problem stringlengths 10 5.15k | answer stringlengths 0 1.23k |
|---|---|
Let \[f(n)=\dfrac{5+3\sqrt{5}}{10}\left(\dfrac{1+\sqrt{5}}{2}\right)^n+\dfrac{5-3\sqrt{5}}{10}\left(\dfrac{1-\sqrt{5}}{2}\right)^n.\]
Then $f(n+1)-f(n-1)$, expressed in terms of $f(n)$, equals: | f(n) |
In $\triangle PQR$, we have $PQ = QR = 46$ and $PR = 40$. Point $M$ is the midpoint of $\overline{QR}$. Find the length of segment $PM$. | \sqrt{1587} |
From the numbers $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, the probability of randomly selecting two different numbers such that both numbers are odd is $\_\_\_\_\_\_\_\_\_$, and the probability that the product of the two numbers is even is $\_\_\_\_\_\_\_\_\_$. | \frac{13}{18} |
Given that vectors $a$ and $b$ satisfy $(2a+3b) \perp b$, and $|b|=2\sqrt{2}$, find the projection of vector $a$ onto the direction of $b$. | -3\sqrt{2} |
Calculate the area of the shape bounded by the lines given by the equations:
$$
\begin{aligned}
& \left\{\begin{array}{l}
x=t-\sin t \\
y=1-\cos t
\end{array}\right. \\
& y=1 \quad (0<x<2\pi, \, y \geq 1)
\end{aligned}
$$ | \frac{\pi}{2} + 2 |
In the triangle shown, what is the positive difference between the greatest and least possible integral values of $x$?
[asy]
defaultpen(linewidth(0.7));
pair a,b,c;
b = (1,2); c = (3,0);
draw(a--b--c--a);
label("$x$",a--b,dir(135)); label("5",b--c,dir(45)); label("6",c--a,S);
[/asy] | 8 |
Bob has a seven-digit phone number and a five-digit postal code. The sum of the digits in his phone number and the sum of the digits in his postal code are the same. Bob's phone number is 346-2789. What is the largest possible value for Bob's postal code, given that no two digits in the postal code are the same? | 98765 |
Given the function $$f(x)=\cos\omega x\cdot \sin(\omega x- \frac {\pi}{3})+ \sqrt {3}\cos^{2}\omega x- \frac { \sqrt {3}}{4}(\omega＞0，x\in\mathbb{R})$$, and the distance from a center of symmetry of the graph of $y=f(x)$ to the nearest axis of symmetry is $$\frac {\pi}{4}$$.
(Ⅰ) Find the value of $\omega$ and the equation of the axis of symmetry for $f(x)$;
(Ⅱ) In $\triangle ABC$, where the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. If $$f(A)= \frac { \sqrt {3}}{4}, \sin C= \frac {1}{3}, a= \sqrt {3}$$, find the value of $b$. | \frac {3+2 \sqrt {6}}{3} |
Find the largest prime factor of $15^3+10^4-5^5$. | 41 |
Adams plans a profit of $10$ % on the selling price of an article and his expenses are $15$ % of sales. The rate of markup on an article that sells for $ $5.00$ is: | 33\frac {1}{3}\% |
Suppose $173\cdot 927\equiv n\pmod{50}$, where $0\le n< 50$.
What is the value of $n$? | 21 |
Given that in square ABCD, AE = 3EC and BF = 2FB, and G is the midpoint of CD, find the ratio of the area of triangle EFG to the area of square ABCD. | \frac{1}{24} |
Given the coefficient of determination R^2 for four different regression models, where the R^2 values are 0.98, 0.67, 0.85, and 0.36, determine which model has the best fitting effect. | 0.98 |
The number $1000!$ has a long tail of zeroes. How many zeroes are there? (Reminder: The number $n!$ is the product of the integers from 1 to $n$. For example, $5!=5\cdot 4\cdot3\cdot2\cdot 1= 120$.) | 249 |
The positive integers are arranged in rows and columns as shown below.
