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 | Jessica needs to convert her height from inches to centimeters for a medical form. Given that there are 10 inches in a foot and 25 centimeters in a foot, what should Jessica specify as her height in centimeters if her height in inches is 65 inches? | 162.5 | 58 | 5 |
math | Consider vectors $\mathbf{u}$ and $\mathbf{x}$ such that $\|\mathbf{u}\| = 5,$ $\|\mathbf{x}\| = 8,$ and $\mathbf{u} \cdot \mathbf{x} = 16$. Find the magnitude of the projection of $\mathbf{u}$ on $\mathbf{x}$ and also calculate $\|\mathbf{u} - \text{proj}_{\mathbf{x}} \mathbf{u}\|$. | \sqrt{21} | 106 | 6 |
math | There are four balls in a bag, which are identical in shape and size. The balls are numbered $1$, $2$, $3$, and $4$ respectively.
(i) Calculate the probability that the product of the numbers of two randomly drawn balls from the bag is not greater than $4$.
(ii) Randomly draw a ball from the bag, its number is $m$, put... | P = \frac{5}{8} | 114 | 9 |
math | Two regular tetrahedra are inscribed in a cube in such a way that four vertices of the cube serve as the vertices of one tetrahedron, and the remaining four vertices of the cube serve as the vertices of the other. What fraction of the volume of the cube is occupied by the volume of the intersection of these tetrahedra? | 1/6 | 72 | 3 |
math | Given that circle M passes through the intersection points of the circles $x^2+y^2+6x-4=0$ and $x^2+y^2+6y-28=0$,
(I) If the center of the circle is on the line $x-2y-3=0$, find the equation of circle M.
(II) If the area of the circle is minimal, find the equation of circle M. | \left(x+ \frac{7}{2}\right)^2+\left(y- \frac{1}{2}\right)^2= \frac{25}{2} | 94 | 37 |
math | The function $y=\log_{a}x$ always satisfies $|y| > 1$ on the interval $[2,+\infty)$. The range of values for $a$ is __________. | \left( \frac{1}{2},1\right) \cup (1,2) | 45 | 21 |
math | Given a sequence $\{a\_n\}$ with its first term $a\_1=1$, the sum of the first $n$ terms denoted by $S\_n$, and satisfying $2a_{n+1}+S_n=2 (n∈N^*)$, find the maximum value of $n$ such that $\frac{1\,001}{1\,000} < \frac{S_{2n}}{S_{n}} < \frac{11}{10}$. | 9 | 112 | 1 |
math | The range of the function $f\left(x\right)=(x^{2}-4x)(e^{x-2}-e^{2-x})+x+1$ on the interval $\left[-1,5\right]$ is $\left[m,M\right]$. Find $m+M$. | 6 | 63 | 1 |
math | A fair, twenty-faced die has numbers $1$ through $9$ and $11$ through $20$ with one blank face. A second fair twenty-faced die has numbers $1$ through $7$ and $9$ through $20$ with one blank face. When these two dice are rolled, what is the probability that the sum of the numbers facing up will be $27$? Express your an... | \frac{1}{40} | 93 | 8 |
math | Let \( f(x) \) be an even function and \( g(x) \) be an odd function, and suppose \( f(x) = -g(x+c) \) where \( c > 0 \). Determine the smallest positive period of the function \( f(x) \). | 4c | 59 | 2 |
math | Let $h(x)$ be a function piecewise defined as \[h(x) = \left\{
\begin{array}{cl}
-x & x\le 0, \\
3x-50 & x>0.
