Unnamed: 0
int64 0
40.3k
| problem
stringlengths 10
5.15k
| ground_truth
stringlengths 1
1.22k
| solved_percentage
float64 0
100
|
|---|---|---|---|
10,000 |
What is the value of $27^3 + 9(27^2) + 27(9^2) + 9^3$?
|
46656
| 44.53125 |
10,001 |
The graph of the function $f(x)=\sin (\omega x+\frac{\pi}{3})$ ($\omega>0$) is shifted to the left by $\frac{\pi}{2}$ units to obtain the curve $C$. If $C$ is symmetric about the $y$-axis, determine the minimum value of $\omega$.
|
\frac{1}{3}
| 68.75 |
10,002 |
Given a hyperbola $C$ with one of its foci on the line $l: 4x-3y+20=0$, and one of its asymptotes is parallel to $l$, and the foci of the hyperbola $C$ are on the $x$-axis, then the standard equation of the hyperbola $C$ is \_\_\_\_\_\_; the eccentricity is \_\_\_\_\_\_.
|
\dfrac{5}{3}
| 99.21875 |
10,003 |
In a 6 by 6 grid of points, what fraction of the larger rectangle's area is inside the shaded right triangle? The vertices of the shaded triangle correspond to grid points and are located at (1,1), (1,5), and (4,1).
|
\frac{1}{6}
| 76.5625 |
10,004 |
Each of the equations \( a x^{2} - b x + c = 0 \) and \( c x^{2} - a x + b = 0 \) has two distinct real roots. The sum of the roots of the first equation is non-negative, and the product of the roots of the first equation is 9 times the sum of the roots of the second equation. Find the ratio of the sum of the roots of the first equation to the product of the roots of the second equation.
|
-3
| 3.125 |
10,005 |
Consider a list of six numbers. When the largest number is removed from the list, the average is decreased by 1. When the smallest number is removed, the average is increased by 1. When both the largest and the smallest numbers are removed, the average of the remaining four numbers is 20. Find the product of the largest and the smallest numbers.
|
375
| 75.78125 |
10,006 |
Given \( \frac{\pi}{4} < \theta < \frac{\pi}{2} \), find the maximum value of \( S = \sin 2\theta - \cos^2 \theta \).
|
\frac{\sqrt{5} - 1}{2}
| 42.96875 |
10,007 |
At the moment when Pierrot left the "Commercial" bar, heading to the "Theatrical" bar, Jeannot was leaving the "Theatrical" bar on his way to the "Commercial" bar. They were walking at constant (but different) speeds. When the vagabonds met, Pierrot proudly noted that he had walked 200 meters more than Jeannot. After their fight ended, they hugged and continued on their paths but at half their previous speeds due to their injuries. Pierrot then took 8 minutes to reach the "Theatrical" bar, and Jeannot took 18 minutes to reach the "Commercial" bar. What is the distance between the bars?
|
1000
| 14.0625 |
10,008 |
A large rectangle measures 15 units by 20 units. One-quarter of this rectangle is shaded. If half of this quarter rectangle is shaded, what fraction of the large rectangle is shaded?
A) $\frac{1}{24}$
B) $\frac{1}{12}$
C) $\frac{1}{10}$
D) $\frac{1}{8}$
E) $\frac{1}{6}$
|
\frac{1}{8}
| 89.84375 |
10,009 |
Define a **valid sequence** 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 — where $A$ 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 valid sequences are there?
|
8748
| 67.1875 |
10,010 |
A round cake is cut into \( n \) pieces with 3 cuts. Find the product of all possible values of \( n \).
|
840
| 53.125 |
10,011 |
A square with an area of one square unit is inscribed in an isosceles triangle such that one side of the square lies on the base of the triangle. Find the area of the triangle, given that the centers of mass of the triangle and the square coincide (the center of mass of the triangle lies at the intersection of its medians).
|
9/4
| 23.4375 |
10,012 |
In the middle of the school year, $40\%$ of Poolesville magnet students decided to transfer to the Blair magnet, and $5\%$ of the original Blair magnet students transferred to the Poolesville magnet. If the Blair magnet grew from $400$ students to $480$ students, how many students does the Poolesville magnet have after the transferring has occurred?
|
170
| 82.8125 |
10,013 |
A university has 120 foreign language teachers. Among them, 50 teach English, 45 teach Japanese, and 40 teach French. There are 15 teachers who teach both English and Japanese, 10 who teach both English and French, and 8 who teach both Japanese and French. Additionally, 4 teachers teach all three languages: English, Japanese, and French. How many foreign language teachers do not teach any of these three languages?
