problem stringlengths 10 5.15k | answer stringlengths 0 1.23k |
|---|---|
Angle $EAB$ is a right angle, and $BE=9$ units. What is the number of square units in the sum of the areas of the two squares $ABCD$ and $AEFG$?
[asy]
draw((0,0)--(1,1)--(0,2)--(-1,1)--cycle);
draw((0,2)--(2,4)--(0,6)--(-2,4)--cycle);
draw((1,1)--(2,4));
draw((-1,1)--(-2,4));
label("A", (0,2), S);
label("B", (1,1), S... | 81 |
Given the equation $x^3 - 12x^2 + 27x - 18 = 0$ with roots $a$, $b$, $c$, find the value of $\frac{1}{a^3} + \frac{1}{b^3} + \frac{1}{c^3}$. | \frac{13}{24} |
There is a wooden stick 240 cm long. First, starting from the left end, a line is drawn every 7 cm. Then, starting from the right end, a line is drawn every 6 cm. The stick is cut at each marked line. How many of the resulting smaller sticks are 3 cm long? | 12 |
Barbara went shopping in a supermarket. She bought 5 packs of tuna for $2 each and 4 bottles of water for $1.5 each. In total, she paid $56 for her shopping. How much did Barbara spend on different than the mentioned goods? | For the tuna Barbara needed to pay 5 * 2 = $<<5*2=10>>10.
For the four bottles of water, she needed to pay 4 * 1.5 = $<<4*1.5=6>>6.
On different goods Barbara spend 56 - 10 - 6 = $<<56-10-6=40>>40.
#### 40 |
For real numbers $t,$ the point
\[(x,y) = \left( \frac{1 - t^2}{1 + t^2}, \frac{2t}{1 + t^2} \right)\]is plotted. All the plotted points lie on what kind of curve?
(A) Line
(B) Circle
(C) Parabola
(D) Ellipse
(E) Hyperbola
Enter the letter of the correct option. | \text{(B)} |
Box $A$ contains 1 red ball and 5 white balls, and box $B$ contains 3 white balls. Three balls are randomly taken from box $A$ and placed into box $B$. After mixing thoroughly, three balls are then randomly taken from box $B$ and placed back into box $A$. What is the probability that the red ball moves from box $A$ to ... | 1/4 |
If $f(x)=\dfrac{x+1}{3x-4}$, what is the value of $f(5)$? | \dfrac{6}{11} |
Chen, Ruan, Lu, Tao, and Yang did push-ups. It is known that Chen, Lu, and Yang together averaged 40 push-ups per person, Ruan, Tao, and Chen together averaged 28 push-ups per person, and Ruan, Lu, Tao, and Yang together averaged 33 push-ups per person. How many push-ups did Chen do? | 36 |
Let $T_n$ be the sum of the reciprocals of the non-zero digits of the integers from 1 to $16^n$ inclusive, considering a hexadecimal system in which digits range from 1 to 15. Find the integer $n$ for which $T_n$ becomes an integer. | 15015 |
Let $\mathcal{S}$ be the set of real numbers that can be represented as repeating decimals of the form $0.\overline{abc}$ where $a, b, c$ are distinct digits. Find the sum of the elements of $\mathcal{S}.$
| 360 |
In the figure, if $A E=3, C E=1, B D=C D=2$, and $A B=5$, find $A G$. | 3\sqrt{66} / 7 |
If $a_0 = \sin^2 \left( \frac{\pi}{45} \right)$ and
\[a_{n + 1} = 4a_n (1 - a_n)\]for $n \ge 0,$ find the smallest positive integer $n$ such that $a_n = a_0.$ | 12 |
Mr Castiel prepared 600 boar sausages on the weekend. He ate 2/5 of them on Monday, half of the remaining on Tuesday and saved the rest for Friday. If he ate 3/4 of the remaining boar sausages on Friday, how many sausages are left? | On Monday, Mr Castiel ate 2/5*600 = <<2/5*600=240>>240 sausages.
The remaining number of sausages after he ate 240 on Monday is 600-240 = <<600-240=360>>360
On Tuesday, he ate half of the remaining sausages, a total of 360/2 = <<360/2=180>>180.
