problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
|---|---|
A ball floats in water in a prism-shaped glass. If we remove the ball, by how much will the water level in the glass sink, if the glass has a regular hexagonal base, each side of which is $5 \mathrm{~cm}$, the floating ball has a radius of $4 \mathrm{~cm}$, and the part of the ball protruding above the water is $1 \mat... | 3.95\mathrm{~} |
Given that $i$ is the imaginary unit, the point corresponding to the complex number $z= \frac {1}{1-i}$ in the complex plane is located in $\text{( } \text{ )}$.
A: the first quadrant
B: the second quadrant
C: the third quadrant
D: the fourth quadrant | A |
## Task 3
Write down
a) the successor of 19
b) the predecessor of 1
c) all numbers that lie between 8 and 12! | 20,0,9,10,11 |
Which of the following propositions is a true statement?
A: Vertical angles are equal.
B: Alternate interior angles are equal.
C: Corresponding angles are complementary.
D: Alternate exterior angles are complementary. | A |
11.2 What is the maximum value that the ratio of the radius of the inscribed circle to the radius of the circumscribed circle of a right triangle can reach? | \sqrt{2}-1 |
9. Use four 4s and some addition, subtraction, multiplication, division signs, and parentheses to write four expressions that equal $3, 4, 5,$ and 6 respectively. | (4+4+4)\div4=3,\quad4+(4-4)\div4=4,\quad(4\times4+4)\div4=5,\quad4+(4+4)\div4=6 |
Task 4. (20 points) To get from the first to the second building of the university, Sasha took a car-sharing vehicle, while Zhenya rented a scooter. Sasha and Zhenya left the first building for the second at the same time, and at the same time, Professor Vladimir Sergeyevich left the second building for the first in a ... | 15,40,60 |
In an isosceles triangle $ABC (AB = BC)$, the altitude $CD$ is drawn. The angle $BAC$ is equal to $\alpha$. The radius of the circle passing through points $A, C$, and $D$ is $R$. Find the area of triangle $ABC$.
# | R^2\operatorname{tg}\alpha |
Exercise 6. Given a point $P$ and a circle $\mathcal{C}$ in the plane, the distance from $P$ to $\mathcal{C}$ is defined as the minimum length $PM$ between $P$ and a point $M$ on the circle $\mathcal{C}$. For example, if $P$ lies on the circle, then the distance from $P$ to $\mathcal{C}$ is zero, and if $P$ is the cent... | 7 |
24. Given that
$$
S=\left|\sqrt{x^{2}+4 x+5}-\sqrt{x^{2}+2 x+5}\right|,
$$
for real values of $x$, find the maximum value of $S^{4}$. | 4 |
Problem 7.2. (15 points) Given an isosceles triangle $A B C(A B=B C)$. On the sides $A B, B C, A C$ points $K, L, M$ are marked respectively such that $\angle A K M=90^{\circ}, \angle B L K=90^{\circ}$ and $K M=K L$. What is the measure of angle $C M L ?$ | 90 |
Among the following functions, the one whose domain and range are the same as the function $y=10^{\lg x}$ is ( )
A: $y=x$
B: $y=\lg x$
C: $y=2^{x}$
D: $y= \dfrac {1}{ \sqrt {x}}$ | D |
4. Let the circle $O: x^{2}+y^{2}=5$ intersect the parabola $C: y^{2}=2 p x(p>0)$ at point $A\left(x_{0}, 2\right)$, and let $AB$ be the diameter of circle $O$. A line passing through $B$ intersects $C$ at two distinct points $D, E$. Then the product of the slopes of lines $AD$ and $AE$ is $\qquad$. | 2 |
Given the function $f(x) = \sin x$, find the equation of the tangent line to $f(x)$ at point A $(0, 0)$. | y = x |
