problem stringlengths 10 5.15k | answer stringlengths 0 1.23k |
|---|---|
A giant spider is discovered. It weighs 2.5 times the previous largest spider, which weighed 6.4 ounces. Each of its legs has a cross-sectional area of .5 square inches. How much pressure in ounces per square inch does each leg undergo? | The spider weighs 6.4*2.5=<<6.4*2.5=16>>16 ounces
So each leg supports a weight of 16/8=<<16/8=2>>2 ounces
That means each leg undergoes a pressure of 2/.5=<<2/.5=4>>4 ounces per square inch
#### 4 |
What is the greatest possible sum of the digits in the base-eight representation of a positive integer less than $1728$? | 23 |
Find the area enclosed by the graph \( x^{2}+y^{2}=|x|+|y| \) on the \( xy \)-plane. | \pi + 2 |
Among all the five-digit numbers formed without repeating any of the digits 0, 1, 2, 3, 4, if they are arranged in ascending order, determine the position of the number 12340. | 10 |
At McDonald's restaurants, we can order Chicken McNuggets in packages of 6, 9, or 20 pieces. (For example, we can order 21 pieces because $21=6+6+9$, but there is no way to get 19 pieces.) What is the largest number of pieces that we cannot order? | 43 |
The area of a square plot of land is 325 square meters. What is the perimeter of the square, in meters? Express your answer in simplest radical form. | 20\sqrt{13} |
Four rectangular strips each measuring $4$ by $16$ inches are laid out with two vertical strips crossing two horizontal strips forming a single polygon which looks like a tic-tack-toe pattern. What is the perimeter of this polygon?
[asy]
size(100);
draw((1,0)--(2,0)--(2,1)--(3,1)--(3,0)--(4,0)--(4,1)--(5,1)--(5,2)--(4,2)--(4,3)--(5,3)--(5,4)--(4,4)--(4,5)--(3,5)--(3,4)--(2,4)--(2,5)--(1,5)--(1,4)--(0,4)--(0,3)--(1,3)--(1,2)--(0,2)--(0,1)--(1,1)--(1,0));
draw((2,2)--(2,3)--(3,3)--(3,2)--cycle);
[/asy] | 80 |
Find the smallest integer satisfying the following conditions:
$\bullet$ I. The sum of the squares of its digits is $85$.
$\bullet$ II. Each digit is larger than the one on its left.
What is the product of the digits of this integer? | 18 |
Will stands at a point \(P\) on the edge of a circular room with perfectly reflective walls. He shines two laser pointers into the room, forming angles of \(n^{\circ}\) and \((n+1)^{\circ}\) with the tangent at \(P\), where \(n\) is a positive integer less than 90. The lasers reflect off of the walls, illuminating the points they hit on the walls, until they reach \(P\) again. (\(P\) is also illuminated at the end.) What is the minimum possible number of illuminated points on the walls of the room? | 28 |
How many four-digit numbers divisible by 17 are also even? | 265 |
Integers a, b, c, d, and e satisfy the following three properties:
(i) $2 \le a < b <c <d <e <100$
(ii) $ \gcd (a,e) = 1 $ (iii) a, b, c, d, e form a geometric sequence.
What is the value of c? | 36 |
For positive integers $m$ and $n$ such that $m+10<n+1$, both the mean and the median of the set $\{m, m+4, m+10, n+1, n+2, 2n\}$ are equal to $n$. What is $m+n$? | 21 |
Compute $(1 + i)^4.$ | -4 |
For what value of $n$ is the five-digit number $\underline{7n933}$ divisible by 33? (Note: the underlining is meant to indicate that the number should be interpreted as a five-digit number whose ten thousands digit is 7, whose thousands digit is $n$, and so on). | 5 |
In a group of people, there are 13 who like apples, 9 who like blueberries, 15 who like cantaloupe, and 6 who like dates. (A person can like more than 1 kind of fruit.) Each person who likes blueberries also likes exactly one of apples and cantaloupe. Each person who likes cantaloupe also likes exactly one of blueberries and dates. Find the minimum possible number of people in the group. | 22 |
An artist spends 30 hours every week painting. If it takes her 3 hours to complete a painting, how many paintings can she make in four weeks? | She completes 30 hours/week / 3 hours/painting = <<30/3=10>>10 paintings/week
So in one month, she paints 10 paintings/week * 4 weeks/month = <<10*4=40>>40 paintings
#### 40 |
Given $\triangle ABC$, where the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and it satisfies $a\cos 2C+2c\cos A\cos C+a+b=0$.
