problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
|---|---|
2. Call a natural number $n$ faithful, if there exist natural numbers $a<b<c$ such that $a$ divides $b, b$ divides $c$ and $n=a+b+c$.
(i) Show that all but a finite number of natural numbers are faithful.
(ii) Find the sum of all natural numbers which are not faithful.
| 65 |
12. As shown in Figure 2, given a circle $\odot O$ with radius $R$ and a line $l$ outside $\odot O$, the projection of $O$ on $l$ is $H$, $O H=d$, and $O H$ intersects $\odot O$ at points $C$ and $D$, with $C H > D H$. Let $P$ and $Q$ be points on $l$, both on the same side of $H$, and $P H = a$, $Q H = b (a > b)$. In ... | 2 a^{2}+2 b^{2}+4 d^{2}+4 R^{2} |
9.3. Solve the equation:
$1+\frac{3}{x+3}\left(1+\frac{2}{x+2}\left(1+\frac{1}{x+1}\right)\right)=x$.
# | 2 |
Given the sets $A=\{x\left|\right.1 \lt x-1\leqslant 3\}$ and $B=\{2,3,4\}$, then $A\cap B=\left(\ \ \right)$
A: $\{2,3,4\}$
B: $\{3,4\}$
C: $\{2,4\}$
D: $\{2,3\}$ | B |
A2 Find the maximum value of $z+x$, if $(x, y, z, t)$ satisfies the conditions:
$$
\left\{\begin{array}{l}
x^{2}+y^{2}=4 \\
z^{2}+t^{2}=9 \\
x t+y z \geq 6
\end{array}\right.
$$
| \sqrt{13} |
Find the largest integer $n$ such that $C_n^0 + C_n^1 + 2C_n^2 + 3C_n^3 + \ldots + nC_n^n < 500$. | n=7 |
7. As shown in Figure 2, in trapezoid
$A B C D$, $A B / / D C$,
$D C=2 A B=2 A D$. If
$B D=6, B C=4$, then
$S_{A B C D}=\ldots \quad\left(S_{A B C D}\right.$ represents
the area of quadrilateral $A B C D$, the same below. | 18 |
5. (3 points) Three circles with radii 1, 1, and \(2 \sqrt{\frac{13-6 \sqrt{3}}{13}}\) are arranged such that the triangle formed by their centers is equilateral with a side length of \(\sqrt{3}\). Find the radius of the circumcircle of the triangle, each vertex of which is the point of intersection of two of these cir... | 4\sqrt{3}-6 |
Let the real number $a \in (0,1)$. Then, the probability that the function $f(x)=x^{2}-(2a+1)x+a^{2}+1$ has a root is $(\quad)$
A: $\dfrac {3}{4}$
B: $\dfrac {2}{3}$
C: $\dfrac {1}{3}$
D: $\dfrac {1}{4}$ | D |
If the solution set of the inequality $ax+b\leqslant 0$ with respect to $x$ is $\{x\left|\right.x\geqslant -1\}$, then the solution set of the inequality $\frac{ax+b}{x-2}＞0$ with respect to $x$ is ( ).
A: $\{x\left|\right.x \lt -1$ or $x \gt 2\}$
B: $\{x\left|\right.-2 \lt x \lt 1\}$
C: $\{x\left|\right.x \lt -2$ o... | D |
2. Calculate $\left(\frac{1}{30}+\frac{1}{35}+\frac{1}{63}\right) \times 2 \frac{1}{7}=$ ? | \frac{1}{6} |
10. (1 mark) How many positive integers less than 500 have exactly 15 positive integer factors?
(1 分) 小於 500 的正整數中有多少個剛好有 15 個正因子? | 3 |
21. Let $P$ be a 30 -sided polygon inscribed in a circle. Find the number of triangles whose vertices are the vertices of $P$ such that any two vertices of each triangle are separated by at least three other vertices of $P$. | 1900 |
In the diagram, each of the five squares is $1 \times 1$. What percentage of the total area of the five squares is shaded?
