Unnamed: 0
int64 0
40.3k
| problem
stringlengths 10
5.15k
| ground_truth
stringlengths 1
1.22k
| solved_percentage
float64 0
100
|
|---|---|---|---|
17,700 |
Let $\{a_n\}$ be an arithmetic sequence, and $S_n$ be the sum of its first $n$ terms, given that $S_5 < S_6$, $S_6=S_7 > S_8$, then the correct conclusion(s) is/are \_\_\_\_\_\_
$(1) d < 0$
$(2) a_7=0$
$(3) S_9 > S_5$
$(4) S_6$ and $S_7$ are both the maximum value of $S_n$.
|
(1)(2)(4)
| 0 |
17,701 |
Find the greatest positive integer $A$ with the following property: For every permutation of $\{1001,1002,...,2000\}$ , the sum of some ten consecutive terms is great than or equal to $A$ .
|
10055
| 88.28125 |
17,702 |
A $3$ by $3$ determinant has three entries equal to $2$ , three entries equal to $5$ , and three entries equal to $8$ . Find the maximum possible value of the determinant.
|
405
| 92.1875 |
17,703 |
Simplify the expression $(-\frac{1}{343})^{-2/3}$.
|
49
| 85.9375 |
17,704 |
Evaluate the expression $\frac{\sqrt{3}\tan 12^\circ - 3}{\sin 12^\circ (4\cos^2 12^\circ - 2)}=\_\_\_\_\_\_\_\_.$
|
-4\sqrt{3}
| 21.875 |
17,705 |
Given that the random variable $X$ follows a normal distribution $N(2, \sigma^2)$, and $P(X \leq 4) = 0.84$, determine the value of $P(X < 0)$.
|
0.16
| 92.96875 |
17,706 |
At the end of a professional bowling tournament, the top 6 bowlers have a playoff. First #6 bowls #5. The loser receives $6^{th}$ prize and the winner bowls #4 in another game. The loser of this game receives $5^{th}$ prize and the winner bowls #3. The loser of this game receives $4^{th}$ prize and the winner bowls #2. The loser of this game receives $3^{rd}$ prize, and the winner bowls #1. The winner of this game gets 1st prize and the loser gets 2nd prize. In how many orders can bowlers #1 through #6 receive the prizes?
|
32
| 86.71875 |
17,707 |
Jeff decides to play with a Magic 8 Ball. Each time he asks it a question, it has a 1/3 chance of giving him a positive answer. If he asks it 7 questions, what is the probability that it gives him exactly 3 positive answers?
|
\frac{560}{2187}
| 6.25 |
17,708 |
Given $(1-2x)^{2017} = a_0 + a_1(x-1) + a_2(x-1)^2 + \ldots + a_{2016}(x-1)^{2016} + a_{2017}(x-1)^{2017}$ ($x \in \mathbb{R}$), find the value of $a_1 - 2a_2 + 3a_3 - 4a_4 + \ldots - 2016a_{2016} + 2017a_{2017}$.
|
-4034
| 58.59375 |
17,709 |
Calculate:
1. $(2\frac{3}{5})^{0} + 2^{-2}\cdot(2\frac{1}{4})^{-\frac{1}{2}} + (\frac{25}{36})^{0.5} + \sqrt{(-2)^{2}}$;
2. $\frac{1}{2}\log\frac{32}{49} - \frac{4}{3}\log\sqrt{8} + \log\sqrt{245}$.
|
\frac{1}{2}
| 83.59375 |
17,710 |
For a permutation $\pi$ of the integers from 1 to 10, define
\[ S(\pi) = \sum_{i=1}^{9} (\pi(i) - \pi(i+1))\cdot (4 + \pi(i) + \pi(i+1)), \]
where $\pi (i)$ denotes the $i$ th element of the permutation. Suppose that $M$ is the maximum possible value of $S(\pi)$ over all permutations $\pi$ of the integers from 1 to 10. Determine the number of permutations $\pi$ for which $S(\pi) = M$ .
