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 | Given a triangle $ABC$ with internal angles $A$, $B$, $C$ opposite to sides $a$, $b$, $c$ respectively, and $A=2C$.
(Ⅰ) If $\triangle ABC$ is an acute triangle, find the range of $\frac{a}{c}$.
(Ⅱ) If $b=1, c=3$, find the area of $\triangle ABC$. | \sqrt{2} | 88 | 5 |
math | Let \( f(x) \) be a function defined on \( \mathbf{R} \). If there exist two distinct real numbers \( x_{1}, x_{2} \in \mathbf{R} \) such that \( f\left(\frac{x_{1}+x_{2}}{2}\right) = \frac{f\left(x_{1}\right) + f\left(x_{2}\right)}{2} \), then the function \( f(x) \) is said to have property \( \mathrm{P} \). Determin... | B | 264 | 2 |
math | Given that point F is the focus of the parabola $y^2=-8x$, O is the origin, point P is a moving point on the directrix of the parabola, and point A is on the parabola with $|AF|=4$, find the minimum value of $|PA|+|PO|$. | 2\sqrt{13} | 71 | 7 |
math | The number of solutions to the equation $a^x = x^2 - 2x - a$ (where $a > 0$ and $a \neq 1$) is to be determined. | 2 | 45 | 1 |
math | The graph of the function $y=g(x)$ is given. For all $x > 5$, it is observed that $g(x) > 0.1$. If $g(x) = \frac{x^2}{Ax^2 + Bx + C}$, where $A, B, C$ are integers, determine $A+B+C$ knowing that the vertical asymptotes occur at $x = -3$ and $x = 4$. | -108 | 95 | 4 |
math | A certain integer has $5$ digits when written in base $16$. The same integer has $d$ digits when written in base $2$. What is the sum of all possible values of $d$? | 74 | 44 | 2 |
math | Given the function $f(x)=|x-3|+|2x-4|-a$.
(I) Solve the inequality $f(x) > 0$ when $a=6$;
(II) If the solution set of the inequality $f(x) < 0$ regarding $x$ is not empty, find the range of the real number $a$. | a > 1 | 78 | 4 |
math | $(1)$ For the polynomial in terms of $x$ and $y$: $4x^{2}y^{m+2}+xy^{2}+\left(n-2\right)x^{2}y^{3}+xy-4$, which is a seventh-degree quadrinomial, find the values of $m$ and $n$;<br/>$(2)$ For the polynomial in terms of $x$ and $y$: $\left(5a-2\right)x^{3}+\left(10a+b\right)x^{2}y-x+2y+7$, which does not contain a cubic... | -2 | 143 | 2 |
math | If the inequality $x^3 + x^2 + a < 0$ holds for all $x \in [0, 2]$, then the range of values for $a$ is ______. | a < -12 | 43 | 5 |
math | The opening ceremony of the 2022 Beijing Winter Olympics was held at the National Stadium in China on the evening of February 4th. The TV broadcast audience reached 316 million people. The number written on the blank line is: ______ people. The closing ceremony was held on the evening of February 20th, with an audience... | 2.4 \text{ billion people} | 91 | 9 |
math | Without using any tables, find the exact value of the product:
\[ P = \cos \frac{\pi}{15} \cos \frac{2\pi}{15} \cos \frac{3\pi}{15} \cos \frac{4\pi}{15} \cos \frac{5\pi}{15} \cos \frac{6\pi}{15} \cos \frac{7\pi}{15}. \] | 1/128 | 100 | 5 |
math | Given that the probabilities of event A, B, and C are $P(A)=0.7$, $P(B)=0.2$, and $P(C)=0.1$, calculate the probability of the event "the drawn product is not a first-class product". | 0.3 | 55 | 3 |
math | In a triangle, one angle measures $45^\circ$, and the triangle is scalene. An exterior angle at one vertex is $135^\circ$. Find the sum of the two possible values for the smallest angle in the triangle. | 90^\circ | 50 | 4 |
math | In $\triangle ABC$, let the sides opposite to angles $A$, $B$, and $C$ be $a$, $b$, and $c$ respectively. Given that $a\cos B=3$ and $b\sin A=4$.
