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 | Calculate $\sqrt[4]{\sqrt[5]{0.00032}}$. Express your answer as a decimal to the nearest thousandth. | 0.669 | 32 | 5 |
math | In triangle $\triangle ABC$, let $a$, $b$, and $c$ be the lengths of the sides opposite to the interior angles $A$, $B$, and $C$, respectively. If $a\cos \left(B-C\right)+a\cos A=2\sqrt{3}c\sin B\cos A$ and $b^{2}+c^{2}-a^{2}=2$, then the area of $\triangle ABC$ is ____. | \frac{\sqrt{3}}{2} | 99 | 10 |
math | Factorize $(x^2-x-6)(x^2+3x-4)+24$. | (x+3)(x-2)(x+\frac{1+\sqrt{33}}{2})(x+\frac{1-\sqrt{33}}{2}) | 22 | 36 |
math | When two stars are very far apart their gravitational potential energy is zero, and when they are separated by a distance $d$, the gravitational potential energy of the system is $U$. Determine the gravitational potential energy of the system when the stars are separated by a distance $2d$. | \frac{U}{2} | 56 | 7 |
math | Given that three roots of $f(x)=x^4+ax^2+bx+c$ are $2$, $-3$, and $5$, what is the value of $a+b+c$? | 79 | 43 | 2 |
math | For \\(\triangle ABC\\), the following propositions are given:
\\(①\\) If \\(\dfrac{\tan A}{\tan B} = \dfrac{a^{2}}{b^{2}}\\), then \\(\triangle ABC\\) must be an isosceles triangle;
\\(②\\) If \\(\dfrac{b^{2} + c^{2} - a^{2}}{a^{2} + c^{2} - b^{2}} = \dfrac{b^{2}}{a^{2}}\\), then \\(\triangle ABC\\) must be an iso... | ①② | 245 | 4 |
math | Daniel and Scott are playing a game where a player wins as soon as he has two points more than his opponent. Both players start at zero points, and points are earned one at a time. If Daniel has a 60% chance of winning each point, what is the probability that he will win the game? | \frac{9}{13} | 64 | 8 |
math | In a positive geometric sequence $\{a_n\}$, it is known that $a_1a_2a_3=4$, $a_4a_5a_6=12$, and $a_{n-1}a_na_{n+1}=324$. Determine the value of $n$. | 14 | 69 | 2 |
math | Consider a sphere $S$ of radius $R$ tangent to a plane $\pi$ . Eight other spheres of the same radius $r$ are tangent to $\pi$ and tangent to $S$ , they form a "collar" in which each is tangent to its two neighbors. Express $r$ in terms of $R$ . | r = R(2 - \sqrt{2}) | 84 | 11 |
math | What is the greatest power of $2$ that is a factor of $10^{1006} - 6^{503}$? | 2^{503} | 32 | 6 |
math | Given the quadratic function $f(x) = x^2 + 2ax + 2a + 1$, it holds that $f(x) \geq 1$ for any $x \in [-1, 1]$. Find the range of the parameter $a$. | a \in [0, +\infty) | 59 | 11 |
math | In the expansion of $\left(a - \dfrac{1}{\sqrt{a}}\right)^7$ the coefficient of $a^{-\frac{1}{2}}$ is: | -21 | 40 | 3 |
math | For which positive integers \( n \) does the polynomial \( p(x) \equiv x^n + (2 + x)^n + (2 - x)^n \) have a rational root? | n = 1 | 40 | 4 |
math | Five students (including A, B, C) are arranged in a row. A must be adjacent to B, and A must not be adjacent to C. The number of different ways to arrange them is _____. (Provide your answer in numerical form) | 36 | 51 | 2 |
math | Let \( f(x) = 3x^5 + 2x^4 - x^3 + 4x^2 - 5x + s \). Find the value of \( s \) such that \( f(3) = 0 \). | -885 | 55 | 4 |
math | Given positive real numbers $x, y, z$ that satisfy $x + y + z = 1$ and $\frac{1}{x} + \frac{1}{y} + \frac{1}{z} = 10$, find the maximum value of $xyz$. | \frac{4}{125} | 60 | 9 |
math | Find the smallest constant \(N\) so that
\[\frac{a^2 + b^2 + c^2}{ab + bc + ca} < N\]
whenever \(a, b,\) and \(c\) are the sides of a triangle. | 1 | 55 | 1 |
math | In triangle \(ABC\), internal and external angle bisectors are drawn from vertex \(C\). The first angle bisector forms an angle of \(40^\circ\) with side \(AB\). What angle does the second angle bisector form with the extension of side \(AB\)? | 50^\circ | 59 | 4 |
math | What is the $21^{\text{st}}$ term of the sequence
$$
1 ; 2+3 ; 4+5+6 ; 7+8+9+10 ; 11+12+13+14+15 ; \ldots ?
