problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
|---|---|
Given $i$ is the imaginary unit, and $|1+ai| = \sqrt{5}$, then the value of the real number $a$ is ( )
A: 1
B: 2
C: 1 or -1
D: 2 or -2 | \text{D} |
Problem 2. A group of adventurers is showing off their loot. It is known that exactly 9 adventurers have rubies; exactly 8 have emeralds; exactly 2 have sapphires; exactly 11 have diamonds. Moreover, it is known that
- if an adventurer has diamonds, then they have either rubies or sapphires (but not both at the same t... | 17 |
Given circle $C$: $(x-1)^{2}+y^{2}=9$, there is a point $P(2,2)$ inside it. A line $l$ passing through point $P$ intersects circle $C$ at points $A$ and $B$.
(1) When line $l$ passes through the center of the circle $C$, find the equation of line $l$;
(2) When chord $AB$ is bisected by point $P$, find the equation of l... | x+2y-6=0 |
Given $a-b=5$ and $ab=-1$, find the value of the algebraic expression $(2a+3b-2ab) - (a+4b+ab) - (3ab+2b-2a)$. | 21 |
If $\overrightarrow{AB}⋅\overrightarrow{AC}={\overrightarrow{AB}^2}=4$, and $|{\overrightarrow{AP}}|=1$, then $|{\overrightarrow{AB}}|=$______, the maximum value of $\overrightarrow{CP}⋅\overrightarrow{AB}$ is ______. | -2 |
Among the following propositions, the correct one(s) is (are):
**(1)** The angle between the plane vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ is $60^{\circ}$, $\overrightarrow{a}=(2,0)$, $|\overrightarrow{b}|=1$, then $|\overrightarrow{a}+\overrightarrow{b}|=\sqrt{7}$.
**(2)** In the triangle $\triangle AB... | (1), (2), (3) |
6. Write all prime numbers that cannot be represented as the sum of two composite numbers. | 2,3,5,7,11 |
3. If the equation about $x$
$$
x^{2}-\left(a^{2}+b^{2}-6 b\right) x+a^{2}+b^{2}+2 a-4 b+1=0
$$
has two real roots $x_{1} 、 x_{2}$ satisfying $x_{1} \leqslant 0 \leqslant x_{2} \leqslant 1$, then the range of $a^{2}+b^{2}+4 a+4$ is $(\quad)$.
(A) $\left[\frac{\sqrt{2}}{2}, \sqrt{5}+2\right]$
(B) $[1,9-4 \sqrt{5}]$
(C)... | D |
2. Given positive integers $a, b$ such that $\frac{a^{2}+b^{2}}{a b-1}=k$ $\left(k \in \mathbf{Z}_{+}\right)$. Find the value of $k$. | 5 |
Problem 11.3. A natural number $n$ is called interesting if $2 n$ is a perfect square, and $15 n$ is a perfect cube. Find the smallest interesting number.
Answer: 1800. | 1800 |
How many divisors does $2013^{13}$ have? (As usual, we count 1 and $2013^{13}$ itself as divisors, but not negative integers.) | 2744 |
Which of the following statements about a pyramid is correct? ( )
A: A quadrilateral pyramid has four edges.
B: A pentagonal pyramid has five faces.
C: A hexagonal pyramid has six vertices.
D: Any pyramid has only one base. | \text{D} |
The standard equation of a parabola with the directrix \\(y=4\\) is ( )
A: \\(x^{2}=16y\\)
B: \\(x^{2}=8y\\)
C: \\(x^{2}=-16y\\)
D: \\(x^{2}=-8y\\) | C |
The coefficient of $x^6$ in the expansion of $(x-2)^8$ is ______. | 112 |
21. The number of positive integer solutions $(a, b, c)$ that satisfy the system of equations
$$
\left\{\begin{array}{l}
a b+b c=44, \\
a c+b c=23
\end{array}\right.
