Unnamed: 0
int64 0
40.3k
| problem
stringlengths 10
5.15k
| ground_truth
stringlengths 1
1.22k
| solved_percentage
float64 0
100
|
|---|---|---|---|
20,700 |
If the graph of the power function $y=f(x)$ passes through the point $\left( -2,-\frac{1}{8} \right)$, find the value(s) of $x$ that satisfy $f(x)=27$.
|
\frac{1}{3}
| 52.34375 |
20,701 |
Given a triangle $\triangle ABC$ with sides $a$, $b$, and $c$ opposite to angles $A$, $B$, and $C$ respectively. The area of the triangle is given by $S= \frac{a^{2}+b^{2}-c^{2}}{4}$ and $\sin A= \frac{3}{5}$.
1. Find $\sin B$.
2. If side $c=5$, find the area of $\triangle ABC$, denoted as $S$.
|
\frac{21}{2}
| 27.34375 |
20,702 |
William is biking from his home to his school and back, using the same route. When he travels to school, there is an initial $20^\circ$ incline for $0.5$ kilometers, a flat area for $2$ kilometers, and a $20^\circ$ decline for $1$ kilometer. If William travels at $8$ kilometers per hour during uphill $20^\circ$ sections, $16$ kilometers per hours during flat sections, and $20$ kilometers per hour during downhill $20^\circ$ sections, find the closest integer to the number of minutes it take William to get to school and back.
*Proposed by William Yue*
|
29
| 40.625 |
20,703 |
In right triangle $ABC$, where $\angle A = \angle B$ and $AB = 10$. What is the area of $\triangle ABC$?
|
25
| 75 |
20,704 |
A square $EFGH$ is inscribed in the region bounded by the parabola $y = x^2 - 6x + 5$ and the $x$-axis. Find the area of square $EFGH$.
\[
\text{[asy]}
unitsize(0.8 cm);
real parab (real x) {
return(x^2 - 6*x + 5);
}
pair E, F, G, H;
real x = -1 + sqrt(3);
E = (3 - x,0);
F = (3 + x,0);
G = (3 + x,-2*x);
H = (3 - x,-2*x);
draw(graph(parab,0,6));
draw(E--H--G--F);
draw((0,0)--(6,0));
label("$E$", E, N);
label("$F$", F, N);
label("$G$", G, SE);
label("$H$", H, SW);
\text{[/asy]}
\]
|
24 - 8\sqrt{5}
| 33.59375 |
20,705 |
Given an arithmetic-geometric sequence $\{a_n\}$ with the sum of its first $n$ terms denoted as $S_n$, if $a_3 - 4a_2 + 4a_1 = 0$, find the value of $\frac{S_8}{S_4}$.
|
17
| 56.25 |
20,706 |
Given that point $P(-4,3)$ lies on the terminal side of angle $\alpha$, find the value of $$\frac{3\sin^{2}\frac{\alpha}{2}+2\sin\frac{\alpha}{2}\cos\frac{\alpha}{2}+\cos^{2}\frac{\alpha}{2}-2}{\sin(\frac{\pi}{2}+\alpha)\tan(-3\pi+\alpha)+\cos(6\pi-\alpha)}.$$
|
-7
| 89.0625 |
20,707 |
Let $b$ and $c$ be real numbers. If the polynomial $x^3 + bx^2 + cx + d$ has exactly one real root and $d = c + b + 1$, find the value of the product of all possible values of $c$.
|
-1
| 8.59375 |
20,708 |
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}
| 42.1875 |
20,709 |
Given the function $y = \lg(-x^2 + x + 2)$ with domain $A$, find the range $B$ for the exponential function $y = a^x$ $(a>0$ and $a \neq 1)$ where $x \in A$.
1. If $a=2$, determine $A \cup B$;
2. If $A \cap B = (\frac{1}{2}, 2)$, find the value of $a$.
|
a = 2
| 48.4375 |
20,710 |
A market survey shows that the sales volume and price of a certain product in the past 50 days are functions of the sales time t(days), and the sales volume approximately satisfies f(t)=−2t+200(1≤t≤50,t∈N). The price for the first 30 days is g(x)=12t+30(1≤t≤30,t∈N), and for the last 20 days is g(t)=45(31≤t≤50,t∈N).
