problem stringlengths 10 5.15k | answer stringlengths 0 1.23k |
|---|---|
A regular dodecagon \( Q_1 Q_2 \dotsb Q_{12} \) is drawn in the coordinate plane with \( Q_1 \) at \( (4,0) \) and \( Q_7 \) at \( (2,0) \). If \( Q_n \) is the point \( (x_n,y_n) \), compute the numerical value of the product
\[
(x_1 + y_1 i)(x_2 + y_2 i)(x_3 + y_3 i) \dotsm (x_{12} + y_{12} i).
\] | 531440 |
The total number of lions in a park is twice the number of leopards in the same park. The number of elephants is half the combined number of lions and leopards. Calculate the total population of the three animals in the park if the number of lions is 200. | If there are 200 lions in the park, which is twice the number of leopards in the park, there are 200/2 = <<200/2=100>>100 leopards in the park.
The total population of lions and leopards in the park is 200 lions + 100 leopards = <<200+100=300>>300
The number of elephants in the park is 1/2 the total population of lions... |
A school arranges for five people, \( A \), \( B \), \( C \), \( D \), and \( E \), to enter into three classes, with each class having at least one person, and \( A \) and \( B \) cannot be in the same class. Calculate the total number of different arrangements. | 114 |
Express this sum as a common fraction: $.\overline{8} + .\overline{2}$ | \frac{10}{9} |
What is the value of the following expression: $\frac{1}{3}-\frac{1}{9}+\frac{1}{27}-\frac{1}{81}+\frac{1}{243}$? Express your answer as a common fraction. | \frac{61}{243} |
Let $x, y$, and $z$ be positive real numbers such that $(x \cdot y)+z=(x+z) \cdot(y+z)$. What is the maximum possible value of $x y z$? | 1/27 |
Let $P(x)$ be a polynomial such that when $P(x)$ is divided by $x-17$, the remainder is $14$, and when $P(x)$ is divided by $x-13$, the remainder is $6$. What is the remainder when $P(x)$ is divided by $(x-13)(x-17)$? | 2x-20 |
In the Cartesian coordinate system $xOy$, the parametric equations of curve $C_{1}$ are $\left\{\begin{array}{l}{x=2+2\cos\varphi}\\{y=2\sin\varphi}\end{array}\right.$ ($\varphi$ is the parameter). Taking the origin $O$ as the pole and the positive half of the $x$-axis as the polar axis, the polar equation of curve $C_... | \frac{3\pi}{4} |
Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfy $|\overrightarrow{a}|=2$, $|\overrightarrow{b}|=\sqrt{2}$, and $\overrightarrow{a}\perp(\overrightarrow{a}+2\overrightarrow{b})$, calculate the projection of $\overrightarrow{b}$ in the direction of $\overrightarrow{a}$. | -1 |
Pete thinks of a number. He doubles it, adds 10, multiplies by 4, and ends up with 120. What was his original number? | 10 |
Two boards, one four inches wide and the other six inches wide, are nailed together to form an X. The angle at which they cross is 60 degrees. If this structure is painted and the boards are separated what is the area of the unpainted region on the four-inch board? (The holes caused by the nails are negligible.) Expres... | 16\sqrt{3} |
How many of the twelve pentominoes pictured below have at least one line of reflectional symmetry?
[asy] unitsize(5mm); defaultpen(linewidth(1pt)); draw(shift(2,0)*unitsquare); draw(shift(2,1)*unitsquare); draw(shift(2,2)*unitsquare); draw(shift(1,2)*unitsquare); draw(shift(0,2)*unitsquare); draw(shift(2,4)*unitsquare)... | 6 |
Marla is mixing a particular shade of lilac that's 70% blue paint, 20% red paint, and the rest white paint. If she adds 140 ounces of blue paint, how many ounces of white paint does she add? | First find the total amount of paint Marla mixes by dividing the amount of blue paint by the percentage of the paint that's blue: 140 ounces / .7 = <<140/.7=200>>200 ounces
Then multiply the total amount of paint by the percentage that's white to find the number of white ounces: 200 ounces * 10% = <<200*10*.01=20>>20 o... |
Let \( A, B, C, \) and \( D \) be positive real numbers such that
\[
\log_{10} (AB) + \log_{10} (AC) = 3, \\
\log_{10} (CD) + \log_{10} (CB) = 4, \\
\log_{10} (DA) + \log_{10} (DB) = 5.
