problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
|---|---|
Given that the derivative of the function $f(x)$ at $x=x_{0}$ is $3$, then $\underset{lim}{Δx→0}\frac{f({x}_{0}+Δx)-f({x}_{0})}{3Δx}=$
A: $3$
B: $1$
C: $2$
D: $\frac{2}{3}$ | B |
Given $a=(\frac{1}{6})^{\frac{1}{2}}$, $b=\log_{6}\frac{1}{3}$, $c=\log_{\frac{1}{6}}\frac{1}{7}$, determine the relationship among $a$, $b$, and $c$.
A: $c > a > b$
B: $a > b > c$
C: $a > c > b$
D: $c > b > a$ | A |
Xiaoqiang's average score in Chinese, Mathematics, and English is 97 points, with a score of 100 in Mathematics. Then, the total score for Chinese and English is ___ points. | 191 |
6. A circle passes through the vertices $A$ and $N$ of triangle $A C N$ and intersects its sides $A C$ and $C N$ at points $B$ and $K$, respectively, different from the vertices of the triangle. The ratio of the area of triangle $B C K$ to the area of triangle $A C N$ is $\frac{1}{4}$.
a) Find the ratio $A N: B K$.
b... | )AN:BK=2,b)NK:AB=2:5 |
Given sets $A = \{x | 2-a \leq x \leq a, a \in \mathbb{R}\}$ and $B = \{x | x \geq 2\}$,
(1) If $a = 3$, find $A \cap B$;
(2) If the universe $U = \mathbb{R}$, and $A \subseteq \complement_U B$, find the range of the real number $a$. | a \in (-\infty, 2) |
Which is the two-digit number that, when divided by the digit in the units place, gives a quotient of 9 and a remainder of 6? | 78 |
Two boys start moving from the same point A on a circular track but in opposite directions. Their speeds are 5 ft. per second and 9 ft. per second. If they start at the same time and finish when they first meet at the point A again, then the number of times they meet, excluding the start and finish, is
$\textbf{(A)}\ ... | 13 |
3. Given the sequence $\left\{a_{n}\right\}$ satisfies:
$$
a_{1}=4, a_{n+1}=2 a_{n}+n^{2} \text {. }
$$
Then the general term formula is $\qquad$ | a_{n}=5\times2^{n}-n^{2}-2n-3 |
We have two types of screws. The combined weight of 3 smaller and 2 larger screws was found to be 319 grams on a scale that measures with gram accuracy. By replacing one smaller screw with a larger one, the scale reads 351 grams. What is the average weight of each screw and what is the accuracy of the results? | 51,83 |
Given that the function f(x) = 2g(x) - x² is an odd function, and g(-1) = -1, find the value of f(1). | 3 |
Problem 6. On the sides \(AB\) and \(BC\) of square \(ABCD\) with side length 10, points \(K\) and \(L\) are marked such that \(AK = CL = 3\). On segment \(KL\), a point \(P\) is chosen, and on the extension of segment \(AB\) beyond point \(B\), a point \(Q\) is chosen such that \(AP = PQ = QL\) (see figure).
a) Prove... | 4 |
$$
\begin{array}{l}
\frac{a^{2}\left(\frac{1}{b}-\frac{1}{c}\right)+b^{2}\left(\frac{1}{c}-\frac{1}{a}\right)+c^{2}\left(\frac{1}{a}-\frac{1}{b}\right)}{a\left(\frac{1}{b}-\frac{1}{c}\right)+b\left(\frac{1}{c}-\frac{1}{a}\right)+c\left(\frac{1}{a}-\frac{1}{b}\right)} \\
=
\end{array}
$$ | a+b+c |
If $xy = 15$ and $x + y = 11$, calculate the value of $x^3 + y^3$. | 836 |
Find the minimum real $x$ that satisfies
$$\lfloor x \rfloor <\lfloor x^2 \rfloor <\lfloor x^3 \rfloor < \cdots < \lfloor x^n \rfloor < \lfloor x^{n+1} \rfloor < \cdots$$ | \sqrt[3]{3} |
Factorization:$(1)x^{2}-4y^{2}$;$(2)3ax^{2}-6ax+3a$. | 3a(x-1)^2 |
Given the following four propositions:
1. If two planes have infinitely many common points, then these two planes coincide;
2. If a plane passes through a perpendicular line of another plane, then these two planes are perpendicular to each other;
3. Two lines that are perpendicular to the same line are parallel to eac... | \text{D} |
Task B-2.2. Points $H$ and $N$ are the feet of the altitudes from vertex $A$ and vertex $B$ of an acute-angled triangle $ABC$. The length of the altitude from vertex $A$ is $5 \sqrt{3} \mathrm{~cm}$, the length of side $\overline{A B}$ is $14 \mathrm{~cm}$, and the measure of the angle between the altitudes $\overline{... | 7 |
7. A container in the shape of a rectangular box is partially filled with $120 \mathrm{~m}^{3}$ of water. The depth of the water is either $2 \mathrm{~m}, 3 \mathrm{~m}$ or $5 \mathrm{~m}$, depending on which side of the box is on the ground.