| Row 1 | 1 |
| Row 2 | 2 | 3 |
| Row 3 | 4 | 5 | 6 |
| Row 4 | 7 | 8 | 9 | 10 |
| Row 5 | 11 | 12 | 13 | 14 | 15 |
| Row 6 | 16 | 17 | 18 | 19 | 20 | 21 |
| ... |
More rows continue to list the positive integers in order, with each new row containing one more integer than the previous row. How many integers less than 2000 are in the column that contains the number 2000? | 16 |
Given the function $f(x)=ax^{2}+bx+c(a\\neq 0)$, its graph intersects with line $l$ at two points $A(t,t^{3}-t)$, $B(2t^{2}+3t,t^{3}+t^{2})$, where $t\\neq 0$ and $t\\neq -1$. Find the value of $f{{'}}(t^{2}+2t)$. | \dfrac {1}{2} |
The integers -5 and 6 are shown on a number line. What is the distance between them? | 11 |
Select 2 different numbers from 1, 3, 5, and 3 different numbers from 2, 4, 6, 8 to form a five-digit number, and determine the total number of even numbers among these five-digit numbers. | 864 |
There are 42 apples in a crate. 12 crates of apples were delivered to a factory. 4 apples were rotten and had to be thrown away. The remaining apples were packed into boxes that could fit 10 apples each. How many boxes of apples were there? | The total number of apples delivered was 42 × 12 = <<42*12=504>>504.
There are 504 - 4 = <<504-4=500>>500 apples were left remaining.
Finally, there are 500 ÷ 10 = <<500/10=50>>50 boxes of apples.
#### 50 |
Points $A$, $B$, $C$, and $D$ lie on a line, in that order. If $AB=2$ units, $BC=5$ units and $AD=14$ units, what is the ratio of $AC$ to $BD$? Express your answer as a common fraction. | \frac{7}{12} |
When $0.76\overline{204}$ is expressed as a fraction in the form $\frac{x}{999000}$, what is the value of $x$? | 761280 |
If $\sec x + \tan x = \frac{5}{2},$ then find $\sec x - \tan x.$ | \frac{2}{5} |
Rotate a square around a line that lies on one of its sides to form a cylinder. If the volume of the cylinder is $27\pi \text{cm}^3$, then the lateral surface area of the cylinder is _________ $\text{cm}^2$. | 18\pi |
Triangle $A B C$ satisfies $\angle B>\angle C$. Let $M$ be the midpoint of $B C$, and let the perpendicular bisector of $B C$ meet the circumcircle of $\triangle A B C$ at a point $D$ such that points $A, D, C$, and $B$ appear on the circle in that order. Given that $\angle A D M=68^{\circ}$ and $\angle D A C=64^{\circ}$, find $\angle B$. | 86^{\circ} |
Let $r$ be the positive real solution to $x^3 + \frac{2}{5} x - 1 = 0.$ Find the exact numerical value of
\[r^2 + 2r^5 + 3r^8 + 4r^{11} + \dotsb.\] | \frac{25}{4} |
If four times the reciprocal of the circumference of a circle equals the diameter of the circle, then the area of the circle is | 1 |
Let $X_1, X_2, \ldots, X_{100}$ be a sequence of mutually distinct nonempty subsets of a set $S$ . Any two sets $X_i$ and $X_{i+1}$ are disjoint and their union is not the whole set $S$ , that is, $X_i\cap X_{i+1}=\emptyset$ and $X_i\cup X_{i+1}\neq S$ , for all $i\in\{1, \ldots, 99\}$ . Find the smallest possible number of elements in $S$ . | \[
|S| \ge 8
\] |
A small square is constructed inside a square of area 1 by dividing each side of the unit square into $n$ equal parts, and then connecting the vertices to the division points closest to the opposite vertices. Find the value of $n$ if the the area of the small square is exactly $\frac1{1985}$. | 32 |
How many total days were there in the years 2005 through 2010? | 2191 |