1. Variant 1.
Find the smallest natural number whose sum of digits is 47. | 299999 |
8,9 A heptagon, three angles of which are equal to $120^{\circ}$, is inscribed in a circle. Can all its sides be of different lengths
# | No |
10. (10 points) $n$ pirates divide gold coins. The 1st pirate takes 1 coin first, then takes $1 \%$ of the remaining coins; then, the 2nd pirate takes 2 coins, then takes $1 \%$ of the remaining coins; the 3rd pirate takes 3 coins, then takes $1 \%$ of the remaining coins; ... the $n$-th pirate takes $n$ coins, then ta... | 9801 |
2. If the graph of the quadratic function $y=(x-3)^{2}-2$ intersects the $x$-axis at two points $\left(x_{1}, 0\right),\left(x_{2}, 0\right)$, then $\frac{\sqrt{x_{1} x_{2}}+x_{1}}{x_{2}+\sqrt{x_{1} x_{2}}}+\frac{\sqrt{x_{1} x_{2}}-x_{2}}{x_{1}-\sqrt{x_{1} x_{2}}}=(\quad)$.
(A) $\frac{12 \sqrt{13}}{13}$
(B) $\frac{10 \... | C |
Task 1. Find a four-digit number with the sum of its digits being 8, where the first digit from the left is obtained by multiplying the second digit by 3, and the fourth digit is obtained by multiplying the third digit by 4. | 6200 |
$1 \cdot 21$ Team A has $2 m$ people, Team B has $3 m$ people. Now, $14-m$ people are drawn from Team A, and $5 m-11$ people are drawn from Team B to participate in the game. How many people are there in Team A and Team B respectively? How many different ways are there to select the participants for the game? | 150 |
For all positive integers $m>10^{2022}$, determine the maximum number of real solutions $x>0$ of the equation $mx=\lfloor x^{11/10}\rfloor$. | 10 |
Given a quadratic function $y=ax^{2}+bx+c\left(a \lt 0\right)$, which intersects the positive half-axis of the $x$-axis at points $A$ and $B$, and the line $y=kx+b$ intersects the positive half-axis of the $y$-axis at point $D$ and intersects the $x$-axis at point $C$ (where $C$ is to the right of $A$ and not coincidin... | C |
Given that the two foci of an ellipse are $F_1(-\sqrt{3}, 0)$ and $F_2(\sqrt{3}, 0)$, and it passes through the point $(1, \frac{\sqrt{3}}{2})$.
(1) Find the equation of the ellipse;
(2) Suppose a line $l$ passing through the point $A(0, 2)$ intersects the ellipse at two distinct points $M$ and $N$ (where $M$ lies ... | \left(\frac{1}{3}, 1\right) |
If $a=2^{0.5}$, $b=\log_{\pi}3$, $c=\log_{2}0.5$, then ( )
A: $a>b>c$
B: $b>a>c$
C: $c>a>b$
D: $b>c>a$ | A |
6. Let $a, b, c, d \in\{-1,0,1\}$. If the ordered array $(a, b, c, d)$ satisfies that $a+b$, $c+d$, $a+c$, and $b+d$ are all distinct, then $(a, b, c, d)$ is called a "good array". The probability that an ordered array $(a, b, c, d)$ is a good array among all possible ordered arrays $(a, b, c, d)$ is $\qquad$ | \frac{16}{81} |
Given that point $P$ is any point on the curve $(x-1)^2+(y-2)^2=9$ with $y \geq 2$, find the minimum value of $x+ \sqrt {3}y$. | 2\sqrt{3} - 2 |
Given the equation about $x$: $mx^{2}-(m+2)x+2=0$.
$(Ⅰ)$ Prove: Regardless of the value of $m$, the equation always has real roots.