$$
is
(A) 0 ;
(B) 1 ;
(C) 2 ;
(D) 3 ;
(E) 4 . | C |
1. (5 points) Arrange 6 consecutive natural numbers in ascending order. If the sum of the last 3 numbers is twice the sum of the first 3 numbers, then the largest number among these 6 numbers is $\qquad$, and the sum of these 6 numbers is $\qquad$. | 7,27 |
9.192. $\sqrt{x^{3}+3 x+4}>-2$, | x\in[-1;\infty) |
The number of false propositions among the following four conclusions is ( )
① Two lines that are perpendicular to the same line are parallel to each other;
② Two lines that are parallel to the same line are parallel to each other;
③ If lines a, b, and c satisfy a∥b and b⊥c, then a⊥c;
④ If lines a and b are s... | B |
1. Let the function $y=\left|\log _{2} \frac{x}{2}\right|$ have the domain $[m, n]$, and the range $[0,2]$. Then the minimum length of the interval $[m, n]$ is $\qquad$ | \frac{3}{2} |
A circle that is tangent to both the circle $x^2+y^2=1$ and the circle $x^2+y^2-8x+12=0$ has its center at ( )
A: On an ellipse
B: On one branch of a hyperbola
C: On a parabola
D: On a circle | \text{B} |
Given sets $A=\{-1,0,1,2\}$ and $B=\{x|x^{2}\leqslant 1\}$, then $A\cap B=\left(\ \ \right)$
A: $\{-1,0,1\}$
B: $\{0,1\}$
C: $\{-1,1\}$
D: $\{0,1,2\}$ | A |
4. Find all positive integers $n$, such that there exists a convex $n$-gon, whose interior angles are consecutive positive integers. | 3,5,9,15,16 |
In class, the teacher wrote the following question on the blackboard. Xiao Cong immediately noticed that one of the questions was wrong. Do you know which question it is?
Use the formula for the difference of squares to factorize the following expressions:
1. $a^{2}-b^{2}$
2. $49x^{2}-y^{2}z^{2}$
3. $-x^{2}-y^{2}$
4. ... | \text{C} |
7. If 653 divides $\overline{a b 2347}$, then $a+b=$
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly. | 11 |
Among the real numbers $-2$, $\sqrt[3]{8}$, $\sqrt{6}$, and $3.14$, the irrational number is ( )
A: $-2$
B: $\sqrt[3]{8}$
C: $\sqrt{6}$
D: $3.14$ | C |
Example 2. Find the logarithmic residue of the function
$$
f(z)=\frac{\operatorname{ch} z}{e^{i z}-1}
$$
with respect to the contour $C:|z|=8$. | 3 |
Given an ellipse $C:\frac{{{x}^{2}}}{3}+\frac{{{y}^{2}}}{{{b}^{2}}}=1$ ($b\gt 0$) with its right focus at $F$, two lines passing through $F$ and tangent to the circle $O:{{x}^{2}}+{{y}^{2}}={{r}^{2}}\left( r\gt 0 \right)$ at points $A$ and $B$, forming a right triangle $\triangle ABF$. It is also known that the maximum... | 2\sqrt{3} |
4 、There is a number, when divided by 3 the remainder is 1, when divided by 5 the remainder is 2, so what is the remainder when this number is divided by 15? | 7 |
3.22. Calculate
$$
\int_{L} \frac{\sin 3 z}{(6 z-\pi)^{3}} d z
$$
where $L$ is the unit circle centered at the origin. | -\frac{i\pi}{24} |
Solve the following equation over the set of integer pairs:
$$
x^{2}-3 y^{2}+2 x y-2 x-10 y+20=0
$$ | (19,-7),(-15,5),(7,5),(-3,-7) |
For the polynomial $6x^5+5x^4+4x^3+3x^2+2x+2002$, given a value for $x$, if Xiaoying calculates the value of this polynomial following the rules of arithmetic operations, she needs to perform 15 multiplications and 5 additions. Xiaoming said, "There is another method, by appropriately adding parentheses, we can keep th... | \text{correct} |
In the isosceles triangle $ABC$, the midpoint of the base $AB$ is $F$, and the orthocenter is $M$. We know that the centroid of the triangle lies on the incircle, and that $FM = \sqrt{6}$. What are the lengths of the sides of the triangle? | =60,=24 |
If $z=(\sin \theta- \frac {3}{5})+i(\cos \theta- \frac {4}{5})$ is a pure imaginary number, then the value of $\tan \theta$ is \_\_\_\_\_. | -\frac {3}{4} |
11.5. For what minimum natural $n$ can $n$ distinct natural numbers $s_{1}, s_{2}, \ldots, s_{n}$ be found such that $\left(1-\frac{1}{s_{1}}\right)\left(1-\frac{1}{s_{2}}\right) \ldots\left(1-\frac{1}{s_{n}}\right)=\frac{7}{66} ?$ | 9 |