(A) $25 \%$
(B) $30 \%$
(C) $35 \%$
(D) $40 \%$
(E) $45 \%$
 | 40 |
$3 \cdot 42$ A certain project, the number of days required for Team A to complete it alone is $a$ times the number of days required for Teams B and C to complete it together; the number of days required for Team B to complete it alone is $b$ times the number of days required for Teams A and C to complete it together; ... | 1 |
The monotonic increasing interval of the function $f(x) = \lg(x^2 - 4x + 3)$ is ( )
A: $(-\infty, 1)$
B: $(-\infty, 2)$
C: $(3, +\infty)$
D: $(2, +\infty)$ | C |
Let $C$ and $D$ be points on the semicircle with center $O$ and diameter $AB$ such that $ABCD$ is a convex quadrilateral. Let $Q$ be the intersection of the diagonals $[AC]$ and $[BD]$, and $P$ be the intersection of the lines tangent to the semicircle at $C$ and $D$. If $m(\widehat{AQB})=2m(\widehat{COD})$ and $|AB|=2... | \frac{2\sqrt{3}}{3} |
4. The great-grandfather has four children, each of his children has four children, and each of those children has four children. How many descendants does the great-grandfather have in total? | 84 |
Determine all the numbers formed by three different and non-zero digits, such that the six numbers obtained by
permuting these digits leaves the same remainder after the division by $4$. | 159 |
Given $0 \lt a \lt 1 \lt b$, if the function $f\left(x\right)=a^{x}+b^{x}$ is monotonically increasing on $\left(0,+\infty \right)$, then the minimum value of $ab$ is ( )
A: $\frac{1}{2}$
B: $3$
C: $2$
D: $1$ | D |
If the inequality $x^2 + |2x - 6| \geq a$ holds for all real numbers $x$, then the maximum value of the real number $a$ is. | 5 |
The set $\{x \mid \cos(\pi\cos x) = 0, x \in [0, \pi]\}$ is equal to __________ (express using enumeration method). | \left\{ \frac{\pi}{3}, \frac{2\pi}{3} \right\} |
Which of the following statements about the inverse proportion function $y=\frac{6}{x}$ is incorrect?
A: The graph passes through the point $(2,3)$.
B: The graph is located in the second and fourth quadrants.
C: The graph is symmetric about the origin.
D: The graph does not intersect the coordinate axes. | \text{B} |
Given vectors $\overrightarrow{a}=(1,-2)$ and $\overrightarrow{b}=(\lambda,1)$, let the angle between vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ be $\theta$, then ()
A: When $\lambda > 2$, $\theta$ is an acute angle
B: When $\lambda < 2$, $\theta$ is an obtuse angle
C: When $\lambda = 2$, $\theta$ is a ri... | ACD |
Calculate $119^2$ using a simplified method. | 14161 |
What is the maximum number of different 3-term arithmetic sequences that can be selected from a sequence of real numbers $a_{1}<a_{2}<\cdots<a_{n}$? | [\frac{(n-1)^{2}}{4}] |
9 If for any real number $x$, there is $f(x)=\log _{a}\left(2+\mathrm{e}^{x-1}\right) \leqslant-1$, then the range of real number $a$ is $\qquad$ . (where $e$ is an irrational number, $e=2.71828 \cdots$ ) | [\frac{1}{2},1) |
Simplify first, then evaluate: $(1-\frac{a}{a-3})\div \frac{{a}^{2}+3a}{{a}^{2}-9}$. Given $a=-6$. | \frac{1}{2} |
Given a set of data in ascending order: $0$, $4$, $x$, $10$, and the median is $5$, then $x$ is equal to:
A: $5$
B: $6$
C: $7$
D: $8$ | B |
Which of the following operations is correct?
A: $\sqrt{2}×\sqrt{3}=\sqrt{6}$
B: ${(2\sqrt{3})^2}=6$
C: $\sqrt{2}+\sqrt{3}=\sqrt{5}$
D: $\sqrt{{{(-2)}^2}}=-2$ | A |
Example 5 In space, there are 2019 points, where no four points are coplanar. Divide them into 30 groups with different numbers of points. For any three different groups, take one point from each as the vertex to form a triangle. How should the points be divided to maximize the number of such triangles?