\end{array}
\right.\] If $b$ is negative, find $b$ such that $h(h(h(15)))=h(h(h(b)))$. | b = -\frac{55}{3} | 85 | 11 |
math | Students A and B participated in a Chinese character typing competition. A can type 10 more characters per minute than B. In the same amount of time, A typed 900 characters, and B typed 840 characters. Let x be the number of characters A types per minute. We can derive an equation from this information. | \frac {900}{x} = \frac {840}{x-10} | 70 | 22 |
math | Determine the constant term in the expansion of $\left(2x+\frac{1}{x}-1\right)^5$. | -161 | 27 | 4 |
math | Find $c$ if $c$ and $d$ are integers such that $x^2 - 2x - 1$ is a factor of $cx^{17} + dx^{16} + 1$. | 987 | 48 | 3 |
math | Find all real values of $x$ that satisfy \[\frac{1}{x^2+1} > \frac{4}{x} + \frac{25}{10}.\] (Give your answer in interval notation.) | (-2, 0) | 50 | 6 |
math | Given an even function $f\left(x\right)$ defined on $R$ satisfies $f\left(x\right)=f\left(-x+4\right)$, $f\left(2024\right)=\frac{1}{{e}^{2}}$, if $f\left(x\right)-{f'}\left(x\right) \gt 0$, then the solution set of the inequality $f\left(x+2\right) \gt e^{x}$ is ______. | (-\infty, -2) | 110 | 8 |
math | Given points $A(x,5-x,2x-1)$ and $B(1,x+2,2-x)$, the minimum value of $|AB|$ is ______. | \frac { \sqrt {35}}{7} | 38 | 12 |
math | If the inequality $|x-4|+|x+3|<a$ has real solutions, determine the range of the real number $a$. | a>7 | 32 | 3 |
math | The line $y = b-x$ with $0 < b < 4$ intersects the $y$-axis at $P$ and the line $x=4$ at $S$. If the ratio of the area of triangle $QRS$ to the area of triangle $QOP$ is 9:25, what is the value of $b$? Express the answer as a decimal to the nearest tenth.
[asy]
draw((0,-3)--(0,5.5),Arrows);
draw((4,-3.5)--(4,5),Arrows... | 2.5 | 305 | 3 |
math | Given the function $f(x)=\frac{2x}{3x+2}$, the sequence ${a_n}$ satisfies $a_1=1$ and $a_{n+1}=f(a_n)$.
1. Find the general formula for the sequence ${a_n}$;
2. (Logic) Let $b_n = a_n a_{n+1}$, and the sum of the first $n$ terms of the sequence ${b_n}$ is $S_n$. If $S_n < \frac{m-2016}{2}$ holds for all positive integ... | S_n = (3n-4)\times 2^n + 4 | 187 | 16 |
math | For each permutation \(a_{1}, a_{2}, \cdots, a_{10}\) of the integers \(1, 2, 3, \cdots, 10\), find the arithmetic mean of the sums \(\left| a_{1} - a_{2} \right| + \left| a_{3} - a_{4} \right| + \left| a_{5} - a_{6} \right| + \left| a_{7} - a_{8} \right| + \left| a_{9} - a_{10} \right|\). | \frac{55}{3} | 132 | 8 |
math | Given that $x_0$ is a root of the function $f(x) = 3^x + \frac{2}{1-x}$, if $x_1 \in (1, x_0)$ and $x_2 \in (x_0, +\infty)$, determine the signs of $f(x_1)$ and $f(x_2)$. | f(x_1) < 0, f(x_2) > 0 | 82 | 17 |
math | Given a complex number $z$ satisfying $(z-1)i=1+i$, calculate the conjugate of $z$. | 2+i | 25 | 2 |
math | An insect lives on the surface of a cube with edges of length 2. It wishes to travel on the surface from the midpoint of one edge to the midpoint of the edge directly opposite on the same face. What is the length of the shortest such trip?
