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 | Form the equation of a circle with center \( O(3, -2) \) and radius \( r = 5 \). | (x - 3)^2 + (y + 2)^2 = 25 | 27 | 18 |
math | The number of sets \\(M\\) that satisfy the condition \\(M \cup \{1\} = \{1,2,3\}\\. | 2 | 34 | 1 |
math | Among the following functions $f(x)$, which satisfies "For any $x_1, x_2 \in (0, +\infty)$, when $x_1 < x_2$, it always holds that $f(x_1) < f(x_2)$"? (Fill in the serial number)
① $f(x) =$ ;
② $f(x) = (x-1)^2$;
③ $f(x) = e^x$;
④ $f(x) = \ln(x+1)$. | ③④ | 120 | 4 |
math | Find the obtuse angle $\alpha$ such that $\sin \alpha (1 + \sqrt{3}\tan 10^\circ) = 1$. | 140^\circ | 33 | 5 |
math | Given that the wage for a carpenter is 50 yuan per person and the wage for a mason is 40 yuan per person, and a labor budget of 2000 yuan is available, determine the relationship between the number of carpenters, x, and the number of masons, y. | 5x+4y\leq200 | 66 | 11 |
math | Given the function $f(x) = \begin{cases} x+6, & x < t \\ x^{2}+2x, & x\geqslant t \end{cases}$, if the range of the function $f(x)$ is $(R)$, then the range of the real number $t$ is _____ . | [-7, 2] | 72 | 6 |
math | If "$0 \lt x \lt 3$" is a sufficient and necessary condition for "$x \gt \log _{2}a$", determine the range of real number $a$. | (0,1] | 39 | 5 |
math | Let \( OP \) be the diameter of the circle \( \Omega \), and let \( \omega \) be a circle with center at point \( P \) and a radius smaller than that of \( \Omega \). The circles \( \Omega \) and \( \omega \) intersect at points \( C \) and \( D \). A chord \( OB \) of the circle \( \Omega \) intersects the second circ... | \sqrt{5} | 116 | 5 |
math | Four friends rent a cottage for a total of £300 for the weekend. The first friend pays half of the sum of the amounts paid by the other three friends. The second friend pays one third of the sum of the amounts paid by the other three friends. The third friend pays one quarter of the sum of the amounts paid by the other... | 65 | 82 | 2 |
math | Each face of a cube is painted with a single narrow stripe from the center of one edge to the center of the opposite edge. Each stripe can be chosen to be either red or blue, and the orientation of the stripe (which edges it connects) is chosen at random and independently for each choice. What is the probability that t... | \frac{3}{512} | 140 | 9 |
math | Given the algebraic expression $\frac{3}{x-2}$, determine the range of real numbers $x$ for which this expression is meaningful. | x \neq 2 | 31 | 6 |
math | Let the function $f(x)=\frac{{e}^{x}}{{x}^{2}}-k\left(\frac{2}{x}+\ln x\right)$ ($k$ is a constant, $e=2.71828\ldots$ is the base of the natural logarithm).
