problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
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Given the proposition "$\exists x\in R$, $x^{2}-ax+1=0$" is false, the range of real number $a$ is ____. | (-2, 2) |
2. For any real number $x$, the value of the function $f(x)=x^{2}-2 x-|x-1-a|-|x-2|+4$ is a non-negative real number, then the range of the real number $a$ is $\qquad$. | -2\leqslant\leqslant1 |
Given the function $f(x) = x^3 - 12x + a$, where $a \geq 16$, which of the following statements is correct?
A: $f(x)$ has exactly one zero point.
B: $f(x)$ has at least two zero points.
C: $f(x)$ has at most two zero points.
D: $f(x)$ definitely has three zero points. | \text{C} |
A unit has 52 employees, and now all employees are randomly numbered from 1, 2, 3, …, 52. If a systematic sampling method is used to draw a sample of size 4, and it is known that employees numbered 6, 32, and 45 are in the sample, then the number of the other employee in the sample is ____. | 19 |
Let $ABCD$ be a square with side length $2$, and let a semicircle with flat side $CD$ be drawn inside the square. Of the remaining area inside the square outside the semi-circle, the largest circle is drawn. What is the radius of this circle? | 4 - 2\sqrt{3} |
The terminal side of the angle $\(-330^{\circ}\)$ is in quadrant $\($ $\)$
A: Quadrant I
B: Quadrant II
C: Quadrant III
D: Quadrant IV | \text{A} |
The slope of the line $x+\sqrt{3}y=0$ is ( ).
A: $-30^{\circ}$
B: $60^{\circ}$
C: $120^{\circ}$
D: $150^{\circ}$ | D |
All integers with absolute value less than $3$ are ____. | -2, -1, 0, 1, 2 |
1A. Let $n$ be a natural number and let $x_{0}=\frac{1}{n}, x_{k}=\frac{1}{n-k}\left(x_{0}+x_{1}+\cdots+x_{k-1}\right)$. Calculate the sum $x_{0}+x_{1}+\cdots+x_{n-1}$. | 1 |
2. The number 98 is written on the board. Every minute, the number is erased and replaced with the product of its digits, increased by 15. What number will be on the board after an hour? | 23 |
The graph of the linear function $y_{1}=kx+b(k\neq 0$,$k$ and $b$ are constants) intersects with the graph of $y_{2}=mx+3(m\neq 0$,$m$ is a constant) at the point $D(1,2)$. Which of the following conclusions are correct? ① The solution to the equation in terms of $x$, $kx+b=mx+3$, is $x=1$; ② For any two distinct point... | B |
7.78 If $a>0, a \neq 1, \quad F(x)$ is an odd function, then the parity of the following function
$$
G(x)=F(x) \cdot\left[\frac{1}{a^{x}-1}+\frac{1}{2}\right]
$$
is
(A) Odd function.
(B) Even function.
(C) Neither odd nor even function.
(D) The parity depends on the specific value of $a$.
(China High School Mathematic... | B |
Each face of a tetrahedron is a triangle with sides $a, b,$c and the tetrahedon has circumradius 1. Find $a^2 + b^2 + c^2$. | 8 |
A1. The function is given by $f(x)=-2(x+3)^{2}+2$. Which statement is correct?
(A) The function is decreasing everywhere.
(B) The vertex is at the point $T(3,2)$.
(C) The function has no real zeros.
(D) The graph is a parabola that is symmetric with respect to the line $x=-3$.
(E) None of the given statements is c... | D |
In an athletics team, there are 20 male athletes and 10 female athletes. Immediately after the competition, a stratified sampling method is used to draw a sample of 6 people from all team members for doping tests. The number of male athletes to be drawn is \_\_\_\_\_\_. | 4 |
Given an arithmetic sequence $\{a_n\}$ satisfies $a_1 = -4$, and $a_4 + a_6 = 16$, then the sum of its first 10 terms $S_{10} =$ （ ）
A: 138
B: 95
C: 23
D: 135 | \text{B} |
A horizontally lying cylinder is fixed by attaching a rib to its side at two points. (The ribs touch the cylinder's side with their upper edges.) What is the radius of the cylinder if the heights of the ribs are 9 and $2 \mathrm{~cm}$, and their distance is $23 \mathrm{~cm} ?$ | 17\mathrm{~} |
## Task 19/82
Solve the equation $25^{x}-30^{x}=36^{x+0.5}$. | -6.03 |
Aurick throws $2$ fair $6$-sided dice labeled with the integers from $1$ through $6$. What is the probability that the sum of the rolls is a multiple of $3$? | \frac{1}{3} |
4. Given the power function $y=x^{m^{2}-m-6}(m \in \mathbf{Z})$ whose graph has no common points with the $x$-axis. Then the range of $m$ is ( ).
