Unnamed: 0
int64 0
40.3k
| problem
stringlengths 10
5.15k
| ground_truth
stringlengths 1
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| solved_percentage
float64 0
100
|
|---|---|---|---|
7,400 |
Select 4 out of the numbers $1, 2, 3, 4, 5$ to form a four-digit number. What is the average value of these four-digit numbers?
|
3333
| 96.875 |
7,401 |
Given that the coordinates of a point on the terminal side of angle $\alpha$ are $(\frac{\sqrt{3}}{2},-\frac{1}{2})$, determine the smallest positive value of angle $\alpha$.
|
\frac{11\pi}{6}
| 88.28125 |
7,402 |
Christina draws a pair of concentric circles. She draws chords $\overline{DE}$, $\overline{EF}, \ldots$ of the larger circle, each chord being tangent to the smaller circle. If $m\angle DEF = 85^\circ$, how many segments will she draw before returning to her starting point at $D$?
|
72
| 25.78125 |
7,403 |
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 $a\cos B=(3c-b)\cos A$.
$(1)$ Find the value of $\cos A$;
$(2)$ If $b=3$, and point $M$ is on the line segment $BC$, $\overrightarrow{AB} + \overrightarrow{AC} = 2\overrightarrow{AM}$, $|\overrightarrow{AM}| = 3\sqrt{2}$, find the area of $\triangle ABC$.
|
7\sqrt {2}
| 0 |
7,404 |
Find a six-digit number where the first digit is 6 times less than the sum of all the digits to its right, and the second digit is 6 times less than the sum of all the digits to its right.
|
769999
| 39.0625 |
7,405 |
There are \( n \) players participating in a round-robin chess tournament, where each player competes exactly once against every other player. The winner of a match earns 3 points, a draw gives each player 1 point, and the loser earns 0 points. If the total sum of all players' points is 120, what is the maximum number of participants in the tournament?
|
11
| 21.875 |
7,406 |
Simplify first, then evaluate: $\left(\frac{{a}^{2}-1}{a-3}-a-1\right) \div \frac{a+1}{{a}^{2}-6a+9}$, where $a=3-\sqrt{2}$.
|
-2\sqrt{2}
| 70.3125 |
7,407 |
Given that \( a \) is an integer, if \( 50! \) is divisible by \( 2^a \), find the largest possible value of \( a \).
|
47
| 100 |
7,408 |
A gardener wants to plant 3 maple trees, 4 oak trees, and 5 birch trees in a row. He will randomly determine the order of these trees. What is the probability that no two birch trees are adjacent?
|
7/99
| 47.65625 |
7,409 |
Determine the sum of the fifth and sixth elements in Row 20 of Pascal's triangle.
|
20349
| 6.25 |
7,410 |
Given that \( f(x) \) is an odd function defined on \( \mathbf{R} \), and for any \( x \in \mathbf{R} \), the following holds:
$$
f(2+x) + f(2-x) = 0.
$$
When \( x \in [-1, 0) \), it is given that
$$
f(x) = \log_{2}(1-x).
$$
Find \( f(1) + f(2) + \cdots + f(2021) \).
|
-1
| 46.875 |
7,411 |
Upon cutting a certain rectangle in half, you obtain two rectangles that are scaled down versions of the original. What is the ratio of the longer side length to the shorter side length?
|
\sqrt{2}
| 81.25 |
7,412 |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. It is given that $2\sin 2A+\sin (A-B)=\sin C$, and $A\neq \frac{\pi}{2}$.
