problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
|---|---|
Calculate $(5-5i) + (-2-i) - (3+4i) = $
A: $-2i$
B: $-10i$
C: $10$
D: $-2$ | \text{B} |
Given propositions $p$: $ \begin{cases} x+2\geqslant 10 \\ x-10\leqslant 0 \\ \end{cases} $ and $q$: $-m\leqslant x\leqslant 1+m$, if $\lnot p$ is a necessary but not sufficient condition for $\lnot q$, then the range of the real number $m$ is _______. | m\geqslant 9 |
The interval of monotonic decrease for the function $y = \log_3 (4 - x^2)$ is ____. | (0, 2) |
To realize the ancient dreams of "a scholar does not need to leave his home to know the world's affairs" and "strategize within the tent and secure victory thousands of miles away," one must rely on ( ).
A: Remote Sensing
B: Global Positioning System
C: Geographic Information Technology
D: Geographic Information Sys... | \text{C} |
Task B-3.4. Helena invited a certain number of children to her birthday party. The age of one child was one eighth of the sum of the ages of the remaining children. This age ratio did not change in later years. How many children were at the party? | 9 |
9. Given the set $U=\{1,2,3,4,5\}, I=\{X \mid X \subseteq U\}$, if two different elements $A, B$ are randomly selected from set $I$, then the probability that $A \cap B$ contains exactly 3 elements is $\qquad$ . | \frac{5}{62} |
If the positive real numbers $x$ and $y$ satisfy $10x+2y+60=xy$, then the minimum value of $xy$ is. | 180 |
## Problem Statement
Calculate the lengths of the arcs of the curves given by equations in a rectangular coordinate system.
$$
y=\frac{e^{x}+e^{-x}}{2}+3, \quad 0 \leq x \leq 2
$$ | \frac{1}{2}(e^{2}-e^{-2}) |
## 8. The Emir and His Chauffeur
Emir Ben Sidi Mohammed always drives from his palace to the airport at the same speed along a magnificent highway that runs through the desert. If his chauffeur increased the average speed of the car by $20 \mathrm{km} /$ h, the Emir would save 2 minutes, and if the chauffeur drove at ... | 20 |
The Titius-Bode law, abbreviated as the "Bode law," is an empirical rule that represents the average distance between each planet and the sun. It was discovered by a German high school teacher, David Titius, in 1766. Later, it was summarized by the director of the Berlin Observatory, Bode, into an empirical formula as ... | D |
In triangle $ABC$, if $\angle C=90^\circ$, $\sin A= \frac{1}{2}$, and $AB=2$, then the perimeter of $\triangle ABC$ is ______. | 3+ \sqrt{3} |
Given the parabola $y=ax^{2}-2ax\left(a\neq 0\right)$.
$(1)$ Find the coordinates of the vertex of this parabola (expressed in terms of $a$);
$(2)$ Two distinct points on the parabola are $\left(-1,m\right)$ and $(x_{0},n)$, if $m=n$, directly determine the value of $x_{0}$;
$(3)$ Points $A(m-1,y_{1})$ and $B(m+2... | m < \frac{1}{2} |
A moving circle P is externally tangent to circle $C_1: x^2+y^2+6x+5=0$ and internally tangent to circle $C_2: x^2+y^2-6x-91=0$. Let the trajectory of the center P of the moving circle be curve C. If point M is any point on curve C, then the maximum value of $|MC_2|$ is \_\_\_\_\_\_. | 9 |
Cyclic quadrilateral $ABCD$ satisfies $\angle ADC = 2 \cdot \angle BAD = 80^\circ$ and $\overline{BC} = \overline{CD}$. Let the angle bisector of $\angle BCD$ meet $AD$ at $P$. What is the measure, in degrees, of $\angle BP D$?