$(1)$ Find the size of angle $C$;
$(2)$ If $b=4\sin B$, find the maximum value of the area $S$ of $\triangle ABC$. | \sqrt{3} |
Consider a regular polygon with $2^n$ sides, for $n \ge 2$ , inscribed in a circle of radius $1$ . Denote the area of this polygon by $A_n$ . Compute $\prod_{i=2}^{\infty}\frac{A_i}{A_{i+1}}$ | \frac{2}{\pi} |
There are exactly $N$ distinct rational numbers $k$ such that $|k|<200$ and $5x^2+kx+12=0$ has at least one integer solution for $x$. What is $N$? | 78 |
Number $a$ is such that $\forall a_1, a_2, a_3, a_4 \in \mathbb{R}$, there are integers $k_1, k_2, k_3, k_4$ such that $\sum_{1 \leq i < j \leq 4} ((a_i - k_i) - (a_j - k_j))^2 \leq a$. Find the minimum of $a$. | 1.25 |
Steve spends 1/3 of the day sleeping, 1/6 of the day in school, 1/12 of the day making assignments, and the rest of the day with his family. How many hours does Steve spend with his family in a day? | Steve spends 24/3 = <<24/3=8>>8 hours sleeping.
He spends 24/6 = <<24/6=4>>4 hours in school.
He spends 24/12 = <<24/12=2>>2 hours making assignments.
He spends 8 + 4 + 2 = <<8+4+2=14>>14 hours either sleeping, going to school, or making assignments.
Hence, Steve spends 24 - 14 = <<24-14=10>>10 hours with his family.
#### 10 |
In a classroom, 34 students are seated in 5 rows of 7 chairs. The place at the center of the room is unoccupied. A teacher decides to reassign the seats such that each student will occupy a chair adjacent to his/her present one (i.e. move one desk forward, back, left or right). In how many ways can this reassignment be made? | 0 |
The ratio of the areas of two squares is $\frac{32}{63}$. After rationalizing the denominator, the ratio of their side lengths can be expressed in the simplified form $\frac{a\sqrt{b}}{c}$ where $a$, $b$, and $c$ are integers. What is the value of the sum $a+b+c$? | 39 |
Given complex numbers $z_1 = \cos\theta - i$ and $z_2 = \sin\theta + i$, the maximum value of the real part of $z_1 \cdot z_2$ is \_\_\_\_\_\_, and the maximum value of the imaginary part is \_\_\_\_\_\_. | \sqrt{2} |
If the six digits 1, 2, 3, 5, 5 and 8 are randomly arranged into a six-digit positive integer, what is the probability that the integer is divisible by 15? Express your answer as a common fraction. | \frac{1}{3} |
In triangle $\triangle ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and $\left(a-c\right)\left(a+c\right)\sin C=c\left(b-c\right)\sin B$.
$(1)$ Find angle $A$;
$(2)$ If the area of $\triangle ABC$ is $\sqrt{3}$, $\sin B\sin C=\frac{1}{4}$, find the value of $a$. | 2\sqrt{3} |
Consider a hyperbola with the equation $x^2 - y^2 = 9$. A line passing through the left focus $F_1$ of the hyperbola intersects the left branch of the hyperbola at points $P$ and $Q$. Let $F_2$ be the right focus of the hyperbola. If the length of segment $PQ$ is 7, then calculate the perimeter of $\triangle F_2PQ$. | 26 |
Tyler weighs 25 pounds more than Sam. Peter weighs half as much as Tyler. If Peter weighs 65 pounds, how much does Sam weigh, in pounds? | Tyler weighs 65*2=<<65*2=130>>130 pounds.