\]
Compute the value of the product \( ABCD \). | 10000 |
An infinite sheet of paper is divided into equal squares, some of which are colored red. In each $2\times3$ rectangle, there are exactly two red squares. Now consider an arbitrary $9\times11$ rectangle. How many red squares does it contain? (The sides of all considered rectangles go along the grid lines.) | 33 |
A game board is constructed by shading two of the regions formed by the altitudes of an equilateral triangle as shown. What is the probability that the tip of the spinner will come to rest in a shaded region? Express your answer as a common fraction. [asy]
import olympiad; size(100); defaultpen(linewidth(0.8));
pair A ... | \frac{1}{3} |
How many $y$-intercepts does the graph of the parabola $x = 2y^2 - 3y + 7$ have? | 0 |
Find the value of $x$ if $\log_8 x = 1.75$. | 32\sqrt[4]{2} |
Find all solutions to the equation $\sqrt{5+2z} = 11$. | 58 |
Given \( f(x)=a \sin ((x+1) \pi)+b \sqrt[3]{x-1}+2 \), where \( a \) and \( b \) are real numbers and \( f(\lg 5) = 5 \), find \( f(\lg 20) \). | -1 |
Three circles, each of radius $3$, are drawn with centers at $(14, 92)$, $(17, 76)$, and $(19, 84)$. A line passing through $(17,76)$ is such that the total area of the parts of the three circles to one side of the line is equal to the total area of the parts of the three circles to the other side of it. What is the ab... | 24 |
A pastry chef is making brownies but is running out of butter. The recipe he is following calls for 2 ounces of butter for every 1 cup of baking mix; however, it allows 2 ounces of coconut oil to be substituted for the 2 ounces of butter if necessary. The chef would like to use as much butter as possible before switc... | Every 2 ounces of butter is enough to cover 1 cup of baking mix, so the chef's remaining 4 ounces of butter will be enough to cover 4/2 = <<4/2=2>>2 cups of baking mix.
This leaves 6 - 2 = <<6-2=4>>4 cups of baking mix that will require the use of coconut oil as a substitute.
Since each cup of baking mix requires 2 oun... |
Given the function $f(x)= \sqrt {3}\cos ( \frac {π}{2}+x)\cdot \cos x+\sin ^{2}x$, where $x\in R$.
(I) Find the interval where $f(x)$ is monotonically increasing.
(II) In $\triangle ABC$, angles $A$, $B$, and $C$ have corresponding opposite sides $a$, $b$, and $c$. If $B= \frac {π}{4}$, $a=2$, and angle $A$ satisfies $... | \frac {3+ \sqrt {3}}{3} |
A box contains 5 white balls and 6 black balls. Five balls are drawn out of the box at random. What is the probability that they all are white? | \dfrac{1}{462} |
Find all the positive perfect cubes that are not divisible by $10$ such that the number obtained by erasing the last three digits is also a perfect cube. | 1331 \text{ and } 1728 |
Two circles of radius \( r \) are externally tangent to each other and internally tangent to the ellipse defined by \( x^2 + 4y^2 = 5 \). The centers of the circles are located along the x-axis. Find the value of \( r \). | \frac{\sqrt{15}}{4} |
Suppose that the roots of $x^3+3x^2+4x-11=0$ are $a$, $b$, and $c$, and that the roots of $x^3+rx^2+sx+t=0$ are $a+b$, $b+c$, and $c+a$. Find $t$.
~ pi_is_3.14 | 23 |
The distance across a country is 8205 kilometers. Amelia started at one end and drove 907 kilometers on Monday and 582 kilometers on Tuesday. How many kilometers does Amelia still have to drive to make it across the country? | Distance driven = 907 + 582 = <<907+582=1489>>1489 km
Distance remaining = 8205 - 1489 = <<8205-1489=6716>>6716 km
Amelia still has 6716 kilometers to drive.
