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40.3k
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100
8,000
Given that the positive integer \( a \) has 15 factors and the positive integer \( b \) has 20 factors, and \( a + b \) is a perfect square, find the smallest possible value of \( a + b \) that meets these conditions.
576
14.0625
8,001
Three people, A, B, and C, are taking an elevator from the 1st floor to the 3rd to 7th floors of a mall. Each floor can accommodate at most 2 people getting off the elevator. How many ways are there for them to get off the elevator?
120
71.875
8,002
Five million times eight million equals
40,000,000,000,000
3.125
8,003
Let \( P \) be an arbitrary point on the graph of the function \( y = x + \frac{2}{x} \) (for \( x > 0 \)). From point \( P \), perpendiculars are drawn to the line \( y = x \) and to the y-axis, with the feet of these perpendiculars labeled as \( A \) and \( B \), respectively. Find the value of \( \overrightarrow{P A} \cdot \overrightarrow{P B} \).
-1
87.5
8,004
Let $A$ be a set of ten distinct positive numbers (not necessarily integers). Determine the maximum possible number of arithmetic progressions consisting of three distinct numbers from the set $A$.
20
54.6875
8,005
Determine the sum of all single-digit replacements for $z$ such that the number ${36{,}z72}$ is divisible by both 6 and 4.
18
80.46875
8,006
Walter gets up at 6:30 a.m., catches the school bus at 7:30 a.m., has 6 classes that last 50 minutes each, has 30 minutes for lunch, and has 2 hours additional time at school. He takes the bus home and arrives at 4:00 p.m. Calculate the total number of minutes Walter spent on the bus.
60
50
8,007
Two candidates participated in an election with \(p+q\) voters. Candidate \(A\) received \(p\) votes and candidate \(B\) received \(q\) votes, with \(p > q\). During the vote counting, only one vote is registered at a time on a board. Let \(r\) be the probability that the number associated with candidate \(A\) on the board is always greater than the number associated with candidate \(B\) throughout the entire counting process. a) Determine the value of \(r\) if \(p=3\) and \(q=2\). b) Determine the value of \(r\) if \(p=1010\) and \(q=1009\).
\frac{1}{2019}
75
8,008
A series of numbers were written: \(100^{100}, 101^{101}, 102^{102}, \ldots, 234^{234}\) (i.e., the numbers of the form \(n^{n}\) for natural \(n\) from 100 to 234). How many of the numbers listed are perfect squares? (A perfect square is defined as the square of an integer.)
71
4.6875
8,009
A student correctly added the two two-digit numbers on the left of the board and got the answer 137. What answer will she obtain if she adds the two four-digit numbers on the right of the board?
13837
23.4375
8,010
Given \( f(x) = 2^x \) and \( g(x) = \log_{\sqrt{2}} (8x) \), find the value of \( x \) that satisfies \( f[g(x)] = g[f(x)] \).
\frac{1 + \sqrt{385}}{64}
43.75
8,011
Calculate the value of the following expressions: (1) $$\sqrt[4]{(3-\pi )^{4}}$$ + (0.008) $$^{- \frac {1}{3}}$$ - (0.25) $$^{ \frac {1}{2}}$$ × ($$\frac {1}{ \sqrt {2}}$$)$$^{-4}$$; (2) $\log_{3}$$\sqrt {27}$$ - $\log_{3}$$\sqrt {3}$$ - $\lg{625}$ - $\lg{4}$ + $\ln(e^{2})$ - $$\frac {4}{3}$$$\lg$$\sqrt {8}$$.
-1
72.65625
8,012
If $\sqrt[3]{3}$ is approximated as $1.442$, calculate the value of $\sqrt[3]{3}-3\sqrt[3]{3}-98\sqrt[3]{3}$.
-144.2
19.53125
8,013
Given vectors $\overrightarrow {m}=(\cos\alpha- \frac { \sqrt {2}}{3}, -1)$, $\overrightarrow {n}=(\sin\alpha, 1)$, and $\overrightarrow {m}$ is collinear with $\overrightarrow {n}$, and $\alpha\in[-\pi,0]$. (Ⅰ) Find the value of $\sin\alpha+\cos\alpha$; (Ⅱ) Find the value of $\frac {\sin2\alpha}{\sin\alpha-\cos\alpha}$.