Find the value of $1006 \sin \frac{\pi}{1006}$. Approximating directly by $\pi=3.1415 \ldots$ is worth only 3 points. | 3.1415875473 |
Given that points P1 and P2 are two adjacent centers of symmetry for the curve $y= \sqrt {2}\sin ωx-\cos ωx$ $(x\in\mathbb{R})$, if the tangents to the curve at points P1 and P2 are perpendicular to each other, determine the value of ω. | \frac{\sqrt{3}}{3} |
A spherical soap bubble lands on a horizontal wet surface and forms a hemisphere of the same volume. Given the radius of the hemisphere is $3\sqrt[3]{2}$ cm, find the radius of the original bubble. | 3 |
Given the function g(n) = log<sub>27</sub>n if log<sub>27</sub>n is rational, and 0 otherwise, find the value of the sum from n=1 to 7290 of g(n). | 12 |
The pages of a book are numbered $1_{}^{}$ through $n_{}^{}$. When the page numbers of the book were added, one of the page numbers was mistakenly added twice, resulting in an incorrect sum of $1986_{}^{}$. What was the number of the page that was added twice? | 33 |
If $\frac{1}{3}$ of $x$ is equal to 4, what is $\frac{1}{6}$ of $x$? | 2 |
What value of $x$ will give the minimum value for $x^2- 10x + 24$? | 5 |
What is the sum of the squares of the coefficients of $4(x^4 + 3x^2 + 1)$? | 176 |
Points A and B are on a circle of radius 7 and AB = 8. Point C is the midpoint of the minor arc AB. What is the length of the line segment AC? | \sqrt{98 - 14\sqrt{33}} |
Find $\tan G$ in the right triangle shown below.
[asy]
pair H,F,G;
H = (0,0);
G = (15,0);
F = (0,8);
draw(F--G--H--F);
draw(rightanglemark(F,H,G,20));
label("$H$",H,SW);
label("$G$",G,SE);
label("$F$",F,N);
label("$17$",(F+G)/2,NE);
label("$15$",G/2,S);
[/asy] | \frac{8}{15} |
A bus with programmers departed from Novosibirsk to Pavlodar. After traveling 70 km, another car with Pavel Viktorovich left Novosibirsk on the same route and caught up with the bus in Karasuk. After that, Pavel traveled another 40 km, while the bus traveled only 20 km in the same time. Find the distance from Novosibirsk to Karasuk, given that both the car and the bus traveled at constant speeds. (Provide a complete solution, not just the answer.) | 140 |
The English alphabet, which has 26 letters, is randomly permuted. Let \(p_{1}\) be the probability that \(\mathrm{AB}, \mathrm{CD}\), and \(\mathrm{EF}\) all appear as contiguous substrings. Let \(p_{2}\) be the probability that \(\mathrm{ABC}\) and \(\mathrm{DEF}\) both appear as contiguous substrings. Compute \(\frac{p_{1}}{p_{2}}\). | 23 |
What is the maximum number of checkers that can be placed on a $6 \times 6$ board so that no three checkers (more precisely, the centers of the cells they occupy) are collinear (in any direction)? | 12 |
A cat has nine lives. A dog has 3 less lives than a cat. A mouse has 7 more lives than a dog. How many lives does a mouse have? | Dog:9-3=<<9-3=6>>6 lives
Mouse:6+7=<<6+7=13>>13 lives
#### 13 |
What is the three-digit (integer) number which, when either increased or decreased by the sum of its digits, results in a number with all identical digits? | 105 |
One of Euler's conjectures was disproved in the 1960s by three American mathematicians when they showed there was a positive integer such that $133^5+110^5+84^5+27^5=n^{5}$. Find the value of $n$.
| 144 |
Given that \(15^{-1} \equiv 31 \pmod{53}\), find \(38^{-1} \pmod{53}\), as a residue modulo 53. | 22 |
The positive integers are grouped as follows:
\( A_1 = \{1\}, A_2 = \{2, 3, 4\}, A_3 = \{5, 6, 7, 8, 9\} \), and so on.
