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 that point $D$ satisfies $\overrightarrow{BC}=2\overrightarrow{CD}$ in triangle $\triangle ABC$, express $\overrightarrow{AD}$. | \frac{3}{2}\overrightarrow{AC} - \frac{1}{2}\overrightarrow{AB} | 34 | 25 |
math | Find the minimum value of
$$
\begin{aligned}
A & =\sqrt{\left(1264-z_{1}-\cdots-z_{n}\right)^{2}+x_{n}^{2}+y_{n}^{2}}+ \\
& \sqrt{z_{n}^{2}+x_{n-1}^{2}+y_{n-1}^{2}}+\cdots+\sqrt{z_{2}^{2}+x_{1}^{2}+y_{1}^{2}}+ \\
& \sqrt{z_{1}^{2}+\left(948-x_{1}-\cdots-x_{n}\right)^{2}+\left(1185-y_{1}-\cdots-y_{n}\right)^{2}}
\end{al... | 1975 | 216 | 4 |
math | From a sequence of natural numbers, all numbers that are squares or cubes of integers have been deleted. Which number is in the 100th position among the remaining numbers? | 112 | 36 | 3 |
math | In triangle $ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and $a=2b$. Also, $\sin A$, $\sin C$, $\sin B$ form an arithmetic sequence.
$(I)$ Find the value of $\cos (B+C)$;
$(II)$ If the area of $\triangle ABC$ is $\frac{8\sqrt{15}}{3}$, find the value of $c$. | 4 \sqrt {2} | 105 | 6 |
math | Given the function $f(x)=x^{2}-2x-a\ln x$, $g(x)=ax$.
$(1)$ Find the extreme values of the function $F(x)=f(x)+g(x)$.
$(2)$ If the inequality $\dfrac {\sin x}{2+\cos x}\leqslant g(x)$ always holds for $x\geqslant 0$, find the range of values for $a$. | [\dfrac {1}{3},+\infty) | 94 | 12 |
math | The graph of $y = f(x)$ is the same described in the original problem. Now, consider the new function defined by
\[ h(x) = a f(bx) + c \]
The graph of $y = h(x)$ is created by shrinking the graph of $y = f(x)$ horizontally by a factor of 4 and vertically by a factor of 1/3, and then shifted up by 1 unit.
Determine th... | \left( \frac{1}{3}, \frac{1}{4}, 1 \right) | 117 | 22 |
math | Find the number of 10-tuples $(x_1, x_2, \dots, x_{10})$ of real numbers such that
\[(1 - x_1)^2 + (x_1 - x_2)^2 + (x_2 - x_3)^2 + \dots + (x_9 - x_{10})^2 + x_{10}^2 = \frac{1}{11}.\] | 1 | 99 | 1 |
math | If the length of the line segment intercepted on line $m$ by two parallel lines $l_1: x - \sqrt{3}y + 1 = 0$ and $l_2: x - \sqrt{3}y + 3 = 0$ is 1, then the size of the angle of inclination of line $m$ is ________. | 120^\circ | 78 | 5 |
math | Given the complex number $z= \frac{1+ \sqrt{3}i}{ \sqrt{3}+i}$ (where $i$ is the imaginary unit), find the complex conjugate of $z$.
