Unnamed: 0
int64 0
40.3k
| problem
stringlengths 10
5.15k
| ground_truth
stringlengths 1
1.22k
| solved_percentage
float64 0
100
|
|---|---|---|---|
9,600 |
On a rectangular sheet of paper, a picture in the shape of a "cross" is drawn from two rectangles \(ABCD\) and \(EFGH\), whose sides are parallel to the edges of the sheet. It is known that \(AB = 9\), \(BC = 5\), \(EF = 3\), and \(FG = 10\). Find the area of the quadrilateral \(AFCH\).
|
52.5
| 79.6875 |
9,601 |
In the sequence \(\left\{a_{n}\right\}\), \(a_{1} = -1\), \(a_{2} = 1\), \(a_{3} = -2\). Given that for all \(n \in \mathbf{N}_{+}\), \(a_{n} a_{n+1} a_{n+2} a_{n+3} = a_{n} + a_{n+1} + a_{n+2} + a_{n+3}\), and \(a_{n+1} a_{n+2} a_{n+3} \neq 1\), find the sum of the first 4321 terms of the sequence \(S_{4321}\).
|
-4321
| 92.1875 |
9,602 |
The diagonals of trapezoid \(ABCD\) intersect at point \(M\). The areas of triangles \(ABM\) and \(CDM\) are 18 and 50 units, respectively. What is the area of the trapezoid?
|
128
| 53.90625 |
9,603 |
A spinner was created by drawing five radii from the center of a circle. The first four radii divide the circle into four equal wedges. The fifth radius divides one of the wedges into two parts, one having twice the area of the other. The five wedges are labeled with the wedge labeled by 2 having twice the area of the wedge labeled by 1. Determine the probability of spinning an odd number.
|
7/12
| 25 |
9,604 |
Six semicircles are evenly arranged along the inside of a regular hexagon with a side length of 3 units. A circle is positioned in the center such that it is tangent to each of these semicircles. Find the radius of this central circle.
|
\frac{3 (\sqrt{3} - 1)}{2}
| 4.6875 |
9,605 |
In a set of 10 programs, there are 6 singing programs and 4 dance programs. The requirement is that there must be at least one singing program between any two dance programs. Determine the number of different ways to arrange these programs.
|
604800
| 62.5 |
9,606 |
If the fractional equation $\frac{3}{{x-2}}+1=\frac{m}{{4-2x}}$ has a root, then the value of $m$ is ______.
|
-6
| 27.34375 |
9,607 |
Find the smallest natural number such that when it is multiplied by 9, the resulting number consists of the same digits in a different order.
|
1089
| 96.09375 |
9,608 |
When $\sqrt[3]{7200}$ is simplified, the result is $c\sqrt[3]{d}$, where $c$ and $d$ are positive integers and $d$ is as small as possible. What is $c+d$?
|
452
| 3.90625 |
9,609 |
A market survey shows that the price $f(x)$ (in yuan) and the sales volume $g(x)$ (in units) of a certain product in the past $20$ days are both functions of time $x$ (in days), and the price satisfies $f(x)=20-\frac{1}{2}|x-10|$, and the sales volume satisfies $g(x)=80-2x$, where $0\leqslant x\leqslant 20$, $x\in N$.
(1) Write the function expression of the daily sales $y$ (in yuan) of the product with time $x$ (in days).
(2) Find the minimum value of the daily sales of the product.
|
600
| 50.78125 |
9,610 |
How many positive integers less than $201$ are multiples of either $6$ or $8$, but not both at once?
|
42
| 28.125 |
9,611 |
Two hunters, $A$ and $B$, went duck hunting. Assume that each of them hits a duck as often as they miss it. Hunter $A$ encountered 50 ducks during the hunt, while hunter $B$ encountered 51 ducks. What is the probability that hunter $B$'s catch exceeds hunter $A$'s catch?
|
1/2
| 80.46875 |
9,612 |
Find the largest six-digit number in which each digit, starting from the third, is the sum of the two preceding digits.
