problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
|---|---|
If $10^{a}=2$, $\log 3=b$, then $\log _{9}8=\_\_\_\_\_\_$. (Express the result in terms of $a$ and $b$). | \frac{3a}{2b} |
Distribute teachers A, B, C, and D among three different schools, with each school receiving at least one teacher, and with the condition that teachers A and B cannot be assigned to the same school. Calculate the number of different distribution methods. | 30 |
Given the power function $f(x) = (m^2 - 4m + 4)x^{m^2 - 6m + 8}$, determine the value(s) of $m$ for which the function is decreasing on $(0,+\infty)$.
A: $1$ or $3$
B: $1$
C: $3$
D: $2$ | C |
2. A square is inscribed in a circle, as shown in the figure. If the area of the circle is $16 \pi \mathrm{cm}^{2}$ and the area of the square is $S \mathrm{~cm}^{2}$, what is the value of $\mathrm{S}$ ? | 32 |
In $\triangle ABC$, the sides $a$, $b$, and $c$ correspond to angles $A$, $B$, and $C$ respectively, and they satisfy the equation $c \cos B + (2a + b) \cos C = 0$.
(1) Find angle $C$;
(2) If $c = \sqrt{3}$, find the maximum area of $\triangle ABC$. | \frac{\sqrt{3}}{4} |
The number of solutions to the equation $|x+1|+|x+9|+|x+2|=1992$ is (γγ)
A: 4
B: 3
C: 2
D: 1 | C |
Given a random variable $\xi$ follows the normal distribution $N(1, \sigma^2)$, and $P(\xi \leq 4) = 0.79$, then $P(-2 \leq \xi \leq 1) =$ οΌγγοΌ
γA.γγγγγ0.21γγγγγγγB.γγγγγ0.58γγγγγγγC.γγγγγ0.42γγγγγγγD.γγγγγ0.29 | D |
To show respect for teachers, on the morning of Teacher's Day, taxi driver Xiao Wang provided free rides for teachers on a road running east to west. If it is defined that east is positive and west is negative, the taxi's journey is as follows (in kilometers): +15, -4, +13, -10, -12, +3, -13, -17.
$(1)$ When Xiao Wang ... | 34.8 \text{ liters} |
The root of the equation $2^{x}+x=4$ lies in the interval (Β Β Β Β Β )
A: $(-1,0)$
B: $(0,1)$
C: $(1,2)$
D: $(2,3)$ | C |
Given the proposition P: For all $x \in \mathbb{R}$, $x > \sin(x)$, the negation of P is ( )
A: $\exists x \in \mathbb{R}, x \leq \sin(x)$
B: $\forall x \in \mathbb{R}, x \leq \sin(x)$
C: $\exists x \in \mathbb{R}, x < \sin(x)$
D: $\forall x \in \mathbb{R}, x < \sin(x)$ | A |
Two people, A and B, jointly solve the system of equations involving $x$ and $y$: $\left\{\begin{array}{l}{ax+5y=15β }\\{4x-by=-2β‘}\end{array}\right.$ A misreads the coefficient $a$ in equation β and obtains the solution to the system as $\left\{\begin{array}{l}{x=-3}\\{y=-1}\end{array}\right.$ B misreads the coefficien... | 0 |
One, (50 points)
As shown in the figure: $A M$ and $A N$ are tangents to circle $O$, with $M$ and $N$ being the points of tangency. $L$ is a point on the minor arc $M N$ different from $M$ and $N$. A line through point $A$ parallel to $M N$ intersects $M L$ and $N L$ at points $P$ and $Q$, respectively. If $S_{\odot O}... | 60 |
1. A nonzero polynomial $f(x)$ with real coefficients has the property that $f(x)=f^{\prime}(x) f^{\prime \prime}(x)$. What is the leading coefficient of $f(x)$ ? | \frac{1}{18} |
20. A voucher code is made up of four characters. The first is a letter: $\mathrm{V}, \mathrm{X}$ or $\mathrm{P}$. The second and third are different digits. The fourth is the units digit of the sum of the second and third digits. How many different voucher codes like this are there?
