Unnamed: 0
int64 0
40.3k
| problem
stringlengths 10
5.15k
| ground_truth
stringlengths 1
1.22k
| solved_percentage
float64 0
100
|
|---|---|---|---|
18,100 |
Fold a piece of graph paper once so that the point (0, 2) coincides with the point (4, 0), and the point (9, 5) coincides with the point (m, n). The value of m+n is ______.
|
10
| 14.84375 |
18,101 |
Let $\{b_k\}$ be a sequence of integers such that $b_1=2$ and $b_{m+n}=b_m+b_n+m^2+n^2,$ for all positive integers $m$ and $n.$ Find $b_{12}.$
|
160
| 16.40625 |
18,102 |
Suppose a sequence $\{a\_n\}$ satisfies $\frac{1}{a\_{n+1}} - \frac{1}{a\_n} = d (n \in \mathbb{N}^*, d$ is a constant), then the sequence $\{a\_n\}$ is called a "harmonic sequence". It is known that the sequence $\{\frac{1}{x\_n}\}$ is a "harmonic sequence", and $x\_1 + x\_2 + ... + x\_{20} = 200$, find the maximum value of $x\_3 x\_{18}$.
|
100
| 21.09375 |
18,103 |
There are 3 different balls to be placed into 5 different boxes, with at most one ball per box. How many methods are there?
|
60
| 67.1875 |
18,104 |
Triangles $ABC$ and $AFG$ have areas $3009$ and $9003,$ respectively, with $B=(0,0), C=(331,0), F=(800,450),$ and $G=(813,463).$ What is the sum of all possible $x$-coordinates of $A$?
|
1400
| 27.34375 |
18,105 |
Given a function $f(x)$ satisfies $f(x) + f(4-x) = 4$, $f(x+2) - f(-x) = 0$, and $f(1) = a$, calculate the value of $f(1) + f(2) + f(3) + \cdots + f(51)$.
|
102
| 67.1875 |
18,106 |
Find all positive integers $x$ such that the product of all digits of $x$ is given by $x^2 - 10 \cdot x - 22.$
|
12
| 99.21875 |
18,107 |
Given the sums of the first n terms of two arithmetic sequences $\{a_n\}$ and $\{b_n\}$ denoted as $S_n$ and $T_n$, respectively, if $\frac {S_{n}}{T_{n}} = \frac {2n}{3n+1}$, calculate the value of $\frac {a_{6}}{b_{6}}$.
|
\frac {11}{17}
| 38.28125 |
18,108 |
A herder has forgotten the number of cows she has, and does not want to count them all of them. She remembers these four facts about the number of cows:
- It has $3$ digits.
- It is a palindrome.
- The middle digit is a multiple of $4$ .
- It is divisible by $11$ .
What is the sum of all possible numbers of cows that the herder has?
|
726
| 45.3125 |
18,109 |
In rectangle $ABCD,$ $P$ is a point on side $\overline{BC}$ such that $BP = 12$ and $CP = 4.$ If $\tan \angle APD = 2,$ then find $AB.$
|
12
| 23.4375 |
18,110 |
Point $(x,y)$ is randomly picked from the rectangular region with vertices at $(0,0),(2010,0),(2010,2011),$ and $(0,2011)$. What is the probability that $x > 3y$? Express your answer as a common fraction.
|
\frac{335}{2011}
| 42.96875 |
18,111 |
If the domain of the function $f(x)=x^{2}$ is $D$, and its range is ${0,1,2,3,4,5}$, then there are \_\_\_\_\_\_ such functions $f(x)$ (answer with a number).
|
243
| 1.5625 |
18,112 |
A $40$ feet high screen is put on a vertical wall $10$ feet above your eye-level. How far should you stand to maximize the angle subtended by the screen (from top to bottom) at your eye?
|
10\sqrt{5}
| 58.59375 |
18,113 |
Given \( a, b > 0 \). Find the maximum values of \( \sqrt{\frac{a}{2a+b}} + \sqrt{\frac{b}{2b+a}} \) and \( \sqrt{\frac{a}{a+2b}} + \sqrt{\frac{b}{b+2a}} \).
|
\frac{2\sqrt{3}}{3}
| 83.59375 |
18,114 |
The coefficient of $x^2$ in the expansion of $(x+1)^5(x-2)$ is $\boxed{5}$.
