problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
|---|---|
If the equation $1-k(x-2e)\cdot\ln x=0$ has two different solutions on $(1,+\infty)$, where $e$ is the base of the natural logarithm, then the range of the real number $k$ is ______. | (-\infty,- \dfrac {1}{e}) |
The solution to the equation $\frac{2}{x}=\frac{3}{x+1}$ is:
A: $x=1$
B: $x=-1$
C: $x=2$
D: $x=-2$ | C |
Problem 9.5. On the base $AC$ of isosceles triangle $ABC (AB = BC)$, a point $M$ is marked. It is known that $AM = 7, MB = 3, \angle BMC = 60^\circ$. Find the length of segment $AC$.
 | 17 |
4. For any positive integer $n>1$, let $P(n)$ denote the largest prime not exceeding $n$. Let $N(n)$ denote the next prime larger than $P(n)$. (For example $P(10)=7$ and $N(10)=11$, while $P(11)=11$ and $N(11)=13$.) If $n+1$ is a prime number, prove that the value of the sum
$$
\frac{1}{P(2) N(2)}+\frac{1}{P(3) N(3)}... | \frac{n-1}{2n+2} |
12. 8 girls and 25 boys stand in a circle, with at least two boys standing between any two girls. Therefore, there are $\qquad$ different arrangements (arrangements that can be made identical by rotating the circle are considered the same). | \frac{16!\cdot25!}{9!} |
Lemma 1 Suppose $\left(x_{i}, y_{i}\right)(i=1,2,3)$ satisfy the equation $x^{2}+y^{2}=1$, then
$$\left\|\begin{array}{lll}
x_{1} & y_{1} & 1 \\
x_{2} & y_{2} & 1 \\
x_{3} & y_{3} & 1
\end{array}\right\| \leqslant \frac{3 \sqrt{3}}{2}$$ | \frac{3 \sqrt{3}}{2} |
2. Determine the least positive value taken by the expression $a^{3}+b^{3}+c^{3}-3 a b c$ as $a, b, c$ vary over all positive integers. Find also all triples $(a, b, c)$ for which this least value is attained.
| 4 |
Given points P and Q are two fixed points on the plane of $\triangle ABC$, and it satisfies $\overrightarrow {PA}+ \overrightarrow {PC}= \overrightarrow {0}$, $2\overrightarrow {QA}+ \overrightarrow {QB}+ \overrightarrow {QC}= \overrightarrow {BC}$. If $|\overrightarrow {PQ}|=λ|\overrightarrow {BC}|$, then the positive... | \lambda= \frac {1}{2} |
Example 3.12 Solve the recurrence relation:
$$
\left\{\begin{array}{l}
a_{n}=8 a_{n-1}-22 a_{n-2}+24 a_{n-3}-9 a_{n-4} \quad(n \geqslant 4), \\
a_{0}=-1, a_{1}=-3, a_{2}=-5, a_{3}=5 .
\end{array}\right.
$$ | a_{n}=2+n-3^{n+1}+n\cdot3^{n} |
The 42nd question: Given infinite real sequences $a_{1}, a_{2}, \ldots$ and $b_{1}, b_{2}, \ldots$, for $n \geq 1$, we have $a_{n+1}+b_{n+1}=\frac{a_{n}+b_{n}}{2}$, $a_{n+1} b_{n+1}=\sqrt{a_{n} b_{n}}$. Suppose $b_{2016}=1$ and $a_{1}>0$, find all possible values of $a_{1}$.
---
The 42nd question: Given infinite real... | 2^{2015} |
Example 3. Calculate $\int_{4}^{9}\left(\frac{2 x}{5}+\frac{1}{2 \sqrt{x}}\right) d x$. | 14 |
The circle that is tangent to both the circle $x^2+y^2=1$ and the circle $x^2+y^2-8x+12=0$ has its center at ( )
A: On an ellipse
B: On one branch of a hyperbola
C: On a parabola
D: On a circle | \text{B} |
Simplify the expression: $110^{2}-109\times 111$. | 1 |
Calculate the Minkowski dimension of the Koch curve. | \frac{\log(4)}{\log(3)} |
5. Given that the angle between vectors $a$ and $b$ is $45^{\circ}$, and
$$
\begin{array}{l}
|a|=4, \\
\left(\frac{1}{2} a+b\right) \cdot(2 a-3 b)=12 .
