problem stringlengths 10 5.15k | answer stringlengths 0 1.23k |
|---|---|
Given $0 < \alpha < \pi$, $\tan\alpha = -2$.
(1) Find the value of $\sin\left(\alpha + \frac{\pi}{6}\right)$;
(2) Calculate the value of $$\frac{2\cos\left(\frac{\pi}{2} + \alpha\right) - \cos(\pi - \alpha)}{\sin\left(\frac{\pi}{2} - \alpha\right) - 3\sin(\pi + \alpha)};$$
(3) Simplify $2\sin^2\alpha - \sin\alpha\cos\a... | \frac{11}{5} |
Triangle \(ABC\) is isosceles with \(AB = AC\) and \(BC = 65 \, \text{cm}\). \(P\) is a point on \(BC\) such that the perpendicular distances from \(P\) to \(AB\) and \(AC\) are \(24 \, \text{cm}\) and \(36 \, \text{cm}\), respectively. Find the area of \(\triangle ABC\). | 2535 |
Given the cubic equation
\[
x^3 + Ax^2 + Bx + C = 0 \quad (A, B, C \in \mathbb{R})
\]
with roots \(\alpha, \beta, \gamma\), find the minimum value of \(\frac{1 + |A| + |B| + |C|}{|\alpha| + |\beta| + |\gamma|}\). | \frac{\sqrt[3]{2}}{2} |
$AB$ and $AC$ are tangents to a circle, $\angle BAC = 60^{\circ}$, the length of the broken line $BAC$ is 1. Find the distance between the points of tangency $B$ and $C$. | \frac{1}{2} |
Let $p$, $q$, and $r$ be the distinct roots of the polynomial $x^3 - 22x^2 + 80x - 67$. It is given that there exist real numbers $A$, $B$, and $C$ such that \begin{equation*} \frac{1}{s^3 - 22s^2 + 80s - 67} = \frac{A}{s-p} + \frac{B}{s-q} + \frac{C}{s-r} \end{equation*} for all $s\not\in\{p,q,r\}$. What is $\tfrac1A+... | 244 |
A box contains 6 cards numbered 1, 2, ..., 6. A card is randomly drawn from the box, and its number is denoted as $a$. The box is then adjusted to retain only the cards with numbers greater than $a$. A second draw is made, and the probability that the first draw is an odd number and the second draw is an even number is... | \frac{17}{45} |
A line passing through point $P(-2,2)$ intersects the hyperbola $x^2-2y^2=8$ such that the midpoint of the chord $MN$ is exactly at $P$. Find the length of $|MN|$. | 2 \sqrt{30} |
The function $y=2\sin(x+\frac{\pi}{3})$ has an axis of symmetry at $x=\frac{\pi}{6}$. | \frac{\pi}{6} |
Let $\mathrm{C}$ be a circle in the $\mathrm{xy}$-plane with a radius of 1 and its center at $O(0,0,0)$. Consider a point $\mathrm{P}(3,4,8)$ in space. If a sphere is completely contained within the cone with $\mathrm{C}$ as its base and $\mathrm{P}$ as its apex, find the maximum volume of this sphere. | \frac{4}{3}\pi(3-\sqrt{5})^3 |
Given the system of equations
\begin{align*}
4x+2y &= c, \\
6y - 12x &= d,
\end{align*}
where \(d \neq 0\), find the value of \(\frac{c}{d}\). | -\frac{1}{3} |
Mikail's birthday is tomorrow. He will be 3 times older than he was when he was three. On his birthday, his parents give him $5 for every year old he is. How much money will they give him? | Mikail is 9 because 3 x 3 = <<3*3=9>>9
He will get $45 because 9 x $5 = $<<9*5=45>>45
#### 45 |
If $cos2α=-\frac{{\sqrt{10}}}{{10}}$, $sin({α-β})=\frac{{\sqrt{5}}}{5}$, and $α∈({\frac{π}{4},\frac{π}{2}})$, $β∈({-π,-\frac{π}{2}})$, then $\alpha +\beta =$____. | -\frac{\pi}{4} |
Casey wants to decorate her toenails and fingernails. First, she wants to do a base coat on each nail, then a coat of paint and finally a coat of glitter on every nail. It takes 20 minutes to apply each coat and 20 minutes for each coat to dry before applying the next one. Assuming Casey's nails will be done when all ... | Casey wants to paint her 10 toenails + 10 fingernails = <<10+10=20>>20 nails in total.
