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 | Determine the value of the expression
\[
\log_3 (50 + \log_3 (50 + \log_3 (50 + \cdots))),
\]
assuming it is positive. | 4 | 45 | 1 |
math | In the expansion of \((x + 2)^{50}\), what is the coefficient of the \(x^3\) term and the constant term? | 2^{50} | 33 | 5 |
math | A group of friends collects $120 to buy flowers for their teacher. Each rose costs $4, and each daisy costs $3. They want to buy at least 5 roses. Find the number of different bouquets they could purchase for exactly $120. | 9 | 57 | 1 |
math |
For a positive integer \( n \), let \( S_{n} \) be the minimum value of \( \sum_{k=1}^{n} \sqrt{(2k-1)^{2} + a_{k}^{2}} \), where \( a_{1}, a_{2}, \cdots, a_{n} \) are positive real numbers whose sum is 17. There exists a unique \( n \) such that \( S_{n} \) is also an integer. Find \( n \). | 12 | 112 | 2 |
math | In the geometric sequence $a_n$, given that $a_5 + a_6 = 4$ and $a_{15} + a_{16} = 16$, determine the value of $a_{25} + a_{26}$. | 64 | 57 | 2 |
math | Given an ellipse in the Cartesian coordinate system $xOy$, its center is at the origin, the left focus is $F(-\sqrt{3},0)$, and the right vertex is $D(2,0)$. Let point $A(1, \frac{1}{2})$.
$(1)$ Find the standard equation of the ellipse;
$(2)$ Given that line $l$ intersects the ellipse and the midpoint of chord $BC$... | 1 + \frac{\sqrt{3}}{2} | 130 | 12 |
math | Analogous to exponentiation, we define the operation of dividing $n$ identical rational numbers (all non-zero) as "dividing operation." For example, $2\div 2\div 2\div 2$ is denoted as $2^{"4"}$, read as "the quotient of $2$ raised to the power of $4$." Generally, we denote $\underset{\underbrace{a÷a÷a÷a}}{n}$ ($a\neq ... | 160 | 276 | 3 |
math | Given that the real numbers $x$, $y$, and $z$ satisfy the equation $x^{2}+2y^{2}+3z^{2}=4$, determine the range of possible values for $T=xy+yz$. | \left[-\frac{2\sqrt{6}}{3}, \frac{2\sqrt{6}}{3}\right] | 51 | 29 |
math | Four new students are to be assigned to three classes: A, B, and C, with at least one student in each class. Student A cannot be assigned to class A. How many different assignment plans are there? | 24 | 44 | 2 |
math | The points on the circle (x-3)^{2}+(y-3)^{2}=4 that are at a distance of 1 from the line 3x+4y-16=0 can be found by solving the equation. Determine the number of solutions. | 3 | 58 | 1 |
math | Two lines intersect an ellipse given by the equation $\frac{x^2}{4} + \frac{y^2}{9} = 1$. Neither line is tangent to the ellipse. Determine the possible number of points of intersection between these two lines and the ellipse. | 2 \text{ or } 4 | 55 | 8 |
math | Determine the solution set for the following inequalities:
(1) $2x^2 - x - 15 < 0$
(2) $\frac{2}{x} > -3$. | (-\infty, -\frac{2}{3}) \cup (0, +\infty) | 44 | 23 |
math | Let real $a$, $b$, and $c$ satisfy $$abc+a+b+c=ab+bc+ca+5.$$ Find the least possible value of $a^2+b^2+c^2$. | 6 | 45 | 1 |
math | When selecting the first trial point using the 0.618 method during the process, if the experimental interval is $[2000, 3000]$, the first trial point $x_1$ should be chosen at ______. | 2618 | 53 | 4 |
math | Lines $p$ and $q$ are parallel. $m\angle E = 150^\circ$, and $m\angle G = 70^\circ$. What is the number of degrees in $m\angle F$?
