problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
|---|---|
Given an arithmetic sequence $\{a_n\}$ satisfies $2a_2 - a_7^2 + 2a_{12} = 0$, and the sequence $\{b_n\}$ is a geometric sequence. If $b_7 = a_7$, then $b_5b_9 =$ （ ）
A: 2
B: 4
C: 8
D: 16 | D |
Cities $A$, $B$, $C$, $D$, and $E$ are connected by roads $\widetilde{AB}$, $\widetilde{AD}$, $\widetilde{AE}$, $\widetilde{BC}$, $\widetilde{BD}$, $\widetilde{CD}$, and $\widetilde{DE}$. How many different routes are there from $A$ to $B$ that use each road exactly once? (Such a route will necessarily visit some citie... | 16 |
Let the line $nx+(n+1)y= \sqrt{2}$ $(n\in\mathbb{N}^{*})$ form a triangle with the two coordinate axes, and let the area of this triangle be $S_{n}$. Find the value of $S_{1}+S_{2}+\ldots+S_{2016}$. | \frac{2016}{2017} |
15. (6 points) The poetry lecture lasted for 2 hours $m$ minutes, and at the end, the positions of the hour and minute hands on the clock were exactly swapped compared to when it started. If $[x]$ represents the integer part of the decimal number $x$, then $[m]=$ $\qquad$ . | 46 |
The coefficient of the term with the highest power of $x$ in the expansion of $(2x-1)(3-2x)^5$ is __________ (answer with a number). | -64 |
5. Drop perpendiculars $D D_{1}, D D_{2}, D D_{3}$ from point $D$ to the planes $S B C$, $S A C$, and $S A B$ respectively. Let $D D_{1}=x, D D_{2}=y, D D_{3}=z$. According to the condition, we form the system of equations
$$
\left\{\begin{array}{l}
y^{2}+z^{2}=5 \\
x^{2}+z^{2}=13 \\
x^{2}+y^{2}=10
\end{array}\right.
... | 27 |
Calculate:$(1)4.7+\left(-2.5\right)-\left(-5.3\right)-7.5$;$(2)18+48\div \left(-2\right)^{2}-\left(-4\right)^{2}\times 5$;$(3)-1^{4}+\left(-2\right)^{2}\div 4\times [5-\left(-3\right)^{2}]$;$(4)(-19\frac{15}{16})×8$ (Solve using a simple method). | -159\frac{1}{2} |
2. The circle with the diameter of the two foci of the ellipse intersects the ellipse at four different points. Connecting these four intersection points and the two foci in sequence forms a regular hexagon. Then the eccentricity of this ellipse is $\qquad$ | \sqrt{3}-1 |
Find the equation of the line that passes through the point (1, 1) and is perpendicular to the line $x-2y+3=0$. | 2x+y-3=0 |
Given the function $f(x)=\left\{\begin{array}{l}-{x}^{2}+2x+3,&x≤2\\ 6+\log_{a}x,&x>2\end{array}\right.$ ($a>0$ and $a\neq 1$), if the range of the function $f(x)$ is $(-\infty, 4]$, then the range of real number $a$ is ( ).
A: $(\frac{\sqrt{2}}{2}, 1)$
B: $[\frac{\sqrt{2}}{2}, 1)$
C: $(1, \sqrt{2}]$
D: $(1, \sqrt{... | B |
(5) Consider all non-empty subsets of the set $S=\{1,2, \cdots, 10\}$. If the number of even numbers in a non-empty subset is no less than the number of odd numbers, this subset is called a "good subset". The number of "good subsets" is ( ) .
(A) 631
(B) 633
(C) 635
(D) 637 | 637 |
A total of $l252$ students participated in a knowledge competition, and a systematic sampling method is used to select a sample of size $50$. How many individuals should be randomly removed from the population?
