problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
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Four. (15 points) The number of elements in set $S$ is denoted as $|S|$, and the number of subsets of set $S$ is denoted as $n(S)$. Given three non-empty finite sets $A$, $B$, and $C$ that satisfy the condition:
$$
\begin{array}{l}
|A|=|B|=2019, \\
n(A)+n(B)+n(C)=n(A \cup B \cup C) .
\end{array}
$$
Determine the maxim... | 2018 |
Evaluate the product \[\left(\sqrt{5}+\sqrt{6}+\sqrt{7}\right)\left(\sqrt{5}+\sqrt{6}-\sqrt{7}\right)\left(\sqrt{5}-\sqrt{6}+\sqrt{7}\right)\left(-\sqrt{5}+\sqrt{6}+\sqrt{7}\right).\] | 104 |
(1) 2011 is such a four-digit number, the sum of its digits is 4; there are $\qquad$ such four-digit numbers whose digits sum to 4. | 20 |
For even positive integer $n$ we put all numbers $1,2,...,n^2$ into the squares of an $n\times n$ chessboard (each number appears once and only once).
Let $S_1$ be the sum of the numbers put in the black squares and $S_2$ be the sum of the numbers put in the white squares. Find all $n$ such that we can achieve $\frac{S... | n = 103k |
32. Sully, the power generator specialist at Monster Power Company, has 7 fingers, and he counts in base 7; Mike has 9 fingers, and he counts in base 9. One day, the company's CEO Henry asked about their ages. Sully said: “My age in base 7 is a three-digit number $(\overline{a b c})_{7}$. ” Mike said: “My age in base 9... | 248 |
The contrapositive of the proposition "If $a^2+b^2=0$, then $a=0$ and $b=0$" is ( )
A: If $a^2+b^2=0$, then $a=0$ and $b\neq0$
B: If $a^2+b^2\neq0$, then $a\neq0$ or $b\neq0$
C: If $a=0$ and $b=0$, then $a^2+b^2\neq0$
D: If $a\neq0$ or $b\neq0$, then $a^2+b^2\neq0$ | D |
Let \\(f(x)\\) be an odd function defined on \\(\mathbb{R}\\), and for any real number \\(x\\), it always holds that \\(f(x+2)=-f(x)\\). When \\(x \in [0,2]\\), \\(f(x)=2x-x^{2}\\). Then, the value of \\(f(0)+f(1)+f(2)+\ldots+f(2018)=\\) \_\_\_\_\_\_\_\_\_\_ | 1 |
A Christmas tree seller sold spruces for 220 Kč, pines for 250 Kč, and firs for 330 Kč. In the morning, he had the same number of spruces, firs, and pines. In the evening, he had sold all the trees and earned a total of 36000 Kč for them.
How many trees did the seller sell that day?
Hint. Count in threes. | 135 |
In the Cartesian coordinate system $xOy$, with $O$ as the origin and the positive half-axis of $x$ as the basis for establishing a polar coordinate system, the polar equation of curve $C$ is $\rho^{2}-4\rho\sin \theta+3=0$, and the parametric equation of line $l$ is $\begin{cases} x=1- \dfrac { \sqrt {2}}{2}t \\ y=3+ \... | \dfrac {\pi}{2} |
Given the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ (where $a > 0$, $b > 0$), its right focus is at point F, the focal distance is 8, and the left vertex is at point A. There is a point B (0, b) on the y-axis such that the dot product of $\overrightarrow{BA}$ and $\overrightarrow{BF}$ equals $2a$. Find the ecce... | 2 |
Given $i$ is the imaginary unit, then $$( \frac {1+i}{1-i})^{2013}$$ equals to （ ）
A: i
B: -i
C: 1
D: -1 | \text{A} |
5. As shown in the figure, the three interior angles of a triangle are $(5 x+3 y)^{\circ} 、(3 x+20)^{\circ}$ and $(10 y+30)^{\circ}$, where $x 、 y$ are positive integers, then $x+y=$ $\qquad$ . | 15 |
Given that $α$ is an acute angle and $\tanα= \frac {4}{3}$, find the value of $\cos(2α+ \frac {π}{2})$
A: $$- \frac {24}{25}$$
B: $$- \frac {16}{25}$$
C: $$\frac {3}{5}$$
D: $$\frac {3}{4}$$ | \text{A} |
Find all real $x$ in the interval $[0, 2\pi)$ such that
\[27 \cdot 3^{3 \sin x} = 9^{\cos^2 x}.\] | x \in \left\{ \frac{7\pi}{6}, \frac{3\pi}{2}, \frac{11\pi}{6} \right\} |
"The Nine Chapters on the Mathematical Art" is an extremely rich ancient Chinese mathematical classic. In the first volume, "Field Measurement," it states: "Now, there is a fan-shaped field with a perimeter of 30 steps and a diameter of 16 steps. How much is the area of the field?" Translated into modern Chinese, it me... | A |
【Question 11】8 different non-zero natural numbers are arranged in ascending order. The average of the first 3 numbers is 9, the average of the 8 numbers is 19, and the average of the last 3 numbers is 29. What is the maximum difference between the second largest and the second smallest number? $\qquad$ _.
