problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
|---|---|
Given three points O(0, 0), R(-2, 1), Q(2, 1), for any point M(x, y) on curve C, it satisfies $$|\overrightarrow{MR} + \overrightarrow{MQ}| = \overrightarrow{OM} \cdot (\overrightarrow{OR} + \overrightarrow{OQ}) + 2$$.
(Ⅰ) Find the equation of curve C;
(Ⅱ) If A and B are any two points on curve C, located on either sid... | 2 |
In how many ways can a square with a side length of 5 units be placed on an $8 \times 8$ chessboard so that each of its vertices falls on the center of a square? (Solutions that can be transformed into each other by reflection or rotation are not considered different.)
Translating the text as requested, while pres... | 4 |
10. (20 points) Draw any line through the fixed point $M(m, 0)$, intersecting the parabola $y^{2}=16 x$ at points $P$ and $Q$. If $\frac{1}{|P M|^{2}}+\frac{1}{|Q M|^{2}}$ is a constant, find all possible values of the real number $m$.
| 8 |
## Task Condition
Calculate the area of the figure bounded by the graphs of the functions:
$$
x=\sqrt{4-y^{2}}, x=0, y=0, y=1
$$ | \frac{\sqrt{3}}{2}+\frac{\pi}{3} |
If -2, a, b, c, -8 form a geometric sequence, then ( )
A: b=4, ac=16
B: b=-4, ac=16
C: b=4, ac=-16
D: b=-4, ac=-16 | \text{B} |
Given the sequence $\{a\_n\}$ satisfies $a\_1=0$, $a\_{n+1}=a\_n+n$, find $a\_{2013}$ \_\_\_\_\_\_. | 2025078 |
Suppose that $N$ is a three digit number divisible by $7$ such that upon removing its middle digit, the remaining two digit number is also divisible by $7$. What is the minimum possible value of $N$?
[i]2019 CCA Math Bonanza Lightning Round #3.1[/i] | 154 |
A company made significant adjustments to its product structure to meet market demand. After the adjustment, the initial profit increased rapidly, but the growth rate slowed down over time. To establish an appropriate function model to reflect the relationship between the company's profit $y$ and the production volume ... | \text{D} |
Which of the following numbers is an irrational number?
A: $\frac{2}{3}$
B: $1.414$
C: $\sqrt{3}$
D: $\sqrt{9}$ | C |
Example 3 If $a_{i}>0(i=1,2, \cdots, n, n \geqslant 2)$, and $\sum_{i=1}^{n} a_{i}=s(s$ is a positive number $)$.
Prove: $\sum_{i=1}^{n} \frac{a_{i}^{k}}{s-a_{i}} \geqslant \frac{s^{k-1}}{(n-1) n^{k-2}}$ (where $k$ is a constant, and $k$ is a positive integer). | \sum_{i=1}^{n} \frac{a_{i}^{k}}{s-a_{i}} \geqslant \frac{s^{k-1}}{(n-1) n^{k-2}} |
In $\triangle ABC$, $a=3$, $A=30^{\circ}$, $B=60^{\circ}$, find the area $S$ of $\triangle ABC$. | \frac {9 \sqrt {3}}{2} |
The monotonic decreasing interval of the function $f(x) = x - \ln x$ is ( )
A: $(0, 1)$
B: $(0, +\infty)$
C: $(1, +\infty)$
D: $(-\infty, 0) \cup (1, +\infty)$ | \text{A} |
In an opaque box, there are $4$ black balls and some white balls. They are identical except for color. After shaking the box evenly, a ball is randomly drawn and its color is noted. The ball is then placed back into the box, and this process is repeated $40$ times. If $10$ of the draws result in black balls, estimate t... | 12 |
Anna and Bella are celebrating their birthdays together. Five years ago, when Bella turned $6$ years old, she received a newborn kitten as a birthday present. Today the sum of the ages of the two children and the kitten is $30$ years. How many years older than Bella is Anna?