What is the volume of the container in $\mathrm{m}^{3}$ ? | 240 |
9.3. Solve the equation:
$1+\frac{3}{x+3}\left(1+\frac{2}{x+2}\left(1+\frac{1}{x+1}\right)\right)=x$.
# | 2 |
If $2x-y=-1$, then the value of $3+4x-2y$ is
A: $5$
B: $-5$
C: $1$
D: $-1$ | C |
Example 2 Let any real numbers $x_{0}>x_{1}>x_{2}>x_{3}>0$. To make $\log _{\frac{x_{0}}{x_{1}}} 1993+\log _{\frac{x_{1}}{x_{2}}} 1993+\log _{\frac{x_{2}}{x_{3}}} 1993 \geqslant$ $k \log _{\frac{x_{0}}{x_{3}}} 1993$ always hold, then the maximum value of $k$ is $\qquad$
(1993, National Competition) | 9 |
The first three terms of a geometric sequence $\{a_n\}$ are $x$, $2x+2$, and $3x+3$, respectively. Find $a_4$. | -\frac{27}{2} |
10. (20 points) Given the sequence $\left\{a_{n}\right\}$ satisfies:
$$
\begin{array}{l}
a_{1}=2 t-3(t \in \mathbf{R}, \text { and } t \neq \pm 1), \\
a_{n+1}=\frac{\left(2 t^{n+1}-3\right) a_{n}+2(t-1) t^{n}-1}{a_{n}+2 t^{n}-1}\left(n \in \mathbf{N}_{+}\right) .
\end{array}
$$
(1) Find the general term formula of the ... | a_{n}=\frac{2\left(t^{n}-1\right)}{n}-1 |
The initial term of an arithmetic sequence $\{a_{n}\}$ is $1$, and the common difference is not $0$. If $a_{2}$, $a_{3}$, and $a_{6}$ form a geometric sequence, then the sum of the first $6$ terms of $\{a_{n}\}$ is ( )
A: $-24$
B: $-3$
C: $3$
D: $8$ | \text{A} |
Given that $z$ is a complex number, $z= \frac{1-2i}{1+i}.$ Find $z \cdot \overline{z}=($ $)$.
A: $\frac{\sqrt{10}}{2}$
B: $\frac{\sqrt{5}}{2}$
C: $1$
D: $\frac{5}{2}$ | D |
Find the equation in general form of the line that passes through the intersection point of the lines $2x-y-3=0$ and $4x-3y-5=0$, and is parallel to the line $2x+3y+5=0$. | 2x+3y-7=0 |
In how many ways can the set $N ={1, 2, \cdots, n}$ be partitioned in the form $p(N) = A_{1}\cup A_{2}\cup \cdots \cup A_{k},$ where $A_{i}$ consists of elements forming arithmetic progressions, all with the same common positive difference $d_{p}$ and of length at least one? At least two?
[hide="Solution"]
[b]Part ... | 2^n - 2 |
The equation of the line that passes through the point $(1,0)$ and is parallel to the line $x-2y-2=0$ is ( )
A: $x-2y-1=0$
B: $x-2y+1=0$
C: $2x+y-2=0$
D: $x+2y-1=0$ | A |
Given that $a, b, c, m, n, p$ are real numbers, and $a^2+b^2+c^2=1$, $m^2+n^2+p^2=1$.