The remaining number of sausages that he saved for Friday after eating 180 ... |
In a multiplication error involving two positive integers $a$ and $b$, Ron mistakenly reversed the digits of the three-digit number $a$. The erroneous product obtained was $396$. Determine the correct value of the product $ab$. | 693 |
Tina is working on her homework when she realizes she's having a hard time typing out her answers on her laptop because a lot of the keys are sticky. She is trying to get her homework done before dinner, though, so she needs to decide if she has time to clean her keyboard first. Tina knows her assignment will only take... | Tina has already cleaned one key so she has 14 left which take 3 minutes each to clean, 14 x 3 = <<14*3=42>>42 minutes to clean all the keyboard keys.
Her assignment will take 10 minutes to complete, so she needs 42 minutes + 10 minutes = 52 minutes total before dinner.
#### 52 |
A square and a regular pentagon have the same perimeter. Let $C$ be the area of the circle circumscribed about the square, and $D$ the area of the circle circumscribed around the pentagon. Find $C/D$.
A) $\frac{25}{128}$
B) $\frac{25(5 + 2\sqrt{5})}{128}$
C) $\frac{25(5-2\sqrt{5})}{128}$
D) $\frac{5\sqrt{5}}{128}$ | \frac{25(5-2\sqrt{5})}{128} |
In a shop, there is a sale of clothes. Every shirt costs $5, every hat $4, and a pair of jeans $10. How much do you need to pay for three shirts, two pairs of jeans, and four hats? | Three shirts cost 3 * 5 = $<<3*5=15>>15.
Two pairs of jeans would cost 2 * 10 = $<<2*10=20>>20.
Four hats would in total cost 4 * 4 = $<<4*4=16>>16.
So for three shirts, two pairs of jeans, and four hats, you would need to pay 15 + 20 + 16 = $<<15+20+16=51>>51.
#### 51 |
In the Cartesian coordinate system, given points A(1, -3), B(4, -1), P(a, 0), and N(a+1, 0), if the perimeter of the quadrilateral PABN is minimal, then find the value of a. | a = \frac{5}{2} |
Given that $\frac {π}{2}＜α＜ \frac {3π}{2}$, points A, B, and C are in the same plane rectangular coordinate system with coordinates A(3, 0), B(0, 3), and C(cosα, sinα) respectively.
(1) If $| \overrightarrow {AC}|=| \overrightarrow {BC}|$, find the value of angle α;
(2) When $\overrightarrow {AC}\cdot \overrightarrow {... | - \frac {5}{9} |
Let $C$ be a point not on line $AE$ and $D$ a point on line $AE$ such that $CD \perp AE.$ Meanwhile, $B$ is a point on line $CE$ such that $AB \perp CE.$ If $AB = 4,$ $CD = 8,$ and $AE = 5,$ then what is the length of $CE?$ | 10 |
The area of the base of a hemisphere is $144\pi$. A cylinder of the same radius as the hemisphere and height equal to the radius of the hemisphere is attached to its base. What is the total surface area of the combined solid (hemisphere + cylinder)? | 576\pi |
How many integers between 1000 and 2000 have all three of the numbers 15, 20 and 25 as factors? | 3 |
A store increased the price of a certain Super VCD by 40% and then advertised a "10% discount and a free 50 yuan taxi fare" promotion. As a result, each Super VCD still made a profit of 340 yuan. What was the cost price of each Super VCD? | 1500 |
For integers $a,b,c$ and $d,$ let $f(x)=x^2+ax+b$ and $g(x)=x^2+cx+d.$ Find the number of ordered triples $(a,b,c)$ of integers with absolute values not exceeding $10$ for which there is an integer $d$ such that $g(f(2))=g(f(4))=0.$ | 510 |
John's camera broke so he decided to rent one for 4 weeks. It was a $5000 camera and the rental fee was 10% of the value per week. His friend who was there when it broke agreed to pay 40% of the rental fee. How much did John pay? | The rental fee is 5000*.1=$<<5000*.1=500>>500 per week
So it cost 500*4=$<<500*4=2000>>2000
His friend pays 2000*.4=$<<2000*.4=800>>800
So he pays 2000-800=$<<2000-800=1200>>1200
#### 1200 |
A company's capital increases by a factor of two each year compared to the previous year after dividends have been paid, with a fixed dividend of 50 million yuan paid to shareholders at the end of each year. The company's capital after dividends were paid at the end of 2010 was 1 billion yuan.