(Ⅰ) When $k\leqslant 0$, find the monotonic intervals of the function $f(x)$;
(Ⅱ) If the function $f(x)$ has two extreme points in the interval $(0,2)$, find the... | (e, \frac{{e}^{2}}{2}) | 128 | 13 |
math | For a positive integer $n$ and nonzero digits $a$, $b$, and $c$, let $A_n$ be the $n$-digit integer each of whose digits is equal to $a$; let $B_n$ be the $2n$-digit integer each of whose digits is equal to $b$, and let $C_n$ be the $3n$-digit integer each of whose digits is equal to $c$. Determine the greatest possibl... | 13 | 132 | 2 |
math | $a$, $b$, and $c$ are real numbers not less than 1, their product is 10, and the product of $a^{\log_a}$, $b^{\log_b}$, and $c^{\log_c}$ is not less than 10. Find $a$, $b$, $c$. | a=b=1, c=10 | 71 | 9 |
math | In a circle, diameter $\overline{EB}$ is parallel to chord $\overline{DC}$. If the angles $AEB$ and $ABE$ are in the ratio of $2 : 3$, determine the degree measure of angle $BCD$. | 36^\circ | 55 | 4 |
math | Let the universal set be $U=\{x\in \mathbb{N}|\sqrt{x}\leqslant 2\}$, $A=\{1,2\}$. Determine the complement of set A with respect to U. | \{0, 3, 4\} | 52 | 11 |
math | A construction team took 20 minutes to repair a 200-meter-long oil pipeline. On average, they repaired meters per minute, and it took minutes to repair 1 meter. | 0.1 | 46 | 3 |
math | How many students are there in our city? The number expressing the quantity of students is the largest of all numbers where any two adjacent digits form a number that is divisible by 23. | 46923 | 38 | 5 |
math | Given the right triangles ABC and ABD, what is the length of segment BC, in units? [asy]
size(150);
pair A, B, C, D, X;
A=(0,0);
B=(0,12);
C=(-16,0);
D=(-35,0);
draw(A--B--D--A);
draw(B--C);
draw((0,1.5)--(-1.5,1.5)--(-1.5,0));
label("$37$", (B+D)/2, NW);
label("$19$", (C+D)/2, S);
label("$16$", (A+C)/2, S);
label("A",... | 20 | 181 | 2 |
math | Given that the larger root of the equation $2002^2x^2 - 2003 \cdot 2001x - 1 = 0$ is $r$, and the smaller root of the equation $2001x^2 - 2002x + 1 = 0$ is $s$, find the value of $r - s$. | \frac{2000}{2001} | 85 | 13 |
math | The distance from point P(1, 2, 2) to the origin is a multiple of 3. | 3 | 24 | 1 |
math | A ball is dropped from a height of 150 feet and rebounds to three-fourths of the distance it fell on each bounce. How many feet will the ball have traveled when it hits the ground the fifth time? | 765.234375 | 45 | 10 |
math | Given $1$ brown tile, $1$ purple tile, $2$ green tiles, and $3$ yellow tiles, calculate the number of distinguishable arrangements in a row from left to right. | 420 | 41 | 3 |
math | Below is a segment of the graph of a quadratic function, $y=p(x)=ax^2+bx+c$:
[asy]
import graph; size(8cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-0.99,xmax=10.5,ymin=-5.5,ymax=5.5;
pen cqcqcq=rgb(0.75,0.75,0.75);
/*grid*/ pen gs=linewidth(0.7)+cqcqcq+linetype("2 ... | 1 | 436 | 1 |
math | Let $P(x)$ be a polynomial defined as $P(x) = 3x^5 - 2x^3 + 5x^2 - 4x + 1$. A new polynomial $Q(x)$ is formed by replacing each nonzero coefficient of $P(x)$ with their mean. Evaluate $P(1)$ and $Q(1)$ and check if they are equal. | 3 | 82 | 1 |
math | The owners of the Luray Caverns in Virginia conduct tours every 20 minutes. Each day in April the first tour is at 9 a.m., and the last tour starts at 6 p.m. How many tours are there per day in April? | 28 | 55 | 2 |
math | The number $2021$ is expressed in the form $2021 = \frac{a_1!a_2!...a_m!}{b_1!b_2!...b_n!}$, where $a_1 \ge a_2 \ge \cdots \ge a_m$ and $b_1 \ge b_2 \ge \cdots \ge b_n$ are positive integers and $a_1 + b_1$ is as small as possible. Determine the value of $|a_1 - b_1|$. | 4 | 123 | 1 |
math | Given the function $$f(x)= \begin{cases} 2^{x}+1, & x\leq1 \\ 1-\log_{2}x, & x>1\end{cases}$$, find the range of $m$ that satisfies the inequality $f(1-m^{2}) > f(2m-2)$. | (-3,1) \cup \left(\frac{3}{2}, +\infty \right) | 75 | 23 |
math | Let \(y=f(x)\) be a decreasing function defined on \((0,+\infty)\) that satisfies \(f(xy)=f(x)+f(y)\), with \(f\left( \frac{1}{3} \right)=1\).
(1) Find the values of \(f(1)\), \(f\left( \frac{1}{9} \right)\), and \(f(9)\).
