problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
|---|---|
22. (5 points)
Jiang Liuer used 30 Go stones to form a circle, with each black stone being adjacent to exactly one white stone, and the white stones adjacent to black stones account for half of the total number of white stones. Therefore, among the 30 stones, the maximum number of black stones is $\qquad$. | 14 |
Let $P$ be the set of all $2012$ tuples $(x_1, x_2, \dots, x_{2012})$, where $x_i \in \{1,2,\dots 20\}$ for each $1\leq i \leq 2012$. The set $A \subset P$ is said to be decreasing if for each $(x_1,x_2,\dots ,x_{2012} ) \in A$ any $(y_1,y_2,\dots, y_{2012})$ satisfying $y_i \leq x_i (1\leq i \leq 2012)$ also belongs t... | \frac{1}{20^{2012}} |
Calculate the value of $$A_{7}^{3} - C_{6}^{4}$$. | 195 |
70. In a football match, each team has 11 players on the field, including 1 goalkeeper who does not participate in the formation of defenders, midfielders, and forwards. It is known that the number of defenders is between 3 to 5, the number of midfielders is between 3 to 6, and the number of forwards is between 1 to 3.... | 8 |
Convert the octal number $135_{8}$ into a binary number:
(A) $1 110 101_{2}$
(B) $1 010 101_{2}$
(C) $111 001_{2}$
(D) $1 011 101_{2}$ | \text{D} |
3. (6 points) Definition: $a \preccurlyeq b=\frac{\mathrm{a}-1}{\mathrm{~b}}$, then $2 \preccurlyeq(3 \preccurlyeq 4)=$ $\qquad$ | 2 |
1. A quadratic polynomial $f(x)=x^{2}+p x+q$, with $p$ and $q$ real numbers, is said to be a double-up polynomial if it has two real roots, one of which is twice the other.
(a) If a double-up polynomial $f(x)$ has $p=-15$, determine the value of $q$.
(b) If $f(x)$ is a double-up polynomial with one of the roots equal t... | 50,20,2,x^{2}-9x+18,x^{2}+\frac{9}{2}x+\frac{9}{2} |
Three, given the system of equations $\left\{\begin{array}{l}x-y=2, \\ m x+y=6 .\end{array}\right.$ If the system has non-negative integer solutions, find the value of the positive integer $m$, and solve the system of equations. | m=1, (x,y)=(4,2); m=3, (x,y)=(2,0) |
1. Find all triples $(f, g, h)$ of injective functions from the set of real numbers to itself satisfying
$$
\begin{aligned}
f(x+f(y)) & =g(x)+h(y) \\
g(x+g(y)) & =h(x)+f(y) \\
h(x+h(y)) & =f(x)+g(y)
\end{aligned}
$$
for all real numbers $x$ and $y$. (We say a function $F$ is injective if $F(a) \neq F(b)$ for any distin... | f(x)=(x)=(x)=x+C |
The value of $\left(2^{3}\right)^{2}-4^{3}$ is
(A) 0
(B) -8
(C) 4
(D) 10
(E) 12 | 0 |
Let $1 \leq n \leq 2021$ be a positive integer. Jack has $2021$ coins arranged in a line where each coin has an $H$ on one side and a $T$ on the other. At the beginning, all coins show $H$ except the nth coin. Jack can repeatedly perform the following operation: he chooses a coin showing $T$, and turns over the coins n... | n = 1011 |
Given vectors $\overrightarrow{a}=(3,1), \overrightarrow{b}=(1,-1),$ and $\overrightarrow{c}=\overrightarrow{a}+k\overrightarrow{b}$. If $\overrightarrow{a}$ is perpendicular to $\overrightarrow{c}$, find the value of $k$. | -5 |
10. A total of 240 students participating in a large group performance stand in a row facing the coach, and they report numbers from left to right as $1,2,3,4$, $5, \cdots$, in sequence. The coach asks all students to remember their reported numbers and perform the following actions: First, all students who reported nu... | 136 |
# 3. (3 points) Find the distance between the curves $y=e^{3 x+11}$ and $y=(\ln x-11) / 3$. | \sqrt{2}(4+\frac{\ln3}{3}) |
1. Given that the real number $a$ satisfies
$$
|2004-a|+\sqrt{a-2005}=a \text {. }
$$
Then, the value of $a-2004^{2}$ is ( ).