(I) Find $\tan B$ and the value of side $a$;
(II) If the area of $\triangle ABC$ is $S=10$, find the perimeter $l$ of $\triangle ABC$. | 10 + 2\sqrt{5} | 99 | 10 |
math | Solve the following problems:
$(1)$ If $\left(x-3\right)\left(x-4\right)=x^{2}+mx+n$, then the value of $m$ is ______, and the value of $n$ is ______;
$(2)$ If $(x+a)(x+b)=x^2-3x+\frac{1}{3}$,
① Find the value of $\left(a-1\right)\left(b-1\right)$;
② Find the value of $\frac{1}{a^2}+\frac{1}{b^2}$. | 75 | 129 | 2 |
math | In the graph of function $f\left(x\right)= \dfrac{1}{2}\sin \left(2\omega x+\varphi \right)$ ($\omega \gt 0$, $|\varphi | \lt \dfrac{\pi }{2}$), which is shifted to the right by $\dfrac{\pi }{12}$ units to obtain the graph of $g\left(x\right)$, the graph of $g\left(x\right)$ is symmetric about the origin. For vector ... | \left[\dfrac{\pi}{6}, \dfrac{2}{3}\pi\right], \left[\dfrac{7}{6}\pi, \dfrac{5}{3}\pi\right] | 417 | 46 |
math | Let \(\mathbf{a}\), \(\mathbf{b}\), and \(\mathbf{c}\) be unit vectors such that the angle between \(\mathbf{a}\) and \(\mathbf{b}\) is \(\alpha\), and the angle between \(\mathbf{c}\) and \(\mathbf{a} \times \mathbf{b}\) is also \(\alpha\). If \(\mathbf{b} \cdot (\mathbf{c} \times \mathbf{a}) = \frac{1}{8}\), find the... | 7.24^\circ | 140 | 6 |
math |
Calculate the areas of the regions bounded by the curves given in polar coordinates.
$$
r=\cos 2 \phi
$$ | \frac{\pi}{2} | 27 | 7 |
math | The circumference of Earth at the equator is approximately $2 \pi \cdot 5000$ miles. If a jet flies once around Earth at an average speed of 650 miles per hour, determine the number of hours of flight. | 48 | 52 | 2 |
math | Write the equations of the tangent and normal to the curve \(x^{2} - 2xy + 3y^{2} - 2y - 16 = 0\) at the point \((1, 3)\). | 7x + 2y - 13 = 0 | 50 | 13 |
math | Let $S$ be a list of positive integers--not necessarily distinct--in which the number $68$ appears. The average (arithmetic mean) of the numbers in $S$ is $56$ . However, if $68$ is removed, the average of the remaining numbers drops to $55$ . What is the largest number that can appear in $S$ ? | 649 | 81 | 3 |
math | One of each tree: poplar, willow, locust, birch, and sycamore is planted in a row with 1 meter between each adjacent tree. The distance between the poplar and the willow is equal to the distance between the poplar and the locust. The distance between the birch and the poplar is equal to the distance between the birch a... | 2 | 99 | 1 |
math | The sequence \(a_n\) is defined by \(a_1 = 20\), \(a_2 = 30\), and \(a_{n+1} = 3a_n - a_{n-1}\). Find all \(n\) for which \(5a_{n+1} \cdot a_n + 1\) is a perfect square. | n = 3 | 78 | 4 |
math | Determine the smallest positive four-digit number that satisfies the following system of congruences:
\begin{align*}
5x &\equiv 25 \pmod{20} \\
3x+10 &\equiv 19 \pmod{7} \\
x+3 &\equiv 2x \pmod{12}
\end{align*} | 1011 | 80 | 4 |
math | Given that the monotonically decreasing function $f(x)$ defined on $(-\infty, 3]$ satisfies $f(1+\sin ^{2}x)\leqslant f(a-2\cos x)$ for all real numbers $x$, determine the range of values for $a$. | (-\infty,-1] | 64 | 7 |
math | On an island, there are knights, who always tell the truth, and liars, who always lie. There are 15 islanders in a room. Each of the islanders in the room made two statements: "Among my acquaintances in this room, there are exactly six liars" and "Among my acquaintances in this room, there are no more than seven knight... | 9 \text{ knights} | 88 | 6 |
