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 the function $f(x)=-\sin ^{2}x+m\sin x+2$, when $x\in\left[ \frac {\pi}{6}, \frac {2\pi}{3}\right]$ the function has a maximum value of $\frac {3}{2}$. Find the value of $m$ at this time. | m=- \frac {1}{2} | 74 | 9 |
math | Maria has been tasked with writing all the possible rearrangements of the letters in her name. If she can write eight rearrangements of her name every minute, how many hours does it take to write all the possible rearrangements of her name? | 0.125 \text{ hours} | 51 | 10 |
math | The solution set of the inequality $$\frac {ax}{x-1} < 1$$ is $\{x|x < b \text{ or } x > 3\}$. Find the value of $a-b$. | -\frac {1}{3} | 47 | 7 |
math | Calculate the sunset time in the city of Springfield, given that the length of daylight on June 15 was 14 hours and 9 minutes and the sunrise was recorded at 5:31 AM. | 7:40 \text{PM} | 43 | 9 |
math | Two infinite geometric progressions are given with a common ratio \( |q| < 1 \), differing only in the sign of their common ratios. Their sums are \( S_{1} \) and \( S_{2} \). Find the sum of the infinite geometric progression formed from the squares of the terms of either of the given progressions. | S_1 S_2 | 71 | 6 |
math | Find all real numbers $x$ such that \[2 \le \frac{x}{2x-5} < 7.\](Give your answer in interval notation.) | (\frac{35}{13}, \frac{10}{3}] | 35 | 17 |
math | Given $a_{1}+a_{2}=1$, $a_{2}+a_{3}=2$, $a_{3}+a_{4}=-3$, $a_{4}+a_{5}=-4$, $a_{5}+a_{6}=5$, $a_{6}+a_{7}=6$, $a_{7}+a_{8}=-7$, $a_{8}+a_{9}=-8$, $\ldots $, $a_{99}+a_{100}=-99$, $a_{100}+a_{1}=-100$, calculate the value of $a_{1}+a_{2}+a_{3}+\ldots +a_{100}$. | -50 | 170 | 3 |
math | If $g(x)$ is a function defined as the remainder when $x^2$ is divided by 13, determine the order of 7 with respect to this function $g$. | 12 | 39 | 2 |
math | The solid shown has a square base of side length \( s \). One pair of opposite top edges parallel to the base are extended such that the distance between them is \( 3s \), and the other pair has a distance of \( s \). The vertical edges have a length of \( s \). Given that \( s = 4\sqrt{2} \), what is the volume of the... | 128\sqrt{2} | 262 | 8 |
math | Construct a rational parameterization of the circle \( x^2 + y^2 = 1 \) by drawing lines through the point \((1,0)\). | \left( \frac{t^2 - 1}{t^2 + 1}, \frac{-2t}{t^2 + 1} \right) | 34 | 36 |
math | Given nonzero real numbers \(a, b, c, d\) and the function \(f(x)=\frac{ax+b}{cx+d}\) for \(x \in \mathbb{R}\) such that \(f(19) = 19\) and \(f(97) = 97\). If for any real number \(x \neq -\frac{d}{c}\), it holds that \(f[f(x)] = x\), find the unique number that is outside the range of \(f(x)\). | 58 | 114 | 2 |
math | Find all functions \( f: \mathbf{R} \rightarrow \mathbf{R} \) such that for any real numbers \( x \) and \( y \), the following holds:
\[ f(f(x) + y) = 2x + f(f(y) - x). \] | f(x) = x + c | 63 | 7 |
math | Regular hexagon $PQRSTU$ is composed of six smaller non-equilateral trapezoids, such as $PQVW$, shown in boldface in the diagram. By connecting alternate vertices, we obtain a larger square $PRCT$, also shown in boldface. Calculate the ratio $[\trapezoid PQVW]/[\square PRCT]$. Assume all sides of the original hexagon a... | \frac{\sqrt{3}}{4} | 94 | 10 |