|
303369
| 66.40625 |
9,613 |
For any positive integer \( n \), let \( f(n) = 70 + n^2 \) and let \( g(n) \) be the greatest common divisor (GCD) of \( f(n) \) and \( f(n+1) \). Find the greatest possible value of \( g(n) \).
|
281
| 78.125 |
9,614 |
An international scientific research cooperation project has been jointly developed and completed by two Americans, one Frenchman, and one Chinese. Now, two individuals are randomly selected to announce the results. The probability that the selected pair includes the Chinese individual is ( ).
|
\frac{1}{2}
| 92.1875 |
9,615 |
Ten points are spaced around at intervals of one unit around a modified $2 \times 2$ square such that each vertex of the square and midpoints on each side of the square are included, along with two additional points, each located midway on a diagonal extension from opposite corners of the square. Two of the 10 points are chosen at random. What is the probability that the two points are one unit apart?
A) $\frac{1}{5}$
B) $\frac{14}{45}$
C) $\frac{5}{18}$
D) $\frac{1}{3}$
E) $\frac{2}{5}$
|
\frac{14}{45}
| 57.8125 |
9,616 |
On a circle, there are 25 points marked, which are colored either red or blue. Some points are connected by segments, with each segment having one end blue and the other red. It is known that there do not exist two red points that are connected to the same number of segments. What is the greatest possible number of red points?
|
13
| 32.8125 |
9,617 |
Given Lara ate $\frac{1}{4}$ of a pie and Ryan ate $\frac{3}{10}$ of the same pie, then Cassie ate $\frac{2}{3}$ of the pie that was left. Calculate the fraction of the original pie that was not eaten.
|
\frac{3}{20}
| 51.5625 |
9,618 |
Solve the equations.
$(3+x) \times 30\% = 4.8$
$5 : x = \frac{9}{2} : \frac{8}{5}$
|
\frac{16}{9}
| 88.28125 |
9,619 |
Given \((1+x-x^2)^{10} = a_0 + a_1 x + a_2 x^2 + \cdots + a_{20} x^{20}\), find \( a_0 + a_1 + 2a_2 + 3a_3 + \cdots + 20a_{20} \).
|
-9
| 32.8125 |
9,620 |
Given that $\cos(\alpha - \beta) = \frac{3}{5}$, $\sin(\beta) = -\frac{5}{13}$, where $\alpha \in \left(0, \frac{\pi}{2} \right)$, $\beta \in \left(-\frac{\pi}{2}, 0 \right)$, find the value of $\sin(\alpha)$.
|
\frac{33}{65}
| 27.34375 |
9,621 |
Given the numbers from $000$ to $999$, calculate how many have three digits in non-decreasing or non-increasing order, including cases where digits can repeat.
|
430
| 1.5625 |
9,622 |
The surface area of the circumscribed sphere of cube \( K_1 \) is twice the surface area of the inscribed sphere of cube \( K_2 \). Let \( V_1 \) denote the volume of the inscribed sphere of cube \( K_1 \), and \( V_2 \) denote the volume of the circumscribed sphere of cube \( K_2 \). What is the ratio \( \frac{V_1}{V_2} \)?
|
\frac{2\sqrt{2}}{27}
| 51.5625 |
9,623 |
In a bag, there are several balls, including red, black, yellow, and white balls. The probability of drawing a red ball is $\frac{1}{3}$, the probability of drawing either a black or a yellow ball is $\frac{5}{12}$, and the probability of drawing either a yellow or a white ball is $\frac{5}{12}$. The probability of drawing a white ball is ______.
|
\frac{1}{4}
| 97.65625 |
9,624 |
There are three novel series Peter wishes to read. Each consists of 4 volumes that must be read in order, but not necessarily one after the other. Let \( N \) be the number of ways Peter can finish reading all the volumes. Find the sum of the digits of \( N \). (Assume that he must finish a volume before reading a new one.)
|
18
| 68.75 |
9,625 |
In the 2009 Stanford Olympics, Willy and Sammy are two bikers. The circular race track has two
lanes, the inner lane with radius 11, and the outer with radius 12. Willy will start on the inner lane,
and Sammy on the outer. They will race for one complete lap, measured by the inner track.