(A) $\{m \mid-2<m<3, m \in \mathbf{Z}\}$
(B) $\{m \mid-2 \leqslant m \leqslant 3, m \in \mathbf{Z}\}$
(C) $\{m \mid-3<m<2, m \in \mathbf{Z}\}$
(D) $\{m \mid-3 \leqslant m \le... | B |
30. There are three lamps, A, B, and C. The red switch controls A and B, the blue switch controls B and C, and the green switch controls A and C. Initially, all three lamps are on. After Peter Pan presses the red switch once, A and B both turn off. Then Peter Pan presses another 19 switches, and in the end, only one la... | C |
Define a monotonic function $f(x)$ on $(0, +\infty)$ that satisfies $f[f(x) - \log_{2}x] = 3$ for all $x > 0$. The number of zeros of $g(x) = f(x) + x - 4$ is \_\_\_\_\_\_. | 1 |
Given the sets $A=\{x|-3<x<6\}$ and $B=\{x|2<x<7\}$, then $A\cap (\complement_{\mathbb{R}} B)=$ ( )
A: $(2,6)$
B: $(2,7)$
C: $(-3,2]$
D: $(-3,2)$ | C |
In the figure below, circle $O$ has two tangents, $\overline{AC}$ and $\overline{BC}$. $\overline{EF}$ is drawn tangent to circle $O$ such that $E$ is on $\overline{AC}$, $F$ is on $\overline{BC}$, and $\overline{EF} \perp \overline{FC}$. Given that the diameter of circle $O$ has length $10$ and that $CO = 13$, ... | \frac{420}{17} |
Given the function $f(x)=\left\{\begin{array}{l}\log_{2}x, x>0\\-sinx, x\leq0\end{array}\right.$, find $f(f(-\frac{\pi}{6}))$. | -1 |
9. To investigate a non-luminous black planet far out in space, Xia Feitian drives a high-speed spacecraft at a speed of 100,000 kilometers per second, with a powerful light on, heading straight towards the black planet. When Xia Feitian has just been traveling for 100 seconds, the spacecraft's instruments receive the ... | 2000 |
Find the real number $t$, such that the following system of equations has a unique real solution $(x, y, z, v)$:
\[ \left\{\begin{array}{cc}x+y+z+v=0\\ (xy + yz +zv)+t(xz+xv+yv)=0\end{array}\right. \] | t \in \left( \frac{3 - \sqrt{5}}{2}, \frac{3 + \sqrt{5}}{2} \right) |
Let $(2x+1)^3 = a_0x^3 + a_1x^2 + a_2x + a_3$, which is an identity in $x$ (i.e., it holds for any value of $x$). Find the value of $a_1 + a_3$. | 13 |
How many ways can the integers from $-7$ to $7$ inclusive be arranged in a sequence such that the absolute value of the numbers in the sequence does not decrease? | 128 |
Subject 3. In the equilateral triangle $\mathrm{ABC}$ with $\mathrm{AB}=6 \mathrm{~cm}$, perpendiculars $\mathrm{A}^{\prime} \mathrm{A} \perp(\mathrm{ABC}), \mathrm{B}^{\prime} \mathrm{B} \perp(\mathrm{ABC})$ are raised from the same side of the plane (ABC), such that $\mathrm{AA}^{\prime}=\mathrm{BB}^{\prime}=6 \sqrt{... | \frac{\sqrt{39}}{8} |
1. The range of the function $f(x)=|x+1|+|x-1|+\sqrt{4-x^{2}}$ is $\qquad$ . | [2+\sqrt{3}, 2 \sqrt{5}] |
4. Calculate: $\frac{1}{2 \cos \frac{2 \pi}{7}}+\frac{1}{2 \cos \frac{4 \pi}{7}}+\frac{1}{2 \cos \frac{6 \pi}{7}}$. | -2 |
Find all values of $n \in \mathbb{N}$ such that $3^{2 n}-2^{n}$ is prime. | 1 |
In the arithmetic sequence $\{a_n\}$, if $a_7 = m$ and $a_{14} = n$, then $a_{21} = \ $ | 2n - m |
The domain of the function $f(x)={(x-\frac{1}{2})}^{\frac{1}{2}}$ is ( ).
