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stringlengths 10
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stringclasses 890
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stringlengths 0
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40.3k
| domain
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|---|---|---|---|---|---|
In an isosceles trapezoid, the bases are 40 and 24, and its diagonals are mutually perpendicular. Find the area of the trapezoid.
|
1024
|
deepscale
| 9,943
| ||
Express the quotient $2213_4 \div 13_4$ in base 4.
|
53_4
|
deepscale
| 25,687
| ||
Each of the nine letters in "STATISTICS" is written on its own square tile and placed in a bag. What is the probability that a tile randomly selected from the bag will have a letter on it that is in the word "TEST"? Express your answer as a common fraction.
|
\frac{2}{3}
|
deepscale
| 32,166
| ||
Among all the four-digit numbers without repeated digits, how many numbers have the digit in the thousandth place 2 greater than the digit in the unit place?
|
448
|
deepscale
| 18,682
| ||
Given that $x+y=12$, $xy=9$, and $x < y$, find the value of $\frac {x^{ \frac {1}{2}}-y^{ \frac {1}{2}}}{x^{ \frac {1}{2}}+y^{ \frac {1}{2}}}=$ ___.
|
- \frac { \sqrt {3}}{3}
|
deepscale
| 19,042
| ||
Consider a square, inside which is inscribed a circle, inside which is inscribed a square, inside which is inscribed a circle, and so on, with the outermost square having side length 1. Find the difference between the sum of the areas of the squares and the sum of the areas of the circles.
|
The ratio of the area of each square and the circle immediately inside it is $\frac{4}{\pi}$. The total sum of the areas of the squares is $1+\frac{1}{2}+\frac{1}{4}+\ldots=2$. Difference in area is then $2-2 \cdot \frac{4}{\pi} = 2 - \frac{\pi}{2}$.
|
2 - \frac{\pi}{2}
|
deepscale
| 4,402
| |
For positive integers $n$, let $c_{n}$ be the smallest positive integer for which $n^{c_{n}}-1$ is divisible by 210, if such a positive integer exists, and $c_{n}=0$ otherwise. What is $c_{1}+c_{2}+\cdots+c_{210}$?
|
In order for $c_{n} \neq 0$, we must have $\operatorname{gcd}(n, 210)=1$, so we need only consider such $n$. The number $n^{c_{n}}-1$ is divisible by 210 iff it is divisible by each of 2, 3, 5, and 7, and we can consider the order of $n$ modulo each modulus separately; $c_{n}$ will simply be the LCM of these orders. We can ignore the modulus 2 because order is always 1. For the other moduli, the sets of orders are $a \in\{1,2\} \bmod 3$, $b \in\{1,2,4,4\} \bmod 5$, $c \in\{1,2,3,3,6,6\} \bmod 7$. By the Chinese Remainder Theorem, each triplet of choices from these three multisets occurs for exactly one $n$ in the range $\{1,2, \ldots, 210\}$, so the answer we seek is the sum of $\operatorname{lcm}(a, b, c)$ over $a, b, c$ in the Cartesian product of these multisets. For $a=1$ this table of LCMs is as follows: $\begin{tabular}{ccccccc} & 1 & 2 & 3 & 3 & 6 & 6 \\ \hline 1 & 1 & 2 & 3 & 3 & 6 & 6 \\ 2 & 2 & 2 & 6 & 6 & 6 & 6 \\ 4 & 4 & 4 & 12 & 12 & 12 & 12 \\ 4 & 4 & 4 & 12 & 12 & 12 & 12 \end{tabular}$ which has a sum of $21+56+28+56=161$. The table for $a=2$ is identical except for the top row, where $1,3,3$ are replaced by $2,6,6$, and thus has a total sum of 7 more, or 168. So our answer is $161+168=329$.
|
329
|
deepscale
| 3,633
| |
In the diagram, if points $ A$ , $ B$ and $ C$ are points of tangency, then $ x$ equals:
[asy]unitsize(5cm);
defaultpen(linewidth(.8pt)+fontsize(8pt));
dotfactor=3;
pair A=(-3*sqrt(3)/32,9/32), B=(3*sqrt(3)/32, 9/32), C=(0,9/16);
pair O=(0,3/8);
draw((-2/3,9/16)--(2/3,9/16));
draw((-2/3,1/2)--(-sqrt(3)/6,1/2)--(0,0)--(sqrt(3)/6,1/2)--(2/3,1/2));
draw(Circle(O,3/16));
draw((-2/3,0)--(2/3,0));
label(" $A$ ",A,SW);
label(" $B$ ",B,SE);
label(" $C$ ",C,N);
label(" $\frac{3}{8}$ ",O);
draw(O+.07*dir(60)--O+3/16*dir(60),EndArrow(3));
draw(O+.07*dir(240)--O+3/16*dir(240),EndArrow(3));
label(" $\frac{1}{2}$ ",(.5,.25));
draw((.5,.33)--(.5,.5),EndArrow(3));
draw((.5,.17)--(.5,0),EndArrow(3));
label(" $x$ ",midpoint((.5,.5)--(.5,9/16)));
draw((.5,5/8)--(.5,9/16),EndArrow(3));
label(" $60^{\circ}$ ",(0.01,0.12));
dot(A);
dot(B);
dot(C);[/asy]
|
$\frac{1}{16}$
|
deepscale
| 14,045
| ||
A function $f$ is defined for all real numbers and satisfies the conditions $f(3+x) = f(3-x)$ and $f(8+x) = f(8-x)$ for all $x$. If $f(0) = 0$, determine the minimum number of roots that $f(x) = 0$ must have in the interval $-500 \leq x \leq 500$.
|
201
|
deepscale
| 28,957
| ||
On the 5 by 5 square grid below, each dot is 1 cm from its nearest horizontal and vertical neighbors. What is the product of the value of the area of square $ABCD$ (in cm$^2$) and the value of the perimeter of square $ABCD$ (in cm)? Express your answer in simplest radical form.