#### 6716 |
There are 100 students in class and the ratio of boys to girls is 3:2. How many more boys than girls are there? | The class was divided into 3 + 2 = <<3+2=5>>5 parts.
So each part is equal to 100/5 = <<100/5=20>>20 students.
Since there are 3 parts for boys, then there are 20 x 3 = <<3*20=60>>60 boys in class.
So there are 100 - 60 = <<100-60=40>>40 girls in class.
Hence, there are 60 - 40 = <<60-40=20>>20 boys more than girls.
##... |
James takes a spinning class 3 times a week. He works out for 1.5 hours each class and burns 7 calories per minute. How many calories does he burn per week? | He works out for 1.5*60=<<1.5*60=90>>90 minutes per class
So he burns 90*7=<<90*7=630>>630 calories per class
That means he burns 630*3=<<630*3=1890>>1890 calories per week
#### 1890 |
What is the last digit of the decimal expansion of $\frac{1}{2^{10}}$? | 5 |
Let $M$ denote the number of $8$-digit positive integers where the digits are in non-decreasing order. Determine the remainder obtained when $M$ is divided by $1000$. (Repeated digits are allowed, and the digit zero can now be used.) | 310 |
Nellie went to a big family party. She played with all the other kids there. She played hide and seek with her 6 sisters and her 8 brothers. Then, she played tag with all 22 of her cousins. How many kids did she play with in all? | Nellie played hide and seek with 6 + 8 = <<6+8=14>>14 kids.
In all, Nellie played with 14 + 22 = <<14+22=36>>36 kids.
#### 36 |
Let $f(x)$ be a function defined on $\mathbb{R}$ with a period of $2$. In the interval $[-1,1)$, $f(x)$ is defined as follows:
$$
f(x)=\begin{cases}
x + a, & -1 \leqslant x < 0, \\
\left| \frac{2}{5} - x \right|, & 0 \leqslant x < 1,
\end{cases}
$$
where $a \in \mathbb{R}$. If $f\left( -\frac{5}{2} \right) = f\left( \... | -\frac{2}{5} |
Holly loves to drink chocolate milk. With breakfast, she drinks 8 ounces of chocolate milk before leaving for work. During her lunch break, she buys a new 64-ounce container of chocolate milk and then drinks 8 ounces of it. With dinner, she drinks another 8 ounces of chocolate milk. If she ends the day with 56 ounc... | Before dinner, she had 56+8=<<56+8=64>>64 ounces of CM.
Before drinking 8 ounces of CM during lunch, she had 64+8=<<64+8=72>>72 ounces of CM.
Before buying 64 ounces of CM, she had 72-64=<<72-64=8>>8 ounces of CM.
And before breakfast, she had 8+8=<<8+8=16>>16 ounces of chocolate milk.
#### 16 |
Laura needs to buy window treatments for 3 windows in her house. She will need to buy a set of sheers and a set of drapes for each window. The sheers cost $40.00 a pair and the drapes cost $60.00 a pair. How much will the window treatments cost? | She needs a set of sheers that cost $40.00 and a set of drapes that cost $60.00 per window so that will cost her 40+60 = $<<40+60=100.00>>100.00
She had 3 windows she needs window treatments for they cost her $100.00 each so 3*100 = $<<3*100=300.00>>300.00
#### 300 |
Using the digits 1, 2, 3, 4, 5, how many even three-digit numbers less than 500 can be formed if each digit can be used more than once? | 40 |
Find the number of positive integers $n \le 1500$ that can be expressed in the form
\[\lfloor x \rfloor + \lfloor 2x \rfloor + \lfloor 4x \rfloor = n\] for some real number $x.$ | 854 |
If $x$ and $\log_{10} x$ are real numbers and $\log_{10} x<0$, then: | $0<x<1$ |
George went to a movie theater to see a movie. He paid $16 for the ticket and decided to buy some nachos. The nachos were half the price of the ticket. How much did George pay in total for the visit to the movie theater? | The nachos cost was 16 / 2 = $<<16/2=8>>8.
