problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
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In the Cartesian coordinate system, the slope of a line is chosen from the set M={1, 3}, and the y-intercept is chosen from the set N={2, 4, 6}. How many different lines can be formed? | 6 |
1. At the intersection of roads $A$ and $B$ (straight lines) is a settlement $C$ (point). Sasha is walking along road $A$ towards point $C$, taking 40 steps per minute with a step length of 65 cm. At the start of the movement, Sasha was 260 meters away from point $C$. Dan is walking towards $C$ along road $B$ at a spee... | 18.1 |
If the tangent at point P on the curve $y=e^{-x}$ is parallel to the line $2x+y+1=0$, then the coordinates of point P are ___. | (x,y)= (-\ln2, 2) |
(2014 Jiangxi Province Key High Schools Alliance Second Mock Exam for Senior High School Students) 21. To us all in China, \_\_\_\_\_\_\_ experience of taking the college entrance examination seems an important way to judge whether one’s life is \_\_\_\_\_\_ complete one.
A: the; a
B: an; a
C: the; the
D: an; t... | \text{A} |
Given the plane vectors $\overrightarrow{a} = (2, 4)$, $\overrightarrow{b} = (-1, 2)$, if $\overrightarrow{c} = \overrightarrow{a} - (\overrightarrow{a} \cdot \overrightarrow{b}) \overrightarrow{b}$, then the magnitude of $\overrightarrow{c}$ is ( )
A: $4 \sqrt{2}$
B: $2 \sqrt{5}$
C: $8$
D: $8 \sqrt{2}$ | D |
Using the completing the square method to solve the equation $x^{2}+7x-5=0$, the correct transformed result is:
A: $(x+\frac{7}{2})^2=\frac{69}{4}$
B: $(x+\frac{7}{2})^2=\frac{29}{4}$
C: $(x-\frac{7}{2})^2=\frac{69}{4}$
D: $(x-\frac{7}{2})^2=\frac{29}{4}$ | A |
In triangle $\triangle ABC$, the opposite sides of $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively, with $\sqrt{3}a\cos B-b\sin A=0$.
$(1)$ Find the measure of angle $B$.
$(2)$ If $b=\sqrt{7}$ and $a+c=5$, find the area of $\triangle ABC$. | \frac{3\sqrt{3}}{2} |
## Task Condition
Find the $n$-th order derivative.
$y=\lg (2 x+7)$ | y^{(n)}=(-1)^{n-1}\cdot\frac{2^{n}\cdot(n-1)!}{\ln10}\cdot(2x+7)^{-n} |
4. As shown in Figure 2, the incircle of $\triangle A B C$ touches sides $A B$, $A C$, and $B C$ at points $E$, $F$, and $G$ respectively. If the product of the lengths of the three sides of $\triangle A B C$ is 64, then the maximum value of $A E \cdot B G \cdot C F$ is ( ).
(A) 6
(B) 8
(C) 10
(D) 16 | B |
Example 6 Find the number of $n$-digit numbers formed by $1,2,3$, where each of 1, 2, and 3 must appear at least once in the $n$-digit number. | 3^{n}-3\cdot2^{n}+3 |
Express the number $4040000$ in scientific notation as ____. | 4.04 \times 10^6 |
6. What is the greatest value that the area of a right triangle can take, one vertex of which coincides with the origin, another lies on the curve $x^{2}+y^{2}=2(x+y)$, and the vertex of the right angle is located on the line $y=x$? In the answer, write the square of the found area.
