Unnamed: 0
int64 0
40.3k
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stringlengths 10
5.15k
| ground_truth
stringlengths 1
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float64 0
100
|
|---|---|---|---|
8,400 |
A sports league consists of 16 teams divided into 2 divisions of 8 teams each. Throughout the season, each team plays every other team in its own division three times and every team in the other division twice. How many games are in a complete season for the league?
|
296
| 93.75 |
8,401 |
A ball thrown vertically upwards has its height above the ground expressed as a quadratic function with respect to its time of motion. Xiaohong throws two balls vertically upwards one after the other, with a 1-second interval between them. Assume the initial height above the ground for both balls is the same, and each reaches the same maximum height 1.1 seconds after being thrown. If the first ball's height matches the second ball's height at $t$ seconds after the first ball is thrown, determine $t = \qquad$ .
|
1.6
| 68.75 |
8,402 |
In triangle $ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and they satisfy the equation $\sin A + \sin B = [\cos A - \cos (π - B)] \sin C$.
1. Determine whether triangle $ABC$ is a right triangle and explain your reasoning.
2. If $a + b + c = 1 + \sqrt{2}$, find the maximum area of triangle $ABC$.
|
\frac{1}{4}
| 54.6875 |
8,403 |
Let \(\alpha\) and \(\beta\) be the two real roots of the quadratic equation \(x^{2} - 2kx + k + 20 = 0\). Find the minimum value of \((\alpha+1)^{2} + (\beta+1)^{2}\), and determine the value of \(k\) for which this minimum value is achieved.
|
18
| 1.5625 |
8,404 |
Given $f(x)= \frac{2x}{x+1}$, calculate the value of the expression $f\left( \frac{1}{2016}\right)+f\left( \frac{1}{2015}\right)+f\left( \frac{1}{2014}\right)+\ldots+f\left( \frac{1}{2}\right)+f(1)+f(2)+\ldots+f(2014)+f(2015)+f(2016)$.
|
4031
| 86.71875 |
8,405 |
Carl wrote a list of 10 distinct positive integers on a board. Each integer in the list, apart from the first, is a multiple of the previous integer. The last of the 10 integers is between 600 and 1000. What is this last integer?
|
768
| 15.625 |
8,406 |
Insert a square into an isosceles triangle with a lateral side of 10 and a base of 12.
|
4.8
| 3.90625 |
8,407 |
For real numbers \( x \) and \( y \) within the interval \([0, 12]\):
$$
xy = (12 - x)^2 (12 - y)^2
$$
What is the maximum value of the product \( xy \)?
|
81
| 29.6875 |
8,408 |
Vasya has a stick that is 22 cm long. He wants to break it into three pieces with integer lengths such that the pieces can form a triangle. In how many ways can he do this? (Ways that result in identical triangles are considered the same).
|
10
| 86.71875 |
8,409 |
In Princess Sissi's garden, there is an empty water reservoir. When water is injected into the reservoir, the drainage pipe will draw water out to irrigate the flowers. Princess Sissi found that if 3 water pipes are turned on, the reservoir will be filled in 30 minutes; if 5 water pipes are turned on, the reservoir will be filled in 10 minutes. Princess Sissi wondered, "If 4 water pipes are turned on, how many minutes will it take to fill the reservoir?"
The answer to this question is $\qquad$ minutes.
|
15
| 75.78125 |
8,410 |
The number 25 is expressed as the sum of positive integers \(x_{1}, x_{2}, \cdots, x_{k}\), where \(k \leq 25\). What is the maximum value of the product of \(x_{1}, x_{2}, x_{3}, \cdots\), and \(x_{k}\)?
|
8748
| 60.9375 |
8,411 |
If the value of the expression $(\square + 121 \times 3.125) \div 121$ is approximately 3.38, what natural number should be placed in $\square$?
|
31
| 15.625 |
8,412 |
In \(\triangle PMO\), \(PM = 6\sqrt{3}\), \(PO = 12\sqrt{3}\), and \(S\) is a point on \(MO\) such that \(PS\) is the angle bisector of \(\angle MPO\). Let \(T\) be the reflection of \(S\) across \(PM\). If \(PO\) is parallel to \(MT\), find the length of \(OT\).