|
-15
| 28.125 |
18,115 |
Kim earned scores of 86, 82, and 89 on her first three mathematics examinations. She is expected to increase her average score by at least 2 points with her fourth exam. What is the minimum score Kim must achieve on her fourth exam to meet this target?
|
94
| 54.6875 |
18,116 |
Let $M$ be the greatest four-digit number whose digits have a product of $24$. Calculate the sum of the digits of $M$.
|
13
| 64.0625 |
18,117 |
$n$ mushroom pickers went into the forest and brought back a total of 200 mushrooms (possibly, some of the pickers did not bring any mushrooms home). A boy named Petya, upon learning this, stated: "Some two of them must have brought the same number of mushrooms!" What is the smallest $n$ for which Petya is certainly right? Don't forget to justify your answer.
|
21
| 12.5 |
18,118 |
If two points are randomly selected from the eight vertices of a cube, the probability that the line determined by these two points intersects each face of the cube is ______.
|
\frac{1}{7}
| 73.4375 |
18,119 |
Calculate $\frac{1}{6} \cdot \frac{2}{7} \cdot \frac{3}{8} \cdot \frac{4}{9} \cdots \frac{94}{99} \cdot \frac{95}{100}$. Express your answer as a common fraction.
|
\frac{1}{75287520}
| 41.40625 |
18,120 |
For each value of $x$, $g(x)$ is defined to be the minimum value of the three numbers $3x + 3$, $\frac{1}{3}x + 1$, and $-\frac{2}{3}x + 8$. Find the maximum value of $g(x)$.
|
\frac{10}{3}
| 47.65625 |
18,121 |
Triangle $PQR$ has vertices $P(0, 6)$, $Q(3, 0)$, $R(9, 0)$. A line through $Q$ cuts the area of $\triangle PQR$ in half. Find the sum of the slope and $y$-intercept of this line.
|
-4
| 35.15625 |
18,122 |
Given the infinite series $1-\frac{1}{2}-\frac{1}{4}+\frac{1}{8}-\frac{1}{16}-\frac{1}{32}+\frac{1}{64}-\frac{1}{128}-\cdots$, find the limiting sum of the series.
|
\frac{2}{7}
| 53.125 |
18,123 |
There are 22 black and 3 blue balls in a bag. Ahmet chooses an integer $ n$ in between 1 and 25. Betül draws $ n$ balls from the bag one by one such that no ball is put back to the bag after it is drawn. If exactly 2 of the $ n$ balls are blue and the second blue ball is drawn at $ n^{th}$ order, Ahmet wins, otherwise Betül wins. To increase the possibility to win, Ahmet must choose
|
13
| 1.5625 |
18,124 |
Given vectors $\overrightarrow {a}$ and $\overrightarrow {b}$ that satisfy $|\overrightarrow {a}|= \sqrt {3}$, $|\overrightarrow {b}|=2$, and $(\overrightarrow {a}- \overrightarrow {b}) \perp \overrightarrow {a}$, find the projection of $\overrightarrow {a}$ on $\overrightarrow {b}$.
|
\frac {3}{2}
| 76.5625 |
18,125 |
Given vectors $\overrightarrow{a} = (4\cos \alpha, \sin \alpha)$, $\overrightarrow{b} = (\sin \beta, 4\cos \beta)$, and $\overrightarrow{c} = (\cos \beta, -4\sin \beta)$, where $\alpha, \beta \in \mathbb{R}$ and neither $\alpha$, $\beta$, nor $\alpha + \beta$ equals $\frac{\pi}{2} + k\pi, k \in \mathbb{Z}$:
1. Find the maximum value of $|\overrightarrow{b} + \overrightarrow{c}|$.
2. When $\overrightarrow{a}$ is parallel to $\overrightarrow{b}$ and perpendicular to $(\overrightarrow{b} - 2\overrightarrow{c})$, find the value of $\tan \alpha + \tan \beta$.
|
-30
| 32.03125 |
18,126 |
In the plane Cartesian coordinate system $xOy$, let two unit vectors in the directions of the $x$-axis and $y$-axis be $\overrightarrow{i}$ and $\overrightarrow{j}$, respectively. Given that $\overrightarrow{OA}=2\overrightarrow{i}+\overrightarrow{j}$ and $\overrightarrow{OB}=4\overrightarrow{i}-3\overrightarrow{j}$.