math | Given that the function $f(x)$ is an even function defined on $\mathbb{R}$, and the odd function $g(x)$ defined on $\mathbb{R}$ passes through the point $(-1, 1)$, and $g(x) = f(x-1)$, find the value of $f(7) + f(8)$. | -1 | 76 | 2 |
math | Arrange the forty natural numbers 1, 2, ..., 40 in any order, and you can always find eight consecutive numbers whose sum is not less than $A$. The maximum value of $A$ is ___. | 164 | 46 | 3 |
math | In triangle \(ABC\), it is known that \(AB = c\), \(BC = a\), \(AC = b\). Point \(O\) is the center of the circle that touches side \(AB\) and the extensions of sides \(AC\) and \(BC\). Point \(D\) is the intersection of ray \(CO\) with side \(AB\). Find the ratio \(\frac{CO}{OD}\). | \frac{a+b}{c} | 87 | 8 |
math | A circle is circumscribed about an equilateral triangle with side lengths of $12$ units each. In the same plane, a square has been drawn with its side length equal to the height of this equilateral triangle. Calculate the area of the circle and the square, expressing the area of the circle in terms of $\pi$. | 108 | 68 | 3 |
math | The function
\[ f(x) = \left\{ \begin{aligned}
x+3 & \quad \text{if } x < 5 \\
\sqrt{x-1} & \quad \text{if } x \ge 5
\end{aligned} \right. \]
has an inverse $f^{-1}$. Find the value of $f^{-1}(-2) + f^{-1}(-1) + \dots + f^{-1}(4) + f^{-1}(5)$. | 44 | 111 | 2 |
math | Given the functions $f(x)=ax^{2}+1(a > 0)$ and $g(x)=x^{3}+bx$.
$(1)$ If the curve $y=f(x)$ and the curve $y=g(x)$ have a common tangent line at their intersection point $(1,c)$, find the values of $a$ and $b$;
$(2)$ When $a^{2}=4b$, find the monotonic intervals of the function $f(x)+g(x)$, and find its maximum value... | h(-\frac{a}{2})=1 | 121 | 11 |
math | The slope angle of the line $y=-\sqrt{3}x+1$ is what? | 120^{\circ} | 21 | 7 |
math | Let's consider two positive real numbers $a$ and $b$, where an operation $a \, \blacktriangle \, b$ is defined such that $(ab) \, \blacktriangle \, b = a(b \, \blacktriangle \, b)$ and $(a \, \blacktriangle \, 1) \, \blacktriangle \, a = a \, \blacktriangle \, 1$ for all $a,b>0$. Additionally, it is given that $1 \, \b... | 2070 | 135 | 4 |
math | The function is $y=2\sin ( \frac {π}{6}-2x)$, where $x\in[0,π]$. Determine the interval(s) within the domain where the function is increasing. | [\frac{π}{3}, \frac{5π}{6}] | 46 | 15 |
math | Given the direction vector of line $l_{1}$ as $\overrightarrow{a}=(2,4,x)$ and the direction vector of line $l_{2}$ as $\overrightarrow{b}=(2,y,2)$, where $|\overrightarrow{a}|=6$ and $\overrightarrow{a}\perp \overrightarrow{b}$, calculate the value of $x+y$. | -3 \; or \; 1 | 84 | 9 |
math | Find the maximum value of the parameter \(a\) for which the equation \((|x-2|+2a)^{2}-3(|x-2|+2a)+4a(3-4a)=0\) has three solutions. Specify the largest value in your answer. | 0.5 | 60 | 3 |
math | Given the function $f(x)=2\sin x\cos x+\sin ^{2}x-\cos ^{2}x.$
$(1)$ Find the interval of monotonic decrease for the function $f(x)$;
$(2)$ If the graph of $f(x)$ is translated to the left by $\dfrac {\pi}{8}$ units and then the x-coordinates are shortened to half of their original length without changing the y-coo... | - \dfrac { \sqrt {5}}{3} | 170 | 13 |
math | The equation $x^2 - 2x = i$ has two complex solutions. Determine the product of their real parts. | \frac{1 - \sqrt{2}}{2} | 26 | 13 |