[asy]unitsize(1cm);
defaultpen(linewidth(0.7));
dot((0,0));
dot((0,1));
dot((0,2));
dot((0,3));
dot((0,4));
dot((1,0));
dot((1,1));
dot((1,2));
dot((1,3));
dot((1,4));
dot((2,0));
dot((2,1));
dot((2,2));
dot((2,3));
dot((2,4));
dot((3,0));
dot((3,1));
dot((3,2));
dot((3,3));
dot((3,4));
dot((4,0));
dot((4,1));
dot((4,2));
dot((4,3));
dot((4,4));
draw((0,3)--(3,4)--(4,1)--(1,0)--cycle);
label("$A$",(3,4),N);
label("$B$",(4,1),E);
label("$C$",(1,0),S);
label("$D$",(0,3),W);
[/asy]
Note that when we say the grid is 5 by 5 we mean that each row and column contains 5 dots!
|
40\sqrt{10}
|
deepscale
| 38,831
| ||
Find the real solutions of $(2 x+1)(3 x+1)(5 x+1)(30 x+1)=10$.
|
$(2 x+1)(3 x+1)(5 x+1)(30 x+1)=[(2 x+1)(30 x+1)][(3 x+1)(5 x+1)]=\left(60 x^{2}+32 x+1\right)\left(15 x^{2}+8 x+1\right)=(4 y+1)(y+1)=10$, where $y=15 x^{2}+8 x$. The quadratic equation in $y$ yields $y=1$ and $y=-\frac{9}{4}$. For $y=1$, we have $15 x^{2}+8 x-1=0$, so $x=\frac{-4 \pm \sqrt{31}}{15}$. For $y=-\frac{9}{4}$, we have $15 x^{2}+8 x+\frac{9}{4}$, which yields only complex solutions for $x$. Thus the real solutions are $\frac{-4 \pm \sqrt{31}}{15}$.
|
\frac{-4 \pm \sqrt{31}}{15}
|
deepscale
| 3,720
| |
In the Cartesian coordinate system, the graphs of the functions $y=\frac{3}{x}$ and $y=x+1$ intersect at the point $\left(m,n\right)$. Evaluate the algebraic expression $\left(m-n\right)^{2}\cdot \left(\frac{1}{n}-\frac{1}{m}\right)$.
|
-\frac{1}{3}
|
deepscale
| 11,753
| ||
Given the planar vectors $\overrightarrow{a}$ and $\overrightarrow{b}$, with magnitudes $|\overrightarrow{a}| = 1$ and $|\overrightarrow{b}| = 2$, and their dot product $\overrightarrow{a} \cdot \overrightarrow{b} = 1$. If $\overrightarrow{e}$ is a unit vector in the plane, find the maximum value of $|\overrightarrow{a} \cdot \overrightarrow{e}| + |\overrightarrow{b} \cdot \overrightarrow{e}|$.
|
\sqrt{7}
|
deepscale
| 9,758
| ||
There are 5 students who signed up to participate in a two-day volunteer activity at the summer district science museum. Each day, two students are needed for the activity. The probability that exactly 1 person will participate in the volunteer activity for two consecutive days is ______.
|
\frac{3}{5}
|
deepscale
| 23,638
| ||
I need to learn vocabulary words for my Spanish exam. There are 500 words and the exam grade is the percentage of these words that I recall correctly. Assuming I will recall correctly the words I learn and assuming my guesses will not yield any points, what is the least number of words I should learn to get at least $85\%$ on the exam?
|
425
|
deepscale
| 38,469
| ||
Given positive numbers $x$ and $y$ satisfying $x^2+y^2=1$, find the maximum value of $\frac {1}{x}+ \frac {1}{y}$.
|
2\sqrt{2}
|
deepscale
| 21,186
| ||
To solve the problem, we need to find the value of $\log_{4}{\frac{1}{8}}$.
A) $-\frac{1}{2}$
B) $-\frac{3}{2}$
C) $\frac{1}{2}$
D) $\frac{3}{2}$
|
-\frac{3}{2}
|
deepscale
| 23,853
| ||
If $e^{i \theta} = \frac{1 + i \sqrt{2}}{2}$, then find $\sin 3 \theta.$
|
\frac{\sqrt{2}}{8}
|
deepscale
| 31,763
| ||
Find all roots of the polynomial $x^3-5x^2+3x+9$. Enter your answer as a list of numbers separated by commas. If a root occurs more than once, enter it as many times as its multiplicity.