- (I) Find the value of $\frac{a}{b}$;
- (II) If $c=2$ and $C= \frac{\pi}{3}$, find the area of $\triangle ABC$.
|
\frac{2 \sqrt {3}}{3}
| 0 |
7,413 |
Inside an isosceles triangle \( ABC \), a point \( K \) is marked such that \( CK = AB = BC \) and \(\angle KAC = 30^\circ\). Find the angle \( AKB \).
|
150
| 42.96875 |
7,414 |
In a tournament, there are 16 chess players. Determine the number of different possible schedules for the first round (schedules are considered different if they differ by the participants of at least one match; the color of the pieces and the board number are not considered).
|
2027025
| 61.71875 |
7,415 |
Angry reviews about the work of an online store are left by $80\%$ of dissatisfied customers (those who were poorly served in the store). Of the satisfied customers, only $15\%$ leave a positive review. A certain online store earned 60 angry and 20 positive reviews. Using this statistic, estimate the probability that the next customer will be satisfied with the service in this online store.
|
0.64
| 22.65625 |
7,416 |
Given the function $y=\left(m+1\right)x^{|m|}+n-3$ with respect to $x$:<br/>$(1)$ For what values of $m$ and $n$ is the function a linear function of $x$?<br/>$(2)$ For what values of $m$ and $n$ is the function a proportional function of $x$?
|
n=3
| 0 |
7,417 |
Determine the area of a triangle with side lengths 7, 7, and 5.
|
\frac{5\sqrt{42.75}}{2}
| 0 |
7,418 |
Given that positive integers \( a, b, c \) (\( a < b < c \)) form a geometric sequence, and
\[ \log_{2016} a + \log_{2016} b + \log_{2016} c = 3, \]
find the maximum value of \( a + b + c \).
|
4066273
| 13.28125 |
7,419 |
99 dwarfs stand in a circle, some of them wear hats. There are no adjacent dwarfs in hats and no dwarfs in hats with exactly 48 dwarfs standing between them. What is the maximal possible number of dwarfs in hats?
|
33
| 63.28125 |
7,420 |
31 cars simultaneously started from one point on a circular track: the first car at a speed of 61 km/h, the second at 62 km/h, and so on (the 31st at 91 km/h). The track is narrow, and if one car overtakes another on a lap, they collide, both go off the track, and are out of the race. In the end, one car remains. At what speed is it traveling?
|
76
| 14.84375 |
7,421 |
Multiply the sum of $158.23$ and $47.869$ by $2$, then round your answer to the nearest tenth.
|
412.2
| 15.625 |
7,422 |
In an acute triangle $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are denoted as $a$, $b$, $c$ respectively. Given $a=4$, $b=5$, and the area of $\triangle ABC$ is $5\sqrt{3}$, find the values of $c$ and $\sin A$.
|
\frac{2\sqrt{7}}{7}
| 7.03125 |
7,423 |
There are $5$ different books to be distributed among three students, with each student receiving at least $1$ book and at most $2$ books. The number of different distribution methods is $\_\_\_\_\_\_\_\_$.
|
90
| 8.59375 |
7,424 |
Let \( x_{1} \) and \( x_{2} \) be the largest roots of the polynomials \( f(x) = 1 - x - 4x^{2} + x^{4} \) and \( g(x) = 16 - 8x - 16x^{2} + x^{4} \), respectively. Find \( \frac{x_{1}}{x_{2}} \).
|
1/2
| 98.4375 |
7,425 |
Let \( x_{1}, x_{2} \) be the roots of the equation \( x^{2} - x - 3 = 0 \). Find \(\left(x_{1}^{5} - 20\right) \cdot \left(3 x_{2}^{4} - 2 x_{2} - 35\right)\).
|
-1063
| 92.96875 |
7,426 |
In a circle centered at $O$, point $A$ is on the circle, and $\overline{BA}$ is tangent to the circle at $A$. Triangle $ABC$ is right-angled at $A$ with $\angle ABC = 45^\circ$. The circle intersects $\overline{BO}$ at $D$. Chord $\overline{BC}$ also extends to meet the circle at another point, $E$. What is the value of $\frac{BD}{BO}$?
A) $\frac{3 - \sqrt{3}}{2}$
B) $\frac{\sqrt{3}}{2}$
C) $\frac{2 - \sqrt{2}}{2}$
D) $\frac{1}{2}$
E) $\frac{\sqrt{2}}{2}$
|
\frac{2 - \sqrt{2}}{2}
| 46.09375 |
7,427 |
The specific heat capacity of a body with mass $m=2$ kg depends on the temperature as follows: $c=c_{0}(1+\alpha t)$, where $c_{0}=150$ J/kg$\cdot^\circ$C is the specific heat capacity at $0^\circ$C, $\alpha=0.05\ ^\circ$C$^{-1}$ is the temperature coefficient, and $t$ is the temperature in degrees Celsius. Determine the amount of heat that needs to be supplied to the body to heat it from $20^\circ$C to $100^\circ$C.
|
96000
| 35.15625 |
7,428 |
The square was cut into 25 smaller squares, of which exactly one has a side length different from 1 (each of the others has a side length of 1).