A 180
B 243
C 270
D 300
E 2700 | 270 |
Calculate $$\sqrt {a \sqrt {a} \sqrt {a}}$$ = \_\_\_\_\_\_. | a |
Given $z_1 = 2 + i$ and $z_2 = 1 + 2i$, then in which quadrant is the point corresponding to the complex number $z = z_2 - z_1$ located?
A: First quadrant
B: Second quadrant
C: Third quadrant
D: Fourth quadrant | \text{B} |
Solve the following equation:
$$
\frac{(1-\log _{6}3)^{2}+\log _{6}2\cdot\log _{6}18}{\log _{6}4}=\_\_\_\_\_\_\_\_.
$$ | 1 |
2. Choose two elements from the set $\{1,2, \cdots, 11\}$ as $m$ and $n$ in the ellipse equation $\frac{x^{2}}{m^{2}}+\frac{y^{2}}{n^{2}}=1$. Then the number of ellipses that can be formed within the rectangular region
$$
B=\{(x, y)|| x|<11,| y \mid<9\}
$$
is ( ).
(A) 43
(B) 72
(C) 86
(D) 9 | B |
\section*{Problem 3 - V01203}
A clock, equipped with a synchronous motor, has ideally uniformly moving hands.
Determine exactly the times at which the hands are positioned such that one hour later, the angle between the hands is of the same size!
(Hint: The relevant angles are less than \(180^{\circ}\).) | \mu_{0}=\sigma_{0}-165\quad\text{or}\quad\mu_{0}=\sigma_{0}+15 |
"The Belt and Road Initiative" is a win-win cooperation route between China and the world. According to statistics, the total population covered by the "Belt and Road" region is approximately 4.4 billion people. In scientific notation, the number "4.4 billion" can be represented as ( ).
A: $44\times 10^{8}$
B: $4.4\t... | C |
The function is given by $f(x) = xe^x - a \ln x$. The tangent to the curve $y = f(x)$ at the point $(1, f(1))$ is parallel to the x-axis.
(β
) Determine the interval of monotonicity for $f(x) = a(x-1)(e^{x}-a)$;
(β
‘) Prove that when $b \leq e$, $f(x) \geq b(x^2 - 2x + 2)$. | f(x) \geq b(x^2 - 2x + 2) |
12. (2004 High School League - Liaoning Preliminary) The sum of the maximum and minimum values of the function $f(x)=\frac{\sqrt{2} \sin \left(x+\frac{\pi}{4}\right)+2 x^{2}+x}{2 x^{2}+\cos x}$ is $\qquad$ . | 2 |
6.155. $a x^{4}-x^{3}+a^{2} x-a=0$.
6.155. $a x^{4}-x^{3}+a^{2} x-a=0$. | x_{1}=\frac{1}{},x_{2}=-\sqrt[3]{}when\neq0;0when=0 |
2. In the plane, there are $h+s$ lines, among which $h$ lines are horizontal, and the $s$ lines satisfy:
(1) None of them are horizontal;
(2) No two of them are parallel;
(3) Any three of these $h+s$ lines do not intersect at the same point.
