times. Suppose that the probability of three heads and five tails is equal to $\frac {1}{25}$ of the probability of five heads and three tails. Let $p = \frac {m}{n}$, where $m$ and $n$ a... | 11 |
Select two different natural numbers from the set $\{1, 2, 3, ..., 8\}$. What is the probability that the greatest common factor of these two numbers is one? Express your answer as a common fraction. | \frac{3}{4} |
A table can seat 6 people. Two tables joined together can seat 10 people. Three tables joined together can seat 14 people. Following this pattern, if 10 tables are arranged in two rows with 5 tables in each row, how many people can sit? | 44 |
In triangle $ABC$, we have $AB=1$ and $AC=2$. Side $\overline{BC}$ and the median from $A$ to $\overline{BC}$ have the same length. What is $BC$? Express your answer in simplest radical form. | \sqrt{2} |
A circular dartboard is divided into regions with various central angles, as shown. The probability of a dart randomly landing in a particular region is $\frac16$. What is the corresponding measure, in degrees, of the central angle of this section of the dartboard? [asy]
unitsize(1.5cm);
defaultpen(linewidth(.7pt));
p... | 60 |
We repeatedly toss a coin until we get either three consecutive heads ($HHH$) or the sequence $HTH$ (where $H$ represents heads and $T$ represents tails). What is the probability that $HHH$ occurs before $HTH$? | 2/5 |
If $f(x)=f(2-x)$ for all $x$, then what line is necessarily an axis of symmetry of the graph of $y=f(x)$? (Give the simplest equation of this line.) | x=1 |
In the expression $x \cdot y^z - w$, the values of $x$, $y$, $z$, and $w$ are 1, 2, 3, and 4, although not necessarily in that order. What is the maximum possible value of the expression? | 161 |
Macy and Piper went to the batting cages. Each token gets you 15 pitches. Macy used 11 tokens and Piper used 17. Macy hit the ball 50 times. Piper hit the ball 55 times. How many pitches did Macy and Piper miss altogether? | Macy used 11 tokens which are worth 15 pitches each, so 11 tokens x 15 pitches = <<11*15=165>>165 pitches.
Piper used 17 tokens x 15 pitches = <<17*15=255>>255 pitches.
Together, Macy and Piper received 165 + 255 = <<165+255=420>>420 pitches.
Macy hit the ball 50 times + Piper’s 55 hits = <<50+55=105>>105 total hits.
O... |
Let $f(n)$ denote the largest odd factor of $n$ , including possibly $n$ . Determine the value of
\[\frac{f(1)}{1} + \frac{f(2)}{2} + \frac{f(3)}{3} + \cdots + \frac{f(2048)}{2048},\]
rounded to the nearest integer. | 1365 |
Given the ellipse C: $$\frac {x^{2}}{25}+ \frac {y^{2}}{9}=1$$, F is the right focus, and l is a line passing through point F (not parallel to the y-axis), intersecting the ellipse at points A and B. l′ is the perpendicular bisector of AB, intersecting the major axis of the ellipse at point D. Then the value of $$\frac... | \frac {2}{5} |
A cuckoo clock strikes the number of times corresponding to the current hour (for example, at 19:00, it strikes 7 times). One morning, Max approached the clock when it showed 9:05. He started turning the minute hand until it moved forward by 7 hours. How many times did the cuckoo strike during this period? | 43 |
Let
$$p(x,y) = a_0 + a_1x + a_2y + a_3x^2 + a_4xy + a_5y^2 + a_6x^3 + a_7x^2y + a_8xy^2 + a_9y^3.$$Suppose that
\begin{align*}
p(0,0) &=p(1,0) = p( - 1,0) = p(0,1) = p(0, - 1)= p(1,1) = p(1, - 1) = p(2,2) = 0.
\end{align*}There is a point $(r,s)$ for which $p(r,s) = 0$ for all such polynomials, where $r$ and $s$ are no... | \left( \frac{5}{19}, \frac{16}{19} \right) |
The points $A$, $B$ and $C$ lie on the surface of a sphere with center $O$ and radius $20$. It is given that $AB=13$, $BC=14$, $CA=15$, and that the distance from $O$ to $\triangle ABC$ is $\frac{m\sqrt{n}}k$, where $m$, $n$, and $k$ are positive integers, $m$ and $k$ are relatively prime, and $n$ is not divisible by t... | 118 |
Given two sets
$$
\begin{array}{l}
A=\{(x, y) \mid |x|+|y|=a, a>0\}, \\
B=\{(x, y) \mid |xy|+1=|x|+|y|\}.
\end{array}
$$
If \( A \cap B \) is the set of vertices of a regular octagon in the plane, determine the value of \( a \). | 2 + \sqrt{2} |
What are the first three digits to the right of the decimal point in the decimal representation of $\left(10^{2005}+1\right)^{11/8}$? | 375 |
Let \( S-ABC \) be a triangular prism with the base being an isosceles right triangle \( ABC \) with \( AB \) as the hypotenuse, and \( SA = SB = SC = 2 \) and \( AB = 2 \). If \( S \), \( A \), \( B \), and \( C \) are points on a sphere centered at \( O \), find the distance from point \( O \) to the plane \( ABC \). | \frac{\sqrt{3}}{3} |
Marta works on her grandparent's farm to raise money for a new phone. So far, she has collected $240. For every hour she works, she receives $10. Her grandmother often gives her tips, and she has collected $50 in tips in total. How many hours has Marta worked on the farm so far? | Ignoring tips, Marta earned $240 - $50 = $<<240-50=190>>190.