Example 1 Given $a, b \in \mathbf{R}$, the equation about $x$
$$
x^{4}+a x^{3}+2 x^{2}+b x+1=0
$$
has one real root. Find the minimum value of $a^{2}+b^{2}$. | 8 |
5. (8 points) On the sides $A B, B C$ and $C A$ of triangle $A B C$, points $C_{1}, A_{1}$ and $B_{1}$ are marked respectively. Segments $A A_{1}, B B_{1}$ and $C C_{1}$ intersect at point $P$. Find the ratio $A P: P A_{1}$, if $A B: A C_{1}=\alpha>1$ and $A C: A B_{1}=\beta>1$. | \frac{1}{\alpha-1}+\frac{1}{\beta-1} |
Given a hyperbola $C:\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1(a > 0,b > 0)$, one of its asymptotes is perpendicular to the line $l:x+\sqrt{3}y=0$, and the distance from one focus of the hyperbola $C$ to the line $l$ is $2$. Determine the standard form equation of the hyperbola $C$. | \frac{x^{2}}{4}-\frac{y^{2}}{12}=1 |
Given the real numbers $a$, $b$, and $c$ that satisfy the inequality $0 < a < b < c < 1$, and $M=2^{a}$, $N=3^{-b}$, $P=\ln c$, determine the relationship among $M$, $N$, and $P$.
A: $P < N < M$
B: $P < M < N$
C: $M < P < N$
D: $N < P < M$ | A |
SG. 3 Let $a \oplus b=a b+10$. If $C=(1 \oplus 2) \oplus 3$, find the value of $C$. | 46 |
4. Given that $P(-2,3)$ is a point on the graph of the inverse proportion function $y=\frac{k}{x}$, $Q$ is a moving point on the branch of the hyperbola in the fourth quadrant. A line is drawn through point $Q$ such that it intersects the hyperbola $y=\frac{k}{x}$ at only one point, and intersects the $x$-axis and $y$-... | 48 |
2. Variant 1.
Nine integers from 1 to 5 are written on the board. It is known that seven of them are not less than 2, six are greater than 2, three are not less than 4, and one is not less than 5. Find the sum of all the numbers. | 26 |
Write down the sequence $a_1=1, a_2, a_3, \ldots, a_n$ ($n \in \mathbb{N}^*$) according to the following rule: if $a_n - 2$ is a natural number that has not been written before, then let $a_{n+1} = a_n - 2$; otherwise, let $a_{n+1} = a_n + 3$. Find the value of $a_6$. | 6 |
Given the function $f(x)= \begin{cases} x^{2}+4x-3, & x < m \\ 4, & x \geqslant m \end{cases}$, if the function $g(x)=f(x)-2x$ has exactly three distinct zeros, then the range of the real number $m$ is \_\_\_\_\_\_. | (1,2] |
For the system of equations $\left\{\begin{array}{l}2x+y=k\\3x+y=3\end{array}\right.$, the solution satisfies $x-2y\geqslant 1$. Find the largest integer value of $k$ that satisfies this condition. | 2 |
$37 \cdot 2$ In the Cartesian coordinate system, a point whose both coordinates are integers is called an integer point. Let the line passing through points $A(-2 \sqrt{2},-1+\sqrt{2}), B(0, \sqrt{2})$ be $l$; and the line passing through points $C\left(0, \frac{3}{2}\right)$, $D\left(\frac{3}{2}, \frac{3}{4}\right)$ b... | B |
1st Putnam 1938 Problem B4 The parabola P has focus a distance m from the directrix. The chord AB is normal to P at A. What is the minimum length for AB? Solution | 3\sqrt{3} |
Let $f(x)=\begin{cases}{(x-a)}^{2}, & x\leqslant 0 \\ x+ \frac{1}{x}+a+4, & x > 0\end{cases}$. If $f(0)$ is the minimum value of $f(x)$, then the range of $a$ is ( )
A: $[-2,3]$
B: $[-2,0]$
C: $[1,3]$
D: $[0,3]$ | \text{D} |
Given that $\overrightarrow{e\_1}$ and $\overrightarrow{e\_2}$ are two non-collinear vectors, $\overrightarrow{AB} = 2\overrightarrow{e\_1} - 8\overrightarrow{e\_2}$, $\overrightarrow{CB} = \overrightarrow{e\_1} + 3\overrightarrow{e\_2}$, and $\overrightarrow{CD} = 2\overrightarrow{e\_1} - \overrightarrow{e\_2}$. If $\... | k = 12 |
6. Let real numbers $x, y$ satisfy
$$
17\left(x^{2}+y^{2}\right)-30 x y-16=0 \text {. }
$$
Then the maximum value of $\sqrt{16 x^{2}+4 y^{2}-16 x y-12 x+6 y+9}$ is ( ).