\end{array}
$$
Then the projection of $\boldsymbol{b}$ onto $\boldsymbol{a}$ is equal to ( ).
(A) $\sqrt{2}$
(B) 4
(C) $4 \sqrt{2}$
(D) 1 | D |
Let the universal set be $\mathbb{R}$, and let the sets be $A=\{x|x\leq3 \text{ or } x\geq6\}$, $B=\{x|-2<x<9\}$.
(1) Find $A\cup B$, $(\mathbb{R}\setminus A) \cap B$;
(2) Given $C=\{x|a<x<a+1\}$, if $C\subseteq B$, find the range of the real number $a$. | [-2, 8] |
$y=(\sin x - \cos x)^2 - 1$ is ( )
A: An even function with a minimum positive period of $2\pi$
B: An odd function with a minimum positive period of $2\pi$
C: An even function with a minimum positive period of $\pi$
D: An odd function with a minimum positive period of $\pi$ | \text{D} |
Given $α∈({0,\frac{π}{2}})$, if $\sin 2α = \cos({\frac{π}{4}-α})$, then the value of $\cos 2\alpha$ is ( )
A: $0$
B: $\frac{1}{2}$
C: $\frac{\sqrt{3}}{2}$
D: $-\frac{\sqrt{3}}{2}$ | A |
Given that the quality index $\xi$ (unit: millimeters) of a batch of components follows a normal distribution $N(25.40, \sigma^2)$, and $P(\xi \geq 25.45) = 0.1$. Now, randomly select $3$ components from this batch. Let $X$ represent the number of components out of these $3$ whose quality index value $\xi$ does not fal... | 0.48 |
[url=http://www.mathlinks.ro/Forum/viewtopic.php?t=107262]IMO 2007 HKTST 1[/url]
Problem 2
Let $A$, $B$ and $C$ be real numbers such that
(i) $\sin A \cos B+|\cos A \sin B|=\sin A |\cos A|+|\sin B|\cos B$,
(ii) $\tan C$ and $\cot C$ are defined.
Find the minimum value of $(\tan C-\sin A)^{2}+(\cot C-\cos B)^{2}$... | 3 - 2\sqrt{2} |
The point $P(-5,4)$ lies in which quadrant of the plane rectangular coordinate system?
A: First quadrant
B: Second quadrant
C: Third quadrant
D: Fourth quadrant | \text{B} |
73. Given three two-digit numbers that increase by 6 in sequence from smallest to largest, and the number of factors of these three numbers also increases by 1 in sequence, then the smallest of the three numbers is $\qquad$ . | 43 |
Consider the function $f(x)=\ln\left(\frac{1-x}{1+x}\right)$. There are the following four propositions regarding this function:
1. The domain of $f(x)$ is $(-\infty, -1) \cup (1, +\infty)$;
2. $f(x)$ is an odd function;
3. $f(x)$ is an increasing function on its domain;
4. For any $x_1, x_2 \in (-1, 1)$, it holds tha... | 2, 4 |
Example 3. Find the minimum value of the function $f(u, v)=(u-v)^{2}+\left(\sqrt{2-u^{2}}\right.$ $\left.-\frac{9}{v}\right)^{2}$. (1983 Putnam Competition) | 8 |
6. When $s$ and $t$ take all real values, the minimum value that $(s+5-3|\cos t|)^{2}+(s-2|\sin t|)^{2}$ can achieve is | 2 |
In triangle $ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively. It is known that $a \neq b$, and $\cos^2A - \cos^2B = \sqrt{3}\sin A\cos A - \sqrt{3}\sin B\cos B$
(I) Find the magnitude of angle $C$;
(II) If $c= \sqrt{3}$, find the range of the perimeter of $\triangl... | (2\sqrt{3}, 3\sqrt{3}) |
Given a triangle $\triangle ABC$ with side lengths $a$, $b$, and $c$ respectively, and given that $a^{2}+b^{2}-c^{2}=ab$, find the measure of $\angle C$. | C = 60^{\circ} |
Problem 3. Consider the sequence
$$
y_{1}=y_{2}=1, y_{n+2}=(4 k-5) y_{n+1}-y_{n}+4-2 k, n \geq 1
$$
Find all integers $k$ such that any term of the sequence is a perfect square.
