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 | Let the complex number z = 2 + i, then calculate |z - \bar{z}|. | 2 | 22 | 1 |
math | Find the sum of all integral values of $c$ with $c \leq 30$ for which the equation $y = x^2 - 9x - c$ has two rational roots. | -28 | 43 | 3 |
math | A phone number is structured as $\text{LMN-OPQ-RSTU}$ where each letter stands for a distinct digit. The digits within each segment are arranged in increasing order: $L < M < N$, $O < P < Q$, $R < S < T < U$. The middle segment $OPQ$ consists of consecutive odd digits; $RSTU$ comprises consecutive even digits; and $L +... | 1 | 105 | 1 |
math | Given a triangle \( \triangle ABC \) with area \( S \) and \(\angle C = \gamma \), find the minimum value of the side \( C \) opposite to \(\angle C\). | 2 \sqrt{S \tan\left(\frac{\gamma}{2}\right)} | 44 | 18 |
math | The sequence $(y_n)$ is defined by $y_1 = 100$ and $y_k = y_{k - 1}^2 + 2y_{k - 1} + 1$ for all $k \ge 2$. Compute
\[
\frac{1}{y_1 + 1} + \frac{1}{y_2 + 1} + \frac{1}{y_3 + 1} + \dotsb.
\] | \frac{1}{101} | 105 | 9 |
math | Find all real $x$ such that \[\left\lfloor 2x \lfloor x \rfloor \right\rfloor = 58.\] | [5.8, 5.9) | 35 | 10 |
math | Consider the numbers $\{24,27,55,64,x\}$ . Given that the mean of these five numbers is prime and the median is a multiple of $3$ , compute the sum of all possible positive integral values of $x$ . | 60 | 58 | 2 |
math | Let $P(n)$ denote the product of the digits of the number $n$. For example, $P(58) = 5 \times 8 = 40$ and $P(319) = 3 \times 1 \times 9 = 27$.
(a) What are the natural numbers less than 1000 whose product of their digits is 12, i.e., the natural numbers $n < 1000$ such that $P(n) = 12$?
(b) How many natural numbers l... | 249 | 201 | 3 |
math | Given the ellipse $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1(a > b > 0)$ with left and right foci $F\_1$ and $F\_2$, a point $P$ moves on the ellipse. The maximum value of $| \overrightarrow{PF}\_{1}|×| \overrightarrow{PF\_{2}}|$ is $m$, and the minimum value of $ \overrightarrow{PF\_{1}}⋅ \overrightarrow{PF\_{2}}$ is... | [\frac{1}{2},1) | 150 | 9 |
math | Let the 800-digit number $n$ be formed by writing 100 $A$s side by side, where $A=20072009$. Determine the remainder when $n$ is divided by 11. | 1 | 53 | 1 |
math | When every vector on the line $y = 3x - 1$ is projected onto a certain vector $\mathbf{w},$ the result is always the vector $\mathbf{p}.$ Find the vector $\mathbf{p}.$ | \begin{pmatrix} 3/10 \\ -1/10 \end{pmatrix} | 53 | 23 |
math | Find all values of the parameter $m$ such that the equations $x^2 = 2^{|x|} + |x| - y - m = 1 - y^2$ have only one root. | m = 0 | 50 | 5 |
math | Determine the smallest positive integer $ n$ such that there exists positive integers $ a_1,a_2,\cdots,a_n$ , that smaller than or equal to $ 15$ and are not necessarily distinct, such that the last four digits of the sum,
\[ a_1!\plus{}a_2!\plus{}\cdots\plus{}a_n!\]
Is $ 2001$ . | 3 | 93 | 1 |
