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 | For each real number $a$ with $0 \leq a \leq 1$, let numbers $x$ and $y$ be chosen independently at random from the intervals $[0, a^2]$ and $[0, 1]$, respectively, and let $Q(a)$ be the probability that
\[\cos^2{(\pi x)} + \cos^2{(\pi y)} < a.\]
What is the maximum value of $Q(a)?$
A) $\frac{1}{2}$
B) $\frac{\sqrt{2}}... | 1 | 140 | 1 |
math | Given vectors $\overset{⇀}{m} =(1,\sqrt{3} )$ and $\overset{⇀}{n} =(\sin x,\cos x)$, define the function $f(x)=\overset{⇀}{m} \cdot\overset{⇀}{n}$.
(1) Find the smallest positive period and the maximum value of the function $f(x)$;
(2) In an acute triangle $\triangle ABC$, let the lengths of the sides opposite angles... | \frac{8}{3} | 160 | 7 |
math | What is the digit in the tens place when $13^{3007}$ is expressed in decimal notation? | 1 | 24 | 1 |
math | Let \( S=\{(x,y) : x\in \{0,1,2,3\}, y\in \{0,1,2,3,4\},\text{ and } (x,y)\ne (0,4)\} \).
Let \( T \) be the set of all right triangles whose vertices are in \( S \). For every right triangle \( t=\triangle{ABC} \) with vertices \( A \), \( B \), and \( C \) in counter-clockwise order and right angle at \( A \), let \... | \frac{9}{2} | 189 | 7 |
math | In the Cartesian coordinate system $xoy$, the parametric equations of curve $C_{1}$ are $\left\{\begin{array}{l}{x=1+t}\\{y=\sqrt{3}t}\end{array}\right.$ (where $t$ is the parameter). Taking the coordinate origin as the pole and the positive $x$-axis as the polar axis, the polar coordinate equation of curve $C_{2}$ is ... | \frac{\sqrt{21}}{2} | 196 | 11 |
math | The equation of the asymptotes of the hyperbola \\(\dfrac{x^2}{9} - \dfrac{y^2}{16} = 1\\) is \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_. | y = \pm \dfrac{4}{3}x | 55 | 13 |
math | Jane and her brother each spin a spinner once. The spinner has six congruent sectors, numbered from 1 to 6. If the non-negative difference of their numbers is less than 4, Jane wins. Otherwise, her brother wins. What is the probability that Jane wins? Express your answer as a common fraction. | \frac{5}{6} | 65 | 7 |
math | A square sheet of paper is folded as follows: the four corners are folded towards the center such that they meet at a single point, forming a new square. Repeating this operation multiple times results in a square with a side length of 3 cm and a thickness of 16 sheets of paper. Find the side length of the original squ... | 12 | 73 | 2 |
math | Let $A = (-3, 1, 5)$, $B = (-4, -2, 3)$, and $C = (-5, -2, 4)$. Compute $\angle ABC$, in degrees. | \cos^{-1} \left(\frac{1}{2\sqrt{7}}\right) \approx 81.79^\circ | 49 | 31 |
math | The base of a pyramid is a rhombus with an acute angle of $30^{\circ}$. The lateral faces are inclined at an angle of $60^{\circ}$ to the base plane. Find the volume of the pyramid if the radius of the circle inscribed in the rhombus is $r$. | \frac{8}{3} r^3 \sqrt{3} | 67 | 15 |
math | Find \(\lim _{x \rightarrow 2}(2-x) \tan \frac{\pi}{4} x\). | \frac{4}{\pi} | 27 | 8 |
math | For a finite sequence \( B = (b_1, b_2, \dots, b_{199}) \) of numbers, the Cesaro sum of \( B \) is defined to be
\[ \frac{T_1 + \cdots + T_{199}}{199}, \]
where \( T_k = b_1 + \cdots + b_k \) and \( 1 \leq k \leq 199 \).
If the Cesaro sum of the 199-term sequence \( (b_1, \dots, b_{199}) \) is 2000, what is the Cesar... | 1992 | 172 | 4 |
math | Let \(x,\) \(y,\) and \(z\) be positive real numbers and let \(k\) be a positive constant. Find the minimum value of
\[
\frac{k \cdot 4z}{2x+y} + \frac{k \cdot 4x}{y+2z} + \frac{k \cdot y}{x+z}.