(2) If \(f(x)-f(2-x)<2\), determine the range of values for \(x\). | \left( \frac{1}{5}, 2 \right) | 116 | 15 |
math | In square $ABCD$, side $AD$ is $s$ centimeters. Let $N$ be a point on $\overline{CD}$ such that $ND = \frac{2}{3} s$ and $NC = \frac{1}{3} s$. Let $O$ be the intersection of $\overline{AC}$ and $\overline{BN}$. Find the ratio of $OC$ to $OA$. | \frac{1}{2} | 91 | 7 |
math | Given points $M(-1,0)$ and $N(1,0)$, the distance from any point on curve $E$ to point $M$ is $\sqrt{3}$ times the distance to point $N$.
(I) Find the equation of curve $E$.
(II) Given that $m \neq 0$, let line $l_{1}: x - my - 1 = 0$ intersect curve $E$ at points $A$ and $C$, and line $l_{2}: mx + y - m = 0$ interse... | y = -x + 3 | 153 | 7 |
math | Given that the function $f(x)$ and its derivative $f'(x)$ have domains of all real numbers, for any $x, y \in R$, it is always true that $f(x+y) + f(x-y) = 2f(x) \cdot f(y)$. Then $f'(x)$ must be a ______ function (fill in "odd, even, neither odd nor even"); if $f(1) = \frac{1}{2}$, then $\sum_{i=1}^{2027}f(i) = \_\_\_... | -1 | 125 | 2 |
math | An \( n \times n \) matrix of integers is called "golden" if, for every row and every column, their union contains all of the numbers \( 1, 2, 3, \ldots, 2n - 1 \). Find all golden matrices (of all sizes). | \left[1\right] | 63 | 7 |
math | Given two arithmetic sequences $\{a_n\}$ and $\{b_n\}$, the sum of the first $n$ terms is denoted by $S_n$ for $\{a_n\}$, and $T_n$ for $\{b_n\}$. If $\frac {S_n}{T_n} = \frac {2n}{3n+1}$, determine the value of $\frac {a_n}{b_n}$. | \frac {2n-1}{3n-1} | 93 | 13 |
math | A marathon is 26 miles and 385 yards. One mile equals 1760 yards.
Leila has run five marathons in her life. If the total distance Leila covered in these marathons is $m$ miles and $y$ yards, where $0 \le y < 1760$, what is the value of $y$? | 165 | 81 | 3 |
math | Given that $P_{1}\left(x_{1}, y_{1}\right), P_{2}\left(x_{2}, y_{2}\right), \cdots, P_{n}\left(x_{n}, y_{n}\right), \cdots$, where $x_{1}=1, y_{1}=0, x_{n+1}=x_{n}-y_{n}, y_{n+1}=x_{n}+y_{n}$ for $n \in \mathbf{Z}_{+}$. Let $a_{n}=\overrightarrow{P_{n} P_{n+1}} \cdot \overrightarrow{P_{n+1} P_{n+2}}$. Find the smallest... | 10 | 181 | 2 |
math | Find the value of $\cos 160^{\circ}\sin 10^{\circ}-\sin 20^{\circ}\cos 10^{\circ}$. | -\dfrac{1}{2} | 40 | 8 |
math | Given that $\{a_n\}$ is an arithmetic sequence with common difference $d$, and $n \in \mathbb{N}$, simplify the polynomial
$$
p_n(x) = a_0 \binom{n}{0} (1-x)^n + a_1 \binom{n}{1} x(1-x)^{n-1} + a_2 \binom{n}{2} x^2 (1-x)^{n-2} + \cdots + a_n \binom{n}{n} x^n
$$ | a_0 + n d x | 118 | 7 |
math | The number whose square is $\frac{4}{9}$ is ______; the number whose cube is $-64$ is ______. | -4 | 28 | 2 |
math | Find the center of mass of a hemisphere
$$
0 \leqslant z \leqslant \sqrt{R^{2}-x^{2}-y^{2}},
$$
where the volume density \( p \) of the material at each point is proportional to the distance to the origin. | \left(0, 0, \frac{2}{5} R\right) | 63 | 19 |
math | A 98 x 98 chessboard has the squares colored alternately black and white in the usual way. A move consists of selecting a rectangular subset of the squares (with boundary parallel to the sides of the board) and changing their color. What is the smallest number of moves required to make all the squares black? | 98 | 66 | 2 |
math | In the set of three-digit numbers, if the digits in the hundreds place, tens place, and ones place can form an arithmetic sequence, determine the total number of arithmetic three-digit numbers. | 45 | 38 | 2 |