First she wants to do a base coat which takes 20 minutes to apply + 20 minutes to dry = <<20+20=40>>40 minutes total for the first coat.
It will take the same amount of time to do a coat of paint and let it dry and to do a coat of gl... |
What is the sum of three consecutive even integers if the sum of the first and third integers is $128$? | 192 |
The Rotokas of Papua New Guinea have twelve letters in their alphabet. The letters are: A, E, G, I, K, O, P, R, S, T, U, and V. Suppose license plates of five letters utilize only the letters in the Rotoka alphabet. How many license plates of five letters are possible that begin with either G or K, end with T, cannot c... | 1008 |
John buys bags of popcorn for $4 and sells them for $8. How much profit does he get by selling 30 bags? | Each bag makes a profit of 8-4=$<<8-4=4>>4
So he makes a total profit of 30*4=$<<30*4=120>>120
#### 120 |
When three positive integers are divided by $47$, the remainders are $25$, $20$, and $3$, respectively.
When the sum of the three integers is divided by $47$, what is the remainder? | 1 |
How many degrees are in the measure of the smaller angle that is formed by the hour-hand and minute-hand of a clock when it is 5 o'clock? | 150^\circ |
Leah has $13$ coins, all of which are pennies and nickels. If she had one more nickel than she has now, then she would have the same number of pennies and nickels. In cents, how much are Leah's coins worth? | 37 |
Given the function $f(x)= \sqrt {x^{2}-4x+4}-|x-1|$:
1. Solve the inequality $f(x) > \frac {1}{2}$;
2. If positive numbers $a$, $b$, $c$ satisfy $a+2b+4c=f(\frac {1}{2})+2$, find the minimum value of $\sqrt { \frac {1}{a}+ \frac {2}{b}+ \frac {4}{c}}$. | \frac {7}{3} \sqrt {3} |
Using the trapezoidal rule with an accuracy of 0.01, calculate $\int_{2}^{3} \frac{d x}{x-1}$. | 0.6956 |
Harry has 3 sisters and 5 brothers. His sister Harriet has $\text{S}$ sisters and $\text{B}$ brothers. What is the product of $\text{S}$ and $\text{B}$? | 10 |
Find $p$ if $12^3=\frac{9^2}3\cdot2^{12p}$. | \frac{1}{2} |
Given that $F$ is the right focus of the hyperbola $C:x^{2}-\frac{y^{2}}{8}=1$, and $P$ is a point on the left branch of $C$, $A(0,6\sqrt{6})$, the minimum perimeter of $\triangle APF$ is $\_\_\_\_\_\_$. | 32 |
Compute $\dbinom{5}{3}$. | 10 |
To express 20 as a sum of distinct powers of 2, we would write $20 = 2^4 + 2^2$. The sum of the exponents of these powers is $4 + 2 = 6$. If 1562 were expressed as a sum of distinct powers of 2, what would be the least possible sum of the exponents of these powers? | 27 |
A rectangle can be divided into $n$ equal squares. The same rectangle can also be divided into $n+76$ equal squares. Find $n$ . | 324 |
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.) | 15 |
A group of 5 children are taken to an amusement park. Only 3 of them were daring enough to get on the Ferris wheel which cost $5 per child. Everyone had a go at the merry-go-round (at $3 per child). On their way home, they bought 2 cones of ice cream each (each cone cost $8). How much did they spend altogether? | 3 children rode the Ferris wheel at $5 reach for a total of 3*$5 = $<<3*5=15>>15
5 children rode the merry-go-round at $3 each for a total of 5*$3 = $<<5*3=15>>15
5 children bought 2 cones of ice cream each at $8 per cone for a total of 5*2*$8 = $<<5*2*8=80>>80
In total, they spent $15+$15+$80 = $<<15+15+80=110>>110
##... |
Cora started reading a 158-page book on Monday, and she decided she wanted to finish it by the end of Friday. She read 23 pages on Monday, 38 pages on Tuesday, and 61 pages on Wednesday. She knows she will have time to read twice as much on Friday as she does on Thursday. How many pages does she have to read on Thursda... | Let P be the number of pages Cora has to read on Thursday.