In which group does 2009 belong? | 45 |
If $e^{i \theta} = \frac{3 + i \sqrt{8}}{4},$ then find $\sin 6 \theta.$ | -\frac{855 \sqrt{2}}{1024} |
A triangle is made of wood sticks of lengths 8, 15 and 17 inches joined end-to-end. Pieces of the same integral length are cut from each of the sticks so that the three remaining pieces can no longer form a triangle. How many inches are in the length of the smallest piece that can be cut from each of the three sticks to make this happen? | 6 |
Given the equation \\((x^{2}-mx+2)(x^{2}-nx+2)=0\\), the four roots of the equation form a geometric sequence with the first term being \\( \frac {1}{2}\\). Find the absolute value of the difference between m and n, i.e., \\(|m-n|\\). | \frac{3}{2} |
Let $(1+x+x^2)^n=a_0 + a_1x+a_2x^2+ \cdots + a_{2n}x^{2n}$ be an identity in $x$. If we let $s=a_0+a_2+a_4+\cdots +a_{2n}$, then $s$ equals: | \frac{3^n+1}{2} |
What is the remainder when $x^4-7x^3+9x^2+16x-13$ is divided by $x-3$? | 8 |
A circle with equation $x^{2}+y^{2}=1$ passes through point $P(1, \sqrt {3})$. Two tangents are drawn from $P$ to the circle, touching the circle at points $A$ and $B$ respectively. Find the length of the chord $|AB|$. | \sqrt {3} |
In triangle \(ABC\), angle \(B\) is \(120^\circ\), and \(AB = 2BC\). The perpendicular bisector of side \(AB\) intersects \(AC\) at point \(D\). Find the ratio \(AD:DC\). | 3/2 |
If $\left( r + \frac{1}{r} \right)^2 = 3,$ then find $r^3 + \frac{1}{r^3}.$ | 0 |
Problem 4. Angel has a warehouse, which initially contains $100$ piles of $100$ pieces of rubbish each. Each morning, Angel performs exactly one of the following moves:
(a) He clears every piece of rubbish from a single pile.
(b) He clears one piece of rubbish from each pile.
However, every evening, a demon sneaks into the warehouse and performs exactly one of the
following moves:
(a) He adds one piece of rubbish to each non-empty pile.
(b) He creates a new pile with one piece of rubbish.
What is the first morning when Angel can guarantee to have cleared all the rubbish from the
warehouse? | 199 |
What is the least five-digit positive integer which is congruent to 6 (mod 17)? | 10,017 |
In a Geometry exam, Madeline got 2 mistakes which are half as many mistakes as Leo. Brent scored 25 and has 1 more mistake than Leo. What is Madeline's score? | Leo got 2 x 2 = <<2*2=4>>4 mistakes.
So, Brent got 4 + 1 = <<4+1=5>>5 mistakes.
This means that 25 + 5 = <<25+5=30>>30 is the perfect score.
Hence, Madeline got a score of 30 - 2 = <<30-2=28>>28.