(i) Find the capital of ... | 2017 |
One interior angle in a triangle measures $50^{\circ}$. What is the angle between the bisectors of the remaining two interior angles? | 65 |
My friend June likes numbers that have an interesting property: they are divisible by 4. How many different pairings of last two digits are possible in numbers that June likes? | 25 |
For certain ordered pairs $(a,b)\,$ of real numbers, the system of equations
\[\begin{aligned} ax+by&=1 \\ x^2 + y^2 &= 50 \end{aligned}\]has at least one solution, and each solution is an ordered pair $(x,y)\,$ of integers. How many such ordered pairs $(a,b)\,$ are there? | 72 |
Let the isosceles right triangle $ABC$ with $\angle A= 90^o$ . The points $E$ and $F$ are taken on the ray $AC$ so that $\angle ABE = 15^o$ and $CE = CF$ . Determine the measure of the angle $CBF$ . | 15 |
Randy had 32 biscuits. His father gave him 13 biscuits as a gift. His mother gave him 15 biscuits. Randy’s brother ate 20 of these biscuits. How many biscuits are Randy left with? | After his father gave him 13 biscuits, Randy has 32 + 13 = <<32+13=45>>45 biscuits.
After his mother gave him 15 biscuits, Randy has 45 + 15 = <<45+15=60>>60 biscuits.
After his brother ate 20 of the biscuits. Randy has 60 – 20 = <<60-20=40>>40 biscuits.
#### 40 |
Given that $x$ and $y$ are positive numbers, $\theta \in \left( \frac{\pi}{4}, \frac{\pi}{2} \right)$, and it satisfies $\frac{\sin\theta}{x} = \frac{\cos\theta}{y}$ and $\frac{\cos^2\theta}{x^2} + \frac{\sin^2\theta}{y^2} = \frac{10}{3(x^2+y^2)}$, determine the value of $\frac{x}{y}$. | \sqrt{3} |
The points $(2, 5), (10, 9)$, and $(6, m)$, where $m$ is an integer, are vertices of a triangle. What is the sum of the values of $m$ for which the area of the triangle is a minimum? | 14 |
In triangle \(ABC\), the side lengths are 4, 5, and \(\sqrt{17}\). Find the area of the region consisting of those and only those points \(X\) inside triangle \(ABC\) for which the condition \(XA^{2} + XB^{2} + XC^{2} \leq 21\) is satisfied. | \frac{5 \pi}{9} |
The positive integers $ a$ and $ b$ are such that the numbers $ 15a \plus{} 16b$ and $ 16a \minus{} 15b$ are both squares of positive integers. What is the least possible value that can be taken on by the smaller of these two squares? | 481^2 |
Let $X=\{2^m3^n|0 \le m, \ n \le 9 \}$ . How many quadratics are there of the form $ax^2+2bx+c$ , with equal roots, and such that $a,b,c$ are distinct elements of $X$ ? | 9900 |
What is the domain of the function $$u(x) = \frac{1}{\sqrt x}~?$$ Express your answer in interval notation. | (0,\infty) |
Calculate the smallest root \(x_0\) of the equation
$$
x^{2}-\sqrt{\lg x +100}=0
$$
(with a relative error of no more than \(10^{-390} \%\)). | 10^{-100} |
In the quadrilateral \(ABCD\), \(AB = 2\), \(BC = 4\), \(CD = 5\). Find its area given that it is both circumscribed and inscribed. | 2\sqrt{30} |
We flip a fair coin 10 times. What is the probability that we get heads in exactly 8 of the 10 flips? | \dfrac{45}{1024} |
Compute $a^2 + b^2 + c^2,$ given that $a,$ $b,$ and $c$ are the roots of \[2x^3 - x^2 + 4x + 10 = 0.\] | -\frac{15}4 |
Given 8 people are sitting around a circular table for a meeting, including one leader, one vice leader, and one recorder, and the recorder is seated between the leader and vice leader, determine the number of different seating arrangements possible, considering that arrangements that can be obtained by rotation are id... | 240 |
Points $A$, $B$, $C$, $D$, and $E$ are located in 3-dimensional space with $AB= BC= CD= DE= EA= 2$ and $\angle ABC = \angle CDE = \angle