|
2\sqrt{183}
| 0.78125 |
8,413 |
Suppose $\alpha,\beta,\gamma\in\{-2,3\}$ are chosen such that
\[M=\max_{x\in\mathbb{R}}\min_{y\in\mathbb{R}_{\ge0}}\alpha x+\beta y+\gamma xy\]
is finite and positive (note: $\mathbb{R}_{\ge0}$ is the set of nonnegative real numbers). What is the sum of the possible values of $M$ ?
|
13/2
| 9.375 |
8,414 |
The teacher wrote a positive number $x$ on the board and asked Kolya, Petya, and Vasya to raise this number to the 3rd, 4th, and 12th powers, respectively. It turned out that Kolya's number had at least 9 digits before the decimal point, and Petya's number had no more than 11 digits before the decimal point. How many digits are there before the decimal point in Vasya's number?
|
33
| 28.125 |
8,415 |
Solve the following equations:
2x + 62 = 248; x - 12.7 = 2.7; x ÷ 5 = 0.16; 7x + 2x = 6.3.
|
0.7
| 98.4375 |
8,416 |
In the diagram, the side \(AB\) of \(\triangle ABC\) is divided into \(n\) equal parts (\(n > 1990\)). Through the \(n-1\) division points, lines parallel to \(BC\) are drawn intersecting \(AC\) at points \(B_i, C_i\) respectively for \(i=1, 2, 3, \cdots, n-1\). What is the ratio of the area of \(\triangle AB_1C_1\) to the area of the quadrilateral \(B_{1989} B_{1990} C_{1990} C_{1989}\)?
|
1: 3979
| 0 |
8,417 |
Three three-digit numbers, with all digits except zero being used in their digits, sum up to 1665. In each number, the first digit was swapped with the last digit. What is the sum of the new numbers?
|
1665
| 52.34375 |
8,418 |
In a $3 \times 3$ table, numbers are placed such that each number is 4 times smaller than the number in the adjacent cell to the right and 3 times smaller than the number in the adjacent cell above. The sum of all the numbers in the table is 546. Find the number in the central cell.
|
24
| 39.84375 |
8,419 |
On a table lie 140 different cards with numbers $3, 6, 9, \ldots, 417, 420$ (each card has exactly one number, and each number appears exactly once). In how many ways can you choose 2 cards so that the sum of the numbers on the selected cards is divisible by $7?$
|
1390
| 78.90625 |
8,420 |
Use the Horner's method to calculate the value of the polynomial $f(x) = 2x^5 - 3x^2 + 4x^4 - 2x^3 + x$ when $x=2$.
|
102
| 14.0625 |
8,421 |
Two circular poles, with diameters of 8 inches and 24 inches, touch each other at a single point. A wire is wrapped around them such that it goes around the entire configuration including a straight section tangential to both poles. Find the length of the shortest wire that sufficiently encloses both poles.
A) $16\sqrt{3} + 32\pi$
B) $16\sqrt{3} + 24\pi$
C) $12\sqrt{3} + 32\pi$
D) $16\sqrt{3} + 16\pi$
E) $8\sqrt{3} + 32\pi$
|
16\sqrt{3} + 32\pi
| 39.0625 |
8,422 |
It is known that $\sin \alpha+\sin \beta=2 \sin (\alpha+\beta)$ and $\alpha+\beta \neq 2 \pi n (n \in \mathbb{Z})$. Find $\operatorname{tg} \frac{\alpha}{2} \operatorname{tg} \frac{\beta}{2}$.
|
\frac{1}{3}
| 60.9375 |
8,423 |
Given a geometric sequence $\{a_n\}$ with a common ratio $q > 1$, $a_1 = 2$, and $a_1, a_2, a_3 - 8$ form an arithmetic sequence. The sum of the first n terms of the sequence $\{b_n\}$ is denoted as $S_n$, and $S_n = n^2 - 8n$.