(A) 2003
(B) 2004
(C) 2005
(D) 2006 | C |
5. A small rabbit eats 2 pieces of cabbage a day, a large rabbit eats 4 pieces of cabbage a day, now raising two large rabbits, one small rabbit, every day $\qquad$ pieces of cabbage need to be prepared. | 10 |
Given the sequence $\{a_n\}$ where $a_1=2$, and $a_n=2a_{n-1}-n+2$ for $n\geq2, n\in\mathbb{N}^*$.
(I) Find $a_2$, $a_3$ and prove that $\{a_n-n\}$ is a geometric sequence.
(II) Let $b_n=\frac{a_n}{2^{n-1}}$, find the sum of the first $n$ terms, denoted as $S_n$, of the sequence $\{b_n\}$. | n+4-\frac{2+n}{2^{n-1}} |
Example 1 Given the parabola $y=x^{2}$, draw a line with a slope of 1 through the origin intersecting the parabola at $P_{1}$, draw a line with a slope of $\frac{1}{2}$ through $P_{1}$ intersecting the parabola at $P_{2}, \cdots \cdots$, generally, draw a line with a slope of $2^{-n}$ through the point $P_{n}$ intersec... | \left(\frac{2}{3}, \frac{4}{9}\right) |
[ Properties of sections ] $[$ Regular tetrahedron $]$
Through the vertex $D$ of a regular tetrahedron $A B C D$ with edge $a$ and the midpoints of edges $A B$ and $A C$, a plane is drawn. Find the area of the resulting section. | \frac{^2\sqrt{11}}{16} |
2. $x_{1}+x_{2}+\cdots+x_{19}+2 x_{20}=3$ has $\qquad$ non-negative integer solutions. | 1349 |
The value of $\cos \frac{7\pi}{6}$ is (Â Â Â ).
A: $\frac{1}{2}$
B: $-\frac{1}{2}$
C: $\frac{\sqrt{3}}{2}$
D: $-\frac{\sqrt{3}}{2}$ | D |
Given that the sum of the first $n$ terms of the sequence $\{a_n\}$ is $S_n=n^2$, and the sequence $\{b_n\}$ satisfies $b_1=a_1$, $b_{n+1}-b_n=a_n$, find the general term formula for the sequence $\{b_n\}$. | n^2-2n+2 |
Given the function $f(x)= \begin{cases} x^{2}-x & \text{for } x \geq 1, \\ x^{2}-3x+2 & \text{for } x < 1, \end{cases}$ find the range of values for the real number $m$ such that the inequality $f(x)\geq mx$ holds true. The options for the range of $m$ are:
A: $[-3-2\sqrt{2}, -3+2\sqrt{2}]$
B: $[-3+2\sqrt{2}, 0]$
C: $[... | C |
5. How many pairs of positive integers $(a, b)$ are there, both not exceeding 10 , such that $a \leq \operatorname{gcd}(a, b) \leq b$ and $a \leq \operatorname{lcm}(a, b) \leq b$ ? | 27 |
1. a) The first four terms of an infinite arithmetic sequence are:
$$
-\frac{3}{2}, \frac{1}{2}, \frac{5}{2}, \frac{9}{2}
$$
Calculate which number is the two hundredth term of this sequence and what is the sum of the first two hundred terms of the sequence.
b) Between the numbers $\frac{1}{2}$ and $\frac{9}{2}$, in... | \frac{793}{2},39500,\frac{\sqrt{3}}{2},\frac{3}{2},\frac{3\sqrt{3}}{2},-\frac{\sqrt{3}}{2},\frac{3}{2},-\frac{3\sqrt{3}}{2} |
Given $(1+i)\sin\theta - (1+i\cos\theta)$ corresponds to a point on the line $x+y+1=0$, find the value of $\tan\theta$. | \frac{1}{2} |
Solve the following equation in the set of real numbers:
$$
(\sqrt{5}+2)^{x}+(\sqrt{5}-2)^{x}=18
$$ | x_{1}=2x_{2}=-2 |
8. Let $A, B, C, D$ be four non-coplanar points in space, and connect an edge between each pair of points with a probability of $\frac{1}{2}$. Whether any two pairs of points are connected is independent of each other. Then the probability that $A$ and $B$ can be connected by (a single edge or a sequence of edges formi... | \frac{3}{4} |
The function $f(x)$ is an odd function defined on the domain $\mathbb{R}$. When $x > 0$, $f(x)=-x+1$. What is the expression for $f(x)$ when $x < 0$?