math | Select three different numbers from the set $\{1, 2, 3, \ldots, 10\}$ such that these three numbers form a geometric sequence. How many such geometric sequences are there? | 6 | 44 | 1 |
math | Let \( A B C D E F \) be a regular hexagon. Let \( G \) be a point on \( E D \) such that \( E G = 3 G D \). If the area of \( A G E F \) is 100, find the area of the hexagon \( A B C D E F \). | 240 | 75 | 3 |
math | When a certain unfair die is rolled, an even number is $3$ times as likely to appear as an odd number. The die is rolled twice. Determine the probability that the sum of the numbers rolled is even. | \frac{5}{8} | 44 | 7 |
math | Given a $4 \times 4$ square grid, where each unit square is painted white or black with equal probability and then rotated $180\,^{\circ}$ clockwise, calculate the probability that the grid becomes entirely black after this operation. | \frac{1}{65536} | 52 | 11 |
math | A ray of light passing through the point \(A = (-2, 8, 10)\), reflects off the plane \(x + y + z = 15\) at point \(B\), and then passes through the point \(C = (4, 4, 7)\). Find the point \(B\). | \left(4, 4, 7\right) | 69 | 13 |
math | Given that \( x \) and \( y \) are real numbers and \( x^2 - xy + y^2 = 1 \), find the maximum value of \( x + 2y \). | \frac{2\sqrt{21}}{3} | 43 | 13 |
math | Call a $4$-digit number geometric if it has $4$ distinct digits which, when read from left to right, form a geometric sequence. Find the difference between the largest and smallest geometric numbers. | 7173 | 42 | 4 |
math | Given that the number $3103$ has four digits, determine the number of different four-digit numbers that can be formed by rearranging these digits. | 9 | 32 | 1 |
math | Let's solve the following system of equations:
$$
\begin{aligned}
& x + y^{2} = a \\
& y + x^{2} = a
\end{aligned}
$$
For which values of \(a\) do we obtain real solutions? | a\geq \frac{3}{4} | 56 | 11 |
math | All three vertices of $\bigtriangleup ABC$ lie on the parabola defined by $y = 2x^2$, with $A$ at the origin and $\overline{BC}$ horizontal. The area of the triangle is $72$. What is the length of $BC$?
**A)** $2 \times \sqrt[3]{18}$
**B)** $2 \times \sqrt[3]{24}$
**C)** $2 \times \sqrt[3]{36}$
**D)** $4 \times \sqrt[3... | 2 \times \sqrt[3]{36} | 140 | 11 |
math | Given vector $\overrightarrow{a} = (2\cos\phi, 2\sin\phi)$, where $\phi \in \left( \frac{\pi}{2}, \pi \right)$, and vector $\overrightarrow{b} = (0, -1)$, find the angle between vector $\overrightarrow{a}$ and $\overrightarrow{b}$. | \frac{3\pi}{2}-\phi | 80 | 11 |
math | Given that there are 4 qualified and 2 defective products, determine the probability of finding the last defective product exactly on the fourth inspection when selectins products one at a time and not returning them after each selection. | \frac{1}{5} | 43 | 7 |
math | The longer leg of a right triangle is 3 feet shorter than three times the length of the shorter leg. The area of the triangle is 90 square feet. What is the length of the hypotenuse, in feet? | \sqrt{829} | 47 | 7 |
math | Suppose that at some point Joe B. has placed 2 black knights on the original board, but gets bored of chess. He now decides to cover the 34 remaining squares with 17 dominos so that no two overlap and the dominos cover the entire rest of the board. For how many initial arrangements of the two pieces is this possible? | 1024 | 73 | 4 |
math | Given the function $f(x)=|x-1|+|x+1|-1$.