$$ | 4641 | 65 | 4 |
math | Given the function $f(x)={\log_{\frac{1}{3}}}({{x^2}-ax-a})$ satisfies the inequality $\frac{{f({{x_2}})-f({{x_1}})}}{{{x_2}-{x_1}}} \gt 0$ for any two distinct real numbers $x_{1}$, $x_{2}\in (-\infty $, $-\frac{1}{2}$), the range of real number $a$ is ______. | [-1, \frac{1}{2}] | 108 | 10 |
math | Find the largest real number $\lambda$ with the following property: for any positive real numbers $p,q,r,s$ there exists a complex number $z=a+bi$ ( $a,b\in \mathbb{R})$ such that $$ |b|\ge \lambda |a| \quad \text{and} \quad (pz^3+2qz^2+2rz+s) \cdot (qz^3+2pz^2+2sz+r) =0. $$ | \sqrt{3} | 114 | 5 |
math | Given that there were five candidates in the school election, and the preliminary results are: Henry: 14 votes, India: 11 votes, Jenny: 10 votes, Ken: 8 votes, Lena: 2 votes, after 90% of the votes had been counted, determine how many students still had a chance of winning the election. | \text{3} | 75 | 5 |
math | Let \( M \) and \( N \) be two points on the Thales' circle of segment \( AB \), distinct from \( A \) and \( B \). Let \( C \) be the midpoint of segment \( NA \), and \( D \) be the midpoint of segment \( NB \). The circle is intersected at the point \( E \) a second time by the line \( MC \), and at point \( F \) by... | 1 | 132 | 1 |
math | Given the function $f(x) = x^3 - 4x^2 + 5x - 4$.
(1) Find the equation of the tangent line to the curve $f(x)$ at the point $(2, f(2))$.
(2) Find the equation of the tangent line of the curve $f(x)$ that passes through point $A(2, -2)$. | y + 2 = 0 | 85 | 7 |
math | The circles \(O_{1}\) and \(O_{2}\) touch the circle \(O_{3}\) with radius 13 at points \(A\) and \(B\) respectively and pass through its center \(O\). These circles intersect again at point \(C\). It is known that \(OC = 12\). Find \(AB\). | 10 | 75 | 2 |
math | How many distinct lines pass through the point $(0, 2016)$ and intersect the parabola $y = x^2$ at two lattice points? (A lattice point is a point whose coordinates are integers.) | 36 | 52 | 2 |
math | A box contains 28 red balls, 20 green balls, 19 yellow balls, 13 blue balls, 11 white balls, and 9 black balls. Calculate the minimum number of balls that must be drawn from the box without replacement to guarantee that at least 15 balls of a single color will be drawn. | 76 | 71 | 2 |
math | Given the equation $x^4 + y^2 - 4y + 4 = 4$, determine the number of ordered pairs of integers $(x, y)$ that satisfy the equation. | 1 | 40 | 1 |
math | In figure 1, \( \triangle ABC \) and \( \triangle EBC \) are two right-angled triangles with \( \angle BAC = \angle BEC = 90^\circ \). Given that \( AB = AC \) and \( EDB \) is the angle bisector of \( \angle ABC \), find the value of \( \frac{BD}{CE} \). | 2 | 84 | 1 |
math | Inside square $ABCD$ with side $s$, quarter-circle arcs with radii $s$ and centers at $A$ and $B$ are drawn. These arcs intersect at a point $X$ inside the square. How far is $X$ from the side of $CD$? | \frac{1}{2} s(2-\sqrt{3}) | 60 | 15 |
math | Mark places six ounces of tea into a ten-ounce cup and eight ounces of milk into a twelve-ounce cup. He then pours one-third of the tea from the first cup into the second cup, stirs thoroughly, and then transfers one-fourth of the mixed liquid from the second cup back to the first cup. What fraction of the liquid in th... | \frac{2}{6.5} | 131 | 9 |
math | Gabi bought two rare songbirds. Later, she sold them both for the same amount. On one bird, she lost 20%, and on the other, she gained 20%, but she still lost a total of 10 Ft. For how much did she buy and sell her birds? | 150 \text{ Ft and } 100 \text{ Ft, sold for } 120 \text{ Ft each} | 63 | 31 |
math | There are four cards, each with one of the numbers $2$, $0$, $1$, $5$ written on them. Four people, A, B, C, and D, each take one card.