\frac {7}{12}
81.25
8,014
At 9:00, a pedestrian set off on a journey. An hour later, a cyclist set off from the same starting point. At 10:30, the cyclist caught up with the pedestrian and continued ahead, but after some time, the bicycle broke down. After repairing the bike, the cyclist resumed the journey and caught up with the pedestrian again at 13:00. How many minutes did the repair take? (The pedestrian's speed is constant, and he moved without stopping; the cyclist's speed is also constant except for the repair interval.)
100
36.71875
8,015
The TV station is broadcasting 5 different advertisements, among which there are 3 different commercial advertisements and 2 different Olympic promotional advertisements. The last advertisement must be an Olympic promotional advertisement, and the two Olympic promotional advertisements cannot be broadcast consecutively. Determine the number of different broadcasting methods.
36
57.03125
8,016
Given the function $f(x)=\frac{1}{2}{f'}(1){x^2}+lnx+\frac{{f(1)}}{{3x}}$, find the value of ${f'}\left(2\right)$.
\frac{33}{4}
32.8125
8,017
Beginner millionaire Bill buys a bouquet of 7 roses for $20. Then, he can sell a bouquet of 5 roses for $20 per bouquet. How many bouquets does he need to buy to "earn" a difference of $1000?
125
7.8125
8,018
A circle is inscribed in a convex quadrilateral \(ABCD\) with its center at point \(O\), and \(AO=OC\). Additionally, \(BC=5\), \(CD=12\), and \(\angle DAB\) is a right angle. Find the area of the quadrilateral \(ABCD\).
60
18.75
8,019
Given $\sin\theta= \frac {m-3}{m+5}$ and $\cos\theta= \frac {4-2m}{m+5}$ ($\frac {\pi}{2} < \theta < \pi$), calculate $\tan\theta$.
- \frac {5}{12}
73.4375
8,020
Find the number of permutations \(a_1, a_2, \ldots, a_{10}\) of the numbers \(1, 2, \ldots, 10\) such that \(a_{i+1}\) is not less than \(a_i - 1\) for \(i = 1, 2, \ldots, 9\).
512
92.1875
8,021
Given an ellipse with its foci on the x-axis and its lower vertex at D(0, -1), the eccentricity of the ellipse is $e = \frac{\sqrt{6}}{3}$. A line L passes through the point P(0, 2). (Ⅰ) Find the standard equation of the ellipse. (Ⅱ) If line L is tangent to the ellipse, find the equation of line L. (Ⅲ) If line L intersects the ellipse at two distinct points M and N, find the maximum area of triangle DMN.
\frac{3\sqrt{3}}{4}
6.25
8,022
In a similar tournament setup, the top 6 bowlers have a playoff. First #6 bowls #5, and the loser gets the 6th prize. The winner then bowls #4, and the loser of this match gets the 5th prize. The process continues with the previous winner bowling the next highest ranked bowler until the final match, where the winner of this match gets the 1st prize and the loser gets the 2nd prize. How many different orders can bowlers #1 through #6 receive the prizes?
32
74.21875
8,023
The sequence $\{a_i\}_{i \ge 1}$ is defined by $a_1 = 1$ and \[ a_n = \lfloor a_{n-1} + \sqrt{a_{n-1}} \rfloor \] for all $n \ge 2$ . Compute the eighth perfect square in the sequence. *Proposed by Lewis Chen*
64
11.71875
8,024
In the Cartesian coordinate system $xOy$, the sum of distances from point $P$ to the two points $(0, -\sqrt{3})$ and $(0, \sqrt{3})$ equals $4$. Let the trajectory of point $P$ be $C$. $(1)$ Write the equation of $C$; $(2)$ Suppose the line $y=kx+1$ intersects $C$ at points $A$ and $B$. For what value of $k$ is $\overrightarrow{OA} \bot \overrightarrow{OB}$? What is the value of $|\overrightarrow{AB}|$ at this time?