| 60^\circ |
227. Find the triangle. The sides and height of a certain triangle are expressed by four consecutive integers. What is the area of this triangle? | 84 |
Given two lines $l_1: y = m$ and $l_2: y = \frac{8}{2m+1}$ ($m > 0$), line $l_1$ intersects the graph of the function $y = |\log_2 x|$ from left to right at points $A$ and $B$, and line $l_2$ intersects the graph of the function $y = |\log_2 x|$ from left to right at points $C$ and $D$. The lengths of the projections o... | 8\sqrt{2} |
11. Given the parabola $C: y^{2}=2 p x(p>0)$ intersects with the moving line $l: y=k x+b(k, b$ are parameters, and $k \neq 0$, $b \neq 0)$ at points $A\left(x_{1}, y_{1}\right)$ and $B\left(x_{2}, y_{2}\right)$, and the origin of the Cartesian coordinate system is $O$. Let the slopes of lines $O A$ and $O B$ be $k_{O A... | (-\frac{2\sqrt{3}}{3}p,0) |
Given two points P and Q on the hyperbola $x^2-\frac{y^2}{2}=1$ that are symmetric about the line $y=-x+b$, and the midpoint of PQ lies on the parabola $y^2=8x$. Find the value of the real number $b$. | 0 \text{ or } 6 |
Putnam 1998 Problem A1 A cone has circular base radius 1, and vertex a height 3 directly above the center of the circle. A cube has four vertices in the base and four on the sloping sides. What length is a side of the cube? Solution | \frac{3\sqrt{2}}{3+\sqrt{2}} |
Three. (25 points) If the pair of positive integers $(a, x)$ satisfies
$$
\sqrt{\frac{a-x}{1+x}}=\frac{a-x^{2}}{1+x^{2}} \neq x \text {, }
$$
find all positive integers $a$ that meet the requirement. | 1 |
8、On a birthday card printed with April 29, a child erased two positive integers $x, y$ between 4 and 29, resulting in a five-digit number $\overline{4 x y 29}$, and found that it is exactly the square of the positive integer $T$ corresponding to their birthday: $\overline{4 x y 29}=T^{2}$; then the number $T$ correspo... | 223 |
2. $n$ is a positive integer, the number of integer solutions to $x^{2}+x y+y^{2}=n$ is ( ).
(A) a multiple of 4
(B) a multiple of 6
(C) a multiple of 2
(D) a multiple of 8 | B |
In the middle of the school year, $40\%$ of Poolesville magnet students decided to transfer to the Blair magnet, and $5\%$ of the original Blair magnet students transferred to the Poolesville magnet. If the Blair magnet grew from $400$ students to $480$ students, how many students does the Poolesville magnet have after... | 170 |
Example 8 Given 4 spheres in space, their radii are $2, 2, 3, 3$, and each sphere is externally tangent to the other 3 spheres. There is another smaller sphere that is externally tangent to these 4 spheres. Find the radius of the smaller sphere. | \frac{6}{11} |
Define $a \otimes b = \begin{cases}
a^{2}+b, & \text{if } a > b \\
a+b^{2}, & \text{if } a \leq b
\end{cases}$. If $a \otimes (-2) = 4$, then $a = \_\_\_\_\_\_$. | \sqrt{6} |
$15 \cdot 34$ For positive real numbers $x$ and $y$ define $x * y=\frac{x \cdot y}{x+y}$, then
(A) “*” is commutative, but not associative.
(B) “*” is associative, but not commutative.
(C) “*” is neither commutative nor associative.
(D) “*” is both commutative and associative.
(E) None of these.
(25th American High Sch... | D |
In the ancient Chinese mathematical classic "The Nine Chapters on the Mathematical Art," there is a passage about the method of exhausting a circle: "When you divide it as thinly as possible, what is lost is negligible. By dividing it again and again until it cannot be divided any further, it merges with the circumfere... | t=\frac{\sqrt{5}+1}{2} |
Which of the following options can form a set?