|
14
| 82.03125 |
10,014 |
Find all values of \( n \in \mathbf{N} \) possessing the following property: if you write the numbers \( n^3 \) and \( n^4 \) next to each other (in decimal system), then in the resulting sequence, each of the 10 digits \( 0, 1, \ldots, 9 \) appears exactly once.
|
18
| 99.21875 |
10,015 |
Martians love dancing dances that require holding hands. In the dance "Pyramid," no more than 7 Martians can participate, each with no more than three hands. What is the maximum number of hands that can be involved in the dance if each hand of one Martian holds exactly one hand of another Martian?
|
20
| 35.15625 |
10,016 |
(Answer in numbers)
From 5 different storybooks and 4 different math books, 4 books are to be selected and given to 4 students, one book per student. How many different ways are there to:
(1) Select 2 storybooks and 2 math books?
(2) Ensure one specific storybook and one specific math book are among the selected?
(3) Ensure at least 3 of the selected books are storybooks?
|
1080
| 83.59375 |
10,017 |
In triangle \( ABC \), the angle bisectors \( AD \) and \( CE \) are drawn. It turns out that \( AE + CD = AC \). Find angle \( B \).
|
60
| 81.25 |
10,018 |
In the diagram, a large circle and a rectangle intersect such that the rectangle halves the circle with its diagonal, and $O$ is the center of the circle. The area of the circle is $100\pi$. The top right corner of the rectangle touches the circle while the other corner is at the center of the circle. Determine the total shaded area formed by the parts of the circle not included in the intersection with the rectangle. Assume the intersection forms a sector.
[Diagram not shown: Assume descriptive adequacy for the composition of the circle and the rectangle.]
|
50\pi
| 16.40625 |
10,019 |
In triangle $ABC$, we have $\angle C = 90^\circ$, $AB = 26$, and $BC = 10$. What is $\sin A$?
|
\frac{5}{13}
| 80.46875 |
10,020 |
Let \( P \) be an arbitrary point on the graph of the function \( y = x + \frac{2}{x} \) (where \( x > 0 \)). From point \( P \), perpendiculars are drawn to the line \( y = x \) and the \( y \)-axis, with the foots of these perpendiculars being points \( A \) and \( B \) respectively. Determine the value of \( \overrightarrow{PA} \cdot \overrightarrow{PB} \).
|
-1
| 77.34375 |
10,021 |
The year 2009 has a unique property: by rearranging the digits of the number 2009, it is impossible to form a smaller four-digit number (numbers do not start with zero). In which future year will this property first repeat again?
|
2022
| 18.75 |
10,022 |
The bending resistance of a beam with a rectangular cross-section is proportional to the product of its width and the square of its height. What should be the width of the beam section, cut from a round log with a diameter of \( 15 \sqrt{3} \), to maximize its bending resistance?
|
15
| 93.75 |
10,023 |
Five fair six-sided dice are rolled. What is the probability that at least three of the five dice show the same value?
|
\frac{23}{108}
| 11.71875 |
10,024 |
In triangle $ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively, and it is given that $$2\sin^{2} \frac {A+B}{2}+\cos2C=1$$
(1) Find the magnitude of angle $C$;
(2) If vector $$\overrightarrow {m}=(3a，b)$$ and vector $$\overrightarrow {n}=(a，- \frac {b}{3})$$, with $$\overrightarrow {m} \perp \overrightarrow {n}$$ and $$( \overrightarrow {m}+ \overrightarrow {n})(- \overrightarrow {m}+ \overrightarrow {n})=-16$$, find the values of $a$, $b$, and $c$.
|
\sqrt {7}
| 0 |
10,025 |
Xiao Hong asked Da Bai: "Please help me calculate the result of $999 \quad 9 \times 999 \quad 9$ and determine how many zeros appear in it."
2019 nines times 2019 nines
Da Bai quickly wrote a program to compute it. Xiao Hong laughed and said: "You don't need to compute the exact result to know how many zeros there are. I'll tell you it's....."
After calculating, Da Bai found Xiao Hong's answer was indeed correct. Xiao Hong's answer is $\qquad$.
|
2018
| 46.875 |
10,026 |
Calculate $(-1)^{45} + 2^{(3^2+5^2-4^2)}$.
|
262143
| 79.6875 |
10,027 |
A local community club consists of four leaders and a number of regular members. Each year, the current leaders leave the club, and every regular member is responsible for recruiting three new members. At the end of the year, four new leaders are elected from outside the club. Initially, there are 20 people in total in the club. How many people will be in the club after four years?
|
4100
| 2.34375 |
10,028 |
Given $f(\alpha)= \frac{\sin (\alpha-3\pi)\cos (2\pi-\alpha)\cdot\sin (-\alpha+ \frac{3}{2}\pi)}{\cos (-\pi-\alpha)\sin (-\pi-\alpha)}$.