|
-1,3,3
|
deepscale
| 36,642
| ||
Let \( F_{1} \) and \( F_{2} \) be the left and right foci of the hyperbola \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1 \) (where \( a > 0 \) and \( b > 0 \)). There exists a point \( P \) on the right branch of the hyperbola such that \( \left( \overrightarrow{OP} + \overrightarrow{OF_{2}} \right) \cdot \overrightarrow{PF_{2}} = 0 \), where \( O \) is the origin. Additionally, \( \left| \overrightarrow{PF_{1}} \right| = \sqrt{3} \left| \overrightarrow{PF_{2}} \right| \). Determine the eccentricity of the hyperbola.
|
\sqrt{3} + 1
|
deepscale
| 15,925
| ||
In a division equation, the dividend is 2016 greater than the divisor, the quotient is 15, and the remainder is 0. What is the dividend?
|
2160
|
deepscale
| 11,349
| ||
On the first day, 1 bee brings back 5 companions. On the second day, 6 bees (1 from the original + 5 brought back on the first day) fly out, each bringing back 5 companions. Determine the total number of bees in the hive after the 6th day.
|
46656
|
deepscale
| 21,116
| ||
Let $S$ denote the sum of all of the three digit positive integers with three distinct digits. Compute the remainder when $S$ is divided by $1000$.
|
680
|
deepscale
| 38,217
| ||
Find the phase shift, vertical shift, and the maximum and minimum values of the graph of \( y = 3\sin\left(3x - \frac{\pi}{4}\right) + 1 \).
|
-2
|
deepscale
| 26,922
| ||
Consider a cube where each pair of opposite faces sums to 8 instead of the usual 7. If one face shows 1, the opposite face will show 7; if one face shows 2, the opposite face will show 6; if one face shows 3, the opposite face will show 5. Calculate the largest sum of three numbers whose faces meet at one corner of the cube.
|
16
|
deepscale
| 27,045
| ||
In a single-elimination tournament, each game is between two players. Only the winner of each game advances to the next round. In a particular such tournament there are 256 players. How many individual games must be played to determine the champion?
|
255
|
deepscale
| 35,456
| ||
What is the sum of the positive factors of 48?
|
124
|
deepscale
| 38,405
| ||
A rectangular chessboard of size \( m \times n \) is composed of unit squares (where \( m \) and \( n \) are positive integers not exceeding 10). A piece is placed on the unit square in the lower-left corner. Players A and B take turns moving the piece. The rules are as follows: either move the piece any number of squares upward, or any number of squares to the right, but you cannot move off the board or stay in the same position. The player who cannot make a move loses (i.e., the player who first moves the piece to the upper-right corner wins). How many pairs of integers \( (m, n) \) are there such that the first player A has a winning strategy?
|
90
|
deepscale
| 15,514
| ||
Nine barrels. In how many ways can nine barrels be arranged in three tiers so that the numbers written on the barrels that are to the right of any barrel or below it are larger than the number written on the barrel itself? An example of a correct arrangement is having 123 in the top row, 456 in the middle row, and 789 in the bottom row. How many such arrangements are possible?
|
42
|
deepscale
| 15,915
| ||
In $\vartriangle ABC, AB=AC=14 \sqrt2 , D$ is the midpoint of $CA$ and $E$ is the midpoint of $BD$ . Suppose $\vartriangle CDE$ is similar to $\vartriangle ABC$ . Find the length of $BD$ .
|
14
|
deepscale
| 20,000
| ||
Let $A=(0,1),$ $B=(2,5),$ $C=(5,2),$ and $D=(7,0).$ A figure is created by connecting $A$ to $B,$ $B$ to $C,$ $C$ to $D,$ and $D$ to $A.$ The perimeter of $ABCD$ can be expressed in the form $a\sqrt2+b\sqrt{5}$ with $a$ and $b$ integers. What is the sum of $a$ and $b$?
|
12
|
deepscale
| 33,308
| ||
Given that line $l$ passes through point $P(-1,2)$ with a slope angle of $\frac{2\pi}{3}$, and the circle's equation is $\rho=2\cos (\theta+\frac{\pi}{3})$:
(1) Find the parametric equation of line $l$;
(2) Let line $l$ intersect the circle at points $M$ and $N$, find the value of $|PM|\cdot|PN|$.
|
6+2\sqrt{3}
|
deepscale
| 9,357
| ||
Given a sequence $\{a_n\}$ with $16$ terms, and $a_1=1, a_8=4$. Let the function related to $x$ be $f_n(x)=\frac{x^3}{3}-a_nx^2+(a_n^2-1)x$, where $n\in \mathbf{N}^*$. If $x=a_{n+1}$ ($1\leqslant n\leqslant 15$) is the extremum point of the function $f_n(x)$, and the slope of the tangent line at the point $(a_{16}, f_8(a_{16}))$ on the curve $y=f_8(x)$ is $15$, then the number of sequences $\{a_n\}$ that satisfy the conditions is ______.
|
1176
|
deepscale
| 28,325
| ||
Given the parametric equation of line $l$ as $$\begin{cases} x= \sqrt {3}+t \\ y=7+ \sqrt {3}t\end{cases}$$ ($t$ is the parameter), a coordinate system is established with the origin as the pole and the positive half of the $x$-axis as the polar axis. The polar equation of curve $C$ is $\rho \sqrt {a^{2}\sin^{2}\theta+4\cos^{2}\theta}=2a$ ($a>0$).
