task_type stringclasses 4
values | problem stringlengths 14 5.23k | solution stringlengths 1 8.29k | problem_tokens int64 9 1.02k | solution_tokens int64 1 1.98k |
|---|---|---|---|---|
math | During the "Golden Week" holiday, the number of visitors to a zoo changes each day over a $7$-day period as shown in the table below (positive numbers indicate more visitors than the previous day, negative numbers indicate fewer visitors than the previous day):
| Date | 1st Day | 2nd Day | 3rd Day | 4th Day | 5th Day ... | 4.08 \times 10^{6} \text{ yuan} | 293 | 17 |
math | For non-negative integers $n$, the function $f(n)$ is defined by $f(0) = 0$, $f(1) = 1$, and $f(n) = f\left(\left\lfloor \frac{1}{2} n \right\rfloor \right) + n - 2\left\lfloor \frac{1}{2} n \right\rfloor$. Find the maximum value of $f(n)$ for $0 \leq n \leq 1997$. | 10 | 113 | 2 |
math | In the Chinese idioms "虚有其表", "表里如一", "一见如故", and "故弄玄虚", each Chinese character represents one of 11 consecutive non-zero natural numbers. Identical characters represent the same number, and different characters represent different numbers. The order of the numbers is such that "表" > "一" > "故" > "如" > "虚". Additionall... | 9 | 120 | 1 |
math | A circle with center $O$ has radius $10$ units, and circle $P$ has radius $4$ units. The circles are externally tangent to each other at point $Q$. Segment $TS$ is the common external tangent to circle $O$ and circle $P$ at points $T$ and $S$, respectively. What is the length of segment $OS$? Express your answer in sim... | 2\sqrt{65} | 88 | 7 |
math | Points $A, B$, and $C$ lie in that order on line $\ell$, such that $AB = 3$ and $BC = 2$. Point $H$ is such that $CH$ is perpendicular to $\ell$. Determine the length $CH$ such that $\angle AHB$ is as large as possible. | \sqrt{10} | 69 | 6 |
math | Suppose $a$ and $b$ are positive integers where $a$ has $4$ factors and $b$ has $a$ factors. If $b$ is divisible by $a$, then what is the least possible value of $b$? | 24 | 53 | 2 |
math | The Sharks initially won 2 out of 3 games, and later played $N$ more games, with a total of $3 + N$ games played. The winning percentage of the Sharks is now no less than 90%, so $\frac{2 + k}{3 + N} \ge 0.9$, where $k$ is the number of the additional games won by the Sharks. | \textbf{1} | 83 | 6 |
math | Given that the radius of a sphere is $24cm$, the height of a cone is equal to the diameter of the sphere, and the surface area of the sphere is equal to the surface area of the cone, what is the volume of the cone in $cm^{3}$? | 12288\pi | 58 | 7 |
math | For every integer $n \ge 2$ let $B_n$ denote the set of all binary $n$ -nuples of zeroes and ones, and split $B_n$ into equivalence classes by letting two $n$ -nuples be equivalent if one is obtained from the another by a cyclic permutation.(for example 110, 011 and 101 are equivalent). Determine the integers ... | n = 2 | 123 | 4 |
math | Write a quadratic equation with one root as $3$: ____. | x^2 - 3x = 0 | 13 | 10 |
math | Define a $\it{good\ word}$ as a sequence of letters that consists only of the letters A, B, C, and D — some of these letters may not appear in the sequence — where A is never immediately followed by B or D, B is never immediately followed by C, C is never immediately followed by A, and D is never immediately followed b... | 512 | 85 | 3 |
math | Let \( S \) be the set of all nonzero real numbers. Define a function \( g : S \to S \) such that
\[ g(x) + g(y) = g\left(\frac{x+y}{g(xy)}\right) \]
for all \( x, y \in S \) with \( x + y \neq 0 \). Determine the number of possible values of \( g(5) \) and their sum. | \frac{1}{5} | 95 | 7 |
math | On a rectangular sheet of paper, a picture in the shape of a "cross" formed by two rectangles $ABCD$ and $EFGH$ is drawn, with their sides parallel to the edges of the sheet. It is known that $AB=9$, $BC=5$, $EF=3$, and $FG=10$. Find the area of the quadrilateral $AFCH$. | 52.5 | 82 | 4 |
math | Given complex numbers $z_{1}=x^{2}-1+(x^{2}-3x+2)i$ and $z_{2}=x+(3-2x)i$, where $x\in\mathbb{R}$.