(I) Prove that $|am+bn+cp| \leq 1$;
(II) If $abc \neq 0$, prove that $\frac{m^4}{a^2} + \frac{n^4}{b^2} + \frac{p^4}{c^2} \geq 1$. | \frac{m^4}{a^2} + \frac{n^4}{b^2} + \frac{p^4}{c^2} \geq 1 |
Four, for what real number $x$ does $y=x^{2}-x+1+$ $\sqrt{2(x+3)^{2}+2\left(x^{2}-5\right)^{2}}$ have a minimum value? What is the minimum value? | 9 |
Given that the angle between the generatrix and the base of a cone is $45^{\circ}$, and the height is $1$, find the length of the generatrix of the cone. | \sqrt{2} |
If the moving straight line $x=a$ intersects with the graph of the functions $f(x)=\sin x$ and $g(x)=\cos x$ at points $M$ and $N$ respectively, find the maximum value of $|MN|$. | \sqrt{2} |
Raskina I.V.
In a colorful family, there were an equal number of white, blue, and striped octopus children. When some blue octopus children turned striped, Dad decided to count the children. There were 10 blue and white children in total, while white and striped children together amounted to 18. How many children are ... | 21 |
Given the function $f(x)=2\sin\frac{x}{2}\cos\frac{x}{2}+2\sqrt{3}\cos^{2}\frac{x}{2}-\sqrt{3}$, where $\triangle ABC$ has sides $a$, $b$, $c$ opposite angles $A$, $B$, $C$, respectively. It is known that $f(A)=\sqrt{3}$, and the circumradius of $\triangle ABC$ is $\sqrt{3}$.
$(1)$ Find the measure of angle $A$.
$(2)$ ... | \frac{9\sqrt{3}}{4} |
A bag contains 6 balls that are identical in shape and size except for their color, with 4 red balls and 2 black balls. Now, 2 balls are drawn from the bag.
(Ⅰ) Calculate the probability that both balls drawn are red;
(Ⅱ) Calculate the probability that the two balls drawn have different colors. | \dfrac {8}{15} |
12. Prime Minister Turtle needs to deliver a batch of pearls to the Dragon King of the East Sea for his birthday. On the first day, he orders "Little Shrimp Soldier" to transport $20 \%$ of the total. On the second day, he orders "General Crab" to transport $80 \%$ of the remaining. It turns out that the second day's t... | 50 |
The function $f(x) = x + \sin x$ ($x \in \mathbb{R}$) is
A: An even function, and is a decreasing function on $(-\infty, +\infty)$
B: An even function, and is an increasing function on $(-\infty, +\infty)$
C: An odd function, and is a decreasing function on $(-\infty, +\infty)$
D: An odd function, and is an increasing ... | \text{D} |
Problem 9.5. Leonid has a white checkered rectangle. First, he painted every other column gray, starting with the leftmost one, and then every other row, starting with the topmost one. All cells adjacent to the border of the rectangle ended up being painted.
How many painted cells could there be in the rectangle if 74... | 301or373 |
The correct interpretation of the assignment statement "x=x+1" is ( )
A: The value of x may be equal to the value of x+1
B: The original value of x is increased by 1, and the resulting value replaces the original value of x
C: This is an incorrect statement
D: After rearranging the terms, this expression can function t... | \text{B} |
15. (10 points) A boat travels 16 kilometers downstream and 8 kilometers upstream in 4 hours; it also travels 12 kilometers downstream and 10 kilometers upstream in the same amount of time. Question: How many hours did it take for the boat to travel 24 kilometers downstream and then return? | 9 |
II. (16 points) Given the sequence $\left\{F_{n}\right\}$ satisfies
$$
\begin{array}{l}
F_{1}=F_{2}=1, \\
F_{n+2}=F_{n+1}+F_{n}\left(n \in \mathbf{Z}_{+}\right) .