A) 1
B) 2
C) $\sqrt{2}$
D) 2$\sqrt{2}$
E) $\frac{3}{2}$ | \sqrt{2} | 89 | 5 |
math | Suppose $A$, $B$, and $C$ are sets such that $|A| = 7$, $|B| = 7$, and $|C| = 6$. If $n(A) + n(B) + n(C) = n(A \cup B \cup C)$, find the minimum possible value of $|A \cap B \cap C|$. | 1 | 82 | 1 |
math | Given the variable $S=\sin\left(\frac{a-b}{3}\pi\right)$, where $a$ is selected from the numbers $\{0, 1, 2, 3\}$ and $b$ is selected from the numbers $\{0, 1, 2\}$, determine the probability that $S \geq 0$. | \frac{3}{4} | 78 | 7 |
math | Determine the number of digits in the value of $2^{15} \times 5^{11}$. | 12 | 25 | 2 |
math | In a quadrilateral $WXYZ$, the angles satisfy $\angle W = 3\angle X = 4\angle Y = 6\angle Z$. Determine the exact degree measure of $\angle W$. | \frac{1440}{7} | 42 | 10 |
math | Given an odd function $y=f\left(x\right)$ defined on $R$, which is strictly decreasing on the interval $\left[0,+\infty \right)$. If for any $x\in R$, we always have $f(kx^{2}+2)+f\left(kx+k\right)\leqslant 0$ holds, then the range of real number $k$ is ______. | [0,+\infty) | 89 | 7 |
math | If $f(x)$ is defined for all integers $x \ge 0,$ $f(1) = 2,$ and
\[f(a + b) = f(a) + f(b) - 2f(a \cdot b)\]for all integers $a,$ $b \ge 0,$ compute $f(1986).$ | 2 | 75 | 1 |
math | Given that points $P$ and $Q$ lie on a plane with $PQ = 10$, determine the number of locations for point $R$ in this plane such that the triangle with vertices $P$, $Q$, and $R$ is a right triangle with an area of $20$ square units and $R$ is equidistant from $P$ and $Q$. | 2 | 82 | 1 |
math | If the consecutive integers from $60$ to $1$ were written as $$6059585756...,$$ what would be the $85^{\text{th}}$ digit to be written? | 1 | 49 | 1 |
math | At the World Meteorological Conference, each participant announced the average monthly temperature in their hometown in turn. All the other participants at that moment recorded the product of the temperatures in their own city and the announced city. A total of 36 positive and 36 negative numbers were recorded. What is... | 3 | 73 | 1 |
math | Consider triangle $DEF$ with vertices $D(1, 6)$, $E(3, -2)$, $F(7, -2)$. A line passing through $E$ bisects the area of triangle $DEF$. Determine the sum of the slope and the $y$-intercept of this line. | -10 | 69 | 3 |
math | Let the graph $G$ have $n$ vertices and not contain $K_3$ as a subgraph. Then the number of edges $|E| \leq \frac{n^2}{4}$. | |E| \leq \frac{n^2}{4} | 45 | 14 |
math | The lateral edges of a triangular pyramid are pairwise perpendicular, with one of them equal to \(a\) and also equal to the sum of the other two. Find the radius of the sphere that touches the base of the pyramid and its lateral faces' extensions. | \frac{a}{2} | 51 | 7 |
math | In a bag, there are 5 white balls and 3 black balls, all identical in size. Calculate the probability of the following events when 4 balls are drawn randomly:
(1) Drawing either 2 or 3 white balls.
(2) Drawing at least one black ball. | \frac{13}{14} | 59 | 9 |
math | Find the number of ordered triples $(x,y,z)$ of real numbers that satisfy
\begin{align*}
x + y - z &= 0, \\
xz - xy + yz &= 27, \\
xyz &= 54.
\end{align*} | 3 | 57 | 1 |
math | A regular polygon has a perimeter of 180 cm and each side measures 15 cm. How many sides does this polygon have? Also, calculate the measure of one internal angle. | 150^\circ | 39 | 5 |
math | Five identical balls roll on a smooth horizontal surface towards each other. The speeds of the first and second balls are \( v_{1} = v_{2} = 0.5 \mathrm{~m/s} \), and the speeds of the others are \( v_{3} = v_{4} = v_{5} = 0.3 \mathrm{~m/s} \). The initial distances between the balls are all equal to \( l = 1 \mathrm{~... | 5 \ \text{seconds} | 132 | 7 |
math | Let
\[f(x) = \frac{x^3 - 6x^2 + 11x - 6}{2x-4}\] and
\[g(x) = \frac{2x^3 - 6x^2 + 5x + d}{x-2}\].
Given that the graphs of $f(x)$ and $g(x)$ have:
- The same vertical asymptote.
- Perpendicular oblique asymptotes intersecting the y-axis.
- Two intersection points, one of which is on the line $x = -1$.
