problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
|---|---|
Let $\{a_{n}\}$ be a sequence which satisfy
$a_{1}=5$ and $a_{n=}\sqrt[n]{a_{n-1}^{n-1}+2^{n-1}+2.3^{n-1}} \qquad \forall n\geq2$
[b](a)[/b] Find the general fomular for $a_{n}$
[b](b)[/b] Prove that $\{a_{n}\}$ is decreasing sequences | a_n = \sqrt[n]{2^n + 3^n} |
1. (8 points) The calculation result of the expression $2016 \times\left(\frac{8}{7 \times 9}-\frac{1}{8}\right)$ is | 4 |
10. (20 points) Point $A$ lies on the line $y=\frac{5}{12} x-7$, and point $B$ lies on the parabola $y=x^{2}$. What is the minimum length of segment $A B$? | \frac{4007}{624} |
Use the Horner's method to calculate the value of the polynomial $f(x)=9x^6+12x^5+7x^4+54x^3+34x^2+9x+1$. The number of multiplication operations required is ______ times, and the number of addition operations required is ______ times. | 6 |
[ Arithmetic. Mental calculation, etc.]
In a row, 37 numbers are written such that the sum of any six consecutive numbers is 29. The first number is 5. What can the last number be?
# | 5 |
6. Solve the equation $\sqrt{\frac{x+2}{x-3}}-6 \sqrt{\frac{x-3}{x+2}}+1=0$. | \frac{14}{3} |
5. [4 points] Find the number of eight-digit numbers, the product of whose digits equals 7000. The answer should be presented as an integer. | 5600 |
Triangle $ABC$ is right angled at $A$. The circle with center $A$ and radius $AB$ cuts $BC$ and $AC$ internally at $D$ and $E$ respectively. If $BD = 20$ and $DC = 16$, determine $AC^2$. | 936 |
28. Let $p_{1}, p_{2}, p_{3}, \ldots, p_{n}$ be distinct prime numbers. How many divisors does the number $q=p_{1} p_{2} p_{3} \ldots p_{n}$ have, including 1 and $q$? | 2^{n} |
Given that the sequence $\{a\_n\}$ is an arithmetic sequence and $\{b\_n\}$ is a geometric sequence, with $a\_1=1$, $b\_1=2$, $a\_2+b\_2=7$, and $a\_3+b\_3=13$.
1. Find the general term formula for both $\{a\_n\}$ and $\{b\_n\}$.
2. If $$c_{n}= \begin{cases} \overset{a_{n},n\text{ is odd}}{b_{n},n\text{ is even}}\end{... | 2n^2 - n + \frac{4^{n+1} - 4}{3} |
find all pairs of relatively prime natural numbers $ (m,n) $ in such a way that there exists non constant polynomial f satisfying \[ gcd(a+b+1, mf(a)+nf(b) > 1 \]
for every natural numbers $ a $ and $ b $ | (m, n) = (1, 1) |
Example 1: In a "level-passing game", the rule is: on the $n$-th level, a die must be rolled $n$ times. If the sum of the points that appear in these $n$ rolls is greater than $\frac{3^{n}}{4}$, then it is considered passing the level; otherwise, it is not considered passing. There are the following statements:
(1) Pas... | B |
For different points A, B, different lines a, b, l, and different planes α, β, which of the following reasoning is incorrect? ( )
A: If A∈a, A∈β, B∈a, B∈β, then a⊂β
B: If A∈α, A∈β, B∈α, B∈β, then α∩β=line AB
C: If l⊄α, A∈l, then A∉α
D: If a∩b=Φ, and a is not parallel to b, then a, b are skew lines | \text{C} |
A2. We have assembled a shape from identical cubes (see the picture). How many cubic decimeters does its volume measure?