1. Find the Cartesian equation of curve $C$.
2. Given point $P(0,4)$, line $l$ intersects curve $C$ at points $M$ and $N$. If $|PM|\cdot|PN|=14$, find the value of $a$.
|
\frac{2\sqrt{21}}{3}
|
deepscale
| 22,553
| ||
For each positive integer $n$, the mean of the first $n$ terms of a sequence is $n$. What is the 2008th term of the sequence?
|
4015
|
deepscale
| 37,249
| ||
A whole number was increased by 2, and its square decreased by 2016. What was the number initially (before the increase)?
|
-505
|
deepscale
| 11,540
| ||
For each positive integer $k$, let $S_k$ denote the increasing arithmetic sequence of integers whose first term is $1$ and whose common difference is $k$. For example, $S_3$ is the sequence $1,4,7,10,\ldots.$ For how many values of $k$ does $S_k$ contain the term $2005$?
|
Suppose that the $n$th term of the sequence $S_k$ is $2005$. Then $1+(n-1)k=2005$ so $k(n-1)=2004=2^2\cdot 3\cdot 167$. The ordered pairs $(k,n-1)$ of positive integers that satisfy the last equation are $(1,2004)$,$(2,1002)$, $(3,668)$, $(4,501)$, $(6,334)$, $(12,167)$, $(167,12)$,$(334,6)$, $(501,4)$, $(668,3)$, $(1002,2)$ and $(2004,1)$, and each of these gives a possible value of $k$. Thus the requested number of values is $12$, and the answer is $\boxed{12}$.
Alternatively, notice that the formula for the number of divisors states that there are $(2 + 1)(1 + 1)(1 + 1) = 12$ divisors of $2^2\cdot 3^1\cdot 167^1$.
|
12
|
deepscale
| 6,822
| |
A university has 120 foreign language teachers. Among them, 50 teach English, 45 teach Japanese, and 40 teach French. There are 15 teachers who teach both English and Japanese, 10 who teach both English and French, and 8 who teach both Japanese and French. Additionally, 4 teachers teach all three languages: English, Japanese, and French. How many foreign language teachers do not teach any of these three languages?
|
14
|
deepscale
| 10,018
| ||
Two dice are made so that the chances of getting an even sum are twice that of getting an odd sum. What is the probability of getting an odd sum in a single roll of these two dice?
(a) \(\frac{1}{9}\)
(b) \(\frac{2}{9}\)
(c) \(\frac{4}{9}\)
(d) \(\frac{5}{9}\)
|
$\frac{4}{9}
|
deepscale
| 26,041
| ||
Let the function \( f(x) = \left| \log_{2} x \right| \). Real numbers \( a \) and \( b \) (where \( a < b \)) satisfy the following conditions:
\[
\begin{array}{l}
f(a+1) = f(b+2), \\
f(10a + 6b + 22) = 4
\end{array}
\]
Find \( ab \).
|
2/15
|
deepscale
| 7,472
| ||
Let the function $f(x)$ be defined on $\mathbb{R}$ and satisfy $f(2-x) = f(2+x)$ and $f(7-x) = f(7+x)$. Also, in the closed interval $[0, 7]$, only $f(1) = f(3) = 0$. Determine the number of roots of the equation $f(x) = 0$ in the closed interval $[-2005, 2005]$.
|
802
|
deepscale
| 10,567
| ||
Given that $x \sim N(-1,36)$ and $P(-3 \leqslant \xi \leqslant -1) = 0.4$, calculate $P(\xi \geqslant 1)$.
|
0.1
|
deepscale
| 29,661
| ||
Let \( x_{1}, x_{2} \) be the roots of the equation \( x^{2} - x - 3 = 0 \). Find \(\left(x_{1}^{5} - 20\right) \cdot \left(3 x_{2}^{4} - 2 x_{2} - 35\right)\).
|
-1063
|
deepscale
| 7,430
| ||
The chord \( AB \) divides the circle into two arcs, with the smaller arc being \( 130^{\circ} \). The larger arc is divided by chord \( AC \) in the ratio \( 31:15 \) from point \( A \). Find the angle \( BAC \).
|
37.5
|
deepscale
| 15,194
| ||
Starting from which number $n$ of independent trials does the inequality $p\left(\left|\frac{m}{n}-p\right|<0.1\right)>0.97$ hold, if in a single trial $p=0.8$?
|
534
|
deepscale
| 24,911
| ||
Given $\sin(\alpha - \beta) = \frac{1}{3}$ and $\cos \alpha \sin \beta = \frac{1}{6}$, calculate the value of $\cos(2\alpha + 2\beta)$.
|
\frac{1}{9}
|
deepscale
| 18,080
| ||
The points $(0,0)\,$, $(a,11)\,$, and $(b,37)\,$ are the vertices of an equilateral triangle. Find the value of $ab\,$.
|
Consider the points on the complex plane. The point $b+37i$ is then a rotation of $60$ degrees of $a+11i$ about the origin, so:
\[(a+11i)\left(\mathrm{cis}\,60^{\circ}\right) = (a+11i)\left(\frac 12+\frac{\sqrt{3}i}2\right)=b+37i.\]
Equating the real and imaginary parts, we have:
\begin{align*}b&=\frac{a}{2}-\frac{11\sqrt{3}}{2}\\37&=\frac{11}{2}+\frac{a\sqrt{3}}{2} \end{align*}
Solving this system, we find that $a=21\sqrt{3}, b=5\sqrt{3}$. Thus, the answer is $\boxed{315}$.