A) $\frac{ \sqrt{3}}{2}- \frac{1}{2}i$
B) $\frac{ \sqrt{3}}{2}+ \frac{1}{2}i$
C) $\sqrt{3}-i$
D) $\sqrt{3}+i$ | \frac{ \sqrt{3}}{2}- \frac{1}{2}i | 111 | 19 |
math | If "$x > a$" is a sufficient but not necessary condition for "$x^{2}-5x+6 \geqslant 0$" to hold, then the range of values for the real number $a$ is ______. | [3, +\infty) | 49 | 8 |
math | Consider a modified octahedron with an additional ring of vertices. There are 4 vertices on the top ring, 8 on the middle ring, and 4 on the bottom ring. An ant starts at the highest top vertex and walks down to one of four vertices on the next level down (the middle ring). From there, without returning to the previous... | \frac{1}{3} | 113 | 7 |
math | Mary and James each sit in a row of 10 chairs. However, chairs number 5 and 8 are broken and cannot be used. They choose their seats at random. What is the probability that they don't sit next to each other? | \frac{11}{14} | 51 | 9 |
math | Jake and Ellie alternately toss their coins until someone gets heads. Jake's coin lands heads with probability $\frac{1}{4}$, and Ellie's coin lands heads with probability $\frac{1}{3}$. Jake goes first. Find the probability that Jake wins. | \frac{1}{2} | 55 | 7 |
math | The arithmetic mean (average) of four numbers is $85$. If the largest of these numbers is $97$, then the mean of the remaining three numbers is | 81.0 | 35 | 4 |
math | A librarian arranges books on shelves in such a way that the top shelf contains 3 books and each lower shelf has three more books than the shelf above it. If the arrangement contains a total of 225 books, how many shelves does the librarian use? | 15 | 54 | 2 |
math | Let $\alpha$ and $\beta$ be the roots of $x^2 + px + 1 = 0,$ and let $\gamma$ and $\delta$ are the roots of $x^2 + qx + 1 = 0.$ Express
\[(\alpha - \gamma)(\beta - \gamma)(\alpha + \delta)(\beta + \delta)\]in terms of $p$ and $q.$ | q^2 - p^2 | 91 | 7 |
math | For a custom-made dress, Emily needs to specify her waist size in centimeters. If there are $12$ inches in a foot and $30.48$ centimeters in a foot, then what size should Emily specify, in centimeters, if her waist size in inches is $28$ inches? | 71.1\ \text{cm} | 66 | 10 |
math | Given the function $f(x)=-\frac{1}{\sqrt{b}}e^{\sqrt{ax}} (a > 0, b > 0)$, the tangent line to its graph at $x=0$ is tangent to the circle $x^2 + y^2 = 1$. Find the minimum value of $2^a + 2^b$. | 2\sqrt{2} | 81 | 6 |
math | Two distinct positive integers from 1 to 100 inclusive are chosen. Let the sum of the integers equal $S$ and the product equal $P$. What is the probability that $P+S$ is two less than a multiple of 7? | \frac{1295}{4950} | 52 | 13 |
math | In triangle $ABC$, $A=60^\circ$, $b=1$, and the area is $\sqrt{3}$. Find the value of $\frac{a+2b-3c}{\sin A+2\sin B-3\sin C}$. | \frac{2\sqrt{39}}{3} | 58 | 13 |
math | Given the function $f(x)=a\sin x- \frac {3}{2}(a\in\mathbb{R})$, if the number of zeros of the function $f(x)$ in $(0,\pi)$ is $2$, then the maximum value of $f(x)$ when $x\in[0, \frac {\pi}{2}]$ is ___. | a- \frac {3}{2} | 79 | 9 |
math | Among prime numbers not exceeding $12$, determine the probability of randomly selecting two different numbers whose sum is an even number. | \frac{3}{5} | 25 | 7 |
math | The probability it will rain on Friday is $40\%$, on Saturday is $50\%$, and on Sunday is $30\%$. Assuming the probability of rain on any given day is independent of the weather on any other day, what is the probability it will rain on all three days, expressed as a percent? | 6\% | 69 | 3 |
math | 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 passing through the point $(1, \frac{3}{2})$. If a point $M(x_0, y_0)$ is on the ellipse $C$, then the point $N(\frac{x_0}{a}, \frac{y_0}{b})$ is called an "ellipse point" of point $M$.
(1) Find the st... | S_{\triangle AOB} = \sqrt{3} | 220 | 13 |
math | Xiaohong bought 2 identical exercise books and 1 pen, spending a total of 12 yuan. The cost of one exercise book is 10% of the cost of one pen. How much does one pen and one exercise book cost respectively? | 1 | 54 | 1 |
math | Let set $M=\{-1, 0, 1\}$, and set $N=\{a, a^2\}$. Find the real number $a$ such that $M \cap N = N$. | -1 | 47 | 2 |
math | Given the function $f(x)=(m^{2}-2m-2)x^{m-1}$ is a power function and monotonically decreasing in $(0,+\infty)$, find $f(3)$. | \frac{1}{9} | 46 | 7 |
math | In $\triangle PQR$, point $S$ is the midpoint of side $PQ$. Point $T$ is on $QR$ such that $QT:TR = 1:3$. Point $U$ is on $PS$ such that $PU:US = 2:1$. If the area of $\triangle STU$ is 12, determine the area of $\triangle PQR$. | 96 | 84 | 2 |
math | How many distinct sequences of four letters can be made from the letters in DYNAMIC if each letter can be used only once and each sequence must begin with D and not end with C? | 60 | 37 | 2 |
math | Given that $\sin(\pi - a) = \frac{4}{5}$, and $a \in (0, \frac{\pi}{2})$, determine the value of $\sin 2a - \cos^2 \frac{a}{2}$. | \frac{4}{25} | 56 | 8 |
math | Let \(a, b, c, d\) be non-zero real numbers. Find the number of real roots of the equation
\[
\begin{vmatrix} x & c+d & -b \\ -c & x & a+d \\ b & -a & x \end{vmatrix} = 0.