$(Ⅱ)$ For what integer values of $m$ does the equation have two distinct positive integer roots. | m = 1 |
Given that $f(x)$ is an even function defined on $[-5,5]$, and $f(3) > f(1)$, determine the correct option:
A: $f(0) f(2)$
D: $f(2) > f(0)$ | \text{B} |
The positional relationship between Circle $O_1$: $(x+1)^2+(y-1)^2=4$ and Circle $O_2$: $(x-2)^2+(y-4)^2=9$ is ( )
A: Tangent internally
B: Tangent externally
C: Intersecting
D: Disjoint | \text{C} |
## Problem Statement
Based on the definition of the derivative, find $f^{\prime}(0)$:
$f(x)=\left\{\begin{array}{c}\operatorname{arctg}\left(x^{3}-x^{\frac{3}{2}} \sin \frac{1}{3 x}\right), x \neq 0 ; \\ 0, x=0\end{array}\right.$ | 0 |
The age difference between two brothers is three years. One year ago, the father's age of these brothers was double the sum of the brothers' ages, and in twenty years, the father's age will be the sum of the ages of these two sons. What is the age of each of the brothers? | x=13,y=10 |
57. (8th grade) Solve the system of equations $\left\{\begin{array}{l}\frac{x^{2}}{y}+\frac{y^{2}}{x}=12 \\ \frac{1}{x}+\frac{1}{y}=\frac{1}{3}\end{array}\right.$. | x_{1}=y_{1}=6,\quadx_{2}=\frac{-3\3\sqrt{5}}{2},y_{2}=\frac{-3\3\sqrt{5}}{2} |
Given the function $f(x)= \sqrt {2}\cos (x- \frac {\pi }{12}),x\in \mathbb{R}$.
$(1)$ Find the value of $f( \frac {\pi }{3})$;
$(2)$ If $\cos \theta = \frac {3}{5},\theta \in ( \frac {3\pi }{2},2\pi )$, find $f(\theta - \frac {\pi }{6})$. | - \frac {1}{5} |
Let $a, b, c, d$ be a permutation of the numbers $1, 9, 8,4$ and let $n = (10a + b)^{10c+d}$. Find the probability that $1984!$ is divisible by $n.$ | \frac{5}{6} |
In the Cartesian coordinate system, the distance from the point $\left(3,-2\right)$ to the origin is ______. | \sqrt{13} |
1. Given $\alpha, \beta \in\left(0, \frac{\pi}{2}\right)$, and
$$
\sin \beta=2 \cos (\alpha+\beta) \cdot \sin \alpha\left(\alpha+\beta \neq \frac{\pi}{2}\right) \text {. }
$$
then the maximum value of $\tan \beta$ is | \frac{\sqrt{3}}{3} |
Task 1. "This is for you to see a bit too early,"- said Baba Yaga to her 33 students and commanded: "Close your eyes!" The right eye was closed by all the boys and a third of the girls. The left eye was closed by all the girls and a third of the boys. How many students still saw what they were not supposed to see yet? ... | 22 |
## 236. How old is Willie?
- Is this problem your teacher gave you? - I asked Willie. - It looks quite boring.
- No, - Willie replied, - I came up with it myself. - See this polynomial? My age is its root. I mean, if you substitute the number of years I was when I last celebrated my birthday for \( x \), the polynomia... | 14 |
3. (15 points) The vertices of the broken line $A B C D E F G$ have coordinates $A(-1 ; -7), B(2 ; 5), C(3 ; -8), D(-3 ; 4), E(5 ; -1), F(-4 ; -2), G(6 ; 4)$.
Find the sum of the angles with vertices at points $B, E, C, F, D$. | 135 |
Given positive numbers $a$ and $b$ satisfying $a^{2}+b^{2}=1$, then ()
A: The maximum value of $a+b$ is $\sqrt{2}$
B: The maximum value of $ab$ is $\frac{1}{2}$
C: The maximum value of $a^{2}+b$ is $1$
D: The minimum value of $\frac{1}{4{a}^{2}}+\frac{1}{{b}^{2}}$ is $2$ | AB |
1. A cake was divided into three parts. If two-thirds are cut from the first part and added to the second, then the third part will be twice as large as the first, but half the size of the second. What fraction of the cake do the cut pieces represent? | \frac{3}{7},\frac{2}{7},\frac{2}{7} |
2. Klárka made a mistake when multiplying two two-digit numbers, and thus her result was 400 less than the correct result. To check, she divided the number she got by the smaller of the multiplied numbers. This time she calculated correctly and got an incomplete quotient of 67 and a remainder of 56. Which numbers did K... | 5775 |
Given an ellipse with the equation $$\frac {x^{2}}{a^{2}}+ \frac {y^{2}}{b^{2}}=1$$ (a > b > 0) and an eccentricity of $$\frac { \sqrt {6}}{3}$$, the left focus of the ellipse is F, and the maximum distance from any point on the ellipse to F is $$\sqrt {6}+2$$. A line l with a non-zero slope intersects the x-axis at po... | \frac { \sqrt {3}}{2} |
In the acute-angled triangle $ABC$ the angle$ \angle B = 30^o$, point $H$ is the intersection point of its altitudes. Denote by $O_1, O_2$ the centers of circles inscribed in triangles $ABH ,CBH$ respectively. Find the degree of the angle between the lines $AO_2$ and $CO_1$. | 45^\circ |
Let $S_n$ be the sum of the first $n$ terms of the sequence $\{a_n\}$. If $2a_n + (-1)^n \cdot a_n = 2^n + (-1)^n \cdot 2^n$ ($n \in \mathbb{N}^*$), then $S_{10}=$ \_\_\_\_\_\_. | \dfrac{2728}{3} |
Find the smallest possible value of the expression
$$
\left\lfloor\frac{a+b+c}{d}\right\rfloor+\left\lfloor\frac{b+c+d}{a}\right\rfloor+\left\lfloor\frac{c+d+a}{b}\right\rfloor+\left\lfloor\frac{d+a+b}{c}\right\rfloor,
$$
where \(a, b, c\) and \(d\) vary over the set of positive integers.
