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 | Define $f(x)$ as the minimum value among $3x+1$, $-\frac{1}{3}x + 2$, and $x + 4$. Find the maximum value of $f(x)$. | \frac{5}{2} | 46 | 7 |
math | The real number \( y \) satisfies \( y^3 - 6y^2 + 11y - 6 < 0. \) Find all possible values of \( y^3 + 6y^2 + 11y + 6. \) | (24, 120) | 58 | 9 |
math | Fill in the 3x3 grid with 9 different natural numbers such that for each row, the sum of the first two numbers equals the third number, and for each column, the sum of the top two numbers equals the bottom number. What is the smallest possible value for the number in the bottom right corner? | 12 | 64 | 2 |
math | Given that $3^a = 4^b = 5^c = 6$, find the value of $\frac {1}{a}+ \frac {1}{b}+ \frac {1}{c}$. | \log_{6} 60 | 48 | 8 |
math | Construct a square if the sum of one of its sides and one of its diagonals is known. | s = \frac{k}{1 + \sqrt{2}} | 20 | 13 |
math | A traffic light repeats the following sequence: green for 45 seconds, yellow for 5 seconds, and red for 40 seconds. Sam randomly chooses a four-second interval to observe the light. What is the probability that the color changes during his observation? | \frac{2}{15} | 53 | 8 |
math | Find the sum $$\frac{2^1}{4^1 - 1} + \frac{2^2}{4^2 - 1} + \frac{2^4}{4^4 - 1} + \frac{2^8}{4^8 - 1} + \cdots.$$ | 1 | 68 | 1 |
math | Given vectors $\overrightarrow {a}$ = (cos($\frac {π}{3}$ - x), -sin(x)) and $\overrightarrow {b}$ = (sin(x + $\frac {π}{6}$), sin(x)), and a function f(x) = $\overrightarrow {a}$ • $\overrightarrow {b}$:
(I) Find the smallest positive period and the monotonically decreasing interval of function f(x);
(II) In triangle ... | \sqrt {3} | 136 | 5 |
math | Let $J(m)$ be the x coordinate of the left end point of the intersection of the graphs of $y=x^3-6$ and $y=mx^2$. For $m \neq 0$, define $s=\frac{J(-m)-J(m)}{m^2}$. What happens to the value of $s$ as $m$ is made arbitrarily close to zero? | 0 | 85 | 1 |
math | If the complex number $z$ satisfies $(3+i)z=2-i$ (where $i$ is the imaginary unit), then $z=$ _______; $|z|=$ _______. | \dfrac{\sqrt{2}}{2} | 40 | 10 |
math | A standard die is rolled eight times. What is the probability that the product of all eight rolls is odd, and their sum is greater than 20? | \frac{1}{512} | 32 | 9 |
math | Let \(ABC\) be a triangle with \(AB=7\), \(BC=9\), and \(CA=4\). Let \(D\) be the point such that \(AB \parallel CD\) and \(CA \parallel BD\). Let \(R\) be a point within triangle \(BCD\). Lines \(\ell\) and \(m\) going through \(R\) are parallel to \(CA\) and \(AB\) respectively. Line \(\ell\) meets \(AB\) and \(BC\) ... | 180 | 178 | 3 |
math | Given that $\alpha$ and $\beta$ are acute angles, $\cos \alpha = \frac{3}{5}$, $\cos (\alpha + \beta) = -\frac{5}{13}$, find $\cos \beta$. | \frac{33}{65} | 51 | 9 |
math | The domain of the function $f(x+1) = x^2 - 2x + 1$ is $[-2, 0]$, then the interval of monotonic decrease for $f(x)$ is \_\_\_\_. | [-1, 1] | 50 | 6 |
math | Let $x$ and $y$ be real numbers such that:
\[
3 < \frac{x - y}{x + y} < 6.