(I) Find the general term formulas for both sequences $\{a_n\}$ and $\{b_n\}$.
(II) Let $c_n = \frac{b_n}{a_n}$. If $c_n \leq m$ holds true for all $n \in \mathbb{N}^*$, determine the minimum value of the real number $m$.
|
\frac{1}{162}
| 35.15625 |
8,424 |
Consider a circle of radius 1 with center $O$ down in the diagram. Points $A$, $B$, $C$, and $D$ lie on the circle such that $\angle AOB = 120^\circ$, $\angle BOC = 60^\circ$, and $\angle COD = 180^\circ$. A point $X$ lies on the minor arc $\overarc{AC}$. If $\angle AXB = 90^\circ$, find the length of $AX$.
|
\sqrt{3}
| 17.1875 |
8,425 |
What is the sum of the digits of \(10^{2008} - 2008\)?
|
18063
| 35.15625 |
8,426 |
A coordinate system and parametric equations problem (4-4):
In the rectangular coordinate system $x0y$, the parametric equations of line $l$ are given by $\begin{cases} x = \frac{1}{2}t \ y = \frac{\sqrt{2}}{2} + \frac{\sqrt{3}}{2}t \end{cases}$, where $t$ is the parameter. If we establish a polar coordinate system with point $O$ in the rectangular coordinate system $x0y$ as the pole, $0x$ as the polar axis, and the same length unit, the polar equation of curve $C$ is given by $\rho = 2 \cos(\theta - \frac{\pi}{4})$.
(1) Find the slope angle of line $l$.
(2) If line $l$ intersects curve $C$ at points $A$ and $B$, find the length of $AB$.
|
\frac{\sqrt{10}}{2}
| 18.75 |
8,427 |
What is the smallest possible perimeter of a triangle with integer coordinate vertices, area $\frac12$ , and no side parallel to an axis?
|
\sqrt{5} + \sqrt{2} + \sqrt{13}
| 0 |
8,428 |
Several consecutive natural numbers are written on the board. It is known that \(48\%\) of them are even, and \(36\%\) of them are less than 30. Find the smallest of the written numbers.
|
21
| 21.09375 |
8,429 |
Given the standard equation of the hyperbola $M$ as $\frac{x^{2}}{4}-\frac{y^{2}}{2}=1$. Find the length of the real axis, the length of the imaginary axis, the focal distance, and the eccentricity of the hyperbola $M$.
|
\frac{\sqrt{6}}{2}
| 82.03125 |
8,430 |
If a computer executes the following program:
(1) Initial values are $x=3, S=0$.
(2) $x=x+2$.
(3) $S=S+x$.
(4) If $S \geqslant 10000$, proceed to step 5; otherwise, go back to step 2.
(5) Print $x$.
(6) Stop.
Then what is the value printed in step 5?
|
201
| 96.875 |
8,431 |
(1) Solve the inequality $$\frac {2x+1}{3-x}≥1$$
(2) Given $x>0$, $y>0$, and $x+y=1$, find the minimum value of $$\frac {4}{x} + \frac {9}{y}$$.
|
25
| 87.5 |
8,432 |
Given that when 81849, 106392, and 124374 are divided by an integer \( n \), the remainders are equal. If \( a \) is the maximum value of \( n \), find \( a \).
|
243
| 41.40625 |
8,433 |
What is the largest 4-digit integer congruent to $7 \pmod{19}$?
|
9982
| 25 |
8,434 |
Find $n$ such that $2^8 \cdot 3^4 \cdot 5^1 \cdot n = 10!$.
|
35
| 45.3125 |
8,435 |
There are 23 socks in a drawer: 8 white and 15 black. Every minute, Marina goes to the drawer and pulls out a sock. If at any moment Marina has pulled out more black socks than white ones, she exclaims, "Finally!" and stops the process.