math | Given $a$, $b$, $c$, $d \in \{-1, 1, 2\}$, determine the largest possible value of $ad - bc$. | 6 | 37 | 1 |
math | A set contains four integers. The six pairwise sums of distinct elements of the set, in no particular order, are $210$, $330$, $290$, $250$, $x$, and $y$. Find the greatest possible value of $x+y$. | 780 | 60 | 3 |
math | If the line $y=2x$ and the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\left(a \gt 0, b \gt 0\right)$ do not have any common points, then the range of eccentricity of the hyperbola is ______. | (1, \sqrt{5}] | 73 | 8 |
math | The sequence $\{2^{n}-1\}$ forms a set $A_{n}=\{1,3,7,…,2^{n}-1\}$ with its first $n$ terms. From the set $A_{n}$, any $k(k=1,2,…,n)$ numbers are chosen, and the sum of all possible products of these $k$ numbers is denoted as $T_{k}$ (if only one number is chosen, the product is defined as the number itself). Let $S_{n... | 2^{\frac {n(n+1)}{2}}-1 | 227 | 15 |
math | Triangle $ABC$ has vertices $A(0,8)$, $B(2,0)$, $C(8,0)$. A vertical line intersects $AC$ at $R$ and $\overline{BC}$ at $S$, forming triangle $RSC$. If the area of $\triangle RSC$ is 12.5, determine the positive difference of the $x$ and $y$ coordinates of point $R$. | 2 | 95 | 1 |
math | Given the function $f(x)=x+2\sin x$ where $x>0$, all the local minimum points are arranged in ascending order to form a sequence $\{a_{n}\}$. Find the value of $\sin (a_{2025})$. | -\frac{\sqrt{3}}{2} | 57 | 10 |
math | Given the function $f(x)=2 \sqrt {3}\sin x\cos x+2\sin ^{2}x$.
$(1)$ If $f(x)=0$, and $x\in\left(- \frac {\pi}{2},\pi\right)$, find the value of $x$;
$(2)$ Move the graph of the function $f(x)$ to the left by $\frac {\pi}{3}$ units, and then stretch all the x-coordinates of the points on the graph by $2$ times (the... | (0,3] | 197 | 5 |
math | The bottoms of two vertical poles are 20 feet apart on a flat ground. One pole is 8 feet tall and the other is 18 feet tall. Simultaneously, the ground between the poles is sloped, with the base of the taller pole being 2 feet higher than the base of the shorter pole due to the slope. Calculate the length in feet of a ... | \sqrt{544} | 95 | 7 |
math | The reciprocal of $-3$ is $\frac{1}{-3}$. | -\frac{1}{3} | 17 | 7 |
math | Given that $F\_1$ and $F\_2$ are the left and right foci of the ellipse $(E)$: $\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1 (a > b > 0)$, $M$ and $N$ are the endpoints of its minor axis, and the perimeter of the quadrilateral $MF\_1NF\_2$ is $4$, let line $(l)$ pass through $F\_1$ intersecting $(E)$ at points $A$... | \frac{2}{3} | 188 | 7 |
math | In the geometric sequence $\{a_n\}$, $a_2$ and $a_6$ are the roots of the equation $x^2-34x+81=0$. Express $a_4$ in terms of the common ratio of the sequence. | 9 | 58 | 1 |
math | Consider a sequence \( x_{n} \) such that \( x_{1} = x_{2} = 1 \), and \( x_{3} = \frac{2}{3} \). Suppose that \( x_{n} = \frac{x_{n-1}^{2} x_{n-2}}{2 x_{n-2}^{2} - x_{n-1} x_{n-3}} \) for all \( n \geq 4 \). Find the least \( n \) such that \( x_{n} \leq \frac{1}{10^{6}} \). | 13 | 134 | 2 |
math | Find all positive integers \(a, b, c\), where \(1 < a < b < c\), such that \((a-1)(b-1)(c-1)\) is a divisor of \(abc - 1\). | (2, 4, 8) | 51 | 9 |
math | Given that the sum of the first $n$ terms of the sequence $\{a\_n\}$ is $S\_n$, and $S\_n = 2a\_n - 2 (n \in \mathbb{N}^*)$.
