problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
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Malmer Pebane's apartment uses a six-digit access code, with leading zeros allowed. He noticed that his fingers leave that reveal which digits were pressed. He decided to change his access code to provide the largest number of possible combinations for a burglar to try when the digits are known. For each number of dis... | 5 |
Suppose the random variable $\xi$ follows a normal distribution $(2, \sigma^2)$, and $P(\xi \leqslant 0) = 0.2$, then $P(0 \leqslant \xi \leqslant 4) =$ ___. | 0.6 |
100 is greater than 80 by ( )
A: 20%
B: 25%
C: 80%
D: 60% | \text{B} |
Given a function $f(x)$ that satisfies $f(x)=f(\pi-x)$, and when $x \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$, $f(x)=e^x + \sin x$, then which of the following is correct?
A: $f(1) < f(2) < f(3)$
B: $f(2) < f(3) < f(1)$
C: $f(3) < f(2) < f(1)$
D: $f(3) < f(1) < f(2)$ | D |
Given the function $f(x)=\begin{cases}
(3-a)^x, & x\leq 2 \\
\log_a(x-1)+3, & x > 2
\end{cases}$ is a monotonically increasing function on $\mathbb{R}$, then the range of $a$ is
A: $(1, \sqrt{3})$
B: $(\sqrt{5}-1, \sqrt{3})$
C: $[3- \sqrt{3},2)$
D: $(1,3- \sqrt{3})$ | \text{C} |
Example 4. Solve the equation:
$$
\sqrt{x^{2}-5 x+1}+\sqrt{x^{2}-6 x+6}=2 \text{. }
$$ | x=5 |
2. The sum of all natural numbers $x$ that make the algebraic expression $y=\frac{x^{2}+11}{x+1}$ an integer is ( ).
(A) 5
(B) 6
(C) 12
(D) 22. | D |
Given the line $l_{1}:y=x+a$ intersects with the line $l_{2}:y=2(x+1)$ and the curve $C:y=x+\ln x$ at points $A$ and $B$ respectively, the minimum distance between points $A$ and $B$ is ( ).
A: $\dfrac{3\sqrt{5}}{5}$
B: $\dfrac{6\sqrt{5}}{5}$
C: $3$
D: $3\sqrt{2}$ | D |
5. A triangular paper piece with a side length of 1 is placed on a regular 12-sided paper piece with a side length of 1, such that one side of the triangle coincides with one side of the 12-sided polygon, and both are flat on the table. Then the number of sides of the resulting convex polygon must not be $(\quad)$.
(A)... | D |
26. Calculate the sum $1^{2}+2^{2}+\ldots+n^{2}$, | \frac{n(n+1)(2n+1)}{6} |
3. In the city of hedgehogs, there are 1000 identical buildings, each with 1000 floors, and exactly one hedgehog lives on each floor. Each resident in the city wears a hat colored in one of 1000 possible colors, and any two residents in the same building have hats of different colors. We say that two hedgehogs are $\ba... | 2\cdot500^{3} |
Determine the range of values for
$$
S=\frac{a}{a+b+d}+\frac{b}{a+b+c}+\frac{c}{b+c+d}+\frac{d}{a+c+d}
$$
where $a, b, c,$ and $d$ are arbitrary positive numbers. | 1<S<2 |
2. Let the edge length of the regular tetrahedron $ABCD$ be 1 meter. A small insect starts from point $A$ and moves according to the following rule: at each vertex, it randomly chooses one of the three edges connected to this vertex and crawls along this edge to another vertex. What is the probability that it returns t... | \frac{7}{27} |
If there is only $1$ complex solution to the equation $8x^3 + 12x^2 + kx + 1 = 0$, what is $k$? | 6 |
8. Given that the volume of tetrahedron $P-ABC$ is $1$, $G$ and $K$ are the centroids of $\triangle ABC$ and $\triangle PBC$ respectively. A line through $G$ intersects $AB$ and $AC$ at points $M$ and $N$. Then the maximum volume of the quadrilateral pyramid $K-MNCB$ is | \frac{5}{27} |
6. (3 points) Xiao Hu placed chess pieces on the grid points of a $19 \times 19$ Go board, first forming a solid rectangular array. Then, by adding 45 more chess pieces, he formed a larger solid rectangular array with one side unchanged. How many chess pieces did Xiao Hu use at most?