$(1)$ Simplify $f(\alpha)$.
$(2)$ If $\alpha$ is an angle in the third quadrant, and $\cos \left(\alpha- \frac{3}{2}\pi\right)= \frac{1}{5}$, find the value of $f(\alpha)$.
|
\frac{2\sqrt{6}}{5}
| 54.6875 |
10,029 |
In the arithmetic sequence $\{a_n\}$, $a_1 > 0$, and $S_n$ is the sum of the first $n$ terms, and $S_9=S_{18}$, find the value of $n$ at which $S_n$ is maximized.
|
13
| 53.125 |
10,030 |
Calculate the value of
\[\left(\left(\left((3+2)^{-1}+1\right)^{-1}+2\right)^{-1}+1\right)^{-1}+1.\]
A) $\frac{40}{23}$
B) $\frac{17}{23}$
C) $\frac{23}{17}$
D) $\frac{23}{40}$
|
\frac{40}{23}
| 78.90625 |
10,031 |
Given in the polar coordinate system, circle $C$: $p=2\cos (\theta+ \frac {\pi}{2})$ and line $l$: $\rho\sin (\theta+ \frac {\pi}{4})= \sqrt {2}$, point $M$ is a moving point on circle $C$. Find the maximum distance from point $M$ to line $l$.
|
\frac {3 \sqrt {2}}{2}+1
| 0 |
10,032 |
Using the four arithmetic operators and parentheses, find a way to combine the numbers 10, 10, 4, and 2 such that the result is 24. What is the arithmetic expression?
|
(2 + 4 \div 10) \times 10
| 0 |
10,033 |
In the Cartesian coordinate system $xOy$, there is a circle $C_{1}$: $(x-2)^{2}+(y-4)^{2}=20$. With the origin $O$ as the pole and the positive half-axis of $x$ as the polar axis, a polar coordinate system is established. For $C_{2}$: $\theta= \frac {\pi}{3}(\rho\in\mathbb{R})$.
$(1)$ Find the polar equation of $C_{1}$ and the Cartesian coordinate equation of $C_{2}$;
$(2)$ If the polar equation of line $C_{3}$ is $\theta= \frac {\pi}{6}(\rho\in\mathbb{R})$, and assuming the intersection points of $C_{2}$ and $C_{1}$ are $O$ and $M$, and the intersection points of $C_{3}$ and $C_{1}$ are $O$ and $N$, find the area of $\triangle OMN$.
|
8+5\sqrt {3}
| 0 |
10,034 |
In a unit cube \( ABCD-A_1B_1C_1D_1 \), let \( O \) be the center of the square \( ABCD \). Points \( M \) and \( N \) are located on edges \( A_1D_1 \) and \( CC_1 \) respectively, with \( A_1M = \frac{1}{2} \) and \( CN = \frac{2}{3} \). Find the volume of the tetrahedron \( OMNB_1 \).
|
11/72
| 6.25 |
10,035 |
Given a square $A B C D$ on a plane, find the minimum of the ratio $\frac{O A + O C}{O B + O D}$, where $O$ is an arbitrary point on the plane.
|
\frac{1}{\sqrt{2}}
| 1.5625 |
10,036 |
When $x \in \left[-\frac{\pi}{3}, \frac{\pi}{3}\right]$, find the minimum value of the function $f(x) = \sqrt{2}\sin \frac{x}{4}\cos \frac{x}{4} + \sqrt{6}\cos^2 \frac{x}{4} - \frac{\sqrt{6}}{2}$.
|
\frac{\sqrt{2}}{2}
| 60.9375 |
10,037 |
What is the value of
$$
(x+1)(x+2006)\left[\frac{1}{(x+1)(x+2)}+\frac{1}{(x+2)(x+3)}+\ldots+\frac{1}{(x+2005)(x+2006)}\right] ?
$$
|
2005
| 73.4375 |
10,038 |
There are a theory part and an experimental part in the assessment of a certain course. Each part of the assessment is graded only as "pass" or "fail", and the whole course assessment is considered "pass" if both parts are passed. The probabilities of passing the theory assessment for A, B, and C are 0.9, 0.8, and 0.7 respectively; while the probabilities of passing the experimental assessment are 0.8, 0.7, and 0.9 respectively. The outcomes of all the assessments are independent.
(Ⅰ) Find the probability that at least two among A, B, and C pass the theory assessment.