If she finishes the book by the end of Friday, she will have read 23 + 38 + 61 + P + 2P = 158 pages.
The pages she has left to read by Thursday are P + 2P = 3P = 158 - 23 - 38 - 61 = pages.
Thus, by Thursday she has 3P = 36 pages left to read.
Therefore, to fin... |
For what is the largest natural number \( m \) such that \( m! \cdot 2022! \) is a factorial of a natural number? | 2022! - 1 |
Two apartment roommates split the rent, utilities, and grocery payments equally each month. The rent for the whole apartment is $1100 and utilities are $114. If one roommate pays $757 in all, how many dollars are groceries for the whole apartment? | Rent plus utilities for the whole apartment is 1100+114 = $<<1100+114=1214>>1214
If one roommate pays $757 in all, the total cost of everything is 757*2 = <<757*2=1514>>1514
The groceries cost 1514-1214 = <<1514-1214=300>>300
#### 300 |
Right triangle $ABC$ is inscribed in circle $W$ . $\angle{CAB}=65$ degrees, and $\angle{CBA}=25$ degrees. The median from $C$ to $AB$ intersects $W$ and line $D$ . Line $l_1$ is drawn tangent to $W$ at $A$ . Line $l_2$ is drawn tangent to $W$ at $D$ . The lines $l_1$ and $l_2$ intersect at ... | 50 |
At a university dinner, there are 2017 mathematicians who each order two distinct entrées, with no two mathematicians ordering the same pair of entrées. The cost of each entrée is equal to the number of mathematicians who ordered it, and the university pays for each mathematician's less expensive entrée (ties broken ar... | 127009 |
A pentagon is inscribed around a circle, with the lengths of its sides being whole numbers, and the lengths of the first and third sides equal to 1. Into what segments does the point of tangency divide the second side? | \frac{1}{2} |
Given \\(|a|=1\\), \\(|b|= \sqrt{2}\\), and \\(a \perp (a-b)\\), the angle between vector \\(a\\) and vector \\(b\\) is ______. | \frac{\pi}{4} |
Given the sets \( A = \{(x, y) \mid |x| + |y| = a, a > 0\} \) and \( B = \{(x, y) \mid |xy| + 1 = |x| + |y| \} \), if the intersection \( A \cap B \) is the set of vertices of a regular octagon in the plane, determine the value of \( a \). | 2 + \sqrt{2} |
What is the measure of angle 4 if $m\angle 1 = 76^{\circ}, m\angle 2 = 27^{\circ}$ and $m\angle 3 = 17^{\circ}$?
[asy]
draw((0,0)--(4,8)--(10,0)--cycle,linewidth(1));
draw((0,0)--(5,3)--(10,0),linewidth(1));
label("2",(1,1.2));
label("1",(4,7.75),S);
label("4",(5,3),S);
label("3",(8,1.8));
[/asy] | 120^{\circ} |
Roman the Tavernmaster has $20 worth of gold coins. He sells 3 gold coins to Dorothy. After she pays him, he has $12. How many gold coins does Roman have left? | Dorothy paid $12 for 3 gold coins so each gold coin is worth $12/3 = $<<12/3=4>>4
The Tavern master originally had $20 worth of gold coins which is $20/$4 = <<20/4=5>>5 gold coins
He gave 3 out of 5 gold coins to Dorothy so he now has 5-3 = <<5-3=2>>2 gold coins left
#### 2 |
What is the greater of the solutions to the equation $x^2 + 15x -54=0$? | 3 |
Find all values of $x$ such that
\[3^x + 4^x + 5^x = 6^x.\] | 3 |
Two alien spacecraft on a sightseeing tour of Earth left New Orleans airport at 3:00 pm to travel the 448-mile distance to Dallas by air. Traveling nonstop, the first spacecraft landed in Dallas at 3:30 pm, while the second spacecraft landed in Dallas thirty minutes later. Assuming both spacecraft traveled at constant... | The first spacecraft flew for 30 minutes, or 30/60=1/2 hour.