A: Students with a higher level of basketball skills in the school
B: Tall trees in the campus
C: All the countries in the European Union in 2012
D: Economically developed cities in China | C |
6. Unlocking the communicator is done by entering a 4-digit numerical code on the touch screen. The arrangement of digits on the keyboard changes after entering the code depending on a random prime number $k$ from 7 to 2017, and the digit $i$ is displayed as $a_{i}$, which is the last digit of the number $i k$. The use... | 3212 |
6. (15 points) Three material points with masses $m_{1}=100$ g, $m_{2}=200$ g, and $m_{3}=400$ g are arranged sequentially on a straight line. The distance between points 1 and 2 is 50 cm. The distance between points 2 and 3 is 200 cm. Determine the distance from point 1 to the center of mass of this system of three po... | 1.57\mathrm{~} |
Quadrilateral $ABCD$ is inscribed into a circle with center $O$. The bisectors of its angles form a cyclic quadrilateral with circumcenter $I$, and its external bisectors form a cyclic quadrilateral with circumcenter $J$. Prove that $O$ is the midpoint of $IJ$. | O \text{ is the midpoint of } IJ |
1. The function $f(x)$ defined on $\mathbf{R}$ satisfies
$$
f\left(\frac{2 a+b}{3}\right)=\frac{2 f(a)+f(b)}{3}(a 、 b \in \mathbf{R}) \text {, }
$$
and $f(1)=1, f(4)=7$.
Then $f(2015)=$ $\qquad$ . | 4029 |
Given proposition \\(p\\): For all \\(x \in \mathbb{R}\\), \\(3^{x} > 0\\), then
A: \\(\neg p\\): There exists an \\(x \in \mathbb{R}\\) such that \\(3^{x} \leqslant 0\\)
B: \\(\neg p\\): For all \\(x \in \mathbb{R}\\), \\(3^{x} \leqslant 0\\)
C: \\(\neg p\\): There exists an \\(x \in \mathbb{R}\\) such that \\(3^{x... | \text{A} |
(1) $\sin 75^{\circ}=$ _______
(2) If the two real roots of the equation $x^{2}+(m-1)x+m^{2}-2=0$ are such that one is less than $-1$ and the other is greater than $1$, then the range of values for the real number $m$ is _______
(3) The graph of the function $f(x)=\log_{a}(x+3)-1$ $(a > 0, a \neq 1)$ always passes th... | \{-1,2\} |
## Task 1 - 330831
At the Sprachfix School in Lernhausen, there are 120 students. Each of them learns at least one of the languages English, Latin, French. The reporter Schreibklug learns the following facts:
(1) For exactly 102 of the 120 students: Each of these 102 students learns at least one of the languages Engl... | 81 |
7. $(5-7,8,9)$ a) In a $3 \times 4$ table, numbers from 1 to 12 need to be arranged such that the difference between any two numbers in the same row is divisible by 3, and the difference between any two numbers in the same column is divisible by 4. An example of such an arrangement is:
| 1 | 4 | 7 | 10 |
| :---: | :--... | No |
4. Let the incircle of $\triangle A B C$ touch side $B C$ at point $D$, dividing $B C$ into segments of lengths $m, n$. Also, $\angle B A C$ $=60^{\circ}$, then the area of $\triangle A B C$ is $\qquad$ | \sqrt{3} m n |
### 7.340
$$
\left\{\begin{array}{l}
x^{\log _{3} y}+2 y^{\log _{3} x}=27 \\
\log _{3} y-\log _{3} x=1
\end{array}\right.
$$ | (3;9),(\frac{1}{9};\frac{1}{3}) |
$8 \cdot 6$ Let the sequence $a_{0}, a_{1}, a_{2}, \cdots$ satisfy
$$a_{0}=a_{1}=11, a_{m+n}=\frac{1}{2}\left(a_{2 m}+a_{2 n}\right)-(m-n)^{2}, m, n \geqslant 0 .$$
Find $a_{45}$. | 1991 |
Problem 8.5. A field was partially planted with corn, oats, and millet. If the remaining part is completely planted with millet, then millet will occupy half of the entire field. If the remaining part is equally divided between oats and corn, then oats will occupy half of the entire field. By what factor will the amoun... | 3 |
6. In $\triangle A B C$, $\angle A$ is an obtuse angle, $O$ is the orthocenter, $A O=B C$,
then the value of $\cos (\angle O B C+\angle O C B)$ is
(A) $-\frac{\sqrt{2}}{2}$;
(B) $\frac{\sqrt{2}}{2}$;
(C) $\frac{\sqrt{3}}{2}$;
(D) $-\frac{1}{2}$. | -\frac{\sqrt{2}}{2} |
Example 2. Find the smallest positive integer $n$, such that in any 9-vertex and $n$-edge graph with edges colored in two colors, there must exist a monochromatic triangle.