$(1)$ If point $P$ lies on the extension of line segment $AB$ and $|\overrightarrow{AP}|=\frac{3}{2}|\overrightarrow{PB}|$, find the coordinates of point $P$.
$(2)$ If point $P$ is the midpoint of line segment $AB$ and $\overrightarrow{OP}$ is perpendicular to $\overrightarrow{OA}+k\overrightarrow{OB}$, find the value of the real number $k$.
|
-\frac{1}{3}
| 86.71875 |
18,127 |
There are 7 people standing in a row. How many different arrangements are there according to the following requirements?
(1) Among them, A, B, and C cannot stand next to each other;
(2) Among them, A and B have exactly one person between them;
(3) A does not stand at the head of the row, and B does not stand at the end of the row.
|
3720
| 84.375 |
18,128 |
Given the sequence $\left\{ a_n \right\}$ such that $a_{n+1}+a_n={(-1)}^n\cdot n$ ($n\in \mathbb{N}^*$), find the sum of the first 20 terms of $\left\{ a_n \right\}$.
|
-100
| 12.5 |
18,129 |
What is the product of the numerator and the denominator when $0.\overline{018}$ is expressed as a fraction in lowest terms?
|
222
| 92.1875 |
18,130 |
How many distinct four-digit numbers are divisible by 5 and have 45 as their last two digits?
|
90
| 93.75 |
18,131 |
The lines $l_1$ and $l_2$ are two tangents to the circle $x^2+y^2=2$. If the intersection point of $l_1$ and $l_2$ is $(1,3)$, then the tangent of the angle between $l_1$ and $l_2$ equals \_\_\_\_\_\_.
|
\frac{4}{3}
| 42.1875 |
18,132 |
The sum of an infinite geometric series is $64$ times the series that results if the first four terms of the original series are removed. What is the value of the series' common ratio?
|
\frac{1}{2}
| 38.28125 |
18,133 |
In $\triangle ABC$, the sides opposite to angles A, B, and C are a, b, and c, respectively. If $C= \frac {\pi}{3}, b= \sqrt {2}, c= \sqrt {3}$, find the measure of angle A.
|
\frac{5\pi}{12}
| 76.5625 |
18,134 |
In triangle $DEF,$ $\cot D \cot F = \frac{1}{3}$ and $\cot E \cot F = \frac{1}{8}.$ Find $\tan F.$
|
12 + \sqrt{136}
| 0 |
18,135 |
The measure of angle $ACB$ is 60 degrees. If ray $CA$ is rotated 630 degrees about point $C$ in a counterclockwise direction, what will be the positive measure of the new acute angle $ACB$, in degrees?
|
30
| 62.5 |
18,136 |
Given that f(x) = |log₃x|, if a and b satisfy f(a - 1) = f(2b - 1), and a ≠ 2b, then the minimum value of a + b is ___.
|
\frac{3}{2} + \sqrt{2}
| 8.59375 |
18,137 |
Evaluate the value of $\frac{(2210-2137)^2 + (2137-2028)^2}{64}$.
|
268.90625
| 42.1875 |
18,138 |
Choose one of the three conditions in $①$ $ac=\sqrt{3}$, $②$ $c\sin A=3$, $③$ $c=\sqrt{3}b$, and supplement it in the following question. If the triangle in the question exists, find the value of $c$; if the triangle in the question does not exist, explain the reason.<br/>Question: Does there exist a $\triangle ABC$ where the internal angles $A$, $B$, $C$ have opposite sides $a$, $b$, $c$, and $\sin A=\sqrt{3}\sin B$, $C=\frac{π}{6}$, _______ $?$<br/>Note: If multiple conditions are selected to answer separately, the first answer will be scored.
|
2\sqrt{3}
| 10.15625 |
18,139 |
Find all solutions to
\[\sqrt{x + 2 - 2 \sqrt{x - 4}} + \sqrt{x + 12 - 8 \sqrt{x - 4}} = 4.\]
|
[13]
| 0 |
18,140 |
Given that there is an extra $32.13 on the books due to a misplaced decimal point, determine the original amount of the sum of money.