A: $2$
B: $3$
C: $4$
D: $5$ | A |
Solve the equations:$(1)3A_x^3=2A_{x+1}^2+6A_x^2$;$(2)C_8^x=C_8^{5x-4}$. | x = 1 \text{ or } x = 2 |
3. If $a$ is a complex number, and $\left(a^{2}-a\right)^{2}=1$, then $a^{3}-2 a^{2}$ can take ( ) different values.
(A)2
(B) 3
(C) 4
(D) 6 | B |
The opposite of $\sqrt{4}$ is ____. | -2 |
An ancient Chinese poem poses a sequence problem: "Gazing afar at the majestic tower with seven levels, red lanterns double in number as one descends, totaling three hundred and eighty-one lanterns. How many lanterns are at the top of the tower?" ( )
A: 5
B: 4
C: 3
D: 2 | \text{C} |
4.28. One of the sides of a pentagon has a length of 30 cm. The lengths of the other sides are expressed as integers and form an arithmetic progression with a difference of 2 cm, and the length of the smallest side does not exceed 7 cm. Find the lengths of the sides of all pentagons that meet these conditions. | 5,7,9,11,30;6,8,10,12,30;7,9,11,13,30 |
2.1. Once, in a company, the following conversation took place:
- We must call Misha immediately! - exclaimed Vanya.
However, no one remembered Misha's phone number.
- I remember for sure that the last three digits of the phone number are consecutive natural numbers, - said Nastya.
- And I recall that the first five... | 7111765 |
7. (6 points) Xiaopang needs 90 seconds to go from the first floor to the third floor. At this speed, it would take him $\qquad$ seconds to go from the second floor to the seventh floor. | 225 |
If the proposition "For all $x \in \mathbb{R}, ax^2 - ax - 2 \leq 0$" is true, then the range of the real number $a$ is ______. | [-8,0] |
The solution set of the inequality system $\left\{\begin{array}{l}{2x-1＜3}\\{x-a＜0}\end{array}\right.$ is $x \lt a$. The range of values for $a$ is ______. | a \leqslant 2 |
10. Now use red, blue, and yellow 3 colors to paint the 6 faces of a cube, so that the faces painted red, blue, and yellow each have 2. Find the number of differently styled colored cubes.
将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻译结果。 | 6 |
4. Determine all complex numbers $z$ and $w$ such that
$$
|z|^{2}+z w+\bar{w}=2+6 i \quad \text { and } \quad|w|^{2}+\overline{z w}+z=2-4 i .
$$ | (z,w)=(-2,2-2i)(z,w)=(-\frac{7+i}{4},\frac{3-9i}{4}) |
8. A broken calculator has the + and $\times$ keys switched. For how many ordered pairs $(a, b)$ of integers will it correctly calculate $a+b$ using the labelled + key? | 2 |
Given sets $A=\left\{ 0,1,2,3 \right\}$ and $B=\left\{ x|{{x}^{2}}-2x-3\geqslant 0 \right\}$, then $A\cup ({{\complement }_{\mathbf{R}}}B)=$（ ）．
A: $(-1,3)$
B: $\left( -1,3 \right]$
C: $(0,3)$
D: $\left( 0,3 \right]$ | \text{B} |
The negation of the proposition "If p, then q" is ( )
A: If p, then $\neg q$
B: If $\neg p$, then $\neg q$
C: If $\neg q$, then $\neg p$
D: If q, then p | \text{A} |
Three. (25 points) Let $p$ be a prime number, and $k$ be a positive integer. When the equation $x^{2}+p x+k p-1=0$ has at least one integer solution, find all possible values of $k$.
Let $p$ be a prime number, and $k$ be a positive integer. When the equation $x^{2}+p x+k p-1=0$ has at least one integer solution, find ... | 1 |
$29 \cdot 60$ Let $g(x)=x^{5}+x^{4}+x^{3}+x^{2}+x+1$. When the polynomial $g\left(x^{12}\right)$ is divided by the polynomial $g(x)$, the remainder is
(A) 6 .
(B) $5-x$.
(C) $4-x+x^{2}$.
(D) $3-x+x^{2}-x^{3}$.
(E) $2-x+x^{2}-x^{3}+x^{4}$.
