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 | Find the equation of the tangent line to the curve $y = -x^3 + 3x^2$ at the point of tangency. | 3x - 1 | 31 | 5 |
math | A plane is drawn through the midpoint of the diagonal of a cube and is perpendicular to this diagonal. Find the area of the resulting cross-section if the edge length of the cube is \(a\). | \frac{3a^2\sqrt{3}}{4} | 40 | 15 |
math | Given that \( a \) is a real number, and for any \( k \in [-1,1] \), when \( x \in (0,6] \), the following inequality is always satisfied:
\[ 6 \ln x + x^2 - 8 x + a \leq k x. \]
Find the maximum value of \( a \). | 6 - 6 \ln 6 | 77 | 8 |
math | If points $A(x_{1}$,$6)$, $B(x_{2}$,$12)$, $C(x_{3}$,$-6)$ are all on the graph of the inverse proportion function $y=\frac{6}{x}$, determine the relationship between $x_{1}$, $x_{2}$, $x_{3}$. | x_{3} < x_{2} < x_{1} | 75 | 14 |
math | The edge lengths of the top and bottom bases of a frustum of a regular triangular pyramid are 3cm and 6cm, respectively, and its height is $\frac{3}{2}$cm. The lateral surface area of the frustum is \_\_\_\_\_\_ cm<sup>2</sup>. | \frac{27 \sqrt{3}}{2} | 66 | 13 |
math | Given the equation in terms of $x$, $x^4 + 2x^3 + (3+k)x^2 + (2+k)x + 2k = 0$ has real roots, and the product of all real roots is $-2$, find the sum of the squares of all real roots. | 5 | 66 | 1 |
math | On the extension of side \(AB\) of triangle \(ABC\) beyond point \(A\), point \(D\) is taken such that \(AD = AB\). On the extension of side \(AC\) of triangle \(ABC\) beyond point \(A\), point \(E\) is taken such that \(AE = AC\). Find the distance between points \(D\) and \(E\) if \(BC = 5\). | 5 | 86 | 1 |
math | The left and right foci of a hyperbola are $F_{1}$ and $F_{2}$, respectively. A line passing through $F_{2}$ intersects the right branch of the hyperbola at points $A$ and $B$. If $\triangle F_{1} A B$ is an equilateral triangle, what is the eccentricity of the hyperbola? | \sqrt{3} | 80 | 5 |
math | Determine all pairs of integers \((x, y)\) such that \(9xy - x^2 - 8y^2 = 2005\). | (63, 58), (-63, -58), (459, 58), (-459, -58) | 36 | 34 |
math | Determine how many $4 \times 4$ arrays containing all the numbers from 1 to 16 can be formed such that each row and column contains numbers in strictly increasing order.
A) 24
B) 36
C) 48
D) 54
E) 60 | 36 | 68 | 2 |
math | The product of a million whole numbers is equal to million. What can be the greatest possible value of the sum of these numbers? | 1999999 | 26 | 7 |
math | In a diagram, there is an equilateral triangle with a side length of $10$ m. Calculate both the perimeter and the height of the triangle. | 5\sqrt{3} | 32 | 6 |
math | Given that $F_{1}$ and $F_{2}$ are two foci of the hyperbola $\frac{x^2}{4}-\frac{y^2}{b^2}=1(b＞0)$, point $A$ is the right vertex of the hyperbola, and $M(x_{0}$,$y_{0})(x_{0} \gt 0$,$y_{0} \gt 0)$ is a point on the asymptote of the hyperbola, satisfying $MF_{1}\bot MF_{2}$. If the parabola with focus at $A$ is $y^{2}... | \sqrt{5} | 164 | 5 |
math | The monotonic increasing interval of the function $f(x) = \left(\frac{1}{2}\right)^{x^2-4}$ is __________. | (-\infty, 0] | 35 | 8 |
math | A subset \( H \) of the set of numbers \(\{1, 2, \ldots, 100\}\) has the property that if an element is in \( H \), then ten times that element is not in \( H \). What is the maximum number of elements that \( H \) can have? | 91 | 70 | 2 |
math | In how many different ways can one place 3 rooks on the cells of a $6 \times 2006$ chessboard such that they don't attack each other? | 20 \cdot 2006 \cdot 2005 \cdot 2004 | 38 | 23 |
math | Let the domain of the function $y=f(x)$ be $D$. If for any $x_{1} \in D$, there exists $x_{2} \in D$ such that $f(x_{1}) \cdot f(x_{2}) = 1$, then the function $f(x)$ is said to have property $M$. The following four statements are given:<br/>① The function $y=x^{3}-x$ does not have property $M$;<br/>② The function $y=\... | ①②③ | 213 | 6 |
math | If real numbers $a$, $b$, $c$ satisfy $a^2+b^2+c^2=9$, then the maximum value of the algebraic expression $(a-b)^2+(b-c)^2+(c-a)^2$ is. | 27 | 53 | 2 |
math | Given the function $f(x) = x^3 + 2x^2 - 4x$.