(A) 1.12
(B) 0.112
(C) 96
(D) 120
(E) None of the above.
 | 0.112 |
Given $2^x = 3$ and $log_{2} \frac {4}{3}=y$, find the value of $x + y$. | 2 |
Given the sets A={(x, y) | $\frac{m}{2}$ ≤ (x-2)² + y² ≤ m²} and B={(x, y) | 2m ≤ x+y ≤ 2m+1}, if A ∩ B ≠ ∅, then the range of the real number m is ( )
A: [$\frac{1}{2}$, 2 + $\sqrt{2}$]
B: [2 - $\sqrt{2}$, 2 + $\sqrt{2}$]
C: [1 + $\frac{\sqrt{2}}{2}$, +∞)
D: ∅ | \text{A} |
Selecting astronauts can be said to be "one in ten thousand". To pass, one must go through five stages: visual inspection, initial inspection, re-inspection, cultural examination, and political review. If students A, B, and C from a certain school all successfully pass the first two stages, the probabilities of passing... | 0.275 |
Calculate: $\int_{0}^{1}\left( \sqrt{2x-x^{2}}-x\right)dx =$ . | \frac{\pi-2}{4} |
In an arithmetic sequence with 20 terms, the sum of the first three terms is 15 and the sum of the last three terms is 12 . What is the sum of the 20 terms in the sequence?
(An arithmetic sequence is a sequence in which each term after the first is obtained from the previous term by adding a constant. For example, $3,... | 90 |
Point $M(x_{0}, y_{0})$ is a point outside the circle $x^{2}+y^{2}=a^{2}$ where $a>0$. The relationship between the line $x_{0}x+y_{0}y=a^{2}$ and the circle is ( )
A: Tangent
B: Intersecting
C: Disjoint
D: Tangent or intersecting | B |
## Problem Statement
Calculate the definite integral:
$$
\int_{0}^{\frac{\pi}{2}} \frac{d x}{(1+\cos x+\sin x)^{2}}
$$ | 1-\ln2 |
Harry Potter would like to purchase a new owl which cost him 2 Galleons, a Sickle, and 5 Knuts. There are 23 Knuts in a Sickle and 17 Sickles in a Galleon. He currently has no money, but has many potions, each of which are worth 9 Knuts. How many potions does he have to exhange to buy this new owl?
[i]2015 CCA Math Bo... | 90 |
For any real numbers $a$, $b$, $c$, $d$, the operation "$\triangle$" between ordered pairs of real numbers $\left(a,b\right)$ and $\left(c,d\right)$ is defined as: $\left(a,b\right)\triangle \left(c,d\right)=\left(ac+bd,ad+bc\right)$. If for any real numbers $u$, $v$, it holds that $\left(u,v\right)\triangle \left(x,y\... | B |
If there exists $x \in [-2, 3]$ such that the inequality $2x - x^2 \geq a$ holds, then the range of values for the real number $a$ is ( )
A. $(-\infty, 1]$ B. $(-\infty, -8]$ C. $[1, +\infty)$ D. $[-8, +\infty)$ | \text{A} |
Let the set $M=\{a| \dfrac {6}{5-a}\in\mathbb{N}\}$, and $a\in\mathbb{Z}$. Represent the set $M$ using the roster method. | \{-1,2,3,4\} |
Question 106, Let the complex number $\mathrm{z}$ be such that the real and imaginary parts of 1 are both greater than 1. If the set of points corresponding to $\mathrm{z}$ on the complex plane is the figure $C$, find the area of figure $C$.
Translate the above text into English, please keep the original text's line b... | \frac{\pi-2}{8} |
232. $\frac{1}{12} \lg ^{2} x=\frac{1}{3}-\frac{1}{4} \lg x$. | x_{1}=0.0001,x_{2}=10 |
3. Gavriil saw a squirrel sitting on a tree branch right in front of him at a distance of 3 m 75 cm through the window. He decided to feed the little animal and threw a nut horizontally at a speed of 2.5 m/s directly towards the squirrel. Will the squirrel be able to catch the nut if it can jump in any direction at a h... | \frac{5\sqrt{5}}{4}<\frac{14}{5} |
# 8. Variant 1
Petya wrote down all positive numbers that divide some natural number $N$. It turned out that the sum of the two largest written numbers is 3333. Find all such $N$. If there are several numbers, write their sum in the answer. | 2222 |
Problem 4. (20 points) A finite increasing sequence $a_{1}, a_{2}, \ldots, a_{n}$ ( $n \geq 3$ ) of natural numbers is given, and for all $k \leq n-2$, the equality $a_{k+2}=3 a_{k+1}-2 a_{k}-1$ holds. The sequence must contain a term $a_{k}=2021$. Determine the maximum number of three-digit numbers, divisible by 25, t... | 36 |
Fomin D.