(1) Write out the daily sales S of the product as a function of time t;
(2) Find the maximum value of the daily sales S.
|
54600
| 2.34375 |
20,711 |
What is the least common multiple of the numbers 1584 and 1188?
|
4752
| 97.65625 |
20,712 |
Define a function $g(x),$ for positive integer values of $x,$ by \[g(x) = \left\{\begin{aligned} \log_3 x & \quad \text{ if } \log_3 x \text{ is an integer} \\ 1 + g(x + 2) & \quad \text{ otherwise}. \end{aligned} \right.\] Compute $g(50).$
|
20
| 2.34375 |
20,713 |
In the Cartesian coordinate system, point O is the origin, and the coordinates of three vertices of the parallelogram ABCD are A(2,3), B(-1,-2), and C(-2,-1).
(1) Find the lengths of the diagonals AC and BD;
(2) If the real number t satisfies $ (\vec{AB}+t\vec{OC})\cdot\vec{OC}=0 $, find the value of t.
|
-\frac{11}{5}
| 85.9375 |
20,714 |
Given that the scores of a math exam follow a normal distribution N(102, 4²), the percentage of scores 114 and above is _______ (Note: P(μ-σ<X≤μ+σ)=0.6826, P(μ-2σ<X≤μ+2σ)=0.9544, P(μ-3σ<X≤μ+3σ)=0.9974).
|
0.13\%
| 27.34375 |
20,715 |
Two strips of width 2 overlap at an angle of 60 degrees inside a rectangle of dimensions 4 units by 3 units. Find the area of the overlap, considering that the angle is measured from the horizontal line of the rectangle.
A) $\frac{2\sqrt{3}}{3}$
B) $\frac{8\sqrt{3}}{9}$
C) $\frac{4\sqrt{3}}{3}$
D) $3\sqrt{3}$
E) $\frac{12}{\sqrt{3}}$
|
\frac{4\sqrt{3}}{3}
| 12.5 |
20,716 |
Given that a positive integer \( A \) can be factorized as \( A = 2^{\alpha} \times 3^{\beta} \times 5^{\gamma} \), where \( \alpha \), \( \beta \), and \( \gamma \) are natural numbers, and given that half of \( A \) is a perfect square, one-third of \( A \) is a perfect cube, and one-fifth of \( A \) is a perfect fifth power of some natural number, determine the minimum value of \( \alpha + \beta + \gamma \).
|
31
| 64.84375 |
20,717 |
Given two vectors $a$ and $b$ in a plane that are orthogonal to each other, with $|a|=2$ and $|b|=1$. Let $k$ and $t$ be real numbers that are not simultaneously zero.
(1) If $x=a+(t-3)b$ and $y=-ka+tb$ are orthogonal, find the functional relationship $k=f(t)$.
(2) Find the minimum value of the function $k=f(t)$.
|
-\frac{9}{16}
| 84.375 |
20,718 |
Thirty-six 6-inch wide square posts are evenly spaced with 6 feet between adjacent posts to enclose a square field. What is the outer perimeter, in feet, of the fence?
|
236
| 18.75 |
20,719 |
Let's call a number palindromic if it reads the same left to right as it does right to left. For example, the number 12321 is palindromic.
a) Write down any five-digit palindromic number that is divisible by 5.
b) How many five-digit palindromic numbers are there that are divisible by 5?
|
100
| 90.625 |
20,720 |
If $f^{-1}(g(x))=x^4-4$ and $g$ has an inverse, find $g^{-1}(f(15))$.
|
\sqrt[4]{19}
| 78.125 |
20,721 |
Given that $a-b=3$, find the value of $1+2b-(a+b)$.
Given that $2^x=3$, find the value of $2^{2x-3}$.
|
\frac{9}{8}
| 17.1875 |
20,722 |
A tetrahedron has a triangular base with sides all equal to 2, and each of its three lateral faces are squares. A smaller tetrahedron is placed within the larger one so that its base is parallel to the base of the larger tetrahedron and its vertices touch the midpoints of the lateral faces of the larger tetrahedron. Calculate the volume of this smaller tetrahedron.