1. Find the general term formula for $\{a\_n\}$.
2. Let $b_{n+1} = 2b\_n - 2^{n+1}$, $b\_1 = 8$, and $T\_n$ is the sum of the first $n$ terms of the sequence $\{b\_n\}$. Find the positive intege... | \frac{2}{3} | 261 | 7 |
math | Given that the median of $18$ integers is $5$, and the $75$th percentile is also $5$, then the minimum possible number of occurrences of $5$ among these $18$ numbers is ______. | 6 | 48 | 1 |
math | In the diagram, the area of triangle $XYZ$ is 36 square units. What is the area of triangle $YZW$?
[asy]
draw((0,0)--(40,0)--(10,18)--(0,0));
dot((0,0));
label("$X$",(0,0),SW);
label("8",(4,0),S);
dot((8,0));
label("$Y$",(8,0),S);
label("32",(24,0),S);
dot((40,0));
label("$W$",(40,0),SE);
dot((10,18));
label("$Z$",(10... | 144 | 168 | 3 |
math | Let \( a \) be a nonzero real number. In the Cartesian coordinate system \( xOy \), the quadratic curve \( x^2 + ay^2 + a^2 = 0 \) has a focal distance of 4. Determine the value of \( a \). | \frac{1 - \sqrt{17}}{2} | 58 | 14 |
math | In a box, there are 6 ballpoint pens: 3 are black, 2 are blue, and 1 is red. Three pens are randomly selected from the box.
(1) How many basic events are there in this experiment? If the 3 black ballpoint pens are labeled as A, B, C, the 2 blue ballpoint pens as d, e, and the 1 red ballpoint pen as x, and a basic eve... | \frac{4}{5} | 146 | 7 |
math | Samantha in her Chemistry class has a different task where she needs to prepare a solution by mixing 0.08 liters of chemical A with 0.04 liters of water, alongside 0.02 liters of chemical B. She plans to create 0.84 liters of this new solution. How many liters of water should she use? | 0.24 | 74 | 4 |
math | The negation of the proposition "$\exists x > 0$, $x^{2}-2x+1 < 0$" is $\forall x > 0$, $x^{2}-2x+1 \geqslant 0$. | \forall x > 0, x^{2}-2x+1 \geqslant 0 | 52 | 22 |
math | The figure below shows line $\ell$ with a regular, infinite, recurring pattern of squares and line segments.
How many of the following four kinds of rigid motion transformations of the plane in which this figure is drawn, other than the identity transformation, will transform this figure into itself?
some rotation ar... | 2 | 95 | 1 |
math | Triangle $AHI$ is equilateral. $\overline{BC}$, $\overline{DE}$ and $\overline{FG}$ are all parallel to $\overline{HI}$ with $AB = BD = DF = FH$. Determine the ratio of the area of trapezoid $FGIH$ to the area of triangle $AHI$ if $\overline{AF} = \frac{5}{6}\overline{AH}$. | \frac{11}{36} | 94 | 9 |
math | For a population of size $m$ (where $m \geq 3$ and $m \in \mathbb{N}$), when selecting a sample size of 3 using systematic sampling, the probability of each individual in the population being selected is $\frac{1}{3}$. Determine the probability of each individual being selected when using stratified sampling to draw th... | \frac{1}{3} | 79 | 7 |
math | There are $n$ cards such that for each $i=1,2, \cdots n$ , there are exactly one card labeled $i$ . Initially the cards are piled with increasing order from top to bottom. There are two operations:
- $A$ : One can take the top card of the pile and move it to the bottom;
- $B$ : One can remove the top card from... | k = 1 | 151 | 5 |
math | Given the function $f(x)=e^{x}-ax-1$ where $a \in \mathbb{R}$:
(I) Discuss the monotonicity of $f(x)$;
(II) Assuming $a > 1$, does there exist a positive real number $x$ such that $f(x) > 0$? If such an $x$ exists, find one that satisfies the condition; if not, explain why. | x = 2\ln a | 92 | 7 |
math | Around the outside of a $6$ by $6$ square, construct four semicircles with the four sides of the square as their diameters. Another square, $EFGH$, has its sides parallel to the corresponding sides of the larger square, and each side of $EFGH$ is tangent to one of the semicircles. Provide the area of square $EFGH$.