The hostess baked a pie for the guests. At the table, there can be either $p$ people or $q$ people ($p$ and $q$ are coprime). Into what minimum number of pieces (not necessarily equal) should the pie be cut in advance so that in any case it can be distributed equally? | p+q-1 |
3. Find the set of values and the smallest period of the function $y=\cos ^{2} 3 x+\sin 6 x$. | [\frac{1}{2}-\frac{\sqrt{5}}{2},\frac{1}{2}+\frac{\sqrt{5}}{2}],\frac{\pi}{3} |
156 Let the function $f(x)$ be defined on the real number set $R$ as an odd function, and another function $g(x)=f(x+2)$ is an even function. Moreover, when $x \in[0,2]$, $f(x)=x$. Then the number of such functions $f(x)$ is ( )
A. 0 (i.e., none)
B. 1
C. More than one, but only a finite number
D. Infinitely many | B |
Task 1. II variant.
The decimal representation of a 2015-digit natural number $N$ contains the digits 5, 6, 7 and no other digits. Find the remainder of the division of the number $N$ by 9, given that the number of fives in the representation of the number is 15 more than the number of sevens. | 6 |
3. Given that $x_{1}, x_{2}, \cdots, x_{2010}$ are all positive real numbers. Then
$$
x_{1}+\frac{x_{2}}{x_{1}}+\frac{x_{3}}{x_{1} x_{2}}+\cdots+\frac{x_{2010}}{x_{1} x_{2} \cdots x_{200}}+\frac{4}{x_{1} x_{2} \cdots x_{2010}}
$$
the minimum value is $\qquad$ | 4 |
Four, (50 points) In a round-robin tournament with $2 n+1$ teams, each team plays exactly one match against every other team, and there are no ties. If three teams $A, B, C$ satisfy: $A$ beats $B, B$ beats $C, C$ beats $A$, then they form a “cyclic triplet”. Find:
(1) the minimum possible number of cyclic triplets;
(2)... | \frac{1}{6} n(n+1)(2 n+1) |
5. In $\triangle A B C$, $\angle B A C=90^{\circ}, A C=A B=4$, point $D$ is inside $\triangle A B C$, and $A D=\sqrt{2}$. Then the minimum value of $B D+C D$ is $\qquad$ . | 2\sqrt{10} |
2. $A B C$ is an acute-angled triangle. $D$ is a point on $B C$ such that $B D: D C=2: 3$, while $E$ is a point on $A C$ such that $A E: E C=3: 4 . A D$ and $B E$ meet at $F$. Find $\frac{A F}{F D} \times \frac{B F}{F E}$.
(1 mark) $A B C$ 是銳角三角形。 $D$ 是 $B C$ 上的一點, 使得 $B D: D C=2: 3 \circ E$ 是 $A C$ 上的一點,使得 $A E: E C=3... | \frac{35}{12} |
Given a triangle $ABC$ with the sides opposite to angles $A$, $B$, $C$ being $a$, $b$, $c$ respectively, if $3a\cos C=2c\cos A$ and $\tan A= \frac {1}{3}$, then the degree measure of angle $B$ is __( )__.
A: $120^{\circ}$
B: $135^{\circ}$
C: $60^{\circ}$
D: $45^{\circ}$ | B |
For the one-variable quadratic equation in $x$, $2x^{2}-3x-1=0$, the constant term of this equation is ( )
A: $2$
B: $-3$
C: $1$
D: $-1$ | D |
7. Given a polynomial $P(x)$, not identically zero. It is known that for all $x$ the identity $(x-2020) \cdot P(x+1)=(x+2021) \cdot P(x)$ holds. How many roots does the equation $P(x)=0$ have? Answer: 4042 | 4042 |
Given the hyperbola $H$: $\frac{x^{2}}{3} - \frac{y^{2}}{6} = 1$, and the moving line $l$ with a slope of $2$ intersects $H$ at points $A$ and $B$. The midpoint of segment $AB$ lies on a fixed line. Determine the equation of this fixed line. The options are:
A: $x + y = 0$
B: $x - y = 0$
C: $x + 2y = 0$
D: $x - 2y = 0$ | B |
Which of the following calculations is correct?