#### 28 |
Let $n$ be the smallest positive integer such that $n$ is divisible by $20$, $n^2$ is a perfect cube, and $n^3$ is a perfect square. What is the number of digits of $n$? | 7 |
A store owner bought 2000 markers at $0.20 each. To make a minimum profit of $200, if he sells the markers for $0.50 each, calculate the number of markers he must sell at least to achieve or exceed this profit. | 1200 |
If the two real roots of the equation (lgx)<sup>2</sup>\-lgx+lg2•lg5=0 with respect to x are m and n, then 2<sup>m+n</sup>\=\_\_\_\_\_\_. | 128 |
An element is randomly chosen from among the first $20$ rows of Pascal's Triangle. What is the probability that the selected element is $1$? | \frac{13}{70} |
If $m$ and $n$ are positive integers with $n > 1$ such that $m^{n} = 2^{25} \times 3^{40}$, what is $m + n$? | 209957 |
Michael has never taken a foreign language class, but is doing a story on them for the school newspaper. The school offers French and Spanish. Michael has a list of all 25 kids in the school enrolled in at least one foreign language class. He also knows that 18 kids are in the French class and 21 kids are in the Spanish class. If Michael chooses two kids at random off his list and interviews them, what is the probability that he will be able to write something about both the French and Spanish classes after he is finished with the interviews? Express your answer as a fraction in simplest form. | \frac{91}{100} |
Find the $r$ that satisfies $\log_{16} (r+16) = \frac{5}{4}$. | 16 |
Given a sequence $\left\{ a_n \right\}$ satisfying $a_1=4$ and $a_1+a_2+\cdots +a_n=a_{n+1}$, and $b_n=\log_{2}a_n$, calculate the value of $\frac{1}{b_1b_2}+\frac{1}{b_2b_3}+\cdots +\frac{1}{b_{2017}b_{2018}}$. | \frac{3025}{4036} |
Given the ellipse $$C: \frac {x^{2}}{a^{2}}+ \frac {y^{2}}{b^{2}}=1(a>b>0)$$ with its left and right foci being F<sub>1</sub> and F<sub>2</sub>, and its top vertex being B. If the perimeter of $\triangle BF_{1}F_{2}$ is 6, and the distance from point F<sub>1</sub> to the line BF<sub>2</sub> is $b$.
(1) Find the equation of ellipse C;
(2) Let A<sub>1</sub> and A<sub>2</sub> be the two endpoints of the major axis of ellipse C, and point P is any point on ellipse C different from A<sub>1</sub> and A<sub>2</sub>. The line A<sub>1</sub>P intersects the line $x=m$ at point M. If the circle with MP as its diameter passes through point A<sub>2</sub>, find the value of the real number $m$. | 14 |
A shooter has a probability of hitting the target of $0.8$ each time. Now, using the method of random simulation to estimate the probability that the shooter hits the target at least $3$ times out of $4$ shots: first, use a calculator to generate random integers between $0$ and $9$, where $0$, $1$ represent missing the target, and $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$ represent hitting the target; since there are $4$ shots, every $4$ random numbers are grouped together to represent the results of $4$ shots. After randomly simulating, $20$ groups of random numbers were generated:
$5727$ $0293$ $7140$ $9857$ $0347$ $4373$ $8636$ $9647$ $1417$ $4698$
$0371$ $6233$ $2616$ $8045$ $6011$ $3661$ $9597$ $7424$ $6710$ $4281$
Based on this, estimate the probability that the shooter hits the target at least $3$ times out of $4$ shots. | 0.75 |
In the Cartesian coordinate system, with the origin as the pole and the positive half-axis of the x-axis as the polar axis, a polar coordinate system is established. The polar equation of curve C is $\rho - 2\cos\theta - 6\sin\theta + \frac{1}{\rho} = 0$, and the parametric equation of line l is $\begin{cases} x=3+ \frac{1}{2}t \\ y=3+ \frac{\sqrt{3}}{2}t \end{cases}$ (t is the parameter).
(1) Find the standard equation of curve C;
(2) If line l intersects curve C at points A and B, and the coordinates of point P are (3, 3), find the value of $|PA|+|PB|$. | 2\sqrt{6} |
Given the function $$f(x)=2\sin x( \sqrt {3}\cos x-\sin x)+1$$, if $f(x-\varphi)$ is an even function, determine the value of $\varphi$. | \frac {\pi}{3} |
Given the expression $2-(-3)-4\times(-5)-6-(-7)-8\times(-9)+10$, evaluate this expression. | 108 |
Given the sequence 1, $\frac{1}{2}$, $\frac{2}{1}$, $\frac{1}{3}$, $\frac{2}{2}$, $\frac{3}{1}$, $\frac{1}{4}$, $\frac{2}{3}$, $\frac{3}{2}$, $\frac{4}{1}$, ..., then $\frac{3}{5}$ is the \_\_\_\_\_\_ term of this sequence. | 24 |
For positive integers $N$ and $k$ define $N$ to be $k$-nice if there exists a positive integer $a$ such that $a^k$ has exactly $N$ positive divisors. Determine the quantity of positive integers smaller than $1500$ that are neither $9$-nice nor $10$-nice. | 1199 |
Loris needs three more books to have the same number as Lamont, who has twice the number of books Darryl has. If Darryl has 20 books, calculate the total number of books the three have. | Since Darryl has 20 books, Lamont, who has twice the number of books Darryl has, has 20*2 = <<20*2=40>>40 books.