| 26 |
Among the following conditions, the one that cannot determine $\overrightarrow{a}$∥$\overrightarrow{b}$ is ( ).
A: $\overrightarrow{a}$∥$\overrightarrow{c}$, $\overrightarrow{b}$∥$\overrightarrow{c}$
B: $\overline{a}=-\overrightarrow{c}$, $\overrightarrow{b}=2\overrightarrow{c}$
C: $\overrightarrow{a}=-2\overline{b}... | D |
2. In the plane, a square $A B C D$ is given. Circle $k$ passes through points $A, B$ and is tangent to line $C D$. Let $M(M \neq B)$ be the intersection of circle $k$ and side $B C$. Determine the ratio $|C M|$ : $:|B M|$. | |CM|:|BM|=1:3 |
$\sqrt[3]{{(\pi-2)}^{3}}=\left(\ \ \right)$
A: $\pi -4$
B: $\pi -3$
C: $\pi -2$
D: $\pi -1$ | C |
Given a circle M that passes through the fixed point (0, 1) and whose center M moves along the parabola $y= \frac {1}{2}x^2$, if the chord intercepted by the x-axis on circle M is $|PQ|$, then the length of the chord $|PQ|$ is ( )
A: $\sqrt {2}$
B: 2
C: 3
D: 4 | \text{B} |
1. (3 points) $20.16 \times 32 + 2.016 \times 680=$ | 2016 |
Simplify the expression $(x+2)(y-2)-2(xy-2)$, where $x=1$, $y=2$. | 0 |
2. Given positive numbers $a$ and $b$, the following propositions are given:
(1) If $a=1, b=1$, then $\sqrt{a b} \leqslant 1$;
(2) If $a=\frac{1}{2}, b=\frac{5}{2}$, then $\sqrt{a b} \leqslant \frac{3}{2}$;
(3) If $a=2, b=3$, then $\sqrt{a b} \leqslant \frac{5}{2}$;
(4) If $a=1, b=5$, then $\sqrt{a b} \leqslant 3$.
Ba... | \left(\frac{13}{2}\right)^{2} |
11.5. On each of ten cards, a real number is written. For each non-empty set of these cards, the sum of all numbers written on the cards in this set was found. It is known that not all of the obtained sums are integers. What is the maximum possible number of integer sums that could have resulted? | 511 |
To demonstrate that the proposition "If $m \gt n$, then $m^{2} \gt n^{2}$" is false, the correct counterexample is:
A: $m=3$, $n=6$
B: $m=6$, $n=3$
C: $m=-3$, $n=-6$
D: $m=-6$, $n=-3$ | \text{C} |
Given vectors $\overrightarrow{a}=(\cos θ, \sin θ)$, $\overrightarrow{b}=( \sqrt {3}, 1)$, find the maximum value of $|\overrightarrow{a}- \overrightarrow{b}|$. | 3 |
A finite collection of positive real numbers (not necessarily distinct) is balanced if each number is less than the sum of the others. Find all $m \geq 3$ such that every balanced finite collection of $m$ numbers can be split into three parts with the property that the sum of the numbers in each part is less than the s... | m \neq 4 |
Find the solution set for the inequality $\frac{x+1}{x+2} < 0$. | (-2,-1) |
Given sets $A=\{x|x^{2}-3x+2 \lt 0\}$ and $B=\{x|x(x-m) \gt 0\}$, if $A\cap B=\varnothing$, what is the range of real number $m$?
