problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
|---|---|
# Task 2. 15 points
Solve the equation
$$
\sqrt{\frac{x^{3}+5}{1+\sqrt{5}}}=x
$$
# | \frac{1+\sqrt{1+4\sqrt{5}}}{2},\sqrt{5} |
\section*{Exercise 2 - 121012}
In a right-angled Cartesian coordinate system, a parabola is given by the equation \(y=x^{2}\).
Provide an equation of a line that does not run parallel to the \(y\)-axis and intersects the parabola at exactly one point \(P\) with the abscissa 3! | 6x-9 |
Given an arithmetic sequence \(\{a_n\}\) with the sum of the first \(n\) terms denoted as \(S_n\), if \(S_3=9\) and \(S_5=30\), then find the value of \(a_7+a_8+a_9\). | 63 |
Find all integers $n$ such that $2^{n}+3$ is a perfect square. Same question with $2^{n}+1$.
## - Solution - | n=0for2^n+3,\;n=3for2^n+1 |
The harmonic mean of a set of non-zero numbers is the reciprocal of the average of the reciprocals of the numbers. What is the harmonic mean of 1, 2, and 4?
$\textbf{(A) }\frac{3}{7}\qquad\textbf{(B) }\frac{7}{12}\qquad\textbf{(C) }\frac{12}{7}\qquad\textbf{(D) }\frac{7}{4}\qquad \textbf{(E) }\frac{7}{3}$ | \frac{12}{7} |
The daily sales volume \\(y\\) (unit: tons) of a certain product and its selling price \\(x\\) (unit: ten thousand yuan/ton, \\(1 < x \leqslant 5\\)) satisfy: when \\(1 < x \leqslant 3\\), \\(y=a(x-4)^2+\frac{6}{x-1}\\) (\\(a\\) is a constant); when \\(3 < x \leqslant 5\\), \\(y=kx+7\\) (\\(k < 0\\)). It is known that ... | 2 |
Find the greatest common divisor and the least common multiple of 336 and 1260. | 5040 |
## Problem Statement
Calculate the definite integral:
$$
\int_{0}^{2} \frac{(4 \sqrt{2-x}-\sqrt{2 x+2}) d x}{(\sqrt{2 x+2}+4 \sqrt{2-x})(2 x+2)^{2}}
$$ | \frac{1}{24}\ln5 |
5. At a fair, there is a very inviting game, as it can be played for free; the winner gets a prize. The agreed prize for the first four games is one coin, for the fifth it is two coins. Nicola has a probability of $\frac{2}{3}$ of winning the prize in each game and decides to play 5 games. What is the probability that ... | 5\cdot(\frac{2}{3})^{5} |
29. $5^{2 x+1}+6^{x+1}>30+5^{x} 30^{x}$. | \frac{1}{2}\log_{5}6<x<\log_{6}5 |
7.2. Find the sum of all three-digit natural numbers that do not contain the digits 0 or 9 in their representation. | 255744 |
I5 (7-1, Yugoslavia) Find all real numbers $x$ in the interval $0 \leqslant x \leqslant 2 \pi$ that satisfy the inequality
$$
2 \cos x \leqslant|\sqrt{1+\sin 2 x}-\sqrt{1-\sin 2 x}| \leqslant \sqrt{2}
$$ | \frac{\pi}{4}\leqslantx\leqslant\frac{7}{4}\pi |
Let $a=\log_{3}2$, $b=\log_{2} \frac{1}{8}$, $c= \sqrt{2}$. Then ( )
A: $a > b > c$
B: $c > b > a$
C: $a > c > b$
D: $c > a > b$ | D |
Tomorrow morning, Li Ming is going to participate in a volunteer activity. To wake up on time, he sets two alarms, A and B. Suppose the probability that alarm A goes off on time is $0.80$, and the probability that alarm B goes off on time is $0.90$. The probability that at least one of the alarms goes off on time is __... | 0.98 |
The maximum distance from a point on the circle $x^{2}+y^{2}-2x-2y+1=0$ to the line $x-y=2$ is ______. | \sqrt{2} + 1 |
[ Arithmetic progression $]$ [ Extreme principle (other) . ] Author: Shapovalov $A . B$. In a $29 \times 29$ table, the numbers $1,2,3, \ldots, 29$ were written, each 29 times. It turned out that the sum of the numbers above the main diagonal is three times the sum of the numbers below this diagonal. Find the number... | 15 |
8. A light ray $l$ emitted from point $A(-3,3)$ is reflected by the $x$-axis. The straight line of the reflected ray is tangent to the circle $x^{2}+y^{2}-4 x-4 y+7=0$. Then the equation of the straight line $l$ is $\qquad$ . | 4x+3y+3=0or3x+4y-3=0 |
Given that the sum of the first $n$ terms of the sequence $\{a_{n}\}$ is $S_{n}=(n-1)^{2}-1$.