(28th American High School Mathematics Examination, 1977) | 6 |
What is the area of the figure in the complex plane enclosed by the origin and the set of all points $\tfrac{1}{z}$ such that $(1-2i)z+(-2i-1)\overline{z}=6i$? | \frac{5\pi}{36} |
6.069. $\left\{\begin{array}{l}x^{-1}+y^{-1}=5 \\ x^{-2}+y^{-2}=13\end{array}\right.$ | (\frac{1}{3};\frac{1}{2})(\frac{1}{2};\frac{1}{3}) |
8. The square of a three-digit number ends with three identical non-zero digits. Write the smallest such three-digit number. | 462 |
9. Given the complex sequence $\left\{z_{n}\right\}$ satisfies: $z_{1}=\frac{\sqrt{3}}{2}, z_{n+1}=\overline{z_{n}}\left(1+z_{n} \mathrm{i}\right)(n=1,2, \cdots)$, where $\mathrm{i}$ is the imaginary unit, find the value of $z_{2021}$. | z_{2021}=\frac{\sqrt{3}}{2}+(\frac{1}{2}+\frac{1}{2^{20200}})\mathrm{i} |
Given a digits {$0,1,2,...,9$} . Find the number of numbers of 6 digits which cantain $7$ or $7$'s digit and they is permulated(For example 137456 and 314756 is one numbers). | 2002 |
Which of the following conditions can determine that the quadrilateral ABCD is a parallelogram? ( )
A: AB=BC; CD=DA
B: AB∥CD; AB=CD
C: AD∥BC; AB=CD
D: AD∥BC; ∠B=∠C | \text{B} |
Calculate: $4\cos 30^{\circ}+(1-\sqrt{2})^{0}-\sqrt{12}+|-2|$. | 3 |
## Condition of the problem
$$
\lim _{n \rightarrow x}\left(\sqrt{\left(n^{2}+1\right)\left(n^{2}-4\right)}-\sqrt{n^{4}-9}\right)
$$ | -\frac{3}{2} |
In the polar coordinate system, given circle $O$: $ρ=\cos θ+\sin θ$ and line $l$: $ρ\sin \left( θ-\frac{π}{4} \right)=\frac{\sqrt{2}}{2}$.
(1) Find the rectangular coordinate equations for circle $O$ and line $l$;
(2) When $θ∈(0,π)$, find one of the polar coordinates of the intersection point(s) of line $l$ and circl... | (1,\frac{π}{2}) |
Example 8 Find the real roots of the equation $\sqrt{x+2 \sqrt{x+2 \sqrt{x+2+\cdots+2 \sqrt{x+2 \sqrt{3 x}}}}}=x$. There are $n$ square roots. | x_{1}=3,x_{2}=0 |
Pertin A.
Find all numbers of the form $2^{n}$ ($n$ is natural) such that when the first digit of their decimal representation is erased, the result is again a power of two. | 32,64 |
12 students are taking a group photo, with 4 in the front row and 8 in the back row. Now the photographer decides to draw 2 people from the back row to adjust to the front row. If the relative order of the other people remains unchanged, the total number of different adjustment methods is ( ).
A: $C_{8}^{2}A_{3}^{2}$
... | C |
12. There are 48 students participating in three sports activities, each person only participates in one activity, the number of participants in each activity is different, and the number of people in each activity has the digit "6". How many people are there in each of the three sports activities? | 6,16,26 |
Given $x=\ln \pi$, $y=\log_{\frac{1}{2}}\pi$, $z=e^{-\frac{1}{2}}$, then ( )
A: $x < y < z$
B: $z < x < y$
C: $z < y < x$
D: $y < z < x$ | D |
Given the function $f(x)=\begin{cases}x^{2}, & x\leqslant 1 \\ x+ \frac{4}{x}-3, & x > 1\end{cases}$, then the range of $f(x)$ is ( )
A: $[1,+\infty)$
B: $[0,+\infty)$
C: $(1,+\infty)$
D: $[0,1)\cup(1,+\infty)$ | \text{B} |
# Problem 6.
B-1
Find the minimum value of the expression $4 x+9 y+\frac{1}{x-4}+\frac{1}{y-5}$ given that $x>4$ and $y>5$. | 71 |
Given the function $f(x) = |x-a|$, if the solution set of the inequality $f(x) \leq 3$ is $\{|x| - 1 \leq x \leq 5\}$.
(Ⅰ) Find the value of the real number $a$.
(Ⅱ) If the inequality $f(3x) + f(x+3) \geq m$ holds true for all real numbers $x$, find the range of the real number $m$. | (-\infty, \frac{5}{3}] |
B1. From the given price list, determine:
(A) How much does $1 \mathrm{~kg}$ of rye bread cost?