Find the point ... | (3, 0) | 151 | 6 |
math | Find the maximal positive integer $n$ , so that for any real number $x$ we have $\sin^{n}{x}+\cos^{n}{x} \geq \frac{1}{n}$ . | n = 8 | 49 | 4 |
math | Given the equation of curve C as y = x ln(x), find the angle of inclination of the tangent line at the point where x = 1 on curve C. | \dfrac{\pi}{4} | 34 | 7 |
math | Using three rectangles to form a larger rectangle (without gaps or overlaps), and given that two of the rectangles have dimensions $3 \mathrm{~cm} \times 8 \mathrm{~cm}$ and $2 \mathrm{~cm} \times 5 \mathrm{~cm}$ respectively, the third rectangle should have dimensions of $\qquad$. | 4 | 73 | 1 |
math | Given a circle with the equation \\(x^{2}+2x+y^{2}=0\\), find the equation of the circle with the same center and a radius of \\(2\\). | (x+1)^2 + y^2 = 4 | 41 | 12 |
math | In triangle $ABC$ , let $I, O, H$ be the incenter, circumcenter and orthocenter, respectively. Suppose that $AI = 11$ and $AO = AH = 13$ . Find $OH$ .
*Proposed by Kevin You* | 10 | 67 | 2 |
math | The vertices of a right-angled triangle lie on the ellipse \(x^2 + 4y^2 = 4\). One vertex, specifically the right-angle vertex, is the centroid of the triangle. Determine the area of this triangle. | 1 | 50 | 1 |
math | Given vectors $\overrightarrow{m} =\left(\begin{matrix} \sqrt{3}\sin \frac{x}{4},1 \end{matrix}\right)$, $\overrightarrow{n} =\left(\begin{matrix}\cos \frac{x}{4},\cos ^{2} \frac{x}{4} \end{matrix}\right)$.
$(1)$ If $\overrightarrow{m} \perp \overrightarrow{n}$, find the value of $\cos \left(\begin{matrix} \frac{2\pi}... | \left(1, \frac{3}{2} \right) | 205 | 15 |
math | Given that line $l\_1$ is tangent to the circle $x^{2}+y^{2}+2y=0$ and is parallel to line $l\_2$: $3x+4y-6=0$, find the equation of line $l\_1$. | 3x+4y-1=0 \text{ or } 3x+4y+9=0 | 60 | 24 |
math | Determine the product of the sum of all numbers of the form $2k+1$ where $k$ ranges from $0$ to $n$, and the sum of the first $n$ even numbers.
- **A)** $n^3(n+1)$
- **B)** $(n+1)^3$
- **C)** $n^4$
- **D)** $(n+1)^3n$ | (n+1)^3n | 91 | 6 |
math | Give the power function $y=(m^{2}-m-1)x^{m^{2}-2m- \frac {1}{3}}$, if it is a decreasing function when $x \in (0, +\infty)$, then the analytic expression of this power function is \_\_\_\_\_\_ . | y = x^{- \frac {1}{3}} | 68 | 11 |
math | Given $\sin x = -\frac{4}{5}$, find the value of $\sin(x+\pi)$. | \frac{4}{5} | 25 | 7 |
math | Let $P(x)$ and $Q(x)$ be distinct, non-constant polynomials such that
\[P(Q(x)) = P(x) Q(x)\]for all $x.$ If $P(1) = P(-1) = 100,$ then find the polynomial $Q(x).$ | x^2 | 66 | 3 |
math | Choose one of the following three conditions to supplement the question below and complete the solution.<br/>①$a+c\sin B=b\cos C$；②$\sqrt{2}a=\sqrt{2}bcosC-c$；③$c+2b\cos B\sin C=0$.<br/>Let the sides opposite angles $A$, $B$, and $C$ of triangle $\triangle ABC$ be $a$, $b$, and $c$, respectively. Given _____.<br/>$(1)$... | \frac{1}{2} | 162 | 7 |
math |
Let \( n \) be a fixed integer, \( n \geq 2 \).
1. Determine the smallest constant \( c \) such that the inequality \[
\sum_{1 \leq i < j \leq n} x_i x_j (x_i^2 + x_j^2) \leq c \left( \sum_{i=1}^n x_i \right)^4
\] holds for all non-negative real numbers \( x_1, x_2, \cdots, x_n \).