- $(1)$ Find the solution set for $f(x)\leqslant x+1$;
- $(2)$ If the inequality $f(x)\geqslant \frac {|a+1|-|2a-1|}{|a|}$ holds for any real number $a\neq 0$, find the range of real numbers $x$. | (-\infty,-2]\cup[2,+\infty) | 98 | 15 |
math | The graph of the function $y=g(x)$ is shown below. For all $x > 5$, it is true that $g(x) > 0.5$. If $g(x) = \frac{x^2}{Dx^2 + Ex + F}$, where $D, E,$ and $F$ are integers, then find $D+E+F$. Assume the function has vertical asymptotes at $x = -3$ and $x = 4$ and a horizontal asymptote below 1 but above 0.5. | -24 | 114 | 3 |
math | Square $ABCD$ has sides of length 2. Set $S$ is the set of all line segments that have length 2 and whose endpoints are on adjacent sides of the square. The midpoints of the line segments in set $S$ enclose a region whose area to the nearest hundredth is $k$. Find $100k$.
| 86 | 74 | 2 |
math | Consider a polynomial whose roots, when plotted in the complex plane, form a square. The polynomial is given by:
\[ z^4 + 4z^3 + (6 - 6i)z^2 + (4 - 8i)z + (1 - 4i) = 0. \]
Find the area of the square formed by plotting these roots in the complex plane. | 2 | 83 | 1 |
math | Let $P(x) = (x-3)(x-4)(x-5)$. Determine how many polynomials $Q(x)$ exist such that there is a polynomial $R(x)$ of degree 3 for which $P(Q(x)) = P(x) \cdot R(x)$. | 6 | 61 | 1 |
math | Given that $\frac{\cos 2x}{\sqrt{2} \cos (x + \frac{\pi}{4})} = \frac{1}{5}$, where $0 < x < \pi$, find the value of $\tan x$. | -\frac{4}{3} | 54 | 7 |
math | Let $S_{n}$ be the sum of the first $n$ terms of the sequence $\{a_{n}\}$, and it is known that $S_n=n^2+n$.
$(1)$ Find the general formula for $\{a_{n}\}$.
$(2)$ Let $T_{n}=a_{1}\cdot {3}^{a_{1}}+a_{2}\cdot {3}^{a_{2}}+\ldots \ldots +a_{n}\cdot {3}^{a_{n}}$, find $T_{n}$. | T_{n} = \frac{1}{32}\left[(8n-1)\cdot 9^{n+1} + 9\right] | 122 | 34 |
math | If the line $ax - by + 2 = 0$ $(a > 0, b > 0)$ passes through the center of the circle ${x}^{2} + {y}^{2} + 4x - 4y - 1 = 0$, find the minimum value of $\frac{2}{a} + \frac{3}{b}$. | 5 + 2 \sqrt{6} | 81 | 9 |
math | Calculate the sides $a, b$, and $c$ of a triangle $ABC$ if the angles $A, B$, and $C$ of the triangle and the area $T$ of the triangle are given. | a = \sqrt{\frac{2T \sin A}{\sin B \sin C}}, \quad b = \sqrt{\frac{2T \sin B}{\sin C \sin A}}, \quad c = \sqrt{\frac{2T \sin C}{\sin A \sin B}} | 45 | 64 |