Sam weighs 130-25=<<130-25=105>>105 pounds.
#### 105 |
Let \( g : \mathbb{R} \to \mathbb{R} \) be a function such that
\[ g(g(x) + y) = g(x) + g(g(y) + g(-x)) - x \] for all real numbers \( x \) and \( y \).
Let \( m \) be the number of possible values of \( g(4) \), and let \( t \) be the sum of all possible values of \( g(4) \). Find \( m \times t \). | -4 |
Given the parametric equation of line $l$ as $$\begin{cases} \left.\begin{matrix}x= \frac {1}{2}t \\ y=1+ \frac { \sqrt {3}}{2}t\end{matrix}\right.\end{cases}$$ (where $t$ is the parameter), and the polar equation of curve $C$ as $\rho=2 \sqrt {2}\sin(\theta+ \frac {\pi}{4})$, line $l$ intersects curve $C$ at points $A$ and $B$, and intersects the $y$-axis at point $P$.
(1) Find the standard equation of line $l$ and the Cartesian coordinate equation of curve $C$;
(2) Calculate the value of $\frac {1}{|PA|} + \frac {1}{|PB|}$. | \sqrt {5} |
The number of diagonals of a regular polygon is subtracted from the number of sides of the polygon and the result is zero. What is the number of sides of this polygon? | 5 |
Complex numbers $p, q, r$ form an equilateral triangle with side length 24 in the complex plane. If $|p + q + r| = 48,$ find $|pq + pr + qr|.$ | 768 |
Let $x,$ $y,$ and $z$ be nonnegative real numbers such that $x + y + z = 2.$ Find the maximum value of
\[(x^2 - xy + y^2)(x^2 - xz + z^2)(y^2 - yz + z^2).\] | \frac{256}{243} |
Consider the line parameterized by
\begin{align*}
x&= 4t + 2,\\
y& = t+2.
\end{align*}Find a vector $\begin{pmatrix}a \\ b \end{pmatrix}$ pointing from the origin to this line that is parallel to $\begin{pmatrix}2 \\1 \end{pmatrix}$. | \begin{pmatrix}6\\3\end{pmatrix} |
Jamestown has 20 theme parks. If Venice has 25 more theme parks than Jamestown, and Marina Del Ray has 50 more theme parks than Jamestown, calculate the number of theme parks present in the three towns. | Since Venice has 25 more theme parks than Jamestown, there are 20+25 = <<20+25=45>>45 theme parks in Venice.
Together, Venice and Jamestown have 45+20 = <<45+20=65>>65 theme parks
With 50 more theme parks than Jamestown, Marina Del Ray has 50+20 = <<50+20=70>>70 theme parks.
The total number of theme parks in the three towns is 70+65 = <<70+65=135>>135
#### 135 |
Cory has $4$ apples, $2$ oranges, and $1$ banana. If Cory eats one piece of fruit per day for a week, and must consume at least one apple before any orange, how many different orders can Cory eat these fruits? The fruits within each category are indistinguishable. | 105 |
If $f(x) = -7x^4 + 3x^3 + x - 5$, and $g(x)$ is a polynomial such that the degree of $f(x) + g(x)$ is 1, then what is the degree of $g(x)$? | 4 |
For how many positive integers $n$ less than or equal to 500 is $$(\sin (t+\frac{\pi}{4})+i\cos (t+\frac{\pi}{4}))^n=\sin (nt+\frac{n\pi}{4})+i\cos (nt+\frac{n\pi}{4})$$ true for all real $t$? | 125 |
Find the maximum value of the expression
$$
\left(x_{1}-x_{2}\right)^{2}+\left(x_{2}-x_{3}\right)^{2}+\ldots+\left(x_{2010}-x_{2011}\right)^{2}+\left(x_{2011}-x_{1}\right)^{2}
$$
where \(x_{1}, \ldots, x_{2011} \in [0, 1]\). | 2010 |
For real numbers $x$, let \[f(x) = \left\{
\begin{array}{cl}
x+2 &\text{ if }x>3, \\
2x+a &\text{ if }x\le 3.