\] | 1 | 68 | 1 |
math | Given the sets $A=\{x|-2\leqslant x\leqslant 5\}$, $B=\{x|m+1\leqslant x\leqslant 2m-1\}$
$(1)$ If $B\subseteq A$, find the range of the real number $m$;
$(2)$ If $A\cap B=\varnothing$, find the range of the real number $m$. | (-\infty,2)\cup(4,+\infty) | 98 | 15 |
math | Solve the equations:<br/>$(1)\left(2x-1\right)^{2}-4x=0$;<br/>$(2)\left(2x-3\right)^{2}=x^{2}$. | x_1 = 3, \quad x_2 = 1 | 50 | 15 |
math | Given $A=\{1, 3, m+2\}$, $B=\{3, m^2\}$, if $B \subseteq A$, then $m=$ ___. | 2 | 41 | 1 |
math | On each square of an $n\times n$ -chessboard, there are two bugs. In a move, each bug moves to a (vertically of horizontally) adjacent square. Bugs from the same square always move to different squares. Determine the maximal number of free squares that can occur after one move.
*(Swiss Mathematical Olympiad 2011, Fin... | n^2 | 85 | 3 |
math |
On the sides $AB$, $BC$, and $AC$ of triangle $ABC$, whose area is 75, points $M$, $N$, and $K$ are respectively located. It is known that $M$ is the midpoint of $AB$, the area of triangle $BMN$ is 15, and the area of triangle $AMK$ is 25. Find the area of triangle $CNK$. | 15 | 91 | 2 |
math | Given the line $x+y+m=0$ and the circle $x^{2}+y^{2}=4$ intersect at two distinct points $A$ and $B$. $O$ is the origin, and $|\overrightarrow{OA}+\overrightarrow{OB}|\geq |\overrightarrow{AB}|$. Determine the range of values for the real number $m$. | (-2\sqrt{2},-2]\cup[2,2\sqrt{2}) | 80 | 20 |
math | A random variable $\zeta$ follows a normal distribution $N(40, \sigma^2)$. If $P(\zeta < 30) = 0.2$, determine the probability $P(30 < \zeta < 50)$. | 0.6 | 58 | 3 |
math | Rationalize the denominator of $\frac{\sqrt{50}}{\sqrt{25} - \sqrt{5}}$. The answer can be written as $\frac{Ax\sqrt{y} + z}{w}$, where $A$, $x$, $y$, $z$, and $w$ are integers, $w$ is positive, and $y$ is not divisible by the square of any prime. Find the minimum possible value of $A+x+y+z+w$. | 13 | 101 | 2 |
math | Given a triangle $ABC$ with internal angles $A$, $B$, and $C$, and vectors $\overrightarrow{m}=(\sqrt{3}\sin A, \sin B)$ and $\overrightarrow{n}=(\cos B, \sqrt{3}\cos A)$, such that $\overrightarrow{m}\cdot\overrightarrow{n}=1+\cos(A+B)$, determine the value of angle $C$. | \frac{2\pi}{3} | 89 | 9 |
math | The solution set of the inequality $\dfrac {1}{x-1}\geqslant -1$ is given by a union of intervals on the real number line. | (-\infty,0]\cup(1,+\infty) | 36 | 15 |
math | Ana's monthly salary was $$2000 in May. In June she received a 20% raise. In July she received a 20% pay cut. | 1920 | 38 | 4 |
math | Find integers $a$, $b$, $c$, and $d$ such that the expression $\cos x + \cos 5x + \cos 9x + \cos 13x + \cos 17x$ can be written in the form \[a \cos bx \cos cx \cos dx \cos ex.\] Then calculate $a + b + c + d + e$. | 22 | 85 | 2 |
math | Given that $\{a\_n\}$ is an arithmetic sequence with a common difference of $d$, and the sum of its first $n$ terms is $S\_n$, where $S\_4 = 2S\_2 + 4$.