What is the maximum number of socks Marina can pull out before she exclaims, "Finally!"? The last sock Marina pulled out is included in the count.
|
17
| 27.34375 |
8,436 |
Calculate the following product:
$$\frac{1}{3}\times9\times\frac{1}{27}\times81\times\frac{1}{243}\times729\times\frac{1}{2187}\times6561\times\frac{1}{19683}\times59049.$$
|
243
| 82.8125 |
8,437 |
Given the ratio of the distance saved by taking a shortcut along the diagonal of a rectangular field to the longer side of the field is $\frac{1}{2}$, determine the ratio of the shorter side to the longer side of the rectangle.
|
\frac{3}{4}
| 83.59375 |
8,438 |
A right cylinder with a height of 8 inches is enclosed inside another cylindrical shell of the same height but with a radius 1 inch greater than the inner cylinder. The radius of the inner cylinder is 3 inches. What is the total surface area of the space between the two cylinders, in square inches? Express your answer in terms of $\pi$.
|
16\pi
| 63.28125 |
8,439 |
A rectangular grid is given. Let's call two cells neighboring if they share a common side. Count the number of cells that have exactly four neighboring cells. There are 23 such cells. How many cells have three neighboring cells?
|
48
| 35.9375 |
8,440 |
The domain of the function \( f(x) \) is \( (0,1) \), and the function is defined as follows:
\[
f(x)=\begin{cases}
x, & \text{if } x \text{ is an irrational number}, \\
\frac{p+1}{q}, & \text{if } x=\frac{p}{q}, \; p, q \in \mathbf{N}^{*}, \; (p, q) = 1, \; p < q.
\end{cases}
\]
Find the maximum value of \( f(x) \) in the interval \(\left( \frac{7}{8}, \frac{8}{9} \right) \).
|
16/17
| 64.0625 |
8,441 |
Find the length of the chord intercepted by the line given by the parametric equations $x = 1 + \frac{4}{5}t$, $y = -1 - \frac{3}{5}t$ and the curve $ρ = \sqrt{2}\cos\left(θ + \frac{π}{4}\right)$.
|
\frac{7}{5}
| 77.34375 |
8,442 |
There are 20 points, each pair of adjacent points are equally spaced. By connecting four points with straight lines, you can form a square. Using this method, you can form _ squares.
|
20
| 24.21875 |
8,443 |
What is the minimum number of participants that could have been in the school drama club if fifth-graders constituted more than $25\%$, but less than $35\%$; sixth-graders more than $30\%$, but less than $40\%$; and seventh-graders more than $35\%$, but less than $45\%$ (there were no participants from other grades)?
|
11
| 89.84375 |
8,444 |
Given several rectangular prisms with edge lengths of $2, 3,$ and $5$, aligned in the same direction to form a cube with an edge length of $90$, how many small rectangular prisms does one diagonal of the cube intersect?
|
66
| 36.71875 |
8,445 |
Given that in the geometric sequence $\{a_n\}$ where all terms are positive, $a_1a_3=16$ and $a_3+a_4=24$, find the value of $a_5$.
|
32
| 92.96875 |
8,446 |
An geometric sequence $\{a_n\}$ has 20 terms, where the product of the first four terms is $\frac{1}{128}$, and the product of the last four terms is 512. The product of all terms in this geometric sequence is \_\_\_\_\_\_.
|
32
| 20.3125 |
8,447 |
Define the function $f(x)$ on $\mathbb{R}$ such that $f(0)=0$, $f(x)+f(1-x)=1$, $f\left(\frac{x}{5}\right)=\frac{1}{2}f(x)$, and for $0 \leq x_1 < x_2 \leq 1$, $f(x_1) \leq f(x_2)$. Find the value of $f\left(\frac{1}{2007}\right)$.
|
\frac{1}{32}
| 28.90625 |
8,448 |
Grandpa is twice as strong as Grandma, Grandma is three times as strong as Granddaughter, Granddaughter is four times as strong as Doggie, Doggie is five times as strong as Cat, and Cat is six times as strong as Mouse. Grandpa, Grandma, Granddaughter, Doggie, and Cat together with Mouse can pull up the Turnip, but without Mouse they can't. How many Mice are needed so that they can pull up the Turnip on their own?
