problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
|---|---|
2. As shown in Figure 1, in a $3 \times 4$ table, one $1 \times 1$ small square in the top-left corner is colored red. Then, the number of rectangles in this table that include the red small square is ( ) .
(A) 11
(B) 12
(C) 13
(D) 14 | B |
Let $p_{1}, p_{2}, \ldots, p_{2005}$ be different prime numbers. Let $\mathrm{S}$ be a set of natural numbers which elements have the property that their simple divisors are some of the numbers $p_{1}, p_{2}, \ldots, p_{2005}$ and product of any two elements from $\mathrm{S}$ is not a perfect square.
What is the maxim... | 2^{2005} |
2. Calculate the sum $f\left(\frac{1}{2020}\right)+f\left(\frac{2}{2020}\right)+\ldots+f\left(\frac{2019}{2020}\right)$, where $f(x)=\frac{4^{x}}{4^{x}+2}, x \in \mathbb{R}$. | 1009.5 |
5. Given the sets $A=\{x \mid 5 x-a \leqslant 0\}, B=\{x \mid 6 x-b>0\}, a, b \in \mathbf{N}$, and $A \cap B \cap \mathbf{N}=\{2,3$, $4\}$, the number of integer pairs $(a, b)$ is $\qquad$. | 30 |
Given that $p$ is a true statement, and $q$ is a false statement, then ( )
A: The statement $p \land q$ is true.
B: The statement $p \lor q$ is false.
C: The statement $\lnot p$ is true.
D: The statement $\lnot q$ is true. | D |
Question 150: Four spheres with a radius of 1 are pairwise externally tangent. Then the volume of the circumscribed regular tetrahedron is $\qquad$ . | \frac{\sqrt{3}(6+\sqrt{6})^{3}}{27} |
B2. Mother decided to save money for her two sons for two weeks. In Andrej's piggy bank, she put 1 cent on the first day, and each subsequent day she put in twice as much as the previous day. In Mih's piggy bank, she put 75 euros at the end of each week.
A How much money did she save for Andrej after one week?
B How ... | 9.22 |
Given the sequence $\{a_n\}$, where $a_1=2$ and $a_n=2a_{n-1}-1$ ($n\geq2$), find the expression for $a_n$. | 2^{n-1}+1 |
3. Xiao Li and Xiao Zhang are running at a constant speed on a circular track. They start from the same place at the same time. Xiao Li runs clockwise and completes a lap every 72 seconds, while Xiao Zhang runs counterclockwise and completes a lap every 80 seconds. At the start, Xiao Li has a relay baton, and each time... | 720 |
9. Given real numbers $a, b, c, d$, and $a \neq b, c \neq d$. If the equations: $a^{2}+a c=2, b^{2}+b c=2, c^{2}+a c=$ $4, d^{2}+a d=4$ all hold, then the value of $6 a+2 b+3 c+2 d$ is $\qquad$. | 0 |
Given that positive real numbers $a$ and $b$ satisfy $a+b=\frac{5}{3}$, then the minimum value of $\frac{4}{{a+2b}}+\frac{9}{{2a+b}}$ is:
A: $6$
B: $5$
C: $12$
D: $10$ | \text{B} |
Russian problems Problem 99 The difference between the longest and shortest diagonals of the regular n-gon equals its side. Find all possible n. Solution | 9 |
Which of the following shapes is the graph of $y=3^x$ symmetric to the graph of $y=3^{-x}$ with respect to?
A: x-axis
B: y-axis
C: The line $y=$
D: Symmetric with respect to the origin | \text{B} |
Given the parabola $y^2=2px$ ($p>0$), and point $M(2, y_0)$ is on the parabola, $|MF|= \frac {5}{2}$.
(1) Find the equation of the parabola.
(2) Let the coordinates of point A be $( \frac {2}{3}, 0)$, find the coordinates of point B on the parabola that is closest to point A and the corresponding distance $|BA|$. | |AB|=\frac{2}{3} |
Problem 9.4. At a round table, 40 knights and 10 samurais are sitting. Exactly 7 knights have a samurai as their right neighbor. What is the maximum number of knights that could be sitting next to two knights? | 32 |
在平面直角坐标系 $xOy$ 中，焦点为 $F(5,0)$ 的抛物线的标准方程是 $\boxed{y^2=20x}$。 | y^2=20x |
Two equal circles in the same plane cannot have the following number of common tangents.