If these $h+s$ lines exactly divide the plane into 1992 regions, find all pair... | (995,1),(176,10),(80,21) |
11.069. The lateral edge of a regular triangular prism is equal to the height of the base, and the area of the section made through this lateral edge and the height of the base is $Q$. Determine the volume of the prism. | Q\sqrt{Q/3} |
1. Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that
$$
f(2 x y)+f(f(x+y))=x f(y)+y f(x)+f(x+y)
$$
for all real numbers $x$ and $y$. | f(x)=0,f(x)=x,f(x)=2-x |
Factorize the following expressions:
(1) $a^3-9a$
(2) $3x^2-6xy+x$
(3) $n^2(m-2)+n(2-m)$
(4) $-4x^2+4xy+y^2$
(5) $a^2+2a-8$ | (a-2)(a+4) |
46. On a library shelf, 15 textbooks are randomly arranged, with five of them in hardcover. The librarian randomly takes three textbooks. Find the probability that at least one of the taken textbooks will be in hardcover (event $A$). | \frac{67}{91} |
Given vectors $\overrightarrow{a} = (\cos\left(\frac{3x}{2}\right), \sin\left(\frac{3x}{2}\right))$ and $\overrightarrow{b} = (\cos\left(\frac{x}{2}\right), -\sin\left(\frac{x}{2}\right))$, with $x \in \left[0, \frac{\pi}{2}\right]$.
1. Determine the dot product $\overrightarrow{a} \cdot \overrightarrow{b}$ and the mag... | \frac{\sqrt{2}}{2} |
Given that the line $x-my+1=0$ intersects the circle $C: \left(x-1\right)^{2}+y^{2}=4$ at points $A$ and $B$, find one value of $m$ that satisfies the condition "$\triangle ABC$ has an area of $\frac{8}{5}$." | 2 |
9. In the sequence $\left\{a_{n}\right\}$, $a_{4}=1, a_{11}=9$, and the sum of any three consecutive terms is 15. Then $a_{2016}=$ $\qquad$ | 5 |
4. Let $x_{1}$ and $x_{2}$ be the largest roots of the polynomials $f(x)=1-x-4 x^{2}+x^{4}$ and $g(x)=16-8 x-16 x^{2}+x^{4}$ respectively. Find $\frac{x_{1}}{x_{2}}$. | 0.5 |
2. Variant 1.
CafΓ© "Buratino" operates 6 days a week with a day off on Mondays. Kolya made two statements: "from April 1 to April 20, the cafΓ© was open for 18 days" and "from April 10 to April 30, the cafΓ© was also open for 18 days." It is known that he was wrong once. How many days was the cafΓ© open from April 1 to A... | 23 |
2. The product of four different numbers, which are roots of the equations
$$
x^{2}+6 b x+c=0 \text{ and } x^{2}+6 c x+b=0
$$
is equal to 1. At least one of the numbers $\mathrm{b}$ and c is an integer. Find $\mathrm{b}+\mathrm{c}$ given that b and c are positive. | 2.5 |
$36 \cdot 12$ If the 120 permutations of $A H S M E$ are considered as ordinary five-letter words and are arranged in alphabetical order, the last letter of the 86th word is
(A) A.
(B) $H$.
(C) $S$.
(D) $M$.
(E) $E$.
(37th American High School Mathematics Examination, 1986) | E |
If $f(n+1) = f(n) + 1$ ($n=1, 2, 3, \ldots$), and $f(1) = 2$, then $f(100) =$ ? | 101 |
6. In triangle $A B C$ with angle $\angle B=120^{\circ}$, the angle bisectors $A A_{1}, B B_{1}, C C_{1}$ are drawn. Find the degree measure of angle $C_{1} B_{1} A_{1}$. | 90 |
In the geometric sequence $\{a_n\}$, the product of the second, third, and fourth terms is $a_{2}a_{3}a_{4}= \frac {27}{64}$, and the common ratio $q=2$. Given that $\{b_n\}$ is an arithmetic sequence and $b_{7}=a_{5}$, find $b_{3}+b_{11}$. | b_3 + b_{11} = 6 |
The hypotenuse $AB$ of the right triangle $ABC$ is a chord of a circle with radius 10. The vertex $C$ lies on the diameter of the circle, which is parallel to the hypotenuse, $\angle A=75^{\circ}$. Find the area of triangle $ABC$. | 40 |
Given that the graph of a quadratic function passes through the point $\left(2,-6\right)$, and the function has a maximum value of $-4$ when $x=1$, find the analytical expression of this quadratic function. | y = -2x^2 + 4x - 6 |
Given the function $f(x)=x\sin x$, where $x\in\mathbb{R}$, then the relationship among $f\left( \frac{\pi}{5} \right)$, $f(1)$, and $f\left( -\frac{\pi}{3} \right)$ is ( ).