Find the area of the original square.
|
49
| 14.0625 |
7,429 |
In a $3 \times 3$ grid (each cell is a $1 \times 1$ square), two identical pieces are placed, with at most one piece per cell. There are ___ distinct ways to arrange the pieces. (If two arrangements can be made to coincide by rotation, they are considered the same arrangement).
|
10
| 82.03125 |
7,430 |
Consider the function $g(x)$ satisfying
\[ g(xy) = 2g(x)g(y) \]
for all real numbers $x$ and $y$ and $g(0) = 2.$ Find $g(10)$.
|
\frac{1}{2}
| 31.25 |
7,431 |
Suppose that the lines \(l_1\) and \(l_2\) are parallel, and on \(l_1\) and \(l_2\) there are 10 points \(A_1, A_2, \dots, A_{10}\) and \(B_1, B_2, \dots, B_{10}\), respectively. The line segments \(A_1 B_1, A_2 B_2, \dots, A_{10} B_{10}\) can divide the strip-shaped area enclosed by \(l_1\) and \(l_2\) into at most how many non-overlapping regions?
|
56
| 58.59375 |
7,432 |
$(2x-1)^{10} = a_0 + a_1x + a_2x^2 + \ldots + a_9x^9 + a_{10}x^{10}$, then $a_2 + a_3 + \ldots + a_9 + a_{10} =$ \_\_\_\_\_\_.
|
20
| 48.4375 |
7,433 |
Three faces of a rectangular box meet at a corner, and the centers of these faces form the vertices of a triangle with side lengths of 4 cm, 5 cm, and 6 cm. What is the volume of the box, in cm^3?
|
90 \sqrt{6}
| 78.90625 |
7,434 |
Calculate the definite integral:
\[
\int_{0}^{\frac{\pi}{4}} \frac{2 \cos x + 3 \sin x}{(2 \sin x - 3 \cos x)^{3}} \, dx
\]
|
-\frac{17}{18}
| 97.65625 |
7,435 |
Xiao Hua plays a certain game where each round can be played several times freely. Each score in a round is one of the numbers $8$, $a$ (a natural number), or $0$. The total score for a round is the sum of all individual scores in that round. Xiao Hua has achieved the following total scores in some rounds: $103, 104, 105, 106, 107, 108, 109, 110$. It is also known that he cannot achieve a total score of $83$. What is the value of $a$?
|
13
| 67.1875 |
7,436 |
Given that \( I \) is the incenter of \( \triangle ABC \), and
\[ 9 \overrightarrow{CI} = 4 \overrightarrow{CA} + 3 \overrightarrow{CB}. \]
Let \( R \) and \( r \) be the circumradius and inradius of \( \triangle ABC \), respectively. Find \(\frac{r}{R} = \).
|
5/16
| 35.9375 |
7,437 |
In a certain exam, 6 questions are randomly selected from 20 questions. If a student can correctly answer at least 4 of these questions, they pass the exam. If they can correctly answer at least 5 of these questions, they achieve an excellent grade. It is known that a certain student can correctly answer 10 of these questions and that they have already passed the exam. The probability that they achieve an excellent grade is \_\_\_\_\_\_.
|
\frac{13}{58}
| 0 |
7,438 |
In the number \( 2016****02** \), each of the 6 asterisks needs to be replaced with any of the digits \( 0, 2, 4, 5, 7, 9 \) (the digits may repeat) so that the resulting 12-digit number is divisible by 15. How many ways can this be done?