$(1)$ If $z_{1}$ is a pure imaginary number, find the value of the real number $x$;
$(2)$ In the complex plane, if the point corresponding to $z_{1}$ is in the fourth quadrant and the point corresponding to $z_{2}$ is ... | 1 < x < \dfrac{3}{2} | 128 | 12 |
math | 3 male students and 3 female students, a total of 6 students, stand in a row. If among the 3 female students, exactly two are adjacent to each other, then the number of different arrangements is \_\_\_\_\_\_. | 432 | 51 | 3 |
math | Elective 4-4: Coordinate System and Parametric Equations
In the Cartesian coordinate system $xOy$, the parametric equation of line $l$ is $\begin{cases}x=1+t\cos \alpha \\ y=2+t\sin \alpha\end{cases}$ (where $t$ is the parameter), in the polar coordinate system (with the same unit length as the Cartesian coordinate sy... | 2\sqrt{7} | 196 | 6 |
math | Compute the sum $\frac{4}{3} + \frac{8}{9} + \frac{18}{27} + \frac{40}{81} + \frac{88}{243} - 5$.
**A** $-\frac{1220}{972}$
**B** $-\frac{610}{486}$
**C** $-\frac{305}{243}$
**D** $-\frac{152.5}{121.5}$
**E** $-\frac{76.25}{60.75}$ | -\frac{305}{243} | 142 | 11 |
math | In a store, there are 21 white shirts and 21 purple shirts hanging in a row. Find the minimum number \( k \) such that, regardless of the initial order of the shirts, it is possible to take down \( k \) white shirts and \( k \) purple shirts so that the remaining white shirts are all hanging consecutively and the remai... | 10 | 85 | 2 |
math | A pyramid $PQRS$ has a square base $QRSU$ and congruent edges $\overline{PQ}, \overline{PR}, \overline{PS},$ and $\overline{PU}$. The angle $\angle QPR = 45^\circ$. Let $\phi$ be the measure of the dihedral angle formed by faces $PQR$ and $PRS$. Given that $\cos \phi = a + \sqrt{b}$, find $a + b$. | a+b=1 | 104 | 4 |
math | Given α ∈ (0, π/2), and √2 cos 2α = sin(α + π/4), determine the value of sin 2α. | \frac{3}{4} | 36 | 7 |
math | Given that points A and B are on the x-axis, and the two circles with centers at A and B intersect at points M $(3a-b, 5)$ and N $(9, 2a+3b)$, find the value of $a^{b}$. | \frac{1}{8} | 57 | 7 |
math | A smaller domino set is comprised only of integers 0 through 6. Each integer within this range pairs with every other integer within the same range exactly once, including itself, to form a complete set. A $\textit{double}$ in this context is a domino where the pair contains the same integer on both squares. What is th... | \frac{1}{3} | 98 | 7 |
math | Given the function f(x) = e^x + ae^-x, where 'a' is a constant. If f(x) is an odd function, find the value of 'a'. If f(x) is an increasing function on R, find the range of 'a'. | a \in (-\infty, 0] | 57 | 11 |
math | Given the sequence $\{a\_n\}$ satisfies $a\_1=1$, and for any $n∈N^∗$, $a_{n+1}=a\_n+n+1$, find the value of $$\frac {1}{a_{1}}+ \frac {1}{a_{2}}+…+ \frac {1}{a_{2017}}+ \frac {1}{a_{2016}}+ \frac {1}{a_{2019}}$$. | \frac{2019}{1010} | 109 | 13 |
math | A circle is positioned inside the parabola with equation \(y = 4x^2\) so that it is tangent to the parabola at two points. How much higher is the center of the circle than the points of tangency? | \frac{1}{8} | 50 | 7 |
math | Observe the following expressions: $1\times 3+1=2^{2}$, $2\times 4+1=3^{2}$, $3\times 5+1=4^{2}$, $4\times 6+1=5^{2}$, $\ldots $,
$(1)$ Please write out the 6th expression according to the above rule: ______;
$(2)$ Please write out the $n$th expression: ______;
$(3)$ Calculate: $(1+\frac{1}{1\times 3})\times (1+\... | \frac{99}{50} | 176 | 9 |
math | If $\log_{36}(x - 6) = \frac{1}{2}$, find $\frac{1}{\log_{x}6}$. | \frac{3}{2} | 35 | 7 |