*Ray Li*
|
40320
| 53.90625 |
17,711 |
Given that $60\%$ of the high school students like ice skating, $50\%$ like skiing, and $70\%$ like either ice skating or skiing, calculate the probability that a randomly selected student who likes skiing also likes ice skating.
|
0.8
| 1.5625 |
17,712 |
The minimum distance from any integer-coordinate point on the plane to the line \( y = \frac{5}{3} x + \frac{4}{5} \) is to be determined.
|
\frac{\sqrt{34}}{85}
| 17.1875 |
17,713 |
Given a sequence $\{a_{n}\}$ that satisfies the equation: ${a_{n+1}}+{({-1})^n}{a_n}=3n-1$ ($n∈{N^*}$), calculate the sum of the first $60$ terms of the sequence $\{a_{n}\}$.
|
2760
| 8.59375 |
17,714 |
Given the sample contains 5 individuals with values a, 0, 1, 2, 3, and the average value of the sample is 1, calculate the standard deviation of the sample.
|
\sqrt{2}
| 96.875 |
17,715 |
A department dispatches 4 researchers to 3 schools to investigate the current status of the senior year review and preparation for exams, requiring at least one researcher to be sent to each school. Calculate the number of different distribution schemes.
|
36
| 57.8125 |
17,716 |
In the Cartesian coordinate system $xOy$, it is known that the line
$$
\begin{cases}
x=-\frac{3}{2}+\frac{\sqrt{2}}{2}l\\
y=\frac{\sqrt{2}}{2}l
\end{cases}
$$
(with $l$ as the parameter) intersects with the curve
$$
\begin{cases}
x=\frac{1}{8}t^{2}\\
y=t
\end{cases}
$$
(with $t$ as the parameter) at points $A$ and $B$. Find the length of the segment $AB$.
|
4\sqrt{2}
| 74.21875 |
17,717 |
Let \(a,\ b,\ c,\ d\) be real numbers such that \(a + b + c + d = 10\) and
\[ab + ac + ad + bc + bd + cd = 20.\]
Find the largest possible value of \(d\).
|
\frac{5 + \sqrt{105}}{2}
| 28.125 |
17,718 |
Given the ellipse $G$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ with an eccentricity of $\frac{\sqrt{6}}{3}$, and its right focus at $(2\sqrt{2}, 0)$. A line $l$ with a slope of $1$ intersects the ellipse $G$ at points $A$ and $B$. An isosceles triangle is formed with $AB$ as the base and $P$ $(-3, 2)$ as the apex.
(1) Find the equation of the ellipse $G$;
(2) Calculate the area of $\triangle PAB$.
|
\frac{9}{2}
| 18.75 |
17,719 |
Find the remainder when $5^{5^{5^5}}$ is divided by 500.
|
125
| 90.625 |
17,720 |
Find the minimum value of the expression \((\sqrt{2(1+\cos 2x)} - \sqrt{3-\sqrt{2}} \sin x + 1) \cdot (3 + 2\sqrt{7-\sqrt{2}} \cos y - \cos 2y)\). If the answer is not an integer, round it to the nearest whole number.
|
-9
| 0.78125 |
17,721 |
In the rectangular coordinate system $(xOy)$, point $P(1, 2)$ is on a line $l$ with a slant angle of $\alpha$. Establish a polar coordinate system with the coordinate origin $O$ as the pole and the positive semi-axis of the $x$-axis as the polar axis. The equation of curve $C$ is $\rho = 6 \sin \theta$.
(1) Write the parametric equation of $l$ and the rectangular coordinate equation of $C$;
(2) Suppose $l$ intersects $C$ at points $A$ and $B$. Find the minimum value of $\frac{1}{|PA|} + \frac{1}{|PB|}$.
|
\frac{2 \sqrt{7}}{7}
| 55.46875 |
17,722 |
A rectangle PQRS has a perimeter of 24 meters and side PQ is fixed at 7 meters. Find the minimum diagonal PR of the rectangle.
|
\sqrt{74}
| 35.9375 |
17,723 |
In $\triangle ABC$, where $C=60 ^{\circ}$, $AB= \sqrt {3}$, and the height from $AB$ is $\frac {4}{3}$, find the value of $AC+BC$.
|
\sqrt {11}
| 0 |
17,724 |
Calculate the number of seven-digit palindromes where the second digit cannot be odd.