points) | 1.6875 |
The domain of \\(f(x)= \sqrt { \frac {2-x}{x-1}}\\) is \\(A\\), and the solution set of the inequality \\(2^{2ax} < 2^{a+x}\\) with respect to \\(x\\) is \\(B\\). Find the range of real numbers \\(a\\) for which \\(A \cap B = A\\). | a < \frac {2}{3} |
2. In the cells of an $11 \times 11$ square, zeros and ones are arranged in such a way that in any figure of four cells in the shape of $\square$, the sum of the numbers is odd. (The figure can be rotated and flipped). What is the minimum number of ones that can be in such an arrangement? | 25 |
12. Distribute $n$ identical (i.e., indistinguishable) footballs to $r$ football teams, so that each team gets at least $s_{1}$ footballs, but no more than $s_{2}\left(r s_{1} \leqslant n \leqslant r s_{2}\right)$ footballs. How many different ways are there to do this? | \sum_{k=0}^{r}(-1)^{k}\binom{r}{k}\binom{n+r-s_{1}r-(s_{2}-s_{1}+1)k-1}{r-1} |
Example 8 Let $Q$ be the set of all rational numbers, find all functions $f$ from $\mathrm{Q}$ to $\mathrm{Q}$ that satisfy the following conditions:
(1) $f(1)=2$;
(2) For any $x, y \in \mathrm{Q}$, $f(x y)=f(x) f(y)-f(x+y)+1$.
| f(x)=x+1 |
Determine the maximal value of $k$ such that the inequality
$$\left(k +\frac{a}{b}\right) \left(k + \frac{b}{c}\right)\left(k + \frac{c}{a}\right)
\le \left( \frac{a}{b}+ \frac{b}{c}+ \frac{c}{a}\right) \left( \frac{b}{a}+ \frac{c}{b}+ \frac{a}{c}\right)$$
holds for all positive reals $a, b, c$. | \sqrt[3]{9} - 1 |
In the Cartesian coordinate plane $xoy$, there is a circle $M$: $(x-a)^{2}+(y+a-3)^{2}=1 (a > 0)$. Point $N$ is any point on circle $M$. If a circle with center $N$ and radius $ON$ has at most one common point with circle $M$, then the range of values for $a$ is _______. | a \geqslant 3 |
Given an integer ${n>1}$, denote by $P_{n}$ the product of all positive integers $x$ less than $n$ and such that $n$ divides ${x^2-1}$. For each ${n>1}$, find the remainder of $P_{n}$ on division by $n$.
[i]Proposed by John Murray, Ireland[/i] | P_n \equiv -1 \pmod{n} \text{ if } n \text{ has a primitive root and 1 otherwise.} |
10.6. Solve in $R$ the equation
$$
\sqrt{2 x^{5}+x^{4}+4 x^{3}+2 x^{2}+2 x+1}+\sqrt{17-2 x+34 x^{2}-4 x^{3}+17 x^{4}-2 x^{5}}=7 x^{2}-8 x+22
$$ | 4 |
Example 10 Find $A^{2}$, where $A$ is the sum of the absolute values of all roots of the following equation:
$$
x=\sqrt{19}+\frac{91}{\sqrt{19}+\frac{91}{\sqrt{19}+\frac{91}{\sqrt{19}+\frac{91}{\sqrt{19}+\frac{91}{x}}}}}
$$
(9th American Invitational Mathematics Examination) | 383 |
11.7. (SFRY, 76). On a plane, a regular hexagon with side $a$ is drawn. For any value of $n \in \mathbf{N}$, greater than 1, construct a segment of length $a / n$ using only a straightedge. | \frac{}{n} |
The function $f(x)=\cos x$ is increasing in the interval $[a,b]$, and $f(a)=-1$, $f(b)=1$. Then, $\cos \frac{a+b}{2}$ equals to
A: $0$
B: $\frac{\sqrt{2}}{2}$
C: $-1$
D: $1$ | A |
We define the operator "$\triangle$" as follows: when $a \gt b$, $a\triangle b=a+b$; when $a\leqslant b$, $a\triangle b=a-b$. Calculate: $(\sqrt{3}\triangle\sqrt{2})+(2\sqrt{3}\triangle3\sqrt{2})=$____. | 3\sqrt{3} - 2\sqrt{2} |
Problem 9.1. Find all real numbers $a$ for which the equation $x^{2}+a x+3 a^{2}-$ $7 a-19=0$ has real roots $x_{1}$ and $x_{2}$ such that
$$
\frac{1}{x_{1}-2}+\frac{1}{x_{2}-2}=-\frac{2 a}{13}
$$
Peter Boyvalenkov | 4 |
If the domain of the function $$f(x)= \sqrt {2^{ax^{2}-2ax-1}-1}$$ is $\mathbb{R}$, then the range of values for the real number $a$ is \_\_\_\_\_\_. | \emptyset |