The total number of books both Darryl and Lamont has is 40+20 = <<40+20=60>>60 books.
To have the same number of books as Lamont, Loris needs 3 more books, meaning she has 40-3 = <<40-3=37>>37 books.
In total, the three has 60+37= <<60+37=97>>97 books
#### 97 |
How many two-digit positive integers $N$ have the property that the sum of $N$ and the number obtained by reversing the order of the digits is a perfect square? | 8 |
Let $\mathbf{u}$ and $\mathbf{v}$ be unit vectors, and let $\mathbf{w}$ be a vector such that $\mathbf{u} \times \mathbf{v} + \mathbf{u} = \mathbf{w}$ and $\mathbf{w} \times \mathbf{u} = \mathbf{v}.$ Compute $\mathbf{u} \cdot (\mathbf{v} \times \mathbf{w}).$ | 1 |
Sally reads 10 pages of a book on weekdays and 20 pages on weekends. If it takes 2 weeks for Sally to finish her book, how many pages that book has? | Two weeks has 5*2=<<5*2=10>>10 weekdays.
Two weeks has 2*2=<<2*2=4>>4 weekends.
On weekdays, she reads 10*10=<<10*10=100>>100 pages in total.
On weekends, she reads 20*4=<<20*4=80>>80 pages in total.
So the book has 100+80=<<100+80=180>>180 pages.
#### 180 |
Regular octagon \( CH I L D R E N \) has area 1. Determine the area of quadrilateral \( L I N E \). | 1/2 |
Given vectors $\overrightarrow {a}$ and $\overrightarrow {b}$ that satisfy $|\overrightarrow {a}|= \sqrt {3}$, $|\overrightarrow {b}|=2$, and $(\overrightarrow {a}- \overrightarrow {b}) \perp \overrightarrow {a}$, find the projection of $\overrightarrow {a}$ on $\overrightarrow {b}$. | \frac {3}{2} |
Rachel is an artist. She posts a speed painting video each week on her Instagram account to promote her work. To save time, she paints and records 4 videos at a time. It takes her about 1 hour to set up her painting supplies and her camera. Then she records herself painting for 1 hour per painting. She takes another 1 hour to clean up. Each video takes 1.5 hours to edit and post. How long does it take Rachel to produce a speed painting video? | Rachel spends 1 + 1 = <<1+1=2>>2 hours setting up and cleaning up.
She paints for 1 hour x 4 paintings = <<1*4=4>>4 hours.
Then she edits 1.5 hours x 4 videos = <<1.5*4=6>>6 hours.
She spends 2 + 4 + 6 = <<2+4+6=12>>12 hours to make 4 videos.
So Rachel spends 12 / 4 = <<12/4=3>>3 hours on each video.
#### 3 |
Let planes \( \alpha \) and \( \beta \) be parallel to each other. Four points are selected on plane \( \alpha \) and five points are selected on plane \( \beta \). What is the maximum number of planes that can be determined by these points? | 72 |
There are many fish in the tank. One third of them are blue, and half of the blue fish have spots. If there are 10 blue, spotted fish, how many fish are there in the tank? | There are 2*10=<<2*10=20>>20 blue fish.
There are 3*20=<<3*20=60>>60 fish.