A: $(-\infty ,0]$
B: $[0,2]$
C: $[2,+\infty)$
D: $[0,1]$ | C |
If the solution to the system of equations $\left\{\begin{array}{l}{2x+y=\square}\\{x-2y=3}\end{array}\right.$ is $\left\{\begin{array}{l}x=5\\ y=\triangle\end{array}\right.$, then the numbers represented by "$\square$" and "$\triangle$" are respectively ( )
A: $14$, $4$
B: $11$, $1$
C: $9$, $-1$
D: $6$, $-4$ | \text{B} |
5. In a certain region, a change of leadership election is being held, and 3 people need to be selected from Jia, Yi, Bing, and Ding to take on 3 different roles. It is stipulated that Jia, Yi, and Bing, who held positions in the previous term, cannot be re-elected to their original roles. The number of different possi... | 11 |
2. 15 students join a summer course. Every day, 3 students are on duty after school to clean the classroom. After the course, it was found that every pair of students have been on duty together exactly once. How many days does the course last for?
(1 mark)
2. 15 students join a summer course. Every day, 3 students are ... | 35 |
Prove that the sum of an odd number of vectors of length 1, of common origin $O$ and all situated in the same semi-plane determined by a straight line which goes through $O,$ is at least 1. | 1 |
If the proposition "There exists an $x \in \mathbb{R}$ such that $x^2 - mx - m < 0$" is false, then the range of values for the real number $m$ is $\_\_\_\_\_\_$. | -4 \leq m \leq 0 |
(5) Let $A_{1}, A_{2}, \cdots, A_{n}$ be $n$ different subsets of the set $S=\{1,2, \cdots, n\}$ $(n \geqslant 4)$. To represent these subsets, construct an $n$ by $n$ array of numbers, where the number in the $i$-th row and $j$-th column is defined as $a_{i j}=\left\{\begin{array}{ll}0, & i \notin A_{j}, \\ 1, & i \in... | C |
22. There is a square pool, as shown in the blue part in the figure below. A lawn 8 meters wide is built around it, and the area of the lawn is 480 square meters. The side length of the square pool is $\qquad$ meters. | 7 |
1. Let the points $A_{i}\left(x_{i}, y_{i}\right), i=1,2,3,4$ be distinct and lie on the hyperbola $x y=1$. If these four points lie on a circle, prove that $x_{1} x_{2} x_{3} x_{4}=1$. | x_{1}x_{2}x_{3}x_{4}=1 |
$n$ boxes initially contain $1,2, \ldots, n$ marbles respectively ( $n \geq 1$ ). Charlotte first adds a marble to each box. Then she adds a marble to each box in which the number of marbles is divisible by 2 , then a marble to each box in which the number of marbles is divisible by 3 , and so on, until she adds a marb... | all\n\such\that\n+1\is\ |
Example 6: On the table, there are 1989 same-sized circular paper pieces placed without overlapping. How many different colors are needed at least to color each circular paper piece so that, no matter how these paper pieces are placed, it is always possible to color them in such a way that any two touching circular pap... | 4 |
10.3. Given a right triangle $A B C$. A point $M$ is taken on the hypotenuse $A C$. Let $K, L$ be the centers of the circles inscribed in triangles $A B M$ and $C B M$ respectively. Find the distance from point $M$ to the midpoint of $K L$, if the radius $R$ of the circle circumscribed around triangle $B K L$ is known. | \frac{R\sqrt{2}}{2} |
Given the functions $f(x)=x^{2}-2x+2$ and $g(x)=-x^{2}+ax+b- \frac {1}{2}$, one of their intersection points is $P$. The tangent lines $l_{1}$ and $l_{2}$ to the functions $f(x)$ and $g(x)$ at point $P$ are perpendicular. Find the maximum value of $ab$. | \frac{9}{4} |
Let's check the following equalities, look for regularities in the table, and based on that, write down the next two rows of the table.