|
4500
| 80.46875 |
17,725 |
Let $1=d_1<d_2<d_3<\dots<d_k=n$ be the divisors of $n$ . Find all values of $n$ such that $n=d_2^2+d_3^3$ .
|
68
| 96.09375 |
17,726 |
Given that $\cos α=-\dfrac{4}{5}\left(\dfrac{π}{2}<α<π\right)$, find $\cos\left(\dfrac{π}{6}-α\right)$ and $\cos\left(\dfrac{π}{6}+α\right)$.
|
-\dfrac{3+4\sqrt{3}}{10}
| 1.5625 |
17,727 |
Let $A = (-4, 0),$ $B=(-3,2),$ $C=(3,2),$ and $D=(4,0).$ Suppose that point $P$ satisfies \[PA + PD = PB + PC = 10.\]Find the $y-$coordinate of $P,$ and express it in its simplest form.
|
\frac{6}{7}
| 1.5625 |
17,728 |
Given positive integers \(a\) and \(b\) are each less than 10, find the smallest possible value for \(2 \cdot a - a \cdot b\).
|
-63
| 89.84375 |
17,729 |
Given that $0 < α < \dfrac{\pi }{2}$, $-\dfrac{\pi }{2} < β < 0$, $\cos \left( \dfrac{\pi }{4}+α \right)=\dfrac{1}{3}$, and $\cos \left( \dfrac{\pi }{4}-\dfrac{\beta }{2} \right)=\dfrac{\sqrt{3}}{3}$, calculate the value of $\cos \left( α +\dfrac{\beta }{2} \right)$.
|
\dfrac{5\sqrt{3}}{9}
| 74.21875 |
17,730 |
The market demand for a certain product over the next four years forms a sequence $\left\{a_{n}\right\}(n=1,2,3,4)$. It is predicted that the percentage increase in annual demand from the first year to the second year is $p_{1}$, from the second year to the third year is $p_{2}$, and from the third year to the fourth year is $p_{3}$, with $p_{1} + p_{2} + p_{3} = 1$. Given the following values: $\frac{2}{7}$, $\frac{2}{5}$, $\frac{1}{3}$, $\frac{1}{2}$, and $\frac{2}{3}$, which can potentially be the average annual growth rate of the market demand for these four years?
|
$\frac{1}{3}$
| 0 |
17,731 |
In the Cartesian coordinate system $xOy$, the sum of distances from point $P$ to the centers of two circles $C_1$ and $C_2$ is equal to 4, where $C_1: x^2+y^2-2\sqrt{3}y+2=0$, $C_2: x^2+y^2+2\sqrt{3}y-3=0$. Let the trajectory of point $P$ be $C$.
(1) Find the equation of $C$;
(2) Suppose the line $y=kx+1$ intersects $C$ at points $A$ and $B$. What is the value of $k$ when $\overrightarrow{OA} \perp \overrightarrow{OB}$? What is the value of $|\overrightarrow{AB}|$ at this time?
|
\frac{4\sqrt{65}}{17}
| 43.75 |
17,732 |
Isabella took 9 math tests and received 9 different scores, each an integer between 88 and 100, inclusive. After each test, she noticed that the average of her test scores was always an integer. Her score on the ninth test was 93. What was her score on the eighth test?
|
96
| 21.09375 |
17,733 |
Given that F<sub>1</sub>(-c, 0) and F<sub>2</sub>(c, 0) are the left and right foci of the ellipse G: $$\frac{x^2}{a^2}+ \frac{y^2}{4}=1 \quad (a>0),$$ point M is a point on the ellipse, and MF<sub>2</sub> is perpendicular to F<sub>1</sub>F<sub>2</sub>, with |MF<sub>1</sub>|-|MF<sub>2</sub>|= $$\frac{4}{3}a.$$
(1) Find the equation of ellipse G;
(2) If a line l with a slope of 1 intersects with ellipse G at points A and B, and an isosceles triangle is formed using AB as the base and vertex P(-3, 2), find the area of △PAB.
|
\frac{9}{2}
| 2.34375 |
17,734 |
Let $\{a_{n}\}$ be an arithmetic sequence with a common difference of $d$, and $d \gt 1$. Define $b_{n}=\frac{{n}^{2}+n}{{a}_{n}}$, and let $S_{n}$ and $T_{n}$ be the sums of the first $n$ terms of the sequences $\{a_{n}\}$ and $\{b_{n}\}$, respectively.