\]
If $\frac{x}{y}$ is an integer, what is its value? | -2 | 52 | 2 |
math | Given \( n \geq 3 \), \(\omega = \cos \frac{2\pi}{n} + \mathrm{i} \sin \frac{2\pi}{n}\) is an \( n \)-th root of unity. Also, \( x_i \) (for \( i = 0, 1, \cdots, n-1 \)) are real numbers with \( x_0 \geq x_1 \geq \cdots \geq x_{n-1} \). Find the necessary and sufficient conditions that \( x_0, x_1, \cdots, x_{n-1} \) m... | x_{0} = x_{1} = x_{2} = \cdots = x_{n-1} | 190 | 25 |
math | You are given a number composed of three different non-zero digits, 7, 8, and a third digit which is not 7 or 8. Find the minimum value of the quotient of this number divided by the sum of its digits. | 11.125 | 50 | 6 |
math | Find the smallest positive real $k$ satisfying the following condition: for any given four DIFFERENT real numbers $a,b,c,d$ , which are not less than $k$ , there exists a permutation $(p,q,r,s)$ of $(a,b,c,d)$ , such that the equation $(x^{2}+px+q)(x^{2}+rx+s)=0$ has four different real roots. | 4 | 95 | 1 |
math | Determine all functions \( f: \mathbb{R} \rightarrow \mathbb{R} \) such that
\[
f(x f(y))+f(f(x)+f(y))=y f(x)+f(x+f(y))
\]
for all \( x, y \in \mathbb{R} \). | f(x) = 0 \text{ or } f(x) = x | 69 | 16 |
math | Determine all functions \( f: \mathbb{Z} \longrightarrow \mathbb{Z} \), where \( \mathbb{Z} \) is the set of integers, such that
\[
f(m+f(f(n)))=-f(f(m+1))-n
\]
for all integers \( m \) and \( n \). | f(n) = -n - 1 | 73 | 9 |
math | From the $10$ numbers $0-9$, select $3$ numbers. Find:<br/>
$(1)$ How many unique three-digit numbers can be formed without repeating any digits?<br/>
$(2)$ How many unique three-digit odd numbers can be formed without repeating any digits? | 320 | 59 | 3 |
math | Find the maximum and minimum values of $ \int_0^{\pi} (a\sin x\plus{}b\cos x)^3dx$ for $ |a|\leq 1,\ |b|\leq 1.$ | \frac{10}{3} \text{ and } -\frac{10}{3} | 53 | 22 |
math | In the diagram, rectangle $ABCD$ is divided into four identical squares. If the area of rectangle $ABCD$ is $400$ square centimeters, what is its perimeter?
[asy]
size(4cm);
pair a = (0, 2); pair b = (2, 2); pair c = (2, 0); pair d = (0, 0);
draw(a--b--c--d--cycle);
draw(shift(1) * (a--d)); draw(shift(1,0) * (a--b));
l... | 80 | 154 | 2 |
math | The boys and girls must sit alternately, and there are 3 boys. The number of such arrangements is the product of the number of ways to choose 3 positions out of a total of 7, and the number of ways to arrange the girls for the remaining spots. | 144 | 56 | 3 |
math | Find the angle between the slant height and the height of a cone, if the lateral surface area is the geometric mean between the area of the base and the total surface area. | \arcsin \left( \frac{\sqrt{5} - 1}{2} \right) | 36 | 23 |
math | A fair coin is tossed 4 times. What is the probability of at least three consecutive heads? | \frac{3}{16} | 20 | 8 |
math | The sum of seven integers is $-1$. What is the maximum number of the seven integers that can be larger than $13$? | 6 | 29 | 1 |
math | Given the line $2x+my-2m+4=0$ and the line $mx+2y-m+2=0$, find the real value of $m$ such that the two lines are parallel. | m=-2 | 46 | 3 |
math | Given the function $f(x)=\sin(2x+\frac{5π}{6})-\cos^2x+1$.
$(1)$ Find the minimum value and the interval of monotonic increase of the function $f(x)$.