(Adapted from ... | 52,53,\cdots,57,59,60,\cdots,82 |
A certain enterprise in our city donated tents of type A and type B, totaling 300 tents worth 260,000 yuan, to the Yushu earthquake disaster area. It is known that each tent of type A costs 800 yuan, and each tent of type B costs 1000 yuan. How many tents of type A and type B were donated, respectively? | 100 |
1. Solve the equation
$$
(\sqrt{3-\sqrt{8}})^{x}+(\sqrt{3+\sqrt{8}})^{x}=6
$$ | x_1=2,x_2=-2 |
2. You are in a completely dark room with a drawer containing 10 red, 20 blue, 30 green, and 40 khaki socks. What is the smallest number of socks you must randomly pull out in order to be sure of having at least one of each color? | 91 |
1. Do there exist ten consecutive natural numbers that are successively divisible by the numbers $9,7,5,3,1,1,3,5,7,9$?
(Jaroslav Zhouf) | 153 |
Given that $\alpha$ and $\beta$ are both acute angles, and $\cos(\alpha+\beta)=\sin(\alpha-\beta)$, then $\tan \alpha=$ ? | 1 |
Given the function $f(x) = mx^2 - mx - 1$. If for all $x \in \mathbb{R}$, $f(x) < 0$ always holds, then the range of the real number $m$ is. | -4 < m \leq 0 |
5. A five-digit number with all distinct digits, if the sum of any two of its digits is not 7, is called a “hidden seven number”. There are $\qquad$ such “hidden seven numbers”.
| 7008 |
Given the function $f(x) = |x - 1| + |x - 2|$.
(I) Find the solution set of the inequality $f(x) < 2$ with respect to $x$;
(II) If the solution set of the inequality $f(x) < a$ about $x$ is non-empty, find the range of values for the real number $a$. | (1, +\infty) |
Given a line $l$ passes through the point $P(2, 1)$ and is parallel to the line $2x - y + 2 = 0$, then the equation of line $l$ is ( )
A: $2x - y - 3 = 0$
B: $x + 2y - 4 = 0$
C: $2x - y - 4 = 0$
D: $x - 2y - 4 = 0$ | \text{A} |
Solve the following system of equations on the set of real numbers:
$$
\begin{aligned}
4 a+b c & =32 \\
2 a-2 c-b^{2} & =0 \\
a+12 b-c-a b & =6
\end{aligned}
$$ | \begin{pmatrix}a_{1}=11.5,&b_{1}=-4,&c_{1}=3.5\\a_{2}=6.5,&b_{2}=1,&c_{2}=6\\a_{3}=14,&b_{3}=6,&c_{3}=-4\\a_{4}=14,&b_{4}=-2,&c_{4} |
Given $A(-1,2)$, $B(-2,4)$, then the slope of line $AB$ is ( )
A: $-2$
B: $- \dfrac {1}{2}$
C: $ \dfrac {1}{2}$
D: $2$ | A |
5. Find the maximum value of the expression $(\sin 3 x+\sin 2 y+\sin z)(\cos 3 x+\cos 2 y+\cos z)$. (15 points) | 4.5 |
A certain physical quantity follows a normal distribution $N(10, \sigma ^{2})$. Which of the following conclusions is incorrect?
A: The smaller the $\sigma$, the greater the probability that the physical quantity falls within $(9.9, 10.1)$ in one measurement.