Given that $a$ and $b$ are real numbers, if $2^{b}=2^{6-a}$, then $a+b=\_\_\_\_\_\_$. | 6 |
59. This year, Xiao Jun is 4 years old, and his father is 31 years old. How many years later will his father's age be 4 times Xiao Jun's age? | 5 |
4th ASU 1964 Problem 13 What is the smallest number of tetrahedrons into which a cube can be partitioned? Solution | 5 |
Given the sets $A=\{-1,0\}$ and $B=\{0,1\}$, then the set $\complement_{A\cup B}(A\cap B)=$
A: $\emptyset$
B: $\{0\}$
C: $\{-1,1\}$
D: $\{-1,0,1\}$ | C |
Given the function $y=f(x)$ whose graph has a tangent line at point $M(1,f(1))$ with the equation $y=\frac{1}{2}x+2$, find $f(1)+f'(1)$. | 3 |
A customer enters a supermarket. The probability that the customer buys bread is .60, the probability that the customer buys milk is .50, and the probability that the customer buys both bread and milk is .30. What is the probability that the customer would buy either bread or milk or both? | 0.80 |
2. If $a, c, d$ are integers, $b$ is a positive integer, and they satisfy $a+b=c, b+c=d, c+d=a$. Then, the maximum value of $a+b+c+d$ is
A. -1
B. -5
C. 0
D. 1 | -5 |
20. A cube has a side length of $n$ units (where $n$ is an integer greater than 2). The entire surface of the cube is painted black, and then the cube is cut into $n^3$ smaller cubes, each with a side length of 1 unit, using planes parallel to the faces of the original cube. If the number of smaller cubes that have exa... | D |
Given the odd function $f(x)$ defined on $\mathbb{R}$, when $x \geqslant 0$, $f(x) = x^2 - 3x$. The solution set of the equation $f(x) = x + 3$ with respect to $x$ is _______. | \{2 + \sqrt{7}, -1, -3\} |
In the cube ABCD-A1B1C1D1, P is a moving point on the base ABCD. If the distance from P to point A1 is equal to the distance from P to the line CC1, then the curve on which point P's trajectory lies is ( )
A: Straight line
B: Circle
C: Parabola
D: Ellipse | \text{A} |
Given $a=\log_{0.5}3$, $b=2^{0.5}$, $c=0.5^{0.3}$, then the relationship among $a$, $b$, and $c$ is ( )
A: $b＞a＞c$
B: $b＞c＞a$
C: $a＞b＞c$
D: $c＞b＞a$ | \text{B} |
91. For the same waterway, ship A needs 2 hours to travel downstream and 3 hours to travel upstream; ship B needs 3 hours to travel downstream, how many hours does it need to travel upstream? | 6 |
Find all positive integers $k$ for which number $3^k+5^k$ is a power of some integer with exponent greater than $1$. | k = 1 |
[ Ratio of areas of similar triangles ]
A line parallel to the base of a triangle divides it into parts whose areas are in the ratio $2: 1$, counting from the vertex. In what ratio does it divide the lateral sides? | (\sqrt{6}+2):1 |
4. As shown in the figure, $P$ is a point on the diagonal $B D_{1}$ of the rectangular prism $A B C D-A_{1} B_{1} C_{1} D_{1}$, and the plane $A P C \parallel$ plane $D A_{1} C_{1}$. If $A A_{1}=2 A D$, then the tangent value of the dihedral angle $P-A B-C$ is $\qquad$ . | 2 |
Ben works quickly on his homework, but tires quickly. The first problem takes him $1$ minute to solve, and the second problem takes him $2$ minutes to solve. It takes him $N$ minutes to solve problem $N$ on his homework. If he works for an hour on his homework, compute the maximum number of problems he can solve. | 10 |
If the product of four consecutive positive integers is 358800 , then the sum of these four integers is
(A) 102
(B) 98
(C) 94
(D) 90
(E) 106 | 98 |