|
3.57
| 60.15625 |
18,141 |
Given right triangle $ABC$ with $\angle A = 90^\circ$, $AC = 3$, $AB = 4$, and $BC = 5$, point $D$ is on side $BC$. If the perimeters of $\triangle ACD$ and $\triangle ABD$ are equal, calculate the area of $\triangle ABD$.
|
\frac{12}{5}
| 25.78125 |
18,142 |
Find the maximum real number \( M \) such that for all real numbers \( x \) and \( y \) satisfying \( x + y \geqslant 0 \), the following inequality holds:
$$
\left(x^{2}+y^{2}\right)^{3} \geqslant M\left(x^{3}+y^{3}\right)(xy - x - y).
|
32
| 12.5 |
18,143 |
In $\triangle ABC$, $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$ respectively. Given that $b^2=ac$ and $a^2-c^2=ac-bc$, find the value of $$\frac{c}{b\sin B}$$.
|
\frac{2\sqrt{3}}{3}
| 42.1875 |
18,144 |
Katherine makes Benj play a game called $50$ Cent. Benj starts with $\$ 0.50 $, and every century thereafter has a $ 50\% $ chance of doubling his money and a $ 50\% $ chance of having his money reset to $ \ $0.50$ . What is the expected value of the amount of money Benj will have, in dollars, after $50$ centuries?
|
13
| 4.6875 |
18,145 |
Given the function $f\left( x \right)=\sin \left( 2x+\phi \right)\left(\left| \phi \right| < \dfrac{\pi }{2} \right)$ whose graph is symmetric about the point $\left( \dfrac{\pi }{3},0 \right)$, and $f\left( {{x}_{1}} \right)+f\left( {{x}_{2}} \right)=0$ when ${{x}_{1}},{{x}_{2}}\in \left( \dfrac{\pi }{12},\dfrac{7\pi }{12} \right)$ $\left( {{x}_{1}}\ne {{x}_{2}} \right)$, find $f\left( {{x}_{1}}+{{x}_{2}} \right)$.
|
-\dfrac{\sqrt{3}}{2}
| 27.34375 |
18,146 |
In the Cartesian coordinate system, $O$ is the origin, and points $A(-1,0)$, $B(0, \sqrt{3})$, $C(3,0)$. A moving point $D$ satisfies $|\overrightarrow{CD}|=1$, then the maximum value of $|\overrightarrow{OA}+ \overrightarrow{OB}+ \overrightarrow{OD}|$ is ______.
|
\sqrt{7}+1
| 12.5 |
18,147 |
How many positive four-digit integers of the form $\_\_90$ are divisible by 90?
|
10
| 77.34375 |
18,148 |
In triangle $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively. Given that $c=2$, $C=\dfrac{\pi }{3}$, and $\sin B=2\sin A$, find the area of $\triangle ABC$.
|
\frac{2\sqrt{3}}{3}
| 89.84375 |
18,149 |
How many times does the digit 9 appear in the list of all integers from 1 to 1000?
|
300
| 100 |
18,150 |
Consider the function $f(x) = \cos(2x + \frac{\pi}{3}) + \sqrt{3}\sin 2x + 2m$, where $x \in \mathbb{R}$ and $m \in \mathbb{R}$.
(I) Determine the smallest positive period of $f(x)$ and its intervals of monotonic increase.
(II) If $f(x)$ has a minimum value of 0 when $0 \leq x \leq \frac{\pi}{4}$, find the value of the real number $m$.
|
-\frac{1}{4}
| 57.03125 |
18,151 |
Given $(x+1)^4(x+4)^8 = a + a_1(x+3) + a_2(x+3)^2 + \ldots + a_{12}(x+3)^{12}$, find the value of $a_2 + a_4 + \ldots + a_{12}$.
|
112
| 9.375 |
18,152 |
Given vectors $\overrightarrow{a} = (\sin x, \cos x)$, $\overrightarrow{b} = (\sin x, \sin x)$, and $f(x) = \overrightarrow{a} \cdot \overrightarrow{b}$
(1) If $x \in \left[-\frac{\pi}{4}, \frac{\pi}{4}\right]$, find the range of the function $f(x)$.