B: The probability that the physical quantity is greater t... | \text{D} |
Given real numbers $a, b \neq 0$, let
$$
x=\frac{a}{|a|}+\frac{b}{|b|}+\frac{a b}{|a b|} \text {. }
$$
Then the sum of the maximum and minimum values of $x$ is $\qquad$ [1] | 2 |
Example 2. Given the circle $x^{2}+y^{2}=4$ and a point $A(2, 0)$ on it, the moving chord $BC$ of this circle always satisfies the condition $\angle BAC = \frac{\pi}{3}$. Try to find the equation of the locus of the centroid of $\triangle ABC$. | \left(x-\frac{2}{3}\right)^{2}+y^{2}=\left(\frac{2}{3}\right)^{2} |
6. Determine all triples $(a, b, c)$ of positive integers such that $a \leq b \leq c$ and
$$
a+b+c+a b+b c+c a=a b c+1
$$
| (2,4,13),(2,5,8),(3,3,7) |
1. (5 points) Calculate: $(3 \div 2) \times(4 \div 3) \times(5 \div 4) \times \cdots \times(2012 \div 2011) \times(2013 \div 2012)$ $=$ . $\qquad$ | \frac{2013}{2} |
Example $7 m(m \geqslant 3)$ people $A_{1}, A_{2}, \cdots, A_{m}$ pass the ball to each other. Starting with $A_{1}$, this is called the 1st pass. After $n(n \geqslant 2)$ passes, the ball returns to $A_{1}$. Try to find the number of all different ways of passing the ball $N$. | c_{n}=\frac{1}{m}\left[(m-1)^{n}+(m-1)(-1)^{n}\right] |
The sum of two factors of the polynomial $x^{5n} + x^n + 1$ when $n = 1$ and $x = 2$ is. | 12 |
Given vectors $\overrightarrow {a}=(\cos\theta，1，\sin\theta)$ and $\overrightarrow {b}=(\sin\theta，1，\cos\theta)$, find the angle between vectors $\overrightarrow {a}+ \overrightarrow {b}$ and $\overrightarrow {a}- \overrightarrow {b}$.
Options:
A: 90°
B: 60°
C: 30°
D: 0° | A |
Each of the letters $\text{W}$, $\text{X}$, $\text{Y}$, and $\text{Z}$ represents a different integer in the set $\{ 1,2,3,4\}$, but not necessarily in that order. If $\dfrac{\text{W}}{\text{X}} - \dfrac{\text{Y}}{\text{Z}}=1$, then the sum of $\text{W}$ and $\text{Y}$ is
$\text{(A)}\ 3 \qquad \text{(B)}\ 4 \qquad \te... | 7 |
What is the value of $\lg 8 + 3\lg 5$?
A: $-3$
B: $-1$
C: $1$
D: $3$ | \text{D} |
10. Xiao Wang and Xiao Li both brought some money to buy the book "Harry - Potter". When they arrived at the bookstore, they found that the money Xiao Wang brought was 6 yuan short of buying 2 copies, and the money Xiao Li brought was 31 yuan short of buying 2 copies. However, the money the two brought together was jus... | 37 |
We know that the following equation has two pairs of roots whose sum is 0. Solve the equation.
$$
x^{6}-2 x^{5}-9 x^{4}+14 x^{3}+24 x^{2}-20 x-20=0
$$ | \sqrt{2},-\sqrt{2},\sqrt{5},-\sqrt{5},1+\sqrt{3},1-\sqrt{3} |
Given the curves $C\_1$ and $C\_2$, where $C\_1$: $\begin{cases} x = -4 + \cos t \\ y = 3 + \sin t \end{cases}$ ($t$ is a parameter) and $C\_2$: $\begin{cases} x = 6\cos \theta \\ y = 2\sin \theta \end{cases}$ ($\theta$ is a parameter).
1. Find the equations of $C\_1$ and $C\_2$ in the standard form and explain what t... | [1 + \sqrt{2}, +\infty) |
A-6. Take any four points on a sphere to form a tetrahedron. What is the probability that the center of the sphere is inside the tetrahedron? (It can be assumed that each point is independently chosen and is uniformly distributed on the sphere) | \frac{1}{8} |
On December 1, 2020, at 10:57 p.m., the Chang'e-5 lunar probe began its powered descent from a distance of 1800 meters above the lunar surface. The 7500 Newton thrust engine gradually reduced the probe's relative longitudinal speed from about 1800 m/s to zero. Twelve minutes later, the probe successfully landed at the ... | C |
5. In the square in Figure 1, 10000 points are randomly thrown. The estimated number of points that fall into the shaded area (the equation of curve $C$ is $x^{2}-y=0$) is ( ).