$(1)$ Find the general formula for $\{a_{n}\}$;
$(2)$ If the sequence $b_{n}=\frac{1}{{a_n}{a_{n+1}}}$, find the sum of the first $n$ terms of the sequence $\{b_{n}\}$, denoted as $T_{n}$. | -\frac{n}{{2n-1}} |
G2.3 Given that $x$ is a positive real number and $x \cdot 3^{x}=3^{18}$. If $k$ is a positive integer and $k<x<k+1$, find the value of $k$. | 15 |
1. The range of the function $f(x)=\sin x+\cos x+\tan x+$ $\arcsin x+\arccos x+\arctan x$ is $\qquad$ . | [-\sin 1+\cos 1-\tan 1+\frac{\pi}{4}, \sin 1+\cos 1+\tan 1+\frac{3 \pi}{4}] |
Given an arithmetic sequence $\{a_{n}\}$ with the sum of the first $n$ terms as $S_{n}$, and it satisfies $a_{3}=8$ and $S_{5}=2a_{7}$.
$(1)$ Find the general formula for the sequence $\{a_{n}\}$;
$(2)$ If the sequence $\{b_{n}\}$ satisfies $b_{n}=a_{n}+2^{n+1}$, find the sum of the first $2n$ terms of the sequence... | T_{2n}=6n^2+n+2^{2n+2}-4 |
Determine the value of the expression $\sin 40^{\circ}\cos 20^{\circ}-\cos 220^{\circ}\sin 20^{\circ}$. | \frac{\sqrt{3}}{2} |
37. Convert $\frac{1}{7}$ to a decimal, then find the sum of the digit in the 2017th position after the decimal point and the digit in the 7102nd position after the decimal point. | 9 |
Let $s_1, s_2, s_3$ be the three roots of $x^3 + x^2 +\frac92x + 9$.
$$\prod_{i=1}^{3}(4s^4_i + 81)$$ can be written as $2^a3^b5^c$. Find $a + b + c$. | 16 |
A2. Two ants start at the same point in the plane. Each minute they choose whether to walk due north, east, south or west. They each walk 1 meter in the first minute. In each subsequent minute the distance they walk is multiplied by a rational number $q>0$. They meet after a whole number of minutes, but have not taken... | 1 |
Let $x_1\leq x_2\leq \cdots\leq x_{100}$ be real numbers such that $|x_1| + |x_2| + \cdots + |x_{100}| = 1$ and $x_1 + x_2 + \cdots + x_{100} = 0$. Among all such $100$-tuples of numbers, the greatest value that $x_{76} - x_{16}$ can achieve is $\tfrac mn$, where $m$ and $n$ are relatively prime positive integers. Find... | 841 |
Inside the ellipse $\frac{x^{2}}{16} + \frac{y^{2}}{4} = 1$, the equation of the line that passes through point M(1, 1) and is bisected by this point is ( )
A: x+4y-5=0
B: x-4y-5=0
C: 4x+y-5=0
D: 4x-y-5=0 | \text{A} |
Given two arithmetic sequences $ \{ a_n \} $ and $ \{ b_n \} $ with the sum of the first $ n $ terms denoted as $ S_n $ and $ T_n $ respectively, and satisfying $ \frac {S_{n}}{T_{n}} = \frac {7n+2}{n+3} $, find the ratio $ \frac {a_{5}}{b_{5}} $. | \frac {65}{12} |
## Problem Statement
Calculate the limit of the function:
$\lim _{x \rightarrow a} \frac{\tan x - \tan a}{\ln x - \ln a}$ | \frac{}{\cos^2} |
Anton, Artem, and Vera decided to solve 100 math problems together. Each of them solved 60 problems. We will call a problem difficult if it was solved by only one person, and easy if it was solved by all three. How much does the number of difficult problems differ from the number of easy ones?