|
\frac{\sqrt{2}}{12}
| 42.96875 |
20,723 |
Find $B^2$, where $B$ is the sum of the absolute values of all roots of the following equation:
\[ x = \sqrt{29} + \frac{121}{{\sqrt{29}+\frac{121}{{\sqrt{29}+\frac{121}{{\sqrt{29}+\frac{121}{{\sqrt{29}+\frac{121}{x}}}}}}}}}.\]
|
513
| 86.71875 |
20,724 |
The solid \( T \) consists of all points \((x,y,z)\) such that \( |x| + |y| \le 2 \), \( |x| + |z| \le 1 \), and \( |z| + |y| \le 1 \). Find the volume of \( T \).
|
\frac{4}{3}
| 11.71875 |
20,725 |
Given that $\tan \beta= \frac{4}{3}$, $\sin (\alpha+\beta)= \frac{5}{13}$, and both $\alpha$ and $\beta$ are within $(0, \pi)$, find the value of $\sin \alpha$.
|
\frac{63}{65}
| 5.46875 |
20,726 |
The function $y = x^2 + 2x - 1$ attains its minimum value on the interval $[0, 3]$.
|
-1
| 95.3125 |
20,727 |
Given that $m$ is an integer and $0 < 3m < 27$, what is the sum of all possible integer values of $m$?
|
36
| 100 |
20,728 |
How many 5-digit numbers beginning with $2$ are there that have exactly three identical digits which are not $2$?
|
324
| 0.78125 |
20,729 |
Rectangle \(ABCD\) has length 9 and width 5. Diagonal \(AC\) is divided into 5 equal parts at \(W, X, Y\), and \(Z\). Determine the area of the shaded region.
|
18
| 20.3125 |
20,730 |
In triangle $\triangle ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, $c$, $\left(a+c\right)\sin A=\sin A+\sin C$, $c^{2}+c=b^{2}-1$. Find:<br/>
$(1)$ $B$;<br/>
$(2)$ Given $D$ is the midpoint of $AC$, $BD=\frac{\sqrt{3}}{2}$, find the area of $\triangle ABC$.
|
\frac{\sqrt{3}}{2}
| 50 |
20,731 |
Given the quadratic equation $x^2 + ax + b = 0$ with roots $r_1$ and $r_2$, find an equation where the roots are three times those of $x^2 + cx + a = 0$ and provide the value of $b/c$.
|
27
| 78.90625 |
20,732 |
A line passing through the focus of the parabola $y^2=4x$ intersects the parabola at points $A(x_1, y_1)$ and $B(x_2, y_2)$. If $|AB|=12$, then $x_1+x_2=$ ___.
|
10
| 75.78125 |
20,733 |
In the Cartesian coordinate system $xOy$, angles $\alpha$ and $\beta$ both start from $Ox$, and their terminal sides are symmetric about the $y$-axis. If the terminal side of angle $\alpha$ passes through the point $(3,4)$, then $\tan (\alpha-\beta)=$ ______.
|
- \dfrac {24}{7}
| 82.8125 |
20,734 |
In the Cartesian coordinate system $xOy$, the graph of the linear function $y=kx+b$ ($k \neq 0$) intersects the positive half-axes of the $x$-axis and $y$-axis at points $A$ and $B$, respectively, and the area of $\triangle OAB$ is equal to $|OA|+|OB|+3$.
(1) Express $k$ in terms of $b$;
(2) Find the minimum value of the area of $\triangle OAB$.
|
7+2\sqrt{10}
| 15.625 |
20,735 |
The union of sets $A$ and $B$ is $A \cup B = \left\{a_{1}, a_{2}, a_{3}\right\}$. When $A \neq B$, the pairs $(A, B)$ and $(B, A)$ are considered different. How many such pairs $(A, B)$ are there?
|
27
| 12.5 |
20,736 |
If $a$, $b$, $c$, and $d$ are real numbers satisfying:
\begin{align*}
a+b+c &= 3, \\
a+b+d &= -2, \\
a+c+d &= 8, \text{ and} \\
b+c+d &= -1,
\end{align*}
what is $ab + cd$?
|
-\frac{190}{9}
| 64.0625 |
20,737 |
A sled loaded with children starts from rest and slides down a snowy $25^\circ$ (with respect to the horizontal) incline traveling $85$ meters in $17$ seconds. Ignore air resistance. What is the coefficient of kinetic friction between the sled and the slope?