What is the square of the distance between Willy and Sammy's starting positions so that they will both race
the same distance? Assume that they are of point size and ride perfectly along their respective lanes
|
265 - 132\sqrt{3}
| 7.03125 |
9,626 |
An escalator has \( n \) visible steps and descends at a constant speed. Two boys, \( A \) and \( Z \), walk down the moving escalator at a steady pace. Boy \( A \) walks twice as many steps per minute as boy \( Z \). \( A \) reaches the bottom after walking 27 steps, and \( Z \) reaches the bottom after walking 18 steps. Find \( n \).
|
54
| 67.1875 |
9,627 |
In a square, 20 points were marked and connected by non-intersecting segments with each other and with the vertices of the square, dividing the square into triangles. How many triangles were formed?
|
42
| 41.40625 |
9,628 |
At HappyTails Training Center, cats can learn to do three tricks: jump, fetch, and spin. Of the cats at the center:
- 40 cats can jump
- 25 cats can fetch
- 30 cats can spin
- 15 cats can jump and fetch
- 10 cats can fetch and spin
- 12 cats can jump and spin
- 5 cats can do all three
- 7 cats can do none
How many cats are in the center?
|
70
| 72.65625 |
9,629 |
Outstanding Brazilian footballer Ronaldinho Gaúcho will be $X$ years old in the year $X^{2}$. How old will he be in 2018, when the World Cup is held in Russia?
|
38
| 41.40625 |
9,630 |
Calculate the lengths of the arcs of curves given by the equations in the rectangular coordinate system.
$$
y=\ln x, \sqrt{3} \leq x \leq \sqrt{15}
$$
|
\frac{1}{2} \ln \frac{9}{5} + 2
| 0 |
9,631 |
Given that $\cos \left(\alpha+ \frac{\pi}{6}\right)= \frac{4}{5}$, find the value of $\sin \left(2\alpha+ \frac{\pi}{3}\right)$.
|
\frac{24}{25}
| 76.5625 |
9,632 |
Given that the cube root of \( m \) is a number in the form \( n + r \), where \( n \) is a positive integer and \( r \) is a positive real number less than \(\frac{1}{1000}\). When \( m \) is the smallest positive integer satisfying the above condition, find the value of \( n \).
|
19
| 78.90625 |
9,633 |
2008 persons take part in a programming contest. In one round, the 2008 programmers are divided into two groups. Find the minimum number of groups such that every two programmers ever be in the same group.
|
11
| 22.65625 |
9,634 |
Find \( k \) such that \((a+b)(b+c)(c+a) = (a+b+c)(ab+bc+ca) + k \cdot abc\).
|
-1
| 89.84375 |
9,635 |
Due to the increasing personnel exchanges between Hefei and Nanjing as a result of urban development, there is a plan to build a dedicated railway to alleviate traffic pressure, using a train as a shuttle service. It is known that the daily round-trip frequency $y$ of the train is a linear function of the number of carriages $x$ it tows each time. If 4 carriages are towed, the train runs 16 times a day, and if 7 carriages are towed, the train runs 10 times a day.
Ⅰ. Find the functional relationship between the daily round-trip frequency $y$ and the number of carriages $x$;
Ⅱ. Find the function equation for the total number $S$ of carriages operated daily with respect to the number of carriages $x$ towed;
Ⅲ. If each carriage carries 110 passengers, how many carriages should be towed to maximize the number of passengers transported daily? Also, calculate the maximum number of passengers transported daily.
|
7920
| 69.53125 |
9,636 |
Find the greatest four-digit number where all digits are distinct, and which is divisible by each of its digits. Zero cannot be used.
|
9864
| 96.875 |
9,637 |
On every kilometer of the highway between the villages Yolkino and Palkino, there is a post with a sign. On one side of the sign, the distance to Yolkino is written, and on the other side, the distance to Palkino is written. Borya noticed that on each post, the sum of all the digits is equal to 13. What is the distance from Yolkino to Palkino?
|
49
| 82.8125 |
9,638 |
In triangle $DEF$, we have $\angle D = 90^\circ$ and $\sin E = \frac{3}{5}$. Find $\cos F$.
|
\frac{3}{5}
| 86.71875 |
9,639 |
Eight numbers \( a_{1}, a_{2}, a_{3}, a_{4} \) and \( b_{1}, b_{2}, b_{3}, b_{4} \) satisfy the equations:
\[
\left\{
\begin{array}{l}
a_{1} b_{1} + a_{2} b_{3} = 1 \\
a_{1} b_{2} + a_{2} b_{4} = 0 \\
a_{3} b_{1} + a_{4} b_{3} = 0 \\
a_{3} b_{2} + a_{4} b_{4} = 1
\end{array}
\right.