# | 20 |
Given that the sum of the first $n$ terms of the sequence $\{a_{n}\}$ is ${S}_{n}=3{n}^{2}+5n$, and the sequence $\{b_{n}\}$ satisfies $b_{1}=80$ and $b_{n}=64b_{n+1}$.
$(1)$ Prove that $\{a_{n}\}$ is an arithmetic sequence.
$(2)$ Determine whether there exist constants $a$ and $b$ such that for all positive intege... | a=\frac{1}{2}, \quad b=12+\log_{2}5 |
Example 7 The roots of the equation $x^{2}+p x+q=0$ are both positive integers, and $p+q=1992$. Then the ratio of the larger root to the smaller root is $\qquad$. | 997 |
64. Let $a, b, c$ be non-zero real numbers, and $a b=3(a+b), b c=4(b+c), a c=5(a+c)$, then $a=$ $\qquad$ $b=$ $\qquad$ , $c=$ $\qquad$ . | \frac{120}{17},\frac{120}{23},\frac{120}{7} |
13. $P Q R S T$ is a regular pentagon. The point $U$ lies on $S T$ such that $\angle Q P U$ is a right angle. What is the ratio of the interior angles in triangle $P U T$ ?
A $1: 3: 6$
B $1: 2: 4$
C $2: 3: 4$
D $1: 4: 8$
E $1: 3: 5$ | 1:3:6 |
Let $f(x) = \frac{4^x}{4^x+2}$,
(1) Prove that $f(x) + f(1-x) = 1$;
(2) Find the sum $f\left(\frac{1}{2014}\right) + f\left(\frac{2}{2014}\right) + \ldots + f\left(\frac{2013}{2014}\right)$. | 1006.5 |
As a reward for working for NIMO, Evan divides $100$ indivisible marbles among three of his volunteers: David, Justin, and Michael. (Of course, each volunteer must get at least one marble!) However, Evan knows that, in the middle of the night, Lewis will select a positive integer $n > 1$ and, for each volunteer, steal ... | 3540 |
Let $n$ be a positive integer with $d$ digits, all different from zero. For $k = 0,. . . , d - 1$, we define $n_k$ as the number obtained by moving the last $k$ digits of $n$ to the beginning. For example, if $n = 2184$ then $n_0 = 2184, n_1 = 4218, n_2 = 8421, n_3 = 1842$. For $m$ a positive integer, define $s_m(n)$ a... | \frac{1}{3} |
Example 4 Given $\cdots 1 \leqslant a \leqslant 1$, the inequality $\left(\frac{1}{2}\right)^{x^{2}+a x}<\left(\frac{1}{2}\right)^{2 x+a-1}$ always holds. Find the range of values for $x$. | x>2 \text{ or } x<0 |
Problem 2. A segment with a length of 89.6 cm is divided into four parts such that $\frac{1}{2}$ of the first part is equal to $\frac{1}{3}$ of the second part, $\frac{1}{3}$ of the second part is equal to $\frac{1}{4}$ of the third part, and $\frac{1}{4}$ of the third part is equal to $\frac{1}{5}$ of the fourth part.... | 12.8\mathrm{~},19.2\mathrm{~},25.6\mathrm{~},32\mathrm{~} |
4. If $a, c$ are both positive integers, and the equations $x^{2}+a x+$ $18=0$ and $x^{2}-a x+c=0$ have a common prime root. Then the value of $c$ is $(\quad$.
(A) -18
(B) -26
(C) -26 or -36
(D) 26 or 36 | C |
In the arithmetic sequence $\{a\_n\}$, $a\_1 + a\_{11} = 8$. The sequence $\{b\_n\}$ is a geometric sequence and $b\_7 = a\_7$. What is the value of $b\_6 \cdot b\_8$?