\end{array}
\right.\]What must the value of $a$ be to make the piecewise function continuous (which means that its graph can be drawn without lifting your pencil from the paper)? | -1 |
Let \( ABC \) be a triangle such that \( AB = 2 \), \( CA = 3 \), and \( BC = 4 \). A semicircle with its diameter on \(\overline{BC}\) is tangent to \(\overline{AB}\) and \(\overline{AC}\). Compute the area of the semicircle. | \frac{27 \pi}{40} |
Harry and Terry are each told to calculate $8-(2+5)$. Harry gets the correct answer. Terry ignores the parentheses and calculates $8-2+5$. If Harry's answer is $H$ and Terry's answer is $T$, what is $H-T$? | -10 |
If set $A=\{-4, 2a-1, a^2\}$, $B=\{a-5, 1-a, 9\}$, and $A \cap B = \{9\}$, then the value of $a$ is. | -3 |
A square 10cm on each side has four quarter circles drawn with centers at the four corners. How many square centimeters are in the area of the shaded region? Express your answer in terms of $\pi$.
[asy]
unitsize (1.5 cm);
draw((-1,-1)--(1,-1)--(1,1)--(-1,1)--cycle);
filldraw(arc((1,1),1,270,180)--arc((-1,1),1,360,270)--arc((-1,-1),1,90,0)--arc((1,-1),1,180,90)--cycle,gray);
[/asy] | 100-25\pi |
Let set $M=\{-1, 0, 1\}$, and set $N=\{a, a^2\}$. Find the real number $a$ such that $M \cap N = N$. | -1 |
The graph of the function $f(x)=\sin(2x+\varphi)$ is translated to the right by $\frac{\pi}{6}$ units, and the resulting graph is symmetric about the origin. Determine the value of $\varphi$. | \frac{\pi}{3} |
How many degrees are in each interior angle of a regular hexagon? | 120^\circ |
There are $2006$ students and $14$ teachers in a school. Each student knows at least one teacher (knowing is a symmetric relation). Suppose that, for each pair of a student and a teacher who know each other, the ratio of the number of the students whom the teacher knows to that of the teachers whom the student knows is at least $t.$ Find the maximum possible value of $t.$ | 143 |
Ruth prepared sandwiches. She ate 1 sandwich and gave 2 sandwiches to her brother. Her first cousin arrived and ate 2 sandwiches. Then her two other cousins arrived and ate 1 sandwich each. There were 3 sandwiches left. How many sandwiches did Ruth prepare? | Ruth's two cousins ate a total of 2 x 1 = <<2*1=2>>2 sandwiches.
So there were 3 + 2 = <<3+2=5>>5 sandwiches left before her two cousins arrived.
Before her first cousin arrived, there were 5 + 2 = <<5+2=7>>7 sandwiches.
Ruth and her brother had 1 + 2 = <<1+2=3>>3 sandwiches altogether.
Therefore, Ruth prepared 7 + 3 = <<7+3=10>>10 sandwiches."