A: $f\left( -\frac{\pi}{3} \right) > f(1) > f\left( \frac{\pi}{5} \right)$
B: $f(1) > f\left( -\frac{\pi}{3} \right) > f\left( \frac{\pi}{5} \rig... | \text{A} |
For the following numbers: $-5$, $0$, $3\frac{1}{2}$, $-0.3$, $10\%$, $9$, which statement is incorrect?
A: $-5$, $0$, $9$ are all integers.
B: Fractions include $3\frac{1}{2}$, $-0.3$, $10\%$.
C: Positive numbers include $3\frac{1}{2}$, $10\%$, $9$.
D: $-0.3$ is a negative rational number, but not a fraction. | D |
6. As shown in Figure 1, in the tetrahedron $P-ABC$, $PA \perp$ plane $ABC$, $\angle ABC=120^{\circ}$, $PA$ $=4$. If the radius of the circumscribed sphere of the tetrahedron $P-$ $ABC$ is $2 \sqrt{2}$, then the tangent value of the angle formed by line $PC$ and plane $ABC$ is $\qquad$ | \frac{2\sqrt{3}}{3} |
3. In September, Vasya's salary was increased by $x$ percent, and in October it was decreased by the same $x$ percent. After these changes, Vasya was surprised to find that his salary had decreased by $36\%$ compared to the original. Find $x$. | 60 |
Given that $\dfrac{\pi }{2} < \alpha < \pi$ and $\tan \alpha - \dfrac{1}{\tan \alpha} = -\dfrac{3}{2}$, find the value of $\tan \alpha$ and evaluate the expression $\dfrac{\cos (\dfrac{3\pi }{2}+\alpha )-\cos (\pi -\alpha )}{\sin (\dfrac{\pi }{2}-\alpha )}$. | \dfrac{\cos (\dfrac{3\pi }{2}+\alpha )-\cos (\pi -\alpha )}{\sin (\dfrac{\pi }{2}-\alpha )} = \tan \alpha + 1 = -2 + 1 = -1 |
1. Divide the set of positive even numbers $\{2,4, \cdots\}$ into groups in ascending order, with the $n$-th group containing $3 n-2$ numbers:
$$
\{2\},\{4,6,8,10\},\{12,14, \cdots, 24\}, \cdots \text {. }
$$
Then 2018 is in the group. | 27 |
2. If the complex number $\frac{z+\frac{1}{3}}{z-\frac{1}{3}}$ is purely imaginary, then $|z|=$ | \frac{1}{3} |
5. From $26=1^{2}+5^{2}=1^{2}+3^{2}+4^{2}$, it can be concluded that 26 can be represented as the sum of squares of at most 3 distinct non-zero natural numbers. Please determine the maximum number of distinct non-zero natural numbers whose squares can sum up to 360? | 360=1^{2}+2^{2}+3^{2}+4^{2}+6^{2}+7^{2}+8^{2}+9^{2}+10^{2} |
$1 \cdot 39$ Find all natural numbers $n, k$ such that $n^{n}$ has $k$ digits, and $k^{k}$ has $n$ digits. | n=k=1,n=k=8,n=k=9 |
Factorize:$(1)7x^{2}-63$;$(2)x^{3}+6x^{2}y+9xy^{2}$. | x(x+3y)^{2} |
3. Given that $\triangle A B C$ is an equilateral triangle, the ellipse $\Gamma$ has one focus at $A$, and the other focus $F$ lies on the line segment $B C$. If the ellipse $\Gamma$ passes exactly through points $B$ and $C$, then its eccentricity is | \frac{\sqrt{3}}{3} |
Subject 3. Let there be a parallelogram $A B C D$ and the points: $Q \in(A B \backslash[A B]$ such that $\frac{B Q}{A Q}=\frac{2}{3}, M \in\left(D A \backslash[A D]\right.$ such that $\frac{A M}{A D}=1$, and $\{P\}=D B \cap M Q$.