(1) Find the intervals where the function is monotonically increasing or decreasing.
(2) Find the extreme values of $f(x)$ on the interval $[-5, 0]$. | 8 | 63 | 1 |
math | Given $p: 1 \leq x < 3$; $q: x^2 - ax \leq x - a$; If $\neg p$ is a sufficient condition for $\neg q$, find the range of the real number $a$. | [1, 3) | 55 | 6 |
math | The digits from 1 to 9 are written in order so that the digit \( n \) is written \( n \) times, forming the block of digits \( 1223334444 \cdots 999999999 \). This block is written 100 times. Calculate the 1953rd digit written. | 6 | 81 | 1 |
math | An art team is rehearsing gymnastics to celebrate the New Year. If 1000 team members are arranged in several rows, with the total number of rows being greater than 16, and each row from the second row onward has one more person than the previous row, how many rows should there be to meet the requirements? How many team... | 28 | 80 | 2 |
math | If \( x \) is a positive number such that
\[
\sqrt{12x} \cdot \sqrt{5x} \cdot \sqrt{7x} \cdot \sqrt{21x} = 21,
\]
find all possible values for \( x \). | \frac{1}{\sqrt{2} \times 5^{1/4}} | 62 | 19 |
math | Given that the ratio of the interior angles of two regular polygons is $5:3$, determine the number of pairs of regular polygons that exist where both have an integer number of sides greater than 2. | 2 | 41 | 1 |
math | A student research group at a school found that the attention index of students during class changes with the listening time. At the beginning of the lecture, students' interest surges; then, their interest remains in a relatively ideal state for a while, after which students' attention begins to disperse. Let $f(x)$ r... | \dfrac {85}{3} | 321 | 8 |
math | On graph paper, a right-angled triangle with legs of length 6 and 10 is shown. Find the total length of the horizontal grid lines inside this triangle. | 27 | 35 | 2 |
math | Given \( f(u) = u^{2} + au + (b-2) \), where \( u = x + \frac{1}{x} \) (with \( x \in \mathbb{R} \) and \( x \neq 0 \)). If \( a \) and \( b \) are real numbers such that the equation \( f(u) = 0 \) has at least one real root, find the minimum value of \( a^{2} + b^{2} \). | 4/5 | 109 | 3 |
math | Given an ellipse $C$ with its center at the origin and eccentricity $e= \frac{\sqrt{2}}{2}$, and one of its foci coincides with the focus of the parabola $y= \frac{1}{4}x^{2}$.
(1) Find the equation of the ellipse $C$;
(2) A moving line $l$ passing through point $S(-\frac{1}{3}, 0)$ intersects the ellipse $C$ at poi... | (1, 0) | 167 | 6 |
math | A $10$-cm-by-$10$-cm square is partitioned as shown. Points $A$ and $B$ are the trisection points (one third from the endpoints) of two opposite sides of the square. What is the area of the shaded region? | \frac{100}{3} \text{ square cm} | 59 | 15 |
math | In $\triangle ABC$, let $A$, $B$, and $C$ be the three interior angles, and $a$, $b$, and $c$ be the sides opposite these angles, respectively. Given that:
\[2 \sqrt{2}\left(\sin ^{2} A-\sin ^{2} C\right) = (a-b) \sin B,\]
and the circumradius of $\triangle ABC$ is $\sqrt{2}$.
(1) Find angle $C$.