math | Given that the domain of the function $f(x)$ is $(-\infty, 0)$, and its derivative $f'(x)$ satisfies $xf'(x) - 2f(x) > 0$, determine the solution set of the inequality $f(x+2023) - (x+2023)^{2}f(-1) < 0$. | (-2024, -2023) | 82 | 12 |
math | (1) Given $f(x)=x^{3}-2f'(1)x$, then $f'(1)=$_______
(2) Given the function $f(x)=a^{2}x-2a+1$, "If $\forall x \in (0,1)$, $f(x) \neq 0$" is a false statement, then the range of $a$ is_______
(3) Given in $\Delta ABC$, $AC=\sqrt{2}$, $BC=\sqrt{6}$, the area of $\Delta ABC$ is $\frac{\sqrt{3}}{2}$. If there exists a p... | (3,+\infty) | 259 | 7 |
math |
Given that \(\alpha\) and \(\beta\) are real numbers, for any real numbers \(x\), \(y\), and \(z\), it holds that
$$
\alpha(x y + y z + z x) \leqslant M \leqslant \beta\left(x^{2} + y^{2} + z^{2}\right),
$$
where \(M = \sum \sqrt{x^{2} + x y + y^{2}} \sqrt{y^{2} + y z + z^{2}}\,\) and \(\sum\) represents cyclic sums. ... | 3 | 149 | 1 |
math | Given positive integers \( x, y, z \) and real numbers \( a, b, c, d \), which satisfy \( x \leq y \leq z \), \( x^a = y^b = z^c = 70^d \), and \( \frac{1}{a} + \frac{1}{b} + \frac{1}{c} = \frac{1}{d} \), determine the relationship between \( x + y \) and \( z \). Fill in the blank with ">", "<" or "=". | = | 120 | 1 |
math | Given that \(\sin \alpha + \cos \alpha = -\frac{3}{\sqrt{5}}\) and \(|\sin \alpha| > |\cos \alpha|\), find the value of \(\operatorname{tg} \frac{\alpha}{2}\). | -\frac{\sqrt{5} + 1}{2} | 58 | 13 |
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 $t \mathbf{a} + u \mathbf{b}.$ Determine the or... | \left(\frac{9}{8}, -\frac{1}{8}\right) | 127 | 19 |
math | The first digit of a string of 2050 digits is a 2. Any two-digit number formed by consecutive digits within this string is divisible by either 17 or 29. What is the largest possible last digit in this string? | 8 | 52 | 1 |
math | Define the length of the intervals \((c, d)\), \([c, d]\), \((c, d]\), and \([c, d)\) as \(d - c\), where \(d > c\). Given real numbers \(a > b\), find the sum of the lengths of the intervals of \(x\) that satisfy \(\frac{1}{x-a} + \frac{1}{x-b} \geq 1\). | 2 | 98 | 1 |
math | A construction company is building a tunnel. After completing $\frac{1}{3}$ of the tunnel at the original speed, they use new equipment, resulting in a $20\%$ increase in construction speed and daily working hours reduced to $80\%$ of the original. The entire tunnel is completed in 185 days. If the new equipment had no... | 180 | 95 | 3 |
math | Given the function $f(x)=2\cos x\sin \left(x+ \frac{\pi}{3}\right)- \sqrt{3}\sin^{2}x+\sin x\cos x$
(Ⅰ) Find the smallest positive period of the function $f(x)$;
(Ⅱ) Find the minimum value of $f(x)$ and the corresponding value of $x$ when the minimum value is attained; | -2 | 88 | 2 |
math | In the first hour of the shift, the master craftsman manufactured 35 parts. He then realized that at his current pace, he would need to stay an extra hour to meet the shift’s quota. By increasing his speed by 15 parts per hour, he completed the quota half an hour before the shift ended. How many parts must the master c... | 210 | 79 | 3 |
math | Thirty percent more than 80 is one-fourth less than what number? | 138.67 | 16 | 6 |