a. Show that $A P Q D$ is a trapezoid.
b. If the area of triangle $M D Q$ is $24 \mathrm... | 8\mathrm{~}^{2} |
Given the function $f(x)=a\ln x+ \dfrac {1}{2}x^{2}-2ax\ (a\in\mathbb{R})$.
$(1)$ Discuss the monotonicity of $f(x)$;
$(2)$ If $f(x)$ has two extreme points, $x_{1}$ and $x_{2}$, with $x_{1} < x_{2}$, prove that: $f(x_{2}) < - \dfrac {3}{2}$. | f(x_{2}) < - \dfrac {3}{2} |
Let $ABCD$ be a convex quadrilateral with positive integer side lengths, $\angle{A} = \angle{B} = 120^{\circ}, |AD - BC| = 42,$ and $CD = 98$. Find the maximum possible value of $AB$. | 69 |
In $\triangle ABC$, if $a\sin B\cos C+c\sin B\cos A= \frac{1}{2}b$, and $a > b$,
(1) Find the measure of $\angle B$;
(2) If $b= \sqrt {13},a+c=4$, find the area of $\triangle ABC$. | \frac{6-3 \sqrt {3}}{4} |
## Task 5 - V00905
In an electrical circuit, a voltage of 120 V is applied. If the resistance is increased by 10 Ohms, the current decreases by 1 Ampere.
What are the current and resistance? | 4,30 |
Given an arithmetic sequence {an} where the sum of the first n terms is S, the common ratio q > 1, and a2 + 1 is the arithmetic mean of a1 and a3, and S3 = 14.
(I) Find the general term formula for the sequence {an}.
(II) Let bn = an β’ log2an, find the sum of the first n terms Tn for the sequence {bn}. | T_n = (n - 1) \cdot 2^{n+1} + 2 |
10 Let $A, B$ be the two foci of a hyperbola, and $C$ a point on the hyperbola. It is known that the three sides of $\triangle A B C$ form an arithmetic sequence, and $\angle A C B=120^{\circ}$. Then the eccentricity of the hyperbola is $\qquad$ | \frac{7}{2} |
If $\alpha$ and $\beta$ are acute angles, and $\alpha + \beta = \frac{\pi}{4}$, then $\left(1+\tan \alpha \right)\left(1+\tan \beta \right)=$____; $\left(1+\tan 1^{\circ}\right)\left(1+\tan 2^{\circ}\right)\left(1+\tan 3^{\circ}\right)\cdots \left(1+\tan 45^{\circ}\right)=$____. | 2^{23} |
Given $P=\{x|a-4 < x < a+4\}$, $Q=\{x|x^{2}-4x+3 < 0\}$, and $x \in P$ is a necessary condition for $x \in Q$, find the range of the real number $a$. | -1 \leqslant a \leqslant 5 |
Given the ellipse $\dfrac {x^{2}}{25}+ \dfrac {y^{2}}{m^{2}}=1(m > 0)$ and the hyperbola $\dfrac {x^{2}}{7}- \dfrac {y^{2}}{n^{2}}=1(n > 0)$ have the same foci, the maximum value of $m+n$ is (γγ).