10.1. Given one hundred numbers: $1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \ldots, \frac{1}{100}$. We will compute 98 differences: $a_{1}=1-\frac{1}{3}, a_{2}=\frac{1}{2}-\frac{1}{4}, \ldots$, $a_{98}=\frac{1}{98}-\frac{1}{100}$. What is the sum of all these differences? | \frac{14651}{9900} |
Calculate: $\sqrt[3]{8}+(\frac{1}{3})^{-1}-2\cos30°+|1-\sqrt{3}|$. | 4 |
Task 12. (16 points)
The budget of the Petrovs consists of the following income items:
- parents' salary after income tax deduction - 56000 rubles;
- grandmother's pension - 14300 rubles;
- son's scholarship - 2500 rubles
Average monthly expenses of the family include:
- utility payments - 9800 rubles;
- food - 210... | 16740 |
1B. In the plane, two sets of parallel lines $p_{1}, p_{2}, \ldots, p_{13}$ and $q_{1}, q_{2}, \ldots, q_{7}$ are given such that the lines from the first set intersect with the lines from the second set. How many parallelograms are determined by these lines? | 1638 |
A sample size of 100 is divided into 10 groups with a class interval of 10. In the corresponding frequency distribution histogram, a certain rectangle has a height of 0.03. What is the frequency of that group? | 30 |
Given that the domains of functions $f(x)$ and $g(x)$ are both $\mathbb{R}$, and $f(x) + g(2-x) = 5$, $g(x) - f(x-4) = 7$. If the graph of $y = g(x)$ is symmetric about the line $x = 2$, $g(2) = 4$, then $\sum _{k=1}^{22}f(k) = (\ \ )$
A: $-21$
B: $-22$
C: $-23$
D: $-24$ | D |
2. Someone says: "There must be a prime number in any seven consecutive integers." Please give an example to show that this statement is wrong.
翻译结果：
2. Someone says: "There must be a prime number in any seven consecutive integers." Please give an example to show that this statement is wrong. | 90,91,92,93,94,95,96 |
Determine the value of $\cos 36^{\circ}\cos 24^{\circ}-\sin 36^{\circ}\sin 24^{\circ}=$ ___. | \dfrac{1}{2} |
The lines $l_1: a_1x+b_1y+1=0$ and $l_2: a_2x+b_2y+1=0$ intersect at the point $(2,3)$. The equation of the line passing through points $A(a_1, b_1)$ and $B(a_2, b_2)$ is __________. | 2x+3y+1=0 |
3. Let $E$ be a given $n$-element set, and $A_{1}, A_{2}, \cdots, A_{k}$ be $k$ distinct non-empty subsets of $E$, satisfying: for any $1 \leqslant i<j \leqslant k$, either $A_{i} \cap A_{j}=\varnothing$ or one of $A_{i}$ and $A_{j}$ is a subset of the other. Find the maximum value of $k$.
(Cold Gangsong, problem contr... | 2n-1 |
Solve the following equation:
$$
\left(11 x^{3}+80\right) \sqrt[3]{x+11}+\left(x^{3}-92\right) \sqrt[3]{92 x-80}=0
$$ | -10,-4,1,2 |
\section*{Problem 6 - 131046}
A regular tetrahedron with vertices \(A, B, C\), and \(D\) and edge length \(a\) is intersected by six pairwise distinct planes, each of which contains exactly one edge of the tetrahedron and the midpoint of the opposite edge.
a) How many sub-bodies are created in total if all cuts are i... | V_{K}=\frac{^{3}}{288}\sqrt{2} |
Given the function $f(x)=|x-a|$.
$(1)$ If $a=1$, solve the inequality $f(x) \geqslant 4-|x+1|$;
$(2)$ If the solution set of the inequality $f(x) \leqslant 1$ is $[0,2]$, and $\frac {1}{m}+ \frac {1}{2n}=a \ (m > 0,n > 0)$, find the minimum value of $mn$. | 2 |
Given that the graph of an inverse proportion function passes through the point (1, 2), then its graph must also pass through which of the following points?
A: (1, -2)
B: (-1, 2)
C: (-2, 1)
D: (-1, -2) | \text{D} |
The points $A(5,-8), B(9,-30)$ and $C(n, n)$ are collinear (that is, lie on the same straight line). What is the value of $n$ ? | 3 |
Example 1 Find the range of the function $y=\sqrt{x-4}+\sqrt{15-3 x}$.