A: $3x+2x=5x^{2}$
B: $\sqrt{9}=±3$
C: $(2x)^{2}=2x^{2}$
D: $2^{-1}=\frac{1}{2}$ | D |
9.065. $\frac{15}{4+3 x-x^{2}}>1$.
9.065. $\frac{15}{4+3 x-x^{2}}>1$. | x\in(-1;4) |
26. In 2075, the Sun rapidly aged and expanded, and humanity initiated the "Wandering Earth" plan, building engines to push Earth out of the solar system and toward a new home—the Alpha Centauri star in the distant universe. After calculations by scientists, the total distance Earth will travel is 4.2 light-years (a li... | 1000 |
Decompose 96 into two positive integer factors whose sum of squares is 208. | 812 |
$13 \cdot 13$ Given 4 weights forming a geometric progression and a two-pan balance, explain how to find the heaviest weight using the balance only twice.
Given 4 weights forming a geometric progression and a two-pan balance, explain how to find the heaviest weight using the balance only twice. | ^{3} |
3. Compute the sum
$$
\frac{1}{1 \cdot 4}+\frac{1}{4 \cdot 7}+\ldots+\frac{1}{(3 n-2)(3 n+1)}
$$ | \frac{n}{3n+1} |
For a certain rectangular prism, the area of the rectangular diagonal section perpendicular to two parallel edges can be one of three possibilities: $t_{1}=60, t_{2}=4 \sqrt{153}, t_{3}=12 \sqrt{10}$. Calculate the surface area and volume of the rectangular prism. | 144 |
18. If the sum of the interior angles of a convex polygon is 5 times the sum of the exterior angles, then this polygon has $\qquad$ sides. | 12 |
Given $f(x)= \begin{cases} 2x-1 & ,(x\geqslant 2)\\ -x^{2}+3x & ,(x < 2) \end{cases}$, then the value of $f(-1)+f(4)$ is ( ).
A: $3$
B: $4$
C: $-7$
D: $-8$ | A |
Find the coefficient of $x^{14}$ in the expansion of $(x- \sqrt{2})^{10}(x+ \sqrt{2})^{10}$. | -960 |
If the complex number $\frac {2-ai}{1+i}$ (where $a\in\mathbb{R}$) is a pure imaginary number ($i$ is the imaginary unit), then $a=$ ____
A: $-2$
B: $-\frac {1}{2}$
C: $\frac {1}{2}$
D: $2$ | D |
Define an odd function $f(x)$ on $\mathbb{R}$ that is a decreasing function. Let $a+b \leq 0$, and consider the following inequalities:
① $f(a)f(-a) \leq 0$; ② $f(b)f(-b) \geq 0$; ③ $f(a) + f(b) \leq f(-a) + f(-b)$; ④ $f(a) + f(b) \geq f(-a) + f(-b)$.
Among these, the correct sequence of inequality numbers is
A.... | \text{B} |
Let $S$ be the set of integers of the form $2^x+2^y+2^z$, where $x,y,z$ are pairwise distinct non-negative integers. Determine the $100$th smallest element of $S$. | 577 |
Which of the following equations is calculated correctly?
A: $3a^{2}+2a^{2}=5a^{4}$
B: $a^{2}+a^{2}=a^{4}$
C: $6a-5a=1$
D: $3a^{2}b-4ba^{2}=-a^{2}b$ | D |
(12) Try to explain the monotonic increasing interval of the function $f(x)=\frac{(x+1)^{2}}{x^{2}+1}$. | [-1,1] |
Example 3 Let $x, y, z>0$ and $4x+3y+5z=1$, find the minimum value of $\frac{1}{x+y}+\frac{1}{y+z}+\frac{1}{z+x}$, and indicate the corresponding values of $x, y, z$. (Mathematical Problems 1625, Mathematics Bulletin) | (1+\sqrt{2}+\sqrt{3})^{2} |
## Task 2 - 320722
Let $A B C D$ be a square, with an area of $25 \mathrm{~cm}^{2}$. A point $E$ lies on the extension of the diagonal $A C$ beyond $C$, such that the segment $A E$ is twice as long as the segment $A C$.