A. 285
B. 171
C. 95
D. 57 | 285 |
A convex quadrilateral's two opposite sides are each divided into 100 equal parts, and the points of the same ordinal number are connected. Among the 100 quadrilaterals thus obtained, the area of the first one is 1, and the area of the last, the 100th one, is 2 square units. What is the area of the original quadrilater... | 150 |
Two bored millionaires, Bilion and Trilion, decide to play a game. They each have a sufficient supply of $\$ 1, \$ 2,\$ 5$, and $\$ 10$ bills. Starting with Bilion, they take turns putting one of the bills they have into a pile. The game ends when the bills in the pile total exactly $\$1{,}000{,}000$, and whoever makes... | 1,000,000 |
6. When $s$ and $t$ take all real values, then
$$
(s+5-3|\cos t|)^{2}+(s-2|\sin t|)^{2}
$$
can achieve the minimum value of $\qquad$ | 2 |
A cube toy has its 6 faces marked with the numbers $1$, $2$, $2$, $3$, $3$, $3$. If the toy is thrown twice in succession, the probability that the sum of the numbers on the top faces is $5$ is ______. | \dfrac{1}{3} |
Given that $\overrightarrow {a}$ and $\overrightarrow {b}$ are both unit vectors with an angle of 120° between them, find the value of $| \overrightarrow {a} + 2 \overrightarrow {b} |$. | \sqrt{3} |
A beginning songwriter always sold CDs with his music after performances. On Thursday, he sold eight identical CDs. The next day, he also offered his new CD, so people could buy the same as on Thursday or the new one. On Saturday, all the listeners wanted the new CD, and the songwriter sold six of them that day. He ear... | 90 |
4. A natural number that reads the same from left to right as it does from right to left is called a "palindromic number". $a, b, c$ are all three-digit "palindromic numbers", and $b=2a+1, c=2b+1$. Then the sum of all possible values of $a$ is $\qquad$ | 372 |
Example 1 As shown in Figure 1, in square $A B C D$, $E$ is a fixed point on $B C$, and $B E=10, E C=14, P$ is a moving point on $B D$. Then the minimum value of $P E+P C$ is $\qquad$
(2006, Zhejiang Province Junior High School Mathematics Competition) | 26 |
In the Cartesian coordinate plane, $O$ is the origin, and two points $A(2,1), B(-1,-2)$ are given. If point $C$ satisfies $\overrightarrow{OC}=s\overrightarrow{OA}+t\overrightarrow{OB}$, and $s+t=1$, then the trajectory equation of point $C$ is ________. | x-y-1=0 |
$17 \cdot 140$ An isosceles triangle with base $\sqrt{2}$, the medians to the two equal sides are perpendicular to each other, then the area of this triangle is
(A) 1.5 .
(B) 2 .
(C) 2.5 .
(D) 3.5 .
(E) 4 .
(10th American High School Mathematics Examination, 1959) | \frac{3}{2} |
4. (7 points) Provide an example of an expression consisting of ones, parentheses, and the signs «+» and «X», such that
- its value is 11
- if in this expression all signs «+» are replaced with signs «X», and signs «X» with signs «+», the result is still 11. | 1+1+1+1+1+1+1+1+1+1+1\times1\times1\times1\times1\times1\times1\times1\times1\times1\times1 |
G4.3 The remainders of an integer when divided by $10,9,8, \ldots, 2$ are $9,8,7, \ldots, 1$ respectively. Find the smallest such an integer. | 2519 |
3. Let $0<x<\pi$, the minimum value of the function $y=\frac{2-\cos x}{\sin x}$ is ( ).
A. 3
B. $\sqrt{3}$
C. 2
D. $2-\sqrt{3}$ | \sqrt{3} |
Example 7 A certain consumer good is priced at 120 yuan per piece. Without the collection of additional taxes, approximately 800,000 pieces are sold annually. If the government levies an additional tax, collecting \( r \) yuan for every 100 yuan sold (called a tax rate of \( r \% \)), then the annual sales volume will ... | 4\leqslantr\leqslant8 |
In a bag, there are 6 white balls and 4 black balls of the same size and shape. Now, balls are randomly drawn from the bag with replacement 3 times, each time getting a white ball scores 0 points, and getting a black ball scores 5 points. Let $X$ be the total score obtained from the 3 draws, then ()
A: The probability... | BC |
Bakayev E.V.
Inside an isosceles triangle $ABC$, a point $K$ is marked such that $AB = BC = CK$ and $\angle KAC = 30^{\circ}$.