math | What is the greatest common divisor of $2^{1025} - 1$ and $2^{1056} - 1$? | 2147483647 | 33 | 10 |
math | Two distinct, real, infinite geometric series each have a sum of $1$ and have the same second term. The third term of one of the series is $1/8$, and the second term of both series can be written in the form $\frac{\sqrt{m}-n}p$, where $m$, $n$, and $p$ are positive integers and $m$ is not divisible by the square of an... | 518 | 101 | 3 |
math | Let $S$ be a set of $n$ distinct real numbers, and let $A_{s}$ be the set of averages of all distinct pairs of elements from $S$. For a given $n \geq 2$, what is the minimum possible number of elements in $A_{s}$? | 2n - 3 | 63 | 5 |
math | In the Cartesian coordinate system $(xOy)$, the parametric equations of curve $C\_1$ are given by $\begin{cases} x=2\cos \alpha +1 \\\\ y=2\sin \alpha \end{cases}$ where $\alpha$ is the parameter. In the polar coordinate system established with the origin $O$ as the pole and the non-negative semi-axis of $x$ as the pol... | \sqrt{11} | 163 | 6 |
math | In triangle $XYZ$, $XY=140$, $XZ=130$, and $YZ=150$. The angle bisector of angle $X$ intersects $\overline{YZ}$ at point $S$, and the angle bisector of angle $Y$ intersects $\overline{XZ}$ at point $T$. Let $U$ and $V$ be the feet of the perpendiculars from $Z$ to $\overline{YT}$ and $\overline{XS}$, respectively. Find... | 70 | 116 | 2 |
math | Find all permutations $a_1, a_2, \ldots, a_9$ of $1, 2, \ldots, 9$ such that \[ a_1+a_2+a_3+a_4=a_4+a_5+a_6+a_7= a_7+a_8+a_9+a_1 \]
and
\[ a_1^2+a_2^2+a_3^2+a_4^2=a_4^2+a_5^2+a_6^2+a_7^2= a_7^2+a_8^2+a_9^2+a_1^2 \] | (2, 9, 4, 5, 1, 6, 8, 3, 7) | 144 | 27 |
math | The diagram shows a quadrilateral \(PQRS\) made from two similar right-angled triangles, \(PQR\) and \(PRS\). The length of \(PQ\) is 3, the length of \(QR\) is 4, and \(\angle PRQ = \angle PSR\).
What is the perimeter of \(PQRS\)? | 22 | 72 | 2 |
math | How many sets of integers \( (a, b, c, d) \) satisfy:
\[ 0 < a < b < c < d < 500, \]
\[ a + d = b + c, \]
and
\[ bc - ad = 93? \]
(The 11th USA Mathematical Talent Search, 1993) | 405 + 465 = 870 | 78 | 13 |
math | Let $\omega$ be a nonreal root of $z^4 = 1.$ Let $b_1, b_2, \dots, b_n$ be real numbers such that
\[
\frac{1}{b_1 + \omega} + \frac{1}{b_2 + \omega} + \dots + \frac{1}{b_n + \omega} = 3 - 4i.
\]
Compute
\[
\frac{2b_1 - 1}{b_1^2 - b_1 + 1} + \frac{2b_2 - 1}{b_2^2 - b_2 + 1} + \dots + \frac{2b_n - 1}{b_n^2 - b_n + 1}.