$(2)$ Let angles $A$, $B$, and $C$ be the three interior angles of $\triangle ABC$. If $\cos B=\frac{1}{3}$ and $f(\frac{C}{2})=-\frac{1}{4}$, find $\sin A$. | \frac{2\sqrt{2} + \sqrt{3}}{6} | 112 | 18 |
math | The half-planes $\alpha_{1}, \alpha_{2}, \ldots, \alpha_{n}$ share a common edge $l$. What is the sum of the dihedral angles $\alpha_{1} \widehat{l \alpha_{2}}, \alpha_{2} \widehat{l \alpha_{3}}, \ldots, \alpha_{n-1} \widehat{l \alpha_{n}}$, $\alpha_{n} \widehat{l \alpha_{1}}$, that together span the entire space? | 2\pi | 111 | 3 |
math | Joy has $30$ thin rods, one each of every integer length from $1$ cm through $30$ cm. She places the rods with lengths $3$ cm, $7$ cm, and $15$ cm on a table. She then wants to choose a fourth rod that she can put with these three to form a quadrilateral with positive area. How many of the remaining rods can she choose... | 17 | 91 | 2 |
math | Find the largest constant $m$, so that for any positive real numbers $a$, $b$, $c$, $d$, and $e$, the inequality
\[\sqrt{\frac{a}{b + c + d + e}} + \sqrt{\frac{b}{a + c + d + e}} + \sqrt{\frac{c}{a + b + d + e}} + \sqrt{\frac{d}{a + b + c + e}} + \sqrt{\frac{e}{a + b + c + d}} > m\]
holds. | 2 | 119 | 1 |
math | Given the function $f(x)=\begin{cases}x, & 0 < x\leqslant 1 \\ 2f(x-1), & x > 1\end{cases}$, then $f\left( \dfrac{3}{2}\right)=$\_\_\_\_\_\_, $f(f(3))=$\_\_\_\_\_\_. | 8 | 83 | 1 |
math | Using the oblique projection method, we get:
① The intuitive diagram of a triangle must be a triangle;
② The intuitive diagram of a square must be a rhombus;
③ The intuitive diagram of an isosceles trapezoid can be a parallelogram;
④ The intuitive diagram of a rhombus must be a rhombus.
Among the above conclu... | ① | 92 | 2 |
math | Observe the following equations and answer:<br/>The first equation: ${a}_{1}=\frac{1}{1×3}=\frac{1}{2}×(1-\frac{1}{3})$;<br/>The second equation: ${a}_{2}=\frac{1}{3×5}=\frac{1}{2}×(\frac{1}{3}-\frac{1}{5})$;<br/>The third equation: ${a}_{3}=\frac{1}{5×7}=\frac{1}{2}×(\frac{1}{5}-\frac{1}{7})$;<br/>The fourth equation:... | \frac{n-1}{3n} | 301 | 9 |
math | Given three points $A$, $B$, $C$ on a plane that satisfy $| \overrightarrow{AB}|= \sqrt {3}$, $| \overrightarrow{BC}|= \sqrt {5}$, $| \overrightarrow{CA}|=2 \sqrt {2}$, determine the value of $\overrightarrow{AB}\cdot \overrightarrow{BC}+ \overrightarrow{BC}\cdot \overrightarrow{CA}+ \overrightarrow{CA}\cdot \overright... | -8 | 109 | 2 |
math | Given plane vectors $\overrightarrow{a}=({1,-2})$ and $\overrightarrow{b}=({4,y})$, if the angle between $\overrightarrow{a}$ and $\overrightarrow{a}+\overrightarrow{b}$ is acute, then the range of $y$ is ______. | \left(-\infty ,-8\right)∪(-8,\frac{9}{2}) | 63 | 21 |
math | A line $l$ passes through the point $P(2,-3)$ and intersects the circle $C: x^{2}+y^{2}+2x+2y-14=0$ to form a chord of length $2\sqrt{7}$. Find the equation of the line $l$. | 5x - 12y - 46 = 0 \text{ or } x = 2 | 67 | 23 |
math | Given the equations of two planes:
$$
A_{1} x + B_{1} y + C_{1} z + D_{1} = 0 \text { and } A_{2} x + B_{2} y + C_{2} z + D_{2} = 0
$$
Find the conditions for parallelism and perpendicularity of these planes. | A_{1}A_{2} + B_{1}B_{2} + C_{1}C_{2} = 0 | 80 | 29 |
math | At a factory, metal discs with a diameter of 1 m are cut. It is known that a disc with a diameter of exactly 1 m weighs exactly 100 kg. During manufacturing, a measurement error occurs, and hence the standard deviation of the radius is 10 mm. Engineer Sidorov believes that a stack of 100 discs will weigh on average 100... | 4 \, \text{kg} | 96 | 8 |
math | Given the sequence \(\{a_{n}\}\) where \(a_{1} = a_{2} = 1\), \(a_{3} = -1\), and \(a_{n} = a_{n-1} a_{n-3}\), find \(a_{1964}\). | -1 | 68 | 2 |
math | Two basketball teams, A and B, compete in a match. It is known that the probability of team A winning each game is 0.6, while the probability for team B is 0.4. Each game must produce a winner, and the competition follows a best-of-three format, meaning the first team to secure two wins is the victor.