(2) Find the maximum... | \frac{3\sqrt{3}}{2} | 125 | 12 |
math | Given the sets \( M = \{x, y, \lg(xy)\} \) and \( N = \{0, |x|, y\} \) with \( M = N \), find the value of \(\left(x + \frac{1}{y}\right) + \left(x^{2} + \frac{1}{y^{2}}\right) + \left(x^{3} + \frac{1}{y^{3}}\right) + \cdots + \left(x^{2001} + \frac{1}{y^{2001}}\right) \). | -2 | 133 | 2 |
math | In triangle ABC, the sides opposite to angles A, B, C are a, b, c, respectively, and $c - a \cos B = \frac{\sqrt{2}}{2} b$.
(I) Find angle A;
(II) If $c = 4\sqrt{2}$ and $\cos B = \frac{7\sqrt{2}}{10}$, find the area of triangle ABC. | 2 | 91 | 1 |
math | A certain company decided to evaluate a certain product under its umbrella. The original price of the product was $25$ yuan per unit, with annual sales of $80,000$ units.
$(1)$ According to market research, if the price is increased by $1$ yuan, the sales volume will decrease by $2,000$ units. To ensure that the tota... | 30 \text{ yuan} | 261 | 7 |
math | Given a circle $C: x^{2}+y^{2}=16$ and a line $l: (a-b)x + (3b-2a)y - a = 0$ where $a$ and $b$ are not both zero. As $a$ and $b$ vary, the minimum value of the chord length intercepted by the circle $C$ and the line $l$ is ____. | 2\sqrt{6} | 88 | 6 |
math | Given the parabola $y=x^{2}-2ax+a^{2}+2a\left(a \gt 0\right)$.
$(1)$ If $a=1$, the coordinates of the vertex of the parabola are ______;
$(2)$ The line $x=m$ intersects with the line $y=2x-2$ at point $P$, and intersects with the parabola $y=x^{2}-2ax+a^{2}+2a$ at point $Q$. If when $m \lt 3$, the length of $PQ$ de... | a \geqslant 2 | 144 | 8 |
math | Given the function $f(x)=\sin 2x$, translate its graph to the right by $\dfrac{\pi}{12}$ units to obtain the graph of the function $y=g(x)$. Then, find the center of symmetry of $y=g(x)$. | \left(\dfrac{\pi}{12},0\right) | 57 | 15 |
math | Line $m$ is parallel to line $n$ and the measure of $\angle 1$ is $\frac{1}{4}$ the measure of $\angle 2$. What is the degree measure of $\angle 5$?
[asy] size(100); defaultpen(linewidth(0.7)+fontsize(10));
path m = (-1.35,0.72)--(0.45,0.72), n = (-1,0)--(1,0), k = (-0.67,1.09)--(0.27,-0.48);
pair A = intersectionpoi... | 36^\circ | 290 | 4 |
math | Let \(a_0 = -3, b_0 = 2\), and for \(n \geq 0\), let
\[ a_{n+1} = a_n + b_n + \sqrt{a_n^2 + b_n^2}, \]
\[ b_{n+1} = a_n + b_n - \sqrt{a_n^2 + b_n^2}. \]
Find \(\frac{1}{a_{10}} + \frac{1}{b_{10}}\). | \frac{1}{3} | 113 | 7 |
math | How many four-digit numbers have the following property? "For each of its digits, when this digit is deleted the resulting three-digit number is a factor of the original number."
A) 5
B) 9
C) 14
D) 19
E) 23 | 14 | 62 | 2 |
math | Frank's teacher asks him to write down five integers such that the median is one more than the mean, and the mode is one greater than the median. Frank is also told that the median is 10. What is the smallest possible integer that he could include in his list?
A 3
B 4
C 5
D 6
E 7 | 4 | 76 | 1 |
math | One hundred fifty people were surveyed. Of these, $120$ indicated they liked Mozart, $105$ indicated they liked Bach, and $45$ indicated they liked Beethoven. What is the minimum number of people surveyed who could have said they liked both Mozart and Bach, but not Beethoven? | 75 | 65 | 2 |
math | Given functions $f(x)=\ln x$ and $g(x)=\frac{1}{2}ax+b$, where $f(x)$ and $g(x)$ have the same tangent line at $x=1$.