|
1237
| 76.5625 |
8,449 |
What is the least positive integer $k$ such that, in every convex 1001-gon, the sum of any k diagonals is greater than or equal to the sum of the remaining diagonals?
|
249750
| 29.6875 |
8,450 |
Five students, A, B, C, D, and E, participated in a labor skills competition. A and B asked about the results. The respondent told A, "Unfortunately, neither you nor B got first place." To B, the respondent said, "You certainly are not the worst." Determine the number of different possible rankings the five students could have.
|
54
| 89.84375 |
8,451 |
Given that $S_{n}$ is the sum of the first $n$ terms of an arithmetic sequence ${a_{n}}$, $S_{1} < 0$, $2S_{21}+S_{25}=0$, find the value of $n$ when $S_{n}$ is minimized.
|
11
| 46.875 |
8,452 |
In the sequence \( \left\{a_{n}\right\} \), if \( a_{k}+a_{k+1}=2k+1 \) (where \( k \in \mathbf{N}^{*} \)), then \( a_{1}+a_{100} \) equals?
|
101
| 34.375 |
8,453 |
Two adjacent faces of a tetrahedron, which are equilateral triangles with side length 3, form a dihedral angle of 30 degrees. The tetrahedron is rotated around the common edge of these faces. Find the maximum area of the projection of the rotating tetrahedron on the plane containing this edge.
|
\frac{9\sqrt{3}}{4}
| 28.125 |
8,454 |
Find the smallest positive integer \( n > 1 \) such that the arithmetic mean of \( 1^2, 2^2, 3^2, \cdots, n^2 \) is a perfect square.
|
337
| 98.4375 |
8,455 |
Given a convex quadrilateral \(ABCD\) with \(\angle C = 57^{\circ}\), \(\sin \angle A + \sin \angle B = \sqrt{2}\), and \(\cos \angle A + \cos \angle B = 2 - \sqrt{2}\), find the measure of angle \(D\) in degrees.
|
168
| 78.90625 |
8,456 |
Let the function \( f(x) = \sin^4 \left( \frac{kx}{10} \right) + \cos^4 \left( \frac{kx}{10} \right) \), where \( k \) is a positive integer. If for any real number \( a \), the set \(\{ f(x) \mid a < x < a+1 \} = \{ f(x) \mid x \in \mathbf{R} \}\), then find the minimum value of \( k \).
|
16
| 14.84375 |
8,457 |
Five people are gathered in a meeting. Some pairs of people shakes hands. An ordered triple of people $(A,B,C)$ is a *trio* if one of the following is true:
- A shakes hands with B, and B shakes hands with C, or
- A doesn't shake hands with B, and B doesn't shake hands with C.
If we consider $(A,B,C)$ and $(C,B,A)$ as the same trio, find the minimum possible number of trios.
|
10
| 63.28125 |
8,458 |
For a natural number \( N \), if at least five out of the nine natural numbers \( 1 \) through \( 9 \) can divide \( N \) evenly, then \( N \) is called a "Five Sequential Number." What is the smallest "Five Sequential Number" greater than 2000?
|
2004
| 96.875 |
8,459 |
For real numbers \( x \) and \( y \) such that \( x + y = 1 \), determine the maximum value of the expression \( A(x, y) = x^4 y + x y^4 + x^3 y + x y^3 + x^2 y + x y^2 \).
|
\frac{7}{16}
| 40.625 |
8,460 |
The coefficient of $x^3y^5$ in the expansion of $(x+y)(x-y)^7$ is __________.
|
14
| 60.9375 |
8,461 |
Triangle \(ABC\) has a right angle at \(B\), with \(AB = 3\) and \(BC = 4\). If \(D\) and \(E\) are points on \(AC\) and \(BC\), respectively, such that \(CD = DE = \frac{5}{3}\), find the perimeter of quadrilateral \(ABED\).
|
28/3
| 29.6875 |
8,462 |
Integers $x$ and $y$ with $x>y>0$ satisfy $x+y+xy=104$. What is $x$?