Sava Grozdev, Svetlozar Doychev | k=1k=3 |
A large rectangle is tiled by some $1\times1$ tiles. In the center there is a small rectangle tiled by some white tiles. The small rectangle is surrounded by a red border which is fi ve tiles wide. That red border is surrounded by a white border which is fi ve tiles wide. Finally, the white border is surrounded by a re... | 350 |
Evaluate the definite integral $\int_{-2}^{2} \sqrt{4 - x^2} dx$. | 2\pi |
2. If positive numbers $x, y$ satisfy $x+3 y=5 x y$, then the minimum value of $3 x+4 y$ is $\qquad$
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly. | 5 |
In $\triangle ABC$, $AB= \sqrt {3}, BC=2AC=2$, the range of real numbers $t$ that satisfy $| \overrightarrow{BA}-t \overrightarrow{BC}|\leqslant \sqrt {3}| \overrightarrow{AC}|$ is ______. | \left[0, \dfrac {3}{2}\right] |
Problem 9.7. On the coordinate plane, points $O(0 ; 0), A(5 ; 0), B(0 ; 4)$ are marked. The line $y=k x+b$ is such that for any point $M$ on this line, the area of the quadrilateral $A O B M$ is 20. What is $k$? | -0.8 |
1764. Out of 2000 parts, 400 were selected, the distribution of which by size is given by the following table:
| Size of parts, mm | $7,95-8,00$ | $8,00-8,05$ | $8,05-8,10$ | $8,10-8,15$ | $8,15-8,20$ | $8,20-8,25$ |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| Number of parts | 12 | 28 | 132 | 150 | 62... | 0.00234 |
Problem 1. Ilya has a one-liter bottle filled with freshly squeezed orange juice and a 19-liter empty jug. Ilya pours half of the bottle's contents into the jug, then adds half a liter of water to the bottle and mixes it thoroughly. Ilya repeats this operation a total of 10 times. After this, he pours everything that r... | 0.17 |
If point M is on line a, and line a is in plane α, then the relationship among M, a, and α can be represented as ( )
A: $M \in a$，$a \in \alpha$
B: $M \in a$，$a \subset \alpha$
C: $M \subset a$，$a \subset \alpha$
D: $M \subset a$，$a \in \alpha$ | \text{B} |
Problem 8.4. Given a square $A B C D$. Point $L$ is on side $C D$ and point $K$ is on the extension of side $D A$ beyond point $A$ such that $\angle K B L=90^{\circ}$. Find the length of segment $L D$, if $K D=19$ and $C L=6$.
^{2}+y^{2}=9 |
Find all positive integers $p$ for which there exists a positive integer $n$ such that $p^n+3^n~|~p^{n+1}+3^{n+1}.$ | 3, 6, 15 |
6.4. Three pirates, Joe, Bill, and Tom, found a treasure containing 110 identical gold coins, and they want to divide them so that each of them gets at least 30 coins. How many ways are there to do this? | 231 |
7. Given $|\boldsymbol{a}|=1,|\boldsymbol{b}|=|c|=2, \boldsymbol{b} \cdot \boldsymbol{c}=0$, $\lambda \in(0,1)$. Then
$$
|a-b+\lambda(b-c)|+\left|\frac{1}{2} c+(1-\lambda)(b-c)\right|
$$
the minimum value is $\qquad$ | \sqrt{5}-1 |
"北", "京", "奥", "运" each represents a digit, and the four-digit number "北京奥运" plus the sum of its digits equals 2008. What is this four-digit number? | 2003 |
Let the set $M=\{-1, 0, 1\}$, $N=\{x|x^2 \leq x\}$, then $M \cap N=$ （ ）
A: $\{0\}$
B: $\{0, 1\}$
C: $\{-1, 1\}$
D: $\{-1, 0, 0\}$ | \text{B} |
Given the function $f(x) = 2\sin^2(\omega x + \frac{\pi}{4}) - \sqrt{3}\cos(2\omega x)$, where $\omega > 0$, has a period of $\pi$.