|
0.40
| 1.5625 |
20,738 |
Given that a child builds towers with $2$ red cubes, $3$ blue cubes, and $4$ green cubes, determine the number of different towers with a height of $8$ cubes that can be built, with one cube left out.
|
1,260
| 0 |
20,739 |
Xiao Ming forgot the last two digits of his WeChat login password. He only remembers that the last digit is one of the letters \\(A\\), \\(a\\), \\(B\\), or \\(b\\), and the other digit is one of the numbers \\(4\\), \\(5\\), or \\(6\\). The probability that Xiao Ming can successfully log in with one attempt is \_\_\_\_\_\_.
|
\dfrac{1}{12}
| 95.3125 |
20,740 |
Given that points A and B are on the x-axis, and the two circles with centers at A and B intersect at points M $(3a-b, 5)$ and N $(9, 2a+3b)$, find the value of $a^{b}$.
|
\frac{1}{8}
| 19.53125 |
20,741 |
A strip of size $1 \times 10$ is divided into unit squares. The numbers $1, 2, \ldots, 10$ are written in these squares. First, the number 1 is written in one of the squares, then the number 2 is written in one of the neighboring squares, then the number 3 is written in one of the squares neighboring those already occupied, and so on (the choice of the first square and the choice of neighbor at each step are arbitrary). In how many ways can this be done?
|
512
| 29.6875 |
20,742 |
Define $g$ by $g(x) = 3x + 2$. If $g(x) = f^{-1}(x) - 5$ and $f^{-1}(x)$ is the inverse of the function $f(x) = cx + d$, find $7c + 3d$.
|
-\frac{14}{3}
| 96.09375 |
20,743 |
Suppose $50x$ is divisible by 100 and $kx$ is not divisible by 100 for all $k=1,2,\cdots, 49$ Find number of solutions for $x$ when $x$ takes values $1,2,\cdots 100$ .
[list=1]
[*] 20
[*] 25
[*] 15
[*] 50
[/list]
|
20
| 32.03125 |
20,744 |
Suppose a cube has a side length of $8$. Its vertices are alternately colored black and green. What is the volume of the tetrahedron whose vertices are the green vertices of the cube?
|
\frac{512}{3}
| 70.3125 |
20,745 |
What is the area of the region defined by the equation $x^2 + y^2 - 10 = 4y - 10x + 4$?
|
43\pi
| 96.875 |
20,746 |
If two lines \( l \) and \( m \) have equations \( y = -2x + 8 \), and \( y = -3x + 9 \), what is the probability that a point randomly selected in the 1st quadrant and below \( l \) will fall between \( l \) and \( m \)? Express your answer as a decimal to the nearest hundredth.
|
0.16
| 48.4375 |
20,747 |
Two fair, eight-sided dice are rolled. What is the probability that the sum of the two numbers showing is less than 12?
|
\frac{49}{64}
| 6.25 |
20,748 |
Complex numbers \( p, q, r \) form an equilateral triangle with a side length of 24 in the complex plane. If \( |p + q + r| = 48 \), find \( |pq + pr + qr| \).
|
768
| 28.125 |
20,749 |
Given the equation of line $l$ is $y=x+4$, and the parametric equation of circle $C$ is $\begin{cases} x=2\cos \theta \\ y=2+2\sin \theta \end{cases}$ (where $\theta$ is the parameter), with the origin as the pole and the positive half-axis of $x$ as the polar axis. Establish a polar coordinate system.
- (I) Find the polar coordinates of the intersection points of line $l$ and circle $C$.
- (II) If $P$ is a moving point on circle $C$, find the maximum value of the distance $d$ from $P$ to line $l$.
|
\sqrt{2}+2
| 3.90625 |
20,750 |
A seafood wholesaler purchased 1000 kg of crucian carp at a price of 10 yuan/kg and cultured them in a lake (assuming that the weight of each crucian carp remains unchanged during the culture period). According to market estimates, the market price of crucian carp after lake culture will increase by 1 yuan/kg for each day of culture. During the culture period (up to a maximum of 20 days), an average of 10 kg of crucian carp will suffer from oxygen deficiency and float to the surface daily. It is assumed that the oxygen-deficient floating crucian carp can be sold out at a price of 5 yuan/kg.