(Ⅰ) Find the prob... | P_2 = 0.352 | 106 | 10 |
math | Given the functions \( f(x) = \frac{x^2 + 4x + 3}{x^2 + 7x + 14} \) and \( g(x) = \frac{x^2 - 5x + 10}{x^2 + 5x + 20} \):
1. Find the maximum value of the function \( f(x) \).
2. Find the maximum value of the function \( g(x)^{f(x)} \). | 9 | 103 | 1 |
math | Given that $M$ is a point inside $\triangle ABC$ and $\overrightarrow{AB} \cdot \overrightarrow{AC} = 4\sqrt{3}$, $\angle BAC = 30^\circ$. If the areas of $\triangle MBC$, $\triangle MCA$, and $\triangle MAB$ are $1$, $x$, and $y$ respectively, find the minimum value of $\frac{y + 4x}{xy}$. | 9 | 98 | 1 |
math | Given that $m>0$ and $n>0$, and vectors $\overrightarrow{a} = (m, 1)$ and $\overrightarrow{b} = (1, n-1)$ are perpendicular to each other, determine the minimum value of $\frac{1}{m} + \frac{2}{n}$. | 3 + 2\sqrt{2} | 70 | 9 |
math | Four distinct integers $a$, $b$, $c$ and $d$ have the property that when added in pairs, the sums 16, 19, 20, 21, 22, and 25 are obtained. What are the four integers in increasing order? (place a comma and then a space between each integer) | 7,9,12,13 | 75 | 9 |
math | When \( 2x^3 - 5x^2 - 12x + 7 \) is divided by \( 2x + 3 \), the quotient is \( x^2 - 4x + 2 \). What is the remainder? | -4x + 1 | 56 | 6 |
math | Given that the line passing through the focus \( F \) of the parabola \( y^2 = 4x \) intersects the parabola at points \( M \) and \( N \), and \( E(m,0) \) is a point on the x-axis, with the extensions of \( M E \) and \( N E \) intersecting the parabola again at points \( P \) and \( Q \) respectively. If the slopes ... | 3 | 149 | 1 |
math | A cube has a side length of $s,$ and its vertices are $A = (0,0,0),$ $B = (s,0,0),$ $C = (s,s,0),$ $D = (0,s,0),$ $E = (0,0,s),$ $F = (s,0,s),$ $G = (s,s,s),$ and $H = (0,s,s).$ A point $P$ inside the cube satisfies $PA = \sqrt{70},$ $PB = \sqrt{97},$ $PC = \sqrt{88},$ and $PE = \sqrt{43}.$ Find the side length $s.$ | 9 | 147 | 1 |
math | If the solution set of the inequality $|ax + 2| < 6$ with respect to $x$ is $(-1, 2)$, find the value of $a$. | -4 | 40 | 2 |
math | An integer $B$ is considered lucky if there exist several consecutive integers, including $B$, that add up to 2023. What is the smallest lucky integer? | -2022 | 36 | 5 |
math | In the expression
$$
1 * 2 * 3 * 4 * 5 = 100
$$
replace each asterisk with an arithmetic operation symbol and arrange the parentheses so that the equation becomes true. | 1 \cdot (2+3) \cdot 4 \cdot 5 = 100 | 47 | 21 |
math | Find all triples of prime numbers \( p, q, \) and \( r \) such that \( \frac{p}{q} = \frac{4}{r+1} + 1 \). | (7,3,2), (5,3,5), (3,2,7) | 43 | 21 |
math | The germination rate of cotton seeds is $0.9$, and the probability of developing into strong seedlings is $0.6$,
$(1)$ If two seeds are sown per hole, the probability of missing seedlings in this hole is _______; the probability of having no strong seedlings in this hole is _______.
$(2)$ If three seeds are sown per ... | 0.936 | 105 | 5 |
math | What is the volume and total surface area in cubic inches and square inches, respectively, of a right rectangular prism if the area of the front, bottom, and side faces are 24, 18, and 12 square inches, respectively? | 108 | 52 | 3 |
math | In a convex quadrilateral, the midpoints of its sides are connected sequentially. The resulting "midpoint" quadrilateral is a rhombus, in which the sides and one of the diagonals are equal to 3. Find the area of the original quadrilateral. | 9\sqrt{3} | 55 | 6 |
math | 1. The quadratic function $f(x)$ satisfies $f(x+1) - f(x) = 3x$ and $f(0) = 1$. Find the analytic expression of $f(x)$.