(B) How much does $\frac{1}{4} \mathrm{~kg}$ of black bread cost?
(C) How much cottage cheese can you get for 200 SIT?
(B) How much do you pay for $\frac{1}{4} \mathrm{~kg}$ of rye bread, 10 dag of edam cheese, and $\fra... | 134 |
\section*{Exercise 2 - 031212}
During the calibration of a dynamometer, the load \(P\) required to deflect the pointer to certain divisions on the scale was measured. The following values were obtained:
\begin{tabular}{cc}
Number of divisions & Load in kp \\
\(N\) & \(P\) \\
\hline 0 & 0 \\
5 & 4.87 \\
10 & 10.52 \\... | 35.15 |
Two similar triangles have a similarity ratio of 3:5. What is the ratio of their corresponding heights? | 3:5 |
For what values of the parameter $a$ is the sum of the squares of the roots of the equation $x^{2}+2 a x+2 a^{2}+4 a+3=0$ the greatest? What is this sum? (The roots are considered with their multiplicity.)
# | 18 |
1. Let the sets be
$$
\begin{aligned}
A & =\{n(n+1) \mid n=1,2, \cdots\}, \\
B & =\{3 m-1 \mid m=1,2, \cdots\} .
\end{aligned}
$$
If the elements of the set $A \cap B$ are arranged in ascending order to form a sequence $\left\{a_{k}\right\}$, then the general term formula for the sequence $\left\{a_{k}\right\}$ is $a_... | 9k^{2}-9k+2 |
Given the real numbers $x$, $y$, and defining a new operation $x \ast y = ax + by + 1$, where $a$, $b$ are constants, and the right side of the equation is addition and multiplication as usual. If $3 \ast 5 = 15$ and $4 \ast 7 = 28$, then $1 \ast 1 = \_\_\_\_$. | -11 |
Let $p, q$ and $r$ be distinct [prime numbers](https://artofproblemsolving.com/wiki/index.php/Prime_number), where $1$ is not considered a prime. Which of the following is the smallest positive [perfect cube](https://artofproblemsolving.com/wiki/index.php/Perfect_cube) having $n=pq^2r^4$ as a [divisor](https://artofpro... | (pqr^2)^3 |
The imaginary part of the complex number $\frac {i}{1-i}$ is ______. | \frac {1}{2} |
If $\sin \theta + \cos \theta = \frac{1}{5}$, find the value of $\sin 2\theta$. | -\frac{24}{25} |
The function $f(x)$ is an odd function defined on the set of real numbers $R$. When $x \geq 0$, $f(x) = \frac{-x}{x+1}$.
$(1)$ Determine the monotonicity of the function $f(x)$ on $\left[0,+\infty \right)$ and provide a proof.
$(2)$ Find the analytical expression of the function $f(x)$ when $x < 0$.
$(3)$ If the ... | k \in [8,+\infty) |
11. In the Cartesian coordinate system $x O y$, for the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)$, the right focus is $F(c, 0)$. If there exists a line $l$ passing through point $F$ intersecting the ellipse at points $A, B$, such that $O A \perp O B$. Find the range of the eccentricity $e=\frac{c}{a}$ ... | [\frac{\sqrt{5}-1}{2},1) |
7. Given the equation of circle $\odot C$ is
$$
x^{2}+y^{2}-4 x-2 y+1=0 \text {, }
$$
the line $y=\left(\tan 10^{\circ}\right) x+\sqrt{2}$ intersects $\odot C$ at points
$A$ and $B$. The sum of the inclination angles of lines $A C$ and $B C$ is
$\qquad$ | 200^{\circ} |
4. Find a polynomial $P(x)$ of degree 2019, for which $P(2018)=2020$ and $P(2014+x)+P(2022-x)=4040$ for all $x$. | P(x)=(x-2018)^{2019}+2020 |
# Problem 1.
In-1 Find $f(1)+f(2)+f(3)+\ldots+f(13)$, if $f(n)=4 n^{3}-6 n^{2}+4 n+13$. | 28743 |
The zeroes of the function $f(x)=x^2-ax+2a$ are integers. What is the sum of the possible values of $a?$
$\textbf{(A)}\ 7\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ 16\qquad\textbf{(D)}\ 17\qquad\textbf{(E)}\ 18$ | 16 |
Regarding the roots of the equation $88(x-2)^2=95$, which of the following judgments is correct?