$\textbf{(A) } 1 \qquad \textbf{(B) } 2 \qqu... | 3 |
4. In a certain city, the fare scheme for traveling by metro with a card is as follows: the first trip costs 50 rubles, and each subsequent trip costs either the same as the previous one or one ruble less. Petya spent 345 rubles on several trips, and then on several subsequent trips - another 365 rubles. How many trips... | 15 |
Let $m$ and $n$ be the numbers obtained from rolling a die consecutively. Given that one of the two numbers is $5$, the probability that the equation $x^2 + mx + n = 0$ has real roots is | \dfrac{7}{11} |
Given that the domain of the function $f(x)$ is $R$, for all $x_{1}$, $x_{2} \in R$, $\frac{f(x_{1})-f(x_{2})}{x_{1}-x_{2}}＜0$ ($x_{1}≠x_{2}$), and the function $g(x) = f(x) - 1$. Also, $g(x)$ is an odd function. The solution set of the inequality $f(m^{2}) + f(2m-3) \gt 2$ is:
A: $(-3,1)$
B: $(-1,3)$
C: $(-\infty ,... | A |
11. (20 points) Let the complex number $z$ satisfy $|z|=1$. Find the minimum possible value of
$$
2021|1+z|+|1+z^{2}|+\cdots+|1+z^{4043}|
$$ | 4042 |
Given that the sequence ${a_{n}}$ is an arithmetic sequence with $a_{11} = \frac{\pi}{2}$, and the function $f(x) = \sin(2x) + 2\cos^2\left(\frac{x}{2}\right)$, let $b_{n} = f(a_{n})$. Calculate the sum of the first 21 terms of the sequence ${b_{n}}$. | 21 |
Let the function $f(x) = |x + \frac{1}{a}| + |x - a + 1|$ ($a > 0$ is a constant).
(Ⅰ) Prove that: $f(x) \geq 1$;
(Ⅱ) If $f(3) < \frac{11}{2}$, find the range of values for $a$. | (2, \frac{13 + 3\sqrt{17}}{4}) |
For real numbers $a$, $b$, and $m$, determine the number of correct statements below:
1. If $am^2 > bm^2$, then $a > b$;
2. If $a > b$, then $a|a| > b|b|$;
3. If $b > a > 0$ and $m > 0$, then $\frac{a+m}{b+m} > \frac{a}{b}$;
4. If $a > b > 0$ and $|\ln a| = |\ln b|$, then $2a + b \in (3, +\infty)$.
A: 1
B: 2
C: 3
D: 4 | D |
6. Given that $\alpha, \beta$ are acute angles, and
$$
\begin{array}{l}
(1+\sin \alpha-\cos \alpha)(1+\sin \beta-\cos \beta) \\
=2 \sin \alpha \cdot \sin \beta .
\end{array}
$$
Then $\alpha+\beta=$ $\qquad$ | \frac{\pi}{2} |
## Task Condition
Find the $n$-th order derivative.
$y=\lg (2 x+7)$ | y^{(n)}=(-1)^{n-1}\cdot\frac{2^{n}\cdot(n-1)!}{\ln10}\cdot(2x+7)^{-n} |
Given the functions $f(x)=x^{2}+2$ and $g(x)=|x-a|-|x-1|$, where $a\in \mathbb{R}$,
(1) If $a=4$, find the solution set of the inequality $f(x) > g(x)$;
(2) If for any $x_{1}, x_{2} \in \mathbb{R}$, the inequality $f(x_{1}) \geq g(x_{2})$ always holds, find the range of the real number $a$. | -1 \leq a \leq 3 |
Given the ellipse $$C: \frac {x^{2}}{25}+ \frac {y^{2}}{9}=1$$, the line $l$ intersects the ellipse $C$ at two distinct points $A$ and $B$. $P$ is the midpoint of chord $AB$.
(1) If the slope of line $l$ is $$\frac {4}{5}$$, find the equation of the trajectory of point $P$.
(2) Does there exist a line $l$ such that... | 12x-15y-25=0 |
$S_n$ is the sum of the first $n$ terms of a geometric sequence $\{a_n\}$, satisfying $S_n = 2a_n - 1$. Find the common ratio $q$ of the sequence $\{a_n\}$. | 2 |
Example 18 (2003 National High School Competition Question) Given that $x$ and $y$ are both in the interval $(-2,2)$, and $xy=-1$, then the minimum value of the function $u=\frac{4}{4-x^{2}}+$ $\frac{9}{9-y^{2}}$ is ( ).