Find the angle $AKB$. | 150 |
Given that $(-3, y_{1})$, $(-15, y_{2})$, $(2, y_{3})$ lie on the inverse proportion function $y=-\frac{a^2}{x}$, the relationship between $y_{1}$, $y_{2}$, $y_{3}$ is ( ).
A: $y_{1} \gt y_{2} \gt y_{3}$
B: $y_{1} \gt y_{3} \gt y_{2}$
C: $y_{3} \gt y_{2} \gt y_{1}$
D: $y_{3} \gt y_{1} \gt y_{2}$ | A |
Let $n$ be a positive integer. A sequence $(a, b, c)$ of $a, b, c \in \{1, 2, . . . , 2n\}$ is called [i]joke [/i] if its shortest term is odd and if only that smallest term, or no term, is repeated. For example, the sequences $(4, 5, 3)$ and $(3, 8, 3)$ are jokes, but $(3, 2, 7)$ and $(3, 8, 8)$ are not. Determine the... | 4n^3 |
4. Solve the inequality
$$
\frac{x(x-2)(x-3)-(x-2)^{2}+1}{(|x-1|-|x-2|) \sqrt{16-x^{2}}} \geq 0
$$
points) | x\in(-4;\frac{3-\sqrt{5}}{2}]\cup(\frac{3}{2};\frac{3+\sqrt{5}}{2}]\cup[3;4) |
2. Let $a, b, c, x, y, z$ all be positive numbers, and $a^{2}+b^{2}+c^{2}=25, x^{2}+y^{2}+z^{2}=36, a x+b y+c z=$ 30 , then $\frac{a+b+c}{x+y+z}=$ $\qquad$ . | \frac{5}{6} |
Problem 9.6. In triangle $A B C$, the angles $\angle B=30^{\circ}$ and $\angle A=90^{\circ}$ are known. On side $A C$, point $K$ is marked, and on side $B C$, points $L$ and $M$ are marked such that $K L=K M$ (point $L$ lies on segment $B M$).
Find the length of segment $L M$, if it is known that $A K=4, B L=31, M C=3... | 14 |
281. Find the six-digit number $\overline{x y 342 z}$, divisible by 396. | 453420413424 |
4.21. The side, bisector, and height of a triangle, all emanating from the same vertex, are equal to 5, 5, and \(2 \sqrt{6}\) cm, respectively. Find the other two sides of the triangle. | 4\frac{8}{21} |
9. In the following diagram, $A B C D$ is a square with $P A=a, P B=2 a$ and $P C=3 a$. Find $\angle A P B$.
(A) $120^{\circ}$
(B) $130^{\circ}$
(C) $135^{\circ}$
(D) $140^{\circ}$
(E) $145^{0}$ | 135 |
2. For what natural numbers $x$ and $y$ is the value of the expression $4 x+\frac{289}{x}+16 y+\frac{529}{y}$ the smallest? In your answer, specify $x$ and $y$ in the correct order separated by a semicolon. | 9;6 |
6. Find all natural numbers $n$ such that the number $2^{n}+n^{2}+25$ is a cube of a prime number. | 6 |
Task 4. (20 points) For the numerical sequence $\left\{x_{n}\right\}$, all terms of which, starting from $n \geq 2$, are distinct, the relation $x_{n}=\frac{x_{n-1}+298 x_{n}+x_{n+1}}{300}$ holds. Find $\sqrt{\frac{x_{2023}-x_{2}}{2021} \cdot \frac{2022}{x_{2023}-x_{1}}}-2023$. | -2022 |
5. The function $f(t)$ with domain $D(f)=[1,+\infty)$ satisfies the equation $f\left(\frac{t^{y}-4^{-3}}{2}\right)=y$ for any $y \geq 0$. For each value of $a \neq 0$, find all solutions to the inequality $f\left(\frac{a}{x-2 a}\right) \leq 1$. Solution: The function $g(y)=\frac{4^{y}+4^{-y}}{2}$ is monotonically incre... | x\in[-\frac{26}{17};-]for>0,x\in[--\frac{26}{17}]for<0 |
[ [ Methods for solving problems with parameters $\quad]$
Given the quadratic trinomial $f(x)=x^{2}+a x+b$. It is known that for any real $x$ there exists a real $y$ such that $f(y)=f(x)+y$. Find the greatest possible value of $a$.