2. For this constant \( c \), det... | \frac{1}{8} | 135 | 7 |
math | Consider the following multiplicative magic square, where each row, column, and diagonal product equals the same value:
$\begin{tabular}{|c|c|c|} \hline 75 & \textit{b} & \textit{c} \\ \hline \textit{d} & \textit{e} & \textit{f} \\ \hline \textit{g} & \textit{h} & 3 \\ \hline \end{tabular}$
All entries are positive i... | 150 | 161 | 3 |
math | For which pairs of numbers \( (x, y) \) is the equation
$$
3 \sin x - 4 \cos x = 4 y^2 + 4 y + 6
$$
satisfied? | (x, y) = \left(-\arccos\left(-\frac{4}{5}\right) + (2k + 1)\pi, -\frac{1}{2} \right) | 48 | 45 |
math | A line passing through the focus $F$ of the parabola $y^2=2px$ ($p>0$) intersects the parabola at points $A$ and $B$. Given that $|AF|=3$ and $|BF|=2$, find the value of $p$. | \frac{12}{5} | 64 | 8 |
math | One less than the reciprocal of a certain number is $\frac{5}{4}$. What is the original number expressed as a common fraction? | -4 | 29 | 2 |
math | If the equation $kx^{2}-2x-3=0$ has two distinct real roots with respect to $x$, then the range of values for $k$ is ____. | k > -\frac{1}{3} \text{ and } k \neq 0 | 39 | 21 |
math | Given a box containing 2 red marbles, 2 green marbles, and 2 yellow marbles, find the probability that Cheryl gets 2 marbles of the same color after Carol takes 2 marbles at random and Claudia takes 2 of the remaining marbles at random. | \frac{1}{5} | 59 | 7 |
math | Determine all positive integers \( n \) such that \( n = [d(n)]^2 \), where \( d(n) \) denotes the number of positive divisors of \( n \) (including 1 and \( n \) itself).
(Canadian Mathematical Olympiad, 1999) | n = 1 | 65 | 4 |
math | What is the least positive integer $n$ such that $n^2 - n$ is divisible by some but not all integer values of $k$ when $1 \le k \le n+1$? | 5 | 44 | 1 |
math | Find the product of the digits of the largest integer satisfying these conditions:
- The sum of the squares of its digits is $45$.
- Each digit is larger than the one to its left. | 18 | 41 | 2 |
math | If $p, q,$ and $r$ are three non-zero integers such that $p + q + r = 26$ and\[\frac{1}{p} + \frac{1}{q} + \frac{1}{r} + \frac{360}{pqr} = 1,\] compute $pqr$. | 576 | 74 | 3 |
math | What is the smallest positive value of $x$ such that $x + 4321$ results in a palindrome and that palindrome is at least 200 more than 4321? | 233 | 43 | 3 |
math | Find the interval in which the solution to the equation $\frac{1}{x-1}-\log_{2}(x-1)=0$ lies. | (2,3) | 32 | 5 |
math | To understand how well students comply with the "People's Republic of China Traffic Safety Law," a survey department conducted the following random survey in a school: Two questions were asked to the respondents: (1) Is your student ID an odd number? (2) Have you ever run a red light at an intersection? The respondents... | 60 | 198 | 2 |
math | The terms $250, b, \frac{81}{50}$ are the first, second, and third terms, respectively, of a geometric sequence. If $b$ is positive, what is the value of $b$? | 9\sqrt{5} | 51 | 6 |
math | Find the larger of the two distinct solutions to the equation $$2x^2 - 14x - 84 = 0.$$ | 14 | 29 | 2 |
math | Paulinho was studying the Greatest Common Divisor (GCD) at school and decided to practice at home. He called $a, b,$ and $c$ the ages of three people living with him. He then performed operations with their prime factors and obtained the greatest common divisors of the three pairs of numbers. A few days later, he forgo... | (15, 60, 20) | 192 | 12 |