#### 10 |
289. A remarkable number. Find a number such that its fractional part, its integer part, and the number itself form a geometric progression. | \frac{1+\sqrt{5}}{2} |
Find the positive value of $t$ that satisfies $ab = t-2i$ given $|a|=2$ and $|b|=\sqrt{26}$. | 10 |
Fully simplify the following expression: $[(2+3+4+5)\div2] + [(2\cdot5+8)\div3]$. | 13 |
Given that the merchant purchased $1200$ keychains at $0.15$ each and desired to reach a target profit of $180$, determine the minimum number of keychains the merchant must sell if each is sold for $0.45$. | 800 |
Let $r_1,$ $r_2,$ and $r_3$ be the roots of
\[x^3 - 3x^2 + 8 = 0.\]Find the monic polynomial, in $x,$ whose roots are $2r_1,$ $2r_2,$ and $2r_3.$ | x^3 - 6x^2 + 64 |
What are the last 8 digits of $$11 \times 101 \times 1001 \times 10001 \times 100001 \times 1000001 \times 111 ?$$ | 19754321 |
Jack makes his own cold brew coffee. He makes it in batches of 1.5 gallons. He drinks 96 ounces of coffee every 2 days. It takes 20 hours to make coffee. How long does he spend making coffee over 24 days? | He makes it in batches of 1.5*128=<<1.5*128=192>>192 ounces
He drinks 96/2=<<96/2=48>>48 ounces of coffee a day
So he drinks the coffee in 192/48=<<192/48=4>>4 days
So in 24 days he makes 24/4=<<24/4=6>>6 batches
That means he spends 6*20=<<6*20=120>>120 hours making coffee
#### 120 |
Compute $\sin 240^\circ$. | -\frac{\sqrt{3}}{2} |
Compute $\arcsin \left( -\frac{\sqrt{3}}{2} \right).$ Express your answer in radians. | -\frac{\pi}{3} |
Kekai is running a sundae booth at the carnival. On Monday, he makes a total of 40 sundaes, and he puts 6 m&ms on each sundae. On Tuesday, he makes a total of 20 sundaes, and he puts 10 m&ms on each sundae. If each m&m pack contains 40 m&ms, how many m&m packs does Kekai use? | On Monday, Kekai uses 40 * 6 = <<40*6=240>>240 m&ms
On Tuesday, Kekai uses 20 * 10 = <<20*10=200>>200 m&ms
Kekai uses a total of 240 + 200 = <<240+200=440>>440 m&ms
Kekai uses a 440 / 40 = <<440/40=11>>11 m&m packs
#### 11 |
Find the units digit of $13 \cdot 41$. | 3 |
Micah drank 1.5 liters of water in the morning. Then she drank three times that much water in the afternoon. How many liters of water did she drink from the morning until the afternoon? | Micah drank 1.5 x 3 = <<1.5*3=4.5>>4.5 liters in the afternoon.
She drank a total of 1.5 + 4.5 = <<1.5+4.5=6>>6 liters of water.
#### 6 |
In the polar coordinate system, the polar equation of curve \\(C\\) is given by \\(\rho = 6\sin \theta\\). The polar coordinates of point \\(P\\) are \\((\sqrt{2}, \frac{\pi}{4})\\). Taking the pole as the origin and the positive half-axis of the \\(x\\)-axis as the polar axis, a Cartesian coordinate system is established.
\\((1)\\) Find the Cartesian equation of curve \\(C\\) and the Cartesian coordinates of point \\(P\\);
\\((2)\\) A line \\(l\\) passing through point \\(P\\) intersects curve \\(C\\) at points \\(A\\) and \\(B\\). If \\(|PA| = 2|PB|\\), find the value of \\(|AB|\\). | 3\sqrt{2} |
How many perfect squares less than 5000 have a ones digit of 4, 5, or 6? | 36 |
Given $f(x)=\sin \left( 2x+ \frac{π}{6} \right)+ \frac{3}{2}$, $x\in R$.