A: $(-∞,\frac{1}{2})$
B: $[\frac{1}{2},+∞)$
C: $\{-∞,-\frac{1}{2}\}$
D: $[-\frac{1}{2},+∞)$ | B |
Given a sequence of positive terms $\{a_{n}\}$ that satisfies: $a_{1}=1$, ${a}_{n}=\frac{{na}_{n+1}^{2}}{n{a}_{n+1}+1}$, then ( )
A: $a_2=\frac{\sqrt{5}-1}{2}$
B: $\{a_{n}\}$ is an increasing sequence
C: $a_{n+1}-a_n>\frac{1}{n+1}$
D: $a_{n+1}<1+\sum_{k=1}^n\frac{1}{k}$ | BCD |
一、(50 分) In $\triangle A B C$, the angle bisectors $B D$ and $C E$ intersect at $I$, with $D$ and $E$ on $A C$ and $A B$ respectively. The perpendicular from $I$ to $D E$ meets $D E$ at $P$. The extension of $P I$ meets $B C$ at $Q$. If $I Q=2 I P^{\prime}$, find $\angle A$. | 60 |
2. Among all positive integers that are multiples of 20, the sum of those that do not exceed 2014 and are also multiples of 14 is | 14700 |
38.4. Find the envelope of the family of lines that cut off from a given right angle a triangle of area $a^{2} / 2$. | xy=\frac{^2}{4} |
5. In an arbitrary triangular pyramid $A B C D$, a section is made by a plane intersecting the edges $A B, D C$, and $D B$ at points $M, N, P$ respectively. Point $M$ divides edge $A B$ in the ratio $A M: M B=2: 3$. Point $N$ divides edge $D C$ in the ratio $D N: N C=3: 5$. Point $P$ divides edge $D B$ in the ratio $D ... | AQ:QC=1:10 |
Given the function $f(x) = 3^x$, which of the following equations holds true?
A: $f(x)f(y) = f(x+y)$
B: $f(x)f(y) = f(xy)$
C: $f(x+y) = f(x) + f(y)$
D: $f(xy) = f(x) + f(y)$ | A |
Given the function $f(x)=\sin ^{2}ωx+ \sqrt {3}\sin ωx\sin (ωx+ \frac {π}{2})$, $(ω > 0)$ with the smallest positive period of $π$, determine the range of $f(x)$ in the interval $[0, \frac {2π}{3}]$.
A: $[0, \frac {3}{2}]$
B: $[-\frac {1}{2}, \frac {3}{2}]$
C: $[-\frac {1}{2},1]$
D: $[-\frac {3}{2}, \frac {1}{2}]$ | A |
Example 2 In the positive term sequence $\left\{a_{n}\right\}$,
$$
a_{1}=10, a_{k+1}=10 \sqrt{a_{k}}(k=1,2, \ldots) \text {. }
$$
Find the general term formula $a_{n}$. | a_{n}=10^{2-(\frac{1}{2})^{n-1}} |
4. Given that $F_{1}, F_{2}$ are the two foci of the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)$, and $A B$ is a chord passing through $F_{1}$, then the perimeter of $\triangle A B F_{2}$ is
A. $2 a$
B. $4 a$
C. $8 a$
D. $2 a+2 b$ | 4a |
15. Let real numbers $a, b$ satisfy $a=x_{1}+x_{2}+x_{3}=$ $x_{1} x_{2} x_{3}, a b=x_{1} x_{2}+x_{2} x_{3}+x_{3} x_{1}$, where $x_{1}, x_{2}, x_{3}>0$. Find the maximum value of $P=\frac{a^{2}+6 b+1}{a^{2}+a}$. | \frac{9+\sqrt{3}}{9} |
If $x$ is the smallest angle in a triangle, then the range of the function $y= \sqrt {2}\sin (x+45 ^{\circ} )$ is $(\quad)$
A: $(0, \dfrac { \sqrt {2}}{2}]$
B: $[\dfrac {1}{2}, \dfrac { \sqrt {2}}{2}]$
C: $(\dfrac {1}{2}, \dfrac { \sqrt {2}}{2}]$
D: $(1, \sqrt {2}]$ | D |
Example 10. (IMO6-1(1)) Find all positive integers $n$ such that $2^{n}-1$ is divisible by 7. | n = 3k |
Find all pairs of non-negative integers $m$ and $n$ that satisfy $$3 \cdot 2^m + 1 = n^2.$$ | (m, n) = (0, 2), (3, 5), (4, 7) |
Given space vectors $\overrightarrow{a}=({1,0,-1})$ and $\overrightarrow{b}=({-1,1,0})$, the projection vector of vector $\overrightarrow{a}$ onto vector $\overrightarrow{b}$ is ( ).