1. Find $\omega$ and the range of the function $f(x)$.
2. If the inequality $|f(x) - m| < 2$ always holds for $x \in \left[\frac{\pi}{4}, \frac{\pi}{2}\right]$, find the range of the real ... | (1, 4) |
In the Cartesian coordinate system, if the point $P\left(x,y\right)$ satisfies $x \lt 0$, $y \gt 0$, then the quadrant it lies in is ( )
A: First quadrant
B: Second quadrant
C: Third quadrant
D: Fourth quadrant | B |
How to calculate the number of divisors of an integer $n$? | \prod_{i=1}^{j}(v_{p_{i}}(n)+1) |
A stationery factory produced 800,000 water-based pens in January. The production increased by 5% each month thereafter. The number of water-based pens produced in April is ( )
A: 926100
B: 972405
C: 882000
D: 270000 | A |
Given vectors $\overrightarrow{m}=(2\sin \frac {x}{4},\cos \frac {x}{2})$ and $\overrightarrow{n}=(\cos \frac {x}{4}, \sqrt {3})$, and the function $f(x)= \overrightarrow{m}\cdot \overrightarrow{n}$:
(1) Find the smallest positive period of $f(x)$;
(2) If $0\leqslant x\leqslant π$, find the maximum and minimum values... | 1 |
Given a geometric sequence $\{a_n\}$ where $a_1=2$, and $a_3+2$ is the arithmetic mean of $a_2$ and $a_4$,
(1) Find the general formula for the sequence $\{a_n\}$.
(2) Let $b_n=a_n \cdot \log_{2}a_n$, find the sum of the first $n$ terms of the sequence $\{b_n\}$, denoted as $T_n$. | T_n=(n-1)\cdot2^{n+1}+2 |
Let the function $f(x)= \begin{cases} 1-x^{2} & (x\leqslant 1)\\ x^{2}+x-2 & (x > 1) \end{cases}$, then the value of $f\left( \frac {1}{f(2)}\right)$ is ( ).
A: $ \frac {15}{16}$
B: $- \frac {27}{16}$
C: $ \frac {8}{9}$
D: $18$ | A |
In equilateral triangle $\triangle ABC$, point $P$ lies on segment $AB$ such that $\overrightarrow{AP} = \lambda \overrightarrow{AB}$. If $\overrightarrow{CP} \cdot \overrightarrow{AB} = \overrightarrow{PA} \cdot \overrightarrow{PB}$, find the value of the real number $\lambda$. | \lambda = \frac{2 - \sqrt{2}}{2} |
Given the complex number ${z}_{1}=1+ \sqrt{3}i $, $|{z}_{2}|=2 $, and $({z}_{1}{z}_{2} )$ is a positive real number, find the complex number ${z}_{2}= $______. | 1- \sqrt{3}i |
Given $\alpha \in (\pi, 2\pi)$, and $\cos\alpha = -\frac{\sqrt{5}}{5}$, then the value of $\tan2\alpha$ is ( )
A: $\frac{3}{4}$
B: $\frac{4}{3}$
C: $-\frac{3}{4}$
D: $-\frac{4}{3}$ | \text{D} |
Given the function $f(x) = e^x + x$, the derivative of the function $f(x)$ is ( )
A: $e^x$
B: $e^x + 1$
C: $\ln x + 1$
D: $e^x + x$ | B |
Let $[x]$ denote the greatest integer less than or equal to $x$. For example, $[1.7]=1$, $[-1.7]=-2$. Based on this definition, perform the following operations:
(1) $[2.3]-[6.3]$
(2) $[4]-[-2.5]$
(3) $[-3.8]\times[6.1]$
(4) $[0]\times[-4.5]$． | 0 |
6. Let the prime $p$ satisfy that there exist positive integers $x, y$ such that $p-1=2 x^{2}, p^{2}-1=2 y^{2}$.
Then the number of primes $p$ that meet the condition is ( ).