1. Find the value of the common difference $d$;
2. If for any $n \in \mathbb{N}^*$, the inequality $S\_n \geq S\_8$ holds, find the range of possible values for $a\_1... | [-8, -7] | 111 | 6 |
math | Given $\triangle XYZ$ in a coordinate plane with vertices $X(0, 10)$, $Y(3, 10)$, and $Z(0, q)$ where $q$ is a positive constant. Determine an expression for the area of $\triangle XYZ$ in terms of $q$. | \frac{3}{2} (10-q) | 65 | 12 |
math | Let the function $f(x) = -\cos^2x - 4t \cdot \sin \frac{x}{2} \cos \frac{x}{2} + 2t^2 - 6t + 2$ ($x \in \mathbb{R}$), where $t \in \mathbb{R}$, and denote the minimum value of $f(x)$ as $g(t)$:
(1) Find the expression for $g(t)$;
(2) When $-1 < t < 1$, to make the equation $g(t) = kt$ have exactly one real root, f... | (-\infty, -8) \cup (-4, +\infty) | 144 | 18 |
math | For quadrilateral \(A B C D\), it is known that \(\angle B A C = \angle C A D = 60^\circ\), and \(A B + A D = A C\). It is also known that \(\angle A C D = 23^\circ\). How many degrees is the angle \(\angle A B C\)? | 83^\circ | 79 | 4 |
math | In $\triangle ABC$, the lengths of the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively, and ${a}^{2}={b}^{2}+{c}^{2}+\sqrt{3}ab$.
(I) Find angle $A$.
(II) Let $a=\sqrt{3}$, and let $S$ be the area of $\triangle ABC$. Find the maximum value of $S+3\cos B\cos C$, and determine the value o... | 3 | 125 | 1 |
math | Three numbers 1, $a$, 2 form a geometric sequence, then the real number $a$ = ______. | \pm \sqrt{2} | 25 | 7 |
math | On an island, there are two tribes: knights and liars. Knights always tell the truth, and liars always lie. One day, 80 people sat at a round table, and each of them declared: "Among the 11 people sitting immediately after me in a clockwise direction, there are at least 9 liars." How many knights are sitting at the rou... | 20 | 86 | 2 |
math | Given vectors $\overrightarrow{m}=(\sqrt{3}\sin \omega x,1)$, $\overrightarrow{n}=({\cos }^{2}\omega x+1)$, let the function $f(x)= \overrightarrow{m}\cdot \overrightarrow{n} +b$.
(Ⅰ) If the graph of the function $f(x)$ is symmetric about the line $x= \frac{\pi}{6}$, and $\omega\in[0,3]$ when, find the intervals of i... | b\in(-2, \frac{\sqrt{3}-3}{2}]\cup\{-\frac{5}{2}\} | 172 | 29 |
math | In the sequence ${a_n}$, $$a_{1}= \frac {2}{3}$$ and $$a_{n+1}= \frac {n}{n+1}a_{n}$$, find the expression for $a_n$. | \frac {2}{3n} | 52 | 8 |
math | The system of equations {x y + y z = 63, x z + y z = 23} has how many positive integer solutions? | 2 | 32 | 1 |
math | In a quadrilateral pyramid \( S A B C D \):
- The lateral faces \( S A B, S B C, S C D, S D A \) have areas 9, 9, 27, and 27 respectively.
- The dihedral angles at the edges \( A B, B C, C D, D A \) are equal.
- The quadrilateral \( A B C D \) is inscribed in a circle, and its area is 36.
Find the volume of the pyram... | 54 | 119 | 2 |
math | Given that the restaurant charges $6.00 for adults and $0.60 for each year of the child's age, and they offer a family discount of $2.00 if more than two family members are dining, if the total bill, after applying the discount, is $13.20, determine the possible age of the child. | 5 | 74 | 1 |
math | How many three-digit numbers are there in which any two adjacent digits differ by 3? | 20 | 18 | 2 |
math | A thousand points form the vertices of a convex polygon with 1000 sides. Inside this polygon, there are another 500 points placed such that no three of these 500 points are collinear. The polygon is triangulated in such a way that all of these 1500 points are vertices of the triangles, and none of the triangles have an... | 1998 | 90 | 4 |
math | Ted's grandfather used his treadmill on 4 days this week. He went 1 mile on Monday at a speed of 6 miles per hour, 2 miles on Tuesday at 4 miles per hour, 1 mile on Thursday at 3 miles per hour, and 2 miles on Friday at 2 miles per hour. Calculate the total time spent on the treadmill at the given speeds and then deter... | 0 | 103 | 1 |
math | Given that set $A$ is the collection of all functions $f(x)$ that satisfy the condition: there exists a real number ${{x}_{0}}$ in the domain such that $f({{x}_{0}}+1)+f({{x}_{0}})=f(1)$.