Note: There is another solution where the point $b+37i$ is a rotation of $-60$ degrees of $a+11i$; however, this triangle is just a reflection of the first triangle by the $y$-axis, and the signs of $a$ and $b$ are flipped. However, the product $ab$ is unchanged.
|
315
|
deepscale
| 6,588
| |
Express $4.\overline{054}$ as a common fraction in lowest terms.
|
\frac{150}{37}
|
deepscale
| 39,209
| ||
Rectangle $ABCD$ has $AB = CD = 3$ and $BC = DA = 5$. The rectangle is first rotated $90^\circ$ clockwise around vertex $D$, then it is rotated $90^\circ$ clockwise around the new position of vertex $C$ (after the first rotation). What is the length of the path traveled by point $A$?
A) $\frac{3\pi(\sqrt{17} + 6)}{2}$
B) $\frac{\pi(\sqrt{34} + 5)}{2}$
C) $\frac{\pi(\sqrt{30} + 5)}{2}$
D) $\frac{\pi(\sqrt{40} + 5)}{2}$
|
\frac{\pi(\sqrt{34} + 5)}{2}
|
deepscale
| 30,480
| ||
What is the 39th number in the row of Pascal's triangle that has 41 numbers?
|
780
|
deepscale
| 34,879
| ||
Square $BCFE$ is inscribed in right triangle $AGD$, as shown in the diagram which is the same as the previous one. If $AB = 36$ units and $CD = 72$ units, what is the area of square $BCFE$?
|
2592
|
deepscale
| 30,939
| ||
A rectangular park is to be fenced on three sides using a 150-meter concrete wall as the fourth side. Fence posts are to be placed every 15 meters along the fence, including at the points where the fence meets the concrete wall. Calculate the minimal number of posts required to fence an area of 45 m by 90 m.
|
13
|
deepscale
| 32,695
| ||
The positive integers \( r \), \( s \), and \( t \) have the property that \( r \times s \times t = 1230 \). What is the smallest possible value of \( r + s + t \)?
|
52
|
deepscale
| 23,332
| ||
Given the function
$$
f(x)=
\begin{cases}
1 & (1 \leq x \leq 2) \\
\frac{1}{2}x^2 - 1 & (2 < x \leq 3)
\end{cases}
$$
define $h(a) = \max\{f(x) - ax \mid x \in [1, 3]\} - \min\{f(x) - ax \mid x \in [1, 3]\}$ for any real number $a$.
1. Find the value of $h(0)$.
2. Find the expression for $h(a)$ and its minimum value.
|
\frac{5}{4}
|
deepscale
| 10,403
| ||
How many ways are there to choose 4 cards from a standard deck of 52 cards, where two cards come from one suit and the other two each come from different suits?
|
158184
|
deepscale
| 23,736
| ||
Let \( p(x) \) be a monic quartic polynomial such that \( p(1) = 1, p(2) = 9, p(3) = 28, \) and \( p(4) = 65. \) Find \( p(5) \).
|
126
|
deepscale
| 28,948
| ||
For a positive integer \( k \), find the greatest common divisor (GCD) \( d \) of all positive even numbers \( x \) that satisfy the following conditions:
1. Both \( \frac{x+2}{k} \) and \( \frac{x}{k} \) are integers, and the difference in the number of digits of these two numbers is equal to their difference;
2. The product of the digits of \( \frac{x}{k} \) is a perfect cube.
|
1998
|
deepscale
| 27,676
| ||
Jenna bakes a $24$-inch by $15$-inch pan of chocolate cake. The cake pieces are made to measure $3$ inches by $2$ inches each. Calculate the number of pieces of cake the pan contains.
|
60
|
deepscale
| 29,840
| ||
A sector with a central angle of 135° has an area of $S_1$, and the total surface area of the cone formed by it is $S_2$. Find the value of $\frac{S_{1}}{S_{2}}$.
|
\frac{8}{11}
|
deepscale
| 23,587
| ||
In a certain country, the airline system is arranged in such a way that any city is connected by airlines with no more than three other cities, and from any city to any other, one can travel with no more than one stopover. What is the maximum number of cities that can be in this country?
|
10
|
deepscale
| 10,534
| ||
Given an ellipse with the equation $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1 (a > b > 0)$ and an eccentricity of $\frac{1}{2}$. $F\_1$ and $F\_2$ are the left and right foci of the ellipse, respectively. A line $l$ passing through $F\_2$ intersects the ellipse at points $A$ and $B$. The perimeter of $\triangle F\_1AB$ is $8$.
(I) Find the equation of the ellipse.
(II) If the slope of line $l$ is $0$, and its perpendicular bisector intersects the $y$-axis at $Q$, find the range of the $y$-coordinate of $Q$.