$(1)$ If $3a_{2}=3a_{1}+a_{3}$ and $S_{3}+T_{3}=21$, find the general formula for $\{a_{n}\}$.
$(2)$ If $\{b_{n}\}$ is an arithmetic sequence and $S_{99}-T_{99}=99$, find $d$.
|
\frac{51}{50}
| 10.9375 |
17,735 |
How many integers $n$ satisfy $(n+2)(n-8) \le 0$?
|
11
| 100 |
17,736 |
For rational numbers $a$ and $b$, define the operation "$\otimes$" as $a \otimes b = ab - a - b - 2$.
(1) Calculate the value of $(-2) \otimes 3$;
(2) Compare the size of $4 \otimes (-2)$ and $(-2) \otimes 4$.
|
-12
| 38.28125 |
17,737 |
In the Cartesian coordinate system $xOy$, the parametric equation of line $l$ is
$$
\begin{cases}
x = -1 + \frac {\sqrt {2}}{2}t \\
y = 1 + \frac {\sqrt {2}}{2}t
\end{cases}
(t \text{ is the parameter}),
$$
and the equation of circle $C$ is $(x-2)^{2} + (y-1)^{2} = 5$. Establish a polar coordinate system with the origin $O$ as the pole and the positive $x$-axis as the polar axis.
(Ⅰ) Find the polar equations of line $l$ and circle $C$.
(Ⅱ) If line $l$ intersects circle $C$ at points $A$ and $B$, find the value of $\cos ∠AOB$.
|
\frac{3\sqrt{10}}{10}
| 23.4375 |
17,738 |
Given the function $f(x) = x^3 - 3x - 1$, if for any $x_1$, $x_2$ in the interval $[-3,2]$, it holds that $|f(x_1) - f(x_2)| \leq t$, then the minimum value of the real number $t$ is ______.
|
20
| 97.65625 |
17,739 |
The polynomial \( f(x) \) satisfies the equation \( f(x) - f(x-2) = (2x-1)^{2} \) for all \( x \). Find the sum of the coefficients of \( x^{2} \) and \( x \) in \( f(x) \).
|
\frac{5}{6}
| 75.78125 |
17,740 |
In the diagram, each of four identical circles touch three others. The circumference of each circle is 48. Calculate the perimeter of the shaded region formed within the central area where all four circles touch. Assume the circles are arranged symmetrically like petals of a flower.
|
48
| 53.125 |
17,741 |
Sophia's age is $S$ years, which is thrice the combined ages of her two children. Her age $M$ years ago was four times the sum of their ages at that time. Find the value of $S/M$.
|
21
| 77.34375 |
17,742 |
In an ${8}$ × ${8}$ squares chart , we dig out $n$ squares , then we cannot cut a "T"shaped-5-squares out of the surplus chart .
Then find the mininum value of $n$ .
|
32
| 45.3125 |
17,743 |
If $\sum_{n = 0}^{\infty}\sin^{2n}\theta = 4$, what is the value of $\sin{2\theta}$?
|
\frac{\sqrt{3}}{2}
| 75.78125 |
17,744 |
In a class of 38 students, we need to randomly select 5 people to participate in a survey. The number of possible selections where student A is chosen but student B is not is ______. (Express the answer numerically)
|
58905
| 100 |
17,745 |
Given that the vertices of triangle $\triangle ABC$ are $A(3,2)$, the equation of the median on side $AB$ is $x-3y+8=0$, and the equation of the altitude on side $AC$ is $2x-y-9=0$.
$(1)$ Find the coordinates of points $B$ and $C$.