$(1)$ Find the value of $a+2b$;
$(2)$ Solve the inequality $f(x) \lt g(x)$. | (0,1) \cup (1,+\infty) | 78 | 14 |
math | Determine the area of the bounded region enclosed by the graph of the equation $y^2 + 4xy + 80|x| = 800$. | 800 | 35 | 3 |
math | Find the sum of all possible sums $a + b$ where $a$ and $b$ are nonnegative integers such that $4^a + 2^b + 5$ is a perfect square. | 9 | 52 | 1 |
math | Given a sequence $\{a_{n}\}$ satisfying ${a}_{n}{a}_{n+2}={a}_{n+1}^{2}$, $n\in{N}^{*}$. If $a_{7}=16$ and $a_{3}a_{5}=4$, then the value of $a_{3}$ is ______. | 1 | 77 | 1 |
math | Given that $a$ is a positive integer and $a = b - 2005$, if the equation $x^2 - ax + b = 0$ has a positive integer solution, what is the minimum value of $a$?
(Hint: First, assume the two roots of the equation are $x_1$ and $x_2$, then…) | 95 | 79 | 2 |
math | The graph of $y = ax^2 + bx + c$ is described, where $a$, $b$, and $c$ are integers. Given that the vertex of this parabola is at $(-2, 3)$ and one point on the graph is $(1, 6)$, determine the value of $a$. | \frac{1}{3} | 70 | 7 |
math | In $\triangle ABC$, where $AB = BC > AC$, let $AH$ and $AM$ be the altitude and median to side $BC$, respectively. Given $\frac{S_{\triangle AMH}}{S_{\triangle ABC}} = \frac{3}{8}$, determine the value of $\cos \angle BAC$. | \frac{1}{4} | 71 | 7 |
math | Determine the sum of the positive integers $a$, $b$, and $c$ if the equation $\sin^2 x + \sin^2 3x + \sin^2 5x + \sin^2 7x = 2$ reduces to $\cos ax \cos bx \cos cx = 0$ for some positive integers $a$, $b$, $c$. | 14 | 82 | 2 |
math | In the Cartesian coordinate plane, vector $\overrightarrow {OA}=(4,1)$, vector $\overrightarrow {OB}=(2,-3)$. If the lengths of the projections of the two vectors on line $l$ are equal, then the slope of line $l$ is \_\_\_\_\_\_. | 3 \text{ or } -\frac{1}{2} | 66 | 14 |
math | Given the parabola $C: y^2 = -4x$ with focus $F$, and point $A(-2, 1)$. Let $P$ be a moving point on the parabola $C$. The minimum value of $|PF| + |PA|$ is \_\_\_\_\_\_. | 3 | 68 | 1 |
math | Given a geometric sequence $\{a_n\}$ with the sum of the first $n$ terms denoted as $S_n$. If $a_1=3$ and $S_2=9$, then $a_n=$ _______ ; $S_n=$ _______ . | 3\cdot(2^n-1) | 58 | 9 |
math | About pentagon $ABCDE$ is known that angle $A$ and angle $C$ are right and that the sides $| AB | = 4$ , $| BC | = 5$ , $| CD | = 10$ , $| DE | = 6$ . Furthermore, the point $C'$ that appears by mirroring $C$ in the line $BD$ , lies on the line segment $AE$ . Find angle $E$ . | 90^\circ | 115 | 4 |
math | Find the coefficient of $x^{79}$ in the expansion of
\[(x - 2^0)(x^2 - 2^1)(x^3 - 2^2) \dotsm (x^{11} - 2^{10})(x^{12} - 2^{11}).\] | 0 | 72 | 1 |
math | Given $- \frac {\pi}{2} < x < 0$, $\sin x+\cos x= \frac {1}{5}$.