|
34
| 61.71875 |
8,463 |
In the expression \((x+y+z)^{2034}+(x-y-z)^{2034}\), the brackets were expanded, and like terms were combined. How many monomials of the form \(x^{a} y^{b} z^{c}\) have a non-zero coefficient?
|
1036324
| 34.375 |
8,464 |
The radius of a sphere that touches all the edges of a regular tetrahedron is 1. Find the edge length of the tetrahedron.
|
2 \sqrt{2}
| 71.875 |
8,465 |
Barney Schwinn noted that his bike's odometer showed a reading of $2332$, a palindrome. After riding for $5$ hours one day and $4$ hours the next day, he observed that the odometer displayed another palindrome, $2552$. Calculate Barney's average riding speed during this period.
|
\frac{220}{9}
| 3.125 |
8,466 |
On a circle of radius 12 with center at point \( O \), there are points \( A \) and \( B \). Lines \( AC \) and \( BC \) are tangent to this circle. Another circle with center at point \( M \) is inscribed in triangle \( ABC \) and touches side \( AC \) at point \( K \) and side \( BC \) at point \( H \). The distance from point \( M \) to line \( KH \) is 3. Find \( \angle AOB \).
|
120
| 32.03125 |
8,467 |
Given that the focus of the parabola $x^{2}=2py$ coincides with the lower focus of the ellipse $\frac{x^{2}}{3}+\frac{y^{2}}{4}=1$, find the value of $p$.
|
-2
| 100 |
8,468 |
Given $A=\{a^{2},a+1,-3\}$ and $B=\{a-3,3a-1,a^{2}+1\}$, if $A∩B=\{-3\}$, find the value of the real number $a$.
|
- \frac {2}{3}
| 93.75 |
8,469 |
Let \(A B C D E\) be a square pyramid of height \(\frac{1}{2}\) with a square base \(A B C D\) of side length \(A B = 12\) (so \(E\) is the vertex of the pyramid, and the foot of the altitude from \(E\) to \(A B C D\) is the center of square \(A B C D\)). The faces \(A D E\) and \(C D E\) meet at an acute angle of measure \(\alpha\) (so that \(0^{\circ}<\alpha<90^{\circ}\)). Find \(\tan \alpha\).
|
\frac{17}{144}
| 19.53125 |
8,470 |
Let \( x_{1}, x_{2}, x_{3}, x_{4}, x_{5} \) be nonnegative real numbers whose sum is 300. Let \( M \) be the maximum of the four numbers \( x_{1} + x_{2}, x_{2} + x_{3}, x_{3} + x_{4}, \) and \( x_{4} + x_{5} \). Find the least possible value of \( M \).
|
100
| 32.03125 |
8,471 |
A cooperative receives apple and grape juice in identical containers and produces an apple-grape drink in identical cans. One container of apple juice is enough for exactly 6 cans of the drink, and one container of grape juice is enough for exactly 10 cans. When the recipe of the drink was changed, one container of apple juice became sufficient for exactly 5 cans of the drink. How many cans of the drink will one container of grape juice be sufficient for now? (The drink is not diluted with water.)
|
15
| 35.9375 |
8,472 |
Three concentric circles have radii $5$ meters, $15$ meters, and $25$ meters respectively. Calculate the total distance a beetle travels, which starts at a point $P$ on the outer circle, moves inward along a radius to the middle circle, traces a one-third arc of the middle circle, then travels radially to the inner circle, follows a half arc of the inner circle, and finally moves directly through the center to the opposite point on the outer circle.
|
15\pi + 70
| 0 |
8,473 |
Given that cosα + 2cos(α + $$\frac{π}{3}$$) = 0, find tan(α + $$\frac{π}{6}$$).
|
3\sqrt{3}
| 91.40625 |
8,474 |
Find the largest \( n \) so that the number of integers less than or equal to \( n \) and divisible by 3 equals the number divisible by 5 or 7 (or both).
|
65
| 72.65625 |
8,475 |
Calculate the sum of the series $(3+13+23+33+43)+(11+21+31+41+51)$.