Therefore, she has worked 190 / 10 = <<190/10=19>>19 hours.
#### 19 |
What is the sum of all of the two-digit primes that are greater than 12 but less than 99 and are still prime when their two digits are interchanged? | 418 |
In a regular quadrilateral pyramid \(P-ABCD\) with a volume of 1, points \(E\), \(F\), \(G\), and \(H\) are the midpoints of segments \(AB\), \(CD\), \(PB\), and \(PC\), respectively. Find the volume of the polyhedron \(BEG-CFH\). | 5/16 |
Connecting the centers of adjacent faces of a cube forms a regular octahedron. What is the volume ratio of this octahedron to the cube? | $\frac{1}{6}$ |
Given the set of $n$ numbers; $n > 1$, of which one is $1 - \frac {1}{n}$ and all the others are $1$. The arithmetic mean of the $n$ numbers is: | 1 - \frac{1}{n^2} |
How many four-digit numbers are composed of four distinct digits such that one digit is the average of any two other digits? | 216 |
Bennett sells window screens. He sold twice as many window screens in February as he sold last month. In February, Bennett sold a fourth of what he sold in March. If Bennet sold 8800 window screens in March, how many screens did Bennett sell from January to March? | In February, Bennett sold 8800/4 = <<8800/4=2200>>2200 window screens.
In January, Bennett sold 2200/2 = <<2200/2=1100>>1100 window screens.
Between January and March, Bennett sold 8800+2200+1100 = <<8800+2200+1100=12100>>12100 window screens.
#### 12100 |
Mike has to get an x-ray and an MRI. The x-ray is $250 and the MRI is triple that cost. Insurance covers 80%. How much did he pay? | The MRI cost 250*3=$<<250*3=750>>750
So between the two things he pays 250+750=$<<250+750=1000>>1000
Insurance covered 1000*.8=$<<1000*.8=800>>800
So he had to pay 1000-800=$<<1000-800=200>>200
#### 200 |
Find the distance from point $M_{0}$ to the plane passing through three points $M_{1}, M_{2}, M_{3}$.
$M_{1}(2, 3, 1)$
$M_{2}(4, 1, -2)$
$M_{3}(6, 3, 7)$
$M_{0}(-5, -4, 8)$ | 11 |
Let $2^x$ be the greatest power of $2$ that is a factor of $144$, and let $3^y$ be the greatest power of $3$ that is a factor of $144$. Evaluate the following expression: $$\left(\frac15\right)^{y - x}$$ | 25 |
Suzie and 5 of her friends decide to rent an Airbnb at Lake Tahoe for 4 days from Thursday to Sunday. The rental rate for weekdays is $420 per day. The weekend rental rate is $540 per day. They all decide to split the rental evenly. How much does each person have to pay? | The cost of the rental for Thursday and Friday comes to $420 * 2 = $<<420*2=840>>840.
The cost of the rental for Saturday and Sunday comes to $540 * 2 = $<<540*2=1080>>1080.
The total cost of the rental comes to $840 + 1080 = $<<840+1080=1920>>1920.
The payment required for each person comes to $1920 / 6 = $<<1920/6=32... |
Bob's favorite number is between $50$ and $100$. It is a multiple of $11$, but not a multiple of $2$. The sum of its digits is a multiple of $3$. What is Bob's favorite number? | 99 |
Given that $α, β ∈ (0, \frac{π}{2})$, and $\frac{\sin β}{\sin α} = \cos(α + β)$,
(1) If $α = \frac{π}{6}$, then $\tan β =$ _______;
(2) The maximum value of $\tan β$ is _______. | \frac{\sqrt{2}}{4} |
The graph shows the total distance Sam drove from 6 a.m to 11 a.m. How many miles per hour is the car's average speed for the period from 6 a.m. to 11 a.m.?
[asy]
unitsize(0.2inch);
draw((0,0)--(5.5,0));
draw((0,0)--(0,8.5));
draw((1,0)--(1,8.5));
draw((2,0)--(2,8.5));
draw((3,0)--(3,8.5));
draw((4,0)--(4,8.5)... | 32 |
What is \( \frac{3}{10} \) more than \( 57.7 \)? | 58 |
Compute the number of functions $f:\{1,2, \ldots, 9\} \rightarrow\{1,2, \ldots, 9\}$ which satisfy $f(f(f(f(f(x)))))=$ $x$ for each $x \in\{1,2, \ldots, 9\}$. | 3025 |
To assess the shooting level of a university shooting club, an analysis group used stratified sampling to select the shooting scores of $6$ senior members and $2$ new members for analysis. After calculation, the sample mean of the shooting scores of the $6$ senior members is $8$ (unit: rings), with a variance of $\frac... | \frac{9}{2} |
Consider a $2 \times n$ grid of points and a path consisting of $2 n-1$ straight line segments connecting all these $2 n$ points, starting from the bottom left corner and ending at the upper right corner. Such a path is called efficient if each point is only passed through once and no two line segments intersect. How m... | \binom{4030}{2015} |
How many positive integers less than $500$ can be written as the sum of two positive perfect cubes? | 26 |
If $\det \mathbf{A} = 2$ and $\det \mathbf{B} = 12,$ then find $\det (\mathbf{A} \mathbf{B}).$ | 24 |
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