$\textbf{(A) \ }1 \qquad \textbf{(B) \ }2 \qquad \textbf{(C) \ }3 \qquad \textbf{(D) \ }4 \qquad \textbf{(E) \ }\text{none of these}$ | \textbf{(A)}\1 |
$3 \cdot 15$ Simplify: $\frac{\csc \alpha}{\sqrt{1+\operatorname{ctg}^{2} \alpha}}-\frac{\cos \alpha}{\sqrt{1-\sin ^{2} \alpha}}$.
(Shanghai Mathematical Competition, China, 1962) | {\begin{pmatrix}0,& |
8.176. $\sin ^{3} x(1+\operatorname{ctg} x)+\cos ^{3} x(1+\operatorname{tg} x)=2 \sqrt{\sin x \cos x}$. | \frac{\pi}{4}(8k+1),k\inZ |
At the school's art festival, there are four entries in the same category, labeled $A$, $B$, $C$, and $D$. Only one of them will win the first prize. Before the winners are announced, four students, named A, B, C, and D, made the following predictions about the four entries:A said: "$A$ will win the first prize";B said... | C |
11. (3 points) The Young Pioneers plan to make some lucky stars to give to the children in the kindergarten. If each person makes 10, they will still be 6 short of completing the plan; if 4 of them each make 8, and the rest each make 12, the plan will be exactly completed. How many lucky stars are planned to be made in... | 116 |
In the book "Nine Chapters on the Mathematical Art," there is a problem as follows: Sending a letter to a place $900$ miles away, if sent by a slow horse, it will arrive 1 day late; if sent by a fast horse, it will arrive 3 days early. It is known that the speed of the fast horse is twice that of the slow horse. Person... | A |
The sequence: $$- \frac {1}{1\times 2}, \frac {1}{2\times 3},- \frac {1}{3\times 4}, \frac {1}{4\times 5}$$, … has a general formula of \_\_\_\_\_\_\_\_\_\_\_\_. | \frac {(-1)^{n}}{n(n+1)} |
Currently, mobile WeChat payment has become a common payment method, greatly loved by consumers. It can be said that "with a mobile phone in hand, travel worry-free." If WeChat receives $22$ yuan, it is recorded as $+22$ yuan. Then, if WeChat spends $10.6$ yuan, it is recorded as: ______ yuan. | -10.6 |
Example 4 Proof: $\sum_{k=0}^{\left[\frac{n}{2}\right]}(-1)^{k} C_{n}^{k} C_{2 n-2 k-1}^{n-1}=1(n \geqslant 1)$. | 1 |
# Problem 5. (3 points)
Vasya has a stick that is 22 cm long. He wants to break it into three pieces with integer lengths and form a triangle from the resulting pieces. In how many ways can he do this? (Ways that result in congruent triangles are considered the same). | 10 |
13. (15 points) In $\triangle A B C$, the internal angles $\angle A$, $\angle B$, and $\angle C$ have opposite sides of lengths $a$, $b$, and $c$ respectively. Given that $c=2$ and $\angle C=\frac{\pi}{3}$. Find
(1) the maximum perimeter of $\triangle A B C$;
(2) the area of $\triangle A B C$ when $2 \sin 2 A+\sin (2 B... | \frac{2\sqrt{3}}{3} |
Let $u$ be a real number. On the coordinate plane, consider two parabolas $C_1: y=-x^2+1,\ C_2: y=(x-u)^2+u$.
The range of $u$ such that $C_1$ and $C_2$ have at least one point of intersection is expressed by $a\leq u\leq b$ for some real numbers $a,\ b$.
(1) Find the values of $a,\ b$.
(2) When $u$ satisfies $a\leq ... | I = \frac{21\pi}{8} |
Points $A, B, C$, and $D$ lie along a line, in that order. If $A B: A C=1: 5$, and $B C: C D=2: 1$, then the ratio $A B: C D$ is equal to
(A) $1: 1$
(B) $1: 2$
(C) $1: 3$
(D) $2: 5$
(E) $3: 5$
Part B: Each correct answer is worth 6. | 1:2 |
15.9. The length of one side of the square $A B C D$ is 4 units. A circle is drawn tangent to $\overline{B C}$ and passing through the vertices $A$ and $D$. Find the area of the circle. | \frac{25\pi}{4} |
For an arithmetic sequence $\{a_n\}$, the sum of the first $n$ terms is $S_n$. If $$\frac {S_{n}}{a_{n}} = \frac {n+1}{2}$$, then among the following conclusions, the correct one is ( )
A: $$\frac {a_{2}}{a_{3}} = 2$$
B: $$\frac {a_{2}}{a_{3}} = \frac {3}{2}$$
C: $$\frac {a_{2}}{a_{3}} = \frac {2}{3}$$
D: $$\frac ... | \text{C} |
Shift the graph of the function $f(x) = 2\sin(\omega x - \frac{\pi}{3})$, ($\omega > 0$) to the left by $\frac{\pi}{3\omega}$ units to obtain the graph of the function $y = g(x)$. If $y = g(x)$ is increasing on the interval $\left[-\frac{\pi}{6}, \frac{\pi}{4}\right]$, then the maximum value of $\omega$ is \_\_\_\_\_\_... | 2 |
Example 17. Solve the equation
$$
\frac{2}{2-\sqrt{x}}+\frac{1}{2}=\frac{4}{2 \sqrt{x}-x}
$$ | 16 |
## Problem Statement
Find the second-order derivative $y_{x x}^{\prime \prime}$ of the function given parametrically.