A: One root is less than 1, and the other is greater than 3.
B: One root is less than -2, and the other is greater than 2.
C: Both roots are less than 0.
D: Both roots are greater than 2. | \text{A} |
## Problem Statement
Calculate the lengths of the arcs of the curves given by the equations in the Cartesian coordinate system.
$$
y=2+\arcsin \sqrt{x}+\sqrt{x-x^{2}}, \frac{1}{4} \leq x \leq 1
$$ | 1 |
Find the number of subsets of $\{1,2,3,\ldots,10\}$ that contain exactly one pair of consecutive integers. Examples of such subsets are $\{\mathbf{1},\mathbf{2},5\}$ and $\{1,3,\mathbf{6},\mathbf{7},10\}.$ | 235 |
A3. In a right triangle $A B C$ with a right angle at $C$, the altitude from vertex $C$ intersects side $A B$ at point $D$. What is the measure of angle $\angle B A C$ if $|C D|=\sqrt{12} \mathrm{~cm}$ and $|A D|=4 \mathrm{~cm}$?
(A) $\alpha \doteq 49,06^{\circ}$
(B) $\alpha \doteq 40,89^{\circ}$
(C) $\alpha \doteq 40,... | 40.89 |
Express $x$ in terms of $y$ in the equation $2x-3y=7$. | x = \frac{7}{2} + \frac{3}{2}y |
[ Isosceles, inscribed, and circumscribed trapezoids ] [ Projections of the bases, sides, or vertices of a trapezoid ]
The height of an isosceles trapezoid, dropped from the vertex of the smaller base to the larger base, divides the larger base into segments that are in the ratio 2:3. How does the larger base relate t... | 5 |
Given a parabola $y^2=4x$, find the $x$-coordinate of point $M$ on the parabola such that the distance $|MF|$ between the point $M$ and the focus $F$ of the parabola is $4$. | 3 |
B2. Consider a circle with diameter $A B$. Point $C$ lies on the line segment $A B$ such that $2 \cdot|A C|=|B C|$. The points $D$ and $E$ lie on the circle so that $C D$ is perpendicular to $A B$ and $D E$ is also a diameter of the circle. Denote the areas of triangles $A B D$ and $C D E$ as $O(A B D)$ and $O(C D E)$.... | 3 |
The left and right foci of the hyperbola $\dfrac {x^{2}}{a^{2}} - \dfrac {y^{2}}{b^{2}} = 1$ ($a > 0, b > 0$) are $F_{1}$ and $F_{2}$, respectively, and its asymptotes are $l_{1}$ and $l_{2}$. A point $P$ in the first quadrant lies on $l_{1}$. If $l_{2} \perp PF_{1}$ and $l_{2} \parallel PF_{2}$, then the eccentricity ... | C |
The binomial coefficients of the third and fourth terms in the expansion of \\((x- \frac {2}{x})^{n}\\) are equal. The area enclosed by the line $y=nx$ and the curve $y=x^{2}$ is \_\_\_\_\_\_. | \frac {125}{6} |
5. As shown in Figure 1, in $\triangle A B C$, $C B>C A, \angle B A C$ $=80^{\circ}, D$ is a point on $A B$ such that $C B-C A=B D$, $I$ is the intersection of the three angle bisectors of $\triangle A B C$. Then $\angle I D A=$ ( ).
(A) $30^{\circ}$
(B) $40^{\circ}$
(C) $50^{\circ}$
(D) $60^{\circ}$ | B |
3. In triangle $A B C \quad A C=18 \text{ cm}, B C=21$ cm. Point $K$ is the midpoint of side $B C$, and point $M$ is the midpoint of side $A B$, point $N$ lies on side $A C$ and $A N=6 \text{ cm}$. Given that $M N=K N$. Find the length of side $A B$. | 15 |
In the regression model of two variables $y$ and $x$, four different models were selected. Their correlation indices $R$ are as follows, among which the model with the best fitting effect is ( )
A: Model 1's correlation index $R=0.21$
B: Model 2's correlation index $R=0.80$
C: Model 1's correlation index $R=0.50$
D: M... | D |
B-4. $p(x)$ is a non-zero polynomial of degree less than 1992, and $p(x)$ has no common factor with $x^{3}-x$. Let
$$
\frac{d^{1992}}{d x^{1992}}\left(\frac{p(x)}{x^{3}-x}\right)=\frac{f(x)}{g(x)},
$$
where $f(x), g(x)$ are polynomials. Find the smallest possible degree of $f(x)$. | 3984 |
1. Let $p$ be a prime number greater than 3, and $p+8$ is also a prime number. Let $S_{1}=p(p+8)$, and $S_{i+1} (i=1,2, \cdots, k-1)$ denote the sum of the digits of $S_{i}$. If $S_{k}$ is a single-digit number, for all possible prime numbers $p$, $S_{k}$ has ( ) different values.