$$
\begin{aligned}
3^{2}+4^{2} & =5^{2} \\
10^{2}+11^{2}+12^{2} & =13^{2}+14^{2} \\
21^{2}+22^{2}+23^{2}+24^{2} & =25^{2}+26^{2}+27^{2} \\
36^{2}+37^{2}+38^{2}+39^{2}+40^{2} & =41^{2}... | \begin{aligned}55^{2}+56^{2}+57^{2}+58^{2}+59^{2}+60^{2}&=61^{2}+62^{2}+63^{2}+64^{2}+65^{2}\\78^{2}+79^{2}+80^{2}+81^{2}+8 |
The value of $(\sqrt{100}-\sqrt{36})^{2}$ is
(A) 16
(B) 256
(C) 8
(D) 1024
(E) 4096 | 16 |
3[ Equilateral (Equiangular) Triangle ]
The angle with vertex $C$ is $120^{\circ}$. A circle of radius $R$ touches the sides of the angle at points $A$ and $B$. Find $A B$. | R |
6. From the 11 positive integers $1,2,3, \cdots, 11$, if 3 different positive integers $a, b, c$ are randomly selected, then the probability that their product $a b c$ is divisible by 4 is $\qquad$. | \frac{20}{33} |
Example 15 (1999 Vietnam Mathematical Olympiad) Solve the system of equations
$$
\left\{\begin{array}{l}
\left(1+4^{2 x-y}\right) \cdot 5^{1-2 x+y}=1+2^{2 x-y+1}, \\
y^{3}+4 x+1+\ln \left(y^{2}+2 x\right)=0 .
\end{array}\right.
$$ | 0,-1 |
Example 2-44 How many regions does $n$ lines divide the plane into? Assume no $=$ lines intersect at a common point, and each pair of lines intersects. | 1+\frac{1}{2}n(n+1) |
In the Cartesian coordinate system $(xOy)$, an ellipse $(C)$ is given by the equation $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ with an eccentricity of $e = \sqrt{\frac{2}{3}}$. The maximum distance from any point on the ellipse to point $Q(0, 2)$ is 3.
1. Find the equation of the ellipse $(C)$.
2. Determine... | \frac{1}{2} |
In a triangular pyramid, the perimeters of all its faces are equal. Find the area of the total surface of this pyramid if the area of one of its faces is $S$.
# | 4S |
## Task B-2.4.
Pinocchio tells the truth on Mondays and Tuesdays, always lies on Saturdays, and on the other days of the week, he either tells the truth or lies. In response to the question "What is your favorite subject at school?" over six consecutive days of the week, he gave the following answers in order: "Histor... | History |
$18 \cdot 63$ rectangle $A B C D$ has $\angle C$ trisected by $C E, C F$, where $E$ is on $A B$ and $F$ is on $A D$, and $A F=2, B E$ $=6$. Then $S_{A B C D}$ is closest to
(A) 110 .
(B) 120
(C) 130 .
(D) 140 .
(E) 150 . | 150 |
Let triangle $ABC$ be a right triangle in the xy-plane with a right angle at $C$. Given that the length of the hypotenuse $AB$ is 60, and that the medians through $A$ and $B$ lie along the lines $y=x+3$ and $y=2x+4$ respectively, find the area of triangle $ABC$. | 400 |
The origin and the point $(1,1)$ have only one point inside the plane region represented by the line $2x-y+a > 0$. The range of values for $a$ is. | (-1,0] |
Given that $a, b \in \mathbb{R}$, which of the following statements is true?
A: If $a \frac{1}{b}$
D: If $a < b$, then $a^n < b^n$ (where $n \in \mathbb{N}^*$, $n \geq 2$) | C |
If $2\sin77°-\sin17°=\lambda\sin73°$, then $\lambda=$ （ ）
A: $\sqrt {3}$
B: 1
C: $-\sqrt {3}$
D: -1 | A |
9.2 The banker leaves home, and at that exact moment, a car from the bank arrives to take him to the bank. The banker and the car always leave at the same time, and the car always travels at a constant speed. One day, the banker left home 55 minutes earlier than usual and, for fun, started walking in the direction oppo... | 12 |
[ Properties and characteristics of an isosceles triangle. ]
The median $A M$ of triangle $A B C$ is perpendicular to its bisector $B K$. Find $A B$, if $B C=12$. | 6 |
## Problem Statement
Find the coordinates of point $A$, which is equidistant from points $B$ and $C$.