DEA = 90^\circ$. The plane of triangle $ABC$ is parallel to $\overline{DE}$. What is the area of triangle $BDE$? | 2 |
A \(10 \times 1\) rectangular pavement is to be covered by tiles which are either green or yellow, each of width 1 and of varying integer lengths from 1 to 10. Suppose you have an unlimited supply of tiles for each color and for each of the varying lengths. How many distinct tilings of the rectangle are there, if at le... | 1022 |
A tractor is dragging a very long pipe on sleds. Gavrila walked along the entire pipe in the direction of the tractor's movement and counted 210 steps. When he walked in the opposite direction, the number of steps was 100. What is the length of the pipe if Gavrila's step is 80 cm? Round the answer to the nearest whole ... | 108 |
How many of the natural numbers from 1 to 800, inclusive, contain the digit 7 at least once? | 62 |
Convert $6351_8$ to base 7. | 12431_7 |
What is the degree of the polynomial $(4 +5x^3 +100 +2\pi x^4 + \sqrt{10}x^4 +9)$? | 4 |
Let $a_1$, $a_2$, $a_3$, $d_1$, $d_2$, and $d_3$ be real numbers such that for every real number $x$, we have
\[
x^8 - x^6 + x^4 - x^2 + 1 = (x^2 + a_1 x + d_1)(x^2 + a_2 x + d_2)(x^2 + a_3 x + d_3)(x^2 + 1).
\]
Compute $a_1 d_1 + a_2 d_2 + a_3 d_3$. | -1 |
Tony decided he wanted to be an astronaut. He went to college for 4 years to get a degree in science. He then went on to get 2 more degrees in other fields for the same period of time. He also got a graduate degree in physics, which took another 2 years. How many years in total did Tony go to school to be an astron... | Tony got 3 degrees which each took 4 years, for 3*4=<<3*4=12>>12 years spent.
He then got his graduate degree, which was another 2 years added to the previous total, for 12+2=<<12+2=14>>14 years in total.
#### 14 |
Except for the first two terms, each term of the sequence $1000, x, 1000 - x,\ldots$ is obtained by subtracting the preceding term from the one before that. The last term of the sequence is the first negative term encountered. What positive integer $x$ produces a sequence of maximum length? | 618 |
There are enough cuboids with side lengths of 2, 3, and 5. They are neatly arranged in the same direction to completely fill a cube with a side length of 90. The number of cuboids a diagonal of the cube passes through is | 65 |
Find the area of the region enclosed by the graph of $|x-60|+|y|=\left|\frac{x}{4}\right|.$ | 480 |
How many unique five-digit numbers greater than 20000, using the digits 1, 2, 3, 4, and 5 without repetition, can be formed such that the hundreds place is not the digit 3? | 78 |
A telephone station serves 400 subscribers. For each subscriber, the probability of calling the station within an hour is 0.01. Find the probabilities of the following events: "within an hour, 5 subscribers will call the station"; "within an hour, no more than 4 subscribers will call the station"; "within an hour, at l... | 0.7619 |
In rectangle $WXYZ$, $P$ is a point on $WY$ such that $\angle WPZ=90^{\circ}$. $UV$ is perpendicular to $WY$ with $WU=UP$, as shown. $PZ$ intersects $UV$ at $Q$. Point $R$ is on $YZ$ such that $WR$ passes through $Q$. In $\triangle PQW$, $PW=15$, $WQ=20$ and $QP=25$. Find $VZ$. (Express your answer as a common fraction... | \dfrac{20}{3} |
Given the following six statements:
(1) All women are good drivers
(2) Some women are good drivers
(3) No men are good drivers
(4) All men are bad drivers
(5) At least one man is a bad driver
(6) All men are good drivers.