(III) Determine if there exists a point $M(m, 0)$ on the $x$-axis such that the $x$-axis bisects $\angle AMB$. If it exists, find the value of $m$; otherwise, explain the reason.
|
m = 4
|
deepscale
| 32,432
| ||
Let the complex numbers \(z\) and \(w\) satisfy \(|z| = 3\) and \((z + \bar{w})(\bar{z} - w) = 7 + 4i\), where \(i\) is the imaginary unit and \(\bar{z}\), \(\bar{w}\) denote the conjugates of \(z\) and \(w\) respectively. Find the modulus of \((z + 2\bar{w})(\bar{z} - 2w)\).
|
\sqrt{65}
|
deepscale
| 8,504
| ||
Find the positive value of $y$ which satisfies
\[\log_7 (y - 3) + \log_{\sqrt{7}} (y^2 - 3) + \log_{\frac{1}{7}} (y - 3) = 3.\]
|
\sqrt{\sqrt{343} + 3}
|
deepscale
| 11,638
| ||
The polynomial $\frac{1}{5}{x^2}{y^{|m|}}-(m+1)y+\frac{1}{7}$ is a cubic binomial in terms of $x$ and $y$. Find the value of $m$.
|
-1
|
deepscale
| 21,684
| ||
Find the largest integer that divides $m^{5}-5 m^{3}+4 m$ for all $m \geq 5$.
|
5!=120.
|
120
|
deepscale
| 3,208
| |
Six consecutive prime numbers have sum \( p \). Given that \( p \) is also a prime, determine all possible values of \( p \).
|
41
|
deepscale
| 11,353
| ||
In the Cartesian coordinate system, the equation of circle $C$ is $(x-2)^2 + y^2 = 4$, with the center at point $C$. Using the origin as the pole and the non-negative half of the $x$-axis as the initial ray, establish a polar coordinate system. The curve $C_1: \rho = -4\sqrt{3}\sin \theta$ intersects with circle $C$ at points $A$ and $B$.
(i) Find the polar equation of line $AB$.
(ii) If a line $C_2$ passing through point $C(2,0)$ with the parametric equations $\begin{cases} x=2+ \frac{\sqrt{3}}{2}t, \\ y=\frac{1}{2}t \end{cases}$ (where $t$ is the parameter) intersects line $AB$ at point $D$ and intersects the $y$-axis at point $E$, calculate the ratio $|CD| : |CE|$.
|
1 : 2
|
deepscale
| 7,722
| ||
Let $ f(x) = x^3 + x + 1$. Suppose $ g$ is a cubic polynomial such that $ g(0) = - 1$, and the roots of $ g$ are the squares of the roots of $ f$. Find $ g(9)$.
|
899
|
deepscale
| 36,558
| ||
Determine the total surface area of a cube if the distance between the non-intersecting diagonals of two adjacent faces of this cube is 8. If the answer is not an integer, round it to the nearest whole number.
|
1152
|
deepscale
| 11,740
| ||
Let $x$ be a positive real number. Define
\[
A = \sum_{k=0}^{\infty} \frac{x^{3k}}{(3k)!}, \quad
B = \sum_{k=0}^{\infty} \frac{x^{3k+1}}{(3k+1)!}, \quad\text{and}\quad
C = \sum_{k=0}^{\infty} \frac{x^{3k+2}}{(3k+2)!}.
\] Given that $A^3+B^3+C^3 + 8ABC = 2014$ , compute $ABC$ .
*Proposed by Evan Chen*
|
183
|
deepscale
| 30,071
| ||
Given that the mean score of the students in the first section is 92 and the mean score of the students in the second section is 78, and the ratio of the number of students in the first section to the number of students in the second section is 5:7, calculate the combined mean score of all the students in both sections.
|
\frac{1006}{12}
|
deepscale
| 18,918
| ||
Six 6-sided dice are rolled. What is the probability that the number of dice showing even numbers and the number of dice showing odd numbers is equal?
|
\frac{5}{16}
|
deepscale
| 34,733
| ||
In the equation $\frac{1}{7} + \frac{7}{x} = \frac{16}{x} + \frac{1}{16}$, what is the value of $x$?
|
112
|
deepscale
| 16,393
| ||
Given that $E$ and $F$ are a chord of the circle $C: x^{2}+y^{2}-2x-4y+1=0$, and $CE\bot CF$, $P$ is the midpoint of $EF$. When the chord $EF$ moves on the circle $C$, there exist two points $A$ and $B$ on the line $l: x-y-3=0$ such that $∠APB≥\frac{π}{2}$ always holds. Determine the minimum value of the length of segment $AB$.
|
4\sqrt{2}
|
deepscale
| 32,905
| ||
Makarla attended two meetings during her $9$-hour work day. The first meeting took $45$ minutes and the second meeting took twice as long. What percent of her work day was spent attending meetings?
|
1. **Convert the work day into minutes**:
Makarla's work day is $9$ hours long. Since there are $60$ minutes in an hour, the total number of minutes in her work day is:
\[
9 \times 60 = 540 \text{ minutes}
\]
2. **Calculate the total time spent in meetings**:
- The duration of the first meeting is $45$ minutes.
- The second meeting took twice as long as the first, so its duration is:
\[
2 \times 45 = 90 \text{ minutes}
\]
- The total time spent in meetings is the sum of the durations of the two meetings:
\[
45 + 90 = 135 \text{ minutes}
\]
3. **Calculate the percentage of the work day spent in meetings**:
- The fraction of the work day spent in meetings is:
\[
\frac{135 \text{ minutes}}{540 \text{ minutes}}
\]
- To convert this fraction to a percentage, multiply by $100\%$:
\[
\frac{135}{540} \times 100\% = 25\%
\]
4. **Conclusion**:
- Makarla spent $25\%$ of her work day in meetings.