(A) 1
(B) 2
(C) 3
(D) 4 | A |
Given the complex number $z = (3 - 4i) \cdot i$, find $|z|$. | 5 |
3. Solution. Since $\left(1-\frac{1}{k^{2}}\right)=\left(1-\frac{1}{k}\right)\left(1+\frac{1}{k}\right)=\frac{k-1}{k} \cdot \frac{k+1}{k}$, then
$$
\begin{aligned}
& \left(1-\frac{1}{4}\right)\left(1-\frac{1}{9}\right)\left(1-\frac{1}{16}\right) \cdots\left(1-\frac{1}{2021^{2}}\right)\left(1-\frac{1}{2022^{2}}\right)=... | \frac{2023}{4044} |
9.61 Given a natural number $n \geqslant 2$. Find the smallest positive number $\lambda$, such that for any positive numbers $a_{1}$, $a_{2}, \cdots, a_{n}$, and any $n$ numbers $b_{1}, b_{2}, \cdots, b_{n}$ in $\left[0, \frac{1}{2}\right]$, if
$$
a_{1}+a_{2}+\cdots+a_{n}=b_{1}+b_{2}+\cdots+b_{n}=1 \text {, }
$$
then ... | \frac{1}{2}(\frac{1}{n-1})^{n-1} |
8.6. All natural numbers from 1 to 20 were divided into pairs, and the numbers in each pair were added. What is the maximum number of the ten resulting sums that can be divisible by 11? Justify your answer. | 9 |
6. Given $x, y \in \mathbf{R}$. If
$$
a(x+y) \leqslant 1+x^{2}+x y+y^{2}
$$
always holds, then the range of values for $a$ is $\qquad$ _. | [-\sqrt{3}, \sqrt{3}] |
(5) Given that the domain of the function $f(x)$ is $[-1,2]$, then the domain of the function $g(x)=f(\sqrt{x})$ is
$\qquad$ . | [0,4] |
3 male students and 3 female students, a total of 6 students, stand in a row. If among the 3 female students, exactly two are adjacent to each other, then the number of different arrangements is \_\_\_\_\_\_. | 432 |
Given that the graph of the power function $y=f(x)$ passes through the point $(\frac{1}{2}, \frac{\sqrt{2}}{2})$, determine the value of $f(4)$. | 2 |
A rectangular wooden block has a square top and bottom, its volume is $576$, and the surface area of its vertical sides is $384$. Find the sum of the lengths of all twelve of the edges of the block. | 112 |
Given point $P(2a-2,a+5)$, solve the following questions:
1. If point $P$ lies on the $x$-axis, find the coordinates of point $P$.
2. If the coordinates of point $Q$ are $(4,5)$ and the line $PQ$ is parallel to the $y$-axis, find the coordinates of point $P$.
3. If point $P$ lies in the second quadrant and the distance... | 2021 |
Think about question 2: In the complex plane, points $A$ and $B$ correspond to the complex numbers $z_{1}=2$, $z_{2}=-3$, respectively. Point $P$ corresponds to the complex number $z$, and the principal value of the argument of $\frac{z-z_{1}}{z-z_{2}}$ is $\varphi$. When point $P$ moves on the upper semicircle with th... | \pi-\arctan\frac{5\sqrt{6}}{12} |
5. The solution set of the inequality $\left|1+x+\frac{x^{2}}{2}\right|<1$ is
A. $\{x \mid-1<x<1\}$
B. $\left\{x \left\lvert\,-\frac{3}{2}<x<0\right.\right\}$
C. $\left\{x \left\lvert\,-\frac{5}{4}<x<0\right.\right\}$
D. $\{x \mid-2<x<0\}$ | D |
Given that the perimeter of right triangle $ABC$ is a constant $2$, find its maximum area. | 3 - 2\sqrt{2} |
$i$ is the imaginary unit. If the complex number $\left(1-2i\right)\left(a+i\right)$ is purely imaginary, then the value of the real number $a$ is:
A: $2$
B: $-2$
C: $\frac{1}{2}$
D: $-\frac{1}{2}$ | B |
A person wakes up from a nap and finds that his clock has stopped. He turns on the radio, wanting to listen to the time signal. The probability that he waits no more than $10$ minutes is $(\quad)$
A: $\dfrac{1}{2}$
B: $\dfrac{1}{4}$
C: $\dfrac{1}{6}$
D: $\dfrac{1}{8}$ | C |
## Problem Statement
Calculate the limit of the function:
$\lim _{x \rightarrow \frac{\pi}{2}}(\sin x)^{\frac{18 \sin x}{\operatorname{ctg} x}}$ | 1 |
Problem 4. Let $\left(a_{n}\right)_{n \in N}$ be a sequence of real numbers with the property that for any $m, n \in N, m \geq n$,