(Here \(\lfloor x\rfloor\) ... | 9 |
The real numbers $c,b,a$ form an arithmetic sequence with $a \geq b \geq c \geq 0$. The quadratic $ax^2+bx+c$ has exactly one root. What is this root?
$\textbf{(A)}\ -7-4\sqrt{3}\qquad\textbf{(B)}\ -2-\sqrt{3}\qquad\textbf{(C)}\ -1\qquad\textbf{(D)}\ -2+\sqrt{3}\qquad\textbf{(E)}\ -7+4\sqrt{3}$ | -2+\sqrt{3} |
## Task $7 / 65$
We are looking for three square numbers $a, b$, and $c$ such that $a-b=b-c=24$. | =49,b=25,=1 |
7.078. $8^{\frac{2}{x}}-2^{\frac{3 x+3}{x}}+12=0$.
7.078. $8^{\frac{2}{x}}-2^{\frac{3 x+3}{x}}+12=0$. | 3;\log_{6}8 |
10. The graph on the right shows the fuel consumption per 100 kilometers for four different types of cars: $U, V, W, X$. If each car has 50 liters of fuel, what is the maximum total distance that these four cars can travel in kilometers? | 1950 |
Given the sets $A=\{x|2\leqslant 2^{x}\leqslant 8\}$, $B=\{x|x > 2\}$, with the universal set $U=\mathbb{R}$.
$(1)$ Find $(\complement_{U}B)\cup A$;
$(2)$ Given the set $C=\{x|1 < x < a\}$, if $C\subseteq A$, find the range of the real number $a$. | (-\infty,3] |
1. Determine the set $A=\left\{(x, y) \in \mathbb{Z} \times \mathbb{Z} \mid x+y=1, x^{2}+2 x+y^{2}=9\right\}$. | {(2,-1),(-2,3)} |
Solve the following inequalities:
$$
\frac{x^{2}-1}{x^{2}+1}<\frac{x^{3}-1}{x^{3}+1}
$$ | x>1\quador\quadx<0 |
Simplify: $$\frac {a+1}{a^{2}-1} \div \frac {a^{2}-4}{a^{2}+a-2} - \frac {1-a}{a-2} = \_ \_ \_ \_ \_ \_.$$ | \frac {a}{a-2} |
In an exam every question is solved by exactly four students, every pair of questions is solved by exactly one student, and none of the students solved all of the questions. Find the maximum possible number of questions in this exam. | 13 |
Given the function $f(x) = ax^3 - bx + 1$ ($a, b \in \mathbb{R}$), if $f(-2) = 1$, then $f(2) = \boxed{1}$. | 1 |
Five, (12 points) As shown in the figure, in isosceles $\triangle ABC$, $AB = AC$, the vertex angle $A = 20^{\circ}$, and a point $D$ is taken on side $AB$ such that $AD = BC$. Find the measure of $\angle BDC$. | 30^{\circ} |
6.077. $\left\{\begin{array}{l}x^{4}+y^{4}=82, \\ x y=3\end{array}\right.$ | (1,3),(3,1),(-1,-3),(-3,-1) |
Given $$\overrightarrow {a} = (2 \sqrt {3}\sin x, \sin x + \cos x)$$ and $$\overrightarrow {b} = (\cos x, \sin x - \cos x)$$, and the function $f(x) = \overrightarrow {a} \cdot \overrightarrow {b}$.