$(1)$ Find the value of $\sin x-\cos x$;
$(2)$ Find the value of $4\sin x\cos x-\cos ^{2}x$. | - \frac {64}{25} | 71 | 10 |
math | Let \( g(x) = x^4 + 20x^3 + 98x^2 + 100x + 25 \). Let \( w_1, w_2, w_3, w_4 \) be the four roots of \( g \). Find the smallest possible value of \( |w_a w_b + w_c w_d| \) where \( \{a, b, c, d\} = \{1, 2, 3, 4\} \). | 10 | 113 | 2 |
math | The interval in which the function $y=\sin 2x$ is monotonically decreasing is $\left[k\pi - \dfrac{\pi}{4}, k\pi + \dfrac{3}{4}\pi\right] (k \in \mathbb{Z})$. | \left[k\pi + \dfrac{\pi}{4}, k\pi + \dfrac{3\pi}{4}\right] | 61 | 30 |
math | Given the parabola $C: y^2 = 4x$ and the point $M(0, 2)$, a line passing through the focus of $C$ with a slope of $k$ intersects $C$ at points $A$ and $B$. If $\overrightarrow{MA} \cdot \overrightarrow{MB} = 0$, then $k = \boxed{\text{\_\_\_\_\_\_}}$. | 1 | 94 | 1 |
math | Let $M$ be the greatest four-digit number whose digits have a product of $24$. Calculate the sum of the digits of $M$. | 13 | 30 | 2 |
math | The range of the inclination angle of the line $x\cos \theta +\sqrt{3}y+2=0$ is __________. | \left[0,\dfrac{\pi}{6} \right]\cup\left[\dfrac{5\pi}{6},\pi\right) | 31 | 33 |
math | Given the function $y=\log_{2}(ax-1)$ is monotonically decreasing on the interval $(-2,-1)$, determine the range of the real number $a$. | (-\infty,-1] | 39 | 7 |
math | 1. Given $\tan \frac{\alpha}{2} = \frac{1}{2}$, find the value of $\sin\left(\alpha + \frac{\pi}{6}\right)$.
2. Given $\alpha \in \left(\pi, \frac{3\pi}{2}\right)$ and $\cos\alpha = -\frac{5}{13}$, $\tan \frac{\beta}{2} = \frac{1}{3}$, find the value of $\cos\left(\frac{\alpha}{2} + \beta\right)$. | -\frac{17\sqrt{13}}{65} | 121 | 15 |
math | In a triangle, there are at least acute angles among the three interior angles, and at most acute angles among the three exterior angles. | 2, 1 | 33 | 4 |
math | Let $g(x)$ be a function piecewise defined as \[g(x) = \left\{
\begin{array}{cl}
-x & x\le 0, \\
2x-40 & x>0.
\end{array}
\right.\] If $a$ is negative, find $a$ so that $g(g(g(11)))=g(g(g(a)))$. | a = -29 | 85 | 5 |
math | After applying the successive discounts of 25%, 15%, and 5% to the bulk order of $\textdollar{15000}$, calculate the sale price with a 30% markup. Then, calculate the sale price after applying a 40% upfront discount to the original price, and compare the results. | 11,809.69 | 72 | 9 |
math | In rectangle $ABCD,$ $P$ is a point on side $\overline{BC}$ such that $BP = 16$ and $CP = 8.$ If $\tan \angle APD = 3,$ then find $AB.$ | 16 | 53 | 2 |
math | In triangle $PQR$, $N$ is the midpoint of $\overline{QR}$, $PQ = 15$, and $PR = 20$. Let $X$ be on $\overline{PR}$, and $Y$ be on $\overline{PQ}$, and let $Z$ be the intersection of $\overline{XY}$ and $\overline{PN}$. If $PX = 3PY$, then find $\frac{XZ}{ZY}$. | 4 | 105 | 1 |
math | One of two parallel lines is tangent to a circle of radius $R$ at point $A$, while the other intersects this circle at points $B$ and $C$. Express the area of triangle $ABC$ as a function of the distance $x$ between the parallel lines. | x \sqrt{2R x - x^2} | 57 | 12 |
math | The parametric equation of curve $C$ is given by $\begin{cases} x=2+3\cos \theta \\ y=1+3\sin \theta \end{cases}$, and the equation of line $l$ is $x-3y+2=0$. Calculate the number of points on curve $C$ that are $\frac{7}{10}\sqrt{10}$ units away from line $l$. | 4 | 93 | 1 |
math | Given that $\sin{2\alpha} = \frac{2}{3}$, find the value of $\cos^2\left(\alpha + \frac{\pi}{4}\right)$. | \frac{1}{6} | 41 | 7 |
math | $4\cos {50}^{0}-\tan {40}^{0}=$_______. | \sqrt{3} | 24 | 5 |
math | The length of the shortest trip from $A$ to $C$ along the edges of a cube shown is the length of 4 edges. How many different 4-edge trips are there from $A$ to $C$?