math | Given that the hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{4} = 1$ ($a > 0$) has an eccentricity of $\frac{\sqrt{5}}{2}$, and the points $F_1$ and $F_2$ are its foci on the left and right side, respectively. Point $P$ has coordinates $(5, y_0)$ and point $Q$ is the point on the hyperbola symmetrical to $P$ with respe... | 6\sqrt{5} | 137 | 6 |
math | Given the function $f(x)=\sin (2x+\varphi)$, if the graph is shifted to the left by $\dfrac {\pi}{6}$ units and the resulting graph is symmetric about the $y$-axis, determine the possible value of $\varphi$. | \dfrac {\pi}{6} | 58 | 7 |
math | Piravena must make a trip from $A$ to $B$, then from $B$ to $C$, and finally from $C$ back to $A$. The cities form a right-angled triangle, with $C$ at $3000\text{ km}$ from $A$, $B$ at $4000\text{ km}$ from $A$, and the right angle at $C$. She starts her trip by taking a bus from $A$ to $C$. Afterwards, she needs to g... | \$573.32 | 211 | 7 |
math | Given the equation $x^{2}-2mx+4=0$, both of its real roots are greater than $1$. Determine the range of the real number $m$. | [2, \frac{5}{2}) | 36 | 10 |
math | How many 4-digit numbers have the property that the units digit is at least twice the tens digit, and the thousands digit is odd? | 1500 | 28 | 4 |
math | Given that the average score for the morning group is 90 and the evening group is 80, and the ratio of the number of students in the morning group to the evening group is $\frac{4}{5}$, calculate the overall average score of the students from both groups. | \frac{760}{9} | 59 | 9 |
math |
In the right triangle \(ABC\), the altitude \(BH\) is drawn to the hypotenuse \(AC\). Points \(X\) and \(Y\) are the centers of the circles inscribed in triangles \(ABH\) and \(CBH\) respectively. The line \(XY\) intersects the legs \(AB\) and \(BC\) at points \(P\) and \(Q\). Find the area of triangle \(BPQ\), given ... | \frac{h^2}{2} | 97 | 9 |
math | Given the non-empty subset $A$ of the set $\{1, 2, 3, 4, 5\}$ has property $P$: if $a \in A$, then $6-a \in A$, determine the number of subsets $A$. | 7 | 56 | 1 |
math | Round 9.874 to the nearest integer, to the nearest tenth, and to the nearest hundredth. | 9.87 | 24 | 4 |
math | Glue two small cuboids, each with a length of 3cm, a width of 2cm, and a height of 1cm, into one large cuboid. Then, cut it into two small cuboids of the same size. The surface area of the final small cuboid could be at most how many square centimeters larger than the surface area of the original small cuboid. | 10 | 81 | 2 |
math | Given four numbers $1$, $2$, $a$, $b$, with a median of $3$ and a mean of $4$, find the value of $ab$. | 36 | 36 | 2 |
math | Each of the twelve letters in "MATHEMATICS" 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 "THEORY"? Express your answer as a common fraction. | \frac{1}{3} | 60 | 7 |
math | Given the function $y=\sin (2x+1)$, determine the direction and magnitude of the horizontal shift required to obtain this graph from the graph of the function $y=\sin 2x$. | \frac{1}{2} | 42 | 7 |
math | Let $p$, $q$, and $r$ be constants, and suppose that the inequality \[\frac{(x-p)(x-q)}{x-r} \le 0\] is true if and only if $x > 2$ or $3 \le x \le 5$. Given that $p < q$, find the value of $p + q + 2r$. | 12 | 82 | 2 |
math | If the monotonically decreasing interval of the function $f(x)=a(x^{3}-x)$ is $\left(- \frac{ \sqrt{3}}{3}, \frac{ \sqrt{3}}{3}\right)$, then the range of values for $a$ is $\_\_\_\_\_\_\_\_\_\_.$ | a > 0 | 72 | 4 |
math | There are four complex numbers $z$ such that
\[ z \overline{z}^3 + \overline{z} z^3 + z \overline{z} = 500,\]
and both the real and imaginary parts of $z$ are integers. These four complex numbers are plotted in the complex plane. Find the area of the quadrilateral formed by these numbers as vertices. | 48 | 85 | 2 |
math | In triangle $DEF,$ $d = 8,$ $e = 15,$ and $f = 17.$ Let $J$ be the incenter.