$$
\left\{\begin{array}{l}
x=\operatorname{ch} t \\
y=\sqrt[3]{\operatorname{sh}^{2} t}
\end{array}\right.
$$ | -\frac{2(3+\operatorname{ch}^{2})}{9\operatorname{sh}^{4}} |
18.104 As shown in the figure, in the right trapezoid, the base $AB=$
13, $CD=8, AD \perp AB$, and $AD=12$, then the distance from $A$ to $BC$ is
(A) 12.
(B) 13.
(C) 10.
(D) $\frac{12 \cdot 21}{13}$.
(E) $\frac{13+8}{2}$.
(China Junior High School Mathematics Correspondence Competition, 1985) | 12 |
9.3. Given a trapezoid $A B C D$, where $A D \| B C, B C=A C=5$, and $A D=6$. The angle $A C B$ is twice the angle $A D B$. Find the area of the trapezoid. | 22 |
3.465 Given that $\operatorname{tg} \alpha=m$, find
$A=\sin ^{2}\left(\frac{\pi}{4}+\alpha\right)-\sin ^{2}\left(\frac{\pi}{6}-\alpha\right)-\cos \frac{5 \pi}{12} \cdot \sin \left(\frac{5 \pi}{12}-2 \alpha\right)$ | \frac{2}{1+^{2}} |
# 7.4. Condition:
On the Christmas tree, there is a garland of 100 bulbs. It is known that the third and fourth bulbs are yellow. Additionally, among any five consecutive bulbs, exactly two are yellow and exactly three are blue. Father Frost can see only part of the garland from behind the tree. Help him determine the... | 97;100-Bluecolor,98;99-Yellowcolor |
3. At the hitmen convention, 1000 participants gathered, each receiving a registration number from 1 to 1000. By the end of the convention, it turned out that all hitmen, except number 1, were killed. It is known that each hitman could only kill hitmen with higher numbers, and the number of his victims could not exceed... | 10 |
## Problem Statement
Calculate the definite integral:
$$
\int_{0}^{4 \sqrt{3}} \frac{d x}{\sqrt{\left(64-x^{2}\right)^{3}}}
$$ | \frac{\sqrt{3}}{64} |
Given the function $f(x)=\ln x+\frac{1}{2}(a-x)^{2}$, where $a\in \mathbb{R}$.
$(1)$ When $a=1$, find the equation of the tangent line to the function $f(x)$ at $(1,f(1))$.
$(2)$ Discuss the monotonicity of the function $f(x)$.
$(3)$ If $f(x)$ has two extreme points $x_{1}$ and $x_{2}$ ($x_{1} \lt x_{2}$) and the... | (\frac{3\sqrt{2}}{2}, \frac{5}{2}) |
2. $f(n)$ is defined on the set of positive integers, and: (1) for any positive integer $n, f[f(n)]=4 n+9$; (2) for any non-negative integer $k, f\left(2^{k}\right)=2^{k+1}+3$. Determine $f(1789)$.