A says: None of the numbers you three have differ by 1 from the number I have.
B says: At least one of the numbers you three have differs by 1 from the number I have.
C says: The numb... | 5120 | 172 | 4 |
math | How many polynomial functions of degree exactly 3 satisfy $f(x^2) = [f(x)]^2 = f(f(x))$ and $f(1) = f(-1)$ | 1 | 40 | 1 |
math | Seven consecutive integers, each increasing by 2 starting from $x$, have an average of $y$. Calculate the average of 7 consecutive integers that start with $y$. | x + 9 | 35 | 4 |
math | In order to save energy and reduce emissions, a certain school in our city is preparing to purchase a certain brand of energy-saving lamps. It is known that $3$ of type $A$ energy-saving lamps and $5$ of type $B$ energy-saving lamps cost a total of $50$ yuan, while $1$ of type $A$ energy-saving lamp and $3$ of type $B$... | W = 1240 \text{ yuan} | 217 | 12 |
math | In the sequence $\{a\_n\}$, $a\_1=2$, $a\_{n+1}=1-a\_n(n∈N^{})$$, $S\_n$ is the sum of the first $n$ terms. Find the value of $S\_{2015}-2S\_{2016}+S\_{2017}$. | 3 | 85 | 1 |
math | Points $P$ and $Q$ are midpoints of two sides of the square. What fraction of the interior of the square is shaded? Express your answer as a common fraction.
[asy]
filldraw((0,0)--(2,0)--(2,2)--(0,2)--(0,0)--cycle,gray,linewidth(1));
filldraw((0,1)--(1,2)--(2,2)--(0,1)--cycle,white,linewidth(1));
label("P",(0,1),W);
l... | \frac{7}{8} | 128 | 7 |
math | A square ABCD has side lengths of 1. Points E and F lie on sides AB and AD respectively, such that AE = AF. If the quadrilateral CDFE has the maximum area, calculate the maximum area of the quadrilateral CDFE. | \frac{5}{8} | 53 | 7 |
math | Given that:
1. \( x \) and \( y \) are integers between 10 and 99, inclusive;
2. \( y \) is the number formed by reversing the digits of \( x \);
3. \( z = |x-y| \).
How many distinct values of \( z \) are possible? | 8 | 69 | 1 |
math | Triangle $ABC$ has $AC = 450$ and $BC = 300$ . Points $K$ and $L$ are located on $\overline{AC}$ and $\overline{AB}$ respectively so that $AK = CK$ , and $\overline{CL}$ is the angle bisector of angle $C$ . Let $P$ be the point of intersection of $\overline{BK}$ and $\overline{CL}$ , and let $M$ be the point on line $B... | 72 | 142 | 2 |
math | What is the greatest integer $x$ such that $|8x^2 - 66x + 21|$ is prime? | 2 | 29 | 1 |
math | Given vectors $\overrightarrow{a} = (4\cos\alpha, \sin\alpha)$, $\overrightarrow{b} = (\sin\beta, 4\cos\beta)$, $\overrightarrow{c} = (\cos\beta, -4\sin\beta)$ (where $\alpha, \beta \in \mathbb{R}$ and $\alpha, \beta, \alpha+\beta$ are not equal to $\frac{\pi}{2} + k\pi, k \in \mathbb{Z}$).
(Ⅰ) Find the maximum value ... | -30 | 198 | 3 |
math | If $m > 0$, $n > 0$, $m+n=1$, and the minimum value of $\frac{t}{m} + \frac{1}{n} (t > 0)$ is $9$, then $t =$ \_\_\_\_\_\_. | t=4 | 60 | 3 |
math | Given that $\overrightarrow {a}$ and $\overrightarrow {b}$ are two unit vectors.
(1) If $|3 \overrightarrow {a}-2 \overrightarrow {b}|=3$, find the value of $|3 \overrightarrow {a}+ \overrightarrow {b}|$.