|
5184
| 58.59375 |
7,439 |
Given the obtuse angle $\alpha$ that satisfies the equation $$\frac {sin\alpha-3cos\alpha}{cos\alpha -sin\alpha }=tan2\alpha$$, find the value of $tan\alpha$.
|
2 - \sqrt{7}
| 32.03125 |
7,440 |
Given the parametric equation of line $l$ as $$\begin{cases} x=t \\ y= \frac { \sqrt {2}}{2}+ \sqrt {3}t \end{cases}$$ (where $t$ is the parameter), if the origin $O$ of the Cartesian coordinate system $xOy$ is taken as the pole and the direction of $Ox$ as the polar axis, and the same unit of length is chosen to establish the polar coordinate system, then the polar equation of curve $C$ is $\rho=2\cos\left(\theta- \frac {\pi}{4}\right)$.
(1) Find the angle of inclination of line $l$ and the Cartesian equation of curve $C$;
(2) If line $l$ intersects curve $C$ at points $A$ and $B$, and let point $P(0, \frac { \sqrt {2}}{2})$, find $|PA|+|PB|$.
|
\frac { \sqrt {10}}{2}
| 0 |
7,441 |
Given the real numbers \( x \) and \( y \) that satisfy
\[ x + y = 3 \]
\[ \frac{1}{x + y^2} + \frac{1}{x^2 + y} = \frac{1}{2} \]
find the value of \( x^5 + y^5 \).
|
123
| 12.5 |
7,442 |
Given the function $f(x) = a\sin(\pi x + \alpha) + b\cos(\pi x + \beta)$, and it is known that $f(2001) = 3$, find the value of $f(2012)$.
|
-3
| 27.34375 |
7,443 |
Given the hyperbola $\dfrac{x^{2}}{a^{2}} - \dfrac{y^{2}}{b^{2}} = 1$ ($a > 0$, $b > 0$) with its right focus at $F(c, 0)$. A circle centered at the origin $O$ with radius $c$ intersects the hyperbola in the first quadrant at point $A$. The tangent to the circle at point $A$ has a slope of $-\sqrt{3}$. Find the eccentricity of the hyperbola.
|
\sqrt{2}
| 67.1875 |
7,444 |
How many numbers of the form $\overline{a b c d a b c d}$ are divisible by 18769?
|
65
| 4.6875 |
7,445 |
Given that the square of a number $y^2$ is the sum of squares of 11 consecutive integers, find the minimum value of $y^2$.
|
121
| 14.84375 |
7,446 |
Let the ellipse $E: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ have an eccentricity $e = \frac{1}{2}$, and the distance from the right focus to the line $\frac{x}{a} + \frac{y}{b} = 1$ be $d = \frac{\sqrt{21}}{7}$. Let $O$ be the origin.
$(1)$ Find the equation of the ellipse $E$;
$(2)$ Draw two perpendicular rays from point $O$ that intersect the ellipse $E$ at points $A$ and $B$, respectively. Find the distance from point $O$ to the line $AB$.
|
\frac{2\sqrt{21}}{7}
| 28.90625 |
7,447 |
Simplify the expression: $\frac{8}{1+a^{8}} + \frac{4}{1+a^{4}} + \frac{2}{1+a^{2}} + \frac{1}{1+a} + \frac{1}{1-a}$ and find its value when $a=2^{-\frac{1}{16}}$.
|
32
| 35.9375 |
7,448 |
Elliot and Emily run a 12 km race. They start at the same point, run 6 km up a hill, and return to the starting point by the same route. Elliot has a 8 minute head start and runs at the rate of 12 km/hr uphill and 18 km/hr downhill. Emily runs 14 km/hr uphill and 20 km/hr downhill. How far from the top of the hill are they when they pass each other going in opposite directions (in km)?