[asy]
size(100); real h = 1.2; currentpen = fontsize(10pt);
draw(Label("$p$",Relative(1)),(0,0)--(1,0),E);
draw(Label("$q$",Relative(1)),(0,-h)--(1,-h),E);
draw((0,-h)--h/2*(cos(150*pi/180... | 110^\circ | 289 | 5 |
math | One notebook, 3 notepads, and 2 pens cost 98 rubles, while 3 notebooks and a notepad cost 36 rubles less than 5 pens. How much does each item cost if the notebook costs an even number of rubles? (Each of these items costs an integer number of rubles.) | 4, 22, 14 | 70 | 9 |
math | Let $ABC$ be a triangle in which ( ${BL}$ is the angle bisector of ${\angle{ABC}}$ $\left( L\in AC \right)$ , ${AH}$ is an altitude of $\vartriangle ABC$ $\left( H\in BC \right)$ and ${M}$ is the midpoint of the side ${AB}$ . It is known that the midpoints of the segments ${BL}$ and ${MH}$ coincides. D... | 60^\circ | 121 | 4 |
math | Find all quadruples of real numbers $(a,b,c,d)$ such that the equalities
\[X^2 + a X + b = (X-a)(X-c) \text{ and } X^2 + c X + d = (X-b)(X-d)\]
hold for all real numbers $X$ .
*Morteza Saghafian, Iran* | (-1, -2, 2, 0), (0, 0, 0, 0) | 80 | 24 |
math | Let \( p, \) \( q, \) and \( r \) be positive real numbers. Find the minimum value of
\[
\frac{5r}{3p + q} + \frac{5p}{q + 3r} + \frac{2q}{p + r}.
\] | 4 | 66 | 1 |
math | Among all pairs of real numbers $(x, y)$ such that $\sin \sin x = \sin \sin y$ with $-10 \pi \le x, y \le 10 \pi$, Oleg randomly selected a pair $(X, Y)$. Compute the probability that $X = Y$. | \frac{1}{20} | 67 | 8 |
math | A combination lock has a 3 number combination, with each number an integer between 0 and 39 inclusive. Call the numbers \( n_{1}, n_{2} \), and \( n_{3} \). If you know that \( n_{1} \) and \( n_{3} \) leave the same remainder when divided by 4, and \( n_{2} \) and \( n_{1}+2 \) leave the same remainder when divided by... | 4000 | 110 | 4 |
math | Let $S_n=1-2+3-4+\cdots +(-1)^{n-1}n$, where $n=1,2,\cdots$. Then $S_{17}+S_{33}+S_{50}$ equals: | 1 | 58 | 1 |
math | If $\frac{720}{2^5\cdot5^9}$ is expressed as a decimal, how many non-zero digits are to the right of the decimal point? | 4 | 37 | 1 |
math | Given a hyperbola with the equation $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ (a > 0, b > 0), one of its asymptotic line equations is y = -$\frac{\sqrt{5}}{2}$x. Determine the eccentricity of the hyperbola. | \frac{3}{2} | 80 | 7 |
math | Let \( x \) and \( y \) be real numbers greater than 1 such that
\[(\log_2 x)^3 + (\log_3 y)^3 + 9 = 9 (\log_2 x)(\log_3 y).\]
Compute \( x^{\sqrt{3}} + y^{\sqrt{3}}. \) | 35 | 76 | 2 |
math | Three complex numbers at the vertices of a square in the complex plane are $2+3i, 1-2i$, and $-2-3i$. Find the fourth vertex of the square.
A) $1+2i$
B) $-1+2i$
C) $2-i$
D) $3+2i$ | -1+2i | 76 | 5 |
math | If two lines $kx-y+1=0$ and $x-ky=0$ intersect, and the intersection point is in the second quadrant, then determine the range of $k$. | (-1,0) | 40 | 5 |
math | Given the numbers from $000$ to $999$, calculate how many have three digits in non-decreasing or non-increasing order, including cases where digits can repeat. | 430 | 38 | 3 |
math | Given that $\{a_n\}$ is an arithmetic sequence, with the first term $a_1 > 0$, $a_5+a_6 > 0$, and $a_5a_6 < 0$, calculate the maximum natural number $n$ for which the sum of the first $n$ terms $S_n > 0$. | 10 | 74 | 2 |
math | In a Cartesian coordinate system, $\vec{e}$ is a unit vector, and vector $\vec{a}$ satisfies $\vec{a} \cdot \vec{e}=2$. Also, $|\vec{a}|^{2} \leqslant 5|\vec{a}+t \vec{e}|$ holds for any real number $t$. Find the range of values of $|\vec{a}|$. | [\sqrt{5}, 2\sqrt{5}] | 90 | 12 |
math | The domain of the function $f(x) = \lg(4-x^2)$ is $(-\infty, -2) \cup (2, +\infty)$. | (-2, 2) | 39 | 6 |
math | A line is drawn through the midpoints of any two edges of the triangular prism $ABC-A_1B_1C_1$. How many such lines are parallel to the plane $ABBA_1$? | 6 | 44 | 1 |
math | Given the sequence $\{a_{n}\}$, $a_{1}=2$, $a_{2}=0$, and ${a}_{n+2}={a}_{n}+2•{(-1)}^{n}$, calculate the sum of the first $2024$ terms of the sequence $\{a_{n}\}$. | 2024 | 73 | 4 |
math | There are 1002 banana candies and 1002 apple candies in a box. Lara takes two candies from the box without looking at the flavor. If $q$ is the probability that the two candies are of different flavors and $p$ is the probability that the two candies are of the same flavor, what is the value of $q - p$?