Determine the area of the quadrilateral $A B E D$ under these conditions! | 50\mathrm{~}^{2} |
## Task 2 - 330842
Let $ABCD$ be a square with given side length $a$. A line parallel to $AB$ intersects the sides $AD$ and $BC$ at $E$ and $F$, respectively. A line parallel to $AD$ inte... | \frac{25}{12} |
Given the complex number $z$ satisfies $z+(1+2i)=10-3i$, then $z=$\_\_\_\_\_\_\_. | 9-5i |
Given the function $f(x) = \log_{a}(x^{2}+2)$, if $f(5) = 3$;
$(1)$ Find the value of $a$;
$(2)$ Find the value of $f(\sqrt{7})$;
$(3)$ Solve the inequality $f(x) < f(x+2)$. | (-1, +\infty) |
In the Cartesian coordinate system $xOy$, a moving point $M(x,y)$ always satisfies the relation $2 \sqrt {(x-1)^{2}+y^{2}}=|x-4|$.
$(1)$ What is the trajectory of point $M$? Write its standard equation.
$(2)$ The distance from the origin $O$ to the line $l$: $y=kx+m$ is $1$. The line $l$ intersects the trajectory of $M... | \frac{3\sqrt{7}}{5} |
On the trip home from the meeting where this AMC$ 10$ was constructed, the Contest Chair noted that his airport parking receipt had digits of the form $ bbcac$, where $ 0 \le a < b < c \le 9$, and $ b$ was the average of $ a$ and $ c$. How many different five-digit numbers satisfy all these properties?
$ \textbf{(A)... | 20 |
Example 5 By Fermat's Little Theorem, for any odd prime $p$, we have $2^{p-1} \equiv 1(\bmod p)$. Question: Does there exist a composite number $n$ such that $2^{n-1} \equiv 1(\bmod n)$ holds? | 341 |
【Question 12】A and B take turns marking numbers from 1 to 17, with the rules: you cannot mark a number that has already been marked; you cannot mark a number that is twice a marked number; you cannot mark a number that is half of a marked number; the player who cannot mark a number loses. Now A marks 8 first, to ensure... | 6 |
The number of zeros of the quadratic function $y=x^{2}+x-1$ is $\boxed{\text{answer}}$. | 2 |
5. Let the complex numbers $Z_{1}, Z_{2}$ satisfy $\left|Z_{1}\right|=\left|Z_{1}+Z_{2}\right|=$ $3,\left|Z_{1}-Z_{2}\right|=3 \sqrt{3}$, then $\log _{3} \mid\left(Z \cdot \overline{Z_{2}}\right)^{2009}+$ $\left(\overline{Z_{1}} Z_{2}\right)^{2009} \mid=$ $\qquad$ | 4018 |
The decimal number corresponding to the binary number $101101_{(2)}$ is $\left( \ \right)$
A: $45$
B: $44$
C: $46$
D: $47$ | A |
$1 \cdot 47$ Let $x^{2}-x+a$ divide $x^{13}+x+90$, determine the positive integer value of $a$.
(24th Putnam Mathematical Competition, 1963) | 2 |
1. Let $\mathbf{P}$ be the set of all prime numbers. Find all functions $f: \mathbf{P} \rightarrow \mathbf{P}$ such that for any $p, q \in \mathbf{P}$, we have
$$
(f(p))^{f(q)} + q^{p} = (f(q))^{f(p)} + p^{q} .
$$ | f(p)=p |
## Task 1
What time is it at the end of the 5th lesson if school starts at 7 o'clock and a lesson lasts $45 \mathrm{~min}$?