\] | 6 | 172 | 1 |
math | Let the curve $y=\dfrac{x+1}{x-1}$ have a tangent at the point $(3,2)$ that is perpendicular to the line $ax+y+1=0$. Find the value of $a$. | -2 | 48 | 2 |
math | Every week, Alice goes to the supermarket and buys the following: $4$ apples at $\$2$ each, $2$ loaves of bread at $\$4$ each, $3$ boxes of cereal at $\$5$ each, $1$ chocolate cake at $\$8$, and a $\$6$ package of cheese. This week the store has a sale where all cereals are $\$1$ off and breads are buy-one-get-one-free... | \$38 | 131 | 3 |
math | China Mobile has announced the $4G$ communication tariff standards, one of which is a package tariff standard: the monthly fee for domestic calls is $39$ yuan (which includes $300$ minutes of voice calls and $30GB/$month), and exceeding the monthly fee of $39$ yuan (i.e., exceeding $300$ minutes) will be charged accord... | y = 0.19x + 39 | 286 | 12 |
math | For a positive integer $n$, let
\[ H_n = 1 + \frac{1}{2} + \frac{1}{3} + \dots + \frac{1}{n} \]
and let \( k \) be a positive integer. Compute
\[ \sum_{n=1}^\infty \frac{1}{(n+k) H_n H_{n+1}}. \] | \frac{1}{1+k} | 88 | 8 |
math | (1) Given an ellipse C: $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ ($a > b > 0$) with an eccentricity of $\frac{1}{2}$, and a circle centered at the origin with the radius equal to the semi-minor axis of the ellipse, which is tangent to the line $\sqrt{7}x - \sqrt{5}y + 12 = 0$. Find the equation of ellipse C;
(... | \frac{x^{2}}{3} - y^{2} = 1 | 193 | 17 |
math | Quadrilateral $EFGH$ is 12 cm by 6 cm. $P$ is the midpoint of $\overline{FG}$, and $Q$ is the midpoint of $\overline{GH}$. A diagonal $\overline{EQ}$ is drawn. Calculate the number of square centimeters in the area of the region $EPQH$. | 45\text{ cm}^2 | 76 | 9 |
math | On a digital watch that displays time in a 24-hour format, with the time shown as hours and minutes, calculate the largest possible sum of the digits in the display. | 24 | 36 | 2 |
math | Given vectors $\overrightarrow{a}=(\cos θ,\sin θ)$ and $\overrightarrow{b}=(2,-1)$.
(1) If $\overrightarrow{a} \perp \overrightarrow{b}$, find the value of $\frac{\sin θ-\cos θ}{\sin θ + \cos θ}$;
(2) If $|\overrightarrow{a}- \overrightarrow{b}|=2$ and $θ \in (0, \frac{π}{2})$, find the value of $\sin (θ+ \frac{π}{4})$... | \frac{7\sqrt{2}}{10} | 130 | 13 |
math | How many unordered pairs of edges of a given regular tetrahedron determine a plane? | 6 | 18 | 1 |
math | A 92-digit natural number $n$ is known to have the following first 90 digits: from the 1st to the 10th digit - all ones, from the 11th to the 20th digit - all twos, and so on, from the 81st to the 90th digit - all nines. Find the last two digits of the number $n$ if it is known that $n$ is divisible by 72. | 36 | 104 | 2 |
math | Find
\[
\sum_{n = 1}^\infty \frac{3^n}{1 + 3^n + 3^{n + 1} + 3^{2n + 1}}.
\] | \frac{1}{4} | 48 | 7 |
math | Given $\cos \alpha = \frac{1}{7}$ and $\cos (\alpha - \beta) = \frac{13}{14}$ with $0 < \beta < \alpha < \frac{\pi}{2}$.
(1) Find the value of $\tan 2\alpha$;
(2) Find the value of angle $\beta$. | \beta = \frac{\pi}{3} | 76 | 10 |
math | Given the even function $f(x) = \sqrt{3}\sin(2x+θ) + \cos(2x+θ)$ where $0 < θ < π$, find the minimum value of $g(x)$ on the interval $\left[-\frac{π}{4}, \frac{π}{6}\right]$. | -2 | 71 | 2 |
math | Given Alice has 30 apples, in how many ways can she share them with Becky and Chris so that each of the three people has at least three apples. | 253 | 33 | 3 |
math | Given the function $$f(x)=2\sin(wx+\varphi+ \frac {\pi}{3})+1$$ where $|\varphi|< \frac {\pi}{2}$ and $w>0$, is an even function, and the distance between two adjacent axes of symmetry of the function $f(x)$ is $$\frac {\pi}{2}$$.