A: $-x-1$
B: $-x+1$
C: $x+1$
D: $x-1$ | A |
If $\sqrt{x-1}+{({y-2})^2}=0$, then $xy=\_\_\_\_\_\_$. | 2 |
(5) The maximum value of the function $y=2 x-5+\sqrt{11-3 x}$ is $\qquad$ . | \frac{65}{24} |
3. Find all integer solutions $a, b, c$ that satisfy $\frac{1}{2}(a+b)(b+c)(c+a)+(a+b+c)^{3}=1-a b c$, and provide necessary proof.
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly. | (1,0,0),(0,1,0),(0,0,1),(2,-1,-1),(-1,2,-1),(-1,-1,2) |
2. Solve in the set of integers the equation: $\sqrt{x^{2}-1900}=\left|\frac{x}{4}-4\right|-\sqrt{2017-x^{2}}$.
( Mathematical Gazette ) | -44 |
In the arithmetic sequence $\{a_n\}$, $a_1+a_5=8$, $a_4=7$, then $a_5=$ （　　）
A: 11
B: 10
C: 7
D: 3 | B |
During a class meeting, a personal talent show is held where 5 students participate, including 3 females and 2 males. If the 2 males cannot appear consecutively, and female A cannot be the first to appear, what is the number of different performance sequences? ( )
A: 24
B: 36
C: 48
D: 60 | D |
Solve the following system of equations:
$$
\begin{array}{ll}
x^{3} y^{2} z=2 & z^{3} u^{2} x=32 \\
y^{3} z^{2} u=8 & u^{3} x^{2} y=8
\end{array}
$$ | \begin{pmatrix}x_{1}=1,&z_{1}=2,&y_{1}=1,&u_{1}=2\\x_{2}=1,&z_{2}=2,&y_{2}=-1,&u_{2}=-2\\x_{3}=-1,&z_{3}=-2,&y_{3}=1,&u_{3}=2\\x_{4} |
Which of the following square roots is of the same type as $2\sqrt{3}$?
A: $\sqrt{8}$
B: $\sqrt{\frac{1}{3}}$
C: $\sqrt{18}$
D: $\sqrt{9}$ | B |
Example 7 Let $V_{1}$ and $V_{2}$ be the volumes of the circumscribed sphere and the inscribed sphere of a regular quadrilateral pyramid, respectively. Try to find the minimum value of $\frac{V_{1}}{V_{2}}$. | 7+5 \sqrt{2} |
\section*{Problem 4 - 261044}
Determine for each natural number \(k \geq 2\) the number of all solutions \((x, y, z, t)\) of the equation \(\overline{x y} + \overline{z t} = \overline{y z}\), where \(x, y, z, t\) are natural numbers with
\[
1 \leq x \leq k-1, \quad 1 \leq y \leq k-1 \quad 1 \leq z \leq k-1 \quad 1 \l... | \frac{1}{2}(k-3)(k-2) |
Task 1. (5 points) Find $\frac{a^{8}+256}{16 a^{4}}$, if $\frac{a}{2}+\frac{2}{a}=3$. | 47 |
31st Swedish 1991 Problem 1 Find all positive integers m, n such that 1/m + 1/n - 1/(mn) = 2/5. | (,n)=(3,10),(4,5),(5,4),(10,3) |
10. (15 points) As shown in Figure 3, three circles intersect to form seven regions. Fill in the numbers $1, 2, 2, 4, 4, 8, 8$ into these seven regions so that the product of the four numbers inside each circle is equal (denoted as $P$). The filling method shown in Figure 4 satisfies the condition, with $P=64$ at this ... | P_{\max}=256,P_{\}=64 |
Given that point $P$ lies on the graph of the function $y=x^{2}$ and point $A$ is $(1,-1)$, determine the maximum value of $\overrightarrow{OP} \cdot \overrightarrow{OA}$. | \frac{1}{4} |
For a positive integer $n$, let $S(n)$ denote its digit sum. Find all positive integers $M$ such that for every positive integer $k$ not exceeding $M$, we have $S(M k)=S(M)$. | 10^{n}-1 |
Problem 12.1. Let $A B C$ be an isosceles triangle such that $A C=B C=1$ and $A B=2 x, x>0$.