2. Given $3f(\frac{1}{x}) + f(x) = x$ $(x \neq 0)$, find the analytic expression of $f(x)$. | \frac{3}{8x} - \frac{x}{8} | 84 | 15 |
math | Calculate using your preferred method:
(1) $42.67-(12.95-7.33)$
(2) $\left[8.4-8.4\times(3.12-3.7)\right]\div0.42$
(3) $5.13\times0.23+8.7\times0.513-5.13$
(4) $6.66\times222+3.33\times556$ | 3330 | 119 | 4 |
math | Let \( S \) be the set of positive real numbers. Let \( f : S \to \mathbb{R} \) be a function such that
\[ f(x) f(y) = f(xy) + 2010 \left( \frac{1}{x} + \frac{1}{y} + 2009 \right) \]
for all \( x, y > 0 \).
Let \( n \) be the number of possible values of \( f(3) \), and let \( s \) be the sum of all possible values of... | 6031 | 139 | 4 |
math | The Evil League of Evil plans to set out from their headquarters at (5,1) to poison two pipes: one along the line \( y = x \) and the other along the line \( x = 7 \). They wish to determine the shortest distance they can travel to visit both pipes and then return to their headquarters. | 4\sqrt{5} | 67 | 6 |
math | Given $y=f(x)$ has an inverse function $y=f^{-1}(x)$, and $y=f(x+2)$, is the inverse function of $y=f^{-1}(x-1)$, then the value of $y=f^{-1}(2010)-f^{-1}(1)$ is. | 4018 | 67 | 4 |
math | A right triangle has sides 15 cm and 20 cm. This triangle is similar to another triangle which has one of its sides as twice the length of a rectangle's shorter side. If the rectangle's shorter side is 30 cm and the longer side is 60 cm, what is the perimeter of the larger triangle? | 240 \text{ cm} | 69 | 8 |
math | Given in $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively, and it is known that $2\sin A\cos B=2\sin C-\sin B$.
(I) Find the angle $A$;
(II) If $a=4\sqrt{3}$ and $b+c=8$, find the area of $\triangle ABC$. | \frac{4\sqrt{3}}{3} | 98 | 12 |
math | A positive number is called $n$-primable if it is divisible by $n$ and each of its digits is a one-digit prime number. Determine how many 5-primable positive integers exist that are less than 1000? | 13 | 52 | 2 |
math | Given $ a_{i} \in \left\{0,1,2,3,4\right\}$ for every $ 0\le i\le 9$ and $6 \sum _{i = 0}^{9}a_{i} 5^{i} \equiv 1\, \, \left(mod\, 5^{10} \right)$ , find the value of $ a_{9} $. | 4 | 95 | 1 |
math | Two thieves stole an open chain with $2k$ white beads and $2m$ black beads. They want to share the loot equally, by cutting the chain to pieces in such a way that each one gets $k$ white beads and $m$ black beads. What is the minimal number of cuts that is always sufficient? | 2 | 75 | 1 |
math | Among 9 consecutive positive odd numbers, what is the maximum number of prime numbers? Answer: . | 7 | 20 | 1 |
math | Given that \(a, b, c\) are integers such that \(abc = 60\), and that the complex number \(\omega \neq 1\) satisfies \(\omega^3 = 1\), find the minimum possible value of \(\left| a + b\omega + c\omega^2 \right|\). | \sqrt{3} | 71 | 5 |
math | Find the maximum value that the expression \(a e k - a f h + b f g - b d k + c d h - c e g\) can take, given that each of the numbers \(a, b, c, d, e, f, g, h, k\) equals \(\pm 1\). | 4 | 68 | 1 |
math | A four-meter gas pipe has rusted in two places. Determine the probability that all three resulting parts can be used as connections to gas stoves, if according to standards, a stove should not be located closer than 1 meter from the main gas pipe. | 1/4 | 52 | 3 |
math | Each letter in the cryptarithm \( A H H A A H : J O K E = H A \) represents a unique decimal digit. Restore the encrypted division. | 377337 : 73 = 5169 | 35 | 16 |
math | Let $a_n$ be the number obtained by writing the integers 1 to $n$ from left to right. For instance, $a_4 = 1234$ and $a_{12} = 123456789101112$. For $1 \le k \le 150$, how many $a_k$ are divisible by both 3 and 5? | 10 | 92 | 2 |