A: $2$
B: $4$
C: $8$
D: $16$ | D |
54. A deck of playing cards has 52 cards. By convention, $A, J, Q, K$ are valued as 1 point, 11 points, 12 points, 13 points respectively. If you draw several cards at random, disregarding suits, to ensure that there are three cards with the same point value, you need to draw at least $\qquad$ cards; to ensure that the... | 27,29 |
In the arithmetic sequence $\{a_n\}$, $a_3=5$ and $a_4+a_8=22$. Find the sum of the first $8$ terms of $\{a_n\}$.
A: $32$
B: $64$
C: $108$
D: $128$ | B |
4. Krt Črt has 5 rooms in his burrow, numbered from 1 to 5. Between some of them, Črt has dug tunnels, so from each room he can slide through several tunnels to any other room. No two tunnels intersect. Each tunnel starts in one room, ends in another (different from the starting one), and does not pass through any room... | (1,4),(2,3),(2,4),(3,4),(4,5) |
Let $f(n)$ denote the largest odd factor of $n$, including possibly $n$. Determine the value of
\[\frac{f(1)}{1} + \frac{f(2)}{2} + \frac{f(3)}{3} + \cdots + \frac{f(2048)}{2048},\]
rounded to the nearest integer. | 1365 |
Let $U=\mathbb{R}$, and define the sets $A=\{x|x^2+3x+2=0\}$, $B=\{x|x^2+(m+1)x+m=0\}$. If $B \subseteq A$, find the value of $m$. | m=1 \text{ or } m=2 |
Given vectors $\overrightarrow{a}=(1,2)$, $\overrightarrow{b}=(x,1)$, and $\overrightarrow{a} \parallel \overrightarrow{b}$, find $|\overrightarrow{b}|$. | \frac{\sqrt{5}}{2} |
If $x \gt y$ holds, then the inequality that does not hold is:
A: $x-1 \gt y-1$
B: $2x \gt x+y$
C: $\frac{x}{2} \gt \frac{y}{2}$
D: $-3x \gt -3y$ | D |
Determine the distance from the point $(2,4,1)$ to the plane $x+2y+3z+3=0$. | \frac{8\sqrt{14}}{7} |
Let $A$ and $B$ be moving points on the lines $y= \frac{ \sqrt{2}}{2}x$ and $y=- \frac{ \sqrt{2}}{2}x$, respectively, with $|AB|=2 \sqrt{2}$. Let $O$ be the coordinate origin, and let moving point $P$ satisfy $\overrightarrow{OP}= \overrightarrow{OA}+ \overrightarrow{OB}$.
(I) Find the trajectory equation $C_{1}$ of mo... | \frac{5y^{2}}{4}- \frac{5x^{2}}{16}=1 |
(i) Find all integers $n \geqslant 1$ such that $n$ divides $2^{n}-1$.
(ii) Find all odd integers $n \geqslant 1$ such that $n$ divides $3^{n}+1$.
## - Solution - | 1 |
The graph of the function $y=a^{x-2016}+2016$ $(a > 0$ and $a \neq 1)$ always passes through the fixed point $\_\_\_\_\_\_\_\_\_.$ | (2016,2017) |
C2. A diagonal of a regular 2006-gon is called odd if its endpoints divide the boundary into two parts, each composed of an odd number of sides. Sides are also regarded as odd diagonals. Suppose the 2006-gon has been dissected into triangles by 2003 nonintersecting diagonals. Find the maximum possible number of isoscel... | 1003 |
Given that the domain of the function $f(x)$ is $R$, and the function $f(2x+1)$ is an odd function with a period of $2$, then which of the following statements is true?
A: The graph of the function $f(x)$ is symmetric with respect to the point $(\frac{1}{2}, 0)$.
B: The graph of the function $f(x)$ is symmetric with ... | B \text{ and } D |
Given the function $f(x)=e^{x}+2x-a$, where $a\in\mathbb{R}$, if there exists a point $(x_{0},y_{0})$ on the curve $y=\sin x$ such that $f(f(y_{0}))=y_{0}$, then the range of the real number $a$ is \_\_\_\_\_\_. | [-1+e^{-1},e+1] |
2. If the planar vectors $\vec{a}=\left(2^{m},-1\right)$ and $\vec{b}=\left(2^{m}-1,2^{m+1}\right)$ are perpendicular, where $m$ is a real number, then the magnitude of $\vec{a}$ is $\qquad$ . | \sqrt{10} |
Steve begins at 7 and counts forward by 3 , obtaining the list $7,10,13$, and so on. Dave begins at 2011 and counts backwards by 5, obtaining the list 2011,2006,2001, and so on. Which of the following numbers appear in each of their lists?