Analysis: The general approach is: squaring, rearranging, isolating the radical, squaring again, and converting to a rational expression for solving. Since $4 \leqslant x \leqslant 5$, a trigonometric substitution can be used. | [1,2] |
6. In the cube $A B C D A_{1} B_{1} C_{1} D_{1}$ with side $a$, point $K$ is the midpoint of edge $B_{1} C_{1}$, point $L$ lies on edge $C_{1} D_{1}$, such that $D_{1} L=2 C_{1} L$, and point $N$ is the midpoint of edge $A A_{1}$. Construct the section of the cube by the plane passing through points $K, L, N$. Describe... | \frac{11^{2}\sqrt{77}}{96} |
In the Cartesian coordinate system, it is known that a point on the $x$-axis is located to the right of the origin and is $3$ units away from the origin. What are the coordinates of this point?
A: $(3,0)$
B: $(0,3)$
C: $(0,-3)$
D: $(-3,0)$ | A |
71.
Let's start with a simple case. A major theft was committed at a warehouse. The criminal (or criminals) transported the stolen goods by car. Suspicion fell on three repeat offenders A, B, and C, who were brought to Scotland Yard for interrogation. The following was established:
1) No one other than A, B, and C w... | A |
8-2. Kolya is older than Tolya, and the age of each of them is a whole number less than 100. If you swap the digits of Kolya's age, you get Tolya's age. Moreover, the difference between the squares of their ages is a square of a whole number. How old is each of them? | Nikolayis65old,Anatolyis56old |
In $\triangle ABC$, the lengths of the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $a^{2}-c^{2}=2b$ and $\sin B=4\cos A\sin C$, find $b$. | b=4 |
8. (10 points) Let for positive numbers $x, y, z$ the following system of equations holds:
$$
\left\{\begin{array}{l}
x^{2}+x y+y^{2}=12 \\
y^{2}+y z+z^{2}=16 \\
z^{2}+x z+x^{2}=28
\end{array}\right.
$$
Find the value of the expression $x y+y z+x z$. | 16 |
Calculate the inverse of 36 modulo 101.
Theorem 2 (Chinese Remainder Theorem (CRT)).
Let $m_{1}, m_{2} \in$ be two coprime integers. Let $r_{1}, r_{2} \in \mathbb{N}$. There exists a unique $r \in$ $\left\{0, \ldots, m_{1} m_{2}-1\right\}$ such that:
$$
x \equiv r_{1}\left[m_{1}\right] \text { and } x \equiv r_{2}\l... | 87 |
2. Equation
$$
x^{2}-31 x+220=2^{x}\left(31-2 x-2^{x}\right)
$$
The sum of the squares of all real roots is $\qquad$ . | 25 |
2. Let the sequence $\left\{a_{n}\right\}$ have the sum of the first $n$ terms $S_{n}=2 a_{n}-1(n=1,2, \cdots)$, and the sequence $\left\{b_{n}\right\}$ satisfies $b_{1}=3, b_{k+1}=$ $a_{k}+b_{k}(k=1,2, \cdots)$. Find the sum of the first $n$ terms of the sequence $\left\{b_{n}\right\}$.
(1996 National High School Comp... | 2^{n}+2n-1 |
8. If the average of $x_{1}, x_{2}, \cdots, x_{10}$ is 6, and the standard deviation is $\sqrt{2}$, then the range of the average of the data $x_{1}, x_{2}, \cdots, x_{5}$ is $\qquad$ . | [6-\sqrt{2},6+\sqrt{2}] |
3.353. $\frac{1}{\sin 10^{\circ}}-\frac{\sqrt{3}}{\cos 10^{\circ}}=4$. | 4 |
10. For $n \in \mathbf{N}$, let $S_{n}$ be the minimum value of $\sum_{k=1}^{n} \sqrt{(2 k-1)^{2}+a_{k}^{2}}$, where $a_{1}, a_{2}, a_{3}, \cdots, a_{n}$ are positive real numbers whose sum is 17. If there exists a unique $n$ such that $S_{n}$ is also an integer, find $n$. | 12 |
4.016. Find four numbers that form a geometric progression, where the second term is 35 less than the first, and the third term is 560 more than the fourth. | 7,-28,112,-448;-\frac{35}{3},-\frac{140}{3},-\frac{560}{3},-\frac{2240}{3} |
5. The words in the circles in the picture should be replaced with the numbers 1, 2, 3, 4, 5, and 6 so that $a+b+c=c+d+e=a+f+e$ holds. All numbers must be used and must not be repeated. What can the sum $a+b+c$ be? For each possible sum, show one corresponding arrangement of the numbers in the picture.