[asy]
size(4cm,4cm);
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a1=(1,1);
b1=(0,1);
c1=(1.6,1.4);
d1=(1,0);
pair e1, f1, g1, h1;
e1=(0,0);
f1=c1-(a1-d1);
g1=b1+(c1-a1);
h1=e1+(... | 12 | 240 | 2 |
math | A circle contains the points \((0, 11)\) and \((0, -11)\) on its circumference and contains all points \((x, y)\) with \(x^{2} + y^{2} < 1\) in its interior. Compute the largest possible radius of the circle. | 61 | 66 | 2 |
math | If the TV factory reduces the cost of a TV by 19% over the course of two years, with the percentage of reduction being the same each year, find this percentage of reduction. | 10\% | 39 | 4 |
math | Given the function $f(x) = \begin{cases} \ln x + b, & x > 1 \\ e^x - 2, & x \leqslant 1 \end{cases}$, if $f(e)=-3f(0)$, then the range of the function $f(x)$ is __________. | (-2, e-2] \cup (2, +\infty) | 73 | 17 |
math | Given $\angle AOB=80^\circ$, with $O$ as the vertex and $OB$ as one side, construct $\angle BOC=20^\circ$. Find the degree measure of $\angle AOC$. | 60^\circ \text{ or } 100^\circ | 47 | 15 |
math | Nine tiles are numbered $1, 2, 3, \cdots, 9,$ respectively. Each of three players randomly selects and keeps three of the tiles, and sums those three values. The probability that all three players obtain an odd sum is $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$
| 17 | 76 | 2 |
math | Suppose $\cos x =0$ and $\cos(x+z)= \frac{1}{2}$. What is the smallest possible positive value of $z,$ in radians? | \frac{\pi}{6} | 36 | 7 |
math | There is a question: "If the value of the algebraic expression $5a+3b$ is $-4$, then what is the value of the algebraic expression $2\left(a+b\right)+4\left(2a+b\right)$?" Tang, who loves to use his brain, solved the problem as follows:
Original expression $=2a+2b+8a+4b=10a+6b=2\left(5a+3b\right)=2\times \left(-4\rig... | -2 | 285 | 2 |
math | Given that rectangle ABCD and right triangle DCE share side DC, the area of rectangle ABCD is twice that of triangle DCE, and side AB = 7 units and side AD = 8 units, find the length of DE. | \sqrt{113} | 49 | 7 |
math | An ellipse has its foci at $(1, 2)$ and $(1, 6)$. Given that it passes through the point $(7, 4)$, its equation can be written in the form \[\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\] where $a, b, h, k$ are constants, and $a$ and $b$ are positive. Find $a+k.$ | 10 | 103 | 2 |
math | In a math test, the scores of four students with IDs $i (i=1,2,3,4)$ are from the set ${90,92,93,96,98}$, and they satisfy $f(1) < f(2) <= f(3) < f(4)$. Calculate the number of all possible situations for these four students' scores. | 15 | 84 | 2 |
math | Given that line $l$ is symmetric to the line $2x - 3y + 4 = 0$ with respect to the line $x = 1$, find the equation of line $l$. | 2x + 3y - 8 = 0 | 44 | 12 |
math | Given the function $f(x)=3x- \dfrac {4}{x}$, where $x\in(1,4)$, the range of values is the interval $D$. If a function value $f(x_{0})$ is taken at random from the interval $D$, determine the probability that $\dfrac {f(x_{0})}{x_{0}}\geqslant 2$. | \dfrac {2}{3} | 87 | 7 |
math | There are six cards, each with one of the numbers $1, 2, 3, 4, 5, 6$. A card is drawn at random, its number is noted, and then it is put back. This process is repeated 4 times. What is the probability that the difference between the maximum and minimum numbers drawn is 5? | \frac{151}{648} | 74 | 11 |
math | Consider the rectangular hyperbola defined by the equation $xy = c^2$, where $c$ is a constant. Find the distance between the foci of this hyperbola. | 2\sqrt{2}c | 38 | 7 |