(33rd IMO, Problem 3) | 33 |
Given the function f(x) = $$\begin{cases} \overset{10^{x-1},x\leq 1}{lgx,x>1}\end{cases}$$, find f(f(10)). | 1 |
Suppose $b > 1$ is a real number where $\log_5 (\log_5 b + \log_b 125) = 2$. Find $log_5 \left(b^{\log_5 b}\right) + log_b \left(125^{\log_b 125}\right).$ | 619 |
Bakayev E.V.
Forty children were holding hands in a circle. Out of them, 22 were holding hands with a boy and 30 were holding hands with a girl. How many girls were in the circle?
# | 24 |
5. Let $x, y$ be real numbers. Then the maximum value of $\frac{2 x+\sqrt{2} y}{2 x^{4}+4 y^{4}+9}$ is . $\qquad$ | \frac{1}{4} |
When we learn about the translation of linear and quadratic function graphs, we know that by shifting the graph of the linear function $y=2x$ up by $1$ unit, we get the graph of $y=2x+1$; by shifting the graph of the quadratic function $y=x^{2}+1$ to the left by $2$ units, we get the graph of $y=(x+2)^{2}+1$. If we shi... | y=\frac{6}{x-3} |
In $\triangle ABC$, $B= \frac {2\pi}{3}, C= \frac {\pi}{6}, a=5$, find the length of the longest side of the triangle. | 5 \sqrt {3} |
Which of the following functions is increasing on the interval $(0, +\infty)$?
A: $y=x^2+1$
B: $y=3-2x$
C: $y= \frac {1}{x}$
D: $y=-x^2+1$ | A |
1. Buses from Moscow to Oryol depart at the beginning of each hour (at 00 minutes). Buses from Oryol to Moscow depart in the middle of each hour (at 30 minutes). The journey between the cities takes 5 hours. How many buses from Oryol will the bus that left from Moscow meet on its way? | 10 |
10-8. In trapezoid $A B C D$, the bases $A D$ and $B C$ are 8 and 18, respectively. It is known that the circumcircle of triangle $A B D$ is tangent to the lines $B C$ and $C D$. Find the perimeter of the trapezoid. | 56 |
6. In the Thrice-Tenth Kingdom, there are 17 islands, each inhabited by 119 people. The inhabitants of the kingdom are divided into two castes: knights, who always tell the truth, and liars, who always lie. During the census, each person was first asked: "Excluding you, are there an equal number of knights and liars on... | 1013 |
4. Let $a>0$ be a real number for which there exist exactly three natural numbers $n$ such that $1<an<2$. How many natural numbers $m$ can there be, for which $2<am<3$? | 2,3,or4 |
11. (16 points) Positive real numbers $x, y, z$ satisfy
$$
\left\{\begin{array}{l}
\frac{2}{5} \leqslant z \leqslant \min \{x, y\}, \\
x z \geqslant \frac{4}{15}, \\
y z \geqslant \frac{1}{5} .
\end{array}\right.
$$
Find the maximum value of $\frac{1}{x}+\frac{2}{y}+\frac{3}{z}$. | 13 |
The solution set of the inequality $\dfrac{2-x}{x+4} > 0$ is ______. | (-4,2) |
Given a point M on the parabola $y^2 = 4x$ whose distance to the focus is 3, find the x-coordinate of point M, denoted as $x = \_\_\_\_\_\_\_$. | 2 |
Two sides of a regular polygon of $n$ sides when extended meet at $28$ degrees. What is smallest possible value of $n$ | 45 |
Let $ABCD$ be a square with side length $4$. Consider points $P$ and $Q$ on segments $AB$ and $BC$, respectively, with $BP=3$ and $BQ=1$. Let $R$ be the intersection of $AQ$ and $DP$. If $BR^2$ can be expressed in the form $\frac{m}{n}$ for coprime positive integers $m,n$, compute $m+n$.