# | \frac{1}{2} |
The sum of the interior angles of a polygon is $1260^{\circ}$. The number of sides of this polygon is ______. | 9 |
3. This system is equivalent to the following system:
$$
\begin{aligned}
& \left\{\begin{array} { l }
{ | \boldsymbol { y } - 3 x | \geq | 2 \boldsymbol { y } + \boldsymbol { x } | } \\
{ - 1 \leq y - 3 \leq 1 }
\end{array} \Leftrightarrow \left\{\begin{array}{l}
(2 \boldsymbol{y}+\boldsymbol{x})^{2}-(\boldsymbol{y}-... | (x;y)=(0;0)or(-2;0) |
Given $E(X) = 10$ and $E(Y) = 3$, then $E(3X + 5Y) =$ （ ）
A: 45
B: 40
C: 30
D: 15 | A |
Given that the moving straight line $l$ passes through point $A(2, -2)$, and the distance from the points on circle $C: x^2 + y^2 - 4y = 0$ to line $l$ is maximum. What is the intercept of line $l$ on the $y$-axis? ( )
A: 2
B: $- \frac {1}{2}$
C: -3
D: 3 | C |
A soccer coach named $C$ does a header drill with two players $A$ and $B$, but they all forgot to put sunscreen on their foreheads. They solve this issue by dunking the ball into a vat of sunscreen before starting the drill. Coach $C$ heads the ball to $A$, who heads the ball back to $C$, who then heads the ball to $... | \frac{10}{19} |
Solve the system of inequalities: $\left\{{\begin{array}{l}{x-3(x-2)≥4}\\{2x+1＜x-1}\end{array}}\right.$, and write down the largest integer solution of this system of inequalities. | -3 |
## Task 4 - 240814
From three congruent isosceles right triangles with given leg length $a$, a trapezoid can be formed.
a) Draw such a trapezoid!
b) Such a trapezoid can be divided into four congruent trapezoids. Draw such a division!
c) Determine the lengths of the parallel sides, the height, and the area of one o... | \frac{3}{8}^{2} |
13. $f(x)$ is a function defined on $\mathbf{R}$ that is odd, and satisfies the following two conditions: (1) For any $x, y \in \mathbf{R}$, $f(x+y)=f(x)+f(y)$; (2) When $x>0$, $f(x)<0$, and $f(1)=-2$. Find the maximum and minimum values of the function $f(x)$ on $[-3,3]$. | 6,-6 |
5. From the set $\{1,2,3, \cdots, 99,100\}$, arbitrarily select $a$, and then from the same set arbitrarily select $b$, the probability that the last digit of $3^{a}+7^{b}$ is 8 equals $\qquad$. | \frac{3}{16} |
The range of the function $y=x^2-2x$ ($-1<x<2$) is ( )
A: $[0,3]$
B: $[1,3]$
C: $[-1,0]$
D: $[-1,3)$ | \text{D} |
Given that $a, b$ are real numbers, the transformation $T$ corresponding to matrix $M= \begin{bmatrix} a & - \frac {1}{2} \\ \frac {1}{2} & b\end{bmatrix}$ transforms the point $(2, 2)$ into point $P′(\sqrt {3}-1, \sqrt {3}+1)$.
(1) Find the values of the real numbers $a, b$;
(2) Find the inverse matrix $N$ of matr... | \begin{bmatrix} \frac { \sqrt {3}}{2} & \frac {1}{2} \\ - \frac {1}{2} & \frac { \sqrt {3}}{2}\end{bmatrix} |
Which of the following calculations is correct?
A: ${(-\sqrt{3})^2}=-3$
B: $\sqrt{12}=2\sqrt{3}$
C: $\sqrt[3]{-1}=1$
D: $(\sqrt{2}+1)(\sqrt{2}-1)=3$ | B |
Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ that satisfy $|\overrightarrow{a}| = 1$, $|\overrightarrow{b}| = 3$, and $\overrightarrow{a} + \overrightarrow{b} = (\sqrt{3}, 1)$, find $|\overrightarrow{a} - \overrightarrow{b}| =$ ___. | 4 |
2. Calculate:
$$
\sin ^{3} 20^{\circ}-\sin ^{3} 100^{\circ}+\sin ^{3} 140^{\circ}=
$$
$\qquad$ | -\frac{3\sqrt{3}}{8} |
12. $\lim _{n \rightarrow \infty}\left(\frac{n+1}{n-1}\right)^{2 n}=$ | e^4 |
Given the set $M=\{1,2\}$, the number of sets $N$ that satisfy the condition $M∪N=\{1,2,6\}$ is $\boxed{\text{answer}}$.