(2) Let the sides opposite the acute angles $A$, $B$, and $C$ of triangle $\triangle ABC$ be $a$, $b$, and $c$, respectively. If $f(B) = 1$, $b = \sqrt{2}$, and $c = \sqrt{3}$, find the value of $a$.
|
\frac{\sqrt{6} + \sqrt{2}}{2}
| 48.4375 |
18,153 |
How many even divisors does $9!$ have?
|
140
| 80.46875 |
18,154 |
The function $f(x)=\frac{1}{3}x^{3}-ax^{2}+4$, and $x=2$ is a local minimum point of the function $f(x)$.
(1) Find the value of the real number $a$;
(2) Find the maximum and minimum values of $f(x)$ on the interval $[-1,3]$.
|
\frac{8}{3}
| 70.3125 |
18,155 |
Compute $156 + 44 + 26 + 74$.
|
300
| 88.28125 |
18,156 |
If $\alpha \in (0, \frac{\pi}{2})$, and $\tan 2\alpha = \frac{\cos \alpha}{2-\sin \alpha}$, calculate the value of $\tan \alpha$.
|
\frac{\sqrt{15}}{15}
| 72.65625 |
18,157 |
Given in the tetrahedron P-ABC, PA is perpendicular to the plane ABC, AB=AC=PA=2, and in triangle ABC, ∠BAC=120°, then the volume of the circumscribed sphere of the tetrahedron P-ABC is \_\_\_\_\_\_.
|
\frac{20\sqrt{5}\pi}{3}
| 17.96875 |
18,158 |
If the complex number $a^2 - 1 + (a - 1)i$ (where $i$ is the imaginary unit) is purely imaginary, calculate the real number $a$.
|
-1
| 16.40625 |
18,159 |
Given the function $$f(x)= \begin{cases} ( \frac {1}{2})^{x} & ,x≥4 \\ f(x+1) & ,x<4\end{cases}$$, find the value of $f(\log_{2}3)$.
|
\frac {1}{24}
| 62.5 |
18,160 |
The Experimental Primary School football team has 42 members, and 17 new members joined. How many members are in the football team now?
|
59
| 100 |
18,161 |
Let $a_n\ (n\geq 1)$ be the value for which $\int_x^{2x} e^{-t^n}dt\ (x\geq 0)$ is maximal. Find $\lim_{n\to\infty} \ln a_n.$
|
-\ln 2
| 74.21875 |
18,162 |
If the graph of the function $y=\log_a(x+m)+n$ passes through the fixed point $(-1, -2)$, then find the value of $m \cdot n$.
|
-4
| 67.96875 |
18,163 |
Let \( a_1, a_2, \dots \) be a sequence of positive real numbers such that
\[ a_n = 7a_{n-1} - n \]for all \( n > 1 \). Find the smallest possible value of \( a_1 \).
|
\frac{13}{36}
| 3.90625 |
18,164 |
Twelve million added to twelve thousand equals
|
12012000
| 93.75 |
18,165 |
Quadrilateral $EFGH$ is a parallelogram. A line through point $G$ makes a $30^\circ$ angle with side $GH$. Determine the degree measure of angle $E$.
[asy]
size(100);
draw((0,0)--(5,2)--(6,7)--(1,5)--cycle);
draw((5,2)--(7.5,3)); // transversal line
draw(Arc((5,2),1,-60,-20)); // transversal angle
label("$H$",(0,0),SW); label("$G$",(5,2),SE); label("$F$",(6,7),NE); label("$E$",(1,5),NW);
label("$30^\circ$",(6.3,2.8), E);
[/asy]
|
150
| 70.3125 |
18,166 |
$44 \times 22$ is equal to
|
$88 \times 11$
| 0 |
18,167 |
On a $123 \times 123$ board, each cell is painted either purple or blue according to the following conditions:
- Each purple cell that is not on the edge of the board has exactly 5 blue cells among its 8 neighbors.
- Each blue cell that is not on the edge of the board has exactly 4 purple cells among its 8 neighbors.
Note: Two cells are neighbors if they share a side or a vertex.
(a) Consider a $3 \times 3$ sub-board within the $123 \times 123$ board. How many cells of each color can there be in this $3 \times 3$ sub-board?