$A(x ; 0 ; 0)$
$B(-2 ; 0 ; 6)$
$C(0 ;-2 ;-4)$ | A(-5;0;0) |
Given vectors $\overrightarrow{m}=(2a\cos x,\sin x)$, $\overrightarrow{n}=(\cos x,b\cos x)$, and the function $f(x)= \overrightarrow{m}\cdot \overrightarrow{n}- \frac { \sqrt {3}}{2}$. The intercept of function $f(x)$ on the $y$-axis is $\frac { \sqrt {3}}{2}$, and the coordinates of the highest point closest to the $y... | \frac {5\pi}{6} |
Calculate the result of $\frac{2x}{x^{2}-9y^{2}}-\frac{1}{x-3y}$ is ____. | \frac{1}{x+3y} |
Given the set $M=\{0,1\}$, and the set $N=\{x|x^{2}+x=0\}$, then the set $M\cup N$ equals to ( ).
A: $0$
B: $\{0\}$
C: $\varnothing$
D: $\{-1,0,1\}$ | D |
$77 a$ is the arithmetic mean of the real numbers $2 x, 2 y$, $\sqrt{|a|}$ is the geometric mean of $x, y$, then the range of the real number $a$ is
A. $(0,4]$
B. $(-\infty, 0) \cup[4,+\infty)$
C. $[-4,0) \cup(0,4]$
D. $(-\infty,-4] \cup(4,+\infty)$ | \leqslant-4or\geqslant4 |
Given $f(x) = x^2 + 2xf'(1)$, where $a$ is a real number, then $f'(a) =$ ? | 2a-4 |
3. In $\triangle A B C$, $\angle B=60^{\circ}, \angle C=75^{\circ}, S_{\triangle A B C}=\frac{1}{2}(3+\sqrt{3}), B C=$ | 2 |
2. Graphically solve the equation $\log _{2}(x-1)+1=-\frac{3}{2} x+4$. Verify the solution by calculation. | 2 |
2. Let the complex number $z \neq 1, z^{11}=1$. Then $z+z^{3}+z^{4}+z^{5}+z^{9}=$ $\qquad$ | \frac{-1 \pm \sqrt{11} \mathrm{i}}{2} |
Given a sequence of numbers $a_{1}$, $a_{2}$, $a_{3}$, $\ldots $, with the following pattern: $a_{2n+1}=a_{n}+a_{n+1}$, $a_{2n}=a_{n}$ (where $n$ is a positive integer). If $a_{1}=1$, then the value of $a_{61}$ is:
A: $9$
B: $10$
C: $11$
D: $12$ | A |
2. Given the set $M=\{(x, y)|| x|+| y \mid \leqslant 1\}$, the set $N=\left\{(x, y)\left|x^{2}+y^{2} \leqslant\right| x|+| y \mid\right\}$, then
A. $M=N$
B. $M \subseteq N$
C. $N \subseteq M$
D. None of the above are correct | B |
If for any $x \in (1, 2]$, $\log_{a}x > (x-1)^2$, then the range of values for $a$ is. | (1, 2) |
2. In the addition shown, each of the letters $D, O, G, C, A$ and $T$ represents a different digit.
$$
\begin{array}{r}
D O G \\
+C A T \\
\hline 1000
\end{array}
$$
What is the value of $D+O+G+C+A+T$ ? | 28 |
Given the function $f(x)= \sqrt{3}\sin x\cos x+\sin^2x$.
(Ⅰ) Find the increasing interval of the function $f(x)$.