$2\left(a_{m+n}+a_{m-n}\right)=a_{2 m}+a_{2 n}$ and $a_{1}=1$. Determine a formula for the general term of the sequence $\left(a_{n}\right)_{n \in N}$. | a_{n}=n^{2} |
28th Putnam 1967 Problem A6 a i and b i are reals such that a 1 b 2 ≠ a 2 b 1 . What is the maximum number of possible 4-tuples (sign x 1 , sign x 2 , sign x 3 , sign x 4 ) for which all x i are non-zero and x i is a simultaneous solution of a 1 x 1 + a 2 x 2 + a 3 x 3 + a 4 x 4 = 0 and b 1 x 1 + b 2 x 2 + b 3 x 3 + b ... | 8 |
8. (10 points) There is a rectangle. If its length and width are both increased by 6 cm, then the area increases by 114 square cm. The perimeter of this rectangle is $\qquad$ cm. | 26 |
# Task 5. (12 points)
On a plane, there are four different circles. We will call an intersection point a point where at least two circles intersect. Find the maximum possible number of intersection points of four circles.
# | 12 |
Which of the following functions is both an odd function and an increasing function?
A: $y=3x$
B: $y=\frac{1}{x}$
C: $y=2x^{2}$
D: $y=-\frac{1}{3}x$ | A |
In a geometric sequence $\{a_{n}\}$ where all terms are positive numbers, if $a_{1}a_{3}=36$ and $a_{4}=54$, then the common ratio $q=\left(\ \ \right)$
A: $3$
B: $9$
C: $\frac{1}{3}$
D: $\frac{1}{9}$ | A |
Given two lines $y=ax-2$ and $y=(2-a)x+1$ are parallel, then $a$ equals ( )
A: -1
B: 2
C: 1
D: 0 | \text{C} |
2. On graph paper, a polygon with a perimeter of 36 is drawn, with its sides running along the grid lines. What is the maximum area it can have? | 81 |
5. The number of common terms (terms with the same value) in the arithmetic sequences $2,5,8, \cdots, 2015$ and $4,9,14, \cdots, 2014$ is $\qquad$. | 134 |
Let $m$, $n$ be two lines in space, and $\alpha$, $\beta$ be two planes in space. Then, among the following options, the incorrect one is ( ).
A: When $n \perp \alpha$, "$n \perp \beta$" is the necessary and sufficient condition for "$\alpha \parallel \beta$" to hold.
B: When $m \subset \alpha$, "$m \perp \beta$" is a... | \text{C} |
Given the functions $f\_1(x)$, $f\_2(x)$, and $h(x)$, if there exist real numbers $a$ and $b$ such that $h(x) = a \cdot f\_1(x) + b \cdot f\_2(x)$, then $h(x)$ is called the harmonious function of $f\_1(x)$ and $f\_2(x)$.
(1) Given the functions $f\_1(x) = x - 1$, $f\_2(x) = 3x + 1$, and $h(x) = 2x + 2$, determine whet... | -\frac{3}{2} \leq t \leq -\frac{4}{3} |
1. For what values of $x$ and $y$ is the equality $x^{2}+(1-y)^{2}+(x-y)^{2}=\frac{1}{3}$ true? | \frac{1}{3},\frac{2}{3} |
Let $f:\mathbb{R}\to\mathbb{R}$ be a twice differentiable function such that $$\frac{1}{2y}\int_{x-y}^{x+y}f(t)\, dt=f(x)\qquad\forall~x\in\mathbb{R}~\&~y>0$$ Show that there exist $a,b\in\mathbb{R}$ such that $f(x)=ax+b$ for all $x\in\mathbb{R}$. | f(x) = ax + b |
12.Cunning manipulations. With the number written on the board, it is allowed to perform the following operations: replace it with its double, or erase its last digit. How can you use these operations to get from the number 458 to the number 14? | 14 |
3. As shown in Figure 3, $\triangle A B C$ is inscribed in $\odot O, B C=a, C A=$ $b, \angle A-\angle B=90^{\circ}$. Then the area of $\odot O$ is | \frac{1}{4} \pi\left(a^{2}+b^{2}\right) |
Folklore
At the market, fish are sold, both large and small. Today, three large and one small fish together cost as much as five large fish did yesterday. And two large and one small fish today cost as much as three large and one small fish did yesterday. Can we determine from this information which is more expensive:... | b_{}+2m_{}=5m_{v} |
1. Given the system of inequalities about $x$
$$
\left\{\begin{array}{l}
3 x-3 \geqslant 6 x+a, \\
x \geqslant 1
\end{array}\right.