\] | 3k | 76 | 2 |
math | $\frac{2\sqrt{6}}{\sqrt{2}+\sqrt{3}+\sqrt{5}}$ equals | \sqrt{2}+\sqrt{3}-\sqrt{5} | 26 | 15 |
math | The monotonically increasing interval of the function $f(x)=\sin x-\sqrt{3}\cos x (x\in[-π,0])$ is $\boxed{\text{answer}}$. | [\frac{-π}{6},0] | 42 | 9 |
math | If the line $4x-3y=0$ is tangent to the circle $x^{2}+y^{2}-2x+ay+1=0$, find the value of the real number $a$. | -1 \text{ or } 4 | 46 | 9 |
math | Anthony made 5 of his first 12 free throw attempts. If he makes 70% of his next 30 attempts, by how many percentage points will he increase his overall success rate percentage? Express your answer to the nearest whole number. | 20\% | 52 | 4 |
math | Let $A$, $M$, and $C$ be positive integers such that $A+M+C=15$. What is the maximum value of \[A\cdot M\cdot C+A\cdot M+M\cdot C+C\cdot A + A + M + C?\] | 215 | 61 | 3 |
math | Given the function $f(x)=\sin (2x+φ)+ \sqrt {3}\cos (2x+φ)$, $(0 < φ < π)$, the graph of the function is shifted to the left by $\frac {π}{4}$ units, and the resulting graph is symmetric about the point $(\frac {π}{2},0)$. Determine the minimum value of the function $g(x)=\cos (x+φ)$ on the interval $[- \frac {π}{2}, \... | \frac{1}{2} | 114 | 7 |
math | Given that Four times Dick's age plus twice Tom's age equals three times Harry's age, and Three times the square of Harry's age is equal to twice the square of Dick's age added to four times the square of Tom's age, and their respective ages are relatively prime to each other, find the sum of the cubes of their ages. | 349 | 70 | 3 |
math | Given a student throws a die $5$ times, recording the number of points each time, with an average of $3$ and a variance of $0.4$, determine the number of times the number $2$ appears. | 1 | 47 | 1 |
math | Given the function $f(x)=x^{3}-ax^{2}-bx$, its graph intersects the $x$-axis at the point $(1,0)$. Determine the extreme values of $f(x)$. | \left(\frac{4}{27}, 0\right) | 45 | 15 |
math | Two athletes start running simultaneously - the first from \( A \) to \( B \), and the second from \( B \) to \( A \). They run at different but constant speeds and meet at a distance of 300 meters from \( A \). After running the track \( AB \) to the end, each of them immediately turns back and meets the other again a... | 500 \text{ meters} | 99 | 8 |
math | In spherical coordinates, what is the equivalent point in the standard spherical coordinate representation for the point \( \left( 5, \frac{9 \pi}{4}, \frac{11 \pi}{7} \right) \)? The answer should be in the form \((\rho, \theta, \phi)\), where \( \rho > 0 \), \( 0 \le \theta < 2 \pi \), and \( 0 \le \phi \le \pi \). | \left( 5, \frac{\pi}{4}, \frac{3 \pi}{7} \right) | 105 | 25 |
math | Among the $n$ angles of a convex $n$-gon, $n-1$ angles are equal to $150^{\circ}$, and the remaining angle is less than $150^{\circ}$. For which values of $n$ is this possible? List all possible answers. | 8, 9, 10, 11 | 65 | 12 |
math | Albert continues making a list, in increasing order, of the positive integers that have a first digit of 2. He writes $2, 20, 21, 22, \ldots$ but by the 1,500th digit, he realizes that the list would contain an infinite number of elements. Find the three-digit number formed by the 1498th, 1499th, and 1500th digits, in ... | 294 | 104 | 3 |