(1) If the seafood wholesaler cultured the crucian carp for x days and then sold all the living crucian carp along with the oxygen-deficient floating crucian carp, a profit of 8500 yuan could be made. How many days should the crucian carp be cultured?
(2) If various expenses of 450 yuan are incurred for each day during the culture period, what is the maximum profit that the seafood wholesaler can make?
|
6000
| 28.90625 |
20,751 |
On the sides \( AB \) and \( AD \) of the square \( ABCD \), points \( E \) and \( F \) are marked such that \( BE : EA = AF : FD = 2022 : 2023 \). The segments \( EC \) and \( FC \) intersect the diagonal of the square \( BD \) at points \( G \) and \( H \) respectively. Find \( \frac{GH}{BD} \).
|
\frac{12271519}{36814556}
| 0 |
20,752 |
Bob buys four burgers and three sodas for $\$5.00$, and Carol buys three burgers and four sodas for $\$5.40$. How many cents does a soda cost?
|
94
| 23.4375 |
20,753 |
All positive integers whose digits add up to 12 are listed in increasing order. What is the eleventh number in that list?
|
147
| 0.78125 |
20,754 |
Two circles with centers $A$ and $B$ intersect at points $X$ and $Y$ . The minor arc $\angle{XY}=120$ degrees with respect to circle $A$ , and $\angle{XY}=60$ degrees with respect to circle $B$ . If $XY=2$ , find the area shared by the two circles.
|
\frac{10\pi - 12\sqrt{3}}{9}
| 1.5625 |
20,755 |
In a diagram, the grid is composed of 1x1 squares. What is the area of the shaded region if the overall width of the grid is 15 units and its height is 5 units? Some parts are shaded in the following manner: A horizontal stretch from the left edge (6 units wide) that expands 3 units upward from the bottom, and another stretch that begins 6 units from the left and lasts for 9 units horizontally, extending from the 3 units height to the top of the grid.
|
36
| 87.5 |
20,756 |
An underground line has $26$ stops, including the first and the final one, and all the stops are numbered from $1$ to $26$ according to their order. Inside the train, for each pair $(x,y)$ with $1\leq x < y \leq 26$ there is exactly one passenger that goes from the $x$ -th stop to the $y$ -th one. If every passenger wants to take a seat during his journey, find the minimum number of seats that must be available on the train.
*Proposed by **FedeX333X***
|
25
| 35.15625 |
20,757 |
A certain school is actively preparing for the "Sunshine Sports" activity and has decided to purchase a batch of basketballs and soccer balls totaling $30$ items. At a sports equipment store, each basketball costs $80$ yuan, and each soccer ball costs $60$ yuan. During the purchase period at the school, there is a promotion for soccer balls at 20% off. Let $m\left(0 \lt m \lt 30\right)$ be the number of basketballs the school wants to purchase, and let $w$ be the total cost of purchasing basketballs and soccer balls.<br/>$(1)$ The analytical expression of the function between $w$ and $m$ is ______;<br/>$(2)$ If the school requires the number of basketballs purchased to be at least twice the number of soccer balls, then the school will spend the least amount when purchasing ______ basketballs, and the minimum value of $w$ is ______ yuan.
|
2080
| 96.875 |
20,758 |
A certain stationery store stipulates that if one purchases 250 or more exercise books at a time (including 250 books), they can pay at the wholesale price; if one purchases fewer than 250 books, they must pay at the retail price. Li, a teacher, went to the store to buy exercise books for the 8th-grade students. If he buys 1 book for each 8th-grade student, he must pay $240$ yuan at the retail price; if he buys 60 more books, he can pay at the wholesale price and will need to pay $260$ yuan.
$(1)$ Find the range of the number of 8th-grade students in the school.
$(2)$ If the amount needed to buy 288 books at the wholesale price is the same as the amount needed to buy 240 books at the retail price, find the number of 8th-grade students in the school.
|
200
| 68.75 |
20,759 |
Find the smallest composite number that has no prime factors less than 15.
|
289
| 82.03125 |
20,760 |
The numbers from 1 to 200, inclusive, are placed in a bag and a number is randomly selected from the bag. What is the probability it is neither a perfect square, a perfect cube, nor a sixth power? Express your answer as a common fraction.
|
\frac{183}{200}
| 63.28125 |
20,761 |
A line \( y = -\frac{2}{3}x + 6 \) crosses the \( x \)-axis at \( P \) and the \( y \)-axis at \( Q \). Point \( T(r,s) \) is on the line segment \( PQ \). If the area of \( \triangle POQ \) is four times the area of \( \triangle TOP \), what is the value of \( r+s \)?
|
8.25
| 0 |
20,762 |
A local community group sells 180 event tickets for a total of $2652. Some tickets are sold at full price, while others are sold at a discounted rate of half price. Determine the total revenue generated from the full-price tickets.