math | Square $ABCD$ has sides of length 2. Set $S$ is the set of all line segments that have length 2 and whose endpoints are on adjacent sides of the square. The midpoints of the line segments in set $S$ enclose a region whose area to the nearest hundredth is $k$. Find $100k$. | 86 | 73 | 2 |
math | Write a program that takes one natural number from each of two sources and returns a third number calculated by raising the number from the first source to the power of the number from the second source. From this resulting value, only the last digit will be sent to the Data Processing Center. Use the following test va... | 4 | 128 | 1 |
math | The base three representation of $x$ is $12112211122211112222$. The first digit (on the left) of the base nine representation of $x$ is | 5 | 49 | 1 |
math | The median of a set of consecutive even integers is 154. If the greatest integer in the set is 168, what is the smallest integer in the set? | 140 | 37 | 3 |
math | In triangle $ABC$, we have $\angle A = 90^\circ$, $BC = 20$, and $\tan C = 3\cos B$. What is $AB$? | \frac{40\sqrt{2}}{3} | 41 | 13 |
math | What is the 150th digit to the right of the decimal point in the decimal representation of $\frac{22}{70}$? | 5 | 31 | 1 |
math | Given an ellipse C: $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$ with $a > b > 0$, draw a perpendicular line from the right focus F<sub>2</sub> to the x-axis that intersects ellipse C at point P. If $\sin\angle PF_1F_2 = \frac{1}{3}$, determine the relationship between $a$ and $b$. | \sqrt{2}b | 100 | 6 |
math | How many 12-step paths are there from $E$ to $G$ which pass through $F$ and then $H$? Assume the points are placed as follows on a grid: $E$ at the top-left corner, $F$ 3 steps right and 2 steps down from $E$, $H$ 2 steps right and 2 steps down from $F$, and $G$ 3 steps right from $H$ at the bottom-right corner. | 60 | 99 | 2 |
math | What is the largest possible median for the five number set \(\{x, y, 4, 3, 7\}\) if \(x\) and \(y\) can be any integers, and \( y = 2x\)? | 4 | 51 | 1 |
math | Given two lines $l_{1}:ax-by-1=0$ and $l_{2}:(a+2)x+y+a=0$, where $a$ and $b$ are non-zero constants.
(1) If $b=0$ and $l_{1}\bot l_{2}$, find the value of the real number $a$.
(2) If $b=2$ and $l_{1}\parallel l_{2}$, find the distance between the lines $l_{1}$ and $l_{2}$. | \frac{11\sqrt{13}}{26} | 115 | 15 |
math | Find the product of two approximate numbers: $0.3862 \times 0.85$. | 0.33 | 23 | 4 |
math | Given $N > 1$, calculate the value of $\sqrt[3]{N\sqrt[3]{N\sqrt[3]{N}}}$ | N^{\frac{13}{27}} | 31 | 11 |
math | Given the function $f(x)=ax-\frac{1}{x}-(a+1)\ln x, a\in R$.
(I) Find the monotonic intervals of the function $f(x)$ when the equation of the tangent line at $x=\frac{1}{2}$ is $4x-y+m=0$.
(II) Determine the number of zeros of the function $g(x)=x[f(x)+a+1]$ when $a>\frac{1}{e}$. | 1 | 104 | 1 |
math | In the Cartesian coordinate system $(xOy)$, the focus of the parabola $y^{2}=2x$ is $F$. If $M$ is a moving point on the parabola, determine the maximum value of $\frac{|MO|}{|MF|}$. | \frac{2\sqrt{3}}{3} | 60 | 12 |
math | Given the speed of sound in the forest is 1100 feet per second and the time elapsed between the flash of lightning and Owl hearing the thunder is 12 seconds, estimate the distance Owl was from the flash of lightning. | 2.5 | 48 | 3 |
math | In the arithmetic sequence $\{a_n\}$, $a_3=6$, $a_8=16$, and $S_n$ is the sum of the first $n$ terms of the geometric sequence $\{b_n\}$, with $b_1=1$ and $4S_1$, $3S_2$, $2S_3$ forming an arithmetic sequence.