math | Given plane vectors $\overrightarrow {a} = (1, m)$, $\overrightarrow {b} = (2, 5)$, $\overrightarrow {c} = (m, 3)$, and $(\overrightarrow {a} + \overrightarrow {c})$ is parallel to $(\overrightarrow {a} - \overrightarrow {b})$, find the value of $m$. | \frac{3 \pm \sqrt{17}}{2} | 85 | 15 |
math | For all positive numbers $x$, $y$, $z$, the product
\[
(2x+2y+2z)^{-1}(x^{-1}+y^{-1}+z^{-1})(xy+yz+xz)^{-1}[2(xy)^{-1}+2(yz)^{-1}+2(xz)^{-1}]
\]
A) $\frac{1}{2xyz}$
B) $\frac{1}{xyz}$
C) $\frac{1}{x^2y^2z^2}$
D) $\frac{1}{2x^2y^2z^2}$ | \frac{1}{x^2y^2z^2} | 135 | 15 |
math | Express 1.1 billion in scientific notation. | 1.1 \times 10^{9} | 10 | 11 |
math | $n$ coins lies in the circle. If two neighbour coins lies both head up or both tail up, then we can flip both. How many variants of coins are available that can not be obtained from each other by applying such operations? | 2 | 49 | 3 |
math | Given the inequality \( k x^{2} - 2 x + 6 k < 0 \) whose solution set is the set of all real numbers, find the range of values for \( k \). | k < -\frac{\sqrt{6}}{6} | 43 | 13 |
math | Elective 4-4: Coordinate System and Parametric Equations
In the plane Cartesian coordinate system $xOy$, the equation of curve $C$ is $x^2-2x+y^2=0$. Taking the origin as the pole and the positive $x$-axis as the polar axis, the polar equation of line $l$ is $\theta =\frac{\pi }{4} (\rho \in \mathbb{R})$.
$(1)$ Write... | 1 | 171 | 1 |
math | Find all ordered integer pairs \((x, y)\) that satisfy the equation \(x^{2} + 2xy + 3y^{2} - 2x + y + 1 = 0\). | (x, y) = (1, 0), (1, -1), (3, -1) | 46 | 23 |
math | Consider an equilateral triangle with a side length of 16 cm. The diameter of a circle lies along one side of this triangle. Calculate the sum of the areas of the two shaded regions, where each region is bounded by an arc and the lines connecting the triangle's vertices and a central point on the circle's diameter. In ... | 99 | 107 | 2 |
math | A solid cube of side length $4$ has a solid cube of side length $2$ removed from each corner. How many edges does the remaining solid have? | 36 | 33 | 2 |
math | In a speech contest held in a class, there are a total of $5$ contestants, including $3$ females (among which is female A) and $2$ males. If the two males cannot appear consecutively, and female A cannot be the first to appear, what is the number of ways for the appearance order? | 60 | 68 | 2 |
math | Observe the characteristics of the following group of equations and explore the pattern:<br/>①$\sqrt{{1}^{3}}=1=1$;<br/>②$\sqrt{{1}^{3}+{2}^{3}}=1+2=3$;<br/>③$\sqrt{{1}^{3}+{2}^{3}+{3}^{3}}=1+2+3=6$;<br/>④$\sqrt{{1}^{3}+{2}^{3}+{3}^{3}+{4}^{3}}=1+2+3+4=10$.<br/>Based on the pattern of the above equations, answer the fo... | 41075 | 355 | 5 |
math | Given \( f(x)=a \sin ((x+1) \pi)+b \sqrt[3]{x-1}+2 \), where \( a \) and \( b \) are real numbers and \( f(\lg 5) = 5 \), find \( f(\lg 20) \). | -1 | 67 | 2 |
math | Given that $x$ and $y$ satisfy the constraints $\begin{cases} x - y - 1 \leqslant 0 \\ 2x - y - 3 \geqslant 0 \end{cases}$, when the objective function $Z = ax + by$ ($a > 0$, $b > 0$) takes the minimum value of $3$ under these constraints, the minimum value of $\frac{2}{a} + \frac{1}{b}$ is | 3 | 108 | 1 |
math | During the draw before the math marathon, the team captains were asked to name the smallest possible sum of the digits in the decimal representation of the number \( n+1 \), given that the sum of the digits of the number \( n \) is 2017. What answer did the captain of the winning team give? | 2 | 67 | 1 |