(Ⅱ) Calculate the probability that all three of them pass the entire course assessment (round the result to three decimal places).
|
0.254
| 59.375 |
10,039 |
A piece of paper with a thickness of $0.1$ millimeters is folded once, resulting in a thickness of $2 \times 0.1$ millimeters. Continuing to fold it $2$ times, $3$ times, $4$ times, and so on, determine the total thickness after folding the paper $20$ times and express it as the height of a building, with each floor being $3$ meters high.
|
35
| 34.375 |
10,040 |
Given \(\alpha \in (0, \pi)\), if \(\sin \alpha + \cos \alpha = \frac{\sqrt{3}}{3}\), then \(\cos^2 \alpha - \sin^2 \alpha = \)
|
-\frac{\sqrt{5}}{3}
| 29.6875 |
10,041 |
In the rectangular coordinate system $(xOy)$, the parametric equations of the curve $C\_1$ are given by $\begin{cases}x=\sqrt{2}\sin(\alpha+\frac{\pi}{4}) \\ y=\sin 2\alpha+1\end{cases}$ (where $\alpha$ is a parameter). In the polar coordinate system with $O$ as the pole and the positive half of the $x$-axis as the polar axis, the curve $C\_2$ has equation $\rho^2=4\rho\sin\theta-3$.
(1) Find the Cartesian equation of the curve $C\_1$ and the polar equation of the curve $C\_2$.
(2) Find the minimum distance between a point on the curve $C\_1$ and a point on the curve $C\_2$.
|
\frac{\sqrt{7}}{2}-1
| 66.40625 |
10,042 |
Given the function \(f(x)=\sin ^{4} \frac{k x}{10}+\cos ^{4} \frac{k x}{10}\), where \(k\) is a positive integer, if for any real number \(a\), it holds that \(\{f(x) \mid a<x<a+1\}=\{f(x) \mid x \in \mathbb{R}\}\), find the minimum value of \(k\).
|
16
| 13.28125 |
10,043 |
Circles of radius 4 and 5 are externally tangent and are circumscribed by a third circle. Calculate the area of the region outside the smaller circles but inside the larger circle.
|
40\pi
| 17.96875 |
10,044 |
Let $\{a_{n}\}$ be a geometric sequence, and let $S_{n}$ be the sum of the first n terms of $\{a_{n}\}$. Given that $S_{2}=2$ and $S_{6}=4$, calculate the value of $S_{4}$.
|
1+\sqrt{5}
| 3.125 |
10,045 |
Given positive numbers $x$ and $y$ satisfying $x+y=2$, calculate the minimum value of $\frac{4}{x+2}+\frac{3x-7}{3y+4}$.
|
\frac{11}{16}
| 6.25 |
10,046 |
Given the function $f(x)$ with the domain $[1, +\infty)$, and $f(x) = \begin{cases} 1-|2x-3|, & 1\leq x<2 \\ \frac{1}{2}f\left(\frac{1}{2}x\right), & x\geq 2 \end{cases}$, then the number of zeros of the function $y=2xf(x)-3$ in the interval $(1, 2017)$ is \_\_\_\_\_\_.
|
11
| 25 |
10,047 |
Given a quadratic function $f(x) = x^2 + mx + n$.
1. If $f(x)$ is an even function and its minimum value is 1, find the expression for $f(x)$.
2. Based on (1), for the function $g(x) = \frac{6x}{f(x)}$, solve the inequality $g(2^x) > 2^x$ for $x$.
3. Let $h(x) = |f(x)|$ and assume that for $x \in [-1, 1]$, the maximum value of $h(x)$ is $M$, such that $M \geq k$ holds true for any real numbers $m$ and $n$. Find the maximum value of $k$.
|
\frac{1}{2}
| 3.90625 |
10,048 |
Josef, Timothy, and Anna play a game. Josef picks an integer between 1 and 1440 inclusive. Timothy states whether the quotient of 1440 divided by Josef's integer is an integer. Finally, Anna adds a twist by stating whether Josef's integer is a multiple of 5. How many integers could Josef pick such that both Timothy's and Anna's conditions are satisfied?
|
18
| 35.15625 |
10,049 |
Evaluate $180 \div \left(12 + 13 \cdot 2\right)$.
|
\frac{90}{19}
| 35.15625 |
10,050 |
What is the smallest positive value of $m$ so that the equation $10x^2 - mx + 180 = 0$ has integral solutions?
|
90
| 7.8125 |
10,051 |
Person A and Person B started working on the same day. The company policy states that Person A works for 3 days and then rests for 1 day, while Person B works for 7 days and then rests for 3 consecutive days. How many days do Person A and Person B rest on the same day within the first 1000 days?