$, $y_{1}=\frac{x_{1}}{1+a}+\frac{a x_{2}}{1+a}, y_{2}=\frac{x_{2}}{1+a}+$ $\frac{a x_{1}}{1+a}$, then the relationship between $x_{1} x_{2} \leftrightarrows y_{1} y_{2}$ is
(A) $x_{1} \cdot x_{2}>y_{1} y_{2}$
(B) $x_{1} x_{2}=y_{1} y_... | x_{1}x_{2}<y_{1}y_{2} |
Find all integers $n \in \mathbb{Z}$ such that $n+1 \mid n^{2}+1$. | -3,-2,0,1 |
(Olimpíada Matemáticos por Diversão 2020)
Find all pairs of integers $(x, y)$ such that:
$$
x^{4}-6 x^{2}+1=7 \cdot 2^{y}
$$ | (3,2) |
5.14. Calculate the limit
$$
\lim _{x \rightarrow 0} \frac{\sqrt[3]{1+\operatorname{arctan} 4 x}-\sqrt[3]{1-\operatorname{arctan} 4 x}}{\sqrt{1-\arcsin 3 x}-\sqrt{1+\operatorname{arctan} 3 x}}
$$ | -\frac{8}{9} |
Given a complex number $z=1+2i$ (where $i$ is the imaginary unit), then $z^{2}=\left(\ \ \right)$
A: $-3+2i$
B: $-3+4i$
C: $5+2i$
D: $5+4i$ | \text{B} |
4-162 Find all integer pairs $(x, y)$ that satisfy the equation $x y=20-3 x+y$.
| (2,14),(18,-2),(0,-20),(-16,-4) |
Question 104, Given complex numbers $\mathrm{z}_{1}, \mathrm{z}_{2}$ satisfy the following conditions: $\left|\mathrm{z}_{1}\right|=2,\left|\mathrm{z}_{2}\right|=3,3 \mathrm{z}_{1}-2 \mathrm{z}_{2}=2-\mathrm{i}$, then $\mathrm{z}_{1} \mathrm{z}_{2}=$ $\qquad$ - | -\frac{18}{5}+\frac{24}{5}i |
Given two circles C1: $(x-3)^2+y^2=1$ and C2: $x^2+(y+4)^2=16$, determine the positional relationship between circles C1 and C2.