(1989 Australian Olympiad Problem) | 3581 |
Four, (50 points) A password lock's password setting involves assigning one of the two numbers, 0 or 1, to each vertex of a regular $n$-sided polygon $A_{1} A_{2} \cdots A_{n}$, and coloring each vertex one of two colors, red or blue, such that for any two adjacent vertices, at least one of the number or color is the s... | 3^n + 1 \text{ if } n \text{ is odd, } 3^n + 3 \text{ if } n \text{ is even} |
In the expansion of $(x^2+3x+2)^5$, the coefficient of $x$ is ( )
A: 800
B: 360
C: 240
D: 160 | C |
In the national fitness movement, cycling is quite popular among the people. Two cycling enthusiasts, A and B, agreed to ride along the same route from point $A$ to point $B$, which is 30 kilometers away. It is known that the average speed of A is 1.2 times the average speed of B. If B rides for 20 minutes first, and t... | 0.3 |
Example 2. In the set of non-negative integers $\overline{\mathbf{Z}}-1$: Define the function $\mathrm{f}(\mathrm{n})$:
$$
\left\{\begin{array}{l}
f(0)=2, f(1)=3, \\
f(k+1)=3 f(k)-2 f(k-1) . \text { where, } k \geqslant 1 .
\end{array}\right.
$$
Try to find the formula for $f(n)$ expressed in terms of $n$. | f(n)=2^{n}+1 |
Solve the application problem by setting up equations: It is known that a Type A tractor takes the same amount of time to plow 18 acres as a Type B tractor takes to plow 24 acres. If Type A and Type B tractors work together, they can plow 7 acres per hour. Find the work efficiency of Type A and Type B tractors separate... | 4 |
9- 118 (1) Factorize $x^{12}+x^{9}+x^{6}+x^{3}+1$.
(2) For any real number $\theta$, prove: $5+8 \cos \theta+4 \cos 2 \theta+\cos 3 \theta \geqslant 0$. | (x^{4}+x^{3}+x^{2}+1)(x^{8}-x^{7}+x^{5}-x^{4}+x^{3}-x+1) |
Given that the radius of the circumcircle of triangle $\triangle ABC$ is 1, with the center O, and $3\vec{OA} + 4\vec{OB} + 5\vec{OC} = \vec{0}$, find the measure of $\angle AOB$. | \angle AOB = \frac{\pi}{2} |
8. Determine the value of the sum
$$
\frac{3}{1^{2} \cdot 2^{2}}+\frac{5}{2^{2} \cdot 3^{2}}+\frac{7}{3^{2} \cdot 4^{2}}+\cdots+\frac{29}{14^{2} \cdot 15^{2}} .
$$ | \frac{224}{225} |
Consider the set
$$
\mathcal{S}=\{(a, b, c, d, e): 0<a<b<c<d<e<100\}
$$
where $a, b, c, d, e$ are integers. If $D$ is the average value of the fourth element of such a tuple in the set, taken over all the elements of $\mathcal{S}$, find the largest integer less than or equal to $D$. | 66 |
Given the inequality about $x$: $|ax-2|+|ax-a| \geq 2$ ($a>0$)
(1) When $a=1$, find the solution set of this inequality;
(2) If the solution set of this inequality is $\mathbb{R}$, find the range of the real number $a$. | a \geq 4 |
(5 points) Among the following propositions, the incorrect one is ( )
A. If plane $\alpha \perp$ plane $\beta$, then there definitely exists a line in plane $\alpha$ that is parallel to plane $\beta$
B. If plane $\alpha$ is not perpendicular to plane $\beta$, then there definitely does not exist a line in plan... | \text{D} |
3. Solve the equation
$2 x+1+\operatorname{arctg} x \cdot \sqrt{x^{2}+1}+\operatorname{arctg}(x+1) \cdot \sqrt{x^{2}+2 x+2}=0$. | -\frac{1}{2} |
13.375 Find a three-digit number, knowing that the number of its tens is the geometric mean of the number of hundreds and units. If in its notation the digits of hundreds and units are swapped and the new number is subtracted from the desired one, then the difference will be equal to 297. | 300or421 |
Let $AEF$ be a triangle with $EF = 20$ and $AE = AF = 21$. Let $B$ and $D$ be points chosen on segments $AE$ and $AF,$ respectively, such that $BD$ is parallel to $EF.$ Point $C$ is chosen in the interior of triangle $AEF$ such that $ABCD$ is cyclic. If $BC = 3$ and $CD = 4,$ then the ratio of areas $\tfrac{[ABCD]}{[AE... | 5300 |
7. (10 points) There are 37 people standing in a line to count off. The first person reports 1, and each subsequent person reports a number that is 3 more than the previous person's number. During the counting, one person made a mistake and reported a number that was 3 less than the previous person's number. The sum of... | 34 |
The limit of the sum of an infinite number of terms in a geometric progression is $\frac {a}{1 - r}$ where $a$ denotes the first term and $- 1 < r < 1$ denotes the common ratio. The limit of the sum of their squares is:
$\textbf{(A)}\ \frac {a^2}{(1 - r)^2} \qquad\textbf{(B)}\ \frac {a^2}{1 + r^2} \qquad\textbf{(C)}\ \... | \frac{^2}{1-r^2} |
Given points $A(3,-1)$ and $B(-1,2)$ are on the same side of the line $ax+2y-1=0$, determine the range of values for the real number $a$. | a \in (1,3) |
If the three line segments with lengths $x^2+4$, $4x$, and $x^2+6$ can form an acute-angled triangle, then the range of values for $x$ is ______. | x > \frac{\sqrt{15}}{3} |
6. Antun has $80 \%$ more stickers than Branko. Branko has $\frac{3}{5}$ of the number of stickers that Darko has. If Branko gave 150 stickers to Darko, then Darko would have 3 times more stickers than Branko. How many stickers do all three of them have together? | 2010 |
Given that the angle between vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ is $120^{\circ}$, $|\overrightarrow{a}|=3$, and $|\overrightarrow{b}|=2$.