$$
the solution is $1 \leqslant x \leqslant 3$. Then $a=$ | -12 |
Given that the sum of the first $n$ terms of the sequence $\{a\_n\}$ is $S\_n$, and $S\_n = 2a_n - 2 (n \in \mathbb{N}^*)$, and the sum of the first $n$ terms of the sequence $\{b\_n\}$ is $T\_n$, and $T\_n = n^2 (n \in \mathbb{N}^*)$.
(I) Find the general term formulas for the sequences $\{a\_n\}$ and $\{b\_n\}$;
(II)... | (2n-3)2^{n+1} + 6 |
The number of correct calculations in the following is ( ).
(1) $a^{4}\div a^{4}=a\left(a\neq 0\right)$, (2) ${(\frac{1}{4}-0.25)^0}=1$; (3) ${(-0.1)^3}=-\frac{1}{{1000}}$; (4) $a^{2}+a^{2}=2a^{4}$; (5) $a^{2}\cdot a^{3}=a^{6}$; (6) $(-3x)^{3}\div 3x=9x^{2}$.
A: $1$ correct
B: $2$ correct
C: $3$ correct
D: $4$ cor... | A |
Problem 11.8. In a $28 \times 35$ table, some $k$ cells are painted red, another $r-$ in pink, and the remaining $s-$ in blue. It is known that
- $k \geqslant r \geqslant s$
- each boundary cell has at least 2 neighbors of the same color;
- each non-boundary cell has at least 3 neighbors of the same color.
What is th... | 28 |
We draw the diagonals of the convex quadrilateral $A B C D$, and then the centroids of the 4 triangles thus formed. What fraction of the area of quadrilateral $A B C D$ is the area of the quadrilateral determined by the 4 centroids? | \frac{2}{9} |
5. Determine how many different solutions the equation has
$$
|| \cdots|||x|-1|-2| \cdots-2016|-2017|=2017
$$
## Second Grade - A Category | 4 |
9. Adam's house number is in exactly one of the following ranges. Which one?
A 123 to 213
B 132 to 231
C 123 to 312
D 231 to 312
E 312 to 321 | E |
2. Given the sets
$$
A=\left\{x \mid x^{2}-3 x+2 \leqslant 0\right\}, B=\left\{x \left\lvert\, \frac{1}{x-3}<a\right.\right\} \text {. }
$$
If $A \subseteq B$, then the range of real number $a$ is
$\qquad$ | \left(-\frac{1}{2},+\infty\right) |
2. Given $x_{1}=1, x_{2}=2, x_{3}=3$ are the three zeros of $f(x)=x^{4}+a x^{3}+b x^{2}+c x+d$, then $f(0)+f(4)=$
$\qquad$ . | 24 |
If the equation of a circle is $x^2+y^2-2x+4y+1=0$, then the coordinates of its center are ______. | (h, k) = (1, -2) |
The coefficients of the quadratic term, linear term, and constant term of the quadratic equation $x^{2}-3x-2=0$ are, respectively, ( ).
A: $1$, $3$, $-2$
B: $1$, $3$, $2$
C: $1$, $-3$, $2$
D: $1$, $-3$, $-2$ | D |
If the parabola $y = x^2 - (k-1)x - k - 1$ intersects the x-axis at points A and B, and its vertex is C, then the minimum area of $\triangle ABC$ is. | 1 |
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