A: Intersecting
B: Separate
C: Internally Tangent
D: Externally Tangent | D |
Wei uses a crowbar to move a large stone. It is known that the resistance and resistance arm are $1000N$ and $0.6m$ respectively. When the effort arm increases from $1.5m$ to $2m$, moving this stone can save _____ $N$ of force. ($($Lever principle: Resistance $\times$ Resistance arm $=$ Effort $\times$ Effort arm$) | 100 |
Improving the vehicle passage capacity of a mountain tunnel can effectively alleviate traffic conditions. Generally, the vehicle flow speed $v$ (unit: kilometers per hour) in the tunnel is a function of the vehicle flow density $x$ (unit: vehicles per kilometer, representing the number of vehicles per kilometer of road... | 3675 |
Given the following sequence of numbers: $1$, $1$, $-1$, $0$, $1$, $1$, $-1$, $0$, $\ldots$, then the $2016$th number is ______. | 0 |
If the function $f(x)=2\sin (\omega x+ \varphi )(\omega \ne 0,\varphi > 0)$ is an even function, find the minimum value of $\varphi$. | \frac{\pi}{2} |
Given that the three sides of a triangle form a geometric sequence, and their common ratio is $q$, the range of $q$ is ( )
A: $(0, \frac {1+ \sqrt {5}}{2})$
B: $( \frac {1- \sqrt {5}}{2}, 1]$
C: $[1, \frac {1+ \sqrt {5}}{2})$
D: $( \frac {-1+ \sqrt {5}}{2}, \frac {1+ \sqrt {5}}{2})$ | \text{D} |
Let $a \in \mathbb{Z}$, and $0 \leq a \leq 13$. If $51^{2012} + a$ is divisible by 13, then $a$ equals to ( )
A: 0
B: 1
C: 11
D: 12 | D |
Given that the domain of the function $f(x)$ is $\mathbb{R}$, and it satisfies $-f(x)=f(-x)$, $f(x)=f(2-x)$, then $f(\log _{2}4+\log _{4}8+\log _{8}16-e^{\ln \frac{5}{6}})=$ $( )$
A: $1$
B: $-1$
C: $\dfrac{3}{2}$
D: $0$ | D |
[ Properties and characteristics of a parallelogram ] Law of Cosines $\quad]$
The sides of a parallelogram are 2 and 4, and the angle between them is $60^{\circ}$. Through the vertex of this angle, lines are drawn passing through the midpoints of the other two sides of the parallelogram. Find the cosine of the angle b... | \frac{5}{2\sqrt{7}} |
How many divisors does $68^{2008}$ have?
Exercise $10 E$ is a set containing $n$ elements. How many pairs $(A, B)$ of two sets are there such that $A \subset B \subset E$? | 3^{n} |
Two players, A and B, are competing in a game. Player A is the younger player, and the rules are as follows: In odd-numbered rounds, the probability of A winning is $\frac{2}{3}$; in even-numbered rounds, the probability of B winning is $\frac{2}{3}$. There are no draws in any round. The game ends when one player has w... | \frac{2968}{729} |
The inclination angle of the line $x-y+1=0$ is ____. | 45^\circ |
Given that $F$ is the right focus of the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ ($a > 0$, $b > 0$), a line $l$ passing through the origin intersects the hyperbola at points $M$ and $N$, and $\overrightarrow{MF} \cdot \overrightarrow{NF} = 0$, the area of $\triangle MNF$ is $ab$. Then, the eccentricity of thi... | \sqrt{2} |
5. $\vec{a}=\left(k^{2}+k-3\right) \vec{i}+(1-k) \vec{j}, \vec{b}=-3 \vec{i}+(k-1) \vec{j}$, vectors $\vec{a}$ and $\vec{b}$ are collinear, $k=$ $\qquad$ | -3,2,1 |
Yakob and Baptiste are playing on a $20 \times 20$ grid where the cells are square and have a side length of 1. The distance between two cells is the distance between their centers. They take turns playing as follows: Yakob places a red stone on a cell, ensuring that the distance between any two cells with red stones i... | 100 |
In the arithmetic sequence $\{a_n\}$, $a_5=0$, $a_{10}=10$.
(Ⅰ) Find the general term formula for the sequence $\{a_n\}$.
(Ⅱ) If the sequence $\{b_n\}$ satisfies $b_n=\left( \frac {1}{2}\right)^{a_n+10}$, find the sum of the first $n$ terms $S_n$ of the sequence $\{nb_n\}$. | \frac{4}{9} - \frac{4+3n}{9\cdot 4^n} |
Given the hyperbola $\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1 (a > 0, b > 0)$, one of its asymptotes intersects the ellipse $\dfrac{x^2}{4} + y^2 = 1$ and the length of the chord obtained is $\dfrac{4\sqrt{3}}{3}$. Then, the eccentricity of this hyperbola is ( ).
A: $\sqrt{2}$
B: $\sqrt{3}$
C: $\dfrac{\sqrt{6}}{2}$
... | B |
1. Find the value of $2019 \times 999-2018 \times 998$ | 3017 |
Cutting a wire of length $a$ into three pieces at random to form a triangle, this event is ( )
A: A certain event
B: An impossible event
C: A random event
D: Indeterminable | C |
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