#### 60 |
Let $p$, $q$, $r$, $s$ be distinct real numbers such that the roots of $x^2 - 12px - 13q = 0$ are $r$ and $s$, and the roots of $x^2 - 12rx - 13s = 0$ are $p$ and $q$. Additionally, $p + q + r + s = 201$. Find the value of $pq + rs$. | -\frac{28743}{12} |
Compute $\arccos \frac{\sqrt{3}}{2}.$ Express your answer in radians. | \frac{\pi}{6} |
How many positive two-digit integers are multiples of 5 and of 7? | 2 |
If the polynomial $x^{3}+x^{10}=a_{0}+a_{1}(x+1)+...+a_{9}(x+1)^{9}+a_{10}(x+1)^{10}$, then $a_{9}=$ \_\_\_\_\_\_. | -10 |
Find the pattern and fill in the blanks:
1. 12, 16, 20, \_\_\_\_\_\_, \_\_\_\_\_\_
2. 2, 4, 8, \_\_\_\_\_\_, \_\_\_\_\_\_ | 32 |
There is a unique positive integer $n$ such that $\log_2{(\log_{16}{n})} = \log_4{(\log_4{n})}.$ What is the sum of the digits of $n?$ | 13 |
A right-angled triangle has an area of \( 36 \mathrm{~m}^2 \). A square is placed inside the triangle such that two sides of the square are on two sides of the triangle, and one vertex of the square is at one-third of the longest side.
Determine the area of this square. | 16 |
Given $y=f(x)+x^2$ is an odd function, and $f(1)=1$, if $g(x)=f(x)+2$, then $g(-1)=$ . | -1 |
In triangle $ABC$, $a=3$, $\angle C = \frac{2\pi}{3}$, and the area of $ABC$ is $\frac{3\sqrt{3}}{4}$. Find the lengths of sides $b$ and $c$. | \sqrt{13} |
Given that $BDEF$ is a square and $AB = BC = 1$, find the number of square units in the area of the regular octagon.
[asy]
real x = sqrt(2);
pair A,B,C,D,E,F,G,H;
F=(0,0); E=(2,0); D=(2+x,x); C=(2+x,2+x);
B=(2,2+2x); A=(0,2+2x); H=(-x,2+x); G=(-x,x);
draw(A--B--C--D--E--F--G--H--cycle);
draw((-x,0)--(2+x,0)--(2+x,2+2x)--(-x,2+2x)--cycle);
label("$B$",(-x,2+2x),NW); label("$D$",(2+x,2+2x),NE); label("$E$",(2+x,0),SE); label("$F$",(-x,0),SW);
label("$A$",(-x,x+2),W); label("$C$",(0,2+2x),N);
[/asy] | 4+4\sqrt{2} |
Compute
\[ e^{2 \pi i/17} + e^{4 \pi i/17} + e^{6 \pi i/17} + \dots + e^{32 \pi i/17}. \] | -1 |
A man travels $m$ feet due north at $2$ minutes per mile. He returns due south to his starting point at $2$ miles per minute. The average rate in miles per hour for the entire trip is: | 48 |
Evaluate $\left\lceil-\sqrt{\frac{49}{4}}\right\rceil$. | -3 |
In quadrilateral $ABCD$, $\angle{BAD}\cong\angle{ADC}$ and $\angle{ABD}\cong\angle{BCD}$, $AB = 8$, $BD = 10$, and $BC = 6$. The length $CD$ may be written in the form $\frac {m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$. | 69 |
Let $a$, $b$, $c$, and $d$ be real numbers with $|a-b|=2$, $|b-c|=3$, and $|c-d|=4$. What is the sum of all possible values of $|a-d|$? | 18 |
Let \(ABCD\) be a trapezoid such that \(AB \parallel CD, \angle BAC=25^{\circ}, \angle ABC=125^{\circ}\), and \(AB+AD=CD\). Compute \(\angle ADC\). | 70^{\circ} |
Given that in quadrilateral $ABCD$, $m\angle B = m \angle C = 120^{\circ}, AB=3, BC=4,$ and $CD=5$, calculate the area of $ABCD$. | 8\sqrt{3} |
Let $g : \mathbb{R} \to \mathbb{R}$ be a function such that
\[g(x) g(y) - g(xy) = x^2 + y^2\] for all real numbers $x$ and $y.$
Let $n$ be the number of possible values of $g(3)$, and let $s$ be the sum of all possible values of $g(3).$ Find $n \times s.$ | 10 |
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