math | Show that if I take 3 integers, I can find 2 of them whose sum is even. Then, if I take 5 integers, I can find 3 of them whose sum is divisible by 3. Then show that if I take 11 integers, I can find 6 of them whose sum is divisible by 6. Find the smallest integer \( n \) such that if I take \( n \) integers, I can alwa... | 35 | 108 | 2 |
math | (2) If $7$ products are sampled from samples with quality index values $m\geqslant 85$ using stratified sampling, and then $3$ products are randomly selected from these $7$ products, find the distribution and mathematical expectation of the number $X$ of products with quality index values $m\in [90, 95)$. | \frac{6}{7} | 79 | 7 |
math | Let $g : \mathbb{R} \to \mathbb{R}$ be a function such that
\[g(g(x - y)) = g(x)g(y) - g(x) + g(y) - 2xy\] for all $x, y$. Find the sum of all possible values of $g(1)$. | 0 | 73 | 1 |
math | Given that $420$ can be written as the sum of an increasing sequence of two or more consecutive positive integers, where the starting term is a perfect square, calculate the number of such possible representations. | 0 | 42 | 1 |
math | Xiao Ming pulls the switches with even numbers and Xiao Cong pulls the switches with numbers divisible by 3, all from switches marked 1 to 100. Calculate the total number of light bulbs that are now on. | 51 | 46 | 2 |
math | Given the line $l$: $y=x+m$, circle $O$: $x^{2}+y^{2}-4=0$, and circle $C$: $x^{2}+y^{2}+2ax-2ay+2a^{2}-4a=0 (0 < a\leqslant 4)$.
1. If $a=3$, circles $O$ and $C$ intersect at points $M$ and $N$. Find the length of the line segment $|MN|$.
2. Line $l$ is tangent to circle $C$ and is below the center of circle $C$. Fin... | m \in [-1, 8 - 4\sqrt{2}] | 156 | 16 |
math | In a rhombus with an acute angle of $30^{\circ}$, a circle is inscribed, and a square is inscribed in the circle. Find the ratio of the area of the rhombus to the area of the square. | 4 | 52 | 1 |
math | The number $1400000000000$ can be expressed in scientific notation. | 1.4\times 10^{12} | 24 | 12 |
math | A landscaping team brought some plane trees to plant on both sides of a road at equal distances. If a tree is planted every 8 meters at both ends of the road, then 8 trees are short. If a tree is planted every 9 meters, then 8 trees are left over. What is the length of this road in meters? | 576 | 70 | 3 |
math | In a school, there are $1200$ notebooks distributed evenly among $30$ boxes. A teacher decides to redistribute these notebooks so that there are now $35$ notebooks in each box. How many notebooks will be left over after redistributing them into as many full boxes as possible? | 10 | 62 | 2 |
math | In triangle $XYZ$, $XY = 25$ and $XZ = 20$. Find the largest possible value of $\tan Y$. | \frac{4}{3} | 31 | 7 |
math | Rectangle \(ABCD\) has length 9 and width 5. Diagonal \(AC\) is divided into 5 equal parts at \(W, X, Y\), and \(Z\). Determine the area of the shaded region. | 18 | 48 | 2 |
math | If $\frac{c}{3}$ is the opposite of $-2d$, and $2a$ is the reciprocal of $-b$, and the distance from the point corresponding to $x$ on the number line to the origin is $9$, find the value of $2ab-6d+c-\frac{x}{3}$. | 2 | 70 | 1 |
math | Given a point $M$ on the ellipse $\frac{{{x}^{{2}}}}{{25}}+ \frac{{{y}^{{2}}}{16}}={1}$, let $F\_{{1}}$ and $F\_{{2}}$ be the foci, and $\angle {{F}\_{{1}}}M{{F}\_{{2}}}=\frac{{ }\,{ }\pi }{6}$. Determine the area of $\triangle MF\_{1}F\_{{2}}$. | 16(2 - \sqrt{3}) | 108 | 10 |
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