(1) Find the minimum positive period of the function $f(x)$;
(2) Find the interval(s) where the function $f(x)$ is monotonically decreasing;
(3) Find the maximum value of the function and the corresponding $x$ value(s). | \frac{5}{2} |
Among five numbers, if we take the average of any four numbers and add the remaining number, the sums will be 74, 80, 98, 116, and 128, respectively. By how much is the smallest number less than the largest number among these five numbers? | 72 |
How many positive integers $n$ less than 100 have a corresponding integer $m$ divisible by 3 such that the roots of $x^2-nx+m=0$ are consecutive positive integers? | 32 |
What is the ratio of $x$ to $y$ if: $\frac{10x-3y}{13x-2y} = \frac{3}{5}$? Express your answer as a common fraction. | \frac{9}{11} |
The perimeter of a rectangle is 48. What is the largest possible area of the rectangle? | 144 |
Determine the smallest integer $B$ such that there exist several consecutive integers, including $B$, that add up to 2024. | -2023 |
What is the units digit of the sum of the nine terms of the sequence $1! + 1, \, 2! + 2, \, 3! + 3, \, ..., \, 8! + 8, \, 9! + 9$? | 8 |
Ryan is learning number theory. He reads about the *Möbius function* $\mu : \mathbb N \to \mathbb Z$ , defined by $\mu(1)=1$ and
\[ \mu(n) = -\sum_{\substack{d\mid n d \neq n}} \mu(d) \]
for $n>1$ (here $\mathbb N$ is the set of positive integers).
However, Ryan doesn't like negative numbers, so he invents his own function: the *dubious function* $\delta : \mathbb N \to \mathbb N$ , defined by the relations $\delta(1)=1$ and
\[ \delta(n) = \sum_{\substack{d\mid n d \neq n}} \delta(d) \]
for $n > 1$ . Help Ryan determine the value of $1000p+q$ , where $p,q$ are relatively prime positive integers satisfying
\[ \frac{p}{q}=\sum_{k=0}^{\infty} \frac{\delta(15^k)}{15^k}. \]
*Proposed by Michael Kural* | 14013 |
Ittymangnark and Kingnook are an Eskimo couple living in the most northern region of the Alaskan wilderness. Together, they live with their child, Oomyapeck. Every day Ittymangnark catches enough fish for the three of them to eat for the day and they split the fish equally between the three of them. But after they have split the fish, they give Oomyapeck all of the eyes, who gives two of the eyes to his dog and eats the rest himself. How many fish will each of them be given to eat if Oomyapeck eats 22 eyes in a day? | If the dog eats two eyes and Oomyapeck eats 22 eyes, then there are a total of 2+22=<<2+22=24>>24 eyes.
With 2 eyes per fish, 24 eyes represent 24/2=<<24/2=12>>12 fish.
If they split that number of fish equally amongst the three of them, then each of them will be given 12/3=<<12/3=4>>4 fish to eat.
#### 4 |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively, and $\cos A= \frac{ \sqrt{6}}{3}$.
(1) Find $\tan 2A$;
(2) If $\cos B= \frac{ 2\sqrt{2}}{3}, c=2\sqrt{2}$, find the area of $\triangle ABC$. | \frac{2\sqrt{2}}{3} |
For any positive integer $n$, let $a_n$ be the $y$-coordinate of the intersection point between the tangent line of the curve $y=x^n(1-x)$ at $x=2$ and the $y$-axis in the Cartesian coordinate system. Calculate the sum of the first 10 terms of the sequence $\{\log_2 \frac{a_n}{n+1}\}$. | 55 |
The product of the first three terms of a geometric sequence is 2, the product of the last three terms is 4, and the product of all terms is 64. Calculate the number of terms in this sequence. | 12 |
Rob has some baseball cards, and a few are doubles. One third of Rob's cards are doubles, and Jess has 5 times as many doubles as Rob. If Jess has 40 doubles baseball cards, how many baseball cards does Rob have? | Rob has 40/5=<<40/5=8>>8 doubles baseball cards.
Rob has 8*3=<<8*3=24>>24 baseball cards.