(I) Find the interval of monotonic decrease for the function $f(x)$.
(II) In triangle ABC, where the sides opposite a... | m \leq -1 |
12. Given $\sin \alpha-\cos \alpha=\frac{1}{2}$. Then the value of $\sin ^{3} \alpha-$ $\cos ^{3} \alpha$ is $\qquad$. | \frac{11}{16} |
If $i$ is the imaginary unit, then the imaginary part of the complex number $i(1+i)$ is
A: 1
B: -1
C: $i$
D: -$i$ | \text{A} |
The graph of the power function $f(x)$ passes through the point $(2,8)$. The expression for $f(x)$ is \_\_\_\_\_\_. | f(x)=x^3 |
6. Calculate $1+\frac{1}{1+2}+\frac{1}{1+2+3}+\frac{1}{1+2+3+4}+\cdots+\frac{1}{1+2+3+\cdots+2019+2020}=$ $\qquad$ | \frac{4040}{2021} |
Which of the following calculations is correct?
A: $x^{3}+x^{3}=x^{6}$
B: $b\cdot b^{3}=b^{4}$
C: $4a^{3}\cdot 2a^{2}=8a^{6}$
D: $5a^{2}-3a^{2}=2$ | B |
Given that the complex number $\frac{2a+2i}{1+i} (a \in R)$ is a purely imaginary number, the complex number $2a+2i$ corresponds to which quadrant in the complex plane?
A: The first quadrant
B: The second quadrant
C: The third quadrant
D: The fourth quadrant | B |
Example 4 Draw the following lines on the coordinate plane $y=k, y=\sqrt{3} x+2 k, y=-\sqrt{3} x+2 k$, where $k=$ $0, \pm 1, \pm 2, \cdots, \pm 10$. These 63 lines can divide the plane into several equilateral triangles. Find the number of equilateral triangles with side length $\frac{2}{\sqrt{3}}$.
(1994 12th American... | 660 |
8-1. A beginner gardener planted daisies, buttercups, and marguerites on their plot. When they sprouted, it turned out that there were 5 times more daisies than non-daisies, and 5 times fewer buttercups than non-buttercups. What fraction of the sprouted plants are marguerites? | 0 |
Given that the line passing through points $A(-1, a)$ and $B(a, 8)$ is parallel to the line $2x - y + 1 = 0$, the value of $a$ is $\boxed{2}$. | 2 |
"The proposition 'There exists an $x \in \mathbb{R}$ such that $x^2 + ax - 4a \leq 0$ is false" is a ( ) for the condition $-16 \leq a \leq 0$.
A: Sufficient but not necessary condition
B: Necessary but not sufficient condition
C: Necessary and sufficient condition
D: Neither sufficient nor necessary condition | A |
In an isosceles triangle $A B C(A C=B C)$, a circle with radius 3 is inscribed. A line $l$ is tangent to this circle and parallel to the line $A C$. The distance from point $B$ to line $l$ is 3. Find the distance between the points where the given circle touches the sides $A C$ and $B C$. | 3\sqrt{3} |
Calculate: $\left(\sqrt {8}+\sqrt {3}\right)\times \sqrt {6}$. | 4\sqrt {3} + 3\sqrt {2} |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively, and $\cos ^{2} \frac {B-C}{2}-\sin B \cdot \sin C = \frac {2- \sqrt {2}}{4}$.