a) Express the inradius $r$ of $\triangle A B C$ as a function of $x$.
b) Find the maximum possible value of $r$.
Oleg Mushkarov | \sqrt{\frac{5\sqrt{5}-11}{2}} |
Given $\sin (\frac{π}{6}+α)=\frac{\sqrt{3}}{3}$, $α∈(-\frac{π}{4},\frac{π}{4})$, find $sin(\frac{π}{3}-α)$. | \frac{\sqrt{6}}{3} |
(a) You are given the expression
$$
1 \diamond 2 \diamond 3 \diamond 4 \diamond 5 \diamond 6 \diamond 7 \text {. }
$$
Determine whether it is possible to replace one of the symbols $\diamond$ with $=$ and the other symbols $\diamond$ with + or - so as to end up with a correct equation. | 2+3+4+5-6-7 |
The absolute value of \\(-3\\) is (Â Â Â Â )
A: \\(-3\\)
B: Â \\(- \\dfrac{1}{3} \\)
C: \\( \\dfrac{1}{3} \\)
D: \\(3\\) | D |
Given the straight line $3x - 4y + 5 = 0$, if the angle of inclination is $\alpha$, then $\sin\alpha = \_\_\_\_\_\_$. | \frac{3}{5} |
If $-\frac{\pi }{2} < \alpha < 0$, then the magnitude of the inclination angle of the line $x\tan \alpha - y = 0$ is ______. | \pi + \alpha |
4. There are circles $\omega_{1}$ and $\omega_{2}$. They intersect in two points, one of which is the point $A$. $B$ lies on $\omega_{1}$ such that $A B$ is tangent to $\omega_{2}$. The tangent to $\omega_{1}$ at $B$ intersects $\omega_{2}$ at $C$ and $D$, where $D$ is the closer to $B$. $A D$ intersects $\omega_{1}$ a... | \frac{4\sqrt{3}}{3} |
Given the proposition $p$: "There exists $x \in \mathbb{R}$, such that $e^{x} - x - 1 \leqslant 0$", then the proposition $\neg p$ is
A: $\forall x \in \mathbb{R}$, $e^{x} - x - 1 > 0$
B: $\forall x \notin \mathbb{R}$, $e^{x} - x - 1 > 0$
C: $\forall x \in \mathbb{R}$, $e^{x} - x - 1 \geqslant 0$
D: $\exists x \in ... | \text{A} |
In a convex $n$-sided polygon, the difference between any two adjacent interior angles is $18^{\circ}$. Try to find the maximum value of $n$.