A) $960
B) $984
C) $1008
D) $1032
|
984
| 4.6875 |
20,763 |
Given an ellipse E: $$\frac {x^{2}}{a^{2}}+ \frac {y^{2}}{b^{2}}$$\=1 (a>b>0) passing through point P ($$\sqrt {3}$$, $$\frac {1}{2}$$) with its left focus at F ($$-\sqrt {3}$$, 0).
1. Find the equation of ellipse E.
2. If A is the right vertex of ellipse E, and the line passing through point F with a slope of $$\frac {1}{2}$$ intersects ellipse E at points M and N, find the area of △AMN.
|
$\frac {2 \sqrt {5}+ \sqrt {15}}{4}$
| 0 |
20,764 |
In $\triangle ABC$, $\tan A= \frac {3}{4}$ and $\tan (A-B)=- \frac {1}{3}$, find the value of $\tan C$.
|
\frac {79}{3}
| 55.46875 |
20,765 |
A math teacher requires Noelle to do one homework assignment for each of the first four homework points she wants to earn; for each of the next four homework points, she needs to do two homework assignments; and so on, so that to earn the $n^{\text{th}}$ homework point, she has to do $\lceil n\div4 \rceil$ homework assignments. Noelle must earn at least 80% of 20 homework points for this semester. What is the smallest number of homework assignments necessary to earn these points?
|
40
| 52.34375 |
20,766 |
Given that $O$ is the coordinate origin, the complex numbers $z_1$ and $z_2$ correspond to the vectors $\overrightarrow{OZ_1}$ and $\overrightarrow{OZ_2}$, respectively. $\bar{z_1}$ is the complex conjugate of $z_1$. The vectors are represented as $\overrightarrow{OZ_1} = (10 - a^2, \frac{1}{a + 5})$ and $\overrightarrow{OZ_2} = (2a - 5, 2 - a)$, where $a \in \mathbb{R}$, and $(z_2 - z_1)$ is a purely imaginary number.
(1) Determine the quadrant in which the point corresponding to the complex number $\bar{z_1}$ lies in the complex plane.
(2) Calculate $|z_1 \cdot z_2|$.
|
\frac{\sqrt{130}}{8}
| 71.875 |
20,767 |
How many different routes are there from point $A$ to point $B$ in a 3x3 grid (where you can only move to the right or down along the drawn segments)?
[asy]
unitsize(0.09inch);
draw((0,0)--(15,0)--(15,15)--(0,15)--cycle);
draw((5,0)--(5,15));
draw((10,0)--(10,15));
draw((0,5)--(15,5));
draw((0,10)--(15,10));
dot((0,15));
dot((15,0));
label("$A$",(0,15),NW);
label("$B$",(15,0),SE);
[/asy]
|
20
| 98.4375 |
20,768 |
Multiply $555.55$ by $\frac{1}{3}$ and then subtract $333.33$. Express the result as a decimal to the nearest hundredth.
|
-148.15
| 0.78125 |
20,769 |
Alice and Bob each arrive at a gathering at a random time between 12:00 noon and 1:00 PM. If Alice arrives after Bob, what is the probability that Bob arrived before 12:45 PM?
|
0.5625
| 3.125 |
20,770 |
If point P is one of the intersections of the hyperbola with foci A(-√10,0), B(√10,0) and a real axis length of 2√2, and the circle x^2 + y^2 = 10, calculate the value of |PA| + |PB|.
|
6\sqrt{2}
| 10.9375 |
20,771 |
Let $A = \{x \mid x^2 - ax + a^2 - 19 = 0\}$, $B = \{x \mid x^2 - 5x + 6 = 0\}$, and $C = \{x \mid x^2 + 2x - 8 = 0\}$.