(a) 0
(b) $\frac... | \frac{1}{2003} | 133 | 10 |
math | Six positive integers are written on the faces of a cube. Each vertex is labeled with the product of the numbers on the three faces adjacent to that vertex. If the sum of the numbers on the vertices is $1512$, and the sum of the numbers on one pair of opposite faces is $8$, what is the sum of the numbers on all the fac... | 38 | 75 | 2 |
math | How many parallelograms with sides of lengths 1 and 2 and angles of $60^{\circ}$ and $120^{\circ}$ can be placed inside a regular hexagon with a side length of 3? | 12 | 49 | 2 |
math | In the Cartesian coordinate system \(xOy\), the set of points \(K=\{(x, y) \mid x, y=-1,0,1\}\). Three points are randomly selected from \(K\). What is the probability that the distance between any two of these three points does not exceed 2? | 5/14 | 67 | 4 |
math | Let the function $f(x)=2x-\cos x$, and $\{a_n\}$ be an arithmetic sequence with a common difference of $\dfrac{\pi}{8}$. If $f(a_1)+f(a_2)+\ldots+f(a_5)=5\pi$, then $\left[f(a_3)\right]^2-a_1a_5=$ \_\_\_\_\_\_. | \dfrac{13\pi^2}{16} | 87 | 13 |
math | Two brothers are practicing writing Chinese characters during the winter vacation. The older brother needs to write 8000 characters, while the younger brother needs to write 6000 characters. The older brother writes 100 more characters per day than the younger brother. They both finish their tasks in the same number of... | \frac{8000}{x} = \frac{6000}{x - 100} | 100 | 26 |
math | If $x_{k+1} = x_k + 1$ for $k=1, 2, \dots, n-1$ and $x_1=2$, find $x_1 + x_2 + \dots + x_n$.
A) $\frac{n(n+1)}{2}$
B) $\frac{n(n+3)}{2}$
C) $n(n+1)$
D) $\frac{n(n+2)}{2}$
E) $\frac{n(n+5)}{2}$ | \frac{n(n+3)}{2} | 116 | 10 |
math | A lattice point in an $xy$-coordinate system is any point $(x, y)$ where both $x$ and $y$ are integers. The graph of $y = mx + 5$ passes through no lattice point with $0 < x \leq 150$ for all $m$ such that $0 < m < b$. What is the maximum possible value of $b$?
**A)** $\frac{1}{150}$
**B)** $\frac{1}{151}$
**C)** $\fra... | \frac{1}{151} | 149 | 9 |
math | Given a hyperbola sharing the same foci with the hyperbola $\dfrac{x^{2}}{16}- \dfrac{y^{2}}{9}=1$ and passing through point $P\left(- \dfrac{ \sqrt{5}}{2},- \sqrt{6} \right)$, find the standard equation of this hyperbola. | x^{2}- \dfrac{y^{2}}{24}=1 | 80 | 17 |
math | Given that $| \overrightarrow{a}|=1$, $| \overrightarrow{b}|=2$, and the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is $120^{\circ}$. Find:
$(1) \overrightarrow{a} \cdot \overrightarrow{b}$;
$(2)( \overrightarrow{a}-3 \overrightarrow{b}) \cdot (2 \overrightarrow{a}+ \overrightarrow{b})$;
$(3)|2 \overri... | 2 \sqrt{3} | 123 | 6 |
math | A fair standard six-sided dice is tossed three times. Given that the sum of the first two tosses equal twice the third toss, what is the probability that at least one "3" is tossed?