math | The coordinates of $A, B,$ and $C$ are $(4,5), (3,2)$, and $(0,k)$ respectively. Find the value of $k$ that minimizes the sum $\sqrt{(0-4)^2 + (k-5)^2} + \sqrt{(0-3)^2 + (k-2)^2}$. | -7 | 77 | 2 |
math | Given that $P$ is a point on the parabola $y^2=4x$, let the distance from $P$ to the directrix be $d_1$, and the distance from $P$ to point $A(1, 4)$ be $d_2$. Find the minimum value of $d_1+d_2$. | 4 | 74 | 1 |
math | There is a moving point \( M \) on the base \( A_{1}B_{1}C_{1}D_{1} \) of the cube \( ABCD - A_{1}B_{1}C_{1}D_{1} \), and \( BM \parallel \) plane \( ADC \). Find the maximum value of \( \tan \angle D_{1}MD \). | \sqrt{2} | 85 | 5 |
math | Given that $α \in (0, \dfrac{\pi}{2})$ and $β \in (\dfrac{\pi}{2}, π)$, with $\cos α = \dfrac{1}{3}$ and $\sin (α + β) = -\dfrac{3}{5}$, find the value of $\cos β$. | \cos β = -\dfrac{4 + 6\sqrt{2}}{15} | 74 | 22 |
math | A person drives z miles due east at a speed of 3 minutes per mile, and then returns to the starting point driving due west at 3 miles per minute. Determine the average speed for the entire round trip in miles per hour. | 36 | 48 | 2 |
math | What is the sum of all positive integers $\nu$ for which $\mathop{\text{lcm}}[\nu, 18] = 90$? | 195 | 34 | 3 |
math | During the "Eleven" holiday, Xiaoming and his parents drove to a scenic spot 300 km away from home. Before departure, the car's fuel tank contained 60 liters of fuel. When they had driven 100 km, they found that there were 50 liters of fuel left in the tank (assuming that the fuel consumption of the car during the jour... | 34 \, \text{liters} | 152 | 10 |
math | $(1)$ Factorization: $x^{2}(x-3)+y^{2}(3-x)$;<br/>$(2)$ Simplify: $\frac{2x}{5x-3}÷\frac{3}{25x^2-9}•\frac{x}{5x+3}$;<br/>$(3)$ Solve the system of inequalities: $\left\{\begin{array}{l}\frac{x-3}{2}+3≥x+1\\ 1-3(x-1)＜8-x\end{array}\right.$. | -2 \lt x \leqslant 1 | 120 | 12 |
math | Line segment $\overline{AB}$ is a diameter of a circle with $AB = 36$. Point $C$, not equal to $A$ or $B$, lies on the circle in such a manner that $\overline{AC}$ subtends a central angle less than $180^\circ$. As point $C$ moves within these restrictions, what is the area of the region traced by the centroid (center ... | 18\pi | 97 | 4 |
math | Given in the geometric sequence $\{a_n\}$, $a_3+a_6=6$, $a_5+a_8=9$, calculate the value of $a_7+a_{10}$. | \frac{27}{2} | 46 | 8 |
math | A rectangular yard has two flower beds in the form of congruent isosceles right triangles. The rest of the yard is a trapezoid. The parallel sides of the trapezoid measure $18$ and $30$ meters. Determine the fraction of the yard occupied by the flower beds. | \frac{1}{5} | 65 | 7 |
math | Let $x,$ $y,$ $z$ be positive real number such that $xyz = \frac{2}{3}.$ Compute the minimum value of
\[x^2 + 6xy + 18y^2 + 12yz + 4z^2.\] | 18 | 62 | 2 |
math | An equilateral triangle $ABC$ is divided by nine lines parallel to $BC$ into ten bands that are equally wide. We colour the bands alternately red and blue, with the smallest band coloured red. The difference between the total area in red and the total area in blue is $20$ $\text{cm}^2$ .