A) ... | 144 | 115 | 3 |
math | A line passing through the left focus $F_1$ of a hyperbola at an inclination of 30° intersects with the right branch of the hyperbola at point $P$. If the circle with the diameter $PF_1$ just passes through the right focus of the hyperbola, determine the eccentricity of the hyperbola. | \sqrt{3} | 73 | 5 |
math | Convert cylindrical coordinates A(2, $\frac{\pi}{6}$, 5) to Cartesian coordinates and find the result. Convert Cartesian coordinates B(-3, $\sqrt{3}$, -$\frac{\pi}{3}$) to cylindrical coordinates and find the result. | \left(2\sqrt{3}, \frac{5\pi}{6}, -\frac{\pi}{3}\right) | 56 | 28 |
math | Find the greatest number \( A \) for which the following statement is true.
No matter how we pick seven real numbers between 1 and \( A \), there will always be two numbers among them whose ratio \( h \) satisfies \( \frac{1}{2} \leq h \leq 2 \). | 64 | 66 | 2 |
math | In each square of a $4$ by $4$ grid, you put either a $+1$ or a $-1$ . If any 2 rows and 2 columns are deleted, the sum of the remaining 4 numbers is nonnegative. What is the minimum number of $+1$ 's needed to be placed to be able to satisfy the conditions | 10 | 84 | 2 |
math | What is the domain of the function $g(x) = \log_3(\log_4(\log_5(\log_6x)))$? | (7776, \infty) | 32 | 10 |
math | Given that the real numbers \( s \) and \( t \) satisfy the equations \( 19s^{2} + 99s + 1 = 0 \) and \( t^{2} + 99t + 19 = 0 \), respectively, and that \( st \neq 1 \), find the value of \( \frac{st + 4s + 1}{t} \). | -5 | 92 | 2 |
math | Players A and B play a dice throwing game, taking turns to throw a fair dice. The game ends when one player rolls a $6$. <br/>$(1)$ Let $X$ be the total number of dice throws by both players. If each player can throw the dice at most twice, find the probability distribution and expectation of $X$ when the game ends; <b... | \frac{1}{6} \times (\frac{5}{6})^{2n-2} | 105 | 22 |
math | Solve the inequality
$$
(2+\sqrt{3})^x + 2 < 3(\sqrt{2-\sqrt{3}})^{2x}
$$
Find the sum of all integer values of \(x\) that satisfy this inequality and belong to the interval \((-20, 53)\). | -190 | 68 | 4 |
math | Five people of heights $65,66,67,68$, and 69 inches stand facing forwards in a line. How many orders are there for them to line up, if no person can stand immediately before or after someone who is exactly 1 inch taller or exactly 1 inch shorter than himself? | 14 | 66 | 2 |
math | In triangle $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. The vectors $\overrightarrow{m}=(b,\sqrt{3}a)$ and $\overrightarrow{n}=(\sin B,\sin 2A)$ are given, and $\overrightarrow{m} \parallel \overrightarrow{n}$.
$(Ⅰ)$ Find the value of angle $A$.