(b) Calculate the number of purple cells on the $123 \times 123$ board.
|
6724
| 10.15625 |
18,168 |
The coefficient sum of the expansion of the binomial ${{\left(\frac{1}{x}-2x^2\right)}^9}$, excluding the constant term, is $671$.
|
671
| 67.1875 |
18,169 |
If three angles \(x, y, z\) form an arithmetic sequence with a common difference of \(\frac{\pi}{2}\), then \(\tan x \tan y + \tan y \tan z + \tan z \tan x = \) ______.
|
-3
| 0 |
18,170 |
Given point $A(3,0)$, $\overrightarrow{EA}=(2,1)$, $\overrightarrow{EF}=(1,2)$, and point $P(2,0)$ satisfies $\overrightarrow{EP}=λ \overrightarrow{EA}+μ \overrightarrow{EF}$, find the value of $λ+μ$.
|
\frac{2}{3}
| 56.25 |
18,171 |
Five packages are delivered to five different houses, with each house receiving one package. If these packages are randomly delivered, what is the probability that exactly three of them are delivered to their correct houses? Express your answer as a common fraction.
|
\frac{1}{12}
| 51.5625 |
18,172 |
Given two arithmetic sequences \\(\{a_n\}\) and \\(\{b_n\}\) with the sum of the first \\(n\) terms denoted as \\(S_n\) and \\(T_n\) respectively. If \\( \dfrac {S_n}{T_n}= \dfrac {2n}{3n+1}\), then \\( \dfrac {a_2}{b_3+b_7}+ \dfrac {a_8}{b_4+b_6}=\) ______.
|
\dfrac {9}{14}
| 25 |
18,173 |
There are 99 children standing in a circle, each initially holding a ball. Every minute, each child with a ball throws their ball to one of their two neighbors. If a child receives two balls, one of the balls is irrevocably lost. What is the minimum amount of time after which only one ball can remain with the children?
|
98
| 82.03125 |
18,174 |
Given $$(1+x)(2-x)^{6}=a_{0}+a_{1}(x-1)+a_{2}(x-1)^{2}+\ldots+a_{7}(x-1)^{7}$$, find the value of $a_{3}$.
|
-25
| 32.8125 |
18,175 |
For certain real values of $p, q, r,$ and $s,$ the equation $x^4+px^3+qx^2+rx+s=0$ has four non-real roots. The product of two of these roots is $17 + 2i$ and the sum of the other two roots is $2 + 5i,$ where $i^2 = -1.$ Find $q.$
|
63
| 3.125 |
18,176 |
If $\tan (\alpha+\beta)= \frac {3}{4}$ and $\tan (\alpha- \frac {\pi}{4})= \frac {1}{2}$, find the value of $\tan (\beta+ \frac {\pi}{4})$.
|
\frac {2}{11}
| 88.28125 |
18,177 |
Given real numbers $x$ and $y$ satisfying $x^{2}+y^{2}-4x-2y-4=0$, find the maximum value of $x-y$.
|
1+3\sqrt{2}
| 85.9375 |
18,178 |
Let $n$ be an integer, and let $\triangle ABC$ be a right-angles triangle with right angle at $C$ . It is given that $\sin A$ and $\sin B$ are the roots of the quadratic equation \[(5n+8)x^2-(7n-20)x+120=0.\] Find the value of $n$
|
66
| 46.875 |
18,179 |
In triangle ABC, the sides opposite to angles A, B, and C are a, b, and c respectively. Given that $(a+c)^2 = b^2 + 2\sqrt{3}ac\sin C$.
1. Find the measure of angle B.
2. If $b=8$, $a>c$, and the area of triangle ABC is $3\sqrt{3}$, find the value of $a$.
|
5 + \sqrt{13}
| 5.46875 |
18,180 |
Find the volume of the region in space defined by
\[|x + 2y + z| + |x + 2y - z| \le 12\]
and \(x, y, z \ge 0.\)
|
54
| 72.65625 |
18,181 |
Calculate $7 \cdot 9\frac{2}{5}$.
|
65\frac{4}{5}
| 70.3125 |
18,182 |
Given a quadratic function $f(x) = ax^2 - 4bx + 1$.
(1) Let set $P = \{-1,1,2,3,4,5\}$ and set $Q = \{-2,-1,1,2,3,4\}$. Randomly select a number from set $P$ as $a$ and from set $Q$ as $b$. Calculate the probability that the function $y = f(x)$ is increasing on the interval $[1,+\infty)$.