#### 24 |
What is the tenth positive integer that is both odd and a multiple of 3? | 57 |
Points \(A\) and \(B\) are connected by two arcs of circles, convex in opposite directions: \(\cup A C B = 117^\circ 23'\) and \(\cup A D B = 42^\circ 37'\). The midpoints \(C\) and \(D\) of these arcs are connected to point \(A\). Find the angle \(C A D\). | 80 |
If $78$ is divided into three parts which are proportional to $1, \frac{1}{3}, \frac{1}{6},$ the middle part is: | 17\frac{1}{3} |
Find \( g\left(\frac{1}{1996}\right) + g\left(\frac{2}{1996}\right) + g\left(\frac{3}{1996}\right) + \cdots + g\left(\frac{1995}{1996}\right) \) where \( g(x) = \frac{9x}{3 + 9x} \). | 997 |
Suppose $f(x),g(x),h(x)$ are all linear functions, and $j(x)$ and $k(x)$ are defined by $$j(x) = \max\{f(x),g(x),h(x)\},$$$$k(x) = \min\{f(x),g(x),h(x)\}.$$This means that, for each $x$, we define $j(x)$ to be equal to either $f(x),$ $g(x),$ or $h(x),$ whichever is greatest; similarly, $k(x)$ is the least of these three values.
Shown below is the graph of $y=j(x)$ for $-3.5\le x\le 3.5$.
Let $\ell$ be the length of the graph of $y=k(x)$ for $-3.5\le x\le 3.5$. What is the value of $\ell^2$?
[asy]
size(150);
real ticklen=3;
real tickspace=2;
real ticklength=0.1cm;
real axisarrowsize=0.14cm;
pen axispen=black+1.3bp;
real vectorarrowsize=0.2cm;
real tickdown=-0.5;
real tickdownlength=-0.15inch;
real tickdownbase=0.3;
real wholetickdown=tickdown;
void rr_cartesian_axes(real xleft, real xright, real ybottom, real ytop, real xstep=1, real ystep=1, bool useticks=false, bool complexplane=false, bool usegrid=true) {
import graph;
real i;
if(complexplane) {
label("$\textnormal{Re}$",(xright,0),SE);
label("$\textnormal{Im}$",(0,ytop),NW);
} else {
label("$x$",(xright+0.4,-0.5));
label("$y$",(-0.5,ytop+0.2));
}
ylimits(ybottom,ytop);
xlimits( xleft, xright);
real[] TicksArrx,TicksArry;
for(i=xleft+xstep; i<xright; i+=xstep) {
if(abs(i) >0.1) {
TicksArrx.push(i);
}
}
for(i=ybottom+ystep; i<ytop; i+=ystep) {
if(abs(i) >0.1) {
TicksArry.push(i);
}
}
if(usegrid) {
xaxis(BottomTop(extend=false), Ticks("%", TicksArrx ,pTick=gray(0.22),extend=true),p=invisible);//,above=true);
yaxis(LeftRight(extend=false),Ticks("%", TicksArry ,pTick=gray(0.22),extend=true), p=invisible);//,Arrows);
}
if(useticks) {
xequals(0, ymin=ybottom, ymax=ytop, p=axispen, Ticks("%",TicksArry , pTick=black+0.8bp,Size=ticklength), above=true, Arrows(size=axisarrowsize));
yequals(0, xmin=xleft, xmax=xright, p=axispen, Ticks("%",TicksArrx , pTick=black+0.8bp,Size=ticklength), above=true, Arrows(size=axisarrowsize));
} else {
xequals(0, ymin=ybottom, ymax=ytop, p=axispen, above=true, Arrows(size=axisarrowsize));
yequals(0, xmin=xleft, xmax=xright, p=axispen, above=true, Arrows(size=axisarrowsize));
}
};
rr_cartesian_axes(-5,5,-5,5);
draw((-3.5,5)--(-2,2)--(2,2)--(3.5,5),red+1.25);
dot((-2,2),red);
dot((2,2),red);
[/asy] | 245 |
What is the ratio of the area of a square inscribed in a semicircle with radius $r$ to the area of a square inscribed in a circle with radius $r$? Express your answer as a common fraction. | \dfrac{2}{5} |
In trapezoid \(ABCD\), \(AD\) is parallel to \(BC\). If \(AD = 52\), \(BC = 65\), \(AB = 20\), and \(CD = 11\), find the area of the trapezoid. | 594 |
Find the center of the circle with equation $x^2 - 2x + y^2 - 4y - 28 = 0$. | (1, 2) |
A cuboid with dimensions corresponding to length twice the cube's edge is painted with stripes running from the center of one edge to the center of the opposite edge, on each of its six faces. Each face's stripe orientation (either horizontal-center or vertical-center) is chosen at random. What is the probability that there is at least one continuous stripe that encircles the cuboid horizontally?