$(Ⅱ)$ If the a... | \sqrt{7} | 213 | 5 |
math | Casper can buy $16$ pieces of red candy, $18$ pieces of green candy, $20$ pieces of blue candy, or $n$ pieces of yellow candy. A piece of yellow candy costs $\30$ cents. Determine the smallest possible value of $n$. | 24 | 61 | 2 |
math | Given the set \( A = \{x \mid (x-2)(x-6) \geqslant 3, x \in \mathbf{Z}, 0 \leqslant x \leq 7\} \), find the number of non-empty subsets of \( A \). | 63 | 66 | 2 |
math | Let $A$ be a set containing $4k$ consecutive positive integers, where $k \geq 1$ is an
integer. Find the smallest $k$ for which the set A can be partitioned into two subsets
having the same number of elements, the same sum of elements, the same sum
of the squares of elements, and the same sum of the cubes of el... | k = 4 | 90 | 5 |
math | A rectangle measures 6 meters by 10 meters. Drawn on each side of the rectangle is a semicircle that has the endpoints of its diameter on the vertices of the rectangle. What percent larger is the area of the large semicircles than the area of the small semicircles? Express your answer to the nearest whole number. | 178\% | 70 | 5 |
math | Given the function $f(x)= \begin{cases} -\frac{1}{2}x+\frac{1}{4},x\in[0,\frac{1}{2}] \\ \frac{x}{x+2},x\in(\frac{1}{2},1] \end{cases}, g(x)=a\cos{\frac{\pi x}{2}}+5-2a (a>0)$. If there exists $x_{1}$, $x_{2}\in[0,1]$, such that $f(x_{1})=g(x_{2})$ holds, then the range of values for the real number $a$ is _____ . | [\frac{7}{3},5] | 144 | 9 |
math | Sindy writes down the positive integers less than 200 in increasing order, but skips the multiples of 10. She then alternately places + and - signs before each of the integers, yielding an expression \( +1-2+3-4+5-6+7-8+9-11+12-\cdots-199 \). What is the value of the resulting expression? | 109 | 87 | 3 |
math | The famous skater Tony Hawk is riding a skateboard (segment $A B$) in a ramp, which is a semicircle with a diameter $P Q$. Point $M$ is the midpoint of the skateboard, and $C$ is the foot of the perpendicular dropped from point $A$ to the diameter $P Q$. What values can the angle $\angle A C M$ take, if it is known tha... | 12 | 102 | 2 |
math | Define the sequence \( b_1, b_2, b_3, \ldots \) by \( b_n = \sum\limits_{k=1}^n \cos{k} \), where \( k \) represents radian measure. Find the index of the 100th term for which \( b_n < 0 \). | 628 | 74 | 3 |
math | In triangle $\triangle ABC$, $b=2\sqrt{3}$, $2a-c=2b\cos C$.
$(1)$ Find $B$;
$(2)$ Find the maximum value of $3a+2c$. | 4\sqrt{19} | 52 | 7 |
math | Given that the center of ellipse $E$ is at the origin $O$, its foci lie on the $x$-axis, and the square formed by the endpoints of the minor axis and the foci of ellipse $E$ has a sum of distances from any point on the ellipse to the two foci equal to $2\sqrt{2}$.
(I) Find the standard equation of ellipse $E$.
(II) I... | \frac{\sqrt{2}}{2} | 124 | 10 |
math | A factory has a fixed daily cost of 20,000 yuan, and the maximum daily production capacity is 360 units. The cost increases by 100 yuan for each unit produced. The revenue function for producing $x$ units of product per day is $R(x) = -\frac{1}{2}x^2 + 400x$. Let $L(x)$ and $P(x)$ represent the daily profit and average... | 95 | 253 | 2 |
math | A standard deck of 52 cards has 13 ranks (Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, King) and 4 suits ($\spadesuit$, $\heartsuit$, $\diamondsuit$, and $\clubsuit$), such that there is exactly one card for any given rank and suit. The deck is randomly arranged. What is the probability that the top two cards drawn ar... | \frac{1}{17} | 111 | 8 |
math | Two people are tossing a coin: one tossed it 10 times, and the other tossed it 11 times.