(I) Determine whether the power function $f(x)={{x}^{-1}}$ belongs to set $A$ and explain the reasoning.
(II) Let $g(x)= \lg \frac... | b=1 | 184 | 3 |
math | There are 4 boys and 3 girls standing in a row:
(1) If 3 people are selected to stand in a row, how many different arrangements are there?
(2) If boy A cannot stand at the head of the line, and girl B cannot stand at the end of the line, how many different arrangements are there?
(3) If the girls must stand together, h... | 1440 | 109 | 4 |
math | A circle with center $C$ is tangent to the positive $x$ and $y$-axes and externally tangent to another circle centered at $(2,0)$ with radius $2$. Determine the radius difference between the largest and smallest possible radii of the circle with center $C$. | 0 | 59 | 1 |
math | Given vectors $\mathbf{a}$ and $\mathbf{b},$ let $\mathbf{p}$ be a vector such that
\[\|\mathbf{p} - \mathbf{b}\| = 3 \|\mathbf{p} - \mathbf{a}\|.\] Among all such vectors $\mathbf{p},$ there exists constants $t$ and $u$ such that $\mathbf{p}$ is at a fixed distance from $t \mathbf{a} + u \mathbf{b}.$ Enter the ordere... | \left(\frac{9}{8}, -\frac{1}{8}\right) | 127 | 19 |
math | If the sum of all integer solutions of the inequality system about $x$ $\left\{{\begin{array}{l}{2x+1>x+a}\\{\frac{x}{2}+1\geq\frac{5}{2}x-9}\end{array}}\right.$ is $14$, then the value of the integer $a$ is ______. | 2 | 79 | 1 |
math | In a new diagram showing the miles traveled by bikers Alberto, Bjorn, and Carlos over a period of 6 hours. The straight lines represent their paths on a coordinate plot where the y-axis represents miles and x-axis represents hours. Alberto's line passes through the points (0,0) and (6,90), Bjorn's line passes through (... | 30 | 122 | 2 |
math | Use arithmetic symbols and parentheses to form an equation with the numbers 5, 7, 8, 8 such that the result is 24. Provide one such equation. | (7 - 5) \times 8 + 8 = 24 | 37 | 17 |
math | Given that $n$ is a positive integer, the graph of the quadratic function $y=2^{2n}x^{2}-6\cdot 2^{n}x+8$ is a parabola. If the length of the segment intercepted by this parabola on the $x$-axis forms a sequence $\{d_{n}\}$, then $\lim_{n→∞}({{d_1}+{d_2}+⋯+{d_n}})=$____. | 2 | 107 | 1 |
math | If $0 \in \{m, m^2 - 2m\}$, then the value of the real number $m$ is. | 2 | 31 | 1 |
math | A circle is inscribed in a triangle with side lengths $10, 15, 19$. Let the segments of the side of length $10$, made by a point of tangency, be $r'$ and $s'$, with $r'<s'$. Find the ratio $r':s'$. | 3:7 | 68 | 3 |
math | Show that there exists an infinite arithmetic progression of natural numbers such that the first term is $16$ and the number of positive divisors of each term is divisible by $5$ . Of all such sequences, find the one with the smallest possible common difference. | 32 | 56 | 2 |
math | Given the set $A=\{-2,0,1,3\}$, in the Cartesian coordinate system, the coordinates of point $M$ are $(x,y)$, satisfying $x\in A$ and $y\in A$.