The statement that negates statement (6) is: | (5) |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are respectively $a$, $b$, $c$. Given that $\cos B = \frac{1}{3}$, $ac = 6$, and $b = 3$.
$(1)$ Find the value of $\cos C$ for side $a$;
$(2)$ Find the value of $\cos (2C+\frac{\pi }{3})$. | \frac{17-56 \sqrt{6}}{162} |
Six green balls and four red balls are in a bag. A ball is taken from the bag, its color recorded, then placed back in the bag. A second ball is taken and its color recorded. What is the probability the two balls are the same color? | \dfrac{13}{25} |
Find the smallest 6-digit palindrome in base 2, that can be expressed as a 4-digit palindrome in a different base. Provide your response in base 2. | 100001_2 |
Alice and Bob play a game on a board consisting of one row of 2022 consecutive squares. They take turns placing tiles that cover two adjacent squares, with Alice going first. By rule, a tile must not cover a square that is already covered by another tile. The game ends when no tile can be placed according to this rule.... | 290 |
In the sequence $\{a_n\}$, $a_{n+1} = 2(a_n - n + 3)$, $a_1 = -1$. If the sequence $\{a_n - pn + q\}$ is a geometric sequence, where $p$, $q$ are constants, then $a_{p+q} = \_\_\_\_\_\_\_\_\_\_\_\_\_.$ | 40 |
The number of natural numbers from 1 to 1992 that are multiples of 3, but not multiples of 2 or 5, is calculated. | 266 |
A rectangle with dimensions $8 \times 2 \sqrt{2}$ and a circle with a radius of 2 have a common center. What is the area of their overlapping region? | $2 \pi + 4$ |
Calculate the area of the parallelogram formed by the vectors $\begin{pmatrix} 4 \\ 2 \\ -3 \end{pmatrix}$ and $\begin{pmatrix} 2 \\ -4 \\ 5 \end{pmatrix}$. | 6\sqrt{30} |
Given the ellipse C: $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$ $(a>b>0)$, the “companion point” of a point M$(x_0, y_0)$ on the ellipse C is defined as $$N\left(\frac{x_0}{a}, \frac{y_0}{b}\right)$$.
(1) Find the equation of the trajectory of the “companion point” N of point M on the ellipse C;
(2) If the “companio... | \sqrt{3} |
The probability of snow for each of the next three days is $\frac{2}{3}$. What is the probability that it will snow at least once during those three days? Express your answer as a common fraction. | \dfrac{26}{27} |
The sequence $\{a_n\}$ satisfies $a_{n+1}+(-1)^n a_n = 2n-1$. Find the sum of the first $80$ terms of $\{a_n\}$. | 3240 |
How many kings can be placed on an $8 \times 8$ chessboard without putting each other in check? | 16 |
If $x$ is a real number and $k$ is a nonnegative integer, recall that the binomial coefficient $\binom{x}{k}$ is defined by the formula
\[
\binom{x}{k} = \frac{x(x - 1)(x - 2) \dots (x - k + 1)}{k!} \, .
\]Compute the value of
\[
\frac{\binom{1/2}{2014} \cdot 4^{2014}}{\binom{4028}{2014}} \, .
\] | -\frac{1} { 4027} |
Given that five students from Maplewood school worked for 6 days, six students from Oakdale school worked for 4 days, and eight students from Pinecrest school worked for 7 days, and the total amount paid for the students' work was 1240 dollars, determine the total amount earned by the students from Oakdale school, igno... | 270.55 |
Assume integers \( u \) and \( v \) satisfy \( 0 < v < u \), and let \( A \) be \((u, v)\). Points are defined as follows: \( B \) is the reflection of \( A \) over the line \( y = x \), \( C \) is the reflection of \( B \) over the \( y \)-axis, \( D \) is the reflection of \( C \) over the \( x \)-axis, and \( E \) i... | 21 |
By a proper divisor of a natural number we mean a positive integral divisor other than 1 and the number itself. A natural number greater than 1 will be called nice if it is equal to the product of its distinct proper divisors. What is the sum of the first ten nice numbers?