(1) If $A = B$, find the value of $a$;
(2) If $B \cap A \neq \emptyset$ and $C \cap A = \emptyset$, find the value of $a$.
|
-2
| 52.34375 |
20,772 |
A sample size of 100 is divided into 10 groups with a class interval of 10. In the corresponding frequency distribution histogram, a certain rectangle has a height of 0.03. What is the frequency of that group?
|
30
| 2.34375 |
20,773 |
Consider a square arrangement of tiles comprising 12 black and 23 white square tiles. A border consisting of an alternating pattern of black and white tiles is added around the square. The border follows the sequence: black, white, black, white, and so on. What is the ratio of black tiles to white tiles in the newly extended pattern?
A) $\frac{25}{37}$
B) $\frac{26}{36}$
C) $\frac{26}{37}$
D) $\frac{27}{37}$
|
\frac{26}{37}
| 50 |
20,774 |
A line passing through the point P(3/2, 1/2) intersects the ellipse x^2/6 + y^2/2 = 1 at points A and B, satisfying PA + PB = 0. If M is any point on the line AB and O is the origin, find the minimum value of |OM|.
|
\sqrt{2}
| 4.6875 |
20,775 |
Given the ratio of the legs of a right triangle is $3: 4$, determine the ratio of the corresponding segments of the hypotenuse created by dropping a perpendicular from the opposite vertex of the right angle onto the hypotenuse.
|
\frac{16}{9}
| 2.34375 |
20,776 |
What is the diameter of the circle inscribed in triangle $DEF$ if $DE = 13,$ $DF = 8,$ and $EF = 15$? Express your answer in simplest radical form.
|
\frac{10\sqrt{3}}{3}
| 92.1875 |
20,777 |
Define a $\it{good\ word}$ as a sequence of letters that consists only of the letters $A$, $B$, $C$, and $D$ --- some of these letters may not appear in the sequence --- and in which $A$ is never immediately followed by $B$, $B$ is never immediately followed by $C$, $C$ is never immediately followed by $D$, and $D$ is never immediately followed by $A$. How many eight-letter good words are there?
|
8748
| 82.03125 |
20,778 |
A shape was cut out from a regular hexagon as shown in the picture. The marked points on both the perimeter and inside the hexagon divide the respective line segments into quarters.
What is the ratio of the areas of the original hexagon to the cut-out shape?
|
4:1
| 0.78125 |
20,779 |
The opposite of the real number $2023$ is
|
-2023
| 98.4375 |
20,780 |
Let $ABC$ be a triangle with $AB=13$ , $BC=14$ , and $CA=15$ . Points $P$ , $Q$ , and $R$ are chosen on segments $BC$ , $CA$ , and $AB$ , respectively, such that triangles $AQR$ , $BPR$ , $CPQ$ have the same perimeter, which is $\frac{4}{5}$ of the perimeter of $PQR$ . What is the perimeter of $PQR$ ?
*2021 CCA Math Bonanza Individual Round #2*
|
30
| 35.15625 |
20,781 |
Given a circle of radius 3, find the area of the region consisting of all line segments of length 6 that are tangent to the circle at their midpoints.
A) $3\pi$
B) $6\pi$
C) $9\pi$
D) $12\pi$
E) $15\pi$
|
9\pi
| 60.9375 |
20,782 |
In triangle $\triangle ABC$, $a$, $b$, $c$ are the opposite sides of the internal angles $A$, $B$, $C$, respectively, and $\sin ^{2}A+\sin A\sin C+\sin ^{2}C+\cos ^{2}B=1$.
$(1)$ Find the measure of angle $B$;
$(2)$ If $a=5$, $b=7$, find $\sin C$.
|
\frac{3\sqrt{3}}{14}
| 75 |
20,783 |
How many distinct four-digit positive integers are there such that the product of their digits equals 18?
|
36
| 83.59375 |
20,784 |
$(1)$ Given $x \gt 0$, $y \gt 0$, and $2x+3y=6$, find the maximum value of $xy$;<br/>$(2)$ Given $x＜\frac{1}{2}$, find the maximum value of $y=2x+\frac{4}{2x-1}$.