(A) 7
(B) $\sqrt{29}$
(C) $\sqrt{19}$
(D) 3 | A |
[ Systems of nonlinear algebraic equations ] Higher degree equations (miscellaneous) $\quad]$
Solve the system of equations:
$x^{3}-y=6$,
$y^{3}-z=6$,
$z^{3}-x=6$. | (2,2,2) |
1. Let the set $X=\{1,2, \cdots, 2017\}$,
$S=\{(x, y, z) \mid x, y, z \in X$, and the condition
$$
x<y<z, y<z<x, z<x<y
$$
holds for exactly one of them $\}$.
If $(x, y, z) \in S$, and $(z, w, x) \in S$, then the correct option is ( ).
(A) $(y, z, w) \in S$, and $(x, y, w) \notin S$
(B) $(y, z, w) \in S$, and $(x, y, w... | B |
$36 \cdot 14$ Given a set $S$ composed of two undefined elements “pib” and “maa”, and the following four axioms: $P_{1}$: Each pib is a set of maa; $P_{2}$: Any two distinct pibs have exactly one common maa; $P_{3}$: Each maa belongs to exactly two pibs; $P_{4}$: There are exactly four pibs.
Consider the three theorem... | E |
Question 70: Let $\mathrm{m}$ be a positive real number, satisfying: for any positive real numbers $\mathrm{x}$ and $\mathrm{y}$, the lengths $\mathrm{x}+\mathrm{y}$, $\sqrt{x^{2}+y^{2}+x y}$, and $m \sqrt{x y}$ of three line segments can form the three sides of some triangle, then the range of values for $m$ is $\qqua... | (2-\sqrt{3},2+\sqrt{3}) |
1. Randomly rolling three identical cubic dice. The probability that the minimum value among the three dice is 3 is $\qquad$ | \frac{37}{216} |
Let's determine the pairs of positive integers $(x, y)$ for which
$$
x^{y}-y^{x}=1
$$ | x_{1}=2,y_{1}=1x_{2}=3,y_{2}=2 |
Example 7 Real numbers $x, y$ satisfy $4 x^{2}-5 x y+4 y^{2}$ $=5$, let $s=x^{2}+y^{2}$. Then $\frac{1}{s_{\text {max }}}+\frac{1}{s_{\text {min }}}=$ $\qquad$ (1993, National High School Mathematics Competition) | \frac{8}{5} |
Given the function $f(x)= \begin{cases} (3a-1)x+4a,&(x < 1) \\ \frac {a}{x},&x\geqslant 1\end{cases}$, if $f(x)$ is a decreasing function on $(-∞,+∞)$, then the range of values of $a$ is $(\ \ \ \ \ )$.
A: $a < \dfrac {1}{3}$
B: $a\leqslant \dfrac {1}{3}$
C: $\dfrac {1}{6}\leqslant a < \dfrac {1}{3}$
D: $0 < a < \d... | C |
(4) A chord $AB$ with an inclination angle of $45^{\circ}$ is drawn through the right focus $F_{2}$ of the ellipse $\frac{x^{2}}{2}+y^{2}=1$. Then $|AB|$ is ( ).
(A) $\frac{2 \sqrt{6}}{3}$
(B) $\frac{4 \sqrt{6}}{3}$
(C) $\frac{4 \sqrt{2}}{3}$
(D) $\frac{4 \sqrt{3}}{3}$ | \frac{4\sqrt{2}}{3} |
Given that the radius of a sphere is $24cm$, the height of a cone is equal to the diameter of the sphere, and the surface area of the sphere is equal to the surface area of the cone, what is the volume of the cone in $cm^{3}$? | 12288\pi |
Given two sets of numbers 3, 7, 11, 15, ... and 5, 8, 11, 14, ..., there are many common numbers between them. For example, 11 is their first common number. What is their 20th common number? | 239 |
Given that the angle between vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ is $\frac{π}{3}$, and $|\overrightarrow{a}|=2$, $|\overrightarrow{b}|=3$, $\overrightarrow{c}=λ\overrightarrow{a}-2\overrightarrow{b}$.