(1) Find the value of $$f( \frac {\pi}{8})$$.
(2) When $x\in(-\frac {\pi}{2}, \frac {... | 2\pi | 145 | 3 |
math | The number of real solutions to the equation $(x^{2006} + 1)(1 + x^2 + x^4 + \ldots + x^{2004}) = 2006x^{2005}$ is ______. | 1 | 57 | 1 |
math | In the Cartesian coordinate system, the equation of circle C is $x^2 + y^2 - 4x = 0$, and its center is point C. Consider the polar coordinate system with the origin as the pole and the non-negative half of the x-axis as the polar axis. Curve $C_1: \rho = -4\sqrt{3}\sin\theta$ intersects circle C at points A and B.
(1)... | 1:2 | 196 | 3 |
math | A cuboid has a length of 5 cm, a width of 4 cm, and a height of 3 cm.
① Among the 6 faces, the area of the smallest face is square centimeters, and the area of the largest face is square centimeters.
② The total length of its edges is cm.
③ Its surface area is square centimeters.
④ Its volume is... | 60 | 107 | 2 |
math | A circle $\Gamma$ is the incircle of $\triangle PQR$ and the circumcircle of $\triangle MNO$. Point $M$ lies on $\overline{QR}$, $N$ on $\overline{PQ}$, and $O$ on $\overline{PR}$. If $\angle P = 50^\circ$, $\angle Q = 70^\circ$, and $\angle R = 60^\circ$, determine the measure of $\angle NMO$. | 60^\circ | 103 | 4 |
math | There are 300 children in the "Young Photographer" club. In a session, they divided into 100 groups of 3 people each, and in every group, each member took a photograph of the other two members in their group. No one took any additional photographs. In total, there were 100 photographs of "boy+boy" and 56 photographs of... | 72 | 105 | 2 |
math | How many triangles can be formed using the vertices of a regular hexacontagon (a 60-sided polygon), avoiding the use of any three consecutive vertices in forming these triangles? | 34160 | 37 | 5 |
math | Given an arithmetic sequence $\{a_{n}\}$ with a common difference equal to the common ratio of a geometric sequence $\{b_{n}\}$, where $a_{1}=1$, $b_{2}=4$, and $b_{3}-a_{6}=-3$. Find:<br/>
$(1)$ The general formulas for $\{a_{n}\}$ and $\{b_{n}\}$;<br/>
$(2)$ Let $\{c_{n}\}$ be a new sequence formed by arranging the t... | 3042 | 156 | 4 |
math | A construction manager calculates that one of his two workers would take 8 hours and the other 12 hours to complete a particular paving job alone. When both workers collaborate, their combined efficiency is reduced by 8 units per hour. The manager decides to assign them both to speed up the job, and it takes them preci... | 192 | 80 | 3 |
math | How many positive integers less than $500$ can be written as the sum of two positive perfect cubes? | 26 | 23 | 2 |
math | $2021$ points are given on a circle. Each point is colored by one of the $1,2, \cdots ,k$ colors. For all points and colors $1\leq r \leq k$ , there exist an arc such that at least half of the points on it are colored with $r$ . Find the maximum possible value of $k$ . | k = 2 | 88 | 4 |
math | If \(8^{3x} = 64\), then \(8^{-x}\) equals \(\frac{1}{8^{x}}\). | \frac{1}{4} | 33 | 7 |
math | Each unit square of a $4 \times 4$ square grid is colored either red, green, or blue. Over all possible colorings of the grid, what is the maximum possible number of L-trominos that contain exactly one square of each color? (L-trominos are made up of three unit squares sharing a corner, as shown below.)