The second spacecraft flew for 30+30=<<30+30=60>>60 minutes, or 1 hour.
Thus the first spacecraft traveled at a speed of 448 miles in 1/2 hour, or 448/(1/2)=896 miles per hour.
The second spacecraft traveled 448 miles in 1 hour, or 448/1=<<448/1=448>>448 mile... |
Seth bought some boxes of oranges. He gave a box to his mother. He then gave away half of the remaining boxes. If Seth has 4 boxes of oranges left, how many boxes did he buy in the first place? | After giving a box to his mom, Seth had 4*2=<<4*2=8>>8 boxes left.
Seth bought 8+1=<<8+1=9>>9 boxes of oranges.
#### 9 |
James buys $3000 worth of stuff from Amazon. He has to return a TV that cost $700 and a bike that cost $500. He also sells another bike that cost 20% more than the bike he returned for 80% of what he bought it for. He then buys a toaster for $100. How much is he out of pocket for everything? | The items he returned were valued at $700 + $500 = $<<700+500=1200>>1200
So far he is out 3000-1200 = <<3000-1200=1800>>1800 after recouping 1200.
An item that is 20% more expensive cost 1 + .2 = 1.2 times as much as the item
So that means the bike he sold cost $500 * 1.2 = $<<500*1.2=600>>600
He sold it for $600 * .8 ... |
Suppose that all four of the numbers \[2 - \sqrt{5}, \;4+\sqrt{10}, \;14 - 2\sqrt{7}, \;-\sqrt{2}\]are roots of the same nonzero polynomial with rational coefficients. What is the smallest possible degree of the polynomial? | 8 |
There are real numbers $a$ and $b$ for which the function $f$ has the properties that $f(x) = ax + b$ for all real numbers $x$, and $f(bx + a) = x$ for all real numbers $x$. What is the value of $a+b$? | -2 |
Compute the product of the roots of the equation \[3x^3 - x^2 - 20x + 27 = 0.\] | -9 |
Find all values of $x$ that satisfy \[\frac{x^2}{x+1} \ge \frac{2}{x-1} + \frac{5}{4}.\] | (-1, 1) \cup [3, \infty) |
There are two coal mines, Mine A and Mine B. Each gram of coal from Mine A releases 4 calories of heat when burned, and each gram of coal from Mine B releases 6 calories of heat when burned. The price per ton of coal at the production site is 20 yuan for Mine A and 24 yuan for Mine B. It is known that the transportatio... | 18 |
How many nondecreasing sequences $a_{1}, a_{2}, \ldots, a_{10}$ are composed entirely of at most three distinct numbers from the set $\{1,2, \ldots, 9\}$ (so $1,1,1,2,2,2,3,3,3,3$ and $2,2,2,2,5,5,5,5,5,5$ are both allowed)? | 3357 |
Given that the area of $\triangle ABC$ is 360, and point $P$ is a point on the plane of the triangle, with $\overrightarrow {AP}= \frac {1}{4} \overrightarrow {AB}+ \frac {1}{4} \overrightarrow {AC}$, then the area of $\triangle PAB$ is \_\_\_\_\_\_. | 90 |
Let $x,$ $y,$ and $z$ be positive real numbers. Find the minimum value of
\[\frac{4z}{2x + y} + \frac{4x}{y + 2z} + \frac{y}{x + z}.\] | 3 |
Let \( x, y, z \) be positive numbers that satisfy the system of equations:
\[
\begin{cases}
x^{2}+xy+y^{2}=27 \\
y^{2}+yz+z^{2}=16 \\
z^{2}+xz+x^{2}=43
\end{cases}
\]
Find the value of the expression \( xy+yz+xz \). | 24 |
Koby and Cherie want to light fireworks. Koby has bought 2 boxes of fireworks while Cherie has just 1 box of fireworks. Koby’s boxes each contain 3 sparklers and 5 whistlers. Cherie’s box has 8 sparklers and 9 whistlers. In total, how many fireworks do Koby and Cherie have? | Each of Koby’s boxes contain 3 sparklers + 5 whistlers = <<3+5=8>>8 fireworks.