$(2)$ Find the area of $\triangle ABC$.
|
\frac{15}{2}
| 10.15625 |
17,746 |
Given that $\tan \alpha = -2$, find the value of the following expressions:
$(1) \frac{\sin \alpha - 3 \cos \alpha}{\sin \alpha + \cos \alpha}$
$(2) \frac{1}{\sin \alpha \cdot \cos \alpha}$
|
-\frac{5}{2}
| 95.3125 |
17,747 |
Two types of steel plates need to be cut into three sizes $A$, $B$, and $C$. The number of each size that can be obtained from each type of steel plate is shown in the table:
\begin{tabular}{|l|c|c|c|}
\hline & Size $A$ & Size $B$ & Size $C$ \\
\hline First type of steel plate & 2 & 1 & 1 \\
\hline Second type of steel plate & 1 & 2 & 3 \\
\hline
\end{tabular}
If we need 15 pieces of size $A$, 18 pieces of size $B$, and 27 pieces of size $C$, find the minimum number of plates $m$ and $n$ of the two types required, so that $m + n$ is minimized.
|
12
| 69.53125 |
17,748 |
$n$ coins are simultaneously flipped. The probability that two or fewer of them show tails is $\frac{1}{4}$. Find $n$.
|
n = 5
| 31.25 |
17,749 |
It is known that the probabilities of person A and person B hitting the target in each shot are $\frac{3}{4}$ and $\frac{4}{5}$, respectively. Person A and person B do not affect each other's chances of hitting the target, and each shot is independent. If they take turns shooting in the order of A, B, A, B, ..., until one person hits the target and stops shooting, then the probability that A and B have shot a total of four times when stopping shooting is ____.
|
\frac{1}{100}
| 42.1875 |
17,750 |
Given that the area of $\triangle ABC$ is $S$, and $\overrightarrow{BA} \cdot \overrightarrow{CA} = S$.
(1) Find the value of $\tan A$;
(2) If $B = \frac{\pi}{4}, c = 6$, find the area of $\triangle ABC$, $S$.
|
12
| 35.15625 |
17,751 |
Let $ABCD$ be a square and $P$ be a point on the shorter arc $AB$ of the circumcircle of the square. Which values can the expression $\frac{AP+BP}{CP+DP}$ take?
|
\sqrt{2} - 1
| 1.5625 |
17,752 |
Let $m\in R$, the moving straight line passing through the fixed point $A$ with equation $x+my-2=0$ intersects the moving straight line passing through the fixed point $B$ with equation $mx-y+4=0$ at point $P\left(x,y\right)$. Find the maximum value of $|PA|\cdot |PB|$.
|
10
| 8.59375 |
17,753 |
795. Calculate the double integral \(\iint_{D} x y \, dx \, dy\), where region \(D\) is:
1) A rectangle bounded by the lines \(x=0, x=a\), \(y=0, y=b\);
2) An ellipse \(4x^2 + y^2 \leq 4\);
3) Bounded by the line \(y=x-4\) and the parabola \(y^2=2x\).
|
90
| 21.875 |
17,754 |
Line $l_{1}$: $mx+2y-3=0$ is parallel to line $l_{2}$: $3x+\left(m-1\right)y+m-6=0$. Find the value of $m$.
|
-2
| 34.375 |
17,755 |
Given that \(x\) is a real number, find the least possible value of \((x+2)(x+3)(x+4)(x+5)+3033\).
|
3032
| 82.03125 |
17,756 |
If $x$ is an integer, find the largest integer that always divides the expression \[(12x + 2)(8x + 14)(10x + 10)\] when $x$ is odd.
|
40
| 15.625 |
17,757 |
There are 10 different televisions, including 3 type A, 3 type B, and 4 type C. Now, 3 televisions are randomly selected from them. If at least two different types are included, calculate the total number of different ways to select them.
|
114
| 88.28125 |
17,758 |
In the triangle $ABC$ , $| BC | = 1$ and there is exactly one point $D$ on the side $BC$ such that $|DA|^2 = |DB| \cdot |DC|$ . Determine all possible values of the perimeter of the triangle $ABC$ .
|
\sqrt{2} + 1
| 6.25 |
17,759 |
Using systematic sampling, \\(32\\) people are selected from \\(960\\) for a questionnaire survey. They are randomly numbered from \\(1\\) to \\(960\\), and then grouped. The number drawn by simple random sampling in the first group is \\(9\\). Among the \\(32\\) people selected, those with numbers in the interval \\([1,450]\\) will take questionnaire \\(A\\), those in the interval \\([451,750]\\) will take questionnaire \\(B\\), and the rest will take questionnaire \\(C\\). How many of the selected people will take questionnaire \\(B\\)?
|
10
| 87.5 |
17,760 |
A psychic is faced with a deck of 36 cards placed face down (four suits, with nine cards of each suit). He names the suit of the top card, after which the card is revealed and shown to him. Then the psychic names the suit of the next card, and so on. The goal is for the psychic to guess as many suits correctly as possible.