math | We call a finite set of different real numbers \( X \) "good" if each number from \( X \) can be represented as the sum of two other different numbers from \( X \). What is the minimum number of elements a good set \( X \) can contain? | 6 | 57 | 1 |
math | Shea and Ara were the same height originally. Shea has grown by 25%, and Ara has grown by one-third the number of inches Shea has grown. If Shea is now 70 inches tall, calculate Ara's current height in inches. | 60.67 | 51 | 5 |
math | Find the root of the equation \(169(157-77x)^{2}+100(201-100x)^{2}=26(77x-157)(1000x-2010)\). | 31 | 62 | 2 |
math | Lev took two natural numbers, added their sum to their product, and obtained 1000 as a result. What numbers could Lev have taken? Find all possible pairs. | \{(6, 142), (10, 90), (12, 76)\} | 37 | 26 |
math | Determine all natural numbers \( n \) that have exactly \( \sqrt{n} \) natural divisors (including 1 and the number \( n \) itself). | 1 \text{ and } 9 | 35 | 8 |
math | A truncated cone has horizontal bases with radii 20 and 5. A sphere is tangent to the top, bottom, and lateral surface of the truncated cone. What is the radius of the sphere? | 10 | 42 | 2 |
math | Find the number of different complex numbers $z$ with the properties that $|z|=1$ and $z^{6!}-z^{5!}$ is a real number. | 1440 | 37 | 4 |
math | Given \( x \in \mathbf{R} \), the sum of the maximum and minimum values of the function \( f(x)=\max \left\{\sin x, \cos x, \frac{\sin x+\cos x}{\sqrt{2}}\right\} \) is equal to? | 1 - \frac{\sqrt{2}}{2} | 65 | 12 |
math | In Terrence's class, there are 48 students. Of these, 15 prefer chocolate pie, 10 prefer apple, and 9 prefer blueberry. One-third of the remaining students prefer cherry pie and the rest prefer lemon pie. For Terrence's pie graph showing this data, how many degrees should he use for cherry pie? | 37.5^{\circ} | 72 | 8 |
math | Five identical squares are arranged in a row and cut by two horizontal lines. The sum of the perimeters of the resulting 15 rectangles is 800 cm. Indicate the side length of the original squares in centimeters. | 20 | 48 | 2 |
math | A set S consists of triangles whose sides have integer lengths and are less than 7, with no side length less than or equal to half of any other side length. Determine the largest number of elements that S can have if no two elements of S are congruent or similar. | 9 | 56 | 1 |
math | In tetrahedron $S\-(ABC)$, $SA$ is perpendicular to plane $ABC$, $\angle BAC=120^{\circ}$, $SA=AC=2$, $AB=1$, find the surface area of the circumscribed sphere of the tetrahedron. | \frac{40\pi}{3} | 64 | 10 |
math | If the sum of the first $3n$ positive integers is $150$ more than the sum of the first $n$ positive integers, then the sum of the first $4n$ positive integers is | 300 | 44 | 3 |
math | Convert the parametric equations \(\left\{\begin{array}{l}x=\frac{\cos \theta}{1+\cos \theta} \\ y=\frac{\sin \theta}{1+\cos \theta}\end{array}\right.\) to their standard form. | y^2 = -2 \left(x - \frac{1}{2} \right) | 57 | 20 |
math | Let $m$ be the smallest integer whose cube root is of the form $n+r$, where $n$ is a positive integer and $r$ is a positive real number less than $1/500$. Find $n$. | 13 | 49 | 2 |