\end{array}
$$
If $F_{a} 、 F_{b} 、 F_{c} 、 F_{d}(a<b<c<d)$ are the side lengths of a convex quadrilateral, find the value of $d-b$. | 2 |
If $n$ is a multiple of $4$, the sum $s=1+2i+3i^2+\cdots+(n+1)i^n$, where $i=\sqrt{-1}$, equals:
$\textbf{(A) }1+i\qquad\textbf{(B) }\frac{1}{2}(n+2)\qquad\textbf{(C) }\frac{1}{2}(n+2-ni)\qquad$
$\textbf{(D) }\frac{1}{2}[(n+1)(1-i)+2]\qquad \textbf{(E) }\frac{1}{8}(n^2+8-4ni)$ | \frac{1}{2}(n+2-ni) |
[ Inequality problems. Case analysis]
Lёsha and Ira live in a house where each floor has 9 apartments (the house has one entrance). The floor number of Lёsha is equal to the apartment number of Ira, and the sum of their apartment numbers is 329. What is the apartment number of Lёsha? | 296 |
Given sets $A=\{1, 2, 5\}$ and $B=\{1, 3, 5\}$, then $A \cap B=$ ? | \{1, 5\} |
The relationship between the electricity bill $y$ (in yuan) and the electricity consumption $x$ (in kilowatt-hours) is given by the equation $y = 0.54x$. Identify the constant in this equation. | 0.54 |
1. (2002 National High School Competition Question) The line $\frac{x}{3}+\frac{y}{4}=1$ intersects the ellipse $\frac{x^{2}}{16}+\frac{y^{2}}{9}=1$ at points $A, B$. A point $P$ on the ellipse makes the area of $\triangle P A B$ equal to 3. How many such $P$ points are there?
A. 1
B. 2
C. 3
D. 4 | B |
7. Among the ten-digit natural numbers in octal, the natural numbers that can be divided by 7 and have all digits as 0 or 5 are $\qquad$ in number. | 84 |
Example 3 As shown in Figure 3, in the acute triangle $\triangle ABC$, it is known that $BE \perp AC$ at point $E$, $CD \perp AB$ at point $D$, $BC=25$, $CE=7$, $BD=15$. If $BE$ and $CD$ intersect at point $H$, connect $DE$, and construct a circle with $DE$ as the diameter, which intersects $AC$ at another point $F$. F... | 9 |
In triangle $ABC$, $AB=AC$. If there is a point $P$ strictly between $A$ and $B$ such that $AP=PC=CB$, then $\angle A =$
[asy]
draw((0,0)--(8,0)--(4,12)--cycle);
draw((8,0)--(1.6,4.8));
label("A", (4,12), N);
label("B", (0,0), W);
label("C", (8,0), E);
label("P", (1.6,4.8), NW);
dot((0,0));
dot((4,12));
dot((8,0));
dot... | 36^\circ |
There are $100$ products numbered from $00$ to $99$. Using the system sampling method, $5$ products are selected for inspection. After grouping, products are sampled from each group at the same interval. If the product number of the $5^{\text{th}}$ group selected is $94$, what is the product number of the $2^{\text{nd}... | 34 |
Determine all prime numbers of the form $1 + 2^p + 3^p +...+ p^p$ where $p$ is a prime number. | p = 2 |
Find all real numbers \(x, y, z\) so that
\[
\begin{aligned}
x^{2} y + y^{2} z + z^{2} & = 0 \\
z^{3} + z^{2} y + z y^{3} + x^{2} y & = \frac{1}{4} \left( x^{4} + y^{4} \right)
\end{aligned}
\] | x=y=z=0 |
Denote by $d(n)$ the number of positive divisors of a positive integer $n$. Find the smallest constant $c$ for which $d(n)\le c\sqrt n$ holds for all positive integers $n$. | \sqrt{3} |
II OM - I - Task 10
A circle is circumscribed around triangle $ ABC $. Given the radius $ R $ of this circle, calculate the radius of the circle passing through the centers of the three excircles (inscribed in the angles) of triangle $ ABC $. | 2R |
Determine the equation of the hyperbola with the focus of the parabola $y^2 = 4x$ as its vertex, and the vertex as its center and an eccentricity of 2. | x^2 - \frac{y^2}{3} = 1 |
When $x=1$, the value of the algebraic expression $\frac{1}{2}ax^{3}-3bx+4$ is $9$. What is the value of this algebraic expression when $x=-1$?
A: $9$
B: $8$
C: $-1$
D: $-9$ | C |
For some positive integer $k$, the repeating base-$k$ representation of the (base-ten) fraction $\frac{7}{51}$ is $0.\overline{23}_k = 0.232323..._k$. What is $k$?