The backs of the cards are asymmetric, and the psychic can see the orientation in which the top card is placed. The psychic's assistant knows the order of the cards in the deck but cannot change it. However, the assistant can place the backs of each card in one of two orientations.
Could the psychic have agreed with the assistant before the assistant knew the order of the cards to ensure guessing the suit of at least a) 19 cards; b) 23 cards?
If you have devised a method to guess a number of cards greater than 19, please describe it.
|
23
| 53.125 |
17,761 |
Given $\sin \alpha - \cos \alpha = \frac{\sqrt{10}}{5}$, $\alpha \in (\pi, 2\pi)$,
$(1)$ Find the value of $\sin \alpha + \cos \alpha$;
$(2)$ Find the value of $\tan \alpha - \frac{1}{\tan \alpha}$.
|
-\frac{8}{3}
| 69.53125 |
17,762 |
In the Cartesian coordinate system $xOy$, the sum of distances from point $P$ to points $F_1(0, -\sqrt{3})$ and $F_2(0, \sqrt{3})$ is equal to 4. Let the trajectory of point $P$ be $C$.
(1) Find the equation of trajectory $C$;
(2) Let line $l: y=kx+1$ intersect curve $C$ at points $A$ and $B$. For what value of $k$ is $|\vec{OA} + \vec{OB}| = |\vec{AB}|$ (where $O$ is the origin)? What is the value of $|\vec{AB}|$ at this time?
|
\frac{4\sqrt{65}}{17}
| 25.78125 |
17,763 |
Given an isosceles triangle $ABC$ satisfying $AB=AC$, $\sqrt{3}BC=2AB$, and point $D$ is on side $BC$ with $AD=BD$, then the value of $\sin \angle ADB$ is ______.
|
\frac{2 \sqrt{2}}{3}
| 28.90625 |
17,764 |
A triangle $\triangle ABC$ satisfies $AB = 13$ , $BC = 14$ , and $AC = 15$ . Inside $\triangle ABC$ are three points $X$ , $Y$ , and $Z$ such that:
- $Y$ is the centroid of $\triangle ABX$
- $Z$ is the centroid of $\triangle BCY$
- $X$ is the centroid of $\triangle CAZ$
What is the area of $\triangle XYZ$ ?
*Proposed by Adam Bertelli*
|
84/13
| 0 |
17,765 |
Given that $\alpha$ is an angle in the second quadrant, simplify $$\frac { \sqrt {1+2\sin(5\pi-\alpha)\cos(\alpha-\pi)}}{\sin\left(\alpha - \frac {3}{2}\pi \right)- \sqrt {1-\sin^{2}\left( \frac {3}{2}\pi+\alpha\right)}}.$$
|
-1
| 81.25 |
17,766 |
Find $AB$ in the triangle below.
[asy]
unitsize(1inch);
pair A,B,C;
A = (0,0);
B = (1,0);
C = (0.5,sqrt(3)/2);
draw (A--B--C--A,linewidth(0.9));
draw(rightanglemark(B,A,C,3));
label("$A$",A,S);
label("$B$",B,S);
label("$C$",C,N);
label("$18$", (A+C)/2,W);
label("$30^\circ$", (0.3,0),N);
[/asy]
|
18\sqrt{3}
| 6.25 |
17,767 |
If the graph of the function $f(x) = (1-x^2)(x^2+ax+b)$ is symmetric about the line $x = -2$, then the maximum value of $f(x)$ is ______.
|
16
| 38.28125 |
17,768 |
Fisherman Vasya caught several fish. He placed the three largest fish, which constitute 35% of the total weight of the catch, in the refrigerator. He gave the three smallest fish, which constitute 5/13 of the weight of the remaining fish, to the cat. Vasya ate all the rest of the caught fish himself. How many fish did Vasya catch?