So in total, Koby has 2 boxes * 8 fireworks per box = <<2*8=16>>16 fireworks.
Cherie’s box of fireworks has 8 sparklers + 9 whistlers = <<8+9=17>>17 fireworks.
Koby and Cherie therefore have a combined total of 16 fireworks from Koby + 17 fi... |
The numbers $x$ and $y$ are inversely proportional. When the sum of $x$ and $y$ is 42, $x$ is twice $y$. What is the value of $y$ when $x=-8$? | -49 |
Given the parabola \( C: x^{2} = 2py \) with \( p > 0 \), two tangents \( RA \) and \( RB \) are drawn from the point \( R(1, -1) \) to the parabola \( C \). The points of tangency are \( A \) and \( B \). Find the minimum area of the triangle \( \triangle RAB \) as \( p \) varies. | 3 \sqrt{3} |
Given sin(x + $\frac{π}{4}$) = $\frac{1}{3}$, find the value of sin4x - 2cos3xsinx = ___. | -\frac{7}{9} |
Little kids were eating candies. Each ate 11 candies less than the rest combined but still more than one candy. How many candies were eaten in total? | 33 |
Let $w_1, w_2, \dots, w_n$ be complex numbers. A line $L$ in the complex plane is called a mean line for the points $w_1, w_2, \dots, w_n$ if $L$ contains points (complex numbers) $z_1, z_2, \dots, z_n$ such that \[\sum_{k = 1}^n (z_k - w_k) = 0.\] For the numbers $w_1 = 32 + 170i$, $w_2 = - 7 + 64i$, $w_3 = - 9 + 200i... | 163 |
Jonathan enjoys walking and running for exercise, and he has three different exercise routines. On Mondays, he walks at 2 miles per hour. On Wednesdays, he walks at 3 miles per hour. And on Fridays, he runs at 6 miles per hour. On each exercise day, he travels 6 miles. What is the combined total time, in hours, he... | On Mondays, he walks 6 miles at 2 miles per hour, for a length of 6/2=<<6/2=3>>3 hours.
On Wednesdays, he walks 6 miles at 3 miles per hour, for a length of 6/3=<<6/3=2>>2 hours.
And on Fridays, he runs 6 miles at 6 miles per hour, for a length of 6/6=<<6/6=1>>1 hour.
The combined total time he spends exercising in a w... |
How many integer pairs $(x,y)$ are there such that \[0\leq x < 165, \quad 0\leq y < 165 \text{ and } y^2\equiv x^3+x \pmod {165}?\] | 99 |
If \(\frac{a}{b} = 5\), \(\frac{b}{c} = \frac{1}{4}\), and \(\frac{c^2}{d} = 16\), then what is \(\frac{d}{a}\)? | \frac{1}{25} |
Given an integer \( n > 4 \), the coefficients of the terms \( x^{n-4} \) and \( xy \) in the expansion of \( (x + 2 \sqrt{y} - 1)^n \) are equal. Find the value of \( n \). | 51 |
Bridget bought a bag of apples at the grocery store. She gave half of the apples to Ann. Then she gave Cassie 3 apples, keeping 4 apples for herself. How many apples did Bridget buy? | 14 |
How many four-digit numbers are composed of four distinct digits such that one digit is the average of any two other digits? | 240 |
Yan is somewhere between his home and the stadium. To get to the stadium he can walk directly to the stadium, or else he can walk home and then ride his bicycle to the stadium. He rides 7 times as fast as he walks, and both choices require the same amount of time. What is the ratio of Yan's distance from his home to ... | \frac{3}{4} |
The polynomial $1-x+x^2-x^3+\cdots+x^{16}-x^{17}$ may be written in the form $a_0+a_1y+a_2y^2+\cdots +a_{16}y^{16}+a_{17}y^{17}$, where $y=x+1$ and the $a_i$'s are constants. Find the value of $a_2$. | 816 |
How many times does the digit 9 appear in the list of all integers from 1 to 700? | 140 |
If $f(x) = \frac{1 + x}{1 - 3x}, f_1(x) = f(f(x)), f_2(x) = f(f_1(x)),$ and in general $f_n(x) = f(f_{n-1}(x)),$ then $f_{1993}(3)=$
| \frac{1}{5} |
There are exactly three integers $x$ satisfying the inequality
\[x^2 + bx + 2 \le 0.\]How many integer values of $b$ are possible? | 2 |