\frac{4\sqrt{65}}{17}
36.71875
8,025
A sequence of two distinct numbers is extended in two ways: one to form a geometric progression and the other to form an arithmetic progression. The third term of the geometric progression coincides with the tenth term of the arithmetic progression. With which term of the arithmetic progression does the fourth term of the geometric progression coincide?
74
21.875
8,026
In \(\triangle ABC\), \(AC = AB = 25\) and \(BC = 40\). \(D\) is a point chosen on \(BC\). From \(D\), perpendiculars are drawn to meet \(AC\) at \(E\) and \(AB\) at \(F\). \(DE + DF\) equals:
24
21.875
8,027
Two riders simultaneously departed from points \( A \) and \( C \) towards point \( B \). Despite the fact that \( C \) was 20 km farther from \( B \) than \( A \) was from \( B \), both riders arrived at \( B \) at the same time. Find the distance from \( C \) to \( B \), given that the rider from \( C \) traveled each kilometer 1 minute and 15 seconds faster than the rider from \( A \), and the rider from \( A \) reached \( B \) in 5 hours.
80
3.125
8,028
Given that $α \in (0, \frac{π}{2})$ and $\sin α - \cos α = \frac{1}{2}$, find the value of $\frac{\cos 2α}{\sin (α + \frac{π}{4})}$. A) $-\frac{\sqrt{2}}{2}$ B) $\frac{\sqrt{2}}{2}$ C) $-1$ D) $1$
-\frac{\sqrt{2}}{2}
64.84375
8,029
A newly designed car travels 4.2 kilometers further per liter of gasoline than an older model. The fuel consumption for the new car is 2 liters less per 100 kilometers. How many liters of gasoline does the new car consume per 100 kilometers? If necessary, round your answer to two decimal places.
5.97
45.3125
8,030
Suppose $n \ge 0$ is an integer and all the roots of $x^3 + \alpha x + 4 - ( 2 \times 2016^n) = 0$ are integers. Find all possible values of $\alpha$ .
-3
26.5625
8,031
The government decided to privatize civil aviation. For each of the 127 cities in the country, the connecting airline between them is sold to one of the private airlines. Each airline must make all acquired airlines one-way but in such a way as to ensure the possibility of travel from any city to any other city (possibly with several transfers). What is the maximum number of companies that can buy the airlines?
63
10.9375
8,032
An apartment and an office are sold for $15,000 each. The apartment was sold at a loss of 25% and the office at a gain of 25%. Determine the net effect of the transactions.
2000
7.8125
8,033
Given the set $$ M=\{1,2, \cdots, 2020\}, $$ for any non-empty subset $A$ of $M$, let $\lambda_{A}$ be the sum of the maximum and minimum numbers in the subset $A$. What is the arithmetic mean of all such $\lambda_{A}$?
2021
90.625
8,034
A parallelogram $ABCD$ has a base $\overline{BC}$ of 6 units and a height from $A$ to line $BC$ of 3 units. Extend diagonal $AC$ beyond $C$ to point $E$ such that $\overline{CE}$ is 2 units, making $ACE$ a right triangle at $C$. Find the area of the combined shape formed by parallelogram $ABCD$ and triangle $ACE$.
18 + 3\sqrt{5}
7.8125
8,035
John has 15 marbles of different colors, including one red, one green, one blue, and one yellow marble. In how many ways can he choose 5 marbles, if at least one of the chosen marbles is red, green, or blue, but not yellow?
1540
2.34375
8,036
Consider a dark rectangle created by merging two adjacent unit squares in an array of unit squares; part as shown below. If the first ring of squares around this center rectangle contains 10 unit squares, how many unit squares would be in the $100^{th}$ ring?
802
57.03125
8,037
Given that both $\alpha$ and $\beta$ are acute angles, and $\sin \alpha = \frac{3}{5}$, $\tan (\alpha - \beta) = -\frac{1}{3}$. (1) Find the value of $\sin (\alpha - \beta)$; (2) Find the value of $\cos \beta$.
\frac{9\sqrt{10}}{50}
26.5625
8,038
A gold coin is worth $x\%$ more than a silver coin. The silver coin is worth $y\%$ less than the gold coin. Both $x$ and $y$ are positive integers. How many possible values for $x$ are there?