math | Given the function $f(x) = \frac{\ln x}{x} - 1$,
(1) Determine the monotonicity of the function $f(x)$;
(2) For $m > 0$, find the maximum value of $f(x)$ on the interval $[m, 2m]$. | f(x)_{\text{max}} = f(e) = \frac{1}{e} - 1 | 68 | 24 |
math | Consider a different cube setup in which there is a vertex $D$ in the middle of the cube, reachable from both $A$ and $B$ by passing through two adjacent vertices first. If $A$ and $B$ are still positioned as in the original problem, how many different 3-edge paths are there from $A$ to $B$ with the condition that all ... | 2 | 87 | 1 |
math | Find the maximum of the expression
$$
|| \ldots|| x_{1}-x_{2}\left|-x_{3}\right|-\ldots\left|-x_{2023}\right|,
$$
where \( x_{1}, x_{2}, \ldots, x_{2023} \) are distinct natural numbers between 1 and 2023. | 2022 | 84 | 4 |
math | Given $n$ new students, where among any 3 students, at least 2 know each other, and among any 4 students, at least 2 do not know each other, find the maximum value of $n$. | 8 | 47 | 1 |
math | It is known that $\sin y = 2 \cos x + \frac{5}{2} \sin x$ and $\cos y = 2 \sin x + \frac{5}{2} \cos x$. Find $\sin 2x$. | -\frac{37}{20} | 54 | 9 |
math | The line $ax+2by+2=0$ is tangent to the circle $x^{2}+y^{2}=2$, and the point of tangency is in the first quadrant. The minimum value of $\dfrac {1}{a^{2}}+ \dfrac {1}{b^{2}}$ is \_\_\_\_\_\_. | \dfrac {9}{2} | 75 | 7 |
math | Given a geometric sequence $\{c_n\}$ satisfying $c_{n+1} + c_n = 10 \cdot 4^{n-1}$ $(n \in \mathbb{N}^*)$, let $S_n$ be the sum of the first $n$ terms of a sequence $\{a_n\}$, where $a_n = \log_2 c_n$,
(Ⅰ) Find $a_n$ and $S_n$;
(Ⅱ) Given another sequence $\{b_n\}$ defined by $b_n = \frac{1}{4S_n - 1}$ and $T_n$ as the ... | m = 2, k = 12 | 219 | 10 |
math | Consider the perpendicular bisector planes of a regular tetrahedron with unit edge length. Into how many parts do these planes divide the tetrahedron, and what is the volume of each part? | 24 \text{ parts, } \frac{\sqrt{2}}{288} \text{ volume each} | 41 | 26 |
math | Given that the set $A=\{x|ax^2+2x+1=0, x\in \mathbb{R}\}$ has only two subsets, then the value of $a$ is \_\_\_\_\_\_. | 0 \text{ or } 1 | 51 | 8 |
math | Misha picked an apple each day for a week and weighed it. Each apple had a different weight, but the weight of each apple was an integer number of grams ranging from 221 grams to 230 grams (inclusive). Misha also calculated the average weight of all the picked apples, and it was always an integer. The apple picked on t... | 230 | 95 | 3 |
math | Find the largest real number $c$ such that \[x_1^2 + x_2^2 + x_3^2 + x_4^2 + x_5^2 \geq cM^2\] whenever $x_1,x_2,x_3,x_4,x_5$ are real numbers such that $x_1+x_2+x_3+x_4+x_5=0$ and $M$ is the median of $x_1,x_2,x_3,x_4,x_5.$ | 2 | 117 | 1 |
math | Given that three circles with centers at $P$, $Q$, and $R$ have radii of $2$, $3$, and $4$, respectively, and are tangent to the same line from below at points $P'$, $Q'$, and $R'$, respectively, and the circles centered at $P$ and $R$ are externally tangent to the circle centered at $Q$, determine the area of triangle... | 6 | 91 | 1 |
math | In the arithmetic sequence $\{a_n\}$, $a_2=2$ and $a_3=4$. Find the value of $a_{10}$.
Options:
A) $12$
B) $14$
C) $16$
D) $18$ | 18 | 63 | 2 |
math | In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively, and $\frac{\cos C}{\cos B} = \frac{3a-c}{b}$.