(Ⅰ) Find the general formulas for the sequences $\{a_n\}$ and $\{b_n\}$;
(Ⅱ) Let $c_n=a_n\cdot b_n$, find th... | T_n=(n-1)\cdot2^{n+1}+2 | 151 | 16 |
math | Let $f : \mathbb{R} \to \mathbb{R}$ be a function such that
\[f(f(x) + y) = f(x^2 - y) + 3f(x) y + 2y\] for all real numbers $x$ and $y.$
Let $n$ be the number of possible values of $f(2),$ and let $s$ be the sum of all possible values of $f(2).$ Find $n \times s.$ | 4 | 108 | 1 |
math | A triangle has sides of lengths $a$, $a+d$, and $a+kd$ where $a, d > 0$ and $k$ is a positive integer. Find the values of $a$, $d$, and $k$ such that the triangle is a right triangle. | 3,1,2 | 60 | 5 |
math | Cindy will randomly select three letters from the word RIVER, three letters from the word STONE, and four letters from the word FLIGHT. What is the probability that she will have all of the letters from the word FILTER? Express your answer as a common fraction. | \frac{3}{125} | 55 | 9 |
math | The range of the inclination angle \\(\theta\\) of the line \\(x\cos \alpha +\sqrt{3}y+2=0\\), determine the interval of possible values for θ. | \left[0,\dfrac{\pi }{6}\right]\cup \left[\dfrac{5\pi }{6},\pi \right) | 44 | 34 |
math | Determine the angles of a triangle with sides 3, $\sqrt{11}$, and $2 + \sqrt{5}$, giving the angles in degrees separated by commas. | 60^\circ, 75^\circ, 45^\circ | 38 | 16 |
math | Sam rolls a fair four-sided die containing the numbers $1,2,3$, and 4. Tyler rolls a fair six-sided die containing the numbers $1,2,3,4,5$, and 6. What is the probability that Sam rolls a larger number than Tyler? | \frac{1}{4} | 59 | 7 |
math | Given the function $y=\cos (x+\frac{π}{3})$, determine the horizontal shift of the graph of the function $y=\sin x$. | \frac{5\pi}{6} | 33 | 9 |
math | Given the function \( f_{1}(x) = \frac{2x - 1}{x + 1} \), for a positive integer \( n \), define \( f_{n+1}(x) = f_{1}\left[ f_{n}(x) \right] \). Find the explicit formula for \( f_{1234}(x) \). | \frac{1}{1-x} | 80 | 8 |
math | Given the set ${2, 5, 8, 11, 14, 17, 20}$, determine the number of different integers that can be expressed as the sum of three distinct members of this set. | 13 | 50 | 2 |
math | Determine the values of the variable \( x \) such that the four expressions \( 2x - 6 \), \( x^2 - 4x + 5 \), \( 4x - 8 \), and \( 3x^2 - 12x + 11 \) differ from each other by the same constant amount, regardless of the order of the expressions. Find all possible integer values of \( x \). | 4 | 93 | 1 |
math | In Pascal's Triangle, each number is the sum of the two numbers directly above it to the left and right. What is the sum of all numbers in Row 12 of Pascal's Triangle? Also, what is the value of the middle number in this row? | 924 | 54 | 3 |
math | Points $A$, $B$, and $C$ are vertices of an isosceles triangle $ABC$ where $AB = AC$ and $BC$ is the base. Points $D$, $E$, and $F$ are the midpoints of $AB$, the midpoint of $AC$, and the foot of the perpendicular from $A$ to $BC$, respectively. How many noncongruent triangles can be drawn using any three of these six... | 4 | 98 | 1 |
math | Find all integer pairs \((x, y)\) such that
\[
3^4 \times 2^3 (x^2 + y^2) = x^3 y^3.
\] | (x, y) = (0, 0), (6, 6), (-6, -6) | 45 | 23 |
math | The sum \( A = \left\lfloor \frac{8}{9} \right\rfloor + \left\lfloor \frac{8^2}{9} \right\rfloor + \cdots + \left\lfloor \frac{8^{2014}}{9} \right\rfloor \) when divided by 63 leaves a remainder of \(\qquad\). (\(\left\lfloor x \right\rfloor\) denotes the greatest integer less than or equal to the real number \( x \)). | 56 | 117 | 2 |
math | The sequence $\{a_n\}$ satisfies $a_1= \frac{1}{2}$, and $a_n \cdot a_{n-1} - 2a_n + a_{n-1} = 0$ $(n \geqslant 2)$, then the largest positive integer $k$ that makes $a_k > \frac{1}{2,017}$ is ______. | 11 | 89 | 2 |
math |
In the Cartesian coordinate system \( xOy \), find the area of the region defined by the inequalities
\[
y^{100}+\frac{1}{y^{100}} \leq x^{100}+\frac{1}{x^{100}}, \quad x^{2}+y^{2} \leq 100.