A) $\frac{1}{9}$
B) $\frac{1}{3}$
C) $\frac{1}{2}$
D) $\frac{2}{3}$
E) $\frac{1}{6}$ | \frac{1}{3} | 90 | 7 |
math | Find the smallest positive integer \( n \) such that in any set of \( n \) irrational numbers, there always exist 3 numbers for which all pairwise sums are still irrational. | 5 | 37 | 1 |
math | For lines $l_{1}: 2x+(m+1)y+4=0$ and $l_{2}: mx+3y-2=0$ to be parallel, find the value of $m$. | -3\text{ or }2 | 46 | 8 |
math | A $200\times 420\times 480$ rectangular solid is made by gluing together $1\times 1\times 1$ cubes. An internal diagonal of this solid passes through the interiors of how many of the $1\times 1\times 1$ cubes? | 1000 | 67 | 4 |
math | Find the second-order derivative \( y_{xx}^{\prime \prime} \) of the function given parametrically.
\[ \left\{\begin{array}{l} x = \sin t \\ y = \ln (\cos t) \end{array}\right. \] | -\frac{1 + \sin^2 t}{\cos^4 t} | 60 | 17 |
math | Find the smallest integer \( t \) such that there exist positive integers \( x_1, \ldots, x_t \) satisfying \( x_1^3 + \ldots + x_t^3 = 2002^{2002} \). | 4 | 57 | 1 |
math | On the radius \( A O \) of a circle with center \( O \), point \( M \) is chosen. On the same side of \( A O \) on the circumference, points \( B \) and \( C \) are chosen such that \( \angle A M B = \angle O M C = \alpha \). Find the length \( B C \), given that the radius of the circle is 15 and \( \cos \alpha = \fra... | 18 | 105 | 2 |
math | Given the probability distribution of the random variable $\xi$ as $p(\xi=k)= \frac{1}{5}(k=2,4,6,8,10)$, determine the value of $D\xi$. | 8 | 48 | 1 |
math | The sides of this parallelogram measure 10 units, $12y-2$, $5x+15$, and 20 units in sequence. What is the value of $x+y$?
[asy]draw((0,0)--(25,0)--(35,30)--(10,30)--cycle);
label("$12y-2$",(18,0),S);
label("20",(30,15),E);
label("10",(12.5,30),N);
label("$5x+15$",(5,15),W);
[/asy] | 2 | 139 | 1 |
math | Let \( ABC \) be a triangle, \( E \) the foot of the altitude from \( B \), and \( F \) the orthogonal projection of \( C \) on the tangent to the circle \( (ABC) \) at \( B \). Show that \( (EF) \) and \( (AB) \) are parallel. | (EF) \parallel (AB) | 71 | 8 |
math | Given \( f(x) = a \sin x + b \sqrt[3]{x} + c \ln \left(x + \sqrt{x^{2} + 1}\right) + 1003 \) (where \( a \), \( b \), and \( c \) are real numbers), and \( f\left(\lg^{2} 10\right) = 1 \), what is \( f(\lg \lg 3) \)? | 2005 | 100 | 4 |
math | Solve in the set of real numbers the equation \[ 3x^3 \minus{} [x] \equal{} 3,\] where $ [x]$ denotes the integer part of $ x.$ | x = \sqrt[3]{\frac{4}{3}} | 46 | 14 |
math | A triangular lattice is formed with seven points arranged as follows: six points form a regular hexagon and one point is at the center. Each point is one unit away from its nearest neighbor. Determine how many equilateral triangles can be formed where all vertices are on this lattice. Assume the points are numbered 1 t... | 12 | 83 | 2 |
math | A child whose age is between 13 and 19 writes his own age after his father's age, creating a four-digit number. The absolute difference between their ages is subtracted from this new number to obtain 4289. What is the sum of their ages?
(Note: From the 22nd Annual USA High School Mathematics Examination, 1971) | 59 | 79 | 2 |
math | Solve the equations:<br/>$(1) 2x^{2}-1=49$;<br/>$(2) (x+3)^{3}=64$. | x=1 | 38 | 3 |
math | A solid box measures 18 cm by 12 cm by 10 cm. A new solid is formed by removing a cube of side 4 cm from each corner of this box. What percent of the original volume is removed? | 23.70\% | 49 | 7 |
math | Given the function $f(x)=x^{3}-3x^{2}$.