(I) Find the value of $\cos B$;
(II) If $b=4\sqrt{2}$, $a=c$, find the value of $\sin(A+\frac{\pi}{6})$. | \frac{3\sqrt{2}+\sqrt{3}}{6} | 97 | 17 |
math | Given $\tan \alpha=3$, find the values of:
$(1)\ \dfrac{3\sin \alpha +2\cos \alpha }{\sin \alpha -4\cos \alpha } \quad\quad\quad\quad (2)\ \dfrac{5{{\cos }^{2}}\alpha -3{{\sin }^{2}}\alpha }{1+{{\sin }^{2}}\alpha }$ | -\frac{11}{5} | 94 | 8 |
math | Dima and Vlad play a game: first, they take turns naming a number from 1 to 97 (Dima goes first, and the numbers must be different). Then, each counts the number of distinct rectangles with integer sides whose perimeter is equal to the named number. The winner is the one with the greater number of rectangles. Which num... | 96 | 112 | 2 |
math | Four vertices of a rectangle include the points $(2, 3)$, $(2, 15)$, and $(13, 3)$. What is the area of the intersection of this rectangular region and the region inside the graph of the equation $(x - 13)^2 + (y - 3)^2 = 16$? | 4\pi | 74 | 3 |
math | A store has 5 bags of flour, each weighing between 25 and 70 kilograms. Determine the minimum number of times the store must use the scale to find the weight of each bag of flour. | 5 | 43 | 1 |
math | The maximum value of the function $f(x) = \frac{-x^{2} + x - 4}{x}$ (where $x > 0$) is _______, and this value occurs when $x$ is equal to _______. | 2 | 50 | 1 |
math | If the algebraic expression $\frac{{\sqrt{x+5}}}{x}$ is meaningful within the real number range, then the range of real number $x$ is ____. | x \geqslant -5 \text{ and } x \neq 0 | 37 | 19 |
math | Given that the terminal side of angle $\alpha$ passes through the point $P\left(\sin \frac {5\pi}{6}, \cos \frac {5\pi}{6}\right)$, then angle $\alpha$ is an angle of the \_\_\_\_\_\_ quadrant, and the smallest positive angle that has the same terminal side as angle $\alpha$ is \_\_\_\_\_\_. | \frac {5\pi}{3} | 86 | 9 |
math | How many positive integers less than $200$ are multiples of $5$, but not multiples of either $10$ or $6$? | 20 | 31 | 2 |
math | Given a regular hexagonal pyramid \( M A B C D E F \). Point \( K \) bisects edge \( B M \). Find the ratio in which the plane \( F E K \) divides edge \( A M \) (at point \( X \)). | 2:1 | 57 | 3 |
math | How many 18-letter arrangements of 6 A's, 6 B's, and 6 C's are there such that no A's appear in the first 6 letters, no B's appear in the next 6 letters, and no C's appear in the last 6 letters?
A) $\sum_{k=0}^{6} \binom{6}{k}^2$
B) $4^6$
C) $\sum_{k=0}^{6} \binom{6}{k}^3$
D) $6^{18}$ | \sum_{k=0}^{6} \binom{6}{k}^3 | 122 | 20 |
math | To meet market demand, a supermarket purchased a brand of zongzi before the arrival of the Dragon Boat Festival on the fifth day of May. The cost price of each box is $40$ yuan, and the supermarket stipulates that the selling price of each box must not be less than $45$ yuan. Based on past sales experience, it was foun... | 8000 | 191 | 4 |
math | Points $A, B, C, D, E, F,$ and $G$ lie, in that order, on $\overline{AG}$, dividing it into six segments, each of length 1. Point $H$ is not on line $AG$. Point $I$ lies on $\overline{HD}$, and point $J$ lies on $\overline{HF}$. The line segments $\overline{IC}, \overline{JE},$ and $\overline{AH}$ are parallel. Find th... | 2 | 116 | 1 |
math | Given that the perimeter of triangle \( \triangle ABC \) is 20, the radius of the inscribed circle is \( \sqrt{3} \), and \( BC = 7 \). Find the value of \( \tan A \). | \sqrt{3} | 51 | 5 |
math | Given positive real numbers $a$ and $b$ with the property that $\sqrt{\log{a}} + \sqrt{\log{b}} + \log \sqrt{a} + \log \sqrt{b} = 100$ and all four terms on the left are positive integers, determine the value of $ab$. | 10^{164} | 70 | 7 |
math | Given the sequence: $\frac{2}{3}, \frac{2}{9}, \frac{4}{9}, \frac{6}{9}, \frac{8}{9}, \frac{2}{27}, \frac{4}{27}, \cdots$, $\frac{26}{27}, \cdots, \frac{2}{3^{n}}, \frac{4}{3^{n}}, \cdots, \frac{3^{n}-1}{3^{n}}, \cdots$. Find the position of $\frac{2018}{2187}$ in the sequence. | 1552 | 132 | 4 |