|
100
| 19.53125 |
10,052 |
A trapezoid inscribed in a circle with a radius of $13 \mathrm{~cm}$ has its diagonals located $5 \mathrm{~cm}$ away from the center of the circle. What is the maximum possible area of the trapezoid?
|
288
| 2.34375 |
10,053 |
Given that $\binom{18}{7}=31824$, $\binom{18}{8}=43758$ and $\binom{18}{9}=43758$, calculate $\binom{20}{9}$.
|
163098
| 86.71875 |
10,054 |
In an isosceles trapezoid, the height is 10, and the diagonals are mutually perpendicular. Find the midsegment (the line connecting the midpoints of the non-parallel sides) of the trapezoid.
|
10
| 54.6875 |
10,055 |
When skipping rope, the midpoint of the rope can be considered to move along the same circle. If Xiaoguang takes 0.5 seconds to complete a "single skip" and 0.6 seconds to complete a "double skip," what is the ratio of the speed of the midpoint of the rope during a "single skip" to the speed during a "double skip"?
(Note: A "single skip" is when the feet leave the ground once and the rope rotates one circle; a "double skip" is when the feet leave the ground once and the rope rotates two circles.)
|
3/5
| 78.125 |
10,056 |
A digit was crossed out from a six-digit number, resulting in a five-digit number. When this five-digit number was subtracted from the original six-digit number, the result was 654321. Find the original six-digit number.
|
727023
| 74.21875 |
10,057 |
Let \( p \) and \( q \) be positive integers such that \( \frac{5}{8} < \frac{p}{q} < \frac{7}{8} \). What is the smallest value of \( p \) so that \( p + q = 2005 \)?
|
772
| 81.25 |
10,058 |
The number of games won by six basketball teams are displayed in the graph, but the names of the teams are missing. The following clues provide information about the teams:
1. The Hawks won more games than the Falcons.
2. The Warriors won more games than the Knights, but fewer games than the Royals.
3. The Knights won more than 30 games.
4. The Squires tied with the Falcons.
How many games did the Warriors win? [asy]
size(150);
defaultpen(linewidth(0.7pt)+fontsize(8));
int i = 1;
draw((0,i*10)--(80,i*10)); ++i;
fill(shift(12,0)*((4,0)--(4,10)--(8,10)--(8,0)--cycle),purple);
draw(shift(12,0)*((4,0)--(4,10)^^(8,0)--(8,10)));
draw((0,i*10)--(80,i*10)); ++i;
fill((4,0)--(4,20)--(8,20)--(8,0)--cycle,purple);
draw((4,0)--(4,20));
draw((8,0)--(8,20));
for(i = 3; i <= 4; ++i)
{
draw((0,i*10)--(80,i*10));
}
fill(shift(24,0)*((4,0)--(4,35)--(8,35)--(8,0)--cycle),purple);
draw(shift(24,0)*((4,0)--(4,35)^^(8,0)--(8,35)));
draw((0,i*10)--(80,i*10)); ++i;
fill(shift(36,0)*((4,0)--(4,40)--(8,40)--(8,0)--cycle),purple);
draw(shift(36,0)*((4,0)--(4,40)^^(8,0)--(8,40)));
draw((0,i*10)--(80,i*10)); ++i;
fill(shift(48,0)*((4,0)--(4,50)--(8,50)--(8,0)--cycle),purple);
draw(shift(48,0)*((4,0)--(4,50)^^(8,0)--(8,50)));
draw((0,i*10)--(80,i*10)); ++i;
fill(shift(60,0)*((4,0)--(4,50)--(8,50)--(8,0)--cycle),purple);
draw(shift(60,0)*((4,0)--(4,50)^^(8,0)--(8,50)));
draw((0,i*10)--(80,i*10));
xaxis(Bottom,0,80,RightTicks(" ",N=6,n=1,Size=2));
yaxis(Left,0,60,LeftTicks(Step=10,Size=2));
yaxis(Right,0,60);
label("Basketball Results",(40,66));
label(rotate(90)*"Number of Wins",(-10,30));
label("Teams",(40,-10));
for(i = 0; i < 6; ++i)
{
label("?",(6+12*i,-4));
}
[/asy]
|
40
| 21.09375 |
10,059 |
Calculate $\frac{1586_{7}}{131_{5}}-3451_{6}+2887_{7}$. Express your answer in base 10.
|
334
| 5.46875 |
10,060 |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given $a\sin A + c\sin C - \sqrt{2}a\sin C = b\sin B$.