A) $\frac{1}{64}$
B) $\frac{1}{32}$
C) $\frac{1}{16}$
D) $\frac{1}{8}$
E) $\frac{1}{4}$ | \frac{1}{16} |
The graph of $xy = 1$ is a hyperbola. Find the distance between the foci of this hyperbola. | 4 |
An ellipse \( \frac{x^2}{9} + \frac{y^2}{16} = 1 \) has chords passing through the point \( C = (2, 2) \). If \( t \) is defined as
\[
t = \frac{1}{AC} + \frac{1}{BC}
\]
where \( AC \) and \( BC \) are distances from \( A \) and \( B \) (endpoints of the chords) to \( C \). Find the constant \( t \). | \frac{4\sqrt{5}}{5} |
The sum of the first $2011$ terms of a geometric sequence is $200$. The sum of the first $4022$ terms is $380$. Find the sum of the first $6033$ terms. | 542 |
How many non-congruent right triangles with positive integer leg lengths have areas that are numerically equal to $3$ times their perimeters? | 7 |
Jane is painting her fingernails. She applies a base coat that takes 2 minutes to dry, two color coats that take 3 minutes each to dry, and a clear top coat that takes 5 minutes to dry. How many minutes total does Jane spend waiting for her nail polish to dry? | First figure out how long both color coats will take to dry: 3 minutes * 2 = <<3*2=6>>6 minutes
Then add up the time for the base coat, top coat, and color coats: 6 minutes + 2 minutes + 5 minutes = <<6+2+5=13>>13 minutes
#### 13 |
What is the least positive integer divisible by each of the first eight positive integers? | 840 |
If $\mathbf{a}$ and $\mathbf{b}$ are vectors such that $\|\mathbf{a}\| = 7$ and $\|\mathbf{b}\| = 11$, then find all possible values of $\mathbf{a} \cdot \mathbf{b}$.
Submit your answer in interval notation. | [-77,77] |
Find the product of the greatest common divisor (gcd) and the least common multiple (lcm) of 225 and 252. | 56700 |
Given integers $a_{1}$, $a_{2}$, $a_{3}$, $a_{4}$, $\ldots $ satisfy the following conditions: $a_{1}=0$, $a_{2}=-|a+1|$, $a_{3}=-|a_{2}+2|$, $a_{4}=-|a_{3}+3|$, and so on, then the value of $a_{2022}$ is ____. | -1011 |
In the diagram below, $BC$ is 8 cm. In square centimeters, what is the area of triangle $ABC$?
[asy]
defaultpen(linewidth(0.7));
draw((0,0)--(16,0)--(23,20)--cycle);
draw((16,0)--(23,0)--(23,20),dashed);
label("8 cm",(8,0),S);
label("10 cm",(23,10),E);
label("$A$",(23,20),N);
label("$B$",(0,0),SW);
label("$C$",(16,0),SE);
[/asy] | 40 |
The numbers 2, 4, 6, and 8 are a set of four consecutive even numbers. Suppose the sum of five consecutive even numbers is 320. What is the smallest of the five numbers? | 60 |
What is the base $2$ representation of $84_{10}$? | 1010100_2 |
How many whole numbers are between $\sqrt[3]{10}$ and $\sqrt[3]{200}$? | 3 |
A right triangle has legs of lengths 126 and 168 units. What is the perimeter of the triangle formed by the points where the angle bisectors intersect the opposite sides? | 230.61 |
Compute the perimeter of the triangle that has area $3-\sqrt{3}$ and angles $45^\circ$ , $60^\circ$ , and $75^\circ$ . | 3\sqrt{2} + 2\sqrt{3} - \sqrt{6} |
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