(A) 1009
(B) 1006
(C) 1003
(D) 1001
(E) 1011 | 1006 |
The hyperbola $C$: $\dfrac{x^{2}}{a^{2}} - \dfrac{y^{2}}{b^{2}} = 1$ $(a > 0, b > 0)$ and the parabola $y^{2} = 4x$ share a common focus $F$, and $P$ is their common point. Let $Q(0,1)$. If $QP \perpendicular QF$, then the eccentricity $e$ of $C$ is ( ).
A: $\sqrt{2} + 1$
B: $\dfrac{\sqrt{5}}{2}$
C: $\sqrt{2}$
D: ... | A |
13.295. A reservoir is supplied by two channels. Water flows out evenly through the first channel, and flows in evenly through the second channel. How many hours will it take for $n$ liters of water to pass through the first channel, if it is known that through the second channel, twice as much water will flow in when ... | \frac{^{2}+n+\sqrt{^{4}+6^{2}n+n^{2}}}{2} |
(Reminder of factorization)
Find all integer pairs $(x, y)$ that are solutions to the equation $x^{2}=7+y^{2}$. | (4,3),(4,-3),(-4,-3),(-4,3) |
5. In decimal, if a positive integer with at least two digits has all its digits, except the leftmost one, smaller than the digit to their left, then it is called a descending positive integer. The total number of such descending positive integers is ( ).
(A) 1001
(B) 1010
(C) 1011
(D) 1013 | 1013 |
4. (20 points) Determine the number of solutions of the equation $a(x-|x|+2)=|x-3|-2$ for each value of the parameter $a$. | For\\\in(-\infty;-1)\-\one\solution;\for\=-1\-\two\solutions;\for\\\in(-1;-0.5)\-\three\solutions;\for\\\in[-0.5;+\infty)\-\two\solutions |
Given $a + a^{-1} = 3$, then $a^2 + a^{-2} = \ $ | 7 |
3. For any natural number $n$, expressed in base 10, let $S(n)$ denote the sum of all digits of $n$. Find all natural numbers $n$ such that $n^{3}=8 S(n)^{3}+6 n S(n)+1$.
| 17 |
Let \\(a \in \mathbb{R}\\), then " \\(a = -2\\) " is the \\(\underline{\hspace{2cm}}\\) condition for "Line \\(l_{1}\\): \\(ax + 2y - 1 = 0\\) is parallel to line \\(l_{2}\\): \\(x + (a + 1)y + 4 = 0\\)".
A: Sufficient but not necessary condition
B: Necessary but not sufficient condition
C: Sufficient and necessar... | A |
Example 3.32. Find the partial derivatives $\frac{\partial z}{\partial x}$ and $\frac{\partial z}{\partial y}$ of the function $z=f(x, y)$, given implicitly by the equation $e^{z^{2}}-$ $-x^{2} y^{2} z^{2}=0$ | \begin{aligned}\frac{\partialz}{\partialx}&=\frac{z}{x(z^{2}-1)}\\\frac{\partialz}{\partialy}&=\frac{z}{y(z^{2}-1)}\end{aligned} |
In the sequence $\{a_n\}$, $$a_{1}= \frac {1}{2019}，a_{n+1}=a_{n}+ \frac {1}{n(n+1)}，(n∈N^*)$$, find the value of $a_{2019}$. | 1 |
We need to conduct the following 3 sampling surveys:
1. Draw 5 bottles from 15 bottles of drinks for food hygiene inspection.
2. A certain middle school in Taizhou has a total of 240 staff members, including 180 general teachers, 24 administrative staff, and 36 logistics staff. To understand the opinions of the staff o... | \text{B} |
Given the function $f(x)= \begin{cases} \sqrt {x},x\geqslant 0 \\ ( \frac {1}{2})^{x},x < 0 \end{cases}$, find $f[f(-4)]=$ \_\_\_\_\_\_. | 4 |
2. Let $k=\int_{0}^{\pi}(\sin x-\cos x) \mathrm{d} x$. If $(1-k x)^{8}=a_{0}+a_{1} x+a_{2} x^{2}+\cdots+a_{8} x^{8}$, then $a_{1}+a_{2}+\cdots+a_{8}=(\quad)$.