|
270
| 24.21875 |
8,476 |
Given the function $f(x)=\cos (\sqrt{3}x+\varphi)$, where $\varphi \in (-\pi, 0)$. If the function $g(x)=f(x)+f'(x)$ (where $f'(x)$ is the derivative of $f(x)$) is an even function, determine the value of $\varphi$.
|
-\frac{\pi }{3}
| 50.78125 |
8,477 |
Among the six-digit numbers formed by the digits 0, 1, 2, 3, 4, 5 without repetition, calculate the number of the numbers that are divisible by 2.
|
312
| 51.5625 |
8,478 |
Calculate the limit of the function:
$$\lim _{x \rightarrow 0} \frac{e^{4 x}-1}{\sin \left(\pi\left(\frac{x}{2}+1\right)\right)}$$
|
-\frac{8}{\pi}
| 93.75 |
8,479 |
A square floor is tiled with a large number of regular hexagonal tiles, which are either blue or white. Each blue tile is surrounded by 6 white tiles, and each white tile is surrounded by 3 white and 3 blue tiles. Determine the ratio of the number of blue tiles to the number of white tiles, ignoring part tiles.
|
1: 2
| 0.78125 |
8,480 |
Six positive numbers, not exceeding 3, satisfy the equations \(a + b + c + d = 6\) and \(e + f = 2\). What is the minimum value of the expression
$$
\left(\sqrt{a^{2}+4}+\sqrt{b^{2}+e^{2}}+\sqrt{c^{2}+f^{2}}+\sqrt{d^{2}+4}\right)^{2}
$$
|
72
| 10.15625 |
8,481 |
Determine the largest square number that is not divisible by 100 and, when its last two digits are removed, is also a square number.
|
1681
| 65.625 |
8,482 |
The Ivanov family consists of three people: a father, a mother, and a daughter. Today, on the daughter's birthday, the mother calculated the sum of the ages of all family members and got 74 years. It is known that 10 years ago, the total age of the Ivanov family members was 47 years. How old is the mother now if she gave birth to her daughter at the age of 26?
|
33
| 0.78125 |
8,483 |
Given $sinα-\sqrt{3}cosα=1$, then the value of $sin({\frac{{7π}}{6}-2α})$ is ______.
|
\frac{1}{2}
| 84.375 |
8,484 |
Let $p$, $q$, $r$, and $s$ be real numbers such that $|p-q|=1$, $|q-r|=5$, and $|r-s|=6$. What is the sum of all possible values of $|p-s|$?
|
24
| 57.03125 |
8,485 |
Given an ellipse $E: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$, there is a point $M(2,1)$ inside it. Two lines $l_1$ and $l_2$ passing through $M$ intersect the ellipse $E$ at points $A$, $C$ and $B$, $D$ respectively, and satisfy $\overrightarrow{AM}=\lambda \overrightarrow{MC}, \overrightarrow{BM}=\lambda \overrightarrow{MD}$ (where $\lambda > 0$, and $\lambda \neq 1$). If the slope of $AB$ is always $- \frac{1}{2}$ when $\lambda$ changes, find the eccentricity of the ellipse $E$.
|
\frac{\sqrt{3}}{2}
| 35.15625 |
8,486 |
A certain fruit store deals with two types of fruits, A and B. The situation of purchasing fruits twice is shown in the table below:
| Purchase Batch | Quantity of Type A Fruit ($\text{kg}$) | Quantity of Type B Fruit ($\text{kg}$) | Total Cost ($\text{元}$) |
|----------------|---------------------------------------|---------------------------------------|------------------------|
| First | $60$ | $40$ | $1520$ |
| Second | $30$ | $50$ | $1360$ |
$(1)$ Find the purchase prices of type A and type B fruits.