\]
Given that \( a_{2} b_{3} = 7 \), find \( a_{4} b_{4} \).
|
-6
| 23.4375 |
9,640 |
Given that circle $C$ passes through points $P(0,-4)$, $Q(2,0)$, and $R(3,-1)$.
$(1)$ Find the equation of circle $C$.
$(2)$ If the line $l: mx+y-1=0$ intersects circle $C$ at points $A$ and $B$, and $|AB|=4$, find the value of $m$.
|
\frac{4}{3}
| 42.1875 |
9,641 |
Given a cube of unit side. Let $A$ and $B$ be two opposite vertex. Determine the radius of the sphere, with center inside the cube, tangent to the three faces of the cube with common point $A$ and tangent to the three sides with common point $B$ .
|
\frac{1}{2}
| 63.28125 |
9,642 |
A positive integer \( N \) and \( N^2 \) end with the same sequence of digits \(\overline{abcd}\), where \( a \) is a non-zero digit. Find \(\overline{abc}\).
|
937
| 72.65625 |
9,643 |
What is the value of $45_{10} + 28_{10}$ in base 4?
|
1021_4
| 100 |
9,644 |
Given the fixed point $M(1,0)$, $A$ and $B$ are two moving points on the ellipse $\frac{x^2}{4}+y^2=1$, and $\overrightarrow{MA} \cdot \overrightarrow{MB}=0$, find the minimum value of $\overrightarrow{AM} \cdot \overrightarrow{AB}$.
|
\frac{2}{3}
| 14.84375 |
9,645 |
Given a sequence $\{a_n\}$ where all terms are positive, and $a_1=2$, $a_{n+1}-a_n= \frac{4}{a_{n+1}+a_n}$. If the sum of the first $n$ terms of the sequence $\left\{ \frac{1}{a_{n-1}+a_n} \right\}$ is $5$, then $n=\boxed{120}$.
|
120
| 79.6875 |
9,646 |
A cube with an edge length of \( n \) (where \( n \) is a positive integer) is painted red on its surface and then cut into \( n^3 \) smaller cubes with an edge length of 1. It is found that the number of small cubes with only one face painted red is exactly 12 times the number of small cubes with two faces painted red. Find the value of \( n \).
|
26
| 82.03125 |
9,647 |
In the diagram, $A$ and $B(20,0)$ lie on the $x$-axis and $C(0,30)$ lies on the $y$-axis such that $\angle A C B=90^{\circ}$. A rectangle $D E F G$ is inscribed in triangle $A B C$. Given that the area of triangle $C G F$ is 351, calculate the area of the rectangle $D E F G$.
|
468
| 1.5625 |
9,648 |
Seryozha and Misha, while walking in the park, stumbled upon a meadow surrounded by linden trees. Seryozha walked around the meadow, counting the trees. Misha did the same, but started at a different tree (although he walked in the same direction). The tree that was the 20th for Seryozha was the 7th for Misha, and the tree that was the 7th for Seryozha was the 94th for Misha. How many trees were growing around the meadow?
|
100
| 66.40625 |
9,649 |
In an arithmetic sequence \(\left\{a_{n}\right\}\), if \(\frac{a_{11}}{a_{10}} < -1\), and the sum of its first \(n\) terms \(S_{n}\) has a maximum value. Then, when \(S_{n}\) attains its smallest positive value, \(n =\) ______ .
|
19
| 47.65625 |
9,650 |
Let the ellipse \\(C: \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1, (a > b > 0)\\) have an eccentricity of \\(\dfrac{2\sqrt{2}}{3}\\), and it is inscribed in the circle \\(x^2 + y^2 = 9\\).