(Ⅰ) Find the intervals of monotonicity for $f(x)$;
(Ⅱ) If the domain of $f(x)$ is $[-1,m]$ and its range is $[-4,0]$, find the maximum value of $m$. | 3 | 72 | 1 |
math | A factory produces a certain type of product with a maximum daily production capacity of 40 units. The ratio of quality products (P) to the total daily production (x, where x is a natural number) is given by the formula P = (4200 - x^2) / 4500. Each quality product generates a profit of 4000 yuan, while each defective ... | -\frac{4}{3} \cdot 30^3 + 3600 \cdot 30 = 72000 \; \text{(yuan)} | 171 | 40 |
math | Two drivers met halfway between cities \( A \) and \( B \). Upon meeting, it was discovered that the first driver from \( A \) had left earlier than the second driver from \( B \) by an amount of time equal to half the time (in hours) it would have taken them to meet if they had left simultaneously from the same points... | \frac{1+\sqrt{5}}{2} | 100 | 12 |
math | Given lines $l_{1}$: $(3+m)x+4y=5-3m$, and $l_{2}$: $2x+(5+m)y=8$ are parallel, find the value of the real number $m$. | -7 | 51 | 2 |
math | Given an ellipse $C$ with its center at the origin and an eccentricity of $\frac{1}{2}$, one of its major axis endpoints is exactly the focus of the parabola $y^2=16x$.
$(1)$ Find the equation of the ellipse $C$;
$(2)$ Given points $P(2,3)$ and $Q(2,-3)$ on the ellipse, let $A$ and $B$ be points on the ellipse locate... | \frac{1}{2} | 187 | 7 |
math | Find the volume of the region in space defined by
\[ |x + y + z| + |x - y + z| \le 10 \]
and $x, y, z \ge 0$. | 62.5 | 46 | 4 |
math | In triangle $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $\frac{{\sin A}}{{\cos B \cos C}} = \frac{{2\sqrt{3}a^2}}{{a^2 + b^2 - c^2}}$.
$(1)$ Find the measure of angle $B$.
$(2)$ If $a+c=2\sqrt{6}\sin C$ and $b=3$, find the area $S$ of triangle $\tri... | S = \frac{3\sqrt{3}}{4} | 125 | 14 |
math | Let the function $y=f(x)$ have its tangent line equation at any point $(x_{0},y_{0})$ on its graph as $y-y_{0}=(3x_{0}^{2}-6x_{0})(x-x_{0})$, and $f(3)=0$. Then, the solution set of the inequality $\frac{x-1}{f(x)}\geqslant 0$ is ______. | \left(-\infty,0\right)\cup(0,1]\cup\left(3,+\infty\right) | 92 | 29 |
math | The deli now offers five kinds of bread, seven kinds of meat, and six kinds of cheese. A sandwich consists of one type of bread, one type of meat, and one type of cheese. Ham, turkey, gouda cheese, chicken, cheddar cheese, and white bread are each offered at the deli. Al refuses to order a sandwich with a ham/cheddar c... | 194 | 112 | 3 |
math | The parabola $y = ax^2 + bx + c$ has a vertex at $(h, k)$ and passes through the point $(0, -k)$, where $k \neq 0$. Determine the value of $b$. | \frac{4k}{h} | 52 | 8 |
math | In the coordinate plane, points with integer values for both coordinates are called lattice points. For a certain lattice point \( P \) and a positive number \( d \), if there are exactly \( k(>0) \) distinct lattice points at a distance \( d \) from \( P \), the range of values for \( k \) is denoted as \( \left\{k_1,... | 8 | 120 | 1 |
math | Distribute 12 different objects among 3 people so that each person receives 4 objects. In how many ways is this possible? | 34650 | 28 | 5 |
math | The basic subscription price of the online streaming service is $15 per month. Calculate the maximum percentage decrease in the number of subscribers that the service can tolerate in order to keep their total income at least the same despite a 20% price increase. | 16.67\% | 51 | 7 |
math | Given that $x$, $y$, and $z$ are positive numbers and they satisfy the equation $x^{2}+y^{2}+z^{2}=1$, find the minimum value of $S= \frac {1+z}{2xyz}$. | 4 | 55 | 1 |
math | Given the function f(x) = xlnx - $\frac{ax^{2}}{2}$ + a - x (a ∈ R).