A: $3$
B: $6$
C: $18$
D: $36$ | B |
A line $l$ passes through point $A(-2,2)$ and has the same y-intercept as the line $y = x + 6$. Find the general form equation of line $l$. | 2x - y + 6 = 0 |
Example 5. Find $\int e^{x} \sin x d x$. | \frac{e^{x}}{2}(\sinx-\cosx)+C |
Given the complex number $z$ that satisfies the equation $z + 2\overline{z} = 3 - i$ (where $i$ is the imaginary unit), find the value of $z$. | z = 1 + i |
Calculate:$(1)y(x+y)+(x+y)(x-y)$;$(2)\left(\frac{2m+1}{m+1}+m-1\right)Γ·\frac{m+2}{m^2+2m+1}$. | m^2 + m |
Given two circles \\(C_{1}\\): \\(x^{2} + y^{2} = r^{2}\\) and \\(C_{2}\\): \\((x - a)^{2} +(y - b)^{2} = r^{2} (r > 0)\\) intersect at two distinct points \\(A(x_{1},y_{1})\\), \\(B(x_{2},y_{2})\\), the following conclusions are given: \\(β a(x_{1} - x_{2})+ b(y_{1} - y_{2})= 0\\); \\(β‘2ax_{1} + 2by_{1} = a^{2} + b^{2}... | \text{D} |
Given that one focus of the ellipse C: $$\frac {x^{2}}{a^{2}}+ \frac {y^{2}}{b^{2}}$$\=1 (a>b>0) coincides with the focus of the parabola y2\=4 $$\sqrt {3}x$$, and the eccentricity is $$\frac { \sqrt {3}}{2}$$.
1. Find the standard equation of the ellipse C.
2. The line l, which does not pass through the origin, inter... | 5 |
8 Let $M=\{1,2, \cdots, 9\}$, and take the product of every two distinct numbers in it. Then the sum of all these products is $\qquad$ . | 870 |
If the graph of $f(x)$ is symmetric about the origin, it is an increasing function on the interval $[3, 7]$ with a maximum value of $5$. Then, $f(x)$ on the interval $[-7, -3]$ is $\underline{\hspace{1cm}}$.
A: an increasing function with a minimum value of $-5$
B: an increasing function with a maximum value of $-5$
C:... | A |
Two walls and the ceiling of a room meet at right angles at point $P.$ A fly is in the air one meter from one wall, eight meters from the other wall, and nine meters from point $P$. How many meters is the fly from the ceiling?
$\text{(A) }\sqrt{13} \qquad \text{(B) }\sqrt{14} \qquad \text{(C) }\sqrt{15} \qquad \text{(D... | 4 |
Given sequences $\{a_{n}\}$ and $\{b_{n}\}$, where $a_{1}=100$, $a_{n+1}=a_n^2$, and $b_{n}=\lg a_{n}$.
$(1)$ Find the general formula for sequence $\{b_{n}\}$.
$(2)$ If $c_{n}=\log _{2}b_{n}+\log _{2}b_{n+1}+\cdots +\log _{2}b_{2n}$, find the sum $S_{n}$ of the first $n$ terms of the sequence $\left\{\frac{1}{c_n}... | S_n=\frac{2n}{3(n+1)} |
13. The number 144 is very special in that it enjoys the mirror property: not only is it true that $144=12^{2}$ (i.e., it is a perfect square), but it is also true that $441=21^{2}$, meaning that reversing the order of its digits results in precisely the square of the number obtained by reversing the digits of its squa... | 12 |
$a=-1$ is a sufficient but not necessary condition for the line $4x-(a+1)y+9=0$ to be perpendicular to the line $(a^2-1)x-ay+6=0$ (γγ)
A: Sufficient but not necessary condition
B: Necessary but not sufficient condition
C: Sufficient and necessary condition
D: Neither sufficient nor necessary condition | \text{A} |
7. The line $l: x+y=t$ intersects the circle $\odot O: x^{2}+y^{2}=20$ at points $A$ and $B$, and $S_{\triangle O A B}$ is an integer. Then the number of all positive integer values of $t$ that satisfy the condition is $\qquad$ . | 2 |
There are $n(n \geqslant 2)$ people each writing a greeting card. First, the cards are collected, and then each person takes one card given by someone else. How many different ways can these $n$ people take the cards? | a_{n}=(n!)\sum_{k=2}^{n}\frac{(-1)^{k}}{k!} |
The distance between the points representing $-5$ and $-1$ on the number line is ( ).