(1) Find $A$;
(2) If $a=4$, find the maximum area of $\triangle ABC$. | 4(\sqrt{2}+1) |
10.1.25 As shown in the figure, in quadrilateral $A B C D, \angle B A C=30^{\circ}, \angle A B D=26^{\circ}, \angle D B C=$ $51^{\circ}, \angle A C D=13^{\circ}$, find $\angle C A D$. | 17 |
Given positive numbers \(x\) and \(y\) satisfying \(x+3y=5xy\), find the minimum value of \(3x+4y\). | 5 |
Test: Given that $a$ is a natural number, a quadratic trinomial with integer coefficients and $a$ as the leading coefficient has two distinct positive roots less than 1. Find the minimum value of $a$.
---
The above text translated into English, preserving the original text's line breaks and format. | 5 |
2. Given $a, b, c > 0$, find the maximum value of the expression
$$
A=\frac{a^{4}+b^{4}+c^{4}}{(a+b+c)^{4}-80(a b c)^{4 / 3}}
$$ | 3 |
7. Define the operation symbol “ * " to mean $a * b=\frac{a+b}{a b}$ (where $a$ and $b$ are not 0).
Given the following two conclusions:
(1) The operation “ * " satisfies the commutative law;
(2) The operation “ * " satisfies the associative law.
Which of the following is correct? ( ).
(A)(1)
(B)(2)
(C)(1)(2)
(D) None... | A |
Given the quadratic function $f(x)=x^{2}+ax+b$ which is symmetric about $x=1$ and whose graph passes through the origin.
(1) Find the analytic expression of this function;
(2) Find the range of the function for $x\in (0,3]$. | [-1,3] |
Determine the coefficient of the $x^{3}$ term in the expansion of $(1-2x)^{5}$. (Express the answer in numeric form) | -80 |
The constant term in the expansion of $(|x|+ \frac {1}{|x|}-2)^{3}$ is
A: -20
B: 19
C: -18
D: 21 | A |
30. The figure below shows two circles with centres $\mathrm{A}$ and $\mathrm{B}$, and a line $\mathrm{L}$ which is a tangent to the circles at $X$ and $Y$. Suppose that $X Y=40 \mathrm{~cm}, A B=41 \mathrm{~cm}$ and the area of the quadrilateral $\mathrm{ABYX}$ is $300 \mathrm{~cm}^{2}$. If $a$ and $b$ denote the area... | 16 |
10.5. The tsar has eight sons, and they are all fools. Every night the tsar sends three of them to guard the golden apples from the firebird. The tsareviches cannot catch the firebird, blame each other for it, and therefore no two of them agree to go on guard duty a second time. For how many nights can this continue at... | 8 |
Given a positive integer $n \geqslant 2$, real numbers $a_{1}, a_{2}, \cdots$, $a_{n}$ satisfy $\sum_{i=1}^{n}\left|a_{i}\right|+\left|\sum_{i=1}^{n} a_{i}\right|=1$. Find the minimum and maximum values of $\sum_{i=1}^{n} a_{i}^{2}$. | \frac{1}{4n} |
Three students took advantage of the summer vacation to participate in work-study activities. One day, they brought watermelons to the farmers' market to sell: the first person brought 10, the second person brought 16, and the third person brought 26. In the morning, they sold several watermelons at the same price (sol... | 4.5, 1.5 |
163. Reduced Share. A father gave his children 6 dollars for entertainment, which was to be divided equally. But two young cousins joined the company. The money was divided equally among all the children, so that each child received 25 cents less than originally intended. How many children were there in total? | 8 |
Given a positive geometric sequence $\{b_n\}$ ($n\in\mathbb{N}^+$) with a common ratio $q>1$, $b_3+b_5=40$, and $b_3b_5=256$, let $a_n=\log_{2}b_n+2$.
1. Prove that the sequence $\{a_n\}$ is an arithmetic sequence;
2. If $c_n= \frac{1}{a_{n}\cdot a_{n+1}}$, find the sum of the first $n$ terms of the sequence $\{c_n\}$,... | \frac{n}{3n+9} |
Given that $\alpha$, $\beta$, and $\gamma$ are three distinct planes, and $l$ is a line, which of the following propositions is correct?