Dima calculated the factorials of all natural numbers from 80 to 99, found the reciprocals of them, and printed the resulting decimal fractions on 20 endless ribbons (for example, the last ribbon had the number \(\frac{1}{99!}=0. \underbrace{00\ldots00}_{155 \text{ zeros}} 10715 \ldots \) printed on it). Sasha wants to... | 155 |
Compute $\cos 135^\circ$. | -\frac{\sqrt{2}}{2} |
How many solutions in integers $x$ and $y$ does the inequality
$$
|x| + |y| < 10
$$
have? | 181 |
A $n$-gon $S-A_{1} A_{2} \cdots A_{n}$ has its vertices colored such that each vertex is colored with one color, and the endpoints of each edge are colored differently. Given $n+1$ colors available, find the number of different ways to color the vertices. (For $n=4$, this was a problem in the 1995 National High School ... | 420 |
Suppose we flip five coins simultaneously: a penny (1 cent), a nickel (5 cents), a dime (10 cents), a quarter (25 cents), and a half-dollar (50 cents). What is the probability that at least 40 cents worth of coins come up heads? | \frac{9}{16} |
In right triangle $DEF$, where $DE=15$, $DF=9$, and $EF=12$ units. What is the distance from $F$ to the midpoint of segment $DE$? | 7.5 |
Given that the sequence \( a_1, a_2, \cdots, a_n, \cdots \) satisfies \( a_1 = a_2 = 1 \) and \( a_3 = 2 \), and for any \( n \in \mathbf{N}^{*} \), it holds that \( a_n \cdot a_{n+1} \cdot a_{n+2} \cdot a_{n+3} = a_n + a_{n+1} + a_{n+2} + a_{n+3} \). Find the value of \( \sum_{i=1}^{2023} a_i \). | 4044 |
Mark constructed a deck that was 30 feet by 40 feet. It cost $3 per square foot. He then paid an extra $1 per square foot for sealant. How much did he pay? | The deck is 30*40=<<30*40=1200>>1200 square feet
He pays 3+1=$<<3+1=4>>4 per square foot
So he paid 1200*4=$<<1200*4=4800>>4800
#### 4800 |
Fred the Four-Dimensional Fluffy Sheep is walking in 4 -dimensional space. He starts at the origin. Each minute, he walks from his current position $\left(a_{1}, a_{2}, a_{3}, a_{4}\right)$ to some position $\left(x_{1}, x_{2}, x_{3}, x_{4}\right)$ with integer coordinates satisfying $\left(x_{1}-a_{1}\right)^{2}+\left... | \binom{40}{10}\binom{40}{20}^{3} |
Simplify
\[\frac{\tan 30^\circ + \tan 40^\circ + \tan 50^\circ + \tan 60^\circ}{\cos 20^\circ}.\] | \frac{8 \sqrt{3}}{3} |
Given a rectangle \(ABCD\). On two sides of the rectangle, different points are chosen: six points on \(AB\) and seven points on \(BC\). How many different triangles can be formed with vertices at the chosen points? | 231 |
Sami finds 3 spiders in the playground. Hunter sees 12 ants climbing the wall. Ming discovers 8 ladybugs in the sandbox, and watches 2 of them fly away. How many insects are remaining in the playground? | The children find 3 + 12 + 8 = <<3+12+8=23>>23 insects.
After the ladybugs fly away, there are 23 - 2 = <<23-2=21>>21 insects.
#### 21 |
Given that the sum of the binomial coefficients in the expansion of $(5x- \frac{1}{\sqrt{x}})^n$ is 64, determine the constant term in its expansion. | 375 |
Given that $α \in \left( \frac{π}{2}, π \right)$ and $3\cos 2α = \sin \left( \frac{π}{4} - α \right)$, find the value of $\sin 2α$. | -\frac{17}{18} |
Seven people arrive to dinner, but the circular table only seats six. If two seatings such that one is a rotation of the other are considered the same, then in how many different ways can we choose six people and seat them at the table? | 840 |
The numbers from 1 to 150, inclusive, are placed in a bag and a number is randomly selected from the bag. What is the probability it is neither a perfect square nor a perfect cube? Express your answer as a common fraction. | \frac{9}{10} |
In the figure below, all corner angles are right angles and each number represents the unit-length of the segment which is nearest to it. How many square units of area does the figure have?