1. Find the range of values for the real number a if the function f(x) has two distinct extreme points.
2. If a = 2, k ∈ N, g(x) = 2 - 2x - x², and the inequality k(x - 2) + g(x) < f(x) always holds when x > 2, find the maximum value of k. | 4 | 112 | 1 |
math | Find the polynomial whose roots satisfy the condition that the sum is double the product of the roots, with the sum of the roots being $2k$ and the product being $k$, and given that one of the roots is $1$. | x^2 - 2x + 1 | 48 | 10 |
math | Given the function $f(x)=|2x-1|-|x-a|$, where $a\leqslant 0$.
$(1)$ When $a=0$, find the solution set for the inequality $f(x) < 1$;
$(2)$ If the area of the triangle formed by the graph of $f(x)$ and the x-axis is greater than $\dfrac{3}{2}$, find the range of values for $a$. | (-\infty, -1) | 98 | 8 |
math | In the quadratic function $y=ax^{2}+bx+c$, the corresponding values of the function value $y$ and the independent variable $x$ are as shown in the table below:
| $x$ | $\cdots $ | $-7$ | $-5$ | $-3$ | $-1$ | $1$ | $\cdots $ |
|-----|-----------|------|------|------|------|-----|-----------|
| $y$ | $\cdots $ | $-9$ | ... | -16 | 165 | 3 |
math | Let \( p(x) = x^2 - x + 1 \). Let \(\alpha\) be a root of \( p(p(p(p(x)))) \). Find the value of
\[
(p(\alpha) - 1) p(\alpha) p(p(\alpha)) p(p(p(\alpha)))
\] | -1 | 67 | 2 |
math | Consider a square quilt block made from sixteen smaller squares, where each smaller square is identical in size. Four of these squares are divided into two right-angled triangles. If two triangles from two different squares are shaded such that they cover half the area of each of these squares, what fraction of the tot... | \frac{1}{8} | 65 | 7 |
math | The new book "Mathematical Wonders" is priced at \$30. A bookstore offers two discounts: a \$5.00 discount and a 25% discount. Determine how much more a shopper will save by choosing the optimal order of discounts. | 125\text{ cents} | 54 | 8 |
math | Snowboarder Gavrila was descending a slope 250 meters high and had a speed of 10 m/s at the bottom of the slope. What fraction of the total mechanical energy lost during the descent went into heating the snowboard with a mass of 6 kg, if it warmed up by 1 degree? The specific heat capacity of the snowboard material is ... | \frac{1}{98} | 100 | 8 |
math | Let $g(x) = x^4 + 16x^3 + 80x^2 + 128x + 64$. Let $z_1, z_2, z_3, z_4$ be the roots of $g$. Find the smallest possible value of $|z_{a}z_{b} + z_{c}z_{d}|$ where $\{a, b, c, d\} = \{1, 2, 3, 4\}$. | 16 | 113 | 2 |
math | The sequence $\{2n+1\}$ ($n\in\mathbb{N}^*$) is arranged sequentially in brackets such that the first bracket contains one number, the second bracket contains two numbers, the third bracket contains three numbers, the fourth bracket contains four numbers, the fifth bracket contains one number, the sixth bracket contain... | 2072 | 95 | 4 |
math | Let the function $f(x)$ be an odd function defined on $\mathbb{R}$, and when $x>0$, $f(x)=x^2+2x+5$.
(1) Find $f(-2)$.
(2) Find the expression for $f(x)$ when $x<0$. | -x^2+2x-5 | 69 | 8 |
math | Given a function $f(x)$ defined on $\mathbb{R}$ that satisfies the equation $f(x)+f(1-x)=1$ for all $x$, and for $x \geqslant 0$ the relation $f\left(\dfrac{x}{3}\right)=\dfrac{1}{2}f(x)$ holds, and for $0 \leqslant x_1 < x_2 \leqslant 1$, $f(x_1)\leqslant f(x_2)$, find the value of $f\left(\dfrac{1}{2018}\right)$. | \dfrac{1}{128} | 137 | 9 |
math | Vasya cut a triangle out of cardboard and numbered its vertices with the numbers 1, 2, and 3. It turns out that if Vasya rotates his triangle 15 times clockwise around the vertex numbered 1 by an angle equal to the angle at this vertex, the triangle returns to its original position. If Vasya rotates his triangle 6 time... | 5 | 171 | 1 |
math | Given that the area of $\triangle ABC$ is $2 \sqrt {3}$, $BC=2$, $C=120^{\circ}$, find the length of side $AB$. | 2 \sqrt {7} | 42 | 6 |
math | For how many integers $a$ with $|a| \leq 2005$ , does the system
$x^2=y+a$
$y^2=x+a$
have integer solutions? | 90 | 50 | 2 |
math | (Full score: 12 points) Form five-digit numbers with the digits 1, 2, 3, 4, and 5 without repeating any digit, and arrange them in ascending order to form a sequence.