1. Find $B$.
2. If $\cos A = \frac{1}{3}$, find $\sin C$.
|
\frac{4 + \sqrt{2}}{6}
| 50.78125 |
10,061 |
Given cos(π/4 - α) = 3/5 and sin(5π/4 + β) = -12/13, where α ∈ (π/4, 3π/4) and β ∈ (0, π/4), find tan(α)/tan(β).
|
-17
| 32.8125 |
10,062 |
Given that $\frac{x}{9}, \frac{y}{15}, \frac{z}{14}$ are all in their simplest forms and their product is $\frac{1}{6}$, find the value of $x+y+z$.
|
21
| 47.65625 |
10,063 |
Given an ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ with eccentricity $e = \frac{\sqrt{6}}{3}$, the distance from the origin to the line passing through points $A(0, -b)$ and $B(a, 0)$ is $\frac{\sqrt{3}}{2}$.
$(1)$ Find the equation of the ellipse;
$(2)$ Given a fixed point $E(-1, 0)$, if the line $y = kx + 2 (k \neq 0)$ intersects the ellipse at points $C$ and $D$, is there a value of $k$ such that the circle with diameter $CD$ passes through point $E$? Please explain your reasoning.
|
\frac{7}{6}
| 14.84375 |
10,064 |
The slope angle of the tangent line to the curve $y= \frac{1}{2}x^{2}$ at the point $(1, \frac{1}{2})$ is
|
\frac{\pi}{4}
| 85.15625 |
10,065 |
Given $\sin\theta + \cos\theta = \frac{1}{5}$, with $\theta \in (0,\pi)$,
1. Find the value of $\tan\theta$;
2. Find the value of $\frac{1+\sin 2\theta + \cos 2\theta}{1+\sin 2\theta - \cos 2\theta}$.
|
-\frac{3}{4}
| 78.125 |
10,066 |
The slope angle of the line passing through points M(-3, 2) and N(-2, 3) is equal to what angle, measured in radians.
|
\frac{\pi}{4}
| 85.15625 |
10,067 |
$$\frac {4}{5} + 9 \frac {4}{5} + 99 \frac {4}{5} + 999 \frac {4}{5} + 9999 \frac {4}{5} + 1 = \_\_\_\_\_\_.$$
|
11111
| 56.25 |
10,068 |
If \( S = \sum_{k=1}^{99} \frac{(-1)^{k+1}}{\sqrt{k(k+1)}(\sqrt{k+1}-\sqrt{k})} \), find the value of \( 1000 S \).
|
1100
| 7.03125 |
10,069 |
A six-digit number 111aaa is the product of two consecutive positive integers \( b \) and \( b+1 \). Find the value of \( b \).
|
333
| 53.125 |
10,070 |
There are 29 ones written on a board. Each minute, Karlson erases any two numbers and writes their sum on the board, then eats a number of candies equal to the product of the two erased numbers. What is the maximum number of candies he could eat in 29 minutes?
|
406
| 89.0625 |
10,071 |
In the Cartesian coordinate system \( xOy \), find the area of the region defined by the inequalities
\[
y^{100}+\frac{1}{y^{100}} \leq x^{100}+\frac{1}{x^{100}}, \quad x^{2}+y^{2} \leq 100.
\]
|
50 \pi
| 53.90625 |
10,072 |
A person flips a coin, where the probability of heads up and tails up is $\frac{1}{2}$ each. Construct a sequence $\left\{a_{n}\right\}$ such that
$$
a_{n}=\left\{
\begin{array}{ll}
1, & \text{if the } n \text{th flip is heads;} \\
-1, & \text{if the } n \text{th flip is tails.}
\end{array}
\right.
$$
Let $S_{n}=a_{1}+a_{2}+\cdots+a_{n}$. Find the probability that $S_{2} \neq 0$ and $S_{8}=2$. Provide your answer in its simplest fractional form.
|
13/128
| 15.625 |
10,073 |
Suppose \(198 \cdot 963 \equiv m \pmod{50}\), where \(0 \leq m < 50\).