(2) If the angle between $\overrightarrow {a}$ and $\overrightarrow {b}$ is $\frac {π}{3}$, find the angle α betwee... | α= \frac {2π}{3} | 143 | 10 |
math | Given the following four propositions:
1. A line outside of a plane can have at most one common point with that plane;
2. If a line $a$ in plane $\alpha$ intersects with a line $b$ in plane $\beta$, then $\alpha$ and $\beta$ intersect;
3. If a line intersects with two parallel lines, then these three lines are coplana... | 1, 2, 3 | 110 | 7 |
math | An urn initially contains two red balls and one blue ball. A supply of extra red and blue balls is available nearby. George performs the following operation five times: he draws a ball from the urn at random and then adds two balls of the same color from the supply before returning all balls to the urn. After the five ... | \frac{1920}{10395} | 158 | 14 |
math | In $\triangle ABC$, the lengths of the sides opposite to angles A, B, and C are $a$, $b$, and $c$ respectively. Given that $a = \sqrt{2}$, $b = \sqrt{3}$, and $A = 45^\circ$, find the measure of angle C. | 15^\circ | 69 | 4 |
math | A large sphere has a volume of $576\pi$ cubic units. A smaller sphere has a volume which is $25\%$ of the volume of the larger sphere. What is the ratio of the radius of the smaller sphere to the radius of the larger sphere? Express your answer as a common fraction. | \frac{1}{\sqrt[3]{4}} | 66 | 12 |
math | Determine the range of the function $y= \frac{1}{x^{2}-4x-2}$. | (-\infty, -\frac{1}{6}] \cup (0, +\infty) | 25 | 23 |
math | Let $\{a\_n\}$ be a geometric sequence with common ratio $q = \sqrt{2}$, and let $S\_n$ denote the sum of the first $n$ terms of the sequence. Define $T\_n = \frac{17S\_n - S\_{2n}}{a\_{n+1}}$, where $n \in \mathbb{N}^*$. Find the value of $m$ for which $T\_m$ is the maximum term of the sequence $\{T\_n\}$. | 4 | 118 | 1 |
math | Let $f : \mathbb{R} \to \mathbb{R}$ be a function such that
\[f(f(x) + y) = f(x^2 - y) + 4f(x) y\]for all real numbers $x$ and $y.$
Let $n$ be the number of possible values of $f(3),$ and let $s$ be the sum of all possible values of $f(3).$ Find $n \times s.$ | 18 | 104 | 2 |
math | Given 2017 lines separated into three sets such that lines in the same set are parallel to each other, what is the largest possible number of triangles that can be formed with vertices on these lines? | 673 * 672^2 | 42 | 10 |
math | Given that $| \overrightarrow{a}|=3 \sqrt {2}$, $| \overrightarrow{b}|=4$, $\overrightarrow{m}= \overrightarrow{a}+ \overrightarrow{b}$, $\overrightarrow{n}= \overrightarrow{a}+λ \overrightarrow{b}$, $ < \overrightarrow{a}, \overrightarrow{b} > =135^{\circ}$, if $\overrightarrow{m} \perp \overrightarrow{n}$, find the v... | -\frac{3}{2} | 116 | 7 |
math | The diagram shows a rectangle dissected into seven non-overlapping squares. Given that the width and the height of the rectangle are relatively prime positive integers, find the perimeter of the rectangle.
[asy]draw((0,0)--(52,0)--(52,37)--(0,37)--(0,0));draw((15,0)--(15,15)--(0,15)); draw((15,22)--(52,22));draw((27,22... | 178 | 182 | 3 |
math | Five congruent rectangles are placed around a regular pentagon. The area of the outer pentagon formed by this arrangement is $5$ times that of the inner regular pentagon. Each rectangle's shorter side lies against a side of the inner pentagon. If the side length of each side of the inner pentagon is $a$, determine the ... | \sqrt{5} - 1 | 122 | 8 |
math | Given proposition $p$: $x^{2}+2mx+(4m-3) > 0$ has the solution set $R$, and proposition $q$: the minimum value of $m+ \frac {1}{m-2}$ is $4$. If only one of $p$ and $q$ is true, find the range of values for $m$. | (1,2] \cup [3,+\infty) | 78 | 14 |
math | Given an ellipse $C$: $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1 \left(a > b > 0\right)$ with eccentricity $\dfrac{\sqrt{3}}{2}$, and it passes through point $A(2,1)$.