math | Given that the sum of the first $n$ terms of the sequence $\{a\_n\}$ is $S\_n=n^2-4n+1$, find the value of $|a\_1|+|a\_2|+...+|a\__{10}|$. | 67 | 63 | 2 |
math | Given that point $P$ is a moving point on the parabola $x^{2}=2y$, and the focus is $F$. If the fixed point is $M(1,2)$, calculate the minimum value of $(|PM|+|PF|)$ when point $P$ moves on the parabola. | \frac{5}{2} | 69 | 7 |
math | Forty-five teams play a tournament in which every team plays every other team exactly once. No ties occur, and each team has a $50 \%$ chance of winning any game it plays. The probability that no two teams win the same number of games is $\frac mn,$ where $m$ and $n$ are relatively prime positive integers. Find $\log_2... | \log_2 n = 949 | 78 | 10 |
math | If the range of the function $f(x) = \log(ax^2+2x+1)$ is $\mathbb{R}$, then the range of possible values for $a$ is ______. | [0, 1] | 43 | 6 |
math | Let $f(x) = ax^3 - x^2 + x + 2$ and $g(x) = \frac{e\ln x}{x}$, for all $x_1 \in (0, 1]$ and $x_2 \in (0, 1]$ such that $f(x_1) \geq g(x_2)$, then the range of the real number $a$ is \_\_\_\_\_\_. | [-2, +\infty) | 98 | 8 |
math | There are \( n \) light bulbs arranged in a row, and some of them are initially turned on. Each minute, the light bulbs that were on the previous minute turn off, while the off light bulbs that were neighbors to exactly one lit bulb the previous minute turn on. For which values of \( n \) is it possible to initially tu... | n \geq 3 | 91 | 6 |
math | A technology company encountered a technical challenge and urgently formed two teams, Team A and Team B, to work on the problem separately for a month. It was decided that the team successfully solving the technical problem at the end of the period would receive a reward. It is known that the probability of Team A solv... | \frac{7}{12} | 212 | 8 |
math | Given that $\overrightarrow{a}$, $\overrightarrow{b}$, and $\overrightarrow{c}$ are unit vectors, the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is $60^{\circ}$, find the maximum value of $(\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c})\cdot\overrightarrow{c}$. | \sqrt{3}+1 | 85 | 7 |
math | Let $n\geq 2$ be an integer. Determine all sets of real numbers $(a_1, a_2,\dots, a_n)$ such that $a_1-2a_2, a_2-2a_3,\dots, a_{n-1}-2a_n, a_n-2a_1$ are rearrangements of $a_1, a_2,\dots, a_n$ . | a_1 = a_2 = \cdots = a_n = 0 | 101 | 17 |
math | Let the three-digit number \( n = abc \). If \( a, b, \) and \( c \) as the lengths of the sides can form an isosceles (including equilateral) triangle, determine the number of such three-digit numbers \( n \). | 165 | 56 | 3 |
math | Given the function $f(x) = (\sin x + \cos x)^2 + \cos 2x$,
(1) Find the minimum positive period of $f(x)$;
(2) Determine the monotonic intervals of $f(x)$ on the interval $\left[0, \frac{\pi}{2}\right]$. | \left[0, \frac{\pi}{8}\right] | 70 | 14 |
math | Let $\boxed{N}$ mean the number of whole number divisors of $N$. For example, $\boxed{3}=2$, because $3$ has two divisors, $1$ and $3.$ Find the value of \[\boxed{\boxed{11}\times\boxed{20}}\] | 12 | 66 | 2 |
math | Let $f(x)=ax^2+bx+c$, where $a$, $b$, and $c$ are integers. Suppose that $f(2)=0$, $30<f(5)<40$, $50<f(6)<60$, and $1000k<f(50)<1000(k+1)$ for some integer $k$. Find the value of $k$. | 7 | 88 | 1 |
math | The hypotenuse of a right triangle measures 15 inches, and one angle is $45^{\circ}$. What is the number of square inches in the area of the triangle? | 112.5 | 40 | 5 |
math | Let $ABCD$ be a trapezoid with $AB\parallel DC$ . Let $M$ be the midpoint of $CD$ . If $AD\perp CD, AC\perp BM,$ and $BC\perp BD$ , find $\frac{AB}{CD}$ .