(A) -1
(B) 0
(C) 1
(D) 256 | B |
Given the set M = {x | -1 < x < 3} and the set N = {x | x < 1}, find the intersection of M and N (M ∩ N) equal to ( ):
A: (1, 3)
B: (-∞, -1)
C: (-1, 1)
D: (-3, 1) | \text{C} |
8. (3 points) In the multiplication equation GrassGreen・FlowerRed $=$ SpringBright, the Chinese characters represent non-zero digits, and different characters represent different digits. Therefore, the smallest four-digit number represented by SpringBright is $\qquad$
将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻译结果。 | 4396 |
Points $X$ and $Y$ are the midpoints of arcs $AB$ and $BC$ of the circumscribed circle of triangle $ABC$. Point $T$ lies on side $AC$. It turned out that the bisectors of the angles $ATB$ and $BTC$ pass through points $X$ and $Y$ respectively. What angle $B$ can be in triangle $ABC$? | 90^\circ |
10. The smallest positive period of the function $y=\sin x-2 \cos 2 x+4 \sin 4 x$ is $\qquad$ | 2\pi |
For a set of data $5$, $3$, $6$, $8$, $3$, which of the following descriptions is correct?
A: The mean is $4$
B: The median is $6$
C: The mode is $5$
D: The variance is $3.6$ | D |
17. What odd digit can the sum of two prime numbers end with, if it is not a single-digit number? | 1,3,5,9 |
6. A. The minimum value of the algebraic expression $\sqrt{x^{2}+4}+\sqrt{(12-x)^{2}+9}$ is | 13 |
Transform the graph of the function $y=\sin x$ by shrinking all the x-coordinates to half of their original values while keeping the y-coordinates unchanged, and then shift the graph to the right by $\dfrac{\pi}{6}$ units. The equation corresponding to this transformed graph is:
A: $y=\sin (2x- \dfrac{\pi}{3})$
B: $y=\... | A |
For a positive integer $n$, let $1 \times 2 \times \cdots \times n=n!$ If $\frac{2017!}{2^{n}}$ is an integer, then the maximum value of $n$ is $\qquad$ . | 2010 |
4. Show that the solution set of the inequality
$$
\sum_{k=1}^{70} \frac{k}{x-k} \geq \frac{5}{4}
$$
is the union of disjoint half-open intervals with the sum of lengths 1988 . | 1988 |
Let's find all integer roots of the following equation: $x+y+z=x y z$. | 1,2,3 |
6th Chinese 1991 Problem A3 10 points are arranged in the plane, so that given any 5, at least 4 lie on a circle. M is the maximum number of points on a circle. What is the minimum possible value of M? Solution | 9 |
Let's write the following sum in a simpler form:
$$
S=\sum_{i=1}^{n} \frac{n-i}{(2 i+1)(n+i+1)} .
$$ | \frac{1}{2}\sum_{k=1}^{n}\frac{1}{k}-\frac{n}{n+1} |
For a fixed positive integer $n \geq 3$ we are given a $n$ $\times$ $n$ board with all unit squares initially white. We define a [i]floating plus [/i]as a $5$-tuple $(M,L,R,A,B)$ of unit squares such that $L$ is in the same row and left of $M$, $R$ is in the same row and right of $M$, $A$ is in the same column and abov... | 4n-4 |
Given the sets $M=\left\{x \mid x^{2}-3x+2=0 \right\}$ and $N=\left\{-2,-1,1,2\right\}$, then $M\cap N =$ . | \{1,2\} |
9. (8th "Hope Cup" Invitational Competition Question) If $a+b+c=1$, what is the maximum value of $\sqrt{3 a+1}+\sqrt{3 b+1}+\sqrt{3 c+1}$? | 3\sqrt{2} |
3. Proposition 甲: Plane $\alpha \perp$ plane $\beta$, plane $\rho \perp$ plane $\gamma$, then plane $\alpha / /$ plane $\gamma$
Proposition 乙: Three non-collinear points on plane $\alpha$ are equidistant from plane $\beta$, then plane $\alpha / /$ plane $\beta$.