[asy]
draw((0,0)--(12,0)--(12,5)--(8,5)--(8,4)--(5,4)
--(5,6)--(0,6)--(0,0));
label("6",(0,3),W);
label("5",(2.5,6),N);
label("... | 62 |
How many ways are there to cut a 1 by 1 square into 8 congruent polygonal pieces such that all of the interior angles for each piece are either 45 or 90 degrees? Two ways are considered distinct if they require cutting the square in different locations. In particular, rotations and reflections are considered distinct. | 54 |
Solve for $R$ if $\sqrt[4]{R^3} = 64\sqrt[16]{4}$. | 256 \cdot 2^{1/6} |
The perpendicular bisectors of the sides of triangle $ABC$ meet its circumcircle at points $A',$ $B',$ and $C',$ as shown. If the perimeter of triangle $ABC$ is 35 and the radius of the circumcircle is 8, then find the area of hexagon $AB'CA'BC'.$
[asy]
unitsize(2 cm);
pair A, B, C, Ap, Bp, Cp, O;
O = (0,0);
A = di... | 140 |
Eight people fit in a row on an airplane, and there are 12 rows. Only 3/4 of the seats in each row are allowed to be seated. How many seats will not be occupied on that plane? | There a total of 8 x 12 = <<8*12=96>>96 seats on that plane.
Only 8 x 3/4 = <<8*3/4=6>>6 seats are allowed to be seated in each row.
So, only 6 x 12 = <<6*12=72>>72 seats are allowed to be seated on the plane.
Therefore, 96 - 72 = <<96-72=24>>24 seats will not be occupied.
#### 24 |
What is the value of $b$ if $-x^2+bx-5<0$ only when $x\in (-\infty, 1)\cup(5,\infty)$? | 6 |
Given that \( a \) is an integer, if \( 50! \) is divisible by \( 2^a \), find the largest possible value of \( a \). | 47 |
There are 12 more green apples than red apples in a bowl. There are 16 red apples. How many apples are there in the bowl? | Green = 16 + 12 = <<16+12=28>>28
Green + Red = 28 + 16 = <<28+16=44>>44
There are 44 apples in the bowl.
#### 44 |
Given the sequence $\{a_n\}$ that satisfies $a_2=102$, $a_{n+1}-a_{n}=4n$ ($n \in \mathbb{N}^*$), find the minimum value of the sequence $\{\frac{a_n}{n}\}$. | 26 |
What is the value of $525^2 - 475^2$? | 50000 |
Let the first term of a geometric sequence be $\frac{3}{4}$, and let the second term be $15$. What is the smallest $n$ for which the $n$th term of the sequence is divisible by one million? | 7 |
Three congruent circles of radius $2$ are drawn in the plane so that each circle passes through the centers of the other two circles. The region common to all three circles has a boundary consisting of three congruent circular arcs. Let $K$ be the area of the triangle whose vertices are the midpoints of those arcs.... | 300 |
Sam has 3 German Shepherds and 4 French Bulldogs. Peter wants to buy 3 times as many German Shepherds as Sam has and 2 times as many French Bulldogs as Sam has. How many dogs does Peter want to have? | Peter wants to have 3 * 3 = <<3*3=9>>9 German Shepherd dogs.
He wants to have 2 * 4 = <<2*4=8>>8 French Bulldogs.
So, Peter wants to have 9 + 8 = <<9+8=17>>17 dogs.
#### 17 |
Given a solid $\Omega$ which is the larger part obtained by cutting a sphere $O$ with radius $4$ by a plane $\alpha$, and $\triangle ABC$ is an inscribed triangle of the circular section $O'$ with $\angle A=90^{\circ}$. Point $P$ is a moving point on the solid $\Omega$, and the projection of $P$ on the circle $O'$ lies... | 10 |
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