$(1)$ Find $|{3\overrightarrow{a}-\overrightarrow{b}}|$;
$(2)$ When $\overrightarrow{b}⊥\overrigh... | 6 |
17. 14 people participate in a Japanese chess round-robin tournament, where each person plays against the other 13 people, and there are no draws in the matches. Find the maximum number of "triangles" (here, a "triangle" refers to a set of three people where each person has one win and one loss).
(2002, Japan Mathemat... | 112 |
11. The sequence $\left\{x_{n}\right\}$ is defined as follows: $x_{1}=\frac{1}{2}, x_{k+1}=x_{k}^{2}+x_{k}$. Then the integer part of the following sum $\frac{1}{x_{1}+1}+\frac{1}{x_{2}+1}+\cdots+\frac{1}{x_{100}+1}$ is . $\qquad$ | 1 |
Given $\xi$ follows the binomial distribution $B(n, p)$, where $E\xi = 8$ and $D\xi = 1.6$, determine the values of $n$ and $p$. | n = 10, \ p = 0.8 |
Given the function $f(x) = x^2 - 2ax - 1 + a$, where $a \in \mathbb{R}$.
$(1)$ If $a=2$, find the minimum value of the function $y = \frac{f(x)}{x}$ $(x > 0)$.
$(2)$ For any $x \in [0,2]$, the inequality $f(x) \leqslant a$ holds, find the range of the real number $a$. | [\frac{3}{4}, +\infty) |
Mathematical operations are endlessly fascinating. For example, when learning about rational numbers, there is an operation $2+2=2\times 2$ where the sum of two rational numbers equals their product. Please find another pair of numbers $a$ and $b\left(a\neq b\right)$ in rational numbers such that $a+b=ab$. The pair you... | \frac{3}{2} \text{ and } 3 |
## Task A-1.2.
On the playground, there are 2014 athletes who have numbers from 1 to 2014 on their jerseys (each number is on exactly one jersey). At the beginning, all of them are standing. At certain time intervals, the coach shouts out all natural numbers from 1 to 2014 in sequence. Athletes whose jersey numbers ar... | 44 |
4. The line $\frac{x}{4}+\frac{y}{3}=1$ intersects the ellipse $\frac{x^{2}}{16}+\frac{y^{2}}{9}=1$ at points $A, B$. There is a point $P$ on the ellipse such that the area of $\triangle P A B$ equals 3. How many such points $P$ are there?
(A) 1
(B) 2
(C) 3
(D) 4 | B |
$11 、 \mathrm{~A}, \mathrm{~B}, \mathrm{C}$ are 3 prime numbers less than 20, $A+B+C=30$, find these three prime numbers | 2,11,17 |
Three, given $x, y \in N$, find the largest $y$ value such that there exists a unique $x$ value satisfying the following inequality:
$$
\frac{9}{17}<\frac{x}{x+y}<\frac{8}{15} \text {. }
$$ | 112 |
Let $S=(x-1)^4+4(x-1)^3+6(x-1)^2+4(x-1)+1$. Then $S$ equals:
$\textbf{(A)}\ (x-2)^4 \qquad \textbf{(B)}\ (x-1)^4 \qquad \textbf{(C)}\ x^4 \qquad \textbf{(D)}\ (x+1)^4 \qquad \textbf{(E)}\ x^4+1$ | C |
Let \\(x,y\\) satisfy the constraint conditions \\(\begin{cases} & x+y-2\geqslant 0 \\\\ & x-y+1\geqslant 0 \\\\ & x\leqslant 3 \end{cases}\\). If the minimum value of \\(z=mx+y\\) is \\(-3\\), then the value of \\(m\\) is . | - \dfrac{2}{3} |
In the polar coordinate system, find the distance from the point $(2, \frac {5π}{6})$ to the line $ρ\sin(θ - \frac {π}{3}) = 4$. | 2 |
$\overrightarrow{m}=(3,-2)$, $\overrightarrow{n}=(1,x)$, if $\overrightarrow{m}⊥(\overrightarrow{m}+\overrightarrow{n})$, then $x=\left(\ \ \right)$
A: $-8$
B: $-6$
C: $6$
D: $8$ | D |
In $\triangle ABC,$ a point $E$ is on $\overline{AB}$ with $AE=1$ and $EB=2.$ Point $D$ is on $\overline{AC}$ so that $\overline{DE} \parallel \overline{BC}$ and point $F$ is on $\overline{BC}$ so that $\overline{EF} \parallel \overline{AC}.$ What is the ratio of the area of $CDEF$ to the area of $\triangle ABC?$
$\te... | \textbf{(A)}\frac{4}{9} |
5. On the sides of a rectangle with length $8 \mathrm{~cm}$ and width $6 \mathrm{~cm}$, points are placed starting from one vertex, each $2 \mathrm{~cm}$ apart. How many different lines do these points determine?