\] | 50 \pi | 83 | 4 |
math | Given a pyramid $P-ABCD$ whose base $ABCD$ is square and whose vertex $P$ is equidistant from $A,B,C$ and $D$, if $AB=1$ and $\angle{APB}=2\theta$, calculate the volume of the pyramid. | \frac{1}{6\sin(\theta)} | 61 | 11 |
math | The admission criteria for a mathematics contest require a contestant to achieve an average score of at least $90\%$ over five rounds to qualify for the final round. Marcy scores $87\%$, $92\%$, and $85\%$ in the first three rounds. What is the minimum average score Marcy must have in the remaining two rounds to qualif... | 93\% | 83 | 4 |
math | The inclination angle of the line $x-y+1=0$ is what value? | \frac{\pi}{4} | 18 | 7 |
math | Let \( \triangle DEF \) be a triangle and \( H \) the foot of the altitude from \( D \) to \( EF \). If \( DE = 60 \), \( DF = 35 \), and \( DH = 21 \), what is the difference between the minimum and the maximum possible values for the area of \( \triangle DEF \)? | 588 | 78 | 3 |
math | A point is chosen at random within a rectangle in the coordinate plane whose vertices are $(0, 0), (3030, 0), (3030, 2020),$ and $(0, 2020)$. The probability that the point is within $d$ units of a lattice point is $\tfrac{3}{4}$. Find the value of $d$ to the nearest tenth. | 0.5 | 91 | 3 |
math | For how many integers \( n \) between 1 and 15 (inclusive) is \(\frac{n}{18}\) a repeating decimal? | 10 | 32 | 2 |
math | A regular m-sided polygon is surrounded by m regular n-sided polygons (without gaps or overlaps). For the case when m = 4 and n = 8, if m = 10, determine the value of n. | 5 | 46 | 1 |
math | A portion of the graph of $f(x)=px^2+qx+r$ is shown. The distance between grid lines on the graph is $1$ unit.
What is the value of $p+q+2r$?
The graph of $y=f(x)$ passes through points $(0, 9)$ and $(1, 6)$. | 15 | 73 | 2 |
math | A merchant increased the original price by 20%, and then sold it at a 20% discount. Calculate the actual discount given to customers by the merchant. | 4\% | 34 | 3 |
math | Let $f(x)$ and $g(x)$ be two monic cubic polynomials, and let $r$ be a real number. Two of the roots of $f(x)$ are $r + 1$ and $r + 7.$ Two of the roots of $g(x)$ are $r + 3$ and $r + 9,$ and
\[f(x) - g(x) = r\]for all real numbers $x.$ Find $r.$ | 32 | 102 | 2 |
math | Let $x$ and $y$ be real numbers such that $x^2 + y^2 + xy = 1.$ Find the maximum value of $3x - 2y.$ | 5 | 40 | 1 |
math | Observe the following equations:
$$
1 = 1 \\
3+5=8 \\
5+7+9=21 \\
7+9+11+13=40 \\
9+11+13+15+17=65 \\
\ldots
$$
Following this pattern, the right side of the seventh equation equals \_\_\_\_\_\_. | 133 | 83 | 3 |
math | Determine the number of angles between $0$ and $2\pi$, excluding angles where $\sec \theta$, $\csc \theta$, or $\cot \theta$ are undefined, such that these three functions form a geometric sequence in some order. | 2 | 52 | 1 |
math | If the fixed charter cost is 10,000 yuan, and the cost of each person's plane ticket is 800 yuan for groups of 20 people or fewer, and decreases by 10 yuan for each additional person for groups with more than 20 people with a maximum of 75 people, determine the maximum profit that the travel agency can make. | 15000 | 80 | 5 |
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