math | Calculate the volume of a spherical layer if the radii of the top and bottom circles are given as \( r_{1} \) and \( r_{2} \), respectively, and the height of the spherical layer is \( h \). | \frac{\pi h}{6} \left(3r_1^2 + 3r_2^2 + h^2 \right) | 49 | 32 |
math | In the sixth, seventh, eighth, and ninth basketball games of the season, a player scored $23$, $14$, $11$, and $20$ points, respectively. Her points-per-game average was higher after nine games than it was after the first five games. If her average after ten games was greater than $18$, what is the least number of poin... | 29 | 88 | 2 |
math | Elective 4-4: Coordinate System and Parametric Equations
In the Cartesian coordinate system $xOy$, the parametric equation of line $l_1$ is $\begin{cases} x=2+t \\ y=kt \end{cases}$ ($t$ is the parameter), and the parametric equation of line $l_2$ is $\begin{cases} x=-2+m \\ y=\frac{m}{k} \end{cases}$ ($m$ is the para... | \sqrt{5} | 233 | 5 |
math | Define the determinant $$\begin{vmatrix} a & b \\ c & d \end{vmatrix}$$ as $ad - bc$. Compute the sum of the determinants: $$\begin{vmatrix} 2 & 4 \\ 6 & 8 \end{vmatrix} + \begin{vmatrix} 10 & 12 \\ 14 & 16 \end{vmatrix} + \ldots + \begin{vmatrix} 2010 & 2012 \\ 2014 & 2016 \end{vmatrix} = \_\_\_\_\_\_.$$ | -2016 | 139 | 5 |
math | Find all real values of $x$ which satisfy
\[\frac{1}{x + 2} + \frac{4}{x + 4} \ge 1.\] | (-2, 1] | 39 | 6 |
math | How many four-character license plates consist of a consonant, followed by a vowel, followed by a consonant, and then a digit? (For this problem, consider Y a vowel.) | 24{,}000 | 38 | 8 |
math | In the rectangular coordinate system $xOy$, the parametric equations of line $l$ are given by $$\begin{cases} x=1+ \frac {3}{5}t \\ y=1+ \frac {4}{5}t\end{cases}$$ ($t$ is the parameter). Establish a polar coordinate system with the coordinate origin as the pole and the positive semi-axis of the $x$-axis as the polar a... | \frac{55}{41} | 218 | 9 |
math | Find the length of side $XZ$ in the right triangle below, where hypotenuse $XY = 13$ and angle $Y = 60^\circ$.
[asy]
unitsize(1inch);
pair P,Q,R;
P = (0,0);
Q= (0.8,0);
R = (0,0.95);
draw (P--Q--R--P,linewidth(0.9));
draw(rightanglemark(Q,P,R,3));
label("$X$",P,W);
label("$Y$",Q,S);
label("$Z$",R,N);
label("$13$",P--Q,... | \frac{13\sqrt{3}}{2} | 151 | 13 |
math | Elijah is curious about how many unique ways he can rearrange the letters of his name. Given that Elijah can write ten rearrangements of his name every minute, calculate how many hours it will take him to write all possible rearrangements of his name. | 1.2 \text{ hours} | 53 | 8 |
math | How many ordered pairs of integers \( (x, y) \) satisfy \( x^2 + y^2 + x = y + 3 \)? | 2 | 32 | 1 |
math | Given the graph of $y=\cos 2x$, obtain the graph of $y=\sin 2x$. | \frac{\pi}{4} | 24 | 7 |
math | What is the domain of the function $f(x) = \log_3(\log_4(\log_5x))$? | (625, \infty) | 28 | 9 |
math | Given a positive integer \( n \geq 3 \), for an \( n \)-dimensional real vector \( \left( x_1, x_2, \cdots, x_n \right) \), if every permutation \( y_1, y_2, \cdots, y_n \) of its elements satisfies \( \sum_{i=1}^{n-1} y_i y_{i+1} \geq -1 \), then the vector \( \left( x_1, x_2, \cdots, x_n \right) \) is called "shining... | -\frac{n-1}{2} | 185 | 8 |
math | A shooter has a 0.9 probability of hitting the target in one shot. He shoots 4 times consecutively, with each shot's outcome independent from the others. The following conclusions are given:
(1) The probability of hitting the target on the 3rd shot is 0.9.