(Ⅰ) Find the equation of ellipse $C$;
(Ⅱ) If $P$, $Q$ are two points on ellipse $C$, and the angle bisector of $\angle PAQ$ is always perpendicular to ... | \dfrac{1}{2} | 152 | 7 |
math | Let $\overline{CH}$ be an altitude of $\triangle ABC$. Let $R\,$ and $S\,$ be the points where the circles inscribed in the triangles $ACH\,$ and $BCH$ are tangent to $\overline{CH}$. If $AB = 13\,$, $AC = 12\,$, and $BC = 5\,$, then $RS\,$ can be expressed as $m/n\,$, where $m\,$ and $n\,$ are relatively prime integer... | 37 | 121 | 2 |
math | Given that
$$\frac{1}{3!18!}+\frac{1}{4!17!}+\frac{1}{5!16!}+\frac{1}{6!15!}+\frac{1}{7!14!}+\frac{1}{8!13!}+\frac{1}{9!12!}+\frac{1}{10!11!} = \frac{M}{1!19!},$$
find the greatest integer that is less than $\frac{M}{100}$. | 262 | 125 | 3 |
math | If the positive numbers \( a \), \( b \), and \( c \) satisfy \( a + 2b + 3c = abc \), then the minimum value of \( abc \) is \(\qquad\). | 9\sqrt{2} | 48 | 6 |
math | Given vectors $\mathbf{a}$ and $\mathbf{b}$, let $\mathbf{p}$ be a vector such that
\[\|\mathbf{p} - \mathbf{a}\| = 3 \|\mathbf{p} - \mathbf{b}\|.\] Among all such vectors $\mathbf{p}$, there exists constants $s$ and $v$ such that $\mathbf{p}$ is at a fixed distance from $s \mathbf{a} + v \mathbf{b}.$ Find the ordered... | \left(\frac{1}{8}, \frac{9}{8}\right) | 126 | 18 |
math | An integer is called "octal" if it is divisible by $8$ or if at least one of its digits is $8$ .
How many integers between $1$ and $100$ are octal? | 27 | 54 | 2 |
math | In triangle $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. It is known that $\sqrt{3}b\sin \left(B+C\right)+a\cos B=c$.
$(1)$ Find the measure of angle $A$.
$(2)$ If $a=6$ and $b+c=6+6\sqrt{3}$, find the area of triangle $\triangle ABC$. | 9\sqrt{3} | 104 | 6 |
math | Given a set of data $x_{1}$, $x_{2}$, $\ldots$, $x_{n}$ with an average of $3$ and a variance of $3$. Each data point is multiplied by $3$ and then subtracted by $2$ to obtain a new set of data $3x_{1}-2$, $3x_{2}-2$, $\ldots$, $3x_{n}-2$. Find the average and variance of the new data. | 27 | 102 | 2 |
math | A binary string is a word containing only $0$ s and $1$ s. In a binary string, a $1-$ run is a non extendable substring containing only $1$ s. Given a positive integer $n$ , let $B(n)$ be the number of $1-$ runs in the binary representation of $n$ . For example, $B(107)=3$ since $107$ in binary is $110101... | 255 | 158 | 3 |
math | Given that $a$ and $b$ are positive real numbers, and the line $(a+1)x+2y-1=0$ is perpendicular to the line $3x+(b-2)y+2=0$, find the minimum value of $\dfrac{3}{a} + \dfrac{2}{b}$. | 25 | 71 | 2 |
math | Given the function $f(x)=x^{2}-2x+a\ln x$.
(I) If $a=2$, find the equation of the tangent line at the point $(1,f(1))$;
(II) Discuss the monotonicity of $f(x)$. | 2x-y-3=0 | 59 | 7 |
math | Given that $49$ of the first $50$ counted were red and $7$ out of every $8$ counted thereafter were red, find the maximum value of $n$, given that $90$% or more of the balls counted were red. | 210 | 55 | 3 |
math | A pedestrian departed from point \( A \) to point \( B \). After walking 8 km, a second pedestrian left point \( A \) following the first pedestrian. When the second pedestrian had walked 15 km, the first pedestrian was halfway to point \( B \), and both pedestrians arrived at point \( B \) simultaneously. What is the ... | 40 | 84 | 2 |
math | There are 5 small balls in a bag, all of the same quality and size, numbered 1, 2, 3, 4, 5. Two players, A and B, play a game. Player A first draws a ball, records the number, puts it back, and then draws another ball. If the sum of the two numbers is even, A wins; otherwise, B wins.