A) $\frac{161}{48}$
B) $\frac{169}{48}$
C) $\frac{173}{48}$
D) $\frac{185}{48}$
|
\frac{169}{48}
| 38.28125 |
7,449 |
Petya and Vasya participated in a bicycle race. All participants started at the same time and finished with different times. Petya finished immediately after Vasya and ended up in tenth place. How many people participated in the race if Vasya was fifteenth from the end?
|
23
| 60.9375 |
7,450 |
Given $a \in \{0, 1, 2\}$ and $b \in \{-1, 1, 3, 5\}$, find the probability that the function $f(x) = ax^2 - 2bx$ is an increasing function on the interval $(1, +\infty)$.
|
\frac{5}{12}
| 26.5625 |
7,451 |
Compute the definite integral:
$$
\int_{1}^{2} \frac{x+\sqrt{3 x-2}-10}{\sqrt{3 x-2}+7} d x
$$
|
-\frac{22}{27}
| 65.625 |
7,452 |
\[
\frac{\left(\left(4.625 - \frac{13}{18} \cdot \frac{9}{26}\right) : \frac{9}{4} + 2.5 : 1.25 : 6.75\right) : 1 \frac{53}{68}}{\left(\frac{1}{2} - 0.375\right) : 0.125 + \left(\frac{5}{6} - \frac{7}{12}\right) : (0.358 - 1.4796 : 13.7)}
\]
|
\frac{17}{27}
| 13.28125 |
7,453 |
Select 3 numbers from the range 1 to 300 such that their sum is exactly divisible by 3. How many such combinations are possible?
|
1485100
| 92.1875 |
7,454 |
A cake has a shape of triangle with sides $19,20$ and $21$ . It is allowed to cut it it with a line into two pieces and put them on a round plate such that pieces don't overlap each other and don't stick out of the plate. What is the minimal diameter of the plate?
|
21
| 29.6875 |
7,455 |
If \( 20 \times 21 \times 22 \times \ldots \times 2020 = 26^{k} \times m \), where \( m \) is an integer, what is the maximum value of \( k \)?
|
165
| 26.5625 |
7,456 |
Alice has six magical pies in her pocket - two that make you grow and the rest make you shrink. When Alice met Mary Ann, she blindly took three pies out of her pocket and gave them to Mary. Find the probability that one of the girls has no growth pies.
|
0.4
| 10.9375 |
7,457 |
Let $ABCD$ be a square and $X$ a point such that $A$ and $X$ are on opposite sides of $CD$ . The lines $AX$ and $BX$ intersect $CD$ in $Y$ and $Z$ respectively. If the area of $ABCD$ is $1$ and the area of $XYZ$ is $\frac{2}{3}$ , determine the length of $YZ$
|
\frac{2}{3}
| 19.53125 |
7,458 |
Find the smallest exact square with last digit not $0$ , such that after deleting its last two digits we shall obtain another exact square.
|
121
| 70.3125 |
7,459 |
Find the minimum value of the expression
$$
\sqrt{x^{2}-2 \sqrt{3} \cdot|x|+4}+\sqrt{x^{2}+2 \sqrt{3} \cdot|x|+12}
$$
as well as the values of $x$ at which it is achieved.
|
2 \sqrt{7}
| 6.25 |
7,460 |
In the rectangular coordinate system \( xOy \), the equation of ellipse \( C \) is \( \frac{x^2}{9} + \frac{y^2}{10} = 1 \). Let \( F \) and \( A \) be the upper focus and the right vertex of ellipse \( C \), respectively. If \( P \) is a point on ellipse \( C \) located in the first quadrant, find the maximum value of the area of quadrilateral \( OAPF \).
|
\frac{3\sqrt{11}}{2}
| 46.875 |
7,461 |
Given that the sides of triangle $\triangle ABC$ opposite to angles $A$, $B$, and $C$ are in arithmetic sequence and $C=2\left(A+B\right)$, calculate the ratio $\frac{b}{a}$.
|
\frac{5}{3}
| 29.6875 |
7,462 |
Four balls of radius $1$ are placed in space so that each of them touches the other three. What is the radius of the smallest sphere containing all of them?
|
\frac{\sqrt{6} + 2}{2}
| 0 |
7,463 |
At the namesake festival, 45 Alexanders, 122 Borises, 27 Vasily, and several Gennady attended. At the beginning of the festival, all of them lined up so that no two people with the same name stood next to each other. What is the minimum number of Gennadys that could have attended the festival?