Find the barycentric coordinates (x, y, z) such that
\[\overrightarrow{J} = x \overrightarrow{D} + y \overrightarrow{E} + z \overrightarrow{F},\]
where $x + y + z = 1.$ | \left( \frac{8}{40}, \frac{15}{40}, \frac{17}{40} \right) | 96 | 32 |
math | A circle inscribed in a right triangle ABC touches the legs CA and CB at points P and Q, respectively. The line PQ intersects a line that passes through the center of the inscribed circle and is parallel to the hypotenuse at point N. M is the midpoint of the hypotenuse. Find the measure of angle MCN. | 90^\circ | 69 | 4 |
math | Given the function $y = x^3 + x^2 + px + q$, determine $q$ such that the smallest possible value of $y$ is zero.
**A)** $-\frac{1}{27}$
**B)** $0$
**C)** $-\frac{2}{27}$
**D)** $\frac{1}{3}$ | -\frac{2}{27} | 78 | 8 |
math | Let $\theta=\frac{2\pi}{2015}$ , and suppose the product \[\prod_{k=0}^{1439}\left(\cos(2^k\theta)-\frac{1}{2}\right)\] can be expressed in the form $\frac{b}{2^a}$ , where $a$ is a non-negative integer and $b$ is an odd integer (not necessarily positive). Find $a+b$ .
*2017 CCA Math Bonanza Tiebreaker Round #3... | 1441 | 121 | 4 |
math | Given the parabola $C$: $y^{2}=4x$, $O$ is the coordinate origin, $F$ is the focus of $C$, and $P$ is a point on $C$. If $\triangle OPF$ is an isosceles triangle, then $|PO|=$ _____. | 1 | 67 | 1 |
math | Given a sequence $\{a_n\}$ whose sum of the first $n$ terms is $S_n$, and $S_n = n^2 - n$, in a positive geometric sequence $\{b_n\}$, $b_2 = a_2$, $b_4 = a_5$.
(1) Find the general formula for $\{a_n\}$ and $\{b_n\}$.
(2) Let $c_n = a_n \times b_n$, find the sum of the first $n$ terms of the sequence $\{c_n\}$, ... | T_n = (n-2) \cdot 2^{n+1} + 4 | 130 | 20 |
math | There are \_\_\_\_\_\_ six-digit numbers composed of three distinct odd numbers, two 2's, and one 0. (Answer with a number) | 3000 | 35 | 4 |
math | For which natural number $k$ does the expression $\frac{k^{2}}{1.001^{k}}$ reach its maximum value? | 2000 | 31 | 4 |
math | Given $f(c)=(c-a)(c-b)$, where $a+b=1-c$ and $c\geqslant 0$, $a\geqslant 0$, $b\geqslant 0$, find the range of $f(c)$. | \left[-\dfrac{1}{8},1\right] | 60 | 15 |
math | If $a^{2x} = \sqrt{2} - 1$, then $\frac{a^{3x} + a^{-3x}}{a^{x} + a^{-x}}$ equals \_\_\_\_\_\_. | 2\sqrt{2} - 1 | 51 | 9 |
math | For all real numbers $x$ except $x=0$ and $x=1$, the function $f(x)$ is defined by
\[f \left( \frac{x}{x - 1} \right) = \frac{1}{x}.\]
Suppose $0\leq t\leq \frac{\pi}{2}$. What is the value of $f(\tan^2 t)$? | \cot^2 t | 89 | 5 |
math | Given the sequence defined by \(a_1=1\) and \(a_k=\left\lfloor\sqrt{a_1 + a_2 + \ldots + a_{k-1}}\right\rfloor\), find \(a_{1000}\). Note: \(\left\lfloor \cdot \right\rfloor\) denotes the floor function, which returns the greatest integer less than or equal to the expression inside. | 31 | 94 | 2 |
math | Given the power function f(x) passes through the point (3, 9), determine the interval of monotonic increase for the function f(x). | [0, +\infty) | 30 | 8 |
math | Which of the following is not the sum of two primes?