A: $1$
B: $3$
C: $2$
D: $4$ | D |
Given a geometric sequence $\{a_n\}$ with a common ratio $q > 1$, and the sum of its first $n$ terms is $S_n$. If $S_4=2S_2+1$, then the minimum value of $S_6$ is. | 2\sqrt{3}+3 |
3. Given $x_{1}=x_{2011}=1$, $\left|x_{n+1}\right|=\left|x_{n}+1\right|(n=1,2, \cdots, 2010)$.
Then $x_{1}+x_{2}+\cdots+x_{2010}=$ $\qquad$ | -1005 |
Given that $A$, $B$, $C$ are the three internal angles of $\triangle ABC$, $\overrightarrow{a}=(\sin B+\cos B,\cos C)$, $\overrightarrow{b}=(\sin C,\sin B-\cos B)$.
(1) If $\overrightarrow{a}\cdot \overrightarrow{b}=0$, find angle $A$;
(2) If $\overrightarrow{a}\cdot \overrightarrow{b}=- \frac {1}{5}$, find $\tan 2A$. | -\frac{24}{7} |
Given the function $f(x)=\ln x+ \frac{7a}{x}$, where $a\in R$.
(1) If the function $y=f(x)$ has exactly one zero in its domain, find the range of values for the real number $a$.
(2) If the minimum value of the function $y=f(x)$ on $[e,e^{2}]$ is $3$, find the value of the real number $a$. ($e$ is the base of the natura... | a= \frac{e^{2}}{7} |
Example 5 Find all real-coefficient polynomials $P(x)$ that satisfy the following relation,
$$
\left(x^{3}+3 x^{2}+3 x+2\right) \cdot P(x-1)=\left(x^{3}-3 x^{2}+3 x-2\right) \cdot P(x) \text {, }
$$
for all real numbers $x$.
(2003 Vietnam Olympiad) | P(x)=(x-1)x(x+1)(x+2)(x^2+x+1) |
7.124. $\log _{5}\left(4^{x}+144\right)-4 \log _{5} 2=1+\log _{5}\left(2^{x-2}+1\right)$ | 2;4 |
8. Given that $A$, $B$, and $C$ are three points on $\odot O$, and
$$
\overrightarrow{A O}=\frac{1}{2}(\overrightarrow{A B}+\overrightarrow{A C}) \text {. }
$$
Then the dot product $\overrightarrow{A B} \cdot \overrightarrow{A C}=$ | 0 |
6・100 Let $n$ positive numbers $x_{1}, x_{2}, \cdots, x_{n}$ sum up to 1. Let $S$ be the largest of the following numbers:
$$
\frac{x_{1}}{1+x_{1}}, \frac{x_{2}}{1+x_{1}+x_{2}}, \cdots, \frac{x_{n}}{1+x_{1}+x_{2}+\cdots+x_{n}} .
$$
Find the possible minimum value of $S$. For what values of $x_{1}, x_{2}, \cdots, x_{n}... | 1-2^{-\frac{1}{n}} |
Given a decreasing arithmetic sequence $\{a_n\}$, where $a_3=-1$ and $a_4$ is the geometric mean between $a_1$ and $-a_6$, if $S_n$ is the sum of the first $n$ terms of the sequence $\{a_n\}$, then the value of $S_7$ is ______. | -14 |
If $x$ is real and $4y^2+4xy+x+6=0$, then the complete set of values of $x$ for which $y$ is real, is:
$\text{(A) } x\le-2 \text{ or } x\ge3 \quad \text{(B) } x\le2 \text{ or } x\ge3 \quad \text{(C) } x\le-3 \text{ or } x\ge2 \quad \\ \text{(D) } -3\le x\le2 \quad \text{(E) } -2\le x\le3$ | x\le-2orx\ge3 |
2. What is the remainder when the number
$1+1 \cdot 2+1 \cdot 2 \cdot 3+1 \cdot 2 \cdot 3 \cdot 4+1 \cdot 2 \cdot 3 \cdot 4 \cdot 5+\ldots+1 \cdot 2 \cdot 3 \cdot 4 \cdot \ldots \cdot 14+1 \cdot 2 \cdot 3 \cdot 4 \cdot \ldots \cdot 14 \cdot 15$
is divided by 72? | 9 |
3. Let $z$ be a complex number, the argument of $z+2$ is $\frac{\pi}{3}$, and the argument of $z-2$ is $\frac{5 \pi}{6}$. Then $z$ equals ( ).