$\textbf{(A) } 13 \qquad\textbf{(B) } 14 \qquad\textbf{(C) } 15 \qquad\textbf{(D) } 16 \qquad\textbf{(E) } 17$ | 16 |
In the diagram, points $P(2,6), Q(2,2)$ and $R(8,5)$ form a triangle. The area of $\triangle P Q R$ is
(A) 24
(B) 14
(C) 21
(D) 12
(E) 16
 | 12 |
Problem 9.5. Point $M$ is the midpoint of side $B C$ of triangle $A B C$, where $A B=17$, $A C=30, B C=19$. A circle is constructed with side $A B$ as its diameter. An arbitrary point $X$ is chosen on this circle. What is the minimum value that the length of segment $M X$ can take?
 $\sqrt{8} + 2\sqrt{3} - (\sqrt{27} - \sqrt{2})$
(2) $(\sqrt{2} - \sqrt{3})^2 + 2\sqrt{\frac{1}{3}} \times 3\sqrt{2}$. | 5 |
The ellipse $\frac{{x}^{2}}{25}+\frac{{y}^{2}}{9}=1$ and $\frac{{x}^{2}}{9-k}+\frac{{y}^{2}}{25-k}=1\left(0 \lt k \lt 9\right)$ have the following properties:
A: The foci are equal.
B: The eccentricities are equal.
C: The lengths of the minor axes are equal.
D: The lengths of the major axes are equal. | \text{A} |
Let $A=\{1,2,3,4,5\}$, $B=\{x|2x \gt 5\}$, then $A\cap B=\_\_\_\_$
A: $\{1,2\}$
B: $\{4,5\}$
C: $\{3,4,5\}$
D: $\{2,3,4,5\}$ | C |
1. The polynomial $P(x)$ with integer coefficients satisfies the condition $P(19)=P(21)=2020$. Find the smallest possible value of $P(0)>0$ under these conditions. | 25 |
Three. (50 points) Given sets $A_{1}, A_{2}, \cdots, A_{n}$ are different subsets of the set $\{1,2, \cdots, n\}$, satisfying the following conditions:
( i ) $i \notin A_{i}$ and $\operatorname{Card}\left(A_{i}\right) \geqslant 3, i=1,2, \cdots, n$;
(ii) $i \in A_{j}$ if and only if $j \notin A_{i}(i \neq j$, and $i, j... | 7 |
Given proposition A: $a+b \neq 4$, and proposition B: $a\neq 1$ and $b\neq 3$, then proposition A is the \_\_\_\_ condition of proposition B.
A. Sufficient but not necessary condition
B. Necessary but not sufficient condition
C. Necessary and sufficient condition
D. Neither sufficient nor necessary condition | D |
Given the functions $f(x) = e^x - 1$, $g(x) = -x^2 + 4x - 3$, if $f(a) = g(b)$, then the range of values for $b$ is ( )
A: $(2-, 2+)$
B: $[2-, 2+]$
C: $[1, 3]$
D: $(1, 3)$ | \text{A} |
Circles $\mathcal{P}$ and $\mathcal{Q}$ have radii $1$ and $4$, respectively, and are externally tangent at point $A$. Point $B$ is on $\mathcal{P}$ and point $C$ is on $\mathcal{Q}$ so that line $BC$ is a common external tangent of the two circles. A line $\ell$ through $A$ intersects $\mathcal{P}$ again at $D$ and in... | 129 |
The formula for converting a Fahrenheit temperature $F$ to the corresponding Celsius temperature $C$ is $C = \frac{5}{9}(F-32).$ An integer Fahrenheit temperature is converted to Celsius, rounded to the nearest integer, converted back to Fahrenheit, and again rounded to the nearest [integer](https://artofproblemsolvin... | 539 |
Given the function $f(x) = (2-a)(x-1) - 2\ln x (a \in \mathbb{R})$.
(1) If the tangent line to the curve $g(x) = f(x) + x$ at the point $(1, g(1))$ passes through the point $(0, 2)$, find the interval where the function $g(x)$ is decreasing;
(2) If the function $y = f(x)$ has no zeros in the interval $\left(0, \frac{... | 2 - 4\ln 2 |
Given a sequence $\{a_n\}$ whose sum of the first $n$ terms is $S_n$, and $S_n = n^2 + 1$, which of the following conclusions is correct? ( )
A: $a_n = 2n - 1$
B: $a_n = 2n + 1$
C: $a_n = \begin{cases} 2 & (n=1) \\ 2n-1 & (n>1) \end{cases}$
D: $a_n = \begin{cases} 2 & (n=1) \\ 2n+1 & (n>1) \end{cases}$ | C |
Given the function $f(x)=4\cos \omega x\cdot\sin (\omega x+ \frac{\pi}{4} )(\omega > 0)$ has the smallest positive period of $\pi$.