So in total George paid 8 + 16 = $<<8+16=24>>24 during the whole visit.
#### 24 |
In a certain warehouse, there are $1335$ boxes, each containing $39$ books.
Melvin's boss orders him to unpack all the books and repack them so that there are $40$ books in each box. After packing as many such boxes as possible, Melvin has how many books left over? | 25 |
In triangle $ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and $a=2b$. Also, $\sin A$, $\sin C$, $\sin B$ form an arithmetic sequence.
$(I)$ Find the value of $\cos (B+C)$;
$(II)$ If the area of $\triangle ABC$ is $\frac{8\sqrt{15}}{3}$, find the value of $c$. | 4 \sqrt {2} |
Let point $P$ be a moving point on the curve $C\_1$: $(x-2)^2 + y^2 = 4$. Establish a polar coordinate system with the coordinate origin $O$ as the pole and the positive semi-axis of the $x$-axis as the polar axis. Rotate point $P$ counterclockwise by $90^{{∘}}$ around the pole $O$ to obtain point $Q$. Denote the traje... | 3 - \sqrt{3} |
Find $\frac{1}{3}+\frac{2}{7}$. | \frac{13}{21} |
Each face of a cube is painted either red or blue, each with probability 1/2. The color of each face is determined independently. What is the probability that the painted cube can be placed on a horizontal surface so that the four vertical faces are all the same color? | \frac{5}{16} |
Find the maximum possible number of diagonals of equal length in a convex hexagon. | 7 |
A circle has an area of $16\pi$ square units. What are the lengths of the circle's diameter and circumference, in units? | 8\pi |
Suppose $\cos Q = 0.6$ in a right triangle $PQR$ where $PQ$ measures 15 units. What is the length of $QR$?
[asy]
pair P,Q,R;
P = (0,0);
Q = (7.5,0);
R = (0,7.5*tan(acos(0.6)));
draw(P--Q--R--P);
draw(rightanglemark(Q,P,R,20));
label("$P$",P,SW);
label("$Q$",Q,SE);
label("$R$",R,N);
label("$15$",Q/2,S);
[/asy] | 25 |
A license plate in a certain state consists of 5 digits, not necessarily distinct, and 3 letters, with the condition that at least one of the letters must be a vowel (A, E, I, O, U). These letters do not need to be next to each other but must be in a sequence. How many distinct license plates are possible if the digits... | 4,989,000,000 |
Natural numbers \(a, b, c\) are chosen such that \(a < b < c\). It is also known that the system of equations \(2x + y = 2035\) and \(y = |x - a| + |x - b| + |x - c|\) has exactly one solution. Find the minimum possible value of \(c\). | 1018 |
A spherical scoop of vanilla ice cream with radius of 2 inches is dropped onto the surface of a dish of hot chocolate sauce. As it melts, the ice cream spreads out uniformly forming a cylindrical region 8 inches in radius. Assuming the density of the ice cream remains constant, how many inches deep is the melted ice cr... | \frac{1}{6} |
Solve the equation
$$
\sin ^{4} x + 5(x - 2 \pi)^{2} \cos x + 5 x^{2} + 20 \pi^{2} = 20 \pi x
$$
Find the sum of its roots that belong to the interval $[-\pi ; 6 \pi]$, and provide the answer, rounding to two decimal places if necessary. | 31.42 |
Sam drove $96$ miles in $90$ minutes. His average speed during the first $30$ minutes was $60$ mph (miles per hour), and his average speed during the second $30$ minutes was $65$ mph. What was his average speed, in mph, during the last $30$ minutes? | 67 |
Real numbers $x, y, z$ satisfy $$x+x y+x y z=1, \quad y+y z+x y z=2, \quad z+x z+x y z=4$$ The largest possible value of $x y z$ is $\frac{a+b \sqrt{c}}{d}$, where $a, b, c, d$ are integers, $d$ is positive, $c$ is square-free, and $\operatorname{gcd}(a, b, d)=1$. Find $1000 a+100 b+10 c+d$. | 5272 |
In how many ways can 81 be written as the sum of three positive perfect squares if the order of the three perfect squares does not matter? | 3 |
Find the smallest positive integer \( n \) such that the mean of the squares of the first \( n \) natural numbers (\( n > 1 \)) is an integer.