What is the area of tri... | 200 | 87 | 3 |
math | Simplify the expression $\dfrac{\sin(2\pi-\alpha)\cos(\pi+\alpha)\cos(\frac{\pi}{2}+\alpha)\cos(\frac{11\pi}{2}-\alpha)}{\cos(\pi-\alpha)\sin(3\pi-\alpha)\sin(-\pi-\alpha)\sin(\frac{9\pi}{2}+\alpha)\tan(\pi+\alpha)}$. | -1 | 89 | 2 |
math | Let $f(x)$ and $g(x)$ be functions satisfying $f(g(x)) = x^4$ and $g(f(x)) = x^6$ for all $x \ge 1.$ If $g(81) = 81,$ compute $[g(3)]^6$. | 81 | 65 | 2 |
math | If the graph of the function $f(x) = |x+m| + |nx+1|$ is symmetric about $x=2$, then the set $\{x | x = m+n\} = \quad$. | \{-4\} | 46 | 5 |
math | Given that $F_1$ and $F_2$ are the common foci of an ellipse and a hyperbola, $P$ is their common point, and $\angle F_1 P F_2 = \frac{\pi}{3}$, find the minimum value of the product of the eccentricities of the ellipse and the hyperbola. | \frac{\sqrt{3}}{2} | 74 | 10 |
math | Given $a, b, c \in (0, 1)$, and $ab + bc + ac = 1$, find the minimum value of $\dfrac{1}{1-a} + \dfrac{1}{1-b} + \dfrac{1}{1-c}$. | \dfrac{9+3\sqrt{3}}{2} | 62 | 14 |
math | Suppose \(a\) and \(b\) are positive integers. Isabella and Vidur both fill up an \(a \times b\) table. Isabella fills it up with numbers \(1, 2, \ldots, ab\), putting the numbers \(1, 2, \ldots, b\) in the first row, \(b+1, b+2, \ldots, 2b\) in the second row, and so on. Vidur fills it up like a multiplication table, ... | 21 | 164 | 2 |
math | Find all integers \( n \) such that \( 3n + 7 \) divides \( 5n + 13 \). | \{ -2, -3, -1 \} | 29 | 12 |
math | Given \( a_{n} = \mathrm{C}_{200}^{n} \cdot (\sqrt[3]{6})^{200-n} \cdot \left( \frac{1}{\sqrt{2}} \right)^{n} \) for \( n = 1, 2, \ldots, 95 \), find the number of integer terms in the sequence \(\{a_{n}\}\). | 15 | 94 | 2 |
math | The sum of \( 2C \) consecutive even numbers is 1170. If \( D \) is the largest of them, find \( D \). | 68 | 35 | 2 |
math | Solve the system of equations:
\[
\begin{cases}
x - y + z = 1 \\
y - z + u = 2 \\
z - u + v = 3 \\
u - v + x = 4 \\
v - x + y = 5
\end{cases}
\] | (0,6,7,3,-1) | 66 | 11 |
math | A finite arithmetic progression \( a_1, a_2, \ldots, a_n \) with a positive common difference has a sum of \( S \), and \( a_1 > 0 \). It is known that if the common difference of the progression is increased by 3 times while keeping the first term unchanged, the sum \( S \) doubles. By how many times will \( S \) incr... | \frac{5}{2} | 107 | 7 |
math | Consider the following lines in the plane: $2y - 3x = 4$, $x + 4y = 1$, and $6x - 4y = 8$. Determine how many points lie at the intersection of at least two of these lines. | 1 | 57 | 1 |
math | Someone used the table of cube numbers of natural numbers to calculate the square root. They calculated the square of 2.5 according to the following scheme. Let's determine the procedure's scheme and whether it is generally valid.
Given:
\[ 3.5^3 = 42.875 \]
\[ 1.5^3 = 3.375 \]
The difference:
\[ d = 39.5 \]
\( d - ... | \frac{(x+k)^3 - (x-k)^3 - 2k^3}{6k} = x^2 | 143 | 27 |
math | In the subtraction shown, \( K, L, M \), and \( N \) are digits. What is the value of \( K+L+M+N \)?
\[
\begin{array}{llll}
5 & K & 3 & L \\
\end{array}
\]
\[
\begin{array}{r}
M & 4 & N & 1 \\
\hline
4 & 4 & 5 & 1 \\
\end{array}
\] | 20 | 102 | 2 |
math | Given the inequality $ax^2+5x-2 > 0$, its solution set is $\left\{ x \mid \frac{1}{2} < x < 2 \right\}$,
$(1)$ Find the value of $a$;
$(2)$ Find the solution set of the inequality $ax^2-5x+a^2-1 > 0$. | \left\{ x \mid -3 < x < \frac{1}{2} \right\} | 81 | 23 |
math | Let $S=\{1,2,4,8,16,32,64,128,256\}$. A subset $P$ of $S$ is called squarely if it is nonempty and the sum of its elements is a perfect square. A squarely set $Q$ is called super squarely if it is not a proper subset of any squarely set. Find the number of super squarely sets. | 5 | 90 | 1 |
math | Given the function $f(x)=a\ln x- \frac{x}{2}$ has an extreme value at $x=2$.