A. $\frac{8}{5}$
B. $\frac{24}{11}$
C. $\frac{12}{7}$
D. $\frac{12}{5}$ | \frac{12}{5} |
In a box, there are six balls marked with the numbers 1 through 6, which are of the same size. If two balls are drawn from the box one after another, their numbers are noted and then the balls are put back. A prize is won if the product of the two numbers is a multiple of 4. With four people participating in the drawin... | \frac{96}{625} |
Given $x+y=5$ and $xy=6$, find $x^{2}+y^{2}=\_\_\_\_\_\_$. | 13 |
[ Arithmetic progression ]
[ Arithmetic. Mental calculation, etc. ]
What is the sum of the digits of all numbers from one to a billion? | 40500000001 |
Distribute 6 identical computers to 5 primary schools, with each school receiving at least one computer. How many different distribution schemes are there? | 5 |
## Problem Statement
Find the coordinates of point $A$, which is equidistant from points $B$ and $C$.
$A(0 ; 0 ; z)$
$B(-18 ; 1 ; 0)$
$C(15 ;-10 ; 2)$ | A(0;0;1) |
19. Find the last two digits of the sum
$$
1^{2}+2^{2}+\ldots+50^{2}-51^{2}-\ldots-100^{2}+101^{2}+\ldots 150^{2}-151^{2}-\ldots 200^{2}+\ldots-2000^{2}+2001^{2}+\ldots+2017^{2}
$$
(i.e., 50 numbers with a plus sign, 50 with a minus sign, and so on.) | 85 |
14. Asymmetric coin. (From 9th grade, 2 points) Billy Bones has two coins - a gold one and a silver one. One of them is symmetric, and the other is not. It is unknown which coin is asymmetric, but it is known that the asymmetric coin lands heads with a probability of \( p = 0.6 \).
Billy Bones tossed the gold coin, an... | \frac{5}{9} |
5. If the positive integer $a$ makes the maximum value of the function $y=f(x)=x+$ $\sqrt{13-2 a x}$ also a positive integer, then this maximum value equals ( . ).
(A) 3
(B) 4
(C) 7
(D) 8 | C |
5. From the 12 edges and the 12 face diagonals of a cube, select $n$ lines such that any two of these lines are skew lines. Then the maximum value of $n$ is ( ).
(A) 3
(B) 4
(C) 5
(D) 6 | B |
Find the value of the $y$-coordinate where the tangent to the curve $y = x^2 + 11$ at the point $P(1, 12)$ intersects the $y$-axis. | 10 |
8、There are three athletes practicing running around a circular track that is 1000 meters long. Athlete A runs one lap every 4 minutes, Athlete B runs one lap every 5 minutes, and Athlete C runs one lap every 6 minutes. Now, the three of them start running clockwise from the same point at the same time. $\qquad$ minute... | 60 |
"x < 2" is a condition for "x^2 - 2x < 0" to be ( )
A: Sufficient but not necessary
B: Necessary but not sufficient
C: Both necessary and sufficient
D: Neither sufficient nor necessary | \text{B} |
16. The solutions to the equation $x^{3}-4\lfloor x\rfloor=5$, where $x$ is a real number, are denoted by $x_{1} x_{2} x_{3} \ldots x_{k}$ for some positive integer $k$. Find $\sum_{i=1}^{k} x_{i}^{3}$. | 10 |
Task B-3.5. A natural number is written on each side of the cube. Each vertex of the cube is marked with a number that is equal to the product of the numbers written on the sides of the cube that meet at that vertex. The sum of the numbers by which the vertices of the cube are thus marked is 3333. Determine the sum of ... | 115 |
38.16. For the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ find the focal point corresponding to the point $\left(a \cos t_{0}, b \sin t_{0}\right)$; find the radius of curvature of the ellipse at this point. | R^{2}=\frac{(^{2}\sin^{2}t_{0}+b^{2}\cos^{2}t_{0})^{3}}{^{2}b^{2}} |
3. What is the largest integer $k$ such that $k+1$ divides
$$
k^{2020}+2 k^{2019}+3 k^{2018}+\cdots+2020 k+2021 ?