$(1)$ Find the value of $\omega$;
$(2)$ Discuss the monotonicity of $f(x)$ in the interval $\left[0, \frac{\pi}{2} \right]$ | 1 |
If $iz=-1+i$, then the point corresponding to the conjugate complex number of complex number $z$ in the complex plane is located in which quadrant?
A: First quadrant
B: Second quadrant
C: Third quadrant
D: Fourth quadrant | D |
Given a sequence $\{a_n\}$, where $a_2=4$ and $a_{n+1}=a_n+2$ $(n\in \mathbb{N}^*)$, let the sum of the first $n$ terms be $S_n$,
(1) Find the general formula for the sequence $\{a_n\}$;
(2) Let $b_n=\frac{1}{S_n}$, find the sum of the first $n$ terms of the sequence $\{b_n\}$, denoted as $T_n$. | \frac{n}{n+1} |
Given the function $g(x) = ax^3 + bx^2 + cx + d$ $(a \neq 0)$, its derivative is $f(x)$, and $a + 2b + 3c = 0$, $f(0)f(1) > 0$. Let $x_1$, $x_2$ be the two roots of the equation $f(x) = 0$, then the range of $|x_1 - x_2|$ is ( )
A: $\left[0, \frac{2}{3}\right)$
B: $\left[0, \frac{4}{9}\right)$
C: $\left(\frac{1}{... | \text{A} |
31. In the diagram below, $A B C D$ is a quadrilateral such that $\angle A B C=$ $135^{\circ}$ and $\angle B C D=120^{\circ}$. Moreover, it is given that $A B=2 \sqrt{3} \mathrm{~cm}$, $B C=4-2 \sqrt{2} \mathrm{~cm}, C D=4 \sqrt{2} \mathrm{~cm}$ and $A D=x \mathrm{~cm}$. Find the value of $x^{2}-4 x$. | 20 |
Given that $a$, $b$, and $c$ are the three side lengths of a triangle, simplify the expression: $|a+c-b|-|a+b+c|+|2b+c|$. | c |
According to many years of meteorological statistics, the probability of rain on June 1st in a certain area is 0.45, and the probability of cloudy weather is 0.20. Therefore, the probability of sunny weather on that day is ( )
A: 0.65
B: 0.55
C: 0.35
D: 0.75 | C |
9.23 How many four-digit numbers, composed of the digits $0,1,2,3,4,5$, contain the digit 3 (the digits in the numbers do not repeat)? | 204 |
Given the equation of line $l$ as $y= \frac {1}{2}x+1$, the slope of line $l$ is $(\ \ \ \ )$
A: $ \frac {1}{2}$
B: $-2$
C: $2$
D: $- \frac {1}{2}$ | A |
1. (10 points) Expression: $2016 \times \frac{1}{1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\frac{1}{32}}$ The calculation result is $\qquad$ _. | 1024 |
Evaluate $\dfrac{\sin 40^{\circ}-\cos 10^{\circ}}{\sin 10^{\circ}-\cos 40^{\circ}}$. | \dfrac{ \sqrt{3}}{3} |
11. Fill in each blank with a number not equal to 1, so that the equation holds. Then, the number of different ways to fill in is $\qquad$.
$$
[\mathrm{A} \times(\overline{1 \mathrm{~B}}+\mathrm{C})]^{2}=\overline{9 \mathrm{DE} 5}
$$ | 8 |
4. The function $y=f(x)$ is defined for all real numbers $x$, has the property $|f(x)| \leqslant 1$, and satisfies the equation $f\left(x_{1}+x_{2}\right)+f\left(x_{1}-x_{2}\right)=2\left[f\left(x_{1}\right)-1\right] \cos 2 x_{2}+2$ for any real numbers $x_{1}$ and $x_{2}$. Then the analytical expression of $y=f(x)$ is... | f(x)=1 |
Given the general term formula of a sequence $a_n$ as follows:
$$
a_n = \begin{cases}
3n + 1, & \text{if } n \text{ is odd} \\
2n - 2, & \text{if } n \text{ is even}
\end{cases}
$$
then the value of $a_2a_3$ is ( )
A: 70
B: 28
C: 20
D: 8 | C |
15. The volume of a cube is $V \mathrm{~cm}^{3}$. The surface area of the cube is $2 V \mathrm{~cm}^{2}$. What is the value of $V$ ?