(A) 1
(B) 2
(C) 3
(D) 4 | A |
(2) Given the set of points on the plane $M=\left\{(x, y) \mid y=\sqrt{2 x-x^{2}}\right\}, N=\{(x, y) \mid y=k(x+1)\}$. When $M \cap N \neq \varnothing$, the range of values for $k$ is ( ).
(A) $\left[-\frac{\sqrt{3}}{3}, \frac{\sqrt{3}}{3}\right]$
(B) $\left[0, \frac{\sqrt{3}}{3}\right]$
(C) $\left[-\frac{\sqrt{3}}{3}... | [0,\frac{\sqrt{3}}{3}] |
1、Calculate: $20140601=13 \times\left(1000000+13397 \times \_\right.$
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly.
1、Calculate: $20140601=13 \times\left(1000000+13397 \times \_\right.$ | 41 |
If $\frac{1}{{\sqrt{x-3}}}$ is meaningful within the real number range, then the range of real number $x$ is ____. | x > 3 |
Given the function $f(x)=\frac{x}{4-{x}^{2}}, x \in (-2,2)$.
$(1)$ Prove: The function $f(x)$ is monotonically increasing on $(0,2)$.
$(2)$ Prove: The function $f(x)$ is an odd function.
$(3)$ Find the solution set of the inequality $f(t)+f(1-2t) > 0$. | (-\frac{1}{2}, 1) |
6. In $\triangle A B C$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given $a^{2}+b^{2}-7 c^{2}=0$, then $\frac{\operatorname{ctg} C}{\operatorname{ctg} A+\operatorname{ctg} B}=$ | 3 |
Which of the following options does not have the same result as $\sqrt[3]{64}\div 4$?
A: $2\times 2^{-1}$
B: $4^{2}\times 4^{3}\times 4^{-5}$
C: $\sqrt{16}\div 2$
D: $3^{0}$ | C |
Given the sets $A=\{x| \frac {x-5}{x+1}\leqslant 0\}$, $B=\{x|x^{2}-2x-m < 0\}$.
$(1)$ When $m=3$, find $A\cap(\complement_{\mathbb{R}}B)$;
$(2)$ If $A\cap B=\{x|-1 < x < 4\}$, find the value of the real number $m$. | 8 |
1. 205 Starting from any three-digit number $n$, we get a new number $f(n)$, which is the sum of the three digits of $n$, the pairwise products of the three digits, and the product of the three digits.
(1) When $n=625$, find $\frac{n}{f(n)}$ (take the integer value).
(2) Find all three-digit numbers $n$ such that $\fra... | 199,299,399,499,599,699,799,899,999 |
Given two circles, circle M: $x^2 + y^2 - 4y = 0$ and circle N: $(x - 1)^2 + (y - 1)^2 = 1$, determine the number of their common tangents.
A: 1
B: 2
C: 3
D: 4 | B |
Task B-3.3. Solve the equation $x^{\log _{5} 6}-5 \cdot 6^{\log _{5} \sqrt{x}}=6$. | 25 |
Where does the following function take its extreme values?