[i]Proposed by Brandon Wang[/i... | 177 |
Given a function $f(x)$ defined on $\mathbb{R}$ such that for any $x, y \in \mathbb{R}$, $f(x+y) = f(x) + f(y)$. When $x > 0$, $f(x) < 0$, and $f(3) = -4$.
$(1)$ Find the value of $f(0)$;
$(2)$ Determine and prove the parity of the function $f(x)$ on $\mathbb{R}$;
$(3)$ Solve the inequality $f(t-1) + f(t) < -8$. | (\frac{7}{2}, +\infty) |
I2.1 If $x, y$ and $z$ are positive real numbers such that $\frac{x+y-z}{z}=\frac{x-y+z}{y}=\frac{-x+y+z}{x}$ and $a=\frac{(x+y) \cdot(y+z) \cdot(z+x)}{x y z}$, find the value of $a$.
I2.2 Let $u$ and $t$ be positive integers such that $u+t+u t=4 a+2$. If $b=u+t$, find the value of $b$. | 10 |
In the November monthly exam for senior high school grade 3 in our school in 2015, the scores in science mathematics followed a normal distribution with a mean of 90 and a standard deviation of σ (σ > 0). The statistical results showed that P(60 ≤ ξ ≤ 120) = 0.8. Assuming that 780 students from our school participated ... | 78 |
A chess tournament is held with the participation of boys and girls. The girls are twice as many as boys. Each player plays against each other player exactly once. By the end of the tournament, there were no draws and the ratio of girl winnings to boy winnings was $7/9$. How many players took part at the tournament? | 33 |
6. The prime numbers $p_{1}, p_{2}, p_{3}$ form an arithmetic sequence with a common difference of 80. The number of such sequences is $(\quad)$.
A. 1
B. 2
C. 3
D. infinitely many | 1 |
## Subject III (7 points)
In an acute-angled triangle, a height and a median constructed from different vertices form an angle with a measure of $60^{\circ}$. Show that the height and the median have the same length.
(Math Gazette No.9/2015) | AD=BM |
13.372 A consignment shop has accepted for sale cameras, watches, fountain pens, and receivers for a total of 240 rubles. The sum of the prices of a receiver and a watch is 4 rubles more than the sum of the prices of a camera and a fountain pen, and the sum of the prices of a watch and a fountain pen is 24 rubles less ... | 18 |
11. As shown in Figure 1, the ellipse $\frac{x^{2}}{4}+\frac{y^{2}}{3}=1$ has left and right foci $F_{1}$ and $F_{2}$. Points $A$ and $B$ are on the upper half of the ellipse, and $F_{1} A \parallel F_{2} B$. Let $B F_{1}$ and $A F_{2}$ intersect at point $P$. Find the maximum value of the y-coordinate of point $P$. | \frac{3}{4} |
Explore the pattern. Observe the following expressions and verification process:
When $n=2$, there is equation ①: $2\times \sqrt{\frac{2}{3}}=\sqrt{2+\frac{2}{3}}$;
When $n=3$, there is equation ②: $3\times \sqrt{\frac{3}{8}}=\sqrt{3+\frac{3}{8}}$;
When $n=4$, there is equation ③: $4\times \sqrt{\frac{4}{15}}=\sq... | n\sqrt{\frac{n}{{n}^{2}-1}} = \sqrt{n + \frac{n}{{n}^{2}-1}} |
5. Let $t$ be a real root of the quadratic equation $a x^{2}+b x+c=0(a \neq 0)$, then the relationship between the discriminant $\Delta=b^{2}-4 a c$ and the quadratic form $M=(2 a t+b)^{2}$ is ( ).
(A) $\Delta>M$
(B) $\Delta=M$.
(C) $\Delta<M$
(D) Cannot be determined
直接输出翻译结果。 | B |
Given $\sin \alpha + \cos \alpha = \sqrt{2}$, the value of $\tan \alpha + \frac{\cos \alpha}{\sin \alpha}$ is ( ).