Thus, the correct answer is $\boxed{25\%}$ or $\boxed{(C)}$.
|
25
|
deepscale
| 466
| |
Let the area of the regular octagon $A B C D E F G H$ be $n$, and the area of the quadrilateral $A C E G$ be $m$. Calculate the value of $\frac{m}{n}$.
|
\frac{\sqrt{2}}{2}
|
deepscale
| 32,702
| ||
Determine all six-digit numbers \( p \) that satisfy the following properties:
(1) \( p, 2p, 3p, 4p, 5p, 6p \) are all six-digit numbers;
(2) Each of the six-digit numbers' digits is a permutation of \( p \)'s six digits.
|
142857
|
deepscale
| 9,388
| ||
A triangle with side lengths $5,7,8$ is inscribed in a circle $C$. The diameters of $C$ parallel to the sides of lengths 5 and 8 divide $C$ into four sectors. What is the area of either of the two smaller ones?
|
Let $\triangle P Q R$ have sides $p=7, q=5, r=8$. Of the four sectors determined by the diameters of $C$ that are parallel to $P Q$ and $P R$, two have angles equal to $P$ and the other two have angles equal to $\pi-P$. We first find $P$ using the law of cosines: $49=25+64-2(5)(8) \cos P$ implies $\cos P=\frac{1}{2}$ implies $P=\frac{\pi}{3}$. Thus the two smaller sectors will have angle $\frac{\pi}{3}$. Next we find the circumradius of $\triangle P Q R$ using the formula $R=\frac{p q r}{4[P Q R]}$, where $[P Q R]$ is the area of $\triangle P Q R$. By Heron's Formula we have $[P Q R]=\sqrt{10(5)(3)(2)}=10 \sqrt{3}$; thus $R=\frac{5 \cdot 7 \cdot 8}{4(10 \sqrt{3})}=\frac{7}{\sqrt{3}}$. The area of a smaller sector is thus $\frac{\pi / 3}{2 \pi}\left(\pi R^{2}\right)=\frac{\pi}{6}\left(\frac{7}{\sqrt{3}}\right)^{2}=\frac{49}{18} \pi$
|
\frac{49}{18} \pi
|
deepscale
| 5,042
| |
For some positive integer \( n \), the number \( 150n^3 \) has \( 150 \) positive integer divisors, including \( 1 \) and the number \( 150n^3 \). How many positive integer divisors does the number \( 108n^5 \) have?
|
432
|
deepscale
| 17,235
| ||
Let $x,$ $y,$ and $z$ be nonnegative numbers such that $x^2 + y^2 + z^2 = 1.$ Find the maximum value of
\[2xy \sqrt{6} + 8yz.\]
|
\sqrt{22}
|
deepscale
| 36,566
| ||
A triangle has three sides of the following side lengths: $7$, $10$, and $x^2$. What are all of the positive integer values of $x$ such that the triangle exists? Separate your answers using commas and express them in increasing order.
|
2, 3, \text{ and } 4
|
deepscale
| 33,037
| ||
My grandpa has 12 pieces of art, including 4 prints by Escher and 3 by Picasso. What is the probability that all four Escher prints and all three Picasso prints will be placed consecutively?
|
\dfrac{1}{660}
|
deepscale
| 19,342
| ||
Determine the number of prime dates in a non-leap year where the day and the month are both prime numbers, and the year has one fewer prime month than usual.
|
41
|
deepscale
| 8,992
| ||
Is
\[f(x) = \frac{5^x - 1}{5^x + 1}\]an even function, odd function, or neither?
Enter "odd", "even", or "neither".
|
\text{odd}
|
deepscale
| 37,116
| ||
Using the differential, calculate to an accuracy of 0.01 the increment of the function \( y = x \sqrt{x^{2} + 5} \) at \( x = 2 \) and \( \Delta x = 0.2 \).
|
0.87
|
deepscale
| 11,386
| ||
Through points \(A(0, 14)\) and \(B(0, 4)\), two parallel lines are drawn. The first line, passing through point \(A\), intersects the hyperbola \(y = \frac{1}{x}\) at points \(K\) and \(L\). The second line, passing through point \(B\), intersects the hyperbola \(y = \frac{1}{x}\) at points \(M\) and \(N\).
What is the value of \(\frac{AL - AK}{BN - BM}\)?
|
3.5
|
deepscale
| 26,684
| ||
What is the smallest integer greater than $-\frac{17}{3}$?
|
-5
|
deepscale
| 21,644
| ||
In the Cartesian coordinate system xOy, the equation of line l is given as x+1=0, and curve C is a parabola with the coordinate origin O as the vertex and line l as the axis. Establish a polar coordinate system with the coordinate origin O as the pole and the non-negative semi-axis of the x-axis as the polar axis.