(Note: The mean of the squares of \( n \) numbers \( a_1, a_2, \cdots, a_n \) is given by \( \sqrt{\frac{a_{1}^2 + a_{2}^2 + \cdots + a_{n}^2}{n}} \).)
(Note: Fifteenth Americ... | 337 |
Mike can type 65 words per minute. Due to a minor accident, Mike cannot use his right hand for a while so that his typing speed is now 20 words less per minute. If he is supposed to type a document with 810 words, how many minutes will it take him to finish typing the document? | After the accident, Mike can only type 65 words/minute - 20 words/minute = <<65-20=45>>45 words/minute.
So, he will be able to finish typing the document in 810 words / 45 words/minute = <<810/45=18>>18 minutes.
#### 18 |
There are two types of camels: dromedary camels with one hump on their back and Bactrian camels with two humps. Dromedary camels are taller, with longer limbs, and can walk and run in the desert; Bactrian camels have shorter and thicker limbs, suitable for walking in deserts and snowy areas. In a group of camels that h... | 15 |
Three distinct integers are chosen uniformly at random from the set $$ \{2021, 2022, 2023, 2024, 2025, 2026, 2027, 2028, 2029, 2030\}. $$ Compute the probability that their arithmetic mean is an integer. | 7/20 |
Triangles $BAD$ and $BDC$ are right triangles with $AB = 12$ units, $BD = 15$ units, and $BC = 17$ units. What is the area, in square units, of quadrilateral $ABCD$?
[asy]
draw((0,0)--(9,0)--(9,0)+8dir(36.87)--(0,12)--cycle,linewidth(1));
draw((0,12)--(9,0),linewidth(1));
label("A",(0,0),SW);
label("B",(0,12),W);
lab... | 114\text{ square units} |
Given a hyperbola $C$ with an eccentricity of $\sqrt {3}$, foci $F\_1$ and $F\_2$, and a point $A$ on the curve $C$. If $|F\_1A|=3|F\_2A|$, then $\cos \angle AF\_2F\_1=$ \_\_\_\_\_\_. | \frac{\sqrt{3}}{3} |
How many different four-digit numbers, divisible by 4, can be made from the digits 1, 2, 3, and 4,
a) if each digit can be used only once?
b) if each digit can be used multiple times? | 64 |
A store sells a type of notebook. The retail price for each notebook is 0.30 yuan, a dozen (12 notebooks) is priced at 3.00 yuan, and for purchases of more than 10 dozen, each dozen can be paid for at 2.70 yuan.
(1) There are 57 students in the ninth grade class 1, and each student needs one notebook of this type. Wh... | 51.30 |
Let the function $f(x)=\ln x-\frac{1}{2} ax^{2}-bx$.
$(1)$ When $a=b=\frac{1}{2}$, find the maximum value of the function $f(x)$;
$(2)$ Let $F(x)=f(x)+\frac{1}{2} x^{2}+bx+\frac{a}{x} (0 < x\leqslant 3)$. If the slope $k$ of the tangent line at any point $P(x_{0},y_{0})$ on its graph is always less than or equal to $... | \frac{1}{2} |
A right triangle has a side length of 21 inches and a hypotenuse of 29 inches. A second triangle is similar to the first and has a hypotenuse of 87 inches. What is the length of the shortest side of the second triangle? | 60\text{ inches} |
Bob grew corn in his garden and is ready to harvest it. He has 5 rows of corn, and each row has 80 corn stalks. About every 8 corn stalks will produce a bushel of corn. How many bushels of corn will Bob harvest? | Bob has 5 rows of corn, each with 80 corn stalks, so he has 5 rows * 80 stalks per row = <<5*80=400>>400 corn stalks.