What is the probability that the second person's coin landed on heads more times than the first person's coin? | \frac{1}{2} | 47 | 7 |
math | Let $g : \mathbb{R} \to \mathbb{R}$ be a function such that
\[g(g(x - y)) = g(x) + g(y) - g(x)g(y) - xy\] for all $x,$ $y.$ Find the sum of all possible values of $g(1).$ | 1 | 72 | 1 |
math | Suppose that $x_1+10=x_2+20=x_3+30=\cdots=x_{100}+1000=x_1+x_2+x_3+\cdots+x_{100}+1010$. Find the value of $\left\lfloor|S|\right\rfloor$, where $S=\sum_{n=1}^{100}x_n$. | 510 | 95 | 3 |
math | Given the line $l_1: ax-y-2=0$ passes through the center of the circle $C: (x-1)^2+y^2=1$.
(1) Find the value of $a$;
(2) Find the equation of the line $l_2$ that passes through the center of the circle $C$ and is parallel to the line $l: x-4y+1=0$. | x-4y-1=0 | 93 | 8 |
math | Given a triangle \( \triangle ABC \) with \( AB = 2 \), \( AC = 1 \), \( BC = \sqrt{7} \), and \( O \) as the circumcenter of \( \triangle ABC \). If \( \overrightarrow{AO} = \lambda \overrightarrow{AB} + \mu \overrightarrow{AC} \), find the value of \( \lambda + \mu \). | \frac{13}{6} | 90 | 8 |
math | I have 12 distinguishable socks in my drawer: 5 black, 3 green, and 4 red. In how many ways can I choose a pair of socks, provided that I must select one sock of each color? | 47 | 48 | 2 |
math | For two functions $f(x)$ and $g(x)$ with the same domain, if there exist real numbers $m$ and $n$ such that $h(x) = mf(x) + ng(x)$, then the function $h(x)$ is said to be generated by the "base functions $f(x), g(x)$".
(Ⅰ) If $h(x) = 2x^2 + 3x + 1$ is generated by the functions $f(x) = x^2 + ax$ and $g(x) = x + b$, wit... | \log_{4}(4^x + 1) - \frac{1}{2}(x - 1) | 231 | 25 |
math | Given that $(xy-2)^2 + (x+y-1)^2$ represents a sum of squares of real numbers, calculate the least possible value. | 2 | 32 | 1 |
math | Team A and Team B are playing in a basketball final, following a best-of-seven series where the first team to win four games wins the championship. Based on the previous match results, Team A's home and away game schedule is "home, home, away, away, home, away, home". It is given that the probability of Team A winning ... | 0.18 | 120 | 4 |
math | In quadrilateral EFGH, EF = 7, FG = 13, GH = 7, HE = 11, and EG is an integer. Calculate the length of EG. | 13 | 41 | 2 |
math | Call a set of integers "widely spacy" if it contains no more than one out of any four consecutive integers. How many subsets of $\{1, 2, 3, \dots, 15\}$, including the empty set, are widely spacy? | 181 | 58 | 3 |
math | Three circles $C_i$ are given in the plane: $C_1$ has diameter $AB$ of length $1$ ; $C_2$ is concentric and has diameter $k$ ( $1 < k < 3$ ); $C_3$ has center $A$ and diameter $2k$ . We regard $k$ as fixed. Now consider all straight line segments $XY$ which have one endpoint $X$ on $C_2$ , one ... | \frac{XB}{BY} = 1 | 172 | 11 |
math | A cuckoo clock is on the wall. At the beginning of every hour, the cuckoo makes a number of "cuckoo" sounds equal to the hour displayed by the hour hand (for example, at 19:00 the cuckoo makes 7 sounds). One morning, Maxim approached the clock when it showed 9:05. He started turning the minute hand until he moved the t... | 43 | 104 | 2 |
math | A basketball player scores 7 baskets in a game, with each basket worth either 1, 2, or 3 points. Find the range of possible total scores for the player. | 15 | 38 | 2 |
math | In $\triangle ABC$, $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$ respectively, and it is given that $a\sin B=-b\sin \left(A+ \frac{\pi}{3}\right)$.