math | Given the matrix M= $$\begin{bmatrix} a & b \\ -1 & 2\end{bmatrix}$$ with two eigenvalues λ<sub>1</sub>\=2, λ<sub>2</sub>\=3. Find the equation of the line l': x-y+2=0 under the transformation of matrix M. | x-3y+12=0 | 75 | 9 |
math | Two given quadratic trinomials differ by exchanging the free term and the second coefficient. The sum of these trinomials has a unique root. What value does this sum take at one? | 18 | 40 | 2 |
math | Given a right triangle with a hypotenuse of length $\sqrt{5}$, and two legs of lengths $x$ and $y$, find the range of values for $x + y$. | (\sqrt{5}, \sqrt{10}] | 40 | 11 |
math | Given any function $y=f(x)$ in the same Cartesian coordinate system, find the type of symmetry exhibited by the graphs of $y=f(x+1)$ and $y=f(-x-1)$. | -1 | 42 | 2 |
math | In triangle \( \triangle ABC \), \( AB = BC = 2 \) and \( AC = 3 \). Let \( O \) be the incenter of \( \triangle ABC \). If \( \overrightarrow{AO} = p \overrightarrow{AB} + q \overrightarrow{AC} \), find the value of \( \frac{p}{q} \). | \frac{2}{3} | 81 | 7 |
math | At 8:08 AM, Xiaoming departs from home on a bicycle. Eight minutes later, his father begins chasing him on a motorcycle. His father catches up to him 4 kilometers away from home, then immediately returns home. Upon arriving home, the father immediately heads out again to chase Xiaoming and catches up to him again exact... | 08:32 | 92 | 5 |
math | A steer initially weighs 300 kilograms. After feeding and care, its weight increases by 15%. With approximately 0.4536 kilograms in a pound, to the nearest whole pound, how much does the steer weigh after the weight increase? | 761\ \text{pounds} | 54 | 10 |
math | Given $sin(\frac{2π}{3}+x)=\frac{3}{5}$, then $cos(\frac{7π}{6}+x)$ is ______. | -\frac{3}{5} | 39 | 7 |
math | In right triangle $ABC$, it is known that $AC=4, BC=1$. $P$ is a moving point on the hypotenuse $AB$ (excluding endpoints), and the distances from $P$ to the two legs are denoted as $d_1$ and $d_2$, respectively. Calculate the minimum value of $\left(\frac{1}{d_1}+\frac{1}{d_2}\right)$. | \frac{9}{4} | 94 | 7 |
math | Petr, Martin, and Jirka were hitting a special target with three fields of different values. Each boy threw a total of ten times and always hit the target. The point scores from the first eight throws were the same for all three boys. In the last two throws, Jirka hit the field with the smallest possible value twice, M... | 12, 17, 22 | 154 | 10 |
math | For a string of \(P\) 's and \(Q\) 's, the value is defined to be the product of the positions of the \(P\) 's. For example, the string \(P P Q P Q Q\) has value \(1 \cdot 2 \cdot 4 = 8\).
Also, a string is called antipalindromic if writing it backwards, then turning all the \(P\) 's into \(Q\) 's and vice versa, prod... | \frac{2005^{1002}}{2004!} | 163 | 20 |
math | Let \( m=30030=2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 13 \) and let \( M \) be the set of its positive divisors which have exactly two prime factors. Determine the minimal integer \( n \) with the following property: for any choice of \( n \) numbers from \( M \), there exist three numbers \( a, b, c \) among them sa... | 11 | 110 | 2 |
math | Given the function $$f(x)=1+x- \frac {x^{3}}{3}+ \frac {x^{5}}{5}- \frac {x^{7}}{7}+ \frac {x^{9}}{9}- \frac {x^{11}}{11}+ \frac {x^{13}}{13},$$ find the smallest integer value of $x$ that makes the inequality $f(x-1)>0$ true. | 0 | 101 | 1 |
math | Two transformations are applied to the complex number $-1 - 2i$:
1. A $60^\circ$ rotation around the origin in the counter-clockwise direction.
2. A dilation, centered at the origin, with scale factor $2$.
What is the resulting complex number? | 2\sqrt{3} - 1 - (2 + \sqrt{3})i | 61 | 19 |
math | A right triangle with integer leg lengths is called "super cool" if the number of square units in its area is equal to three times the number of units in the sum of the lengths of its legs. What is the sum of all the different possible areas of super cool right triangles? | 471 | 57 | 3 |
math | The angles of a hexagon are in the ratio of 2:3:3:4:5:6. Calculate the measure in degrees of the largest angle. | \frac{4320^\circ}{23} | 34 | 13 |
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