1. Find the probability that A wins... | \frac{1}{5} | 116 | 7 |
math | In triangle $ABC$, $\angle C = 90^\circ$, $AC = 5$ and $BC = 12$. Point $D$ is on $\overline{AB}$, point $E$ is on $\overline{BC}$, and point $F$ is on $\overline{AC}$ such that $\angle FED = 90^\circ$. If $DE = 5$ and $DF = 3$, then what is the length of $BD$? | BD = 10 | 105 | 5 |
math | For a natural number $A$, define the product of its digits in base 10 as $p(A)$. Find all values of $A$ that satisfy the equation $A = 1.5 \cdot p(A)$. | 48 | 48 | 2 |
math | A standard deck of 52 cards is randomly arranged. What is the probability that the top four cards are all from the same suit? | \frac{2860}{270725} | 28 | 15 |
math | I have four apples and twelve oranges. If a fruit basket must contain at least one piece of fruit and must include at least two oranges, how many kinds of fruit baskets can I make? (The apples are identical and the oranges are identical. A fruit basket consists of some number of pieces of fruit, and it doesn't matter h... | 55 | 75 | 2 |
math | The bottoms of two vertical poles are 20 feet apart on a gently sloping ground where one base is 3 feet higher than the other. One pole is 8 feet tall and the other is 18 feet tall. How long, in feet, is a wire stretched from the top of the shorter pole to the top of the taller pole? | \sqrt{569} | 72 | 7 |
math | Given a $2022 \times 2022$ board, Olya and Masha take turns to color $2 \times 2$ squares on it using red and blue. They agreed that each cell can be painted no more than once in blue and no more than once in red. Cells painted blue and then red (or vice versa) become purple. Once all cells are painted, the girls count... | 2020 \cdot 2020 | 191 | 11 |
math | If \( 2A99561 \) is equal to the product when \( 3 \times (523 + A) \) is multiplied by itself, find the digit \( A \). | 4 | 44 | 1 |
math | Given the function $f(x)$, if the function ${f'}(x_{0})$ exists, determine the limit of the expression $\frac{f({x}_{0}+h)-f({x}_{0}-h)}{h}$ as $h$ approaches $0$ infinitely. | 2f'(x_0) | 61 | 7 |
math | A large rectangle consists of three identical squares and three identical small rectangles. The perimeter of a square is 24, and the perimeter of a small rectangle is 16. What is the perimeter of the large rectangle?
The perimeter of a figure is the sum of the lengths of all its sides. | 52 | 61 | 2 |
math | Given four propositions, determine the number of true propositions. | 0 | 11 | 1 |
math | Given the function $f(x)=ax^3+bx$.
(Ⅰ) If the tangent line to the function $f(x)$ at $x=3$ is parallel to the line $24x-y+1=0$, and the function $f(x)$ has an extremum at $x=1$, find the function $f(x)$ and its interval of monotonic decrease.
(Ⅱ) If $a=1$, and the function $f(x)$ is decreasing on the interval $[-1,1]... | b\leqslant -3 | 121 | 8 |
math | In triangle $ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively. It is given that $2a= \sqrt{3}c\sin A-a\cos C$.