A: $({-\frac{1}{2},0,\frac{1}{2}})$
B: $({\frac{{\sqrt{2}}}{2},-\frac{{\sqrt{2}}}{2},0})$
C: $(1,-1,0)$
D: $({\frac{1}{2},-\frac{1}{2... | D |
Under which condition does the following equality sequence hold:
$$
\frac{a+\frac{a b c}{a-b c+b}}{b+\frac{a b c}{a-a c+b}}=\frac{a-\frac{a b}{a+2 b}}{b-\frac{a b}{2 a+b}}=\frac{\frac{2 a b}{a-b}+a}{\frac{2 a b}{a-b}-b}=\frac{a}{b}
$$ | 0,\quadb\neq0,\quad\neq1 |
Given that the function $f(x)$ is an even function on $\mathbb{R}$ and that for $x \geq 0$, $f(x+2) = f(x)$ holds. If $f(x) = x^2 - x + 1$ when $x \in [0, 2)$, then the value of $f(-2014) + f(1)$ is ( ).
A: $-2$
B: $-1$
C: $1$
D: $2$ | D |
Given an arithmetic sequence ${a_n}$ with the sum of its first $n$ terms denoted as $S_n$, where $S_{10} = -2$ and $S_{20} = 16$, find the value of $n$ when $S_n$ reaches its minimum.
A: 4
B: 5
C: 6
D: 7 | C |
Determine the value of $2^{4}\left(1+\frac{1}{2}+\frac{1}{2^{2}}+\frac{1}{2^{3}}+\frac{1}{2^{4}}\right)$. | 31 |
$\begin{array}{l}\text { 1. } \lim _{n \rightarrow \infty} \frac{1}{\sqrt[3]{n}}\left(\frac{1}{1+\sqrt[3]{2}+\sqrt[3]{4}}+ \\ \frac{1}{\sqrt[3]{4}+\sqrt[3]{6}+\sqrt[3]{9}}+ \\ \left.\frac{1}{\sqrt[3]{(n-1)^{2}}+\sqrt[3]{n(n-1)}+\sqrt[3]{n^{2}}}\right)=\end{array}$ | 1 |
## Problem Statement
Calculate the lengths of the arcs of the curves given by equations in a rectangular coordinate system.
$$
y=\ln \left(x^{2}-1\right), 2 \leq x \leq 3
$$ | 1+\ln\frac{3}{2} |
2. Simplify the expression:
$$
2\left(x y^{-1}-1\right)^{-p}\left(x^{2} y^{-2}-1\right)^{p}-\left(\frac{x+y}{y}\right)^{p}
$$
For $x=-2, y=-\frac{1}{2}$ and $p=-3$, calculate the value of the expression. | \frac{1}{125} |
## Task 4 - 050514
Gerd, Fred, Heinz, and Werner are on their way to school. Fred is still three times as far away from the school as Gerd. Heinz has four times the distance left to the school that Gerd has to cover.
Werner still has to walk $2.4 \mathrm{~km}$ to the school; this is twice the length of Fred's distanc... | 36 |
Variance is a measure of the degree of data fluctuation. For a set of data $x_{1}$, $x_{2}$, $x_{3}$, $\ldots$, $x_{n}$, the variance can be calculated using the following formula: $s^{2}=\frac{1}{n}[(x_{1}-5)^{2}+(x_{2}-5)^{2}+(x_{3}-5)^{2}+\ldots +(x_{n}-5)^{2}]$, where "5" represents the ( ).