(Ⅱ) In triangle $ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. The angle bisector of $A$ intersects $BC$ at $D$, $f(A)= \frac{3}{2}$, $AD= \sqrt{2}BD=2$, find $\... | \frac{ \sqrt{6}- \sqrt{2}}{4} |
3. If point $P\left(x_{0}, y_{0}\right)$ is such that the chord of contact of the ellipse $E: \frac{x}{4}+y^{2}=1$ and the hyperbola $H: x^{2}-\frac{y^{2}}{4}=1$ are perpendicular to each other, then $\frac{y_{0}}{x_{0}}=$ $\qquad$ | \pm 1 |
[ Trigonometric Identities. ]
Show that $\sin 36^{\circ}=1 / 4 \sqrt{10-2 \sqrt{5}}$.
# | \sin36=\frac{1}{4}\sqrt{10-2\sqrt{5}} |
The equation of the tangent line to the curve $y=x^3-3x^2+1$ at point $P(0,1)$ is ( )
A: $y=x+1$
B: $y=1$
C: $x=0$
D: Does not exist | \text{B} |
1. Let real numbers $a$, $b$, $c$ satisfy
$$
\begin{array}{l}
a^{2}-b c-2 a+10=0, \\
b^{2}+b c+c^{2}-12 a-15=0 .
\end{array}
$$
Then the range of values for $a$ is | 1 \leqslant a \leqslant 5 |
A particle projected vertically upward reaches, at the end of $t$ seconds, an elevation of $s$ feet where $s = 160 t - 16t^2$. The highest elevation is:
$\textbf{(A)}\ 800 \qquad \textbf{(B)}\ 640\qquad \textbf{(C)}\ 400 \qquad \textbf{(D)}\ 320 \qquad \textbf{(E)}\ 160$ | 400 |
Given a sequence of positive terms $\{a_{n}\}$ that satisfies $a_{m}\cdot a_{n}=2a_{m+n}$, where $m$, $n\in N^{*}$, and $a_{1}=1$, find the sum of the first $5$ terms of the sequence, denoted as $S_{5}=\left(\ \ \right)$
A: $\frac{31}{32}$
B: $\frac{63}{32}$
C: $\frac{31}{16}$
D: $\frac{47}{32}$ | C |
The domain of the function $y=\log_{2}(x^{2}-2x-3)$ is ( )
A: $(-\infty, -1) \cup (3, +\infty)$
B: $[-1, 3]$
C: $(-\infty, -1] \cup [3, +\infty)$
D: $(-1, 3)$ | A |
Given the inequality $|x+1|+|x-1|\leqslant 2$, let $M$ be the solution set.
(I) Find the set $M$;
(II) If $x\in M$, $|y|\leqslant \dfrac {1}{6}$, and $|z|\leqslant \dfrac {1}{9}$, prove that $|x+2y-3z|\leqslant \dfrac {5}{3}$. | \dfrac {5}{3} |
Problem 9.3. Find all four-digit positive integers $m$ less than 2005 for which there exists a positive integer $n<m$, such that $m n$ is a perfect square and $m-n$ has at most three distinct positive divisors.
Peter Boyvalenkov, Ivailo Kortezov | 1764,1600,1369,1296,1156,1900,1377 |
A2. Polona wants to draw three lines that intersect at the same point (see figure), such that the angles between them satisfy $\beta=2 \alpha$ and $\alpha=3 \gamma$. How many degrees must the angle $\alpha$ be?
(A) 18
(B) 36
(C) 50
(D) 54
(E) 60 | 54 |
5. Given the number $5300 \ldots 0035$ (100 zeros). It is required to replace some two zeros with non-zero digits so that after the replacement, the resulting number is divisible by 495. In how many ways can this be done? | 22100 |
Given that the even function f(x) is monotonically decreasing in the interval $[0, +\infty)$, if f(x-2) > f(3), then the range of x is _____. | (-1, 5) |
Simplify the following expressions:
① $$\overrightarrow {AB}+ \overrightarrow {BC}+ \overrightarrow {CA}$$;
② $$\overrightarrow {AB}- \overrightarrow {AC}+ \overrightarrow {BD}- \overrightarrow {CD}$$;
③ $$\overrightarrow {FQ}+ \overrightarrow {QP}+ \overrightarrow {EF}$$\- $$\overrightarrow {EP}$$
④ $$\overrig... | D |
6.116. $\left\{\begin{array}{l}\sqrt[4]{u}-\sqrt[4]{v}=1 \\ \sqrt{u}+\sqrt{v}=5\end{array}\right.$ | (16;1) |
$30 \cdot 31$ makes $n^{5}-5 n^{3}+4 n+7$ a perfect square for natural numbers $n$ with the value
(A) There is exactly one.