12
42.1875
8,039
A baker adds 0.45 kilograms of flour to a mixing bowl that already contains 2 3/4 kilograms of flour. How many kilograms of flour are in the bowl now?
3.20
4.6875
8,040
If the binomial coefficient of only the fourth term in the expansion of $(x^{2} - \frac {1}{2x})^{n}$ is the largest, then the sum of all the coefficients in the expansion is $\boxed{\text{answer}}$.
\frac {1}{64}
60.15625
8,041
A certain senior high school has a total of 3200 students, with 1000 students each in the second and third grades. A stratified sampling method is used to draw a sample of size 160. The number of first-grade students that should be drawn is ______ .
60
96.875
8,042
Let $n$ be a positive integer, and let $b_0, b_1, \dots, b_n$ be a sequence of real numbers such that $b_0 = 25$, $b_1 = 56$, $b_n = 0$, and $$ b_{k+1} = b_{k-1} - \frac{7}{b_k} $$ for $k = 1, 2, \dots, n-1$. Find $n$.
201
10.15625
8,043
If line $l_1: ax+2y+6=0$ is parallel to line $l_2: x+(a-1)y+(a^2-1)=0$, then the real number $a=$ .
-1
35.9375
8,044
A man asks: "How old are you, my son?" To this question, I answered: "If my father were seven times as old as I was eight years ago, then one quarter of my father's current age would surely be fourteen years now. Please calculate from this how many years weigh upon my shoulders!"
16
88.28125
8,045
Find the measure of the angle $$ \delta=\arccos \left(\left(\sin 2907^{\circ}+\sin 2908^{\circ}+\cdots+\sin 6507^{\circ}\right)^{\cos 2880^{\circ}+\cos 2881^{\circ}+\cdots+\cos 6480^{\circ}}\right) $$
63
2.34375
8,046
Two teachers are taking a photo with 3 male students and 3 female students lined up in a row. The teachers can only stand at the ends, and the male students cannot be adjacent. How many different ways are there to arrange the photo?
288
67.1875
8,047
Convert the binary number $1110011_2$ to its decimal equivalent.
115
100
8,048
A rectangular table of size \( x \) cm by 80 cm is covered with identical sheets of paper of size 5 cm by 8 cm. The first sheet is placed in the bottom-left corner, and each subsequent sheet is placed one centimeter higher and one centimeter to the right of the previous one. The last sheet is placed in the top-right corner. What is the length \( x \) in centimeters?
77
21.875
8,049
The sum of the first four terms of an arithmetic progression, as well as the sum of the first seven terms, are natural numbers. Furthermore, its first term \(a_1\) satisfies the inequality \(a_1 \leq \frac{2}{3}\). What is the greatest value that \(a_1\) can take?
9/14
2.34375
8,050
In the isosceles trapezoid $ABCD$, $AD \parallel BC$, $\angle B = 45^\circ$. Point $P$ is on the side $BC$. The area of $\triangle PAD$ is $\frac{1}{2}$, and $\angle APD = 90^\circ$. Find the minimum value of $AD$.
\sqrt{2}
37.5
8,051
Given circle $C$: $x^{2}+y^{2}+8x+ay-5=0$ passes through the focus of parabola $E$: $x^{2}=4y$. The length of the chord formed by the intersection of the directrix of parabola $E$ and circle $C$ is $\_\_\_\_\_\_$.
4 \sqrt{6}
92.96875
8,052
Given that \(0 \leq a_{k} \leq 1 \) for \(k=1,2, \ldots, 2020\), and defining \(a_{2021} = a_{1}\), \(a_{2022} = a_{2}\), find the maximum value of \(\sum_{k=1}^{2020}\left(a_{k} - a_{k+1}a_{k+2}\right)\).
1010
43.75
8,053
In how many ways can a barrel with a capacity of 10 liters be emptied using two containers with capacities of 1 liter and 2 liters?