$(1)$ Please list all the coordinates of point $M$;
$(2)$ Calculate the probability that point $M$ is not on the $y$-axis;
$(3)$ Calculate the probability that point $M$ fal... | \dfrac{3}{16} | 132 | 8 |
math | Xiao Li plans to spend 31 yuan to buy ballpoint pens priced at 2 yuan, 3 yuan, and 4 yuan each, with at least one of each kind. Calculate the maximum and minimum number of pens she can buy. | 9 | 51 | 1 |
math | A certain medicine underwent two price reductions, with the price per box decreasing from $50 to $32. The average percentage reduction each time is ____. | 20\% | 32 | 4 |
math | When a biased coin is flipped five times, the probability of getting heads exactly twice is equal to that of getting heads exactly three times. Find the probability, expressed as a fraction $\frac{p}{q}$ in lowest terms, that the coin shows heads exactly four times out of the five flips. | \frac{5}{32} | 60 | 8 |
math | Given the graph of $y = mx + 3$ passes through no lattice point with $0 < x \le 50$ for all $m$ such that $\frac{1}{3} < m < b$, find the maximum possible value of $b$. | \frac{17}{50} | 56 | 9 |
math | The function \[f(x) = \left\{ \begin{aligned} x-3 & \quad \text{ if } x < 5 \\ \sqrt{x} & \quad \text{ if } x \ge 5 \end{aligned} \right.\] has an inverse $f^{-1}.$ Find the value of $f^{-1}(0) + f^{-1}(1) + \dots + f^{-1}(9).$ | 291 | 97 | 3 |
math | Square \(ABCD\) has side length 1, and points \(P\) and \(Q\) are located on sides \(AB\) and \(AD\) respectively. If the perimeter of \(\triangle APQ\) is 2, find the measure of \(\angle PCQ\) in degrees. | 45^\circ | 60 | 4 |
math | Find the domain of the expression $\frac{\sqrt{x-3}}{\sqrt{7-x} \cdot (x-1)}$. | [3, 7) | 28 | 6 |
math | Three pirates divided the diamonds they acquired during the day in the evening: Bill and Sam each received twelve diamonds, and the rest were given to John, who did not know how to count. During the night, Bill stole one diamond from Sam, Sam stole one diamond from John, and John stole one diamond from Bill. As a resul... | 9 \text{ diamonds} | 111 | 6 |
math | If the fractional equation $\frac{3}{{x-2}}+1=\frac{m}{{4-2x}}$ has a root, then the value of $m$ is ______. | -6 | 42 | 2 |
math | Given \(f(x) = x^5 + 2x^3 + 3x^2 + x + 1\), when applying the Horner's method to calculate the value at \(x = 3\), calculate the value of \(v_3\). | 36 | 57 | 2 |
math | Consider that Henry's little brother now has 10 identical stickers and 5 identical sheets of paper. How many ways can he distribute all the stickers on the sheets of paper, if only the number of stickers on each sheet matters and no sheet can remain empty? | 126 | 53 | 3 |
math | In a tug-of-war competition, class 3, class 4, and class 5 were victorious in the preliminary rounds. Three judges made predictions about the champion: Judge A said: "The champion will not be class 3, nor class 4"; Judge B said: "The champion will not be class 3, it must be class 5"; Judge C said: "The champion will no... | 3 | 136 | 1 |
math | Given an arithmetic sequence $\{a_n\}$ that satisfies $(a_1+a_2)+(a_2+a_3)+\ldots+(a_n+a_{n+1})=2n(n+1)$ for $n\in\mathbb{N}^*$.
$(1)$ Find the general formula for the sequence $\{a_n\}$.
$(2)$ Find the sum of the first $n$ terms, $S_n$, of the sequence $\left\{ \frac{a_n}{2^{n-1}} \right\}$. | 6-(2n+3)\cdot\left(\frac{1}{2}\right)^{n-1} | 119 | 24 |
math | Given positive numbers \(a, b, c, x, y, z\) that satisfy the equations \(cy + bz = a\), \(az + cx = b\), and \(bx + ay = c\), find the minimum value of the function
\[ f(x, y, z) = \frac{x^2}{1 + x} + \frac{y^2}{1 + y} + \frac{z^2}{1 + z}. \] | 1/2 | 98 | 3 |
math | A square flower bed with an area of approximately 80 square meters is being planned. Calculate the approximate length of its side. | 8.9 | 26 | 3 |
math | How many positive three-digit integers with each digit greater than 6 are divisible by 12? | 1 | 20 | 1 |
math | What is the largest divisor by which $29 \cdot 14$ leaves the same remainder when divided by $13511, 13903,$ and $14589$? | 98 | 46 | 2 |
math | A sealed horizontal cylindrical vessel of length \( l \) is divided into two parts by a movable partition. On one side of the partition, there is 1 mole of oxygen, and on the other side, there is 1 mole of helium and 1 mole of oxygen, with the partition being initially in equilibrium. At a certain moment, the partition... | \frac{l}{6} | 107 | 6 |
math | Given a quadratic function \( f(x) = a x^{2} + b x + c \) where \( a, b, c \in \mathbf{R} \) and \( a \neq 0 \), the following conditions are satisfied:
1. \( f(x-4) = f(2-x) \) for all \( x \in \mathbf{R} \), and \( f(x) \geq x \);
2. \( f(x) \leq \left(\frac{x+1}{2}\right)^{2} \) for \( x \in (0, 2) \);
3. The minimu... | 9 | 217 | 1 |
math | Given an ellipse $\frac {x^{2}}{a^{2}}+ \frac {y^{2}}{b^{2}}=1$ (where $a>b>0$) with eccentricity $e=\frac {\sqrt {2}}{2}$, point $P(1,\frac {\sqrt {2}}{2})$ lies on this ellipse.