[i]Proposed by Nathan Ramesh | \frac{\sqrt{2}}{2} | 82 | 10 |
math | When dividing a two-digit number by 6, the remainder is the digit in its tens place. When dividing the same number by 10, the quotient is 3, and the remainder is the digit in the units place. Find all such two-digit numbers. | 33, 39 | 54 | 6 |
math | Sector $OAB$ is a third of a circle of radius 5 cm. A circle is drawn inside this sector, tangent at three points. Find the number of centimeters in the radius of the inscribed circle. Express your answer in simplest radical form. | \frac{5(\sqrt{3}-1)}{2} \text{ centimeters} | 53 | 20 |
math | Given $r$ as the resulting number when both the base and the exponent of $a^b$ are doubled, if $r$ equals the product of $a^{2b}$ and $y^{2b}$, where all variables are positive, determine the value of $y$.
A) $1$
B) $2$
C) $4^{b}$
D) $2^{2b}$
E) $a^{b}$ | 2 | 96 | 1 |
math | The solution set of the inequality $\dfrac{x}{{x}^{2}-8x+15}\geqslant 2$ is . | \left[ \dfrac{5}{2},3\right) \cup (5,6] | 32 | 22 |
math | The car engine operates with a power of \( P = 60 \text{ kW} \). Determine the car's speed \( v_0 \) if, after turning off the engine, it stops after traveling a distance of \( s = 450 \text{ m} \). The force resisting the car's motion is proportional to its speed. The mass of the car is \( m = 1000 \text{ kg} \). | 30 | 95 | 2 |
math | Arrange 25 numbers in a 5 by 5 grid:
\(a\_{11}\) \(a\_{12}\) \(a\_{13}\) \(a\_{14}\) \(a\_{15}\)
\(a\_{21}\) \(a\_{22}\) \(a\_{23}\) \(a\_{24}\) \(a\_{25}\)
\(a\_{31}\) \(a\_{32}\) \(a\_{33}\) \(a\_{34}\) \(a\_{35}\)
\(a\_{41}\) \(a\_{42}\) \(a\_{43}\) \(a\_{44}\) \(a\_{45}\)
\(a\_{51}\) \(a\_{52}... | -11 | 310 | 3 |
math | Given the function $f\left(x\right)=\sin \left(2x+\frac{π}{6}\right)$, its graph is shifted to the left by $\frac{π}{4}$ units and then down by $1$ unit to obtain the graph of function $g\left(x\right)$. Find one of the symmetry centers of $g\left(x\right)$. | \left(\frac{π}{6},-1\right) | 83 | 14 |
math | Compare the magnitude of $0.4^4$, $1$, and $4^{0.4}$. | 0.4^4 < 1 < 4^{0.4} | 23 | 16 |
math | A book with 63 pages numbered 1 to 63 has its pages renumbered in reverse, from 63 to 1. For how many pages do the new page number and old page number share the same units digit? | 13 | 50 | 2 |
math | $(1)$ Given the function $f(x) = |x+1| + |2x-4|$, find the solution to $f(x) \geq 6$;<br/>$(2)$ Given positive real numbers $a$, $b$, $c$ satisfying $a+2b+4c=8$, find the minimum value of $\frac{1}{a} + \frac{1}{b} + \frac{1}{c}$. | \frac{11+6\sqrt{2}}{8} | 98 | 15 |
math | For certain real numbers $a$, $b$, and $c$, the polynomial \[g(x) = x^3 + ax^2 + x + 10\]has three distinct roots, and each root of $g(x)$ is also a root of the polynomial \[f(x) = x^4 + x^3 + bx^2 + 100x + c.\]What is $f(1)$? | -7007 | 92 | 5 |
math | Let the function $f(x)=(2x^{2}-4ax)\ln x$, where $a\in\mathbb{R}$.
$(1)$ When $a=1$, find the equation of the tangent line to the curve $y=f(x)$ at the point $(1,f(1))$.
$(2)$ If for any $x\in[1,+\infty)$, $f(x)+x^{2}-a > 0$ always holds, find the range of the real number $a$. | a < 1 | 110 | 4 |
math | In a regular tetrahedron the centers of the four faces are the vertices of a smaller tetrahedron. The ratio of the volume of the smaller tetrahedron to that of the larger is $m/n$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
| 28 | 66 | 2 |
math | All positive integers whose binary representations (excluding leading zeroes) have at least as many 1's as 0's are put in increasing order. Compute the number of digits in the binary representation of the 200th number. | 9 | 47 | 1 |
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