|
10
| 43.75 |
17,769 |
Given \( a \in \mathbb{R} \), the complex number \( z = \frac{(a-i)(1+i)}{i} \), if \( \bar{z} = z \), then find the value of \( a \).
|
-1
| 98.4375 |
17,770 |
A class is scheduled to have 6 classes in one day: Chinese, Mathematics, Politics, English, PE, and Art. It is required that the Mathematics class is scheduled within the first 3 periods, and the PE class cannot be scheduled in the first period. The number of different scheduling methods is ______. (Answer in digits).
|
312
| 3.125 |
17,771 |
A five-digit integer is chosen at random from all possible positive five-digit integers. What is the probability that the number's units digit is an even number and less than 6? Express your answer as a common fraction.
|
\frac{3}{10}
| 89.84375 |
17,772 |
A square with a perimeter of 36 is inscribed in a square with a perimeter of 40. What is the greatest distance between a vertex of the inner square and a vertex of the outer square?
A) $\sqrt{101}$
B) $9\sqrt{2}$
C) $8\sqrt{2}$
D) $\sqrt{90}$
E) $10$
|
9\sqrt{2}
| 10.9375 |
17,773 |
Two teachers and 4 students need to be divided into 2 groups, each consisting of 1 teacher and 2 students. Calculate the number of different arrangements.
|
12
| 23.4375 |
17,774 |
Cookie Monster now finds a bigger cookie with the boundary described by the equation $x^2 + y^2 - 8 = 2x + 4y$. He wants to know both the radius and the area of this cookie to determine if it's enough for his dessert.
|
13\pi
| 57.8125 |
17,775 |
Given that the random variable $X$ follows a normal distribution $N(1,4)$, and $P(0 \leq X \leq 2) = 0.68$, find $P(X > 2)$.
|
0.16
| 75.78125 |
17,776 |
Five identical right-angled triangles can be arranged so that their larger acute angles touch to form a star. It is also possible to form a different star by arranging more of these triangles so that their smaller acute angles touch. How many triangles are needed to form the second star?
|
20
| 26.5625 |
17,777 |
Given a geometric sequence $\{a_n\}$, where $a_1+a_2+a_3=4$ and $a_2+a_3+a_4=-2$, find the value of $a_3+a_4+a_5+a_6+a_7+a_8$.
|
\frac{7}{8}
| 78.125 |
17,778 |
Evaluate $\frac{5}{a+b}$ where $a=7$ and $b=3$.
A) $\frac{1}{2}$
B) $1$
C) $10$
D) $-8$
E) Meaningless
|
\frac{1}{2}
| 25 |
17,779 |
Calculate $3.6 \times 0.25 + 1.5$.
|
2.4
| 100 |
17,780 |
Given that $\sin(α + \sqrt{2}\cos(α) = \sqrt{3})$, find the value of $\tan(α)$.
A) $\frac{\sqrt{2}}{2}$
B) $\sqrt{2}$
C) $-\frac{\sqrt{2}}{2}$
D) $-\sqrt{2}$
|
\frac{\sqrt{2}}{2}
| 29.6875 |
17,781 |
How many distinct arrangements of the letters in the word "basics" are there, specifically those beginning with a vowel?
|
120
| 64.0625 |
17,782 |
Given a sequence $\{a_n\}$ whose general term formula is $a_n = -n^2 + 12n - 32$, and the sum of the first $n$ terms is $S_n$, then for any $n > m$ (where $m, n \in \mathbb{N}^*$), the maximum value of $S_n - S_m$ is __.
|
10
| 40.625 |
17,783 |
In $\triangle XYZ$, $\angle XYZ = 30^\circ$, $XY = 12$, and $XZ = 8$. Points $P$ and $Q$ lie on $\overline{XY}$ and $\overline{XZ}$ respectively. What is the minimum possible value of $YP + PQ + QZ$?