$$
\sin \left(\cos ^{2} x\right)+\sin \left(\sin ^{2} x\right)
$$ | k\pi/4(k=0,\1,\2,\ldots) |
In the Cartesian coordinate system $xOy$, the eccentricity of the hyperbola $\frac {x^{2}}{4} - \frac {y^{2}}{3} = 1$ is _____. | \frac {\sqrt {7}}{2} |
Let the universal set $U=\{1,2,3,4,5\}$, $A=\{1,2,3\}$, $B=\{2,3,4\}$, then $\complement_U(A\cap B)=$ ( )
A: $\{2,3\}$
B: $\{1,4,5\}$
C: $\{4,5\}$
D: $\{1,5\}$ | \text{B} |
Let $A$ be a non-empty set of positive integers. Suppose that there are positive integers $b_{1}$, $\cdots$, $b_{n}$ and $c_{1}$, $\cdots$, $c_{n}$ such that [list] [*] for each $i$ the set $b_{i}A+c_{i}=\{b_{i}a+c_{i}\vert a \in A \}$ is a subset of $A$, [*] the sets $b_{i}A+c_{i}$ and $b_{j}A+c_{j}$ are disjoint when... | \sum_{i=1}^n \frac{1}{b_i} \leq 1 |
In triangle $\triangle ABC$, if $\dfrac{1}{\tan B}+\dfrac{1}{\tan C}=\dfrac{1}{\tan A}$, then the range of $\cos A$ is:
A: $\left(0,\dfrac{1}{3}\right]$
B: $[\dfrac{1}{3}$, $1)$
C: $\left(0,\dfrac{2}{3}\right]$
D: $[\dfrac{2}{3}$, $1)$ | D |
$(1)$ Given $|a|=5$, $|b|=3$, and $a \gt 0$, $b \gt 0$, find the value of $a+b$;$(2)$ Given $|a-2|+|b-3|+|c-4|=0$, find the value of $a+b+c$. | 9 |
I was given two two-digit natural numbers. Then I rounded both of them to the tens. Determine which numbers I had, given that at the same time:
- the difference of the rounded numbers is the same as the difference of the original numbers,
- the product of the rounded numbers is 184 greater than the product of the orig... | 1626 |
Given the set $M=\{-1,0,1,2\}$ and $N=\{x|x^{2}-ax \lt 0\}$, if $M\cap N=\{1,2\}$, then one possible value of $a$ is ( )
A: $0$
B: $1$
C: $2$
D: $3$ | D |
3. Let $p$ be a real number and $c \neq 0$ an integer such that
$$
c-0.1<x^{p}\left(\frac{1-(1+x)^{10}}{1+(1+x)^{10}}\right)<c+0.1
$$
for all (positive) real numbers $x$ with $0<x<10^{-100}$. (The exact value $10^{-100}$ is not important. You could replace it with any "sufficiently small number".)
Find the ordered pair... | (-1,-5) |
5. In a room, there are several red, green, blue, yellow, and white socks. To ensure that 10 pairs of socks (two socks of the same color make a pair) are taken out of the room, the minimum number of socks that should be taken out is ( ).
(A) 23. (B)24. (C)25. (D) 26. | B |
## Task B-3.3.
For which natural numbers $n$ is the value of the expression $\frac{n^{2}-4 n+4}{n+1}$ an integer? | n\in{2,8} |
4・68 In a competition with $n(n \geqslant 2)$ participants lasting $k$ days, each participant's score each day is exactly $1,2, \cdots, n$ (no two participants have the same score), and after $k$ days, each participant's total score is 26. Try to find all possible values of $(n, k)$.
| (n,k)=(25,2),(12,4),(3,13) |
4. Determine all triples $(x, y, z)$ of natural numbers such that
$$
x y z + x y + y z + z x + x + y + z = 243
$$ | (1,1,60),(1,60,1),(60,1,1) |
Given the universal set $U=\mathbb{R}$, set $M=\{x|0\leqslant x < 5\}$, $N=\{x|x\geqslant 2\}$, then $(\complement_{U}N)\cap M=$ ()
A: $\{x|0\leqslant x < 2\}$
B: $\{x|0 < x\leqslant 2\}$
C: $\{x|0 < x < 2\}$
D: $\{x|0\leqslant x\leqslant 2\}$ | A |
(IMO 1975, Problem 6)
Let $n$ be a positive integer. Find all polynomials $f$ in two variables such that:
1. for all real numbers $t, x, y: f(t x, t y)=t^{n} f(x, y)$;
2. for all real numbers $a, b, c: f(a+b, c)+f(b+c, a)+f(c+a, b)=0$;
3. $f(1,0)=1$. | f(X,Y)=(X-2Y)(X+Y)^{n-1} |
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