NATIONAL MATHEMATICS COMPETITIONPrimošten, April 4-6, 2016.6th grade-elementary school | 63 |
3. Place a regular tetrahedron with volume 1 inside a cube, then the minimum volume of this cube is $\qquad$ | 3 |
In $\triangle ABC$, $AB= \frac{ \sqrt{6}}{2}$, $A=45^{\circ}$, $C=60^{\circ}$, find the length of $BC$. | BC=1 |
Shift the graph of the function $y=\sin 2x$ to the left by $\varphi$ ($\varphi > 0$) units so that it coincides with the graph of the function $$y=\cos\left(2x- \frac{\pi}{3}\right)$$, then the minimum value of $\varphi$ is \_\_\_\_\_\_. | \frac{\pi}{12} |
Problem 7. For what values of the parameter $a$ does the equation $x^{3}-14 x^{2}+a x-27=0$ have three distinct real roots that form a geometric progression? | 42 |
1. If real numbers $a, b, c$ satisfy $|a|+a=0,|a b|=a b,|c|$ $-c=0$, then the algebraic expression $\sqrt{b^{2}}-|a+b|-\sqrt{c^{2}-2 b c+b^{2}}$ $+|a-c|$ simplifies to ( ).
(A) $2 c-b$ or $b$
(B) $2 c-b$
(C) $b$
(D) -6 | A |
# 2. Solve the inequality:
$$
\{x\}([x]-1)<x-2,
$$
where $[x]$ and $\{x\}$ are the integer and fractional parts of the number $\boldsymbol{x}$, respectively (9 points).
# | x\geq3 |
5. Four friends (from 6th grade, 1 point). Masha, Nina, Lena, and Olya are friends. They are all of different heights, but the difference is very small - it's hard to tell by eye. One day they decided to find out who is taller and who is shorter. It turned out that Nina is shorter than Masha, and Lena is taller than Ol... | \frac{1}{6} |
Determine all functions $f : \mathbb {N} \rightarrow \mathbb {N}$ such that $f$ is increasing (not necessarily strictly) and the numbers $f(n)+n+1$ and $f(f(n))-f(n)$ are both perfect squares for every positive integer $n$. | f(x) = x^2 + x |
14. A unit spent 500,000 yuan to purchase a piece of high-tech equipment. According to the tracking survey of this model of equipment: after the equipment is put into use, if the maintenance and repair costs are averaged to the first day, the conclusion is: the maintenance and repair cost on the $x$-th day is
$$
\left[... | 2000 |
Given a set $A \subset \mathbb{N}^{*}$, defined as follows: the number of elements in set $A$ is $n$, and $n \geq 2$. If $B=\{z|z=x+y, x \in A, y \in A, x \neq y\}$, then set $B$ is called the derived sum set of set $A$.
$(Ⅰ)$ When $A_{1}=\{1,2,3,4\}$ and $A_{2}=\{1,2,4,7\}$, write down the derived sum sets of sets $... | 9 |
14. Let $\left\{a_{n}\right\}$ be a sequence of positive numbers, with the sum of its first $n$ terms being $S_{n}$, and for all natural numbers $n$, we have
$$
S_{n}=\frac{3 n+1}{2}-\frac{n}{2} a_{n} \text {. }
$$
(1) Write down the first three terms of the sequence $\left\{a_{n}\right\}$;
(2) Find the general term fo... | S_{n}^{\prime}=\frac{n(n+1)(n+2)(n+3)}{8} |
6. Given a pyramid $S-ABCD$ with a height of $\frac{\sqrt{2}}{4}$, the base is a square with side length 1, and points $S, A, B, C, D$ are all on the same sphere with a radius of 1. Then the maximum length of the lateral edge $SA$ is $\qquad$ | \frac{\sqrt{6+2 \sqrt{7}}}{2} |
Which of the following events is a certain event?