| 182 |
A bag contains 2 red balls, 3 white balls, and 4 yellow balls. If 4 balls are randomly selected from the bag, what is the probability that the selection includes balls of all three colors? | 4/7 |
Point $P$ lies on the diagonal $AC$ of square $ABCD$ with $AP > CP$. Let $O_{1}$ and $O_{2}$ be the circumcenters of triangles $ABP$ and $CDP$ respectively. Given that $AB = 12$ and $\angle O_{1}PO_{2} = 120^{\circ}$, then $AP = \sqrt{a} + \sqrt{b}$, where $a$ and $b$ are positive integers. Find $a + b$.
| 96 |
When two fair 8-sided dice (labeled from 1 to 8) are tossed, the numbers \(a\) and \(b\) are obtained. What is the probability that the two-digit number \(ab\) (where \(a\) and \(b\) are digits) and both \(a\) and \(b\) are divisible by 4? | \frac{1}{16} |
In triangle $XYZ,$ points $G,$ $H,$ and $I$ are on sides $\overline{YZ},$ $\overline{XZ},$ and $\overline{XY},$ respectively, such that $YG:GZ = XH:HZ = XI:IY = 2:3.$ Line segments $\overline{XG},$ $\overline{YH},$ and $\overline{ZI}$ intersect at points $S,$ $T,$ and $U,$ respectively. Compute $\frac{[STU]}{[XYZ]}.$ | \frac{9}{55} |
Find $b^2$ if the foci of the ellipse $\frac{x^2}{25} + \frac{y^2}{b^2} = 1$ and the foci of the hyperbola
\[\frac{x^2}{196} - \frac{y^2}{121} = \frac{1}{49}\] coincide. | \frac{908}{49} |
A square piece of paper, 4 inches on a side, is folded in half vertically. Both layers are then cut in half parallel to the fold. Three new rectangles are formed, a large one and two small ones. What is the ratio of the perimeter of one of the small rectangles to the perimeter of the large rectangle? | \frac{5}{6} |
A flower bouquet contains pink roses, red roses, pink carnations, and red carnations. One third of the pink flowers are roses, three fourths of the red flowers are carnations, and six tenths of the flowers are pink. What percent of the flowers are carnations? | 70 |
All students at Adams High School and at Baker High School take a certain exam. The average scores for boys, for girls, and for boys and girls combined, at Adams HS and Baker HS are shown in the table, as is the average for boys at the two schools combined. What is the average score for the girls at the two schools com... | 84 |
Given that in triangle PQR, side PR = 6 cm and side PQ = 10 cm, point S is the midpoint of QR, and the length of the altitude from P to QR is 4 cm, calculate the length of QR. | 4\sqrt{5} |
Two of the roots of
\[ax^3 + (a + 2b) x^2 + (b - 3a) x + (8 - a) = 0\]are $-2$ and 3. Find the third root. | \frac{4}{3} |
In the cube $ABCDEFGH$ with opposite vertices $C$ and $E,$ $J$ and $I$ are the midpoints of segments $\overline{FB}$ and $\overline{HD},$ respectively. Let $R$ be the ratio of the area of the cross-section $EJCI$ to the area of one of the faces of the cube. What is $R^2?$ | \frac{9}{4} |
John decides to install a ramp in his house. He needs to get permits which cost $250. After that, he hires a contractor which costs $150 an hour and the guy works for 3 days at 5 hours per day. He also pays an inspector 80% less to make sure it is OK. How much was the total cost? | The contractor works for 3*5=<<3*5=15>>15 hours
That means he charged 150*15=$<<150*15=2250>>2250
The inspector charged 2250*.8=$1800 less
So the inspector charged 2250-1800=$<<2250-1800=450>>450
So the total amount charged was 250+450+2250=$<<250+450+2250=2950>>2950
#### 2950 |
Javier has a wife and 3 children. They have 2 dogs and 1 cat. Including Javier, how many legs are there in total in Javier’s household? | For the humans in the house, there are 5 * 2 = <<5*2=10>>10 legs
Javier has a total of 2 + 1 = <<2+1=3>>3 four-legged pets
For the pets in the house, there are 3 * 4 = <<3*4=12>>12 legs
There are a total of 10 + 12 = <<10+12=22>>22 legs
#### 22 |
Given that the slant height of a cone is 2, and its net is a semicircle, what is the area of the cross section of the axis of the cone? | \sqrt{3} |
Let $a$ be the sum of the numbers: $99 \times 0.9$ $999 \times 0.9$ $9999 \times 0.9$ $\vdots$ $999\cdots 9 \times 0.9$ where the final number in the list is $0.9$ times a number written as a string of $101$ digits all equal to $9$ .