$(1)$ Find $C$;
$(2)$ If $c= \sqrt{3}$, find the maximum value of the area $S$ of triangle $ABC$. | \dfrac{\sqrt{3}}{4} | 96 | 10 |
math | For any given real numbers $x$, $y$, $z$ ($z \neq 0$ and $z \neq 6$), let $P(x, y)$ be a point on the $xy$-plane, and the maximum value among the distances from $P$ to three points $A(z, z)$, $B(6-z, z-6)$, and $C(0, 0)$ be denoted as $g(x, y, z)$. Find the minimum value of $g(x, y, z)$. | 3 | 118 | 1 |
math | In triangle \( A B C \) with the side ratio \( A B: A C = 5:4 \), the angle bisector of \( \angle B A C \) intersects side \( B C \) at point \( L \). Find the length of segment \( A L \), given that the length of the vector \( 4 \cdot \overrightarrow{A B} + 5 \cdot \overrightarrow{A C} \) is 2016. | 224 | 101 | 3 |
math | Given three planes $\alpha$, $\beta$, and $\gamma$, where $\beta \perp \gamma$ and $\alpha$ intersects with $\gamma$ but not perpendicularly, the lines a, b, c are within planes $\alpha$, $\beta$, $\gamma$ respectively. Among the following propositions: ① Any line $b \subset \beta$ is perpendicular to $\gamma$; ② Any l... | ④⑥ | 194 | 4 |
math | The sum of the dimensions of a rectangular prism is the sum of the number of edges, corners, and faces, where the dimensions are 2 units by 3 units by 4 units. Calculate the resulting sum. | 26 | 44 | 2 |
math | In the interval $[0, p]$, the number of solutions to the trigonometric equation $\cos 7x = \cos 5x$ is ______. | 7 | 35 | 1 |
math | $P(x)$ is a polynomial of degree $3n$ such that
\begin{eqnarray*} P(0) = P(3) = \cdots &=& P(3n) = 2, \\ P(1) = P(4) = \cdots &=& P(3n-2) = 1, \\ P(2) = P(5) = \cdots &=& P(3n-1) = 0, \quad\text{ and }\\ && P(3n+1) = 730.\end{eqnarray*}
Determine $n$. | n = 4 | 136 | 4 |
math | Given a Ferris wheel with a radius of 30 feet, if it makes one complete revolution every 90 seconds, determine the time in seconds for a rider to move from the bottom of the wheel to a point 15 feet above the bottom. | 30 | 53 | 2 |
math | Given that $A$, $B$, $C$ are the three internal angles of $\triangle ABC$, vector $\overrightarrow{m}=(\cos A+1, \sqrt {3})$, $\overrightarrow{n}=(\sin A,1)$, and $\overrightarrow{m}/\!/ \overrightarrow{n}$;
$(1)$ Find angle $A$;
$(2)$ If $\dfrac {1+\sin 2B}{\cos \;^{2}B-\sin \;^{2}B}=-3$, find $\tan C$. | \dfrac {8+5 \sqrt {3}}{11} | 119 | 15 |
math | Triangle $PQR$ has vertices $P(1, 6)$, $Q(3, -2)$, and $R(9, -2)$. A line through $R$ cuts the area of $\triangle PQR$ in half; find the sum of the slope and $y$-intercept of this line. | \frac{18}{7} | 70 | 8 |
math | Given $a$, $b$, $c$, $d$, $e$ are 5 elements taken from the set $\{1, 2, 3, 4, 5\}$ without repetition, calculate the probability that $abc+de$ is an odd number. | \frac{2}{5} | 58 | 7 |
math | In an $h$-meter race, Sunny is exactly $d$ meters ahead of Windy when Sunny finishes the race. The next time they race, Sunny sportingly starts $d$ meters behind Windy, who is at the starting line. Both runners run at the same constant speed as they did in the first race. Calculate the distance by which Sunny is ahead ... | \frac{d^2}{h} | 85 | 9 |
math | How many times do we need to flip a coin so that with an error probability of less than 0.01, we can state that the frequency of getting heads will be between 0.4 and 0.6? | 166 | 47 | 3 |
math | Given the $R^2$ values for four different regression models are $0.95$, $0.70$, $0.55$, and $0.30$, determine the model with the best fitting effect. | 0.95 | 48 | 4 |
math | Given the hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 (a > 0, b > 0)$, its left and right foci are $F\_1$ and $F\_2$ respectively. An equilateral triangle $PF\_1F\_2$ intersects the hyperbola at points $M$ and $N$. If $M$ and $N$ are the midpoints of the line segments $PF\_1$ and $PF\_2$, find the eccentri... | \sqrt{3} + 1 | 127 | 8 |
math | An old clock's minute and hour hands overlap every 66 minutes of standard time. Calculate how much the old clock's 24 hours differ from the standard 24 hours. | 12 | 38 | 2 |
math | Circle $B$ has radius $6\sqrt{7}$ . Circle $A$ , centered at point $C$ , has radius $\sqrt{7}$ and is contained in $B$ . Let $L$ be the locus of centers $C$ such that there exists a point $D$ on the boundary of $B$ with the following property: if the tangents from $D$ to circle $A$ intersect circl... | 168\pi | 158 | 5 |
math | Three pirates divided the diamonds they collected during the day: Bill and Sam got twelve each, and the rest went to John, who did not know how to count. At night, Bill stole one diamond from Sam, Sam stole one diamond from John, and John stole one diamond from Bill. As a result, the average mass of Bill's diamonds dec... | 9 | 109 | 1 |
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