A: minimum value
B: m... | B |
Factorize: $-2x^{2}+4xy-2y^{2}=\_\_\_\_\_\_.$ | -2\left(x-y\right)^{2} |
Which of the following functions is an even function and is increasing on the interval $(0,+∞)$?
A: $y=x^{-2}$
B: $y=x^{2}+3x+2$
C: $y=\ln x$
D: $y=3^{|x|}$ | D |
For the function $y=f(x)$ with domain $D$, the correct statement(s) is(are) (only the number of the statement is required):
① If $f(-1)=f(1)$ and $f(-2)=f(2)$, then $y=f(x)$ is an even function on $D$;
② If $f(-1)<f(0)<f(1)<f(2)$, then $y=f(x)$ is an increasing function on $D$;
③ If $f'(2)=0$, then $y=f(x)$ must ha... | 4 |
4. Kolya has 440 identical cubes with a side length of 1 cm. Kolya assembled a rectangular parallelepiped from them, with all edges having a length of at least 5 cm. Find the total length of all the edges of this parallelepiped | 96 |
Problem 1. Let $n$ be a natural number. Find the least natural number $k$ for which there exist $k$ sequences of 0 's and 1 's of length $2 n+2$ with the following property: any sequence of 0 's and 1 's of length $2 n+2$ coincides in at least $n+2$ positions with some of these $k$ sequences.
| 4 |
10.389 In an isosceles triangle $A B C \quad A B=B C=25$ cm and $A C=14$ cm. Calculate the radius of the circle that touches $B C$ at point $D$, the foot of the altitude $A D$, and passes through the midpoint of $A C$. | \frac{175}{48} |
$14 \cdot 21$ Three balls marked $1, 2, 3$ are placed in a jar. A ball is drawn from the jar, its number is recorded, and then it is returned to the jar. This process is repeated three times. Each ball has an equal chance of being drawn each time. If the sum of the recorded numbers is 6, what is the probability that th... | \frac{1}{7} |
1. Find all values of $x$, for each of which one of the three given numbers $\log _{x}\left(x-\frac{13}{6}\right)$, $\log _{x-\frac{13}{6}}(x-3)$, and $\log _{x-3} x$ is equal to the product of the other two. | \frac{11}{3},\frac{3+\sqrt{13}}{2} |
Determine the truth values of the following propositions:
Proposition $p$: There exist real numbers $α$ and $β$ such that $\sin (α+β)=\sin α+\sin β$.
Proposition $q$: $\log_{a}2+\log_{2}a \geqslant 2$ where $a > 0$ and $a \neq 1$.
Which of the following compound propositions is true?
A: $p \lor q$ (either $p$ or $q$ is... | A |
8. Find the value using multiple methods: $\cos \frac{\pi}{7}-\cos \frac{2 \pi}{7}+\cos \frac{3 \pi}{7}$. | \frac{1}{2} |
$2 \cdot 82$ Given $|x-1|+|x-5|=4$, then the range of $x$ is
(A) $1 \leqslant x \leqslant 5$.
(B) $x \leqslant 1$.
(C) $1<x<5$.
(D) $x \geqslant 5$.
(Mathematics League of Junior High Schools in Guangzhou, Wuhan, Fuzhou, and other five cities, 1985) | 1\leqslantx\leqslant5 |
In July 2019, the Liangzhu Ancient City Site in China was approved to be included in the World Heritage List, marking the international recognition of the 5,000-year history of Chinese civilization. The Liangzhu Ancient City Site is an example of early urban civilization of mankind, witnessing the 5,000-year history of... | 6876 |
Given the function $f(x)=x^{3}+ax$ is an increasing function on $[1,+\infty)$, what is the minimum value of $a$?