(A) $-\sqrt{3}+i$
(B) $-1+\sqrt{3} i$
(C) $-\frac{\sqrt{3}}{2}+\frac{1}{2} i$
(D) $-\frac{1}{2}+\frac{\sqrt{3}}{2} i$ | B |
Given that $\frac {1}{\cos\alpha}$ and $\tan\alpha$ are the two roots of the equation $x^2+3x+m=0$, find the value of the real number $m$. | \frac {20}{9} |
For any positive integer $n$, let $D_n$ denote the set of all positive divisors of $n$, and let $f_i(n)$ denote the size of the set
$$F_i(n) = \{a \in D_n | a \equiv i \pmod{4} \}$$
where $i = 1, 2$.
Determine the smallest positive integer $m$ such that $2f_1(m) - f_2(m) = 2017$. | 2 \cdot 5^{2016} |
Given the proposition $p: \alpha = \frac{π}{4}$ and the proposition $q: \sin \alpha = \cos \alpha$, then $p$ is ( )
A: Sufficient but not necessary condition
B: Necessary but not sufficient condition
C: Necessary and sufficient condition
D: Neither sufficient nor necessary condition | A |
A line passing through the focus of the parabola $y^2=2px$ ($p > 0$) intersects the parabola at points $P(x_1, y_1)$ and $Q(x_2, y_2)$. If $x_1 + x_2 = 3p$, then the length of $PQ$ is $\boxed{\_\_\_\_\_\_\_\_}$. | 4p |
## 26. Moving Walkway
When I walk on a moving walkway ${ }^{1}$ in one direction, I cover it in 6 s. In the opposite direction, I cover it in 6 min, which is the time it usually takes me to walk a distance of $500 \mathrm{~m}$.
What is the length of the moving walkway?[^5] | 16.39 |
Example 5 Let $f(x)$ be a function defined on $\mathbf{R}$, for any $x \in \mathbf{R}$, there are $\hat{j}(x+3) \leqslant f(x)+3$ and $f(x+2) \geqslant f(x)+2$, let $g(x)=f(x)-x$.
(1) Prove that $g(x)$ is a periodic function; (2) If $f(994)=992$, find $f(2008)$.
Translate the above text into English, please keep the o... | 2006 |
Let the arithmetic sequence $\{a_n\}$ and the geometric sequence $\{b_n\}$ both have a first term of $1$, and both have a common difference and common ratio of $2$, respectively. Then $a_{b_{2}}+a_{b_{3}}+a_{b_{4}}=$ ()
A: $24$
B: $25$
C: $26$
D: $27$ | B |
3. In square $A B C D$, points $E$ and $F$ are on sides $A D$ and $B C$ respectively, such that $E F / / A B$; points $G$ and $H$ are on sides $A B$ and $D C$ respectively, such that $G H / / A D$; $E F$ intersects $G H$ at point $K$. If the area of rectangle $K F C H$ is twice the area of rectangle $A G K E$, find the... | 45 |
78. There is a book, Dan Dan read 4 pages on the first day, and every day after that, she read 1 more page than the previous day. In the first 3 days, she read $\qquad$ pages in total. | 15 |
In a polygon, each exterior angle measures $40^{\circ}$. The sum of the interior angles of this polygon is ______. | 1260^{\circ} |
If $m < 0$, then the absolute value of the difference between $m$ and its opposite number is ( )
A: 0
B: m
C: $-2m$
D: $2m$ | C |
Suppose that $f(x)=\frac{2 x+1}{x-2}$ and that $y=g(x)$ is a linear function. If $f^{-1}(g(2))=7$ and $g^{-1}(f(1))=\frac{4}{5}$, what is the $x$-intercept of $y=g(x) ?$ | \frac{7}{5} |
$(1)$ Calculate: $40+(\frac{1}{6}-\frac{2}{3}+\frac{3}{4})×12$;
$(2)$ Calculate: $-1^{2}\times \left(-5\right)+[\left(-3\right)^{2}+2\times \left(-5\right)]$. | 4 |
[ Chessboards and chess pieces $]$ [ Examples and counterexamples. Constructions ] [ Evenness and oddness
Authors: Pechkovsky A.N., Itenberg I.