|
49
| 16.40625 |
7,464 |
If there are 3 identical copies of the Analects and 6 different modern literary masterpieces in the classroom, and 3 books are selected from these 9 books, determine the number of different ways to make the selection.
|
42
| 27.34375 |
7,465 |
Given that $$cos(α- \frac {π}{6})-sinα= \frac {2 \sqrt {3}}{5}$$, find the value of $$cos(α+ \frac {7π}{6})$$.
|
- \frac {2 \sqrt {3}}{5}
| 1.5625 |
7,466 |
Numbers between $200$ and $500$ that are divisible by $5$ contain the digit $3$. How many such whole numbers exist?
|
24
| 0 |
7,467 |
Let the function \( f(x) = \left| \log_{2} x \right| \). Real numbers \( a \) and \( b \) (where \( a < b \)) satisfy the following conditions:
\[
\begin{array}{l}
f(a+1) = f(b+2), \\
f(10a + 6b + 22) = 4
\end{array}
\]
Find \( ab \).
|
2/15
| 28.90625 |
7,468 |
How many cubic centimeters are in the volume of a cone having a diameter of 12 cm and a slant height of 10 cm?
|
96 \pi
| 66.40625 |
7,469 |
Petya and Vasya took a math test. Petya answered $80\%$ of all the questions correctly, while Vasya answered exactly 35 questions correctly. The number of questions both answered correctly is exactly half the total number of questions. No one answered 7 questions. How many questions were on the test?
|
60
| 70.3125 |
7,470 |
A man buys a house for $15,000 and wants to achieve a $6\%$ return on his investment while incurring a yearly tax of $450$, along with an additional $200$ yearly for owner's insurance. The percentage he sets aside from monthly rent for maintenance remains $12\frac{1}{2}\%$. Calculate the monthly rent.
|
147.62
| 46.875 |
7,471 |
What is the largest integer \( k \) such that \( k+1 \) divides
\[ k^{2020} + 2k^{2019} + 3k^{2018} + \cdots + 2020k + 2021? \
|
1010
| 86.71875 |
7,472 |
A swimming pool is in the shape of a circle with diameter 60 ft. The depth varies linearly along the east-west direction from 3 ft at the shallow end in the east to 15 ft at the diving end in the west but does not vary at all along the north-south direction. What is the volume of the pool, in cubic feet (ft³)?
|
8100 \pi
| 55.46875 |
7,473 |
During an underwater archaeological activity, a diver needs to dive $50$ meters to the bottom of the water for archaeological work. The oxygen consumption consists of the following three aspects: $(1)$ The average diving speed is $x$ meters/minute, and the oxygen consumption per minute is $\frac{1}{100}x^{2}$ liters; $(2)$ The working time at the bottom of the water is between $10$ and $20$ minutes, and the oxygen consumption per minute is $0.3$ liters; $(3)$ When returning to the water surface, the average speed is $\frac{1}{2}x$ meters/minute, and the oxygen consumption per minute is $0.32$ liters. The total oxygen consumption of the diver in this archaeological activity is $y$ liters.
$(1)$ If the working time at the bottom of the water is $10$ minutes, express $y$ as a function of $x$;
$(2)$ If $x \in [6, 10]$, the working time at the bottom of the water is $20$ minutes, find the range of total oxygen consumption $y$;
$(3)$ If the diver carries $13.5$ liters of oxygen, how many minutes can the diver stay underwater at most (round to the nearest whole number)?
|
18
| 28.125 |
7,474 |
Given a regular polygon with $n$ sides. It is known that there are $1200$ ways to choose three of the vertices of the polygon such that they form the vertices of a **right triangle**. What is the value of $n$ ?
|
50
| 72.65625 |
7,475 |
Given the sequence $a_1, a_2, \ldots, a_n, \ldots$, insert 3 numbers between each pair of consecutive terms to form a new sequence. Determine the term number of the 69th term of the new sequence.
|
18
| 81.25 |
7,476 |
Calculate the sum $C_{3}^{2}+C_{4}^{2}+C_{5}^{2}+\ldots+C_{19}^{2}$.