(A) 5000
(B) 6667
(C) 7500
(D) 7854 | 6667 |
Given the function $f(x)$ satisfies $f(x+3)=f(x)$, when $x \in [-1,2]$, $f(x)=x+1$, then $f(2017)=$ ( )
A: $1$
B: $-1$
C: $2$
D: $2017$ | C |
I4.1 Given that $x$ and $y$ are real numbers such that $|x|+x+y=10$ and $|y|+x-y=10$. If $P=x+y$, find the value of $P$. | 4 |
Given points A(2,1), B(-2,3), and C(0,1), find the length of the median on side BC of triangle ABC. | \sqrt{10} |
3. The sum of the digits of a three-digit number is 18. The first digit is equal to one-eighth of the number formed by the other two digits, and the last digit is equal to one-eighth of the number formed by the first two digits. Calculate this number.
(8 points) | 648 |
Example 3. Simplify: $\sqrt{(a-5)^{2}}$ | |a-5| |
The coordinates of the point $P(-3,4)$ with respect to the $y$-axis are ( )
A: $(-3,-4)$
B: $(3,-4)$
C: $(3,4)$
D: $(-4,3)$ | C |
Given $A = \sqrt{2} - 1$, $B = \sqrt{3} - \sqrt{2}$, $C = \sqrt{4} - \sqrt{3}$:
(I) Compare the sizes of $A$ and $B$, as well as $B$ and $C$ separately (only state the results, no need to provide the proof);
(II) Based on the results from (I), conjecture the relationship between $\sqrt{k} - \sqrt{k-1}$ and $\sqrt{k+1} ... | \sqrt{k} - \sqrt{k-1} > \sqrt{k+1} - \sqrt{k} |
Example 1 In an $8 \times 8$ chessboard, how many ways are there to select 56 squares such that: all the black squares are selected, and each row and each column has exactly seven squares selected? ${ }^{[1]}$
(2014, Irish Mathematical Olympiad) | 576 |
In the Cartesian coordinate system, point $P(1,2)$, point $Q$ is below the $x$-axis, and $PQ$ is parallel to the $y$-axis. If $PQ=5$, then the coordinates of point $Q$ are ( ).
A: $(-4,2)$
B: $(6,2)$
C: $(1,-3)$
D: $(1,7)$ | C |
Example 25 Let $x, y, z \in(0,1)$, satisfying:
$$\sqrt{\frac{1-x}{y z}}+\sqrt{\frac{1-y}{z x}}+\sqrt{\frac{1-z}{x y}}=2,$$
Find the maximum value of $x y z$. | \frac{27}{64} |
Yasmine makes her own chocolate beverage by mixing volumes of milk and syrup in the ratio $5: 2$. Milk comes in $2 \mathrm{~L}$ bottles and syrup comes in $1.4 \mathrm{~L}$ bottles. Yasmine has a limitless supply of full bottles of milk and of syrup. Determine the smallest volume of chocolate beverage that Yasmine can ... | 19.6\mathrm{~L} |
Which of the following options is correct?
A: $M(a_{1}$, $b_{1})$, $N(a_{2}$, $b_{2})$ are two points on the inverse proportion function $y=\frac{\sqrt{3}}{x}$, and $a_{1} \lt a_{2}$, then $b_{1} \gt b_{2}$.
B: Arcs of equal length are equal arcs.
C: Arcs corresponding to equal central angles in equal circles are eq... | D |
The following procedure provides an approximate value (segment) for the side of a regular 19-sided polygon inscribed in a circle. The midpoint of the ray inclined at $60^{\circ}$ to the radius $O A$ is $B$, its projection on $O A$ is $C$, the projection of $C$ on $A B$ is $D$, and the projection of $D$ on $O A$ is $E$.... | 457' |
\section*{Problem 1 - 241041}
Determine all ordered pairs \((x ; y)\) of integers for which \(\sqrt{x}+\sqrt{y}=\sqrt{1985}\) holds! | (0,1985)(1985,0) |
23. Given that $x$, $y$, and $z$ are positive integers less than 10, and satisfy $(100 x+10 y+z)^{2}=(x+y+z)^{5}$. Then the value of $x^{2}+y^{2}+z^{2}$ is ( ).