A) 5
B) 7
C) 9
D) 11
E) 13 | 11 | 38 | 2 |
math | Given the function $f(x)= \begin{cases} x+2 & (x\leqslant -1) \\ x^{2} & (-1 < x < 2) \\ 2x & (x\geqslant 2) \end{cases}$, find the value of $f( \frac {1}{f(2)})=$ \_\_\_\_\_\_, and if $f(x)=3$, find the value of $x=$ \_\_\_\_\_\_. | x= \sqrt {3} | 107 | 7 |
math | Given that the eccentricity of the ellipse $C:\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a > b > 0)$ is $\frac{\sqrt{2}}{2}$, and it passes through the point $P(\sqrt{2},1)$. The line $y=\frac{\sqrt{2}}{2}x+m$ intersects the ellipse at two points $A$ and $B$.
(1) Find the equation of the ellipse $C$; (2) Find the maxi... | \sqrt{2} | 128 | 5 |
math | A mother purchases 5 blue plates, 2 red plates, 2 green plates, and 1 orange plate. How many ways are there for her to arrange these plates for dinner around her circular table if she doesn't want the 2 green plates to be adjacent? | 588 | 55 | 3 |
math | Given lines $l\_1$: $x+ay+6=0$ and $l\_2$: $(a-2)x+3y+2a=0$, the necessary and sufficient condition for $l\_1$ to be parallel to $l\_2$ is $a=$ _____ . | -1 \text{ or } 3 | 62 | 9 |
math | Students A, B, and C are running for the president of the student council at their school. During the election, 1500 valid votes were received, and the results of the first 1000 votes were as follows: A received 350 votes, B received 370 votes, and C received 280 votes. To ensure that A wins the election with the most ... | 261 | 104 | 3 |
math | A bus arrives randomly between 3:30 pm and 4:30 pm, waits for 40 minutes, and then departs. If Sara also arrives randomly between 3:30 pm and 4:30 pm, what is the probability that the bus will still be there when she arrives? | \frac{2}{3} | 66 | 7 |
math | How many zeros are in the expansion of $99,\!999,\!998^2$? | 8 | 25 | 1 |
math | The real numbers $a_i$ ($i=1,2,3,4,5,6$) satisfy $(a_2-a_1)^2+(a_3-a_2)^2+(a_4-a_3)^2+(a_5-a_4)^2+(a_6-a_5)^2=1$. Find the maximum value of $(a_5+a_6)-(a_1+a_4)$. | 2\sqrt{2} | 93 | 6 |
math | Given that \(x\) is real and \(x^3 + \frac{1}{x^3} = 116\), find the value of \(x + \frac{1}{x}\). | 4 | 44 | 1 |
math | Write the integers from 1 to 6 on the six faces of a cube so that any pair of consecutive numbers - including the pair 6 and 1 - are on adjacent faces. How many different arrangements are possible? (Two arrangements are not considered different if one can be transformed into the other by some symmetry of the cube - suc... | 2 | 97 | 1 |
math | If $a$ and $b$ are opposite numbers ($b$ is not $0$), $c$ and $d$ are reciprocals, and $m$ is a natural number with an absolute value less than $2$, then $m-cd+\frac{a+b}{2023}+\frac{a}{b}=\_\_\_\_\_\_$. | -1 \text{ or } -2 | 80 | 9 |
math | Two dice are thrown, and the number of points facing up on each die are represented by $m$ and $n$ respectively. Let $\overset{→}{a}=(m,n)$. The probability that $|\overset{→}{a}|<5$ is _________. | \dfrac{13}{36} | 60 | 9 |
math | There are $6$ light bulbs in a box, of which $2$ are defective and $4$ are non-defective. If two bulbs are drawn randomly with replacement, find the probability of the following events:
$(1)$ Both drawn bulbs are defective;
$(2)$ One drawn bulb is defective and the other is non-defective;
$(3)$ At least one of the draw... | \dfrac{8}{9} | 84 | 7 |
math | \(8.469 \sin^{4} x + 2 \cos^{3} x + 2 \sin^{2} x - \cos x + 1 = 0\). | x = \pi(2k + 1) | 42 | 11 |
math | The infinite sequence \(T = \{t_1, t_2, t_3, \ldots\}\) is defined by \(t_1 = 3\) and \(t_n = 3^{t_{n-1}}\) for each integer \(n > 1\). What is the remainder when \(t_{100}\) is divided by 7? | 6 | 81 | 1 |
math | Observing the temperatures recorded in Cesenatico during the December and January, Stefano noticed an interesting coincidence: in each day of this period, the low temperature is equal to the sum of the low temperatures the preceeding day and the succeeding day.