|
1139
| 58.59375 |
7,477 |
If two different properties are randomly selected from the five types of properties (metal, wood, water, fire, and earth) where metal overcomes wood, wood overcomes earth, earth overcomes water, water overcomes fire, and fire overcomes metal, determine the probability that the two selected properties do not overcome each other.
|
\dfrac{1}{2}
| 33.59375 |
7,478 |
Given $f(\alpha) = \frac {\sin(\pi-\alpha)\cos(2\pi-\alpha)\tan(-\alpha+\pi)}{-\tan(-\alpha -\pi )\cos( \frac {\pi}{2}-\alpha )}$:
1. Simplify $f(\alpha)$.
2. If $\alpha$ is an angle in the third quadrant and $\cos(\alpha- \frac {3\pi}{2}) = \frac {1}{5}$, find the value of $f(\alpha)$.
|
\frac {2\sqrt{6}}{5}
| 88.28125 |
7,479 |
Calculate the value of $\sin 68^{\circ} \sin 67^{\circ} - \sin 23^{\circ} \cos 68^{\circ}$.
|
\frac{\sqrt{2}}{2}
| 97.65625 |
7,480 |
Let the arithmetic sequence $\{a_n\}$ satisfy: the common difference $d\in \mathbb{N}^*$, $a_n\in \mathbb{N}^*$, and any two terms' sum in $\{a_n\}$ is also a term in the sequence. If $a_1=3^5$, then the sum of all possible values of $d$ is .
|
364
| 96.875 |
7,481 |
A four-meter gas pipe has rusted in two places. Determine the probability that all three resulting pieces can be used as connections to gas stoves, given that regulations require a stove to be at least 1 meter away from the main gas pipeline.
|
1/8
| 19.53125 |
7,482 |
The point $P$ $(4,5)$ is reflected over the $y$-axis to $Q$. Then $Q$ is reflected over the line $y=-x$ to $R$. What is the area of triangle $PQR$?
|
36
| 0 |
7,483 |
In a region, three villages \(A, B\), and \(C\) are connected by rural roads, with more than one road between any two villages. The roads are bidirectional. A path from one village to another is defined as either a direct connecting road or a chain of two roads passing through a third village. It is known that there are 34 paths connecting villages \(A\) and \(B\), and 29 paths connecting villages \(B\) and \(C\). What is the maximum number of paths that could connect villages \(A\) and \(C\)?
|
106
| 0 |
7,484 |
Let $a$, $b$, and $c$ be solutions of the equation $x^3 - 6x^2 + 11x - 6 = 0$. Compute $ \frac{ab}{c} + \frac{bc}{a} + \frac{ca}{b}$.
|
\frac{49}{6}
| 85.9375 |
7,485 |
In rectangle \(ABCD\), \(E\) and \(F\) are chosen on \(\overline{AB}\) and \(\overline{CD}\), respectively, so that \(AEFD\) is a square. If \(\frac{AB}{BE} = \frac{BE}{BC}\), determine the value of \(\frac{AB}{BC}\).
|
\frac{3 + \sqrt{5}}{2}
| 65.625 |
7,486 |
Given that Lucas makes a batch of lemonade using 200 grams of lemon juice, 100 grams of sugar, and 300 grams of water. If there are 25 calories in 100 grams of lemon juice and 386 calories in 100 grams of sugar, and water has no calories, determine the total number of calories in 200 grams of this lemonade.
|
145
| 1.5625 |
7,487 |
In the trapezoid \(ABCD\), the lengths of the bases \(AD = 24\) cm and \(BC = 8\) cm, and the diagonals \(AC = 13\) cm, \(BD = 5\sqrt{17}\) cm are known. Calculate the area of the trapezoid.
|
80
| 4.6875 |
7,488 |
\(a, b, c\) are distinct positive integers such that \(\{a+b, b+c, c+a\} = \left\{n^2, (n+1)^2, (n+2)^2\right\}\), where \(n\) is a positive integer. What is the minimum value of \(a^2 + b^2 + c^2\)?