1. How many items are there in this sequence?
2. What is the 96th item in this sequence? | 45321 | 72 | 5 |
math | Some middle school students in a city participated in a mathematics invitational competition, which consisted of 6 problems. It is known that each problem was solved by exactly 500 students, but for any two students, there is at least one problem that neither of them solved. What is the minimum number of middle school ... | 1000 | 74 | 4 |
math | Write down the numbers 1, 2, 3, 4, 5, …, 997, 998, 999 in the order of natural numbers to form a very large number 123456789101112…997998999. The sum of all the digits in this number is ______. | 13500 | 86 | 5 |
math | In the plane rectangular coordinate system $xOy$, a ray of light is emitted from the point $\left(2,3\right)$, reflected by the $y$-axis, and tangent to the circle $x^{2}-6x+y^{2}+4y+12=0$. Determine the slope of the line where the reflected light lies. | -\frac{4}{3} \text{ or } -\frac{3}{4} | 76 | 20 |
math | Find the non-negative real solutions of the system of equations:
$$
\left\{\begin{array}{l}
x + y + z = 3xy, \\
x^2 + y^2 + z^2 = 3xz, \\
x^3 + y^3 + z^3 = 3yz
\end{array}\right.
$$ | (1, 1, 1) | 76 | 9 |
math | Given the function $f(x)=|x+ \frac {4}{x}-m|+m$.
- (I) When $m=0$, find the minimum value of the function $f(x)$;
- (II) If the function $f(x)\leqslant 5$ holds true for all $x\in[1,4]$, find the range of the real number $m$. | (-\infty, \frac {9}{2}] | 87 | 12 |
math | Let $f(x)$ be a function defined on $\mathbb{R}$ such that for any real number $x$, it always satisfies $f(x+2)=f(-x)$, $f(x)=-f(4-x)$, and when $x\in[0,2]$, $f(x)=2x-x^2$.
(1) Find the analytic expression of $f(x)$ when $x\in[2,4]$;
(2) Calculate the sum $f(0)+f(1)+f(2)+\cdots+f(2019)$. | 0 | 126 | 1 |
math | **A park is designed with two circular walking tracks. The smaller track has a diameter of 15 meters. If the diameter of the larger track is 20 meters, what percent increase in area results from the larger track compared to the smaller one?** | 77.78\% | 53 | 7 |
math | The rectangular prism $ABCD-A_1B_1C_1D_1$ has length, width, and height of $a$, $b$, and $c$, respectively, with $a > b > c$. The shortest path from $A$ to $C_1$ along the surface is ______. | \sqrt{a^2 + (b + c)^2} | 66 | 14 |
math | A bakery prepares a large $24$-inch by $30$-inch tray of brownies. Each brownie is cut into pieces that measure $3$ inches by $4$ inches. Calculate the number of brownie pieces obtained from the tray. | 60 | 54 | 2 |
math | Determine the largest constant $K\geq 0$ such that $$ \frac{a^a(b^2+c^2)}{(a^a-1)^2}+\frac{b^b(c^2+a^2)}{(b^b-1)^2}+\frac{c^c(a^2+b^2)}{(c^c-1)^2}\geq K\left (\frac{a+b+c}{abc-1}\right)^2 $$ holds for all positive real numbers $a,b,c$ such that $ab+bc+ca=abc$ .
*Proposed by Orif Ibrogimov (Czech Technical Uni... | 18 | 150 | 2 |
math | Given the function $f(x)=\ln (x+1)- \frac{ax}{x+1}-x$, where $a\in R$.
(I) Find the monotonic intervals of the function $f(x)$ when $a > 0$.
(II) If there exists $x > 0$ such that $f(x)+x+1 < - \frac{x}{x+1}$ ($a\in Z$) holds, find the minimum value of $a$. | 5 | 103 | 1 |
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