The difference between any two adjacent interior angles of a convex $n$-sided polygon is $18^{\circ}$. Try to find the maximum value of $n$. | 38 |
Let $a_1$, $a_2$, $\ldots\,$, $a_{2019}$ be a sequence of real numbers. For every five indices $i$, $j$, $k$, $\ell$, and $m$ from 1 through 2019, at least two of the numbers $a_i$, $a_j$, $a_k$, $a_\ell$, and $a_m$ have the same absolute value. What is the greatest possible number of distinct real numbers in the giv... | 8 |
Task B-2.7. Let $A B C$ be a right triangle with a right angle at vertex $C$. The height $C N$ is drawn to the side $A B$ and from point $N$ we draw the height $N P$ to the side $B C$ in triangle $B C N$. If the lengths of the legs are $|A C|=3$ and $|B C|=4$, what is the length of the height $N P$? | \frac{48}{25} |
An isosceles right triangle with legs of length $8$ is partitioned into $16$ congruent triangles as shown. The shaded area is
$\text{(A)}\ 10 \qquad \text{(B)}\ 20 \qquad \text{(C)}\ 32 \qquad \text{(D)}\ 40 \qquad \text{(E)}\ 64$ | 20 |
## Problem Statement
Calculate the indefinite integral:
$$
\int \ln \left(x^{2}+4\right) d x
$$ | x\cdot\ln(x^{2}+4)-2x+4\operatorname{arctg}\frac{x}{2}+C |
10,11 |
The numbers $1,2, \ldots, k^{2}$ are arranged in a square table
| 1 | 2 | $\ldots$ | $k$ |
| :---: | :---: | :---: | :---: |
| $k+1$ | $k+2$ | $\ldots$ | $2 k$ |
| $\ldots$ | $\ldots$ | $\ldots$ | $\ldots$ |
| $(k-1) k+1$ | $\ldots$ | $\ldots$ | $k^{2}$ |
An arbitrary number is written down, after which the ... | \frac{1}{2}k(k^2+1) |
Convert $-sinα + \sqrt{3}cosα$ into the form of $A\sin(α + φ)$ where $A > 0$ and $φ ∈ (0, 2π)$. | 2\sin\left(α + \frac{2π}{3}\right) |
11. Express 36 as the product of three natural numbers greater than 1 (the order of the three numbers does not matter), how many different ways are there? | 3 |
6. A line $l$ passing through the origin intersects the hyperbola $x y=-2 \sqrt{2}$ at points $P$ and $Q$, where point $P$ is in the second quadrant. Now, fold the upper and lower half-planes to form a right dihedral angle, at which point $P$ falls to point $P^{\prime}$. Then the minimum length of the line segment $P^{... | D |
Let $\{a_{n}\}$ be a sequence with the sum of its first $n$ terms denoted as $S_{n}$. It is known that $S_{n+2}+3S_{n}=4S_{n+1}-2a_{n}$, $a_{1}=1$, $a_{2}=3$.
$(1)$ Prove that the sequence $\{a_{n+1}-2a_{n}\}$ is an arithmetic sequence.
$(2)$ Let $({a}_{n}+1){b}_{n}=\frac{n+2}{{n}^{2}+n}$, $T_{n}$ be the sum of the... | T_{n}=1-\frac{1}{(n+1)\cdot2^{n}} |
13.333. A ball falls from a height of 2 m 43 cm and, upon hitting the ground, bounces back up, each time reaching $2 / 3$ of the height from which it falls again. After how many bounces will the ball rise to a height of 32 cm? | 5 |
6.351 Solve the equations $x^{3}-6 x^{2}-39 x-10=0$ and $x^{3}+x^{2}-20 x-50=0$, given that one of the roots of the first equation is twice one of the roots of the second equation. | x_{1}=10,x_{2,3}=-2\\sqrt{3};5 |
If $a^{x}=2$ and $a^{y}=3$, then $a^{2x+y}=$____. | 12 |
In the geometric sequence $\{a_n\}$, if $a_4 = 5$, then $a_1a_7 = \_\_\_\_\_\_$. | 25 |
5. Given $a_{1}=1, a_{2}=3$,
$$
a_{n+2}=(n+3) a_{n+1}-(n+2) a_{n} \text {, }
$$
when $m \geqslant n$, $a_{m}$ is divisible by 9. Then the minimum value of $n$ is $\qquad$ . | 5 |
12. After a coordinate paper is folded once, point $A(0,4)$ overlaps with point $B(8,0)$. If point $C(6,8)$ overlaps with point $D(m$, $n)$, then $m+n=$ | 14.8 |
10. The monotonic decreasing interval of the function $y=\arcsin \left(2-x^{2}\right)$ is | [1,\sqrt{3}] |
Problem 4. On the math test, the teacher set 30 problems. The number of problems correctly solved by a student is 20 more than the number of problems not solved. For each correctly solved problem, the student gets 8 points, and for each unsolved or incorrectly solved problem, the student loses 3 points. How many points... | 185 |