$(1)$ Find the value of the real number $a$;
$(2)$ When $x > 1$, $f(x)+ \frac{k}{x} < 0$ always holds, find the range of the real number $k$. | (-\infty, \frac{1}{2}] | 80 | 12 |
math | The highest degree term of the polynomial $-\frac{4}{5}x^{2}y+\frac{2}{3}x^{4}y^{2}-x+1$ is ______, and the coefficient of the linear term is ______. | -1 | 52 | 2 |
math | Given the function $y=x^{2}+1$, find:
$(1)$ The equation of the tangent line at the point $(1,2)$;
$(2)$ The equation of the tangent line passing through the point $(1,1)$. | 4x-y-3=0 | 53 | 7 |
math | Let $g(n)$ be a function that, given an integer $n$, returns an integer $k$, where $k$ is the smallest possible integer such that $k!$ is divisible by $n$. If $n$ is a multiple of 24, what is the smallest value of $n$ such that $g(n) > 24$? | n = 696 | 76 | 6 |
math | Given a function $f(x) = \ln x + \frac{k}{x}$, where $k \in \mathbb{R}$.
1. If the tangent line to the curve $y=f(x)$ at the point $(e, f(e))$ is perpendicular to the line $x-2=0$, find the interval where $f(x)$ is strictly decreasing and its minimum value (where $e$ is the base of the natural logarithm).
2. If for any... | \left[\frac{1}{4}, +\infty\right) | 147 | 16 |
math | Each of $2011$ boxes in a line contains two red marbles, and for $1 \le k \le 2011$, the box in the $k\text{th}$ position also contains $k+1$ white marbles. Liam begins at the first box and successively draws a single marble at random from each box, in order. He stops when he first draws a red marble. Let $Q(n)$ be the... | 62 | 164 | 2 |
math | Define the munificence of a polynomial $p(x)$ as the maximum value of $|p(x)|$ on the interval $-1 \le x \le 1.$ For a monic cubic polynomial of the form $p(x) = x^3 + bx^2 + cx + d$, find the smallest possible munificence. | 1 | 71 | 1 |
math | Arrange 6 balls in a line, where balls 1, 2, and 3 are black and balls 4, 5, and 6 are white. If you swap balls 2 and 5, the 6 balls become alternately black and white. Now, given 20 balls in a line, where balls 1 to 10 are black and balls 11 to 20 are white, what is the minimum number of swaps needed to arrange these ... | 5 | 111 | 1 |
math | Four French people, two representatives from the Republic of Côte d'Ivoire, three English people, and four Swedish people gathered at a conference to discuss forestry issues. At the opening ceremony, participants from the same country sat next to each other on a 13-seat bench. Later, at the working sessions, they sat a... | 41472 | 95 | 5 |
math | There are 3 different pairs of shoes in a shoe cabinet. If one shoe is picked at random from the left shoe set of 6 shoes, and then another shoe is picked at random from the right shoe set of 6 shoes, calculate the probability that the two shoes form a pair. | \frac{1}{3} | 59 | 7 |
math | Find $x$ such that $\lceil x \rceil \cdot x = 168$. Express $x$ as a decimal. | 12.9231 | 30 | 7 |
math | A rectangular photograph with dimensions 30 cm and 4 dm was enlarged multiple times to create a rectangular billboard. The area of the billboard is 48 square meters. What are its length and width? | 6 \text{ m} \text{ (width)}, 8 \text{ m} \text{ (length)} | 42 | 25 |
math | Given the function $f(x)= \begin{cases} 2x+a,x < 1 \\ -x-2a,x\geqslant 1\\ \end{cases}$, find the value of $a$ such that $f(1-a)=f(1+a)$. | -\frac{3}{4} | 62 | 7 |
math | Given the function $f(x)=2\cos^2(\omega x)-1+2\sqrt{3}\cos(\omega x)\sin(\omega x)$ $(0 < \omega < 1)$, the line $x= \frac{\pi}{3}$ is an axis of symmetry for the graph of $f(x)$.