$$ | 1010 |
For thousands of years, the working people of our country have summarized rich experiences of "observing clouds to predict the weather" in production practice based on the changes in the shape, direction, speed, thickness, color, etc. of clouds. They have compiled these experiences into proverbs, such as "Clouds hookin... | \text{D} |
In an opaque black bag, there are several white and red balls, totaling $10$ balls. These balls are identical except for their colors. After mixing the balls, one ball is randomly drawn each time, the color is recorded, the ball is put back, the balls are mixed again, and another ball is drawn. After conducting a large... | A |
Given the ellipse $\Gamma$: $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 (a > b > 0)$ with its right focus $F(3, 0)$, and upper and lower vertices $A$ and $B$, respectively. The line $AF$ intersects the ellipse $\Gamma$ at another point $M$. If the line $BM$ intersects the $x$-axis at the point $N(12, 0)$, find the e... | \frac{1}{2} |
Define a new operation $*$, with the rule $a*b=\frac{1}{a}+\frac{1}{b}$, for example: $2*3=\frac{1}{2}+\frac{1}{3}=\frac{5}{6}$. Then the value of $3*\left(-4\right)$ is ______. | \frac{1}{12} |
From a bag containing 3 red balls and 2 white balls, if you draw 3 balls at random, and event A is defined as "at least one of the three drawn balls is white," then what is the complementary event to A?
A: 1 white ball and 2 red balls
B: 2 white balls and 1 red ball
C: 3 red balls
D: At least one red ball | C |
Problem 4. In a singing competition, a Rooster, a Crow, and a Cuckoo participated. Each member of the jury voted for one of the three performers. The Woodpecker calculated that there were 59 judges in total, and that the Rooster and the Crow received a total of 15 votes, the Crow and the Cuckoo received 18 votes, and t... | 13 |
In triangle $ABC$ inscribed in a circle with center $P$ and radius $1$, and $3\overrightarrow{PA} + 4\overrightarrow{PB} + 5\overrightarrow{PC} = \overrightarrow{0}$, the length of side $AB$ of $\triangle ABC$ is ______. | \sqrt{2} |
Find all pairs of integers $x, y$ such that $2 x^{2}-6 x y+3 y^{2}=-1$. | \\frac{(2+\sqrt{3})^{n}+(2-\sqrt{3})^{n}}{2},\quadx\\frac{(2+\sqrt{3})^{n}-(2-\sqrt{3})^{n}}{2\sqrt{3}} |
4. The sum of 74 (not necessarily distinct) real numbers from the interval $\langle 4,10\rangle$ is 356. Determine the greatest possible value of the sum of their squares.
(Zdeněk Pezlar) | 2024 |
Example 1.8.1. Suppose that $a, b, c$ are three positive real numbers satisfying
$$(a+b+c)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)=13 .$$
Find the minimum value of
$$P=\left(a^{2}+b^{2}+c^{2}\right)\left(\frac{1}{a^{2}}+\frac{1}{b^{2}}+\frac{1}{c^{2}}\right) .$$ | 33 |
2. 10 people go to the bookstore to buy books, it is known that: (1) each person bought three types of books; (2) any two people have at least one book in common.
How many people at most could have bought the book that was purchased by the fewest people? | 5 |
5. In a certain table tennis singles competition, it was originally planned that each pair of players would play exactly one match, but after 3 players each played 2 matches, they withdrew. As a result, only 50 matches were played in total. Then, the number of matches played between these 3 players is
A. 0
B. 1
C. 2
D.... | 1 |
$8 \cdot 135$ Let $a_{0}>0$ and
$$
a_{n+1}=\frac{a_{n}^{2}-1}{n+1}, n \geqslant 0 .
$$
Prove that there exists a real number $a>0$, such that
(i) if $a_{0} \geqslant a$, then as $n \rightarrow+\infty$, $a_{n} \rightarrow+\infty$.