(1) Find the standard equation of the ellipse.
(2) If a line $l$ is tangent to the circle $O: x^{2}+y^{2}=1$ and intersects the ellipse at ... | \frac {\sqrt {2}}{2} | 148 | 10 |
math | Let S<sub>n</sub> be the sum of the first n terms of the sequence {a<sub>n</sub>}, satisfying a<sub>n</sub><sup>2</sup>+1=2a<sub>n</sub>S<sub>n</sub>, and a<sub>n</sub> > 0, then a<sub>100</sub>\=\_\_\_\_\_\_． | 10-3 \sqrt {11} | 89 | 10 |
math | Let $a$ and $b$ be positive real numbers with $a \ge b$. Consider $\rho$ to be the maximum possible value of $\frac{a}{b}$ for which the system of equations
$$
a^2 + y^2 = (a - x)^2 + (b - y)^2 = b^2 - x^2 + y^2
$$
has a solution in $(x, y)$ satisfying $0 \leq x < a$ and $0 \leq y < b$. Find $\rho^2.$ | 2 | 117 | 1 |
math | Let the function $f(x)= \sqrt {x^{2}-3x+2}$ have a domain of set $A$, and the solution set of the inequality $(x-a)(x-3a)\leqslant 0$ be set $B$ (where $a\in\mathbb{R}$, and $a > 0$).
(I) When $a=1$, find the set $A\cap B$;
(II) When $A\cap B=B$, find the range of the real number $a$. | (0, \frac {1}{3}]\cup[2,+\infty) | 117 | 19 |
math | A $2.00 \text{L}$ balloon at $20.0^{\circ} \text{C}$ and $745 \text{mmHg}$ floats to an altitude where the temperature is $10.0^{\circ} \text{C}$ and the air pressure is $700 \text{mmHg}$ . Calculate the new volume of the balloon. | 2.06 \text{ L} | 94 | 9 |
math | Given the line $ax + y - 2 + a = 0$, the intercepts on both coordinate axes are equal. Find the real number $a$. | 1 | 33 | 1 |
math | Evaluate $\left(ax + \frac{y}{b} \right)^{-1} \left[(ax)^{-1} + \left(\frac{y}{b} \right)^{-1} \right]$. | (axy)^{-1} | 46 | 6 |
math | Given a right prism $ABC-A_{1}B_{1}C_{1}$ with height $3$, whose base is an equilateral triangle with side length $1$, find the volume of the conical frustum $B-AB_{1}C$. | \frac{\sqrt{3}}{4} | 54 | 10 |
math | Let the random variable $X$ have the probability distribution $P(X= \frac {k}{5})=ak$ for $k=1,2,3,4,5$.
1. Determine the value of the constant $a$.
2. Calculate $P(X\geq \frac {3}{5})$.
3. Calculate $P(\frac {1}{10}<X< \frac {7}{10})$. | \frac {2}{5} | 94 | 7 |
math | Let $g(x)$ be a function defined piecewise as follows:
\[ g(x) = \left\{
\begin{array}{cl}
-x & \text{if } x\leq 0, \\
2x - 45 & \text{if } x>0.
\end{array}
\right. \]
If $a$ is negative, find $a$ such that $g(g(g(11))) = g(g(g(a)))$. | a = -34 | 99 | 5 |
math | Points $A$, $B$, $C$, and $D$ are midpoints of the respective sides of a large square. Consider the triangle formed by points $A$, $B$, and $C$. If the larger square has area $128$, what is the area of the triangle formed by $A$, $B$, and $C$? | 16 | 73 | 2 |
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