(1) Find the value of $\omega$;
(2) The graph of the function $y=g(x)$ is obtained by stretching the horizontal coordinates of the points ... | \sin(\alpha) = \frac{4\sqrt{3}-3}{10} | 188 | 20 |
math | The hyperbola and the ellipse, both centered at the origin \\(O\\) and symmetric with respect to the coordinate axes, have a common focus. Points \\(M\\) and \\(N\\) are the two vertices of the hyperbola. If \\(M\\), \\(O\\), and \\(N\\) divide the major axis of the ellipse into four equal parts, then the ratio of the ... | 2 | 99 | 1 |
math | In the Cartesian coordinate system $xoy$, the parametric equations of line $\mathit{l}$ are
$$
\begin{cases}
x= \frac {1}{2}t \\
y= \frac { \sqrt {2}}{2}+ \frac { \sqrt {3}}{2}t
\end{cases}
$$
($t$ is the parameter). If we establish a polar coordinate system with the origin $O$ of the Cartesian coordinate system $xoy$ ... | \frac {\sqrt {10}}{2} | 179 | 11 |
math | If the graph of the function $f(x)=(x^{2}-4)(x^{2}+ax+b)$ is symmetric about the line $x=-1$, find the values of $a$ and $b$, and the minimum value of $f(x)$. | -16 | 55 | 3 |
math | Let S<sub>k</sub> = 1/(k+2) + 1/(k+3) + 1/(k+4) + ... + 1/(2k-1), where k ≥ 3, k ∈ N*. Find the value of S<sub>k+1</sub>. | S_k + \frac{1}{2k} + \frac{1}{2k+1} - \frac{1}{k+2} | 66 | 32 |
math | On the AMC 8 contest Billy answers 13 questions correctly, answers 7 questions incorrectly and doesn't answer the last 5. What is his score? | 13 | 33 | 2 |
math | In a country, coins have the following thicknesses: Type A, 2.1 mm; Type B, 1.8 mm; Type C, 1.2 mm; Type D, 2.0 mm. If a stack of these coins is exactly 18 mm high, how many coins are in the stack? | 9 \text{ coins of Type D} | 69 | 9 |
math | To understand the audience's evaluation of the skit "Pit" in the 2023 CCTV Spring Festival Gala, a certain organization randomly selected 10 audience members to rate it, as shown in the table below:
| Audience Number | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
|-----------------|-----|-----|-----|---... | 0.63 | 247 | 4 |
math | Solve in $\mathbb{Z}$ the following equations:
(i) $x-1 \mid x+3$.
(ii) $x+2 \mid x^{2}+2$ | \{-8, -5, -4, -3, -1, 0, 1, 4\} | 41 | 26 |
math | Given that the vision data recorded by the five-point recording method satisfies the equation $L=5+\lg V$, and a student's vision test data using the decimal recording method is $0.8$, calculate the student's vision data using the five-point recording method. | 4.9 | 54 | 3 |
math | Given an arithmetic sequence $\{a_{n}\}$ with a non-zero common difference, where $a_{2}$ is the geometric mean of $a_{1}$ and $a_{4}$, and $a_{5}+a_{6}=11$.<br/>$(Ⅰ)$ Find the general formula for the sequence $\{a_{n}\}$;<br/>$(Ⅱ)$ Choose one of the following conditions as known to find the sum of the first $n$ terms ... | S_{n} = \frac{2n^2 + 2n}{2n+1} | 195 | 22 |
math | For how many integers $n$ where $2 \le n \le 100$ is $\binom{n}{2}$ odd? | 50 | 30 | 2 |
math | Given two lines $l_{1}$: $(a-1)x+2y+1=0$, $l_{2}$: $x+ay+1=0$, find the value of $a$ that satisfies the following conditions:
$(1) l_{1} \parallel l_{2}$
$(2) l_{1} \perp l_{2}$ | \frac{1}{3} | 78 | 7 |
math | Consider the infinite series: $1 - \frac{1}{3} - \frac{1}{9} + \frac{1}{27} - \frac{1}{81} - \frac{1}{243} + \frac{1}{729} - \cdots$. Let $T$ be the limiting sum of this series. Find $T$.
**A)** $\frac{3}{26}$
**B)** $\frac{15}{26}$
**C)** $\frac{27}{26}$
**D)** $\frac{1}{26}$
**E)** $\frac{40}{26}$ | \frac{15}{26} | 145 | 9 |
math | A coin is flipped ten times, and the sequence of heads and tails occurring is recorded. However, the sequence must start with at least two heads. How many distinct sequences are possible? | 256 | 37 | 3 |
math | Given the function $f(x)=ax^{3}-3x^{2}+1-\frac{3}{a}$, determine the range of values for the real number $a$ such that $f(x)$ has 3 zeros. | (-1,0)\cup(3,4) | 49 | 11 |
math | Rectangle $ABCD$ has sides $AB=4$ and $AD=3$. Triangle $ADC$ is folded along $AC$ to form triangle $AD'C$ such that plane $AD'C$ is perpendicular to plane $ABC$. Point $F$ is the midpoint of $AD'$, and point $E$ is a point on $AC$. Consider the following statements:
1. There exists a point $E$ such that $EF\parallel$... | 1, 3 | 192 | 4 |
math | In a right triangle with integer length sides, the hypotenuse has length 65 units. How many units is the length of the shorter leg, assuming a unique solution exists? | 25 | 37 | 2 |
math | Let $S_n$ be the sum of the first $n$ terms of the sequence $\{a_n\}$, given that $a_1=3$, $a_{n+1}=2S_n+3$.