(ii) if $a_{0}<a$, then as $n \rightarrow+\infty$, $a_{n} \rightarrow 0$. | 2 |
Let the isosceles right triangle $ABC$ with $\angle A= 90^o$ . The points $E$ and $F$ are taken on the ray $AC$ so that $\angle ABE = 15^o$ and $CE = CF$. Determine the measure of the angle $CBF$. | 15^\circ |
Let the sum of the first $n$ terms of the sequence $\{a_{n}\}$ be $S_{n}$. If $\frac{1}{2}≤\frac{{a}_{n+1}}{{a}_{n}}≤2(n∈{N}^{*})$, then the sequence $\{a_{n}\}$ is called a "compact sequence".$(1)$ If ${a}_{n}=\frac{{n}^{2}+2n}{{4}^{n}}$, determine whether $\{a_{n}\}$ is a "compact sequence", and explain the reason;$(... | [\frac{1}{2}, 1] |
Find the equation of the circle passing through three points P(2, 2), M(5, 3), and N(3, -1). | x^2 + y^2 - 8x - 2y + 12 = 0 |
The maximum value of the quadratic function $y=-2x^{2}+9$ is ______. | 9 |
Define a function $f$ on $\mathbb{R}$ that satisfies $f(x+1)=2f(x)$. If for $0 \leqslant x \leqslant 1$, $f(x)=x(1-x)$, then find $f(x)$ for $-1 \leqslant x \leqslant 0$. | - \dfrac{1}{2}x(x+1) |
In [convex](https://artofproblemsolving.com/wiki/index.php/Convex_polygon) [hexagon](https://artofproblemsolving.com/wiki/index.php/Hexagon) $ABCDEF$, all six sides are congruent, $\angle A$ and $\angle D$ are [right angles](https://artofproblemsolving.com/wiki/index.php/Right_angle), and $\angle B, \angle C, \angle E,... | 46 |
For how many integers $x$ will the following polynomial be a perfect square?
$$
4 x^{4}-12 x^{3}+17 x^{2}-6 x-14
$$ | 2 |
## Task 4 - 221224
Let $n \neq 0$ be a natural number. On a circle, there are $2 n$ pairwise distinct points $P_{1}, P_{2}, \ldots, P_{2 n}$ given.
We are looking for the number $A_{n}$ of all different ways to draw a set of $n$ chords such that the following conditions are met:
Each chord connects one of the points... | 42 |
Example 36 (2005 Slovenia National Team Selection Exam) Find all pairs of positive integers $(m, n)$ such that $m^{2}-4 n$ and $n^{2}-4 m$ are both perfect squares. | (4,4),(5,6),(6,5) |
Given the specific proposition _P_: $x \in \mathbb{R}, 2x + 1 \leq 0$, the negation of proposition _P_ is ( )
A: $x \in \mathbb{R}, 2x + 1 > 0$
B: "$x \in \mathbb{R}, 2x + 1 > 0$
C: $x \in \mathbb{R}, 2x + 1 \geq 0$
D: "$x \in \mathbb{R}, 2x + 1 \geq 0$ | \text{B} |
7. An Unusual Client. A certain man brought 1000 dollars in one-dollar bills and 10 empty bags to the bank and, addressing the clerk, said:
- I would appreciate it if you could distribute this money into the bags in such a way that any amount of money I need can always be given out in one or several bags without openi... | 1,2,4,8,16,32,64,128,256,489 |
8. Given a positive integer $n(n \geqslant 2)$. Find the minimum value of $|X|$, such that for any $n$ two-element subsets $B_{1}$, $B_{2}, \cdots, B_{n}$ of the set $X$, there exists a subset $Y$ of $X$ satisfying:
(1) $|Y|=n$;
(2) for $i=1,2, \cdots, n$, we have $\left|Y \cap B_{i}\right| \leqslant 1$.
Here, $|A|$ de... | 2n-1 |
4. Given complex numbers $z_{1}$ and $z_{2}$ satisfy $\arg \frac{z_{1}}{z_{2}}=\frac{5 \pi}{6}$, and $\left|z_{1}\right|=\left|z_{1}+z_{2}\right|$. Then the minimum value of $\frac{\left|2 z_{1}+t z_{2}\right|}{\left|z_{2}\right|}$ is $\qquad$ | \frac{\sqrt{3}}{3} |
16. Through the right focus $F$ of the ellipse $\frac{x^{2}}{3}+\frac{y^{2}}{2}=1$, draw two perpendicular chords $A B$ and $C D$. Let the midpoints of $A B$ and $C D$ be $M$ and $N$ respectively.