(A) 21
(B) 23
(C) 29
(D) 33
(E) 37 | C |
Calculate: $\tan(18^\circ-x)\tan(12^\circ+x)+ \sqrt{3}[\tan(18^\circ-x)+\tan(12^\circ+x)]$. | 1 |
There are 6 people standing in a line. How many different arrangements are there such that persons A and B are not next to each other? (Answer with a number) | 480 |
Given the function $f(x) = e^{x} - \frac{1}{e^{x}} + x$, where $e$ is the base of the natural logarithm, if the real number $a$ satisfies $f(\log_{2}a) - f(\log_{0.5}a) \leqslant 2f(1)$, determine the range of values for $a$.
A: $(-\infty, \frac{1}{2}) \cup (2, +\infty)$
B: $(0, \frac{1}{2}] \cup [2, +\infty)$
C: $[\f... | D |
7.5. Zya decided to buy a crumblik. In the store, they also sold kryambliks. Zya bought a kryamblik and received coupons worth $50\%$ of the cost of the purchased kryamblik. With these coupons, he was able to pay $20\%$ of the cost of the crumblik. After paying the remaining amount, he bought the crumblik as well. By w... | 20 |
Let $a$, $b$, and $c$ be pairwise distinct positive integers, which are side lengths of a triangle. There is a line which cuts both the area and the perimeter of the triangle into two equal parts. This line cuts the longest side of the triangle into two parts with ratio $2:1$. Determine $a$, $b$, and $c$ for which the ... | 504 |
16. Every sixth bus in the bus park of the aluminum plant is equipped with an air conditioner. After the plant director ordered to install it on 5 more buses, a quarter of the buses had an air conditioner. How many buses are there in the park of the plant, if each bus is equipped with only one air conditioner? | 60 |
Let $\{a_n\}$ be a sequence with $S_n$ as the sum of its first $n$ terms, and assume the equation $x^2 - a_n x - a_n = 0$ has one root equal to $S_n - 1$ for $n=1,2,3,\ldots$.
(1) Find $a_1$ and $a_2$;
(2) Conjecture the general term formula for the sequence $\{S_n\}$. | \frac{n}{n+1} |
Among the following propositions, the false proposition is ( ).
A: A quadrilateral with two pairs of opposite sides equal is a parallelogram
B: A quadrilateral with diagonals perpendicular to each other is a rhombus
C: A parallelogram with a right angle is a rectangle
D: A rhombus with equal diagonals is a square | B |
5. In an equilateral triangle $ABC$ with side length $a$, points $M, N, P, Q$ are positioned as shown in the figure. It is known that $MA + AN = PC + CQ = a$. Find the measure of angle $NOQ$.
 | 60 |
Among the following propositions:
(1) Complementary events are necessarily mutually exclusive events;
(2) If $A$ and $B$ are two random events, then $P(A \cup B) = P(A) + P(B)$;
(3) If events $A$, $B$, $C$ are mutually exclusive, then $P(A) + P(B) + P(C) = 1$;
(4) If events $A$ and $B$ satisfy $P(A) + P(... | A |
Given sets $A=\{x\left|\right.-1\leqslant x\leqslant 2\}$ and $B=\{x\left|\right.x \gt 0\}$, then $A\cup B=\left(\ \ \right)$
A: $\{x\left|\right.x\leqslant 2\}$
B: $\{x\left|\right.x\geqslant -1\}$
C: $\{x\left|\right.x \gt 1\}$
D: $\{x\left|\right.x \gt 0\}$ | B |
In triangle $K L M$, the ratio of the radii of the circumscribed and inscribed circles is 3. The inscribed circle touches the sides of triangle $K L M$ at points $A, B$, and $C$. Find the ratio of the area of triangle $K L M$ to the area of triangle $A B C$.