A: $-3$
B: $-2$
C: $2$
D: $3$ | A |
The solution set of the inequality $x(9-x) > 0$ is ( )
A: $(0,9)$
B: $(9,+\infty)$
C: $(-\infty,9)$
D: $(-\infty,0) \cup (9,+\infty)$ | A |
The increasing interval of the function $y=x-2\sin x$ on $[0,\pi]$ is ______. | [\frac{\pi}{3},\pi] |
Given the sets $M=\{y|y=-x^2+1\}$ and $P=\{x|y=2x+1\}$, the relationship between set $M$ and set $P$ is ( )
A: $M=P$
B: $P \in M$
C: $M \subsetneq P$
D: $P \subsetneq M$ | C |
6. $\underbrace{2 \times 2 \times \ldots \times 2}_{20 \uparrow 2}-1$
The unit digit of the result is $\qquad$ | 5 |
Determine the integer solutions $(a, b)$ of
$$
\frac{1}{a}-\frac{1}{b}=\frac{1}{37}
$$
## Sophie Germain Factorization
## Proposition
Let $a, b$ be integers. Then,
$$
a^{4}+4 b^{4}=\left(a^{2}+2 b^{2}+2 a b\right)\left(a^{2}+2 b^{2}-2 a b\right)
$$
## Proof
It suffices to write $a^{4}+4 b^{4}=a^{4}+4 a^{2} b^{2}... | (,b)=(36,1332) |
A1. At the beginning of a gambling game, Ali, Bente, and Chris had money in the ratio 11:8:5. At the end of the game, the same amount of money was distributed in the ratio $4: 3: 2$. Which statement is true?
(A) Ali lost, Bente lost, and Chris won
(B) Ali won, Bente lost, and Chris won
(C) Ali won, Bente lost, and C... | D |
Variant 10.6.1. A pair of natural numbers $(a, p)$ is called good if the number $a^{3}+p^{3}$ is divisible by $a^{2}-p^{2}$, and $a>p$.
(a) (1 point) Indicate any possible value of $a$ for which the pair $(a, 13)$ is good.
(b) (3 points) Find the number of good pairs for which $p$ is a prime number less than 20. | 24 |
In quadrilateral $A B C D$, two non-intersecting circles are placed such that one of them touches sides $A B, B C$ and $C D$, while the other touches sides $A B, A D$ and $C D$. Line $M N$ intersects sides $A B$ and $C D$ at points $M$ and $N$ respectively and is tangent to both circles. Find the distance between the c... | \sqrt{r^{2}+(p-)^{2}} |
In a massive school which has $m$ students, and each student took at least one subject. Let $p$ be an odd prime. Given that:
(i) each student took at most $p+1$ subjects. \\
(ii) each subject is taken by at most $p$ students. \\
(iii) any pair of students has at least $1$ subject in common. \\
Find the maximum possib... | p^2 |
Problem 2. (Option 1). Given an acute triangle $\mathrm{ABC}(\mathrm{AB}=\mathrm{BC})$ and $\mathrm{BC}=12$. $A N \perp B C$. On the side $\mathrm{BC}$, a point $M$ (M lies between B and $\mathrm{N}$) is marked such that $\mathrm{AN}=\mathrm{MN}$ and $\angle \mathrm{BAM}=\angle \mathrm{NAC}$. Find $\mathrm{BN}$. | 6\sqrt{3} |
**Text**: From a group of 5 students, 4 are to be selected to participate in high school competitions for "Mathematics", "Physics", and "Chemistry". Each subject must have at least one participant, and each student can only participate in one subject. How many different arrangements are there? (Answer with a number.) | 180 |
C2. The integers $a, b, c, d, e, f$ and $g$, none of which is negative, satisfy the following five simultaneous equations:
$$
\begin{array}{l}
a+b+c=2 \\
b+c+d=2 \\
c+d+e=2 \\
d+e+f=2 \\
e+f+g=2 .
\end{array}
$$
What is the maximum possible value of $a+b+c+d+e+f+g$ ? | 6 |
Given that $\dfrac{1+ai}{2-i}$ is a pure imaginary number, the value of the real number $a$ is ( ).
A: $2$
B: $-2$
C: $-\dfrac{1}{2}$
D: $\dfrac{1}{2}$ | A |
3. In how many different ways can four friends Ankica, Gorana, Ivana, and Melita divide 11 identical fries among themselves, so that each gets at least two? The fries cannot be cut into smaller pieces. | 20 |
6. Let $a_{1}, a_{2}, \cdots, a_{2014}$ be a permutation of the positive integers $1,2, \cdots$, 2014. Define
$$
S_{k}=a_{1}+a_{2}+\cdots+a_{k}(k=1,2, \cdots, 2014) .
$$
Then the maximum number of odd numbers in $S_{1}, S_{2}, \cdots, S_{2014}$ is . $\qquad$ | 1511 |
Given the universal set $U = \mathbb{R}$, set $P=\{x|x(x-2)\geqslant 0\}$, $M=\{x|a \lt x \lt a+3\}$.