First break: $10 \mathrm{~min}$
Second break (milk break): $15 \mathrm{~min}$
Third break (yard break): $20 \mathrm{~min}$
Fourth break: $10 \mathrm{~min}$ | 11:40 |
Huanggang City has many scenic spots, with 10 $4A$-level scenic areas alone. A certain unit plans to organize a tour to three nearby scenic areas: Turtle Peak Mountain, Tiantang Village, and Hong'an Red Scenic Area for their five outstanding employees. Each employee is limited to visiting one scenic spot, and at least ... | D |
3. Let $\Varangle A O B, \Varangle B O C, \Varangle C O D, \Varangle D O E, \Varangle E O A$ be angles around point $O$ such that: $m(\Varangle B O C)=m(\Varangle A O B)+n^{0}, m(\Varangle C O D)=m(\Varangle B O C)+n^{0}, m(\Varangle D O E)=m(\Varangle C O D)+n^{0}$, $m(\nless E O A)=m(\nless D O E)+n^{0}, n \in \mathb... | 48 |
Given the power function $f(x) = k \cdot x^a$ passes through the point $(3, \sqrt{3})$, then $k+a=$ \_\_\_\_\_\_. | \frac{3}{2} |
## Problem 3
A tetrahedron has at most one edge longer than 1 . What is the maximum total length of its edges?
| 5+\sqrt{3} |
Calculate: $3\sqrt{2}+|\begin{array}{l}{1-\sqrt{2}}\\{}\end{array}|+\sqrt[3]{8}$. | 4\sqrt{2} + 1 |
7. As shown in Figure 3, quadrilateral $ABCD$ is a right trapezoid $\left(\angle B=\angle C=90^{\circ}\right)$, and $AB=BC$. If there exists a point $M$ on side $BC$ such that $\triangle AMD$ is an equilateral triangle, then the value of $\frac{CD}{AB}$ is $\qquad$ | \sqrt{3}-1 |
4. Given $O$ is the origin, point $A$ in the first quadrant lies on the graph of the inverse proportion function $y=\frac{1}{x}(x>0)$, and point $B$ in the second quadrant lies on the graph of the inverse proportion function $y=-\frac{4}{x}$ $(x<0)$, and $O A \perp O B$. Then the value of $\tan \angle A B O$ is ( ).
(A... | \frac{1}{2} |
## Task B-1.3.
Let $a, b, c$ and $d$ be real numbers different from zero such that $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}=0$ and $a+b+c+d+a b c d=0$. Calculate $\left(\frac{1}{a b}-\frac{1}{c d}\right)\left(\frac{1}{c}+\frac{1}{d}\right)$. | -1 |
1. The quadratic trinomial
$$
x^{2}+a x+b(a 、 b \in \mathbf{N})
$$
has real roots, and $a b=2^{2011}$. Then the number of such quadratic trinomials is $\qquad$. | 1341 |
Given propositions $p$: $∀α∈R$, $sin(π-α)≠-sinα$, and $q$: $∃x∈[0,+∞)$, $sinx > x$, the correct conclusion is ($\ \ \ \ $).
A: $¬p∨q$ is a true proposition
B: $p∨q$ is a true proposition
C: $¬p∧q$ is a true proposition
D: $q$ is a true proposition | A |
Four. (50 points) Let $n$ be a positive integer, and let the planar point set be
$$
S=\{(x, y) \mid x, y \in\{0,1, \cdots, n\}, x+y \neq 0\} \text {. }
$$
Question: What is the minimum number of lines in the plane whose union can contain $S$, but not include the point $(0,0)$? | 2n |
Let the domain of the function $f\left(x\right)$ be $R$. $f\left(x-1\right)$ is an odd function, and $f\left(x+1\right)$ is an even function. When $x\in \left(-1,1\right)$, $f\left(x\right)=-e^{x}$, then ( )
A: $f\left(3\right)=-1$
B: $f\left(-2\right)=-1$
C: $f\left(x+6\right)$ is an odd function
D: $f\left(2x\rig... | D |
Problem 9.8. Young entomologist Dima is observing two grasshoppers. He noticed that when a grasshopper starts jumping, it jumps 1 cm, then after a second, 2 cm, then another second, 3 cm, and so on.
Initially, both grasshoppers were in the same place. One of them started jumping, and after a few seconds, the second on... | 10,15,30 |
4. If the numerator of a fraction is reduced by $8\%$, and the denominator is increased by $8\%$, the new fraction will be 2 less than the original fraction. Determine the value of the original fraction. | \frac{27}{2} |
The cost of a phone call from City A to City B for $t$ minutes is given by the function $g(t)=1.06\times(0.75[t]+1)$, where $t > 0$, and $[t]$ denotes the smallest integer greater than or equal to $t$. Then, the cost of a phone call from City A to City B for $5.5$ minutes is $(\quad)$.