89
25
8,054
Calculate the limit of the function: $$ \lim_{x \rightarrow 1} (2-x)^{\sin \left(\frac{\pi x}{2}\right) / \ln (2-x)} $$
e
92.1875
8,055
In the geometric sequence $\{a_n\}$, $a_2a_3=5$ and $a_5a_6=10$. Calculate the value of $a_8a_9$.
20
46.09375
8,056
How many five-digit natural numbers are divisible by 9, where the last digit is greater than the second last digit by 2?
800
20.3125
8,057
In triangle $ABC$, $D$ is on segment $BC$ such that $BD:DC = 3:2$. Point $E$ bisects $\angle BAC$. Given $BD = 45$ units and $DC = 30$ units, find the length of $BE$.
45
17.96875
8,058
To assess the shooting level of a university shooting club, an analysis group used stratified sampling to select the shooting scores of $6$ senior members and $2$ new members for analysis. After calculation, the sample mean of the shooting scores of the $6$ senior members is $8$ (unit: rings), with a variance of $\frac{5}{3}$ (unit: rings$^{2}$). The shooting scores of the $2$ new members are $3$ rings and $5$ rings, respectively. What is the variance of the shooting scores of these $8$ members?
\frac{9}{2}
37.5
8,059
Point \( D \) lies on side \( CB \) of right triangle \( ABC \left(\angle C = 90^{\circ} \right) \), such that \( AB = 5 \), \(\angle ADC = \arccos \frac{1}{\sqrt{10}}, DB = \frac{4 \sqrt{10}}{3} \). Find the area of triangle \( ABC \).
15/4
14.0625
8,060
Let $N$ denote the number of all natural numbers $n$ such that $n$ is divisible by a prime $p> \sqrt{n}$ and $p<20$ . What is the value of $N$ ?
69
33.59375
8,061
The only prime factors of an integer $n$ are 2 and 3. If the sum of the divisors of $n$ (including itself) is $1815$ , find $n$ .
648
46.875
8,062
In a $5 \times 18$ rectangle, the numbers from 1 to 90 are placed. This results in five rows and eighteen columns. In each column, the median value is chosen, and among the medians, the largest one is selected. What is the minimum possible value that this largest median can take? Recall that among 99 numbers, the median is such a number that is greater than 49 others and less than 49 others.
54
3.90625
8,063
A truncated pyramid has a square base with a side length of 4 units, and every lateral edge is also 4 units. The side length of the top face is 2 units. What is the greatest possible distance between any two vertices of the truncated pyramid?
\sqrt{32}
0
8,064
For a positive integer \( n \), let \( s(n) \) denote the sum of its digits, and let \( p(n) \) denote the product of its digits. If the equation \( s(n) + p(n) = n \) holds true, then \( n \) is called a coincidence number. What is the sum of all coincidence numbers?
531
96.09375
8,065
Let $[x]$ represent the greatest integer less than or equal to the real number $x$. Define the sets $$ \begin{array}{l} A=\{y \mid y=[x]+[2x]+[4x], x \in \mathbf{R}\}, \\ B=\{1,2, \cdots, 2019\}. \end{array} $$ Find the number of elements in the intersection $A \cap B$.
1154
1.5625
8,066
The sides of a triangle are 5, 6, and 7. Find the area of the orthogonal projection of the triangle onto a plane that forms an angle equal to the smallest angle of the triangle with the plane of the triangle.
\frac{30 \sqrt{6}}{7}
21.09375
8,067
In how many ways can you select two letters from the word "УЧЕБНИК" such that one of the letters is a consonant and the other is a vowel?
12
61.71875
8,068
Find the largest value of the parameter \(a\) for which the equation: \[ (|x-2| + 2a)^2 - 3(|x-2| + 2a) + 4a(3-4a) = 0 \] has exactly three solutions. Indicate the largest value in the answer.
0.5
14.0625
8,069
Choose one digit from 0, 2, 4, and two digits from 1, 3, 5 to form a three-digit number without repeating digits. The total number of different three-digit numbers that can be formed is (    ) A 36      B 48       C 52       D 54
48
50.78125
8,070
An electric company installed a total of 402 poles along both sides of the road, with a distance of 20 meters between each adjacent pole. Later, all poles were replaced, and only 202 poles were installed. The distance between each adjacent pole after the replacement is $\qquad$ meters.