A: Tossing a coin with the head facing down
B: The sum of two sides of a triangle is greater than the third side
C: The sum of the three interior angles of a triangle is less than $180^{\circ}$
D: Drawing a red ball from a box containing black balls | \text{B} |
Ricsi, Dénes, and Attila often play ping pong against each other, with two of them standing on one side. Dénes and Attila win against Ricsi three times as often as they lose; Dénes wins against Ricsi and Attila as often as he loses; finally, Attila wins against Ricsi and Dénes twice as often as he loses. Recently, they... | \frac{15}{16} |
(9) (14 points) Given the sequence $\left\{a_{n}\right\}$ satisfies $\frac{a_{n+1}+a_{n}-1}{a_{n+1}-a_{n}+1}=n\left(n \in \mathbf{N}^{*}\right)$, and $a_{2}=6$.
(1) Find the general term formula of the sequence $\left\{a_{n}\right\}$;
(2) Let $b_{n}=\frac{a_{n}}{n+c}\left(n \in \mathbf{N}^{*}\right)$, where $c$ is a no... | 4-\frac{n+2}{2^{n-1}} |
The function $f : \mathbb{R}\to\mathbb{R}$ satisfies $f(x^2)f^{\prime\prime}(x)=f^\prime (x)f^\prime (x^2)$ for all real $x$. Given that $f(1)=1$ and $f^{\prime\prime\prime}(1)=8$, determine $f^\prime (1)+f^{\prime\prime}(1)$. | 6 |
In the Cartesian coordinate system $xOy$, there is a point $M$ on the hyperbola $\frac{x^{2}}{4} - \frac{y^{2}}{12} = 1$. If the x-coordinate of point $M$ is 3, then the distance from $M$ to the right focus of the hyperbola is __________. | 4 |
7. A line segment connecting two different points on an ellipse is called a chord of the ellipse. Then, the equation of the locus of the midpoint of a chord of length $\sqrt{5}$ on the ellipse $\frac{x^{2}}{4}+y^{2}=1$ is $\qquad$ | x^{4}+20x^{2}y^{2}+64y^{4}+x^{2}-44y^{2}=0 |
1. Given $A=\left\{x \mid x^{2}-2 x-3>0\right\}, B=\left\{x \mid x^{2}+a x+b \leqslant 0\right\}$. If $A \cup$ $B=\mathbf{R}, A \cap B=(3,4]$, then the correct conclusion is $(\quad)$.
A. $a=3, b=4$
B. $a=3, b=-4$
C. $a=-3, b=4$
D. $a=-3, b=-4$ | D |
Problem 11.6. Given the set of numbers $\{-1,-2,-3, \ldots,-26\}$. On the board, all possible subsets of this set were written, each containing at least 2 numbers. For each written subset, the product of all numbers belonging to that subset was calculated. What is the sum of all these products? | 350 |
Given $(\sqrt{3} + i) \cdot z = -i$, where $i$ is the imaginary unit, the point corresponding to the complex number $z$ is located in which quadrant of the complex plane?
A: First quadrant
B: Second quadrant
C: Third quadrant
D: Fourth quadrant | C |
Given the function $f(x-1) = x^2 - 2x + 2$, then $f(x) = \ $ | x^2 + 1 |
8. Let $a>1$. If the equation $a^{x}=x$ has no real roots, then the range of the real number $a$ is $\qquad$ . | >\mathrm{e}^{\frac{1}{e}} |
Given that the sum of the first $n$ terms of a sequence $\{a_{n}\}$ is ${S}_{n}={n}^{2}$.
$(1)$ Prove that the sequence $\{a_{n}\}$ is an arithmetic sequence.
$(2)$ Let ${T}_{n}=\frac{1}{{a}_{1}{a}_{2}}+\frac{1}{{a}_{2}{a}_{3}}+…+\frac{1}{{a}_{n}{a}_{n+1}}$, find the expression for $T_{n}$. | \frac{n}{2n+1} |
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