# | 6 |
The value range of the function $y=x^2-4x+3$, where $x \in [-1, 1]$, is ( )
A: $[-1, 0]$
B: $[0, 8]$
C: $[-1, 8]$
D: $[3, 8]$ | \text{B} |
Four, in isosceles $\triangle A B C$, the base $B C$ lies on the line $x+y=0$, the vertex $\mathrm{A}(2,3)$, and it is known that one of its equal sides $\mathrm{AB}$ is parallel to the line $\mathbf{x}-4 y-6=0$. Find the equation of the other equal side $\mathrm{AC}$. | 4 x-y-5=0 |
How many integers $n$ are there such that $\frac{n}{3}$ and $3 n$ are both integers between 1 and 1000 (inclusive)?
Only a numerical answer is expected here. | 111 |
1. Given $a b c \neq 0$, and $a+b+c=0$. Then the value of the algebraic expression $\frac{a^{2}}{b c}+\frac{b^{2}}{c a}+\frac{c^{2}}{a b}$ is ( ).
(A) 3
(B) 2
(C) 1
(D) 0 | 3 |
How many even whole numbers lie between $3^{2}$ and $3^{3}$ ?
(A) 9
(B) 4
(C) 6
(D) 10
(E) 17 | 9 |
Regarding the function $y=\tan (2x-\frac{π}{3})$, the correct statement is $(\quad\quad)$
A: The minimum positive period is $π$
B: It is an odd function
C: It is monotonically decreasing in the interval $(-\frac{1}{12}π, \frac{5}{12}π)$
D: $(\frac{5}{12}π,0)$ is one of its graph's symmetry centers | D |
Problem 4. In a right triangle $KLM$, a circle is constructed on the leg $KM$ as its diameter, which intersects the hypotenuse $KL$ at point $G$. A tangent to the circle is drawn through point $G$, intersecting the leg $ML$ at point $F$. Find the length of $FL$, if $KG=5$ and $LG=4$. | 3 |
Given $n\in\mathbb{N}^*$, the area of the circle $C_n: \left(x - \frac{1}{n}\right)^2 + (y - 1)^2 = \frac{4^{n+1}-1}{4^n+1}$ is denoted as $S_n$. Calculate $\lim_{n \to +\infty} S_n$. | 4\pi |
Insert three numbers \(a\), \(b\), and \(c\) between \(-1\) and \(7\) such that \(-1\), \(a\), \(b\), \(c\), \(7\) form an arithmetic sequence. Calculate the value of \(a + b + c\). | a + b + c = 9 |
30. Several families lived in one house. In total, there are more children in these families than adults: there are more adults than boys; more boys than girls; and more girls than families. There are no childless families, and no families have the same number of children. Each girl has at least one brother and at most... | 3 |
Example 8 In the quadratic equation $x^{2}+z_{1} x+z_{2}+m=0$ about $x$, $z_{1}, z_{2}, m$ are all complex numbers, and $z_{1}^{2}-$ $4 z_{2}=16+20 \mathrm{i}$. Let the two roots of this equation be $\alpha, \beta$, satisfying $|\alpha-\beta|=2 \sqrt{7}$, find the maximum and minimum values of $|m|$. | ||_{\text{max}}=\sqrt{41}+7,\||_{\text{}}=7-\sqrt{41} |
Example 1 Let non-negative real numbers $a, b, c$ satisfy $a+b+c=1$. Find the extremum of $S=\frac{1}{1+a}+\frac{1}{1+b}+\frac{1}{1+c}$. | \frac{9}{4} \leqslant S \leqslant \frac{5}{2} |
## Task A-1.3. (4 points)
If $a+b=4$ and $a^{2}+b^{2}=14$, determine $a^{3}+b^{3}$. | 52 |
3. The difference of two natural numbers is 5 times less than their sum and 24 times less than their product. Find these numbers. | 12,8 |
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