A: $5.83$ yuan
B: $5.25$ yuan ... | A |
1. If real numbers $x, y$ satisfy $x^{2}+y^{2}+x y=1$, then the maximum value of $x+y$ is $\qquad$ | \frac{2\sqrt{3}}{3} |
(5) Definition: If there exists a constant $k$, such that for any two different real numbers $x_{1}, x_{2}$ in the domain $D$, the following holds: $\left|f\left(x_{1}\right)-f\left(x_{2}\right)\right| \leqslant k\left|x_{1}-x_{2}\right|$, then $f(x)$ is said to satisfy the Lipschitz condition on $D$.
(1) Give an examp... | \frac{1}{2} |
The lengths of the sides of a rectangle are all integers less than 10 (unit: cm), and these four lengths can form a four-digit number. The thousand's digit and the hundred's digit of this four-digit number are the same, and this four-digit number is a perfect square. Find the area of this rectangle. | 28 \text{ cm}^2 |
In some countries, automobile fuel efficiency is measured in liters per $100$ kilometers while other countries use miles per gallon. Suppose that 1 kilometer equals $m$ miles, and $1$ gallon equals $l$ liters. Which of the following gives the fuel efficiency in liters per $100$ kilometers for a car that gets $x$ miles ... | \frac{100lm}{x} |
In the Cartesian coordinate system $xOy$, the curve $C_{1}$ passes through the point $P(a,1)$, with its parametric equations given by
$$
\begin{cases}
x=a- \frac { \sqrt {2}}{2}t \\
y=1+ \frac { \sqrt {2}}{2}t
\end{cases}
(t \text{ is the parameter, } a\in\mathbb{R})
$$
Taking the origin as the pole and the positive ... | \frac {33}{2} |
Given the equation $x^{2}-2x+m-1=0$.
$(1)$ If the equation has two equal real roots, find the value of $m$;
$(2)$ If one real root of the equation is $5$, find the other root of the equation. | x_2 = -3 |
Given that the vertex of the parabola $C$ is at the coordinate origin, its axis of symmetry is $x=-1$, and the line $l$ intersects with the parabola $C$ at points $A$ and $B$. If the midpoint of line segment $AB$ is $(2,1)$, find the equation of line $l$.
A: $y=2x-3$
B: $y=-2x+5$
C: $y=-x+3$
D: $y=x-1$ | A |
The intersection point $P$ of two lines $l_{1}: x-2y+4=0$ and $l_{2}: x+y-2=0$, the distance from $P$ to the line $l_{3}: 3x-4y+5=0$ is ______. | \dfrac {3}{5} |
Given proposition $p$: $\exists x_{0} \in \mathbb{R}$, $2x_{0}+1 \leqslant 0$, the negation of proposition $p$ is ( )
A: $\exists x_{0} \in \mathbb{R}$, $2x_{0}+1 > 0$
B: $\forall x \in \mathbb{R}$, $2x+1 > 0$
C: $\exists x_{0} \in \mathbb{R}$, $2x_{0}+1 \leqslant 0$
D: $\forall x \in \mathbb{R}$, $2x+1 \geqslant 0$ | B |
A plane intersects each side of the quadrilateral $ABCD$ at an internal point. Following a circumferential direction, we write down the ratio in which each intersection point divides the side it lies on. What will be the product of the four ratios?
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Translate the above text into English, please keep the original t... | 1 |
Define a function $f(x)$ on $\mathbb{R}$ that is both an odd function and a periodic function with a period of 2. Then, $f(1)$ equals \_\_\_\_\_\_. | 0 |
10. (3 points) Cars A and B both depart from point A, pass through point B, and head to point C. The distance between A and B is equal to the distance between B and C. Car B's speed is 80% of Car A's speed. It is known that Car B departs 11 minutes earlier than Car A, but stops for 7 minutes at point B, while Car A con... | 27 |
Suppose $m$ and $n$ are positive integers with $m<n$. The value of $\frac{m+3}{n+3}$ will be
(A) equal to 1
(B) equal to 3
(C) less than the value of $\frac{m}{n}$
(D) greater than the value of $\frac{m}{n}$
(E) equal to the value of $\frac{m}{n}$ | D |
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