A: $6$
B: $5$
C: $4$
D: $3$ | C |
Problem 11.4. In a sports school, 55 people are training, each of whom is either a tennis player or a chess player. It is known that there are no four chess players who would have the same number of friends among the tennis players. What is the maximum number of chess players that can train in this school? | 42 |
In the geometric sequence ${a_{n}}$, if $a_{1}a_{2}a_{3}=-8$, then $a_{2}$ is equal to $($ $)$.
A: $-\frac{8}{3}$
B: $-2$
C: $\pm \frac{8}{3}$
D: $\pm 2$ | \text{B} |
Someone puts 5000 francs into a savings account, how much money will they have at the beginning of the 20th year, if starting from the third year, they add 350 francs at the beginning of every second year? $p=4$. | 15107 |
Given that the sequences $\{a_{n}\}$ and $\{b_{n}\}$ satisfy ${b_{n+1}}{a_n}+{b_n}{a_{n+1}}={({-2})^n}+1,{b_n}=\frac{{3+{{({-1})}^{n-1}}}}{2}({nβ{N^*}})$, and $a_{1}=2$, find $a_{2n}$. | \dfrac{1 - {4}^{n}}{2} |
If the real part and the imaginary part of the complex number $z=a+i$ are equal, then the real number $a=$ οΌγγοΌ
A: -1
B: 1
C: -2
D: 2 | B |
2. (10 points) It is known that the distance between $A$ and $B$ is 300 meters. Two people, Jia and Yi, start from $A$ and $B$ respectively at the same time, walking towards each other, and meet at a point 140 meters from $A$; if Yi walks 1 meter more per second, then the meeting point is 180 meters from $B$. What was ... | 3\frac{1}{5} |
Express the set using the listing method: $M=\{m| \frac {10}{m+1} \in \mathbb{Z}, m \in \mathbb{Z}\}=$ \_\_\_\_\_\_ . | \{-11, -6, -3, -2, 0, 1, 4, 9\} |
Balkan Olympiads 2009
Find all functions $f: \mathbb{N}^{*} \rightarrow \mathbb{N}^{*}$ such that for all positive integers $m$ and $n$,
$$
f\left(f^{2}(m)+2 f^{2}(n)\right)=m^{2}+2 n^{2}
$$ | f(n)=n |
\section*{Task 1 - 211011}
A traveler covered the first part of a business trip by car and the rest by train.
When he had driven the car part of the journey and exactly one fifth of the train part, he realized that at that point he had covered exactly one third of the total distance. Later, when he had covered exactl... | 120\mathrm{~} |
If the monomials $6x^{m}y^{2}$ and $3x^{3}y^{n}$ are like terms, then $m+n=\_\_\_\_\_\_$. | 5 |
Let the universal set be $U = \mathbb{R}$, set $A=\{x | -2 \lt x \lt 5\}$, $B=\{x | 2-a \lt x \lt 1+2a\}$.