40
82.03125
8,071
Construct a square \(A B C D\) with side length \(6 \text{ cm}\). Construct a line \(p\) parallel to the diagonal \(A C\) passing through point \(D\). Construct a rectangle \(A C E F\) such that vertices \(E\) and \(F\) lie on the line \(p\). Using the given information, calculate the area of rectangle \(A C E F\).
36
13.28125
8,072
If $a$, $b$, $c$, $d$, $e$, and $f$ are integers for which $512x^3 + 125 = (ax^2 + bx + c)(dx^2 + ex + f)$ for all $x$, then what is $a^2+b^2+c^2+d^2+e^2+f^2$?
6410
0.78125
8,073
The set $\{[x] + [2x] + [3x] \mid x \in \mathbb{R}\} \mid \{x \mid 1 \leq x \leq 100, x \in \mathbb{Z}\}$ has how many elements, where $[x]$ denotes the greatest integer less than or equal to $x$.
67
0
8,074
In the trapezoid \(ABCD\) with bases \(AD \parallel BC\), the diagonals intersect at point \(E\). Given the areas \(S(\triangle ADE) = 12\) and \(S(\triangle BCE) = 3\), find the area of the trapezoid.
27
42.96875
8,075
In a class at school, all students are the same age, except seven of them who are 1 year younger and two of them who are 2 years older. The sum of the ages of all the students in this class is 330. How many students are in this class?
37
68.75
8,076
If the functions \( f(x) \) and \( g(x) \) are defined on \( \mathbf{R} \), and \( f(x-y)=f(x)g(y)-g(x)f(y) \), with \( f(-2)=f(1) \neq 0 \), what is \( g(1) + g(-1) \)?
-1
71.875
8,077
Given that the cross-section of a cylinder is a square and the height of the cylinder is equal to the diameter of a sphere, calculate the ratio of the total surface area of the cylinder to the surface area of the sphere.
\frac{3}{2}
99.21875
8,078
How many perfect squares less than 20,000 can be represented as the difference of squares of two integers that differ by 2?
70
87.5
8,079
An isosceles triangle and a rectangle both have perimeters of 60 inches. The rectangle's length is twice its width. What is the ratio of the length of one of the equal sides of the triangle to the width of the rectangle? Express your answer as a common fraction.
\frac{5}{2}
6.25
8,080
In triangle $\triangle ABC$, $\sin A = \frac{3}{5}$, $\tan B = 2$. Find the value of $\tan 2\left(A+B\right)$.
\frac{44}{117}
72.65625
8,081
In a right cone with a base radius of \(15\) cm and a height of \(30\) cm, a sphere is inscribed. The radius of the sphere can be expressed as \(b\sqrt{d} - b\) cm. Find the value of \(b + d\).
12.5
10.9375
8,082
Given that a ship travels in one direction and Emily walks parallel to the riverbank in the opposite direction, counting 210 steps from back to front and 42 steps from front to back, determine the length of the ship in terms of Emily's equal steps.
70
64.0625
8,083
Find the last 3 digits of \(1 \times 3 \times 5 \times 7 \times \cdots \times 2005\).
375
87.5
8,084
Square $PQRS$ has sides of length 1. Points $M$ and $N$ are on $\overline{QR}$ and $\overline{RS},$ respectively, so that $\triangle PMN$ is equilateral. A square with vertex $Q$ has sides that are parallel to those of $PQRS$ and a vertex on $\overline{PM}.$ The length of a side of this smaller square is $\frac{d-\sqrt{e}}{f},$ where $d, e,$ and $f$ are positive integers and $e$ is not divisible by the square of any prime. Find $d+e+f.$
12
27.34375
8,085
In triangle \( \triangle ABC \), \( |AB| = 13 \), \( |BC| = 14 \), \( |CA| = 15 \), an internal point \( P \) satisfies \[ \overrightarrow{BP} \cdot \overrightarrow{CA} = 18 \text{ and } \overrightarrow{CP} \cdot \overrightarrow{BA} = 32. \] What is \( \overrightarrow{AP} \cdot \overrightarrow{BC} \)?