What is the value of \(m\)?
|
24
| 58.59375 |
10,074 |
If $M = 1764 \div 4$, $N = M \div 4$, and $X = M - N$, what is the value of $X$?
|
330.75
| 44.53125 |
10,075 |
Given that a geometric sequence $\{a_n\}$ consists of positive terms, and $(a_3, \frac{1}{2}a_5,a_4)$ form an arithmetic sequence, find the value of $\frac{a_3+a_5}{a_4+a_6}$.
|
\frac{\sqrt{5}-1}{2}
| 87.5 |
10,076 |
Find the largest five-digit number whose digits' product equals 120.
|
85311
| 88.28125 |
10,077 |
In triangle $ABC,$ angle bisectors $\overline{AD}$ and $\overline{BE}$ intersect at $P.$ If $AB = 8,$ $AC = 6,$ and $BC = 4,$ find $\frac{BP}{PE}.$
|
\frac{3}{2}
| 0 |
10,078 |
Given $x$ and $y \in \mathbb{R}$, if $x^2 + y^2 + xy = 315$, find the minimum value of $x^2 + y^2 - xy$.
|
105
| 83.59375 |
10,079 |
Determine the largest multiple of 36 that consists of all even and distinct digits.
|
8640
| 38.28125 |
10,080 |
Ivan Petrovich wants to save money for his retirement in 12 years. He decided to deposit 750,000 rubles in a bank account with an 8 percent annual interest rate. What will be the total amount in the account by the time Ivan Petrovich retires, assuming the interest is compounded annually using the simple interest formula?
|
1470000
| 19.53125 |
10,081 |
There are three pairs of real numbers $(x_1,y_1)$, $(x_2,y_2)$, and $(x_3,y_3)$ that satisfy both $x^3 - 3xy^2 = 2017$ and $y^3 - 3x^2y = 2016$. Compute $\left(2 - \frac{x_1}{y_1}\right)\left(2 - \frac{x_2}{y_2}\right)\left(2 - \frac{x_3}{y_3}\right)$.
|
\frac{26219}{2016}
| 1.5625 |
10,082 |
Given the positive integer \( A = \overline{a_{n} a_{n-1} \cdots a_{1} a_{0}} \), where \( a_{n}, a_{n-1}, \ldots, a_{0} \) are all non-zero and not all equal (with \( n \) being a positive integer), consider the following cyclic permutations of \( A \):
$$
\begin{array}{l}
A_{1}=\overline{a_{n-1} \cdots a_{1} a_{0} a_{n}}, \\
A_{2}=\overline{a_{n-2} \cdots a_{1} a_{0} a_{n} a_{n-1}}, \\
\cdots \cdots \\
A_{k}=\overline{a_{n-k} a_{n-k-1} \cdots a_{0} a_{n} \cdots a_{n-k+1}}, \\
\cdots \cdots \\
A_{n}=\overline{a_{0} a_{n} \cdots a_{1}}
\end{array}
$$
Determine \( A \) such that \( A \) divides each \( A_{k} \) for \( k=1,2, \cdots, n \).
|
142857
| 49.21875 |
10,083 |
Given a function $f(x)$ that satisfies the functional equation $f(x) = f(x+1) - f(x+2)$ for all $x \in \mathbb{R}$. When $x \in (0,3)$, $f(x) = x^2$. Express the value of $f(2014)$ using the functional equation.
|
-1
| 32.8125 |
10,084 |
Ilya Muromets encounters the three-headed Dragon, Gorynych. Each minute, Ilya chops off one head of the dragon. Let $x$ be the dragon's resilience ($x > 0$). The probability $p_{s}$ that $s$ new heads will grow in place of a chopped-off one ($s=0,1,2$) is given by $\frac{x^{s}}{1+x+x^{2}}$. During the first 10 minutes of the battle, Ilya recorded the number of heads that grew back for each chopped-off one. The vector obtained is: $K=(1,2,2,1,0,2,1,0,1,2)$. Find the value of the dragon's resilience $x$ that maximizes the probability of vector $K$.
|
\frac{1 + \sqrt{97}}{8}
| 1.5625 |
10,085 |
In a company, employees have a combined monthly salary of $10,000. A kind manager proposes to triple the salaries of those earning up to $500, and to increase the salaries of others by $1,000, resulting in a total salary of $24,000. A strict manager proposes to reduce the salaries of those earning more than $500 to $500, while leaving others' salaries unchanged. What will the total salary be in this case?
|
7000
| 51.5625 |
10,086 |
Let $a \bowtie b = a + \sqrt{b + \sqrt{b + \sqrt{b + \ldots}}}$. If $5 \bowtie x = 12$, find the value of $x$.
|
42
| 96.09375 |
10,087 |
Given the relationship between shelf life and storage temperature is an exponential function $y = ka^x$, where milk has a shelf life of about $100$ hours in a refrigerator at $0°C$, and about $80$ hours in a refrigerator at $5°C$, determine the approximate shelf life of milk in a refrigerator at $10°C$.
|
64
| 100 |
10,088 |
Given a moving point $M$ whose distance to the point $F(0,1)$ is equal to its distance to the line $y=-1$, the trajectory of point $M$ is denoted as $C$.
(1) Find the equation of the trajectory $C$.