\\((1)\\) Find the equation of ellipse \\(C\\).
\\((2)\\) A line \\(l\\) (not perpendicular to the x-axis) passing through point \\(Q(1,0)\\) intersects the ellipse at points \\(M\\) and \\(N\\), and intersects the y-axis at point \\(R\\). If \\(\overrightarrow{RM} = \lambda \overrightarrow{MQ}\\) and \\(\overrightarrow{RN} = \mu \overrightarrow{NQ}\\), determine whether \\(\lambda + \mu\\) is a constant, and explain why.
|
-\dfrac{9}{4}
| 76.5625 |
9,651 |
How many integers from 100 through 999, inclusive, do not contain any of the digits 0, 1, 8, or 9?
|
216
| 92.1875 |
9,652 |
Given the point $M(m, m^2)$ and $N(n, n^2)$, where $m$ and $n$ are the two distinct real roots of the equation $\sin\theta \cdot x^2 + \cos\theta \cdot x - 1 = 0 (\theta \in R)$. If the maximum distance from a point on the circle $O: x^2 + y^2 = 1$ to the line $MN$ is $d$, and the positive real numbers $a$, $b$, and $c$ satisfy the equation $abc + b^2 + c^2 = 4d$, determine the maximum value of $\log_4 a + \log_2 b + \log_2 c$.
|
\frac{3}{2}
| 11.71875 |
9,653 |
Given a convex hexagon $A B C D E F$ with all six side lengths equal, and internal angles $\angle A$, $\angle B$, and $\angle C$ are $134^{\circ}$, $106^{\circ}$, and $134^{\circ}$ respectively. Find the measure of the internal angle $\angle E$.
|
134
| 17.1875 |
9,654 |
The wavelength of red light that the human eye can see is $0.000077$ cm. Please round the data $0.000077$ to $0.00001$ and express it in scientific notation as ______.
|
8 \times 10^{-5}
| 32.8125 |
9,655 |
There are 10 sheikhs each with a harem of 100 wives standing on the bank of a river along with a yacht that can hold $n$ passengers. According to the law, a woman must not be on the same bank, on the yacht, or at any stopover point with a man unless her husband is present. What is the smallest value of $n$ such that all the sheikhs and their wives can cross to the other bank without breaking the law?
|
10
| 13.28125 |
9,656 |
Suppose that $a$ and $ b$ are distinct positive integers satisfying $20a + 17b = p$ and $17a + 20b = q$ for certain primes $p$ and $ q$ . Determine the minimum value of $p + q$ .
|
296
| 64.0625 |
9,657 |
Given a triangle \(ABC\) with an area of 2. Points \(P\), \(Q\), and \(R\) are taken on the medians \(AK\), \(BL\), and \(CN\) of the triangle \(ABC\) respectively, such that \(AP : PK = 1\), \(BQ : QL = 1:2\), and \(CR : RN = 5:4\). Find the area of the triangle \(PQR\).
|
1/6
| 5.46875 |
9,658 |
A point is randomly thrown onto the segment [6, 11], and let \( k \) be the resulting value. Find the probability that the roots of the equation \( \left(k^{2}-2k-15\right)x^{2}+(3k-7)x+2=0 \) satisfy the condition \( x_{1} \leq 2x_{2} \).
|
1/3
| 48.4375 |
9,659 |
Given real numbers \( x \) and \( y \in (1, +\infty) \), and \( x y - 2 x - y + 1 = 0 \). Find the minimum value of \( \frac{3}{2} x^{2} + y^{2} \).
|
15
| 52.34375 |
9,660 |
Consider a 4x4 grid of points (equally spaced). How many rectangles, of any size, can be formed where each of its four vertices are points on this grid?
|
36
| 17.96875 |
9,661 |
A circle is circumscribed around a unit square \(ABCD\), and a point \(M\) is selected on the circle.