14
21.09375
8,086
On the extensions of the medians \(A K\), \(B L\), and \(C M\) of triangle \(A B C\), points \(P\), \(Q\), and \(R\) are taken such that \(K P = \frac{1}{2} A K\), \(L Q = \frac{1}{2} B L\), and \(M R = \frac{1}{2} C M\). Find the area of triangle \(P Q R\) if the area of triangle \(A B C\) is 1.
25/16
0
8,087
Let function $G(n)$ denote the number of solutions to the equation $\cos x = \sin nx$ on the interval $[0, 2\pi]$. For each integer $n$ greater than 2, what is the sum $\sum_{n=3}^{100} G(n)$?
10094
16.40625
8,088
If $k \in [-2, 2]$, find the probability that for the value of $k$, there can be two tangents drawn from the point A(1, 1) to the circle $x^2 + y^2 + kx - 2y - \frac{5}{4}k = 0$.
\frac{1}{4}
0
8,089
On a line, 5 points \( P, Q, R, S, T \) are marked in that order. It is known that the sum of the distances from \( P \) to the other 4 points is 67, and the sum of the distances from \( Q \) to the other 4 points is 34. Find the length of segment \( PQ \).
11
7.8125
8,090
Given the numbers \(5^{1971}\) and \(2^{1971}\) written consecutively, what is the number of digits in the resulting number?
1972
98.4375
8,091
Starting with a pair of adult rabbits, the number of pairs of adult rabbits after one year can be calculated using the recurrence relation of the Fibonacci sequence, and the result is given by $F(13).
233
80.46875
8,092
In the right parallelopiped $ABCDA^{\prime}B^{\prime}C^{\prime}D^{\prime}$ , with $AB=12\sqrt{3}$ cm and $AA^{\prime}=18$ cm, we consider the points $P\in AA^{\prime}$ and $N\in A^{\prime}B^{\prime}$ such that $A^{\prime}N=3B^{\prime}N$ . Determine the length of the line segment $AP$ such that for any position of the point $M\in BC$ , the triangle $MNP$ is right angled at $N$ .
27/2
19.53125
8,093
Given $\sin (α- \frac{π}{3})= \frac{3}{5} $, where $a∈(\frac{π}{4}, \frac{π}{2})$, find $\tan α=\_\_\_\_\_\_\_\_\_$.
- \frac{48+25 \sqrt{3}}{11}
32.03125
8,094
An infinite geometric series has a first term of \( 512 \) and its sum is \( 8000 \). What is its common ratio?
0.936
9.375
8,095
For the positive integer \( n \), if the expansion of \( (xy - 5x + 3y - 15)^n \) is combined and simplified, and \( x^i y^j \) (where \( i, j = 0, 1, \ldots, n \)) has at least 2021 terms, what is the minimum value of \( n \)?
44
32.8125
8,096
Find the sum of all integers $x$ , $x \ge 3$ , such that $201020112012_x$ (that is, $201020112012$ interpreted as a base $x$ number) is divisible by $x-1$
32
56.25
8,097
Compute $$ \lim _{h \rightarrow 0} \frac{\sin \left(\frac{\pi}{3}+4 h\right)-4 \sin \left(\frac{\pi}{3}+3 h\right)+6 \sin \left(\frac{\pi}{3}+2 h\right)-4 \sin \left(\frac{\pi}{3}+h\right)+\sin \left(\frac{\pi}{3}\right)}{h^{4}} $$
\frac{\sqrt{3}}{2}
17.96875
8,098
There are nine parts in a bag, including five different genuine parts and four different defective ones. These parts are being drawn and inspected one by one. If the last defective part is found exactly on the fifth draw, calculate the total number of different sequences of draws.
480
6.25
8,099
Given \(a, b, c, x, y, z \in \mathbf{R}_{+}\) satisfying the equations: \[ cy + bz = a, \quad az + cx = b, \quad bx + ay = c, \] find the minimum value of the function \[ f(x, y, z) = \frac{x^2}{1 + x} + \frac{y^2}{1 + y} + \frac{z^2}{1 + z}. \]
\frac{1}{2}
68.75