8 corn stalks will produce a bushel of corn, so Bob will harvest 400 corn stalks / 8 corn stalks per bushel = <<400/8=50>>50 bushels of corn.
#### 50 |
What is the minimum number of straight cuts needed to cut a cake in 100 pieces? The pieces do not need to be the same size or shape but cannot be rearranged between cuts. You may assume that the cake is a large cube and may be cut from any direction. | 11 |
Given a parallelepiped \(A B C D A_{1} B_{1} C_{1} D_{1}\). Point \(X\) is chosen on edge \(A_{1} D_{1}\) and point \(Y\) is chosen on edge \(B C\). It is known that \(A_{1} X = 5\), \(B Y = 3\), and \(B_{1} C_{1} = 14\). The plane \(C_{1} X Y\) intersects the ray \(D A\) at point \(Z\). Find \(D Z\). | 20 |
Find the domain of the function
\[f(x) = \sqrt{1 - \sqrt{2 - \sqrt{3 - x}}}.\] | [-1,2] |
Point P is any point on the surface of the circumscribed sphere of a cube ABCD-A1B1C1D1 with edge length 2. What is the maximum volume of the tetrahedron P-ABCD? | \frac{4(1+\sqrt{3})}{3} |
A line through the points $(2, -9)$ and $(j, 17)$ is parallel to the line $2x + 3y = 21$. What is the value of $j$? | -37 |
15 balls numbered 1 through 15 are placed in a bin. Joe produces a list of four numbers by performing the following sequence four times: he chooses a ball, records the number, and places the ball back in the bin. Finally, Joe chooses to make a unique list by selecting 3 numbers from these 4, and forgetting the order in... | 202500 |
We write the equation on the board:
$$
(x-1)(x-2) \ldots (x-2016) = (x-1)(x-2) \ldots (x-2016) .
$$
We want to erase some of the 4032 factors in such a way that the equation on the board has no real solutions. What is the minimal number of factors that need to be erased to achieve this? | 2016 |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively, and $c=2$, $2\sin A= \sqrt {3}a\cos C$.
(1) Find the measure of angle $C$;
(2) If $2\sin 2A+ \sin (2B+C)= \sin C$, find the area of $\triangle ABC$. | \dfrac {2 \sqrt {3}}{3} |
In rectangle $LMNO$, points $P$ and $Q$ quadruple $\overline{LN}$, and points $R$ and $S$ quadruple $\overline{MO}$. Point $P$ is at $\frac{1}{4}$ the length of $\overline{LN}$ from $L$, and point $Q$ is at $\frac{1}{4}$ length from $P$. Similarly, $R$ is $\frac{1}{4}$ the length of $\overline{MO}$ from $M$, and $S$ is... | 0.75 |
Three fair, six-sided dice are rolled. What is the probability that the sum of the three numbers showing is less than 16? | \frac{103}{108} |
If $(ax+b)(bx+a)=26x^2+\Box\cdot x+26$, where $a$, $b$, and $\Box$ are distinct integers, what is the minimum possible value of $\Box$, the coefficient of $x$? | 173 |
Robie bought 3 bags of chocolates. She gave the 2 bags to her brothers and sisters then bought another 3 bags. How many bags of chocolates were left? | She was left with 3 - 2= <<3-2=1>>1 bag of chocolate after she gave 2 bags to her brother and sisters.
Therefore, the total number of bags of chocolates left after buying another 3 is 1 + 3= <<1+3=4>>4.
#### 4 |
If $x$ and $y$ are positive integers with $x+y=31$, what is the largest possible value of $x y$? | 240 |
A circle with radius 4 cm is tangent to three sides of a rectangle, as shown. The area of the rectangle is twice the area of the circle. What is the length of the longer side of the rectangle, in centimeters? Express your answer in terms of $\pi$.