$(1)$ Find the general formula for the sequence $\{a_n\}$.
$(2)$ Let $b_n=(2n-1)a_n$, find the sum of the first $n$ terms of the sequence $\{b_n\}$, denoted as $T_n$. | T_n=3+(n-1)\cdot3^{n+1} | 108 | 16 |
math | For the "Skillful Hands" club, Pavel needs to cut several identical pieces of wire (the length of each piece is an integer number of centimeters). Initially, Pavel took a wire piece of 10 meters and was able to cut only 15 pieces of the required length. Then, Pavel took a piece that was 40 centimeters longer, but it wa... | 66 | 100 | 2 |
math | fantasticbobob is proctoring a room for the SiSiEyMB with $841$ seats arranged in $29$ rows and $29$ columns. The contestants sit down, take part $1$ of the contest, go outside for a break, and come back to take part $2$ of the contest. fantasticbobob sits among the contestants during part $1$ , also goes ou... | 421 | 167 | 3 |
math | Given circles of radius $5$ and $8$ are drawn on a sheet of paper, find the number of different values of $k$, where $k\geq 0$ represents the number of tangent lines that can be drawn without allowing the circles to overlap. | 3 | 55 | 1 |
math | Determine the force of water pressure on the wall of a lock, which is $20 \mathrm{~m}$ long and $5 \mathrm{~m}$ high (assuming the lock is filled with water to the top). | 2.45 \times 10^6 \, \text{N} | 48 | 18 |
math | We have selected 7 subsets of a finite set, each containing 3 elements, such that any two distinct elements are contained in exactly one of the selected subsets.
a) How many elements does the set have?
b) What is the maximum number of selected subsets that can be chosen such that no three of them contain the same ele... | 4 | 68 | 1 |
math | How many perfect squares less than 5000 have a ones digit of 1, 2 or 4? | 28 | 25 | 2 |
math | What digit can the number \( f(x) = \lfloor x \rfloor + \lfloor 3x \rfloor + \lfloor 6x \rfloor \) end with, where \( x \) is any positive real number? Here, \( \lfloor x \rfloor \) denotes the greatest integer less than or equal to \( x \). | 0, 1, 3, 4, 6, 7 | 78 | 16 |
math | A particle is placed on the parabola $y = x^2 - x - 6$ at a point $P$ whose $y$-coordinate is $6$. Find the horizontal distance traveled by the particle, defined as the numerical value of the difference in the $x$-coordinates of $P$ and $Q$, where $Q$ is the nearest point on the parabola with a $y$-coordinate of $-6$. | 4 | 94 | 1 |
math | Given a circle M with its center on the x-axis and a radius of 1, the line $l: y=3x-1$ is intersected by circle M, producing a chord of length $\frac{2\sqrt{15}}{5}$, and the center of circle M is below line $l$.