What is the maximum value that the product \(MA \cdot MB \cdot MC \cdot MD\) can take?
|
0.5
| 3.125 |
9,662 |
How many solutions does the equation
\[ x^{2}+y^{2}+2xy-1988x-1988y=1989 \]
have in the set of positive integers?
|
1988
| 94.53125 |
9,663 |
Approximate the increase in the volume of a cylinder with a height of \( H = 40 \) cm and a base radius of \( R = 30 \) cm when the radius is increased by \( 0.5 \) cm.
|
1200\pi
| 11.71875 |
9,664 |
Let \( x < 0.1 \) be a positive real number. Consider the series \( 4 + 4x + 4x^2 + 4x^3 + \ldots \), and the series \( 4 + 44x + 444x^2 + 4444x^3 + \ldots \). Suppose that the sum of the second series is four times the sum of the first series. Compute \( x \).
|
3/40
| 46.09375 |
9,665 |
A tour group has three age categories of people, represented in a pie chart. The central angle of the sector corresponding to older people is $9^{\circ}$ larger than the central angle for children. The percentage of total people who are young adults is $5\%$ higher than the percentage of older people. Additionally, there are 9 more young adults than children. What is the total number of people in the tour group?
|
120
| 50.78125 |
9,666 |
A conference lasted for 2 days. On the first day, the sessions lasted for 7 hours and 15 minutes, and on the second day, they lasted for 8 hours and 45 minutes. Calculate the total number of minutes the conference sessions lasted.
|
960
| 69.53125 |
9,667 |
Given that the vertex of the parabola C is O(0,0), and the focus is F(0,1).
(1) Find the equation of the parabola C;
(2) A line passing through point F intersects parabola C at points A and B. If lines AO and BO intersect line l: y = x - 2 at points M and N respectively, find the minimum value of |MN|.
|
\frac {8 \sqrt {2}}{5}
| 0 |
9,668 |
Given a geometric sequence $\left\{a_{n}\right\}$ with real terms, and the sum of the first $n$ terms is $S_{n}$. If $S_{10} = 10$ and $S_{30} = 70$, calculate the value of $S_{40}$.
|
150
| 61.71875 |
9,669 |
In triangle $ABC$ with sides $AB=9$, $AC=3$, and $BC=8$, the angle bisector $AK$ is drawn. A point $M$ is marked on side $AC$ such that $AM : CM=3 : 1$. Point $N$ is the intersection point of $AK$ and $BM$. Find $KN$.
|
\frac{\sqrt{15}}{5}
| 10.9375 |
9,670 |
In square \(ABCD\), an isosceles triangle \(AEF\) is inscribed such that point \(E\) lies on side \(BC\) and point \(F\) lies on side \(CD\), and \(AE = AF\). The tangent of angle \(AEF\) is 3. Find the cosine of angle \(FAD\).
|
\frac{2 \sqrt{5}}{5}
| 20.3125 |
9,671 |
Given \(\alpha, \beta \in \left(0, \frac{\pi}{2}\right)\), \(\sin \beta = 2 \cos (\alpha + \beta) \cdot \sin \alpha \left(\alpha + \beta \neq \frac{\pi}{2}\right)\), find the maximum value of \(\tan \beta\).
|
\frac{\sqrt{3}}{3}
| 46.875 |
9,672 |
Let $Q$ be the product of the first $50$ positive odd integers. Find the largest integer $m$ such that $Q$ is divisible by $5^m.$
|
12
| 85.9375 |
9,673 |
Given the parabola $C$: $y^{2}=2px(p > 0)$ with focus $F$ and passing through point $A(1,-2)$.
$(1)$ Find the equation of the parabola $C$;
$(2)$ Draw a line $l$ through $F$ at an angle of $45^{\circ}$, intersecting the parabola $C$ at points $M$ and $N$, with $O$ being the origin. Calculate the area of $\triangle OMN$.
|
2\sqrt{2}
| 88.28125 |
9,674 |
A flock of geese is flying, and a lone goose flies towards them and says, "Hello, a hundred geese!" The leader of the flock responds, "No, we are not a hundred geese! If there were as many of us as there are now, plus the same amount, plus half of that amount, plus a quarter of that amount, plus you, goose, then we would be a hundred geese. But as it is..." How many geese were in the flock?