$4 \cdot 172$ Find all positive integer solutions $(x, y, z, n)$ of the equation
$$
x^{2 n+1}-y^{2 n+1}=x y z+2^{2 n+1}
$$
satisfying the conditions $n \geqslant 2, z \leqslant 5 \cdot 2^{2 n}$. | (3,1,70,2) |
3. Determine the largest prime number whose all digits are distinct such that any permutation of its digits yields another prime number. | 97 |
One, (20 points) Given real numbers $a, b, c, d$ satisfy $2a^2 + 3c^2 = 2b^2 + 3d^2 = (ad - bc)^2 = 6$. Find the value of $\left(a^2 + \dot{b}^2\right)\left(c^2 + d^2\right)$. | 6 |
Observe the following equations:The $1$st equation: $a_{1}=\frac{1}{1+\sqrt{2}}=\sqrt{2}-1$,The $2$nd equation: $a_{2}=\frac{1}{\sqrt{2}+\sqrt{3}}=\sqrt{3}-\sqrt{2}$,The $3$rd equation: $a_{3}=\frac{1}{\sqrt{3}+2}=2-\sqrt{3}$,The $4$th equation: $a_{4}=\frac{1}{2+\sqrt{5}}=\sqrt{5}-2$,$\ldots $Following the above patte... | \sqrt{n+1}-1 |
Given vectors $\overset{⇀}{a}$ and $\overset{⇀}{b}$ that satisfy $|\overset{⇀}{a}| = 4$, $|\overset{⇀}{b}| = 3$, and the angle between $\overset{⇀}{a}$ and $\overset{⇀}{b}$, denoted by $θ$, is $60^{\circ}$.
(1) Find the value of $(\overset{⇀}{a} + 2\overset{⇀}{b}) \cdot (2\overset{⇀}{a} - \overset{⇀}{b})$;
(2) Find t... | 7 |
The sum of the first $n$ terms of a geometric sequence is 48, and the sum of the first $2n$ terms is 60. Then, the sum of the first $3n$ terms is \_\_\_\_\_\_. | 63 |
Given an arithmetic sequence $\{a_n\}$, the sum of the first $n$ terms is $S_n$. If $a_6 = -3$ and $S_6 = 12$, then $a_5$ equals (　　)
A: $-3$
B: $-1$
C: $1$
D: $4$ | \text{B} |
Let set $S = \{y | y = 3^x, x \in \mathbb{R}\}$, and $T = \{y | y = x^2 + 1, x \in \mathbb{R}\}$. Then, $S \cap T =$ （ ）
A: $\emptyset$
B: $S$
C: $T$
D: $\{(0,1)\}$ | C |
The set $\{1,2, \ldots, 100\}$ has a subset $H$ with the property that the tenfold of any element in $H$ is not in $H$. What is the maximum number of elements $H$ can have? | 91 |
1. Among the real numbers $99, \frac{\sqrt{3}}{2}, 0, \sqrt{2}-1, \sqrt{64}, \sqrt[3]{25}, \pi, \frac{7}{9}$, there are $\qquad$ irrational numbers. | 4 |
5. There is a moving line $y=\frac{2}{t}(x-t-1)$ (where $t$ is a parameter), and a parabola $y^{2}=4 x$. What is the locus of the midpoint of the line segment joining the two points of intersection of the line and the parabola? Prove that this locus is tangent to the line $x+y+1=0$.
---
The translation is provided as... | 2 x-1=(y+1)^{2} |
11.1. The numbers $2^{2019}$ and $5^{2019}$ are written consecutively. How many digits are written in total? | 2020 |
6.342 Let $S_{n}=\alpha^{n}+\beta^{n}$, where $\alpha$ and $\beta$ are the roots of the equation $a x^{2}+b x+c=0$. Find the relationship between $S_{n}, S_{n+1}, S_{n+2}$. | S_{n+2}=\frac{-(b\cdotS_{n+1}+\cdotS_{n})}{} |
10. Given that the largest angle of an isosceles triangle is 4 times the smallest angle, then the difference between the largest and smallest angles is $(\quad)$ degrees. | 90 |
A4. If we halve the straight angle, divide the half into thirds, the third into fifths, and the fifth into sixths, the sixth of the fifth of the third of the half of the straight angle measures:
(A) $1^{\circ}$
(B) $2^{\circ}$
(C) $30^{\circ}$
(D) $1^{\prime}$
(E) $30^{\prime}$ | 1 |
Given that the vertex of angle $\alpha$ is at the origin, the initial side coincides with the non-negative half-axis of the $x$-axis, and the terminal side is in the third quadrant intersecting the unit circle at point $P\left(-\frac{{\sqrt{5}}}{5},m\right)$, then $\sin \alpha =\left(\ \ \right)$
A: $-\frac{{\sqrt{5}}... | C |
If the tangent line to the curve $f(x)=x^{4}-4x$ at point $A$ is parallel to the $x$-axis, then the coordinates of point $A$ are (　　).