(B) There are finitely many, but more than one.
(C) There are infinitely many.
(D) None exist.
(China Fujian Province Fuzhou City High School Mathematics Competition, 1990) | D |
Given that the lines $l_{1}$: $ax+2y+3a-2=0$ and $l_{2}$: $x+\left(a+1\right)y+4=0$ are parallel, the value of the real number $a$ is ____. | 1 |
## Task 36/78
Choose any 1979-digit natural number $a$ that is divisible by 9, and form the cross sum $z$ from the cross sum $y$ of the cross sum $x$ of $a$. How large is $z$? | 9 |
2. If
$$
\begin{array}{l}
a+b-2 \sqrt{a-1}-4 \sqrt{b-2} \\
=3 \sqrt{c-3}-\frac{1}{2} c-5,
\end{array}
$$
then, the value of $a+b+c$ is $\qquad$ | 20 |
The monotonic increasing interval of the function $f(x)=\log_{\frac{1}{2}}(x^2-4)$ is _______. | (-\infty,-2) |
(1) Let $\left(\frac{\sqrt{3}}{2}+\frac{x}{2} \mathrm{i}\right)^{2008}=f(x)+\mathrm{i} g(x)$, where $f(x) 、 g(x)$ are both real-coefficient polynomials, then the sum of the coefficients of $f(x)$ is ( ).
(A) $-\frac{\sqrt{3}}{2}$
(B) $\frac{\sqrt{3}}{2}$
(C) $-\frac{1}{2}$
(D) $\frac{1}{2}$ | -\frac{1}{2} |
Given an arithmetic sequence $\{a_{n}\}$ with the sum of the first $n$ terms being $S_{n}$, and the solution set of the inequalities $(S_{n}-2022)(S_{n+1}-2022) \lt 0$ and $(S_{n}-2023)(S_{n+1}-2023) \lt 0$ with respect to the positive integer $n$ is denoted as $M$. Proposition $\alpha$: The number of elements in the s... | B |
3. In the Caribbean, loggerhead turtles lay three million eggs in twenty thousand nests. On average, how many eggs are in each nest?
A 15
B 150
C 1500
D 15000
E 150000 | 150 |
If $xy\cdot \left(\ \ \right)=-x^{3}y^{2}$, then the algebraic expression that should be filled in the parentheses is ( ).
A: $xy$
B: $-xy$
C: $-x^{2}y$
D: $-y$ | C |
Given the sequence $\sqrt{3}, 3, \sqrt{15}, \ldots, \sqrt{3(2n-1)}, \ldots$, then $9$ is the $\boxed{\text{th}}$ term of this sequence.
A: $12$
B: $13$
C: $14$
D: $15$ | C |
An insurance company's revenues increased by $25 \%$ and its expenses by $15 \%$ compared to the previous year in 2006. The company's profit (revenue - expenses) increased by 40\%. What percentage of the revenues were the expenses in 2006? | 55.2 |
Given that $a \in \mathbb{R}$, compare the size of $\frac{1}{1-a}$ and $1+a$. | \frac{1}{1-a} < 1+a |
Given that the line $x - y + a = 0$ intersects the circle $x^2 + y^2 + 2x - 4y + 2 = 0$ at points A and B, and $|AB| = 2$, find the value(s) of $a$.
A: $-1$ or $1$
B: $1$ or $5$
C: $-1$ or $3$
D: $3$ or $5$ | B |
Given the curve $y=\frac{1}{3}x^{3}+\frac{4}{3}$.
(1) Find the equation of the tangent line to the curve at $x=2$;
(2) Find the equation of the tangent line to the curve passing through the point $(2,\,4)$. | x-y+2=0 |
Let $m$ and $n$ be the points obtained from two consecutive rolls of a dice, and let vectors $\overrightarrow{a} = (m, n)$ and $\overrightarrow{b} = (1, -1)$. The probability that the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is acute is $\_\_\_\_\_\_\_\_\_.$ | \frac{5}{12} |
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