1. Find the polar coordinate equations for line l and curve C respectively.
2. Point A is a moving point on curve C in the first quadrant, and point B is a moving point on line l in the second quadrant. If ∠AOB=$$\frac {π}{4}$$, find the maximum value of $$\frac {|OA|}{|OB|}$$.
|
\frac { \sqrt {2}}{2}
|
deepscale
| 26,677
| ||
Select the shape of diagram $b$ from the regular hexagonal grid of diagram $a$. There are $\qquad$ different ways to make the selection (note: diagram $b$ can be rotated).
|
72
|
deepscale
| 26,781
| ||
Given that $A$, $B$, and $C$ are three fixed points on the surface of a sphere with radius $1$, and $AB=AC=BC=1$, the vertex $P$ of a cone $P-ABC$ with a height of $\frac{\sqrt{6}}{2}$ is also located on the same spherical surface. Determine the area of the planar region enclosed by the trajectory of the moving point $P$.
|
\frac{5\pi}{6}
|
deepscale
| 25,526
| ||
Given $m=(\sqrt{3}\sin \omega x,\cos \omega x)$, $n=(\cos \omega x,-\cos \omega x)$ ($\omega > 0$, $x\in\mathbb{R}$), $f(x)=m\cdot n-\frac{1}{2}$ and the distance between two adjacent axes of symmetry on the graph of $f(x)$ is $\frac{\pi}{2}$.
$(1)$ Find the intervals of monotonic increase for the function $f(x)$;
$(2)$ If in $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively, and $b=\sqrt{7}$, $f(B)=0$, $\sin A=3\sin C$, find the values of $a$, $c$ and the area of $\triangle ABC$.
|
\frac{3\sqrt{3}}{4}
|
deepscale
| 18,210
| ||
Mr. Earl E. Bird gets up every day at 8:00 AM to go to work. If he drives at an average speed of 40 miles per hour, he will be late by 3 minutes. If he drives at an average speed of 60 miles per hour, he will be early by 3 minutes. How many miles per hour does Mr. Bird need to drive to get to work exactly on time?
|
Let's denote the distance Mr. Bird needs to travel to work as $d$ miles and the time he should ideally take to reach on time as $t$ hours.
#### Step 1: Establish equations based on given conditions
- If he drives at 40 mph and is 3 minutes late, the time taken is $t + \frac{3}{60} = t + \frac{1}{20}$ hours. Thus, the equation for distance is:
\[
d = 40 \left(t + \frac{1}{20}\right)
\]
- If he drives at 60 mph and is 3 minutes early, the time taken is $t - \frac{3}{60} = t - \frac{1}{20}$ hours. Thus, the equation for distance is:
\[
d = 60 \left(t - \frac{1}{20}\right)
\]
#### Step 2: Equate the two expressions for $d$ to find $t$
Setting the two expressions for $d$ equal to each other:
\[
40 \left(t + \frac{1}{20}\right) = 60 \left(t - \frac{1}{20}\right)
\]
Expanding both sides:
\[
40t + 2 = 60t - 3
\]
Solving for $t$:
\[
20t = 5 \implies t = \frac{5}{20} = \frac{1}{4} \text{ hours}
\]
#### Step 3: Calculate the distance $d$
Substitute $t = \frac{1}{4}$ back into one of the original equations for $d$:
\[
d = 40 \left(\frac{1}{4} + \frac{1}{20}\right) = 40 \left(\frac{5}{20} + \frac{1}{20}\right) = 40 \left(\frac{6}{20}\right) = 40 \times 0.3 = 12 \text{ miles}
\]
#### Step 4: Determine the correct speed to arrive on time
Now, we know $d = 12$ miles and $t = \frac{1}{4}$ hours. The speed $r$ required to travel 12 miles in $\frac{1}{4}$ hours is:
\[
r = \frac{d}{t} = \frac{12}{\frac{1}{4}} = 12 \times 4 = 48 \text{ mph}
\]
Thus, Mr. Bird needs to drive at $\boxed{\textbf{(B)}\ 48}$ miles per hour to get to work exactly on time.
|
48
|
deepscale
| 1,106
| |
Calculate the surface area of the part of the paraboloid of revolution \( 3y = x^2 + z^2 \) that is located in the first octant and bounded by the plane \( y = 6 \).
|
\frac{39 \pi}{4}
|
deepscale
| 9,163
| ||
Ben rolls four fair 10-sided dice, each numbered with numbers from 1 to 10. What is the probability that exactly two of the dice show a prime number?
|
\frac{216}{625}
|
deepscale
| 35,426
| ||
How many ways are there to put 7 balls in 2 boxes if the balls are distinguishable but the boxes are not?
|
64
|
deepscale
| 21,305
| ||
A number is composed of 6 millions, 3 tens of thousands, and 4 thousands. This number is written as ____, and when rewritten in terms of "ten thousands" as the unit, it becomes ____ ten thousands.
|
603.4
|
deepscale
| 20,476
| ||
Convert the binary number $101110_2$ to an octal number.
|
56_8
|
deepscale
| 18,292
| ||
What is the smallest prime divisor of $5^{23} + 7^{17}$?
|
2
|
deepscale
| 37,815
| ||
A parallelogram $ABCD$ is inscribed in the ellipse $\frac{x^{2}}{4}+y^{2}=1$, where the slope of the line $AB$ is $k_{1}=1$. Determine the slope of the line $AD$.
|
-\frac{1}{4}
|
deepscale
| 31,444
| ||
Suppose the graph of $y=f(x)$ includes the points $(1,5),$ $(2,3),$ and $(3,1)$.
Based only on this information, there are two points that must be on the graph of $y=f(f(x))$. If we call those points $(a,b)$ and $(c,d),$ what is $ab+cd$?
|
17
|
deepscale
| 33,346
|
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