(2) The probability of hitting the target exactly 3 times is $0... | 1, 3 | 121 | 4 |
math | If I roll a fair, regular six-sided die five times, what is the probability that I will roll the number $1$ at least twice and the number $2$ at least twice? | \frac{5}{324} | 39 | 9 |
math | Given the hyperbola \( C_1: 2x^2 - y^2 = 1 \) and the ellipse \( C_2: 4x^2 + y^2 = 1 \). If \( M \) and \( N \) are moving points on the hyperbola \( C_1 \) and ellipse \( C_2 \) respectively, such that \( OM \perp ON \) and \( O \) is the origin, find the distance from the origin \( O \) to the line \( MN \). | \frac{\sqrt{3}}{3} | 116 | 10 |
math | Two circles are tangent at a single point O. The first circle has a radius of 7 inches and the second has a radius of 3 inches. Two bugs start crawling at the same time from point O; one crawling along the larger circle at 4π inches per minute, the other along the smaller circle at 3π inches per minute. Determine how l... | 7 | 85 | 1 |
math | Teacher Lin wants to choose 2 different letters from $\{A,B,C\}$ and 3 different numbers from $\{1,3,5,7\}$ to form the last five digits of a license plate number in the form of $J\times \times \times \times \times$, with the condition that the numbers chosen must not be adjacent to each other. Calculate the number of ... | 144 | 86 | 3 |
math | A multiplication table of the numbers 1 to 10 is shown. What is the sum of all the odd products in the complete table? | 625 | 29 | 3 |
math | Given the vectors $\overrightarrow{m}=(2\sin \omega x, \cos ^{2}\omega x-\sin ^{2}\omega x)$ and $\overrightarrow{n}=( \sqrt {3}\cos \omega x,1)$, where $\omega > 0$ and $x\in R$. If the minimum positive period of the function $f(x)= \overrightarrow{m}\cdot \overrightarrow{n}$ is $\pi$,
(I) Find the value of $\omega$.... | -\frac{3}{2} | 163 | 7 |
math | Let \( r \) be the number that results when both the base and the exponent of \( 2^b \) are doubled, where \( b > 0 \) is an integer. If \( r \) equals the product of \( 2^b \) and \( x^b \) where \( x > 0 \), find the value of \( x \). | 8 | 79 | 1 |
math | Let \( n \) be a positive integer. Determine the smallest positive integer \( k \) such that it is possible to mark \( k \) cells on a \( 2n \times 2n \) board so that there exists a unique partition of the board into \( 1 \times 2 \) and \( 2 \times 1 \) dominoes, none of which contains two marked cells. | 2n | 87 | 2 |
math | Given the function $f(x) = \log_a(a^{2x} - 2a^x - 2)$ where $a > 1$, determine the range of $x$ for which $f(x) > 0$. | (\log_a 3, +\infty) | 50 | 11 |
math | Three members of the Euclid Middle School girls' softball team had the following conversation.
Ashley: I just realized that our uniform numbers are all $2$-digit primes.
Bethany : And the sum of your two uniform numbers is the date of my birthday earlier this month.
Caitlin: That's funny. The sum of your two uniform nu... | 11 | 105 | 2 |
math | A point has rectangular coordinates $(-3, -4, 5)$ and spherical coordinates $(\rho, \theta, \phi)$. Find the rectangular coordinates of the point with spherical coordinates $(\rho, \theta + \pi, -\phi)$. | (3, 4, 5) | 54 | 9 |
math | The roots of the equation \( f(x) \equiv x^{2} + p x + q = 0 \) are \( x_{1} \) and \( x_{2} \), which are numbers different from unity. What is the quadratic equation whose roots are
\[ y_{1} = \frac{x_{1} + 1}{x_{1} - 1} \quad \text{and} \quad y_{2} = \frac{x_{2} + 1}{x_{2} - 1} \quad? \] | (1 + p + q)y^2 + 2(1 - q)y + (1 - p + q) = 0 | 118 | 28 |
math | The point that is $4$ units away from the origin on the number line represents the number $x$ such that $|x|=4$. | 4 \text{ or } -4 | 30 | 8 |
math | Determine the range of values of the base $b$ for which the number $144_b$ is the square of an integer. | b > 4 | 29 | 4 |
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