|
-3
| 59.375 |
20,785 |
For $ n \in \mathbb{N}$ , let $ f(n)\equal{}1^n\plus{}2^{n\minus{}1}\plus{}3^{n\minus{}2}\plus{}...\plus{}n^1$ . Determine the minimum value of: $ \frac{f(n\plus{}1)}{f(n)}.$
|
8/3
| 32.8125 |
20,786 |
Let \(ABCD\) be a trapezium with \(AD\) parallel to \(BC\) and \(\angle ADC = 90^\circ\). Given that \(M\) is the midpoint of \(AB\) with \(CM = \frac{13}{2} \text{ cm}\) and \(BC + CD + DA = 17 \text{ cm}\), find the area of the trapezium \(ABCD\) in \(\text{cm}^2\).
|
30
| 15.625 |
20,787 |
At a certain crosswalk, the pedestrian signal alternates between red and green lights, with the red light lasting for $40s$. If a pedestrian arrives at the crosswalk and encounters a red light, the probability that they need to wait at least $15s$ for the green light to appear is ______.
|
\dfrac{5}{8}
| 92.1875 |
20,788 |
From the numbers $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, the probability of randomly selecting two different numbers such that both numbers are odd is $\_\_\_\_\_\_\_\_\_$, and the probability that the product of the two numbers is even is $\_\_\_\_\_\_\_\_\_$.
|
\frac{13}{18}
| 92.96875 |
20,789 |
Given that $\log_{10}\sin x + \log_{10}\cos x= -2$ and that $\log_{10}(\sin x+\cos x)=\frac{1}{2}(\log_{10}m-2)$, find $m$.
|
102
| 90.625 |
20,790 |
Determine the coefficient of $x^{8}$ in the expansion of \\((x^{3}+ \frac{1}{2 \sqrt {x}})^{5}\\).
|
\frac{5}{2}
| 97.65625 |
20,791 |
For any real number $t$ , let $\lfloor t \rfloor$ denote the largest integer $\le t$ . Suppose that $N$ is the greatest integer such that $$ \left \lfloor \sqrt{\left \lfloor \sqrt{\left \lfloor \sqrt{N} \right \rfloor}\right \rfloor}\right \rfloor = 4 $$ Find the sum of digits of $N$ .
|
24
| 57.03125 |
20,792 |
Calculate \(3^{18} \div 27^2\) and multiply the result by 7. Write your answer as an integer.
|
3720087
| 5.46875 |
20,793 |
China was the first country in the world to use negative numbers. Li Heng, in the book "Fa Jing" written during the Warring States period, already used negative numbers. If the year 500 BC is written as $-500$ years, then the year 2024 AD should be written as ______ years.
|
+2024
| 0 |
20,794 |
Given a set of data $(1)$, $(a)$, $(3)$, $(6)$, $(7)$, its average is $4$, what is its variance?
|
\frac{24}{5}
| 10.15625 |
20,795 |
Consider a $10\times10$ checkerboard with alternating black and white squares. How many distinct squares, with sides on the grid lines of the checkerboard (horizontal and vertical) and containing at least 6 black squares, can be drawn on the checkerboard?
|
140
| 10.9375 |
20,796 |
Let \(a\), \(b\), and \(c\) be real numbers such that \(9a^2 + 4b^2 + 25c^2 = 4\). Find the maximum value of
\[6a + 3b + 10c.\]
|
\sqrt{41}
| 82.03125 |
20,797 |
Find the least possible sum of two bases, $c$ and $d$, such that the numeral $29$ in base $c$ represents the same number as $92$ in base $d$, where $c$ and $d$ are positive integers.
|
13
| 13.28125 |
20,798 |
Given the function $f(x)=\sqrt{3}\sin x\cos x-{\cos }^2x$.
$(1)$ Find the smallest positive period of $f(x)$;
$(2)$ If $f(x)=-1$, find the value of $\cos \left(\dfrac{2\pi }{3}-2x\right)$.
|
-\dfrac{1}{2}
| 55.46875 |
20,799 |
The South China tiger is a first-class protected animal in our country. To save the species from the brink of extinction, the country has established a South China tiger breeding base. Due to scientific artificial cultivation, the relationship between the number of South China tigers $y$ (individuals) and the breeding time $x$ (years) can be approximately described by $y=a\log_{2}(x+1)$. If there were 20 tigers in the first year of breeding (2012), then by 2015, it is predicted that there will be approximately how many tigers?
|
46
| 22.65625 |
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