[asy]
draw((... | 18 | 292 | 2 |
math | Given the following four propositions:
\\(①\\) A symmetry axis of the function \\(y=2\sin(2x- \frac{\pi}{3})\\) is \\(x= \frac{5\pi}{12}\\);
\\(②\\) The graph of the function \\(y=\tan x\\) is symmetric about the point \\((\frac{\pi}{2},0)\\);
\\(③\\) The sine function is increasing in the first quadrant;
\\(④\... | ①② | 208 | 4 |
math | Calculate $\sqrt{60x} \cdot \sqrt{12x} \cdot \sqrt{63x}$ . Express your answer in simplest radical form in terms of $x$.
Note: When entering a square root with more than one character, you must use parentheses or brackets. For example, you should enter $\sqrt{14}$ as "sqrt(14)" or "sqrt{14}". | 36x \sqrt{35x} | 88 | 10 |
math | Find the value of $b$ that satisfies the equation $243_b + 156_b = 411_b$. | 10 | 29 | 2 |
math | If for any $x\in A$, we have $\frac{1}{x}\in A$, then the set $A$ is called a "harmonious" set. What is the probability of "harmonious" sets among all non-empty subsets of the set $M=\{-1,0, \frac{1}{3}, \frac{1}{2},1,2,3,4\}$? | \frac{1}{17} | 88 | 8 |
math | Find the smallest integer $n$ such that each subset of $\{1,2,\ldots, 2004\}$ with $n$ elements has two distinct elements $a$ and $b$ for which $a^2-b^2$ is a multiple of $2004$. | 1003 | 64 | 4 |
math | Let $a_1=24$ and form the sequence $a_n$ , $n\geq 2$ by $a_n=100a_{n-1}+134$ . The first few terms are $$ 24,2534,253534,25353534,\ldots $$ What is the least value of $n$ for which $a_n$ is divisible by $99$ ? | 88 | 116 | 2 |
math | Calculate the limit of the function:
$$\lim _{x \rightarrow 1} \frac{\sqrt[3]{1+\ln ^{2} x}-1}{1+\cos \pi x}$$ | \frac{2}{3 \pi^2} | 44 | 11 |
math | Given real numbers $x$ and $y$ that satisfy the equation $x - \sqrt{x+1} = \sqrt{y+3} - y$, find the maximum value of $x+y$. | 4 | 43 | 1 |
math | Evaluate the expression $\frac{2015^3 - 2 \cdot 2015^2 \cdot 2016 + 3 \cdot 2015 \cdot 2016^2 - 2016^3 + 1}{2015 \cdot 2016}$. | 0 | 75 | 1 |
math | In a pond there are $n \geq 3$ stones arranged in a circle. A princess wants to label the stones with the numbers $1, 2, \dots, n$ in some order and then place some toads on the stones. Once all the toads are located, they start jumping clockwise, according to the following rule: when a toad reaches the stone label... | \left\lceil \frac{n}{2} \right\rceil | 184 | 15 |
math | The square of an integer is 140 greater than three times the integer itself. What is the sum of all integers for which this is true? | 4 | 31 | 1 |
math | Given a rectangular garden measuring $a$ by $b$ meters, with $a$ and $b$ being positive integers where $b > a$, an area is set aside for planting, which is a rectangle with sides parallel to the sides of the garden, and a path of width $2$ meters surrounds the planting area within the garden and covers one-third of the... | 3 | 96 | 1 |
math | The center of the upper base of a regular quadrilateral prism and the midpoints of the sides of the lower base serve as the vertices of a pyramid inscribed in the prism, with a volume equal to \( V \). Find the volume of the prism. | 6V | 52 | 2 |
math | The numbers in the sequence $101$, $104$, $109$, $116$,$\ldots$ are of the form $a_n=100+n^2$, where $n=1,2,3,\ldots$ For each $n$, let $d_n$ be the greatest common divisor of $a_n$ and $a_{n+1}$. Find the maximum value of $d_n$ as $n$ ranges through the positive integers. | 401 | 106 | 3 |
math | When $555_{10}$ is expressed in this base, it has 4 digits, in the form ABAB, where A and B are two different digits. What base is it? | 6 | 42 | 1 |
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