|
36
| 100 |
9,675 |
Simplify and evaluate the following expressions:
\\((1){{(\\dfrac{9}{4})}^{\\frac{1}{2}}}{-}{{({-}2017)}^{0}}{-}{{(\\dfrac{27}{8})}^{\\frac{2}{3}}}\\)
\\((2)\\lg 5+{{(\\lg 2)}^{2}}+\\lg 5\\bullet \\lg 2+\\ln \\sqrt{e}\\)
|
\frac{3}{2}
| 55.46875 |
9,676 |
900 cards are inscribed with all natural numbers from 1 to 900. Cards inscribed with squares of integers are removed, and the remaining cards are renumbered starting from 1.
Then, the operation of removing the squares is repeated. How many times must this operation be repeated to remove all the cards?
|
59
| 25.78125 |
9,677 |
The equation
$$
(x-1) \times \ldots \times(x-2016) = (x-1) \times \ldots \times(x-2016)
$$
is written on the board. We want to erase certain linear factors so that the remaining equation has no real solutions. Determine the smallest number of linear factors that need to be erased to achieve this objective.
|
2016
| 66.40625 |
9,678 |
Given a triangle $\triangle ABC$ with internal angles $A$, $B$, $C$ and their corresponding opposite sides $a$, $b$, $c$. $B$ is an acute angle. Vector $m=(2\sin B, -\sqrt{3})$ and vector $n=(\cos 2B, 2\cos^2 \frac{B}{2} - 1)$ are parallel.
(1) Find the value of angle $B$.
(2) If $b=2$, find the maximum value of the area $S_{\triangle ABC}$.
|
\sqrt{3}
| 80.46875 |
9,679 |
A circle with center P and radius 4 inches is tangent at D to a circle with center Q, located at a 45-degree angle from P. If point Q is on the smaller circle, what is the area of the shaded region? Express your answer in terms of $\pi$.
|
48\pi
| 9.375 |
9,680 |
In triangle $ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively. Given that $2a\cos A=c\cos B+b\cos C$.
1. Find the value of $\cos A$.
2. If $a=1$ and $\cos^{2}\frac{B}{2}+\cos^{2}\frac{C}{2}=1+\frac{\sqrt{3}}{4}$, find the length of side $c$.
|
\frac { \sqrt {3}}{3}
| 0 |
9,681 |
Milly has twelve socks, three of each color: red, blue, green, and yellow. She randomly draws five socks. What is the probability that she has exactly two pairs of socks of different colors and one sock of a third color?
|
\frac{9}{22}
| 23.4375 |
9,682 |
Given complex numbers \( z_{1}, z_{2}, z_{3} \) such that \( \left|z_{1}\right| \leq 1 \), \( \left|z_{2}\right| \leq 1 \), and \( \left|2 z_{3}-\left(z_{1}+z_{2}\right)\right| \leq \left|z_{1}-z_{2}\right| \). What is the maximum value of \( \left|z_{3}\right| \)?
|
\sqrt{2}
| 3.125 |
9,683 |
A line with slope equal to $-1$ and a line with slope equal to $-2$ intersect at the point $P(2,5)$. What is the area of $\triangle PQR$?
|
6.25
| 0 |
9,684 |
If $a$, $b$, $c$, and $d$ are real numbers satisfying:
\begin{align*}
a+b+c &= 3, \\
a+b+d &= 9, \\
a+c+d &= 24, \text{ and} \\
b+c+d &= 15,
\end{align*}
what is $ab + cd$?
|
98
| 85.15625 |
9,685 |
From a square with a side length of $6 \text{ cm}$, identical isosceles right triangles are cut off from each corner so that the area of the square is reduced by $32\%$. What is the length of the legs of these triangles?
|
2.4
| 96.09375 |
9,686 |
Given three integers \( x, y, z \) satisfying \( x + y + z = 100 \) and \( x < y < 2z \), what is the minimum value of \( z \)?
|
21
| 26.5625 |
9,687 |
Esther and Frida are supposed to fill a rectangular array of 16 columns and 10 rows with the numbers 1 to 160. Esther fills it row-wise so that the first row is numbered 1, 2, ..., 16 and the second row is 17, 18, ..., 32 and so on. Frida fills it column-wise, so that her first column has 1, 2, ..., 10, and the second column has 11, 12, ..., 20 and so on. Comparing Esther's array with Frida's array, we notice that some numbers occupy the same position. Find the sum of the numbers in these positions.
|
322
| 21.09375 |
9,688 |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are respectively $a$, $b$, $c$. Given that $\cos B = \frac{1}{3}$, $ac = 6$, and $b = 3$.