(2) Suppose the point $(a, b)$ is a random point within the region defined by $\begin{cases} x+y-8\leqslant 0, \\ x > 0, \\ y > 0 \end{cases}$. Calculate the probability that the function $y = f(x)$ is increasing on the interval $[1,+\infty)$.
|
\dfrac{1}{3}
| 42.1875 |
18,183 |
What is the sum of all integer solutions to $4<(x-3)^2<64$?
|
30
| 42.1875 |
18,184 |
Xiao Wang's scores in three rounds of jump rope were 23, 34, and 29, respectively. Xiao Wang's final score is.
|
86
| 1.5625 |
18,185 |
Given the function $f(x)=2\sin ωx (ω > 0)$, find the minimum value of $ω$ such that the minimum value in the interval $[- \frac {π}{3}, \frac {π}{4}]$ is $(-2)$.
|
\frac {3}{2}
| 69.53125 |
18,186 |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively, and satisfy the vectors $\overrightarrow{m}=(\cos A,\cos B)$, $\overrightarrow{n}=(a,2c-b)$, and $\overrightarrow{m} \parallel \overrightarrow{n}$.
(I) Find the measure of angle $A$;
(II) If $a=2 \sqrt {5}$, find the maximum area of $\triangle ABC$.
|
5 \sqrt {3}
| 0 |
18,187 |
From a deck of cards, 5 spades, 4 clubs, and 6 hearts, totaling 15 cards, are drawn. If drawing $m$ cards such that all three suits are present is a certain event, then the minimum value of $m$ is \_\_\_\_\_\_\_\_\_.
|
12
| 10.15625 |
18,188 |
Denis has cards with numbers from 1 to 50. How many ways are there to choose two cards such that the difference of the numbers on the cards is 11, and their product is divisible by 5?
The order of the selected cards does not matter: for example, selecting cards with numbers 5 and 16, as well as selecting cards with numbers 16 and 5, is considered the same way.
|
15
| 83.59375 |
18,189 |
Liam read for 4 days at an average of 42 pages per day, and for 2 days at an average of 50 pages per day, then read 30 pages on the last day. What is the total number of pages in the book?
|
298
| 47.65625 |
18,190 |
Given that $θ$ is a real number, if the complex number $z=\sin 2θ-1+i( \sqrt {2}\cos θ-1)$ is a purely imaginary number, then the imaginary part of $z$ is _______.
|
-2
| 70.3125 |
18,191 |
How many three-digit numbers are there in which the hundreds digit is greater than the tens digit which is in turn greater than the ones digit?
|
120
| 59.375 |
18,192 |
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}
| 53.90625 |
18,193 |
Two hundred people were surveyed. Of these, 150 indicated they liked Beethoven, and 120 indicated they liked Chopin. Additionally, it is known that of those who liked both Beethoven and Chopin, 80 people also indicated they liked Vivaldi. What is the minimum number of people surveyed who could have said they liked both Beethoven and Chopin?
|
80
| 42.1875 |
18,194 |
A square flag features a green cross of uniform width, and a yellow square in the center, against a white background. The cross is symmetric with respect to each of the diagonals of the square. Suppose the entire cross (including the green arms and the yellow center) occupies 49% of the area of the flag. If the yellow center itself takes up 4% of the area of the flag, what percent of the area of the flag is green?
|
45\%
| 63.28125 |
18,195 |
Given two lines $l_{1}$: $x+my+6=0$, and $l_{2}$: $(m-2)x+3y+2m=0$, if the lines $l_{1}\parallel l_{2}$, then $m=$_______.
|
-1
| 30.46875 |
18,196 |
Calculate $6^6 \cdot 3^6$.
|
34012224
| 83.59375 |
18,197 |
Let \(a\), \(b\), \(c\) be distinct complex numbers such that
\[
\frac{a+1}{2 - b} = \frac{b+1}{2 - c} = \frac{c+1}{2 - a} = k.
\]
Find the sum of all possible values of \(k\).
|
1.5
| 0.78125 |
18,198 |
Given real numbers \\(x\\) and \\(y\\) satisfy the equation \\((x-3)^{2}+y^{2}=9\\), find the minimum value of \\(-2y-3x\\) \_\_\_\_\_\_.
|
-3\sqrt{13}-9
| 26.5625 |
18,199 |
Let $p$ and $q$ be real numbers, and suppose that the roots of the equation \[x^3 - 8x^2 + px - q = 0\] are three distinct positive integers. Compute $p + q.$
|
27
| 67.96875 |
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