(Ⅰ) Find $A$;
(Ⅱ) If the area of $\triangle ABC$, $S= \frac{ \sqrt{3}}{4}{c}^{2}$, find the value of $\sin C$. | \sin C= \frac{ \sqrt{7}}{14} | 110 | 16 |
math | The vertex of the parabola $x^2 = 4y$ is at point $A$ with a vertical coordinate of $4$. Calculate the distance between point $A$ and the focus of the parabola. | 3 | 47 | 1 |
math | There are 28 students in a class. On March 8th, each boy gave each girl one flower - a tulip, a rose, or a daffodil. How many roses were given if it is known that the number of roses is 4 times the number of daffodils, but 3 times less than the number of tulips? | 44 | 76 | 2 |
math | Given vectors $$\overrightarrow {a} = (1+\cos\alpha, \sin\alpha)$$, $$\overrightarrow {b} = (1-\cos\beta, \sin\beta)$$, $$\overrightarrow {c}=(1,0)$$, with $\alpha \in (0,\pi)$, $\beta \in (\pi,2\pi)$, the angle between vector $$\overrightarrow {a}$$ and $$\overrightarrow {c}$$ is $\theta_1$, the angle between vector $... | (12,8 \sqrt {3}] | 234 | 10 |
math | The number 1,268,000,000 can be expressed in scientific notation. | 1.268\times 10^{9} | 23 | 13 |
math | The sequence $a_1,a_2,\dots,a_{13}$ is a geometric sequence with $a_1=a$ and common ratio $r$ , where $a$ and $r$ are positive integers. Given that $$ \log_{2015}a_1+\log_{2015}a_2+\dots+\log_{2015}a_{13}=2015, $$ find the number of possible ordered pairs $(a,r)$ . | 26^3 | 116 | 4 |
math | Two numbers between $0$ and $1$ on a number line are to be chosen at random. What is the probability that the second number chosen will exceed the first number chosen by a distance greater than $\frac 14$ unit on the number line? Express your answer as a common fraction. | \frac{9}{32} | 61 | 8 |
math | Given is a simple graph with $239$ vertices, such that it is not bipartite and each vertex has degree at least $3$ . Find the smallest $k$ , such that each odd cycle has length at most $k$ . | k = 3 | 57 | 4 |
math | In the country of Logicville, the car license plates are formatted with four characters. The first character is from the modified vowel set (A, E, I, O, U, Y), the second and third characters are two different letters chosen from the 21 consonants (excluding Y), and the fourth character is a two-digit number (from 00 t... | \frac{1}{252,000} | 171 | 13 |
math | Among the following expressions, the correct serial numbers are:
① $0.7^{-0.3} > 0.7^{-0.4}$
② $\ln2 > \ln e$
③ $1.01^{-2} > 1.01^{-3}$
④ $\log_{2}2 < \log_{2}3$
⑤ $a^{1.3} < a^{2.5}$
⑥ $\log_{a}e < \log_{a}2$ (where $0 < a < 1$) | ③④⑥ | 125 | 6 |
math | I had $\$50$ in allowance money and spent it as indicated in the pie graph shown. How many dollars did I spend on games?
[asy]
size(150);
pair A, B, C, D, O, W, X, Y, Z;
O=(0,0);
A=(.707,.707);
B=(-.966,.259);
C=(-.707,-.707);
D=(.342,-.940);
draw(Circle(O, 1));
draw(O--A);
draw(O--B);
draw(O--C);
draw(O--D);
W=(-.1,.... | 7.5 | 250 | 3 |
math | By what number must we multiply ${ }^{4} \log 8$ to obtain ${ }^{32} \log 8$? | \frac{2}{5} | 30 | 7 |
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