$(Ⅰ)$ Find the complement of set $P$ with respect to $U$.
$(Ⅱ)$ If $a=1$, find the intersection of sets $P$ and $M$.
$(Ⅲ)$ If $\complement _{U}P\subseteq M$, find the range of real number $a$. | [-1, 0] |
4. The function $f(x)$ that satisfies the equation
$$
f(x)+(x-2) f(1)+3 f(0)=x^{3}+2 \quad (x \in \mathbf{R})
$$
is $f(x)=$ . $\qquad$ | x^{3}-x+1 |
1.010. $\frac{\left(5 \frac{4}{45}-4 \frac{1}{6}\right): 5 \frac{8}{15}}{\left(4 \frac{2}{3}+0.75\right) \cdot 3 \frac{9}{13}} \cdot 34 \frac{2}{7}+\frac{0.3: 0.01}{70}+\frac{2}{7}$. | 1 |
Given the complex number $z= \frac {2i}{1+i}$ (where $i$ is the imaginary unit), the conjugate of $z$ is
A: $-1+i$
B: $-1-i$
C: $1+i$
D: $1-i$ | D |
5. In $\triangle A B C$, $A B=1, B C=2$, then the range of $\angle C$ is
A. $\left(0, \frac{\pi}{6}\right]$
B. $\left(\frac{\pi}{4}, \frac{\pi}{2}\right)$
C. $\left(\frac{\pi}{6}, \frac{\pi}{3}\right)$
D. $\left(0, \frac{\pi}{2}\right)$ | A |
Given a complex number $z$ satisfies $z(1-i)^{2}=1+i$ (where $i$ is the imaginary unit), then $|z|$ is
A: $\dfrac {1}{2}$
B: $\dfrac { \sqrt {2}}{2}$
C: $\sqrt {2}$
D: $1$ | B |
6.214. $\left\{\begin{array}{l}x+y+z=0, \\ 2 x+3 y+z=0, \\ (x+1)^{2}+(y+2)^{2}+(z+3)^{2}=14 .\end{array}\right.$ | (0;0;0),(2;-1;-1) |
Example 9 Let $a, b, c \in \mathbf{R}^{+}$, and $a b c=1$, find the maximum value of $\frac{1}{a+b+1}+\frac{1}{b+c+1}+\frac{1}{c+a+1}$. | 1 |
If $\lg x - \lg y = a$, then $\lg\left(\frac{x}{2}\right)^3 - \lg\left(\frac{y}{2}\right)^3 = \_\_\_\_\_$. | 3a |
To investigate the vision of students in a class of 48 students in a middle school, it is planned to use systematic sampling to select 4 students for the survey. If student number 7 is selected in the first group of numbers 1 to 12, the numbers of the other 3 students selected are ( )
A: 17, 27, 37
B: 18, 27, 36
... | D |
In a certain school this semester, there were four math tests. The scores (in points) of Li Na in the four tests were $80$, $70$, $90$, $70$, and the scores of Wang Yue in the four tests were $80$, $a (a \geq 70)$, $70$, $90$. If the median of Li Na's scores is $5$ points less than the median of Wang Yue's scores, whic... | D |
1. In the wallet, there are six different coins of cents and euros (5 cents, 10 cents, 20 cents, 50 cents, $1 €$ and $2 €$). Three coins are drawn simultaneously and without looking. In how many different ways can the coins be drawn so that their total value is greater than 85 cents? | 16 |
9. (14 points) Given the sequence $\left\{a_{n}\right\}$ with the general term
$$
a_{n}=1+2+\cdots+n\left(n \in \mathbf{N}_{+}\right) \text {, }
$$
take all multiples of 3 from this sequence to form a new sequence $b_{1}, b_{2}, \cdots, b_{m}, \cdots$. Find the sum of the first $2 m$ terms of the sequence $\left\{b_{m... | \frac{3}{2} m(m+1)(2 m+1) |
Given an arithmetic sequence $\{a\_n\}$ with the sum of the first $n$ terms denoted as $S\_n$, and a common difference of $d$. If $\frac{S_{2017}}{2017} - \frac{S_{17}}{17} = 100$, find the value of $d$. | \frac{1}{10} |
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