(Ⅰ) Find the equation of circle M;
(Ⅱ) Let A(0, t), B(0, t+4) (-3≤t≤-1), and draw tangents to circle M from... | 6 | 144 | 1 |
math | Given $y^2 = 16x$ and point $A(1, 2)$, with $P$ being a point on the parabola and $F$ the focus of the parabola, find the minimum value of $|PF| + |PA|$. | 5 | 60 | 1 |
math | Using the digits 1, 2, 3, and 4, calculate the number of four-digit numbers that can be formed without repeating any digit. | 24 | 32 | 2 |
math | Given $θ∈\left( \dfrac {π}{2},π\right)$, $\dfrac {1}{\sin θ}+ \dfrac {1}{\cos θ}=2 \sqrt {2}$, then $\sin \left(2θ+ \dfrac {π}{3}\right)=$ \_\_\_\_\_\_ . | \dfrac {1}{2} | 78 | 7 |
math | In triangle $\triangle DEF$, point $L$ is located on $EF$ such that $DL$ is an altitude of $\triangle DEF$. If $DE = 14,$ $EL = 9$, and $EF = 17$, what is the area of $\triangle DEF$? | \frac{17 \sqrt{115}}{2} | 61 | 15 |
math | A circle is tangent to a circle with a radius of 2, and the distance between their centers is 5. Calculate the radius of this circle. | 3 \text{ or } 7 | 31 | 8 |
math | Ben throws six identical darts. Each hits one of five identical dartboards on the wall. After throwing the six darts, he lists the number of darts that hit each board, from greatest to least. How many different lists are possible? | 11 | 50 | 2 |
math | If the five numbers in a sample set $a$, $99$, $b$, $101$, $c$ exactly form an arithmetic sequence, then the standard deviation of this sample is $\_\_\_\_\_$. | \sqrt{2} | 47 | 5 |
math | From 4 integers, any 3 are selected to find their average. Then, the sum of this average and the remaining integer is calculated. This yields 4 numbers: $8$, $12$, $10 \frac{2}{3}$, and $9 \frac{1}{3}$. Find the sum of the original 4 integers. | 30 | 74 | 2 |
math | A rectangular piece of paper $A B C D$ is folded and flattened such that triangle $D C F$ falls onto triangle $D E F$, with vertex $E$ landing on side $A B$. Given that $\angle 1 = 22^{\circ}$, find $\angle 2$. | 44 | 64 | 2 |
math | If for $x\in (0,\frac{\pi }{2})$, the inequality $\frac{1}{{{\sin }^{2}}x}+\frac{p}{{{\cos }^{2}}x}\geqslant 9$ always holds, then the range of the positive real number $p$ is _______ | [4,+\infty) | 71 | 7 |
math | $(1)$ Calculate: $\sqrt{2}\sin45{}°-(-\frac{1}{2})^{-2}-(-1)^{2023}$;<br/>$(2)$ Simplify first, then evaluate: $(\frac{x+2}{x}-\frac{x-1}{x-2})÷\frac{x-4}{x^2-4x+4}-1$, where $x$ is the square root of $4$. | 1 | 98 | 1 |
math | (The full score of this question is 12 points)
Given the functions $f(x) = -x^2 + 8x$, $g(x) = 6\ln x + m$.
(1) Find the maximum value $h(t)$ of $f(x)$ in the interval $[t, t+1]$.
(2) Does there exist a real number $m$ such that the graph of $y = f(x)$ and the graph of $y = g(x)$ have exactly three different inte... | (7, 15 - 6\ln3) | 132 | 13 |
math | Given a circle $C$ that passes through two points $A(-2,1)$ and $B(5,0)$, and the center of the circle $C$ is on the line $y=2x$.
(1) Find the equation of the circle $C$;
(2) A moving line $l$: $(m+2)x+(2m+1)y-7m-8=0$ passes through a fixed point $M$, and a line $m$ with a slope of $1$ passes through the point $M$, i... | \sqrt{82} | 140 | 6 |
math | Given that the area of an equilateral triangle $ABC$ is 1, and $P$ is a point inside $\triangle ABC$, with the areas of $\triangle PAB$, $\triangle PBC$, and $\triangle PCA$ being equal, then the total number of points $P$ that satisfy the condition is $\boxed{1}$; the area of $\triangle PAB$ is $\boxed{\frac{1}{3}}$. | \frac{1}{3} | 89 | 7 |
math | Given that the complex number $z$ satisfies $$\frac {z-i}{z-2}=i$$, find the imaginary part of the complex number $z$. | -\frac{1}{2} | 34 | 7 |
math | Given that both $\alpha$ and $\beta$ are acute angles, and $\sin \alpha = \frac{3}{5}$, $\tan (\alpha - \beta) = -\frac{1}{3}$.
(1) Find the value of $\sin (\alpha - \beta)$;
(2) Find the value of $\cos \beta$. | \frac{9\sqrt{10}}{50} | 74 | 14 |
math | Given right triangle \( ABC \), with \( AB = 4 \), \( BC = 3 \), and \( CA = 5 \). Circle \(\omega\) passes through \( A \) and is tangent to \( BC \) at \( C \). What is the radius of \(\omega\)? | \frac{25}{8} | 64 | 8 |
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