A: If $\alpha \perp \beta$ and $l \perp \beta$, then $l \parallel \alpha$
B: If $l \perp \alpha$ and $l \parallel \beta$, then $\alpha \perp \beta$
C: If there are two points on ... | \text{B} |
In a geometric sequence $\{a_{n}\}$, for all $n\in N^{*}$, $a_{n} \lt 0$, and $a_{3}+a_{7}\geqslant 2a_{5}$, then the common ratio $q$ of the sequence $\{a_{n}\}$ is $\_\_\_\_\_\_$. | 1 |
77. Point $A(4,0), B(0,3)$ together with point $C$ form a right triangle with side lengths $3,4,5$. If point $C$ lies on the graph of the inverse proportion function $y=\frac{k}{x}$, then the number of possible values for $k$ is $\qquad$. | 3 |
2. Given $M=\left\{(x, y) \mid x^{2}+2 y^{2}=3\right\}, N=\{(x, y) \mid y=m x+b\}$. If for all $m \in \mathbf{R}$, we have $M \cap N$ $\neq \varnothing$, then the range of values for $b$ is
(A) $\left[-\frac{\sqrt{6}}{2}, \frac{\sqrt{6}}{2}\right]$
(B) $\left(-\frac{\sqrt{6}}{6}, \frac{\sqrt{6}}{2}\right)$
(C) $\left(-... | [-\frac{\sqrt{6}}{2},\frac{\sqrt{6}}{2}] |
Find the largest integer $n$ which equals the product of its leading digit and the sum of its digits. | 48 |
Given an ellipse M, whose axes of symmetry align with the coordinate axes, and a parabola with equation $x^{2}=-4\sqrt{2}y$, share a focus with the ellipse M. Additionally, given that point A $(1, \sqrt{2})$ lies on ellipse M,
(I) find the equation of ellipse M;
(II) given a line l with direction vector $(1, \sqrt{... | \sqrt{2} |
The maximum value of the function $f(x)= \frac {\sqrt {x}}{x+1}$ is ( )
A: $ \frac {2}{5}$
B: $ \frac {1}{2}$
C: $ \frac {\sqrt {2}}{2}$
D: $1$ | B |
In the arithmetic sequence $\{a_n\}$, $a_1=1$, $a_{2n}=2a_n+1$ ($n\in\mathbb{N}^*$), and $S_n$ is the sum of the first $n$ terms of the sequence $\{a_n\}$.
(Ⅰ) Find $a_n$ and $S_n$;
(Ⅱ) Let the sequence $\{b_n\}$ satisfy $\frac{b_1}{a_1} + \frac{b_2}{a_2} + \ldots + \frac{b_n}{a_n} = 1 - \frac{1}{2^n}$ ($n\in\mathbb{N}... | T_n=3- \frac{2n+3}{2^n} |
How many numbers less than four digits (from 0 to 9999) are not divisible by 3, 5, or 7? | 4571 |
The function $y=a^{x-1}$ ($a>0$ and $a\neq 1$) always passes through the fixed point ( )
A: $(0,1)$
B: $(1,1)$
C: $(1,0)$
D: $(2,1)$ | B |
Let $p = 1+\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+\frac{1}{2^4}+\frac{1}{2^5}. $ For nonnegative reals $x, y,z$ satisfying $(x-1)^2 + (y-1)^2 + (z-1)^2 = 27,$ find the maximum value of $x^p + y^p + z^p.$ | 40.4 |
5. Positive integers $m, n$ satisfy $m<n$, and $\frac{1}{m^{2}+m}+\frac{1}{(m+1)^{2}+(m+1)}+\cdots+\frac{1}{n^{2}+n}=\frac{1}{2021}$, then the minimum value of $m+n$ is . $\qquad$ | 359 |
Example 3. Find the differential of the function $y=e^{x}\left(x^{2}+3\right)$. Calculate the value of the differential at the point $x=0$. | 3 |
15. Real numbers $x_{1}, x_{2}, \cdots, x_{2001}$ satisfy $\sum_{k=1}^{2000}\left|x_{k}-x_{k+1}\right|=2001$, let $y_{k}=\frac{1}{k}\left(x_{1}+\right.$ $\left.x_{2}+\cdots+x_{k}\right), k=1,2, \cdots, 2001$. Find the maximum possible value of $\sum_{k=1}^{2000}\left|y_{k}-y_{k+1}\right|$. | 2000 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.