[asy]
import graph;
draw((0,0)--(30,0)--(30,20)--(0,20)--cycle);
draw(C... | 4\pi |
Determine the sum of the real numbers \( x \) for which \(\frac{2 x}{x^{2}+5 x+3}+\frac{3 x}{x^{2}+x+3}=1\). | -4 |
\( A, B, C \) are positive integers. It is known that \( A \) has 7 divisors, \( B \) has 6 divisors, \( C \) has 3 divisors, \( A \times B \) has 24 divisors, and \( B \times C \) has 10 divisors. What is the minimum value of \( A + B + C \)? | 91 |
Triangle $ABC$ is inscribed in a circle of radius 2 with $\angle ABC \geq 90^\circ$, and $x$ is a real number satisfying the equation $x^4 + ax^3 + bx^2 + cx + 1 = 0$, where $a=BC$, $b=CA$, $c=AB$. Find all possible values of $x$. | $x = -\frac{1}{2} (\sqrt6 \pm \sqrt 2)$ |
A parallelogram has 2 sides of length 20 and 15. Given that its area is a positive integer, find the minimum possible area of the parallelogram. | 1 |
Find $r$ if $3(r-7) = 4(2-2r) + 4$. | 3 |
Use \((a, b)\) to represent the greatest common divisor of \(a\) and \(b\). Let \(n\) be an integer greater than 2021, and \((63, n+120) = 21\) and \((n+63, 120) = 60\). What is the sum of the digits of the smallest \(n\) that satisfies the above conditions? | 15 |
A standard deck of 52 cards is randomly arranged. What is the probability that the top three cards are $\spadesuit$, $\heartsuit$, and $\spadesuit$ in that sequence? | \dfrac{78}{5100} |
Sofia has forgotten the passcode of her phone. She only remembers that it has four digits and that the product of its digits is $18$ . How many passcodes satisfy these conditions? | 36 |
Given that $\{1, a, \frac{b}{a}\} = \{0, a^2, a+b\}$, find the value of $a^{2017} + b^{2017}$. | -1 |
Decompose $\frac{1}{4}$ into unit fractions. | \frac{1}{8}+\frac{1}{12}+\frac{1}{24} |
Given that angle DEF is a right angle and the sides of triangle DEF are the diameters of semicircles, the area of the semicircle on segment DE equals $18\pi$, and the arc of the semicircle on segment DF has length $10\pi$. Determine the radius of the semicircle on segment EF. | \sqrt{136} |
Given the function $f(x) = e^{\sin x + \cos x} - \frac{1}{2}\sin 2x$ ($x \in \mathbb{R}$), find the difference between the maximum and minimum values of the function $f(x)$. | e^{\sqrt{2}} - e^{-\sqrt{2}} |
Find the number of functions $f : \mathbb{R} \to \mathbb{R}$ such that
\[f(x + f(y)) = x + y\]for all real numbers $x$ and $y.$ | 1 |
Given the function $f(x)=\sqrt{3}\cos (\frac{\pi }{2}+x)\bullet \cos x+\sin^{2}x$, where $x\in R$.
(I) Find the interval where $f(x)$ is monotonically increasing.
(II) In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively. If $B=\frac{\pi }{4}$, $a=2$ and angle $A$ satisfies $f(A... | \frac{3+\sqrt{3}}{3} |
How many 5-digit positive numbers contain only odd numbers and have at least one pair of consecutive digits whose sum is 10? | 1845 |
What is the smallest positive integer with exactly 12 positive integer divisors? | 72 |
Given $f(\sin \alpha + \cos \alpha) = \sin \alpha \cdot \cos \alpha$, determine the domain of $f(x)$ and the value of $f\left(\sin \frac{\pi}{6}\right)$. | -\frac{3}{8} |
How many numbers between $1$ and $2500$ are integer multiples of $4$ or $5$ but not $15$? | 959 |
What common fraction (that is, a fraction reduced to its lowest terms) is equivalent to $0.1\overline{35}$? | \frac{67}{495} |
Alice and Bob stand atop two different towers in the Arctic. Both towers are a positive integer number of meters tall and are a positive (not necessarily integer) distance away from each other. One night, the sea between them has frozen completely into reflective ice. Alice shines her flashlight directly at the top of ... | 7,15 |
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