Given that the low temperatures in December $3$ and Janu... | -3 | 108 | 2 |
math | Simplify completely: $$\sqrt[3]{40^3+60^3+80^3}$$. | 20\sqrt[3]{99} | 26 | 10 |
math | Given the function $f(x)=e^{2x}-1-2x-kx^{2}$,
(I) Find the monotonic intervals of $f(x)$ when $k=0$;
(II) Find the range of $k$ such that $f(x) \geqslant 0$ holds for all $x \geqslant 0$;
(III) Compare the magnitude relationship between $\frac{e^{2n}-1}{e^{2}-1}$ and $\frac{2n^{3}+n}{3}$ $(n \in \mathbb{N}^*)$ and prov... | \frac{e^{2n}-1}{e^{2}-1}\geqslant \frac{2n^{3}}{3}+ \frac{n}{3} | 173 | 38 |
math | There are \( n \) vectors in space such that any pair of them forms an obtuse angle. What is the maximum possible value of \( n \)? | 4 | 32 | 1 |
math | The sum of all integers between 50 and 350 that end in 1 is | 5880 | 20 | 4 |
math | Compute the definite integral:
$$
\int_{\pi / 2}^{\pi} 2^{4} \cdot \sin ^{6} x \cos ^{2} x \, dx
$$ | \frac{5\pi}{16} | 46 | 10 |
math | Given the function f(x) = ae^x - ln(x) - 1, if x = 1 is the extreme point of f(x), find the value of a and the interval where f(x) is monotonically increasing. | (1, +\infty) | 49 | 8 |
math | Add the square of the smallest area from squares of size $1 \times 1, 2 \times 2,$ and $3 \times 3,$ such that the number of squares of each size is the same. | 14 | 46 | 2 |
math | Isabella uses one-foot cubical blocks to build a rectangular fort that is $12$ feet long, $10$ feet wide, and $5$ feet high. The floor and the four walls are all one foot thick. How many blocks does the fort contain? | 280 | 56 | 3 |
math | The maximum value of the function $f(x)=\sqrt{3}\sin2x-2\cos^2x$ on the interval $[0,\frac{π}{2}]$ is $\sqrt{3}$. | 1 | 47 | 1 |
math | Given that $\overrightarrow{a}$, $\overrightarrow{b}$, and $\overrightarrow{c}$ are all unit vectors, and $\overrightarrow{a} \cdot \overrightarrow{b} = 0$, $( \overrightarrow{a} - \overrightarrow{c} ) \cdot ( \overrightarrow{b} - \overrightarrow{c} ) \leqslant 0$, find the maximum value of $| \overrightarrow{a} + \ove... | 1 | 116 | 1 |
math | Given the water tower in real life is 60 meters high and the spherical top holds 200,000 liters of water, and Logan wants his model's water tower to hold 0.2 liters of water, determine the height of Logan's model tower. | 0.6 | 57 | 3 |
math | Given an odd function $f(x)$ defined on $\mathbb{R}$ with derivative $f'(x)$, for all $x\in (-\infty,0]$, it always holds that $xf(x) < f(-x)$. Define $F(x)=xf(x)$. Find the range of real numbers $x$ that satisfy $F(3) > F(2x-1)$. | (-1, 2) | 86 | 6 |
math | In $\triangle ABC$, $\angle ACB$ is an obtuse angle, $AC=BC=1$, and $\overrightarrow {CO}=x \overrightarrow {CA}+y \overrightarrow {CB}$ with $x+y=1$. The minimum value of the function $f(m)=| \overrightarrow {CA}-m \overrightarrow {CB}|$ is $\frac { \sqrt {3}}{2}$. Find the minimum value of $| \overrightarrow {CO}|$. | \frac {1}{2} | 104 | 7 |
math | Given $f(x) = \begin{cases} 1 & \text{for } x \in [0,1], \\ x-3 & \text{for } x \notin [0,1], \end{cases}$ find the range of values of $x$ for which $f(f(x))=1$. | [0,1] \cup [3,4] \cup \{7\} | 69 | 19 |
math | Given the parabola $x^{2}=4y$ with focus $F$ and point $A(-1,8)$, and $P$ is a point on the parabola, determine the minimum value of $|PA|+|PF|$. | 9 | 55 | 1 |