|
1297
| 50 |
7,489 |
Let \( x, y, z, u, v \in \mathbf{R}_{+} \). Determine the maximum value of \( f = \frac{xy + yz + zu + uv}{2x^2 + y^2 + 2z^2 + u^2 + 2v^2} \).
|
1/2
| 71.875 |
7,490 |
Given the function $f(x)=2\sin(\omega x+\varphi)$, where $(\omega > 0, |\varphi| < \frac{\pi}{2})$, the graph passes through the point $B(0,-1)$, and is monotonically increasing on the interval $\left(\frac{\pi}{18}, \frac{\pi}{3}\right)$. Additionally, the graph of $f(x)$ coincides with its original graph after being shifted to the left by $\pi$ units. If $x_{1}, x_{2} \in \left(-\frac{17\pi}{12}, -\frac{2\pi}{3}\right)$ and $x_{1} \neq x_{2}$, and $f(x_{1}) = f(x_{2})$, calculate $f(x_{1}+x_{2})$.
|
-1
| 28.90625 |
7,491 |
Find the angle of inclination of the tangent line to the curve $y=\frac{1}{3}x^3-5$ at the point $(1,-\frac{3}{2})$.
|
\frac{\pi}{4}
| 22.65625 |
7,492 |
Let the function $y=f\left(x\right)$ have domain $D$, and all points on its graph be above the line $y=t$. If the function $f\left(x\right)=\left(x-t\right)e^{x}$ has the domain $R$ and is a "$\left(-\infty ,+\infty \right)-t$ function", determine the largest integer value of the real number $t$.
|
-1
| 63.28125 |
7,493 |
Given that $\sin \alpha$ is a root of the equation $5x^{2}-7x-6=0$, find:
$(1)$ The value of $\frac {\cos (2\pi-\alpha)\cos (\pi+\alpha)\tan ^{2}(2\pi-\alpha)}{\cos ( \frac {\pi}{2}+\alpha)\sin (2\pi-\alpha)\cot ^{2}(\pi-\alpha)}$.
$(2)$ In $\triangle ABC$, $\sin A+ \cos A= \frac { \sqrt {2}}{2}$, $AC=2$, $AB=3$, find the value of $\tan A$.
|
-2- \sqrt {3}
| 1.5625 |
7,494 |
Find the fraction \(\frac{p}{q}\) with the smallest possible natural denominator for which \(\frac{1}{2014} < \frac{p}{q} < \frac{1}{2013}\). Enter the denominator of this fraction in the provided field.
|
4027
| 90.625 |
7,495 |
The length of the escalator is 200 steps. When Petya walks down the escalator, he counts 50 steps. How many steps will he count if he runs twice as fast?
|
80
| 53.90625 |
7,496 |
Consider an arithmetic sequence $\{a_n\}$ with the sum of its first $n$ terms denoted as $S_n$. Given that $a_1=9$, $a_2$ is an integer, and $S_n \leq S_5$, find the sum of the first 9 terms of the sequence $\{\frac{1}{a_n a_{n+1}}\}$.
|
-\frac{1}{9}
| 3.90625 |
7,497 |
Given a cone with a base radius of $1$ and a height of $\sqrt{3}$, both the apex of the cone and the base circle are on the surface of a sphere $O$, calculate the surface area of this sphere.
|
\frac{16\pi}{3}
| 62.5 |
7,498 |
Given the sequence \(\left\{a_{n}\right\}\) with the general term
\[ a_{n} = n^{4} + 6n^{3} + 11n^{2} + 6n, \]
find the sum of the first 12 terms \( S_{12} \).
|
104832
| 55.46875 |
7,499 |
Suppose $f(x), g(x), h(x), p(x)$ are linear equations where $f(x) = x + 1$, $g(x) = -x + 5$, $h(x) = 4$, and $p(x) = 1$. Define new functions $j(x)$ and $k(x)$ as follows:
$$j(x) = \max\{f(x), g(x), h(x), p(x)\},$$
$$k(x)= \min\{f(x), g(x), h(x), p(x)\}.$$
Find the length squared, $\ell^2$, of the graph of $y=k(x)$ from $x = -4$ to $x = 4$.
|
64
| 14.0625 |
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