A: $-1$
B: $-2$
C: $\frac{1}{2}$
D: $2$ | D |
The value of $x$ that satisfies $\log_{2^x} 3^{20} = \log_{2^{x+3}} 3^{2020}$ can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. | 103 |
Solve the following system of equations:
$$
\begin{aligned}
x+y+z & =12 \\
x^{2}+y^{2}+z^{2} & =230 \\
x y & =-15
\end{aligned}
$$ | (15,-1,-2),(-1,15,-2),(3,-5,14),(-5,3,14) |
9. Given real numbers $x, y, z$ satisfy: $x \geqslant y \geqslant z, x+y+z=1, x^{2}+y^{2}+z^{2}=3$. Find the range of real number $x$.
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly. | [1,\frac{5}{3}] |
The distance between the line $x+2y-5=0$ and the line $x+2y=0$ is _______. | \sqrt{5} |
We have assembled a large cube with $n$ edge length using $n^{3}$ unit cubes. Is there an $n$ value for which the number of small cubes cut by the space diagonals of the large cube is exactly half the number of small cubes not cut by the space diagonals? | 3 |
\section*{Problem 4 - 051044}
Calculate the difference \(D\) between the sum of the squares of all even natural numbers \(\leq 100\) and the sum of the squares of all odd natural numbers \(<100\)! | 5050 |
## Problem Statement
Calculate the definite integral:
$$
\int_{0}^{2} \frac{x^{4} \cdot d x}{\sqrt{\left(8-x^{2}\right)^{3}}}
$$ | 10-3\pi |
Let the random variable $\xi$ have the following probability distribution: $P(\xi=k)=\frac{c}{k+1}$ for $k=0,1,2,3$. Find the value of $c$. | \frac{12}{25} |
GS. 4 If the mean, mode and median of the data $30,80,50,40, d$ are all equal, find the value of $d$. | 50 |
Calculate the value of $|-3| \times 2$.
A: 6
B: -6
C: $\pm$6
D: -1 | A |
Example 3 Let the quadratic function $f(x)=a x^{2}+b x+c(a, b, c \in \mathbf{R}, a \neq 0)$ satisfy the conditions:
(1) For $x \in \mathbf{R}$, $f(x-4)=f(2-x)$, and $f(x) \geqslant x$;
(2) For $x \in(0,2)$, $f(x) \leqslant\left(\frac{x+1}{2}\right)^{2}$;
(3) The minimum value of $f(x)$ on $\mathbf{R}$ is 0.
Find the l... | 9 |
10. (12 points) 19 gardeners went to plant trees, 4 of them went to plant trees on Street $A$, and the remaining 15 went to plant trees on Street $B$. In the evening, when they returned to the dormitory, Gardener A said, “Although we are fewer in number, we took the same amount of time as you.” Gardener B said, “Althou... | 57 |
Let $a_1,a_2,\ldots$ be a bounded sequence of reals. Is it true that the fact
$$\lim_{N\to\infty}\frac1N\sum_{n=1}^Na_n=b\enspace\text{ and }\enspace\lim_{N\to\infty}\frac1{\log N}\sum_{n=1}^N\frac{a_n}n=c$$implies $b=c$? | b = c |
In a certain [circle](https://artofproblemsolving.com/wiki/index.php/Circle), the [chord](https://artofproblemsolving.com/wiki/index.php/Chord) of a $d$-degree arc is $22$ centimeters long, and the chord of a $2d$-degree arc is $20$ centimeters longer than the chord of a $3d$-degree arc, where $d < 120.$ The length of... | 174 |
Solve the following equation:
$$
4^{2 x+1}-\sqrt{29 \cdot 4^{2 x+1}+2^{2 x+1}+2 \cdot 8^{2 x}}=21 \cdot 2^{2 x}-4
$$ | x_1=\frac{3}{2},x_2=-\frac{3}{2} |
9. In a $3 \times 3$ table, all natural numbers from 1 to 9 are written. It is known that: the product of the numbers in the top row is 84; the product of the numbers in the left column is 16; the sum of the numbers in the middle row is equal to the sum of the numbers in the bottom row; the product of the numbers on on... | 4 |
Example 6 Let $n$ be a given positive integer, $6 \mid n$. In an $n \times n$ chessboard, each square is filled with a positive integer, and the numbers in the squares of the $i$-th row from left to right are $(i-1) n+1,(i-1) n+2, \cdots,(i-1) n+n$. Now, take any 2 adjacent (sharing a common edge) squares, add 1 to one... | \frac{n^2(n^2 - 1)}{6} |