A: $(-1,2)$
B: $$(1,-3)$$
C: $(1,0)$
D: $(1,5)$ | B |
Given $f(x) = |x-1| + |x+2|$.
1. Solve the inequality $f(x) \geq 5$;
2. If the inequality $f(x) > a^2 - 2a - 5$ holds for any $x \in \mathbb{R}$, find the range of $a$. | (-2, 4) |
In a right-angled triangle, the center $O$ of the inscribed circle is such that the half of the hypotenuse closer to $O$ is seen at a right angle from $O$. What is the ratio of the sides of the triangle? | 3:4:5 |
## Problem Statement
Calculate the limit of the function:
$\lim _{x \rightarrow \pi} \frac{\sin ^{2} x-\tan ^{2} x}{(x-\pi)^{4}}$ | -1 |
1. Find all triples of real numbers $(x, y, z)$ such that
$$
\left\{\begin{array}{l}
x^{2}+y^{2}+25 z^{2}=6 x z+8 y z, \\
3 x^{2}+2 y^{2}+z^{2}=240 .
\end{array}\right.
$$ | (x,y,z)=(-6,-8,-2),(6,8,2) |
Given the sets $M=\{x|x^2-2x-3\leq0, x\in \mathbb{R}\}$ and $N=\{x||x|<2, x\in \mathbb{R}\}$, then $M\cap N$ equals ( )
A: $\emptyset$
B: $\{x|-1\leq x<2\}$
C: $\{x|-2\leq x<-1\}$
D: $\{x|2\leq x<3\}$ | \text{B} |
Let the function $f$ be defined on the set of ordered pairs of positive integers, and satisfy:
1. $f(x, x) = x$;
2. $f(x, y) = f(y, x)$;
3. $(x+y)f(x, y) = yf(x, x+y)$;
Then the value of $f(12, 16) + f(16, 12)$ is (　　)
A: 24
B: 48
C: 64
D: 96 | D |
Three. (50 points) In a $\left(2^{n}-1\right) \times\left(2^{n}-1\right)(n$ $\geqslant 2)$ grid, each cell is filled with 1 or -1. If the number in any cell is equal to the product of the numbers in the cells that share an edge with it, then this filling method is called "successful". Find the total number of "successf... | 1 |
Given a quadratic equation in $x$, $mx^{2}-(m+2)x+\frac{m}{4}=0$ has two distinct real roots $x_{1}$ and $x_{2}$. If $\frac{1}{{x}_{1}}+\frac{1}{{x}_{2}}=4m$, then the value of $m$ is ______. | 2 |
Problem N1. Find all positive integers $n$ such that $n 2^{n+1}+1$ is a perfect square.
| n=0n=3 |
Given the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ ($a > 0, b > 0$) has an eccentricity of $2$, then the equation of the asymptotes of the hyperbola is (　　)
A: $\sqrt{2}x \pm y = 0$
B: $x \pm y = 0$
C: $2x \pm y = 0$
D: $\sqrt{3}x \pm y = 0$ | D |
$i$ is the imaginary unit. If $\left(1+2i\right)\overline{z}=4+3i$, the imaginary part of $z$ is ____. | 1 |
If the distance from a point $P$ on the ellipse $\frac{x^2}{144} + \frac{y^2}{36} = 1$ to one of its foci $F_1$ is equal to $10$, then the distance from point $P$ to the other focus $F_2$ is ______. | 14 |
If the curve $y=\ln x+ax^{2}-2x$ (where $a$ is a constant) does not have any tangent lines with a negative slope, then the range of the real number $a$ is \_\_\_\_\_\_. | \left[ \frac{1}{2}, +\infty \right) |
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