math | The area of the lunar crescent shape bounded by the portion of the circle of radius 5 and center (0,0), the portion of the circle with radius 2 and center (0,2), and the line segment from (0,0) to (5,0). | \frac{21\pi}{4} | 58 | 10 |
math | Given a bag containing cards labeled with numbers $1$, $2$, and $3$ each, a card is drawn each time, the number is recorded, and then the card is put back. The drawing stops when all three numbered cards have been drawn. Determine the probability that the drawing stops exactly after $5$ draws. | \frac{14}{81} | 66 | 9 |
math | The base of a triangular piece of paper $ABC$ is $15\text{ cm}$ long. The paper is folded down over the base, with the crease $DE$ parallel to the base of the paper. The area of the triangle that projects below the base is $25\%$ that of the area of triangle $ABC$. Calculate the length of $DE$, in cm. | 7.5\text{ cm} | 82 | 8 |
math | (Ⅰ) Given that the central angle $\alpha$ of sector OAB is $120^\circ$ and the radius $r=6$, find the length of arc AB and the area of the sector;
(Ⅱ) Given that the perimeter of a sector is 20cm, what should be the central angle for it to have the maximum area, and what is the maximum area? | 25 | 84 | 2 |
math | In a regular dodecagon \(ABCDEFGHIJKL\), \(M\) and \(N\) are the midpoints of \(\overline{CD}\) and \(\overline{GH}\) respectively. Compute \([ABCM]/[EFGHM]\). | \frac{2}{3} | 56 | 7 |
math | Given that $| \overrightarrow{a}|=| \overrightarrow{b}|=| \overrightarrow{c}|=1$, and $ \overrightarrow{a}+ \overrightarrow{b}+ \sqrt {3} \overrightarrow{c}=0$, find the value of $ \overrightarrow{a} \overrightarrow{b}+ \overrightarrow{b} \overrightarrow{c}+ \overrightarrow{c} \overrightarrow{a}$. | \dfrac {1}{2}- \sqrt {3} | 102 | 12 |
math | Given that sequence ${a_{n}}$ is an equal product sequence, and $a_{1}=1$, $a_{2}=2$, common product $K=8$, find the value of $a_{1}+a_{2}+a_{3}+…+a_{12}$. | 28 | 65 | 2 |
math | Let the function $f(x) = 2x^3 - 3(a+1)x^2 + 6ax + 8$, where $a \in \mathbb{R}$.
1. If $f(x)$ has an extreme value at $x=3$, find the value of the constant $a$.
2. If $f(x)$ is increasing on $(-\infty, 0)$, find the range of values for $a$. | [0, +\infty) | 98 | 8 |
math | Given the ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) \((a>b>0)\) with eccentricity \(\frac{1}{2}\), where \(F_{1}\) and \(F_{2}\) are the left and right foci respectively, a line passing through \(F_{2}\) intersects the ellipse at points \(A\) and \(B\). If the maximum area of \(\triangle F_{1}AB\) is 6, fin... | \frac{x^2}{8} + \frac{y^2}{6} = 1 | 121 | 21 |
math | Let $m$ and $n$ be positive integers where $m$ has $d$ digits in base ten and $d\leq n$ . Find the sum of all the digits (in base ten) of the product $(10^n-1)m$ . | 9n | 67 | 4 |
math | Given rational numbers $a$, $b$, $c$ satisfy $a - b + c = 3$ and $a^2 + b^2 + c^2 = 3$, calculate the value of $a^3 + b^3 + c^3$. | 1 | 57 | 1 |
math | Given that $\{a_n\}$ is an arithmetic sequence, and $a_7 - 2a_4 = -1$, $a_3 = 0$, determine the common difference $d$. | -\frac{1}{2} | 43 | 7 |
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