A) $\sqrt{154}$
B) $\sqrt{208 + 96\sqrt{3}}$
C) $16$
D) $\sqrt{208}$
|
\sqrt{208 + 96\sqrt{3}}
| 39.0625 |
17,784 |
The lengths of a pair of corresponding medians of two similar triangles are 10cm and 4cm, respectively, and the sum of their perimeters is 140cm. The perimeters of these two triangles are , and the ratio of their areas is .
|
25:4
| 7.8125 |
17,785 |
The perimeter of a semicircle with an area of ______ square meters is 15.42 meters.
|
14.13
| 16.40625 |
17,786 |
Given the polar equation of circle $E$ is $\rho=4\sin \theta$, with the pole as the origin and the polar axis as the positive half of the $x$-axis, establish a Cartesian coordinate system with the same unit length (where $(\rho,\theta)$, $\rho \geqslant 0$, $\theta \in [0,2\pi)$).
$(1)$ Line $l$ passes through the origin, and its inclination angle $\alpha= \dfrac {3\pi}{4}$. Find the polar coordinates of the intersection point $A$ of $l$ and circle $E$ (point $A$ is not the origin);
$(2)$ Line $m$ passes through the midpoint $M$ of segment $OA$, and line $m$ intersects circle $E$ at points $B$ and $C$. Find the maximum value of $||MB|-|MC||$.
|
2 \sqrt {2}
| 0 |
17,787 |
Four mathletes and two coaches sit at a circular table. How many distinct arrangements are there of these six people if the two coaches sit opposite each other?
|
24
| 79.6875 |
17,788 |
A certain school arranges $5$ senior high school teachers to visit $3$ schools for exchange and learning. One school has $1$ teacher, one school has $2$ teachers, and one school has $2$ teachers. The total number of different arrangements is _______.
|
90
| 46.09375 |
17,789 |
How many positive divisors does the number $360$ have? Also, calculate the sum of all positive divisors of $360$ that are greater than $30$.
|
1003
| 0 |
17,790 |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are denoted as $a$, $b$, $c$ respectively, and $a^{2}$, $b^{2}$, $c^{2}$ form an arithmetic sequence. Calculate the maximum value of $\sin B$.
|
\dfrac{ \sqrt {3}}{2}
| 0 |
17,791 |
If $128^3 = 16^x$, what is the value of $2^{-x}$? Express your answer as a common fraction.
|
\frac{1}{2^{5.25}}
| 0 |
17,792 |
The sum of the first $n$ terms of an arithmetic sequence $\{a_n\}$ is $S_n$, and it is given that $S_4=20$, $S_{n-4}=60$, and $S_n=120$. Find the value of $n$.
|
12
| 68.75 |
17,793 |
A traffic light cycles as follows: green for 45 seconds, then yellow for 5 seconds, and then red for 50 seconds. Mark chooses a random five-second interval to observe the light. What is the probability that the color changes during his observation?
|
\frac{3}{20}
| 61.71875 |
17,794 |
Given $A=\{-3,-2,-1,0,1,2,3\}$, $a$ and $b$ are elements of $A$. How many cases are there where $|a| \lt |b|$?
|
18
| 91.40625 |
17,795 |
Given that the populations of three communities $A$, $B$, and $C$ are 600, 1200, and 1500 residents respectively, and if 15 people are drawn from community $C$, determine the total number of people drawn in the sample.
|
33
| 86.71875 |
17,796 |
What is the maximum value of $\frac{(3^t - 2t)t}{9^t}$ for integer values of $t$?
**A)** $\frac{1}{8}$
**B)** $\frac{1}{10}$
**C)** $\frac{1}{12}$
**D)** $\frac{1}{16}$
**E)** $\frac{1}{18}$
|
\frac{1}{8}
| 57.8125 |
17,797 |
Add $26_7 + 245_7.$ Express your answer in base 7.
|
304_7
| 82.8125 |
17,798 |
Given the function $f(x)=\tan(\omega x+\phi)$ $(\omega>0, 0<|\phi|<\frac{\pi}{2})$, where two adjacent branches of the graph intersect the coordinate axes at points $A(\frac{\pi}{6},0)$ and $B(\frac{2\pi}{3},0)$. Find the sum of all solutions of the equation $f(x)=\sin(2x-\frac{\pi}{3})$ for $x\in [0,\pi]$.
|
\frac{5\pi}{6}
| 38.28125 |
17,799 |
Given that 8 balls are randomly and independently painted either red or blue with equal probability, find the probability that exactly 4 balls are red and exactly 4 balls are blue, and all red balls come before any blue balls in the order they were painted.
|
\frac{1}{256}
| 44.53125 |
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