Given an infinite grid paper with a cell side equal to one. The distance between two cells is defined as the length of the shortest path of a rook from one cell to another (t... | 4 |
28. Five little sheep go on an adventure, they need to cross a log bridge. Lazy Sheep says: “I am timid, I won't walk at the front.” Pretty Sheep says: “I want to follow right behind Happy Sheep.” Boiling Sheep says: “Either I or Warm Sheep will be the last, protecting everyone.” The number of ways they can line up to ... | 8 |
96. Incorrect scales. When the pudding was placed on one pan of the scales, they showed 4 g more than $\frac{9}{11}$ of its true weight. When it was placed on the other pan, the scales showed 48 g more than in the first case. What is the true weight of the pudding?
=\ln x- \frac{1}{2}ax^{2}+(1-a)x+1$, where $a\in R$.
(I) Find the monotonic intervals of the function $f(x)$;
(II) Let $g(x)=f(x)+ax- \frac{13}{2}$, if $a=2$, and the positive real numbers $x_{1}$, $x_{2}$ satisfy $g(x_{1})+g(x_{2})+x_{1}x_{2}=0$, find the minimum value of $x_{1}+x_{2}$. | 3 |
Given $$\frac {\pi}{4} < \alpha < \frac {\pi}{2}$$, and $\sin 2\alpha = \frac {24}{25}$, then $\cos\alpha - \sin\alpha =$ （ ）
A: $$\frac {1}{5}$$
B: \- $$\frac {1}{5}$$
C: ± $$\frac {1}{5}$$
D: ± $$\frac {7}{5}$$ | \text{B} |
Among the following functions, the one whose graph passes through the origin is ( )
A: $y=\frac{-x+1}{\pi}$
B: $y=x+1$
C: $y=-x$
D: $y=3-x$ | C |
Three students $A, B$ and $C$ are traveling from a location on the National Highway No.$5$ on direction to Hanoi for participating the HOMC $2018$. At beginning, $A$ takes $B$ on the motocycle, and at the same time $C$ rides the bicycle. After one hour and a half, $B$ switches to a bicycle and immediately continues the... | 100 \text{ km} |
Iterations $\quad]$ [ Binary Number System ]
A flock of geese was flying. On each lake, half of the geese and another half-goose landed. The rest continued flying. All the geese landed on $n$ lakes.
How many geese were there in the flock? | 2^{n}-1 |
The zeros of the function $f(x) = x^3 - 16x$ are ( )
A: (0, 0)
B: 0, 4
C: (-4, 0), (0, 0), (4, 0)
D: -4, 0, 4 | D |
23. (8 points) Given $\sin \left(x+20^{\circ}\right)=\cos (x$ $+10^{\circ})+\cos \left(x-10^{\circ}\right)$. Find $\operatorname{tg} x$. | \sqrt{3} |
Simplify first, then evaluate: $3(2a^2b-3ab^2-1)-2(3a^2b-4ab^2+1)-1$, where $a=2$, $b=-1$. | -8 |
Given vectors $\overrightarrow{a}=({\cos\frac{{3x}}{2},\sin\frac{{3x}}{2}})$, $\overrightarrow{b}=({\cos\frac{x}{2},-\sin\frac{x}{2}})$, and $x∈[{0,\frac{π}{2}}]$.$(1)$ When $\overrightarrow{a}⊥\overrightarrow{b}$, find the value of $x$;$(2)$ When $|\overrightarrow{a}+\overrightarrow{b}|≥1$, find the range of $x$;$(3)$... | \frac{\sqrt{2}}{2} |
Given two points A and B moving on the lines $y=\frac{\sqrt{3}}{3}x$ and $y=-\frac{\sqrt{3}}{3}x$ respectively, the length of segment $AB$ is $2\sqrt{3}$. Let P be the midpoint of AB.
(1) Determine the equation of the trajectory $C$ of the moving point P.
(2) Through the point $Q(1, 0)$, draw a line $l$ (not perpendicu... | \lambda + \mu = -\frac{9}{4} |
Highimiy and.
In Anchuria, a day can be either clear, when the sun shines all day, or rainy, when it rains all day. And if today is not like yesterday, the Anchurians say that the weather has changed today. Once, Anchurian scientists established that January 1 is always clear, and each subsequent day in January will b... | 2047 |
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