$(1)$ Find the value of $\cos C$ for side $a$;
$(2)$ Find the value of $\cos (2C+\frac{\pi }{3})$.
|
\frac{17-56 \sqrt{6}}{162}
| 14.84375 |
9,689 |
Michael, David, Evan, Isabella, and Justin compete in the NIMO Super Bowl, a round-robin cereal-eating tournament. Each pair of competitors plays exactly one game, in which each competitor has an equal chance of winning (and there are no ties). The probability that none of the five players wins all of his/her games is $\tfrac{m}{n}$ for relatively prime positive integers $m$ , $n$ . Compute $100m + n$ .
*Proposed by Evan Chen*
|
1116
| 3.90625 |
9,690 |
A bag contains 4 identical balls, numbered 0, 1, 2, and 2. Player A draws a ball and puts it back, then player B draws a ball. If the number on the drawn ball is larger, that player wins (if the numbers are the same, it's a tie). What is the probability that player B draws the ball numbered 1, given that player A wins?
|
\frac{2}{5}
| 6.25 |
9,691 |
A point is randomly selected on a plane, where its Cartesian coordinates are integers with absolute values less than or equal to 4, and all such points are equally likely to be chosen. Find the probability that the selected point is at most 2 units away from the origin.
|
\frac{13}{81}
| 69.53125 |
9,692 |
We marked the midpoints of all sides and diagonals of a regular 1976-sided polygon. What is the maximum number of these points that can lie on a single circle?
|
1976
| 80.46875 |
9,693 |
A new city is constructed in the form of 12 rectangular blocks of houses, divided by streets; at each corner of the block and at every street intersection, there is a mailbox. The blocks either touch each other along entire sides or share only a single corner.
Assuming that there are 37 segments of streets enclosing the city blocks, i.e., 37 intervals between mailboxes, how many mailboxes are there in this city?
|
26
| 22.65625 |
9,694 |
Find the sum of the digits of all counting numbers less than 1000.
|
13500
| 97.65625 |
9,695 |
Petra has 49 blue beads and one red bead. How many beads must Petra remove so that 90% of her beads are blue?
|
40
| 29.6875 |
9,696 |
Calculate the limit of the numerical sequence:
$$\lim _{n \rightarrow \infty} \frac{\sqrt{n+6}-\sqrt{n^{2}-5}}{\sqrt[3]{n^{3}+3}+\sqrt[4]{n^{3}+1}}$$
|
-1
| 89.0625 |
9,697 |
In acute triangle $\triangle ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and $a\sin B = \frac{1}{2}b$.
$(1)$ Find angle $A$;
$(2)$ If $b+c=4\sqrt{2}$ and the area of $\triangle ABC$ is $2$, find $a$.
|
2\sqrt{3}-2
| 25.78125 |
9,698 |
Given the function $f(x)=\sin (\omega x+\varphi)$ $(0 < \omega < 3,0 < \varphi < \pi)$, if $x=-\frac{\pi}{4}$ is a zero of the function $f(x)$, and $x=\frac{\pi}{3}$ is an axis of symmetry for the graph of the function $f(x)$, then the value of $\omega$ is \_\_\_\_.
|
\frac{6}{7}
| 76.5625 |
9,699 |
Given $\cos\alpha= \frac {4}{5}$, $\cos\beta= \frac {3}{5}$, $\beta\in\left(\frac {3\pi}{2}, 2\pi\right)$, and $0<\alpha<\beta$, calculate the value of $\sin(\alpha+\beta)$.
|
-\frac{7}{25}
| 97.65625 |
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