Then ( ).
(A) Proposition 甲 is true, Proposition 乙 is tr... | D |
Given that the mean of $x_1, x_2, \ldots, x_n$ is 10 and the standard deviation is 2, then the mean and standard deviation of $2x_1-1, 2x_2-1, \ldots, 2x_n-1$ are respectively ( )
A: 19 and 2
B: 19 and 3
C: 19 and 4
D: 19 and 8 | \text{C} |
Given the function $f(x)=\ln x+ \frac{ax}{x+1}(a∈R)$
(1) If the function $f(x)$ is monotonically increasing in the interval $(0,4)$, find the range of values for $a$;
(2) If the graph of the function $y=f(x)$ is tangent to the line $y=2x$, find the value of $a$. | a=4 |
4. Vasya and his father are collecting mushrooms in the forest. Father told Vasya: “Go ahead along this straight path. I will quickly check the glade and catch up with you.” The father's step is $20 \%$ longer than Vasya's. They both walk at a constant speed and do not change the length of their steps. There is a perio... | 16 |
1. In the field of real numbers, solve the system of equations
$$
\begin{aligned}
& \sqrt{x^{2}-y}=z-1, \\
& \sqrt{y^{2}-z}=x-1, \\
& \sqrt{z^{2}-x}=y-1 .
\end{aligned}
$$ | (x,y,z)=(1,1,1) |
Given that the derivative of the function $f\left(x\right)$ for $x\in R$ is ${f'}\left(x\right)$, and it satisfies $f\left(3\right)=7$, ${f'}\left(x\right) \lt 2$, find the solution set of $f\left(x\right) \lt 2x+1$. | (3, +\infty) |
6. Let's call the distance between numbers the absolute value of their difference. It is known that the sum of the distances from twenty-nine consecutive natural numbers to some number $a$ is 1624, and the sum of the distances from these same twenty-nine numbers to some number $b$ is 1305. Find all possible values of $... | =52,=108,=7 |
9. A chess piece starts from point $A$ in the figure, and each step is determined by the number that appears when a die is rolled. If the result of the roll is 1 or 2, the chess piece remains stationary; if the result is 3 or 4, the chess piece moves up one square; if the result is 5 or 6, the chess piece moves right o... | \frac{40}{81} |
1. Let's call a natural number $n$ squareable if the numbers from 1 to $n$ can be arranged in such an order that each member of the sequence, when added to its position, results in a perfect square. For example, the number 5 is squareable, as the numbers can be arranged as: 32154, in which $3+1=2+2=1+3=4$ and $5+4=4+5=... | 915 |
The function $f(x)=x^{3}-ax^{2}-bx+a^{2}$ has an extremum of $10$ at $x=1$. Find $a+b=$ __________. | 7 |
Given that when $x \lt 0$, the function of the inverse proportion $y=\frac{k}{x}(x≠0)$ decreases as the independent variable increases, then regarding the quadratic equation in $x$, $x^{2}-2x+1-k=0$, the situation of the roots is:
A: Two distinct real roots
B: No real roots
C: Two equal real roots
D: Depends on the... | A |
From the five numbers $1$, $2$, $3$, $4$, $5$, if any two numbers are selected, the probability that both numbers are even is ( ).
A: $\frac{1}{10}$
B: $\frac{1}{5}$
C: $\frac{2}{5}$
D: $\frac{3}{5}$ | \text{A} |
B4. Determine all integers $a$ for which $\frac{a}{1011-a}$ is an even integer. | \in{1010,1012,1008,1014,674,1348,0,2022} |
3.48. The height of the cone is $H$, the angle between the slant height and the height is $\alpha$. A smaller cone is inscribed in this cone such that the vertex of the smaller cone coincides with the center of the base of the larger cone, and the corresponding slant heights of both cones are perpendicular to each othe... | \frac{1}{3}\piH^{3}\sin^{4}\alpha\cos^{2}\alpha |
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