$(2)$ After selling all the fruits purchased in the first two batches, the fruit store decides to reward customers by launching a promotion. In the third purchase, a total of $200$ $\text{kg}$ of type A and type B fruits are bought, and the capital invested does not exceed $3360$ $\text{元}$. Of these, $m$ $\text{kg}$ of type A fruit and $3m$ $\text{kg}$ of type B fruit are sold at the purchase price, while the remaining type A fruit is sold at $17$ $\text{元}$ per $\text{kg}$ and type B fruit is sold at $30$ $\text{元}$ per $\text{kg}$. If all $200$ $\text{kg}$ of fruits purchased in the third batch are sold, and the maximum profit obtained is not less than $800$ $\text{元}$, find the maximum value of the positive integer $m$.
|
22
| 2.34375 |
8,487 |
Egorov decided to open a savings account to buy a car worth 900,000 rubles. The initial deposit is 300,000 rubles. Every month, Egorov plans to add 15,000 rubles to his account. The bank offers a monthly interest rate of $12\%$ per annum. The interest earned each month is added to the account balance, and the interest for the following month is calculated on the new balance. After how many months will there be enough money in the account to buy the car?
|
29
| 3.90625 |
8,488 |
In how many different ways can a chess king move from square $e1$ to square $h5$, if it is only allowed to move one square to the right, upward, or diagonally right-upward?
|
129
| 49.21875 |
8,489 |
Find the number ot 6-tuples $(x_1, x_2,...,x_6)$ , where $x_i=0,1 or 2$ and $x_1+x_2+...+x_6$ is even
|
365
| 91.40625 |
8,490 |
Point \( F \) is the midpoint of side \( BC \) of square \( ABCD \). A perpendicular \( AE \) is drawn to segment \( DF \). Find the angle \( CEF \).
|
45
| 89.84375 |
8,491 |
The greatest common divisor of natural numbers \( m \) and \( n \) is 1. What is the greatest possible value of \(\text{GCD}(m + 2000n, n + 2000m) ?\)
|
3999999
| 65.625 |
8,492 |
How many different three-letter sets of initials are possible using the letters $A$ through $J$, where no letter is repeated in any set?
|
720
| 78.125 |
8,493 |
Given a parabola $y = ax^2 + bx + c$ ($a \neq 0$) with its axis of symmetry on the left side of the y-axis, where $a, b, c \in \{-3, -2, -1, 0, 1, 2, 3\}$. Let the random variable $X$ represent the value of $|a-b|$. Calculate the expected value $E(X)$.
|
\frac{8}{9}
| 0.78125 |
8,494 |
A number \( n \) has a sum of digits equal to 100, while \( 44n \) has a sum of digits equal to 800. Find the sum of the digits of \( 3n \).
|
300
| 50.78125 |
8,495 |
The extensions of a telephone exchange have only 2 digits, from 00 to 99. Not all extensions are in use. By swapping the order of two digits of an extension in use, you either get the same number or the number of an extension not in use. What is the highest possible number of extensions in use?
(a) Less than 45
(b) 45
(c) Between 45 and 55
(d) More than 55
(e) 55
|
55
| 65.625 |
8,496 |
Among the positive integers less than 1000, there are how many numbers that are perfect squares but not perfect cubes?
|
28
| 41.40625 |
8,497 |
The positive five-digit integers that use each of the five digits $1,$ $2,$ $3,$ $4,$ and $5$ exactly once are ordered from least to greatest. What is the $50^{\text{th}}$ integer in the list?
|
31254
| 39.0625 |
8,498 |
Given the function $f\left(x+ \frac {1}{2}\right)= \frac {2x^{4}+x^{2}\sin x+4}{x^{4}+2}$, calculate the value of $f\left( \frac {1}{2017}\right)+f\left( \frac {2}{2017}\right)+\ldots+f\left( \frac {2016}{2017}\right)$.
|
4032
| 35.15625 |
8,499 |
Let the complex numbers \(z\) and \(w\) satisfy \(|z| = 3\) and \((z + \bar{w})(\bar{z} - w) = 7 + 4i\), where \(i\) is the imaginary unit and \(\bar{z}\), \(\bar{w}\) denote the conjugates of \(z\) and \(w\) respectively. Find the modulus of \((z + 2\bar{w})(\bar{z} - 2w)\).
|
\sqrt{65}
| 28.90625 |
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