(1) Prove that the line $M N$ must pass through a fixed point, and find this fixed point;
(2) If the slopes of chords $A B$... | \frac{4}{25} |
Teacher Kadrnozkova was buying tickets at the cash register of the zoo for her students and for herself. The ticket for an adult was more expensive than for a schoolchild, but not more than twice as much. Teacher Kadrnozkova paid a total of 994 CZK. Teacher Hnizdo had three more students with him than his colleague, so... | 25 |
Zaslavsky A.A.
In triangle $A B C$, the bisectors $B B_{1}$ and $C C_{1}$ are drawn. It is known that the center of the circumcircle of triangle $B B_{1} C_{1}$ lies on the line $A C$. Find the angle $C$ of the triangle. | 120 |
8.4. Sasha is a coffee lover who drinks coffee at work, so every morning before work, he prepares a thermos of his favorite drink according to a strict ritual. He brews 300 ml of the strongest aromatic coffee in a Turkish coffee pot. Then Sasha repeats the following actions 6 times: he pours 200 ml from the pot into th... | 0.2 |
Given $f(x)=(x^{2}+2x-1)e^{2-x}$, find $f′(x)$\_$\_\_\_\_\_\_\_\_\_\_\_$. | f'(x) = (3-x^{2})e^{2-x} |
Example 3 Choose 1 young man and 1 young woman from each of the four groups to perform a circle dance, with the rule that men and women must alternate, and two people from the same group cannot be adjacent. How many different team formation variations are there? | 12 |
2. Find all positive integers $n$ such that $\phi(n)$ has the value
a) 1
d) 6
b) 2
e) 14
c) 3
f) 24 . | a) 1,2 \quad b) 3,4,6 \quad c) \text{no solution} \quad d) 7,9,14,18 \quad e) \text{no solution} \quad f) 35,39,45,52,56,70,72,78,84,90 |
2. Find the pattern and fill in the number:
$$
100,81,64,49 \text {, }
$$
$\qquad$ $, 25,16,9,4,1$ . | 36 |
The coordinates of the foci of the ellipse \\(mx^{2}+ny^{2}+mn=0(m < n < 0)\\) are \\((\\) \\()\\)
A: \\((0,± \sqrt {m-n})\\)
B: \\((± \sqrt {m-n},0)\\)
C: \\((0,± \sqrt {n-m})\\)
D: \\((± \sqrt {n-m},0)\\) | C |
7. The sum of all positive integers $n$ that satisfy $\frac{1}{4}<\sin \frac{\pi}{n}<\frac{1}{3}$ is $\qquad$ . | 33 |
10. An arbitrary circle and the curve $y=\sin x$ can intersect at
(A) at most 2 points;
(B) at most 4 points;
(C) at most 6 points;
(D) at most 8 points;
(E) more than 16 points. | E |
5. Let $x=0.82^{0.5}, y=\sin 1, z=\log 3 \sqrt{7},$ the size relationship of $x, y, z$ is $(\quad)$.
A. $x<y<z$
B. $y<z<x$
C. $z<x<y$
D. $z<y<x$ | B |
Two lines $l_1: ax + (1+a)y = 3$ and $l_2: (a+1)x + (3-2a)y = 2$ are perpendicular to each other. The value of $a$ is （ ）
A: 3
B: -1
C: -1 or 3
D: 0 or 3 | \text{C} |
In a triangle, two of the interior angles are 50° and 80°, respectively. The degree measure of the third angle is ( )
A: 80°
B: 50°
C: 65°
D: Cannot be determined | B |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. If $b-\frac{1}{2}c=a\cos C$.