(2) Let $P$ be a point on the line $l: x-y-2=0$. Construct two tangents $PA$ and $PB$ from point $P$ to the curve $C$.
(i) When the coordinates of point $P$ are $\left(\frac{1}{2},-\frac{3}{2}\right)$, find the equation of line $AB$.
(ii) When point $P(x_{0},y_{0})$ moves along the line $l$, find the minimum value of $|AF|\cdot|BF|$.
|
\frac{9}{2}
| 25.78125 |
10,089 |
Given that there are $3$ red cubes, $4$ blue cubes, $2$ green cubes, and $2$ yellow cubes, determine the total number of different towers with a height of $10$ cubes that can be built, with the condition that the tower must always have a yellow cube at the top.
|
1,260
| 0 |
10,090 |
Find the difference between the largest integer solution of the equation \(\lfloor \frac{x}{3} \rfloor = 102\) and the smallest integer solution of the equation \(\lfloor \frac{x}{3} \rfloor = -102\).
|
614
| 64.0625 |
10,091 |
Compute the prime factorization of \(1007021035035021007001\). (You should write your answer in the form \(p_{1}^{e_{1}} p_{2}^{e_{2}} \ldots p_{k}^{e_{k}}\), where \(p_{1}, \ldots, p_{k}\) are distinct prime numbers and \(e_{1}, \ldots, e_{k}\) are positive integers.)
|
7^7 * 11^7 * 13^7
| 0 |
10,092 |
A food factory has made 4 different exquisite cards. Each bag of food produced by the factory randomly contains one card. If all 4 different cards are collected, a prize can be won. Xiaoming buys 6 bags of this food at once. What is the probability that Xiaoming will win the prize?
|
195/512
| 17.96875 |
10,093 |
Out of two hundred ninth-grade students, $80\%$ received excellent grades on the first exam, $70\%$ on the second exam, and $59\%$ on the third exam. What is the minimum number of students who could have received excellent grades on all three exams?
|
18
| 58.59375 |
10,094 |
The number of integers between 1 and 100 that are not divisible by 2, 3, or 5 is \( (\ \ \ ).
|
26
| 94.53125 |
10,095 |
In $ \triangle ABC$ points $ D$ and $ E$ lie on $ \overline{BC}$ and $ \overline{AC}$ , respectively. If $ \overline{AD}$ and $ \overline{BE}$ intersect at $ T$ so that $ AT/DT \equal{} 3$ and $ BT/ET \equal{} 4$ , what is $ CD/BD$ ?
[asy]unitsize(2cm);
defaultpen(linewidth(.8pt));
pair A = (0,0);
pair C = (2,0);
pair B = dir(57.5)*2;
pair E = waypoint(C--A,0.25);
pair D = waypoint(C--B,0.25);
pair T = intersectionpoint(D--A,E--B);
label(" $B$ ",B,NW);label(" $A$ ",A,SW);label(" $C$ ",C,SE);label(" $D$ ",D,NE);label(" $E$ ",E,S);label(" $T$ ",T,2*W+N);
draw(A--B--C--cycle);
draw(A--D);
draw(B--E);[/asy]
|
$ \frac {4}{11}$
| 0 |
10,096 |
In the country of Taxonia, each person pays as many thousandths of their salary in taxes as the number of tugriks that constitutes their salary. What salary is most advantageous to have?
(Salary is measured in a positive number of tugriks, not necessarily an integer.)
|
500
| 64.0625 |
10,097 |
In acute triangle $ABC$, $\sin A=\frac{2\sqrt{2}}{3}$. Find the value of $\sin^{2}\frac{B+C}{2}+\cos (3\pi -2A)$.
|
\frac{13}{9}
| 27.34375 |
10,098 |
Given a wire of length \(150 \mathrm{~cm}\) that needs to be cut into \(n (n>2)\) smaller pieces, with each piece being an integer length of at least \(1 \mathrm{~cm}\). If any 3 pieces cannot form a triangle, what is the maximum value of \(n\)?
|
10
| 75 |
10,099 |
Teacher Xixi and teacher Shanshan are teachers in the senior and junior classes of a kindergarten, respectively. Teacher Xixi prepared a large bag of apples to distribute to her students, giving exactly 3 apples to each child; teacher Shanshan prepared a large bag of oranges to distribute to her students, giving exactly 5 oranges to each child. However, they mistakenly took each other's bags. In the end, teacher Xixi distributed 3 oranges to each child, but was short of 12 oranges; teacher Shanshan distributed 6 apples to each child, using up all the apples. How many apples did teacher Xixi prepare?
|
72
| 57.8125 |
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