Find the sum of the digits in the number $a$ . | 891 |
A coordinate system is established with the origin as the pole and the positive half of the x-axis as the polar axis. Given the curve $C_1: (x-2)^2 + y^2 = 4$, point A has polar coordinates $(3\sqrt{2}, \frac{\pi}{4})$, and the polar coordinate equation of line $l$ is $\rho \cos (\theta - \frac{\pi}{4}) = a$, with poin... | 2\sqrt{2} |
A point is chosen at random from within a circular region. What is the probability that the point is closer to the center of the region than it is to the boundary of the region? | \frac{1}{4} |
Mason opens the hood of his car and discovers that squirrels have been using his engine compartment to store nuts. If 2 busy squirrels have been stockpiling 30 nuts/day and one sleepy squirrel has been stockpiling 20 nuts/day, all for 40 days, how many nuts are in Mason's car? | First find the number of nuts the two busy squirrels gather per day: 2 squirrels * 30 nuts/day/squirrel = <<2*30=60>>60 nuts/day
Then add the number the sleepy squirrel gathers each day: 60 nuts/day + 20 nuts/day = <<60+20=80>>80 nuts/day
Then multiply the number of nuts gathered daily by the number of days to find the... |
Find the sum of all angles $x \in [0^\circ, 360^\circ]$ that satisfy
\[\sin^5 x - \cos^5 x = \frac{1}{\cos x} - \frac{1}{\sin x}.\] | 270^\circ |
Find the number of sequences $a_{1}, a_{2}, \ldots, a_{10}$ of positive integers with the property that $a_{n+2}=a_{n+1}+a_{n}$ for $n=1,2, \ldots, 8$, and $a_{10}=2002$. | 3 |
What is the remainder when $11065+11067+11069+11071+11073+11075+11077$ is divided by $14$? | 7 |
Three dice with faces numbered 1 through 6 are stacked as shown. Seven of the eighteen faces are visible, leaving eleven faces hidden (back, bottom, between). The total number of dots NOT visible in this view is | 21 |
A triangle has vertices at coordinates (2,2), (5,6) and (6,2). What is the number of units in the length of the longest side of the triangle? | 5 |
Every bedtime, Juwella reads a book. Three nights ago, she read 15 pages. Two nights ago she read twice that many pages, while last night she read 5 pages more than the previous night. She promised to read the remaining pages of the book tonight. If the book has 100 pages, how many pages will she read tonight? | Juwella read 15 x 2 = <<15*2=30>>30 pages two nights ago.
She read 30 + 5 = <<30+5=35>>35 pages last night.
She read a total of 15 + 30 + 35 = <<15+30+35=80>>80 pages for three nights.
Thus, she needs to read 100 - 80 = <<100-80=20>>20 pages tonight to finish the book.
#### 20 |
You have three shirts and four pairs of pants. How many outfits consisting of one shirt and one pair of pants can you make? | 12 |
Let \( z_{1} \) and \( z_{2} \) be complex numbers such that \( \left|z_{1}\right|=3 \), \( \left|z_{2}\right|=5 \), and \( \left|z_{1} + z_{2}\right|=7 \). Find the value of \( \arg \left(\left( \frac{z_{2}}{z_{1}} \right)^{3}\right) \). | \pi |
If $a$, $b$, and $c$ are positive integers such that $\gcd(a,b) = 168$ and $\gcd(a,c) = 693$, then what is the smallest possible value of $\gcd(b,c)$? | 21 |
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