A 8
B 16
C 27
D 64
E 128 | 27 |
With three different digits, all greater than $0$, six different three-digit numbers are formed. If we add these six numbers together the result is $4.218$. The sum of the three largest numbers minus the sum of the three smallest numbers equals $792$. Find the three digits. | 8, 7, 4 |
The normal vector of the line $3x+2y+5=0$ is $(a, a-2)$. The value of $a$ is ( )
A: 3
B: 6
C: -3
D: -6 | B |
A8. How many three-digit integers less than 1000 have exactly two different digits in their representation (for example, 232 , or 466 )? | 243 |
Let $ABC$ be a triangle with $\angle A = 60^o$. The points $M,N,K$ lie on $BC,AC,AB$ respectively such that $BK = KM = MN = NC$. If $AN = 2AK$, find the values of $\angle B$ and $\angle C$.
by Mahdi Etesami Fard | \angle B = 75^\circ \quad \text{and} \quad \angle C = 45^\circ |
The equation of a circle with center A(1, -2) and tangent to the line $x-3y+3=0$ is ( )
A: $(x-1)^2+(y+2)^2= \sqrt{10}$
B: $(x-1)^2+(y+2)^2=10$
C: $(x+1)^2+(y-2)^2= \sqrt{10}$
D: $(x+1)^2+(y-2)^2=10$ | \text{B} |
15. Find the real numbers $a, b, c$ that satisfy the following inequalities: $4(ab + bc + ca) - 1 \geqslant a^2 + b^2 + c^2 \geqslant 3(a^3 + b^3 + c^3) \cdot$ (2005 Australian National Training Team Problem) | a=b=c=\frac{1}{3} |
# 4. CONDITION
The sequence of numbers $\mathrm{a}_{1}, \mathrm{a}_{2}, \mathrm{a}_{3}, \ldots, \mathrm{a}_{\mathrm{n}}, \ldots$ satisfies the relations $\mathrm{a}_{\mathrm{n}}=\mathrm{a}_{\mathrm{n}-1} \cdot \mathrm{a}_{\mathrm{n}-3}$ for $\mathrm{n}=4,5,6, \ldots$ Find $\mathrm{a}_{2019}$, given that $\mathrm{a}_{1... | -1 |
Example 4 In the arithmetic sequence $\left\{a_{n}\right\}$, $a_{l}=\frac{1}{a}, a_{m}=\frac{1}{b}, a_{n}=\frac{1}{c}$, try to find the value of $(l-m) a b+$ $(m-n) b c+(n-l) c a$. | 0 |
Given points P(1, t) and Q(t^2, t-1) that both lie above the line x + y - 1 = 0, find the range of values for t. | (1, +\infty) |
Given $x^{2}-2x-3=0$, find $2x^{2}-4x+12$. | 18 |
Example 3 A closed figure formed by the intersection of a plane with the surface of a cube is called the "section figure" of the cube. In a cube $A B C D-A_{1} B_{1} C_{1} D_{1}$ with edge length 1, $E$ is the midpoint of $A B$, and $F$ is the midpoint of $C C_{1}$. Then the perimeter of the section figure passing thro... | A |
3. (17 points) The cross-section of a regular triangular pyramid passes through the midline of the base and is perpendicular to the base. Find the area of the cross-section if the side of the base is 8 and the height of the pyramid is 12. | 18 |
Given $\frac{a}{b}=2$, find $\frac{a}{a-b}$. | 2 |
7. Given that the parabola $P$ has the center of the ellipse $E$ as its focus, $P$ passes through the two foci of $E$, and $P$ intersects $E$ at exactly three points, then the eccentricity of $E$ is equal to | \frac{2\sqrt{5}}{5} |
The school invited 12 parents of 6 students, and asked 4 of these parents to talk about their child's education. If exactly one pair of the chosen 4 are a couple, how many different ways can these 4 parents be selected? | 240 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.