$(1)$ Find angle $A$;
$(2)$ If $4(b+c)=3bc$ and $a=2\sqrt{3}$, find the area $S$ of $\triangle ABC$. | 2\sqrt{3} |
10.1. Solve the equation: $\sqrt{x+\sqrt{2 x-1}}+\sqrt{x-\sqrt{2 x-1}}=\sqrt{2}$. | \frac{1}{2}\leqx\leq1 |
Given $\left(m-4\right)^{2}+|n+3|=0$, find the value of $n^{m}$. | 81 |
The interval containing the zero of the function $$f(x)=x^{\frac{1}{2}}-2+\log_{2}x$$ is ( )
A: (0, 1)
B: (1, 2)
C: (2, 3)
D: (3, +∞) | \text{B} |
The negation of the proposition "$\forall x\in \left[0,+\infty \right)$, $x^{3}+x\geqslant 0$" is ( ).
A: $\forall x\in \left(-\infty ,0\right)$, $x^{3}+x \lt 0$
B: $\forall x\in \left(-\infty ,0\right)$, $x^{3}+x\geqslant 0$
C: $\exists x\in \left[0,+\infty \right)$, $x^{3}+x \lt 0$
D: $\exists x\in \left[0,+\inft... | C |
The negation of the proposition "∀x∈R, |x|+x^4≥0" is ( )
A: ∀x∈R, |x|+x^4<0
B: ∀x∈R, |x|+x^4<0
C: ∃x₀∈R, |x₀|+x₀^4≥0
D: ∃x₀∈R, |x₀|+x₀^4<0 | D |
Given sets $A=\{x|y=\ln (-x^{2}-2x+8)\}$ and $B=\{x|\log _{2}x \lt 1\}$, then $A\cap B$ is equal to:
A: $(-4,2)$
B: $[-4,2)$
C: $(0,2)$
D: $(0,2]$ | C |
For the polynomial $x-y-z-m-n$, after adding parentheses arbitrarily and simplifying the resulting expression, which only contains subtraction operations, it is called an "addition operation." For example: $\left(x-y\right)-\left(z-m-n\right)=x-y-z+m+n$, $x-y-\left(z-m\right)-n=x-y-z+m-n$, $\ldots$. Given the following... | D |
2. If
$$
\begin{array}{l}
a+b-2 \sqrt{a-1}-4 \sqrt{b-2} \\
=3 \sqrt{c-3}-\frac{1}{2} c-5,
\end{array}
$$
then, the value of $a+b+c$ is $\qquad$ | 20 |
Approximate the number $566.1234$ using rounding to the nearest whole number. | 566 |
4. Find the largest value of $3 k$ for which the following equation has a real root:
$$
\sqrt{x^{2}-k}+2 \sqrt{x^{3}-1}=x \text {. }
$$ | 4 |
4. It is known that $3 \operatorname{tg}^{2} \gamma+3 \operatorname{ctg}^{2} \gamma+\frac{2}{\sin ^{2} \gamma}+\frac{2}{\cos ^{2} \gamma}=19$. Find the value of the expression $\sin ^{4} \gamma-\sin ^{2} \gamma$.
Answer. $-\frac{1}{5}$. | -\frac{1}{5} |
The number of intersection points between the graph of the function $f(x) = 2 \ln x$ and the graph of the function $g(x) = x^2 - 4x + 5$ is ______. | 2 |
The line $l$ passes through the intersection point $P$ of $l1: x + y - 2 = 0$ and $l2: x - y - 4 = 0$, and also passes through the midpoint $Q$ of the segment $AB$, where $A(-1,3)$, $B(5,1)$. The equation of the line $l$ is ( ▲ )
(A) $3x - y - 8 = 0$ (B) $3x + y + 8 = 0$
(C) $3x + y - 8 = 0$ (D) $3x - y + 8 =... | \text{C} |
5. (10 points) The integer part of the expression $\frac{50}{11}+\frac{55}{12}+\frac{60}{13}+\frac{65}{14}+\frac{70}{15}+\frac{75}{16}+\frac{80}{17}+\frac{85}{18}+\frac{90}{19}+\frac{95}{20}$ is | 46 |
3. The AC-2016 calculator can perform two operations: taking the cube root and calculating the tangent. Initially, the number $2^{-243}$ was entered into the calculator. What is the minimum number of operations required to obtain a number greater than 1? | 7 |
The parabola with its vertex at the origin, its axis of symmetry along the y-axis, and its focus on the line $3x-4y-24=0$ has the standard equation of __ . | x^2=-24y |
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