## Task B-4.4.
Determine all complex numbers $z$ for which:
$$
|z+i z|=2, \quad \operatorname{Re}\left(z^{4}\right)=-2 \quad \text { and } \quad \frac{3 \pi}{2}<\arg (z)<2 \pi
$$ | z_{1}=\frac{\sqrt{2}}{2}-i\frac{\sqrt{6}}{2},\quadz_{2}=\frac{\sqrt{6}}{2}-i\frac{\sqrt{2}}{2} |
Simplify: $\frac{{x}^{2}+1}{x-1}-\frac{2x}{x-1}$. | x-1 |
The point $(-1, a^{2}+1)$ must lie in which quadrant?
A: First quadrant
B: Second quadrant
C: Third quadrant
D: Fourth quadrant | \text{B} |
Given an arithmetic sequence $\{a_n\}$ and a geometric sequence $\{b_n\}$ with the sum of the first $n$ terms denoted as $S_n$ and $T_n$ ($n \in \mathbb{N}^*$) respectively. If $S_n = \frac{3}{2}n^2 + \frac{1}{2}n$, $b_1 = a_1$, and $b_2 = a_3$, then $a_n = \_\_\_\_\_\_$, $T_n = \_\_\_\_\_\_$. | \frac{2}{3}(4^n - 1) |
Given two points P(a, b) and Q(c, d) lying on the line $y = mx + k$, express the magnitude of the vector $\vec{PQ}$ in terms of a, c, and m. | |\vec{PQ}| = |a-c| \sqrt{1 + m^2} |
Calculate:$(1){(\sqrt{5})}^{2}+|-3|-{(π+\sqrt{3})}^{0}$;$(2)\left\{\begin{array}{l}5x-10≤0\\ x+3＞-2x\end{array}\right.$. | -1 < x \leqslant 2 |
At the market, tomatoes cost a yuan per kilogram and cabbage costs b yuan per kilogram. The school cafeteria buys 30 kg of tomatoes and 50 kg of cabbage, in total costing _____ yuan. | 30a + 50b |
29. The numbers $1, 2, 3, 4, 5, 6, 7, 8, 9, 0$ that we use for counting are called Arabic numerals, and they have become the universal symbols for numbers used around the world. Arabic numerals were actually invented by the ancient $\qquad$.
A. Indians
B. China
C. Babylon
D. Arabs | A |
In a mathematics competition, 1000 students are numbered as follows: 0001, 0002, 0003, ..., 1000. A sample of size 50 is to be drawn from these students using systematic sampling method, which divides them into 50 parts. If the first part consists of numbers 0001, 0002, ..., 0020 and number 0010 is randomly selected fr... | 810 |
Let complex number $z$ satisfy $(1+i)z=-3+i$ (where $i$ is the imaginary unit), then $|z|=$_______. | \sqrt{5} |
Subject 3. Let $a, b \in \mathbb{C}$ and the function $f: \mathbb{C} \rightarrow \mathbb{C}$, defined by $f(z)=z^{2}+a|z|+b$.
i) Determine $a, b \in \mathbb{C}$ such that $f(1)=f(2)=0$.
ii) For $a, b$ determined in the previous point, find all complex numbers $z$ such that $f(z)=0$.
Mathematical Gazette 1980 | z=1orz=2 |
Given a sequence $\{a_{n}\}$ where $a_{1}=1$, $a_{2}=2$, and $a_{n+2}-a_{n}=\left(-1\right)^{n}+2$, find the sum of the first $30$ terms of the sequence $\{a_{n}\}$. | 465 |
A4. For the numbers $a=\log _{3} x$ and $b=\log _{4} x$, the following holds:
(A) $a1$
(B) $a>b$ for every $x>1$
(C) $a \neq b$ for every $x>0$
(D) $a \log 4=b \log 3$ for every $x>0$
(E) $a b=\log _{12} x$ for every $x>0$ | B |
Given that the polynomial $ax^3+bx^2-47x-15$ is divisible by both $3x+1$ and $2x-3$, find the value of $a+b$. | 26 |
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