$(1)$ If $a=3$, find $A\cup B$ and $A\cap B$;
$(2)$ If $x\in A$ is a necessary condition for $x\in B$, find the range of real number $a$. | (-\infty, 2] |
In the right-angled triangle $ABC$, the lengths of the legs $AC=3$ and $BC=4$. Point $A$ was moved to point $A_{1}$ parallel to $BC$, then point $B$ was moved to point $B_{1}$ parallel to $A_{1}C$, and finally, point $C$ was moved to point $C_{1}$ parallel to $A_{1}B_{1}$, such that the resulting triangle $A_{1}B_{1}C_... | 12 |
7. Given the sequence $\left\{a_{n}\right\}$ satisfies
$$
a_{1}=1, a_{n}+a_{n+1}=-n^{2} \text {. }
$$
then $a_{15}=$ $\qquad$ | -104 |
7.4 On the table, there are 2020 boxes, some of which contain candies, while the others are empty. On the first box, it is written: "All boxes are empty." On the second: "At least 2019 boxes are empty." On the third: "At least 2018 boxes are empty," and so on, up to the 2020th, which says: "At least one box is empty." ... | 1010 |
Let $i$ be the imaginary unit, and $\overline{z}$ represent the conjugate of the complex number $z$. If $\overline{z}= \frac {4-5i}{i}$, then $(3-i)z=$γγ()
A: $11+17i$
B: $11-17i$
C: $-11+17i$
D: $-11-17i$ | C |
4.91 Solve the equation $\cos ^{n} x-\sin ^{n} x=1$. Here $n$ is any given natural number. | {\begin{pmatrix}k\pi&(n\text{iseven}),\\2k\pi\text{or}2k\pi-\frac{\pi}{2}&(n\text{isodd}),\end{pmatrix}.} |
Given that $y=f(x)$ is a differentiable function on $\mathbb{R}$, when $x\neq 0$, $f''(x)+ \frac{f(x)}{x} > 0$, the number of zeros of the function $g(x)=f(x)+ \frac{1}{x}$ with respect to $x$ is
A: $0$ or $2$
B: $2$
C: $1$
D: $0$ | D |
In the geometric sequence $\\{a\_n\\}$, $a\_1 + a\_2 = 40$, $a\_3 + a\_4 = 60$, find $a\_7 + a\_8 = ( \ \ )$
A: $80$
B: $90$
C: $100$
D: $135$ | D |
In the Cartesian coordinate system $xOy$, establish a polar coordinate system with $O$ as the pole and the positive semi-axis of the $x$ axis as the polar axis. The polar equation of line $l$ is $\rho(\cos\theta + \sin\theta) = 1$. The circle has the following parametric equations:
$$
\begin{cases}
x=1+r\cos\theta \\
... | r = \frac{\sqrt{2}}{2} |
3. Find all mappings $f: \mathbf{Z} \rightarrow \mathbf{Z}$, such that for any $m, n \in \mathbf{Z}$, we have
$$
f(f(m+n))=f(m)+f(n) .
$$ | f(n)=n+(\in{Z})orf(n)=0 |
Let the sequence $\{a_n\}$ satisfy $a_1+3a_2+\ldots+(2n-1)a_n=2n$.
(1) Find the general formula for $\{a_n\}$.
(2) Calculate the sum of the first $n$ terms of the sequence $\left\{ \frac{a_n}{2n+1} \right\}$. | \frac{2n}{2n+1} |
1.7 The expression $\frac{2 x^{2}-x}{(x+1)(x-2)}-\frac{4+x}{(x+1)(x-2)}$ cannot be evaluated for $x=-1$ or $x=$ 2, because zero cannot be a divisor. For other values of $x$,
(A) this expression can have many different values.
(B) this expression has only the value 2.
(C) this expression has only the value 1.
(D) this e... | B |
Let $f(x) = x^2 - m \ln(2x + 1)$, where $x \in (- \frac{1}{2}, 1]$, and $m > 0$.
(I) Find the range of values of the real number $m$ such that the function $f(x)$ is decreasing on the interval $(- \frac{1}{2}, 1]$;
(II) Determine whether the function $f(x)$ has a minimum value. If it exists, find the value of $x$ when ... | 1 |
41. For a real number $x$, the symbol $[\mathrm{x}]$ represents the greatest integer not greater than $x$, for example $[0.3]=0,[\pi]=3,[-2.5]=-3$. If $\left[\frac{x+4}{10}\right]=5$, then the maximum value of $\left[\frac{6 x}{5}\right]$ is . $\qquad$ | 67 |
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