$(1)$ Find the value of $({2\overrightarrow{a}+\overrightarrow{b}})\cdot({\overrightarrow{a}-2\overrightarrow{b}})$;
$(2)$ Find the value of $|{2\overrightarro... | 2\sqrt{7} |
Let $S_n$ be the sum of the first $n$ terms of the sequence $\{a_n\}$, and $a_1=-1$, $a_{n+1}=S_nS_{n+1}$. Find $S_n$. | -\frac{1}{n} |
You are writing a multiple choice test and on one question you guess and pick an answer at random. If there are five possible choices $(A, B, C, D, E)$, what is the probability that you guessed correctly?
(A) $\frac{1}{5}$
(B) $\frac{5}{5}$
(C) $\frac{4}{5}$
(D) $\frac{2}{5}$
(E) $\frac{3}{5}$ | \frac{1}{5} |
One, Select (6 points per question, total 36 points)
1. The number of solutions to the equation $\left(\frac{3}{19}\right)^{x}+\left(\frac{5}{19}\right)^{x}+\left(\frac{11}{19}\right)^{x}=$ $2 \sqrt{x-1}$ is (
(A) 0
(B) 1
(C) 2
(D) Infinite | B |
2. In $\square A B C D$, $\angle B<90^{\circ}, A B<B C$. From point $D$ draw tangents to the circumcircle $\Gamma$ of $\triangle A B C$, the points of tangency are $E$ and $F$. It is known that $\angle E D A=\angle F D C$. Find $\angle A B C$ | 60^{\circ} |
Example 4. Determine the non-negative roots of the equation $\sin x=\frac{1}{100} x$ | 32 |
4. The twelve zodiac signs are divided into four categories: Fire, Earth, Air, and Water, with 3 zodiac signs in each category.
The magic school has 20 students, and their zodiac signs are as follows:
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|}
\hline Category & \multicolumn{3}{|c|}{ Fire } & \multicolumn{3}{|c|}{ Ear... | 968 |
In a Cartesian coordinate plane, the number of lines that are at a distance of 2 from point A (3, 0) and at a distance of 3 from point B (0, 4) is __________. | 3 |
1. Find all integers $n$ for which $n^{4}-3 n^{2}+9$ is a prime number. | n\in{-2,-1,1,2} |
Example 41 (12th CMO Question) Let real numbers $x_{1}, x_{2}, \cdots, x_{1997}$ satisfy the following conditions:
(1) $-\frac{1}{\sqrt{3}} \leqslant x_{i} \leqslant \sqrt{3}(i=1,2, \cdots, 1997)$;
(2) $x_{1}+x_{2}+\cdots+x_{1997}=-318 \sqrt{3}$, find the maximum value of $x_{1}^{12}+x_{2}^{12}+\cdots+x_{1997}^{12}$. | 189548 |
Triangle $XYZ$ is inside square $KLMN$ shown below so that its vertices each lie on three different sides of the square. It is known that:
$\bullet$ The area of square $KLMN$ is $1$.
$\bullet$ The vertices of the triangle divide three sides of the square up into these ratios:
$KX : XL = 3 : 2$
$KY : YN = 4 : 1$
$NZ : Z... | \frac{11}{50} |
1. Given 10 pairwise distinct positive integers $a_{1}$, $a_{2}, \cdots, a_{10}$ that satisfy the conditions
$$
\begin{array}{l}
a_{2}=a_{1}+a_{5}, a_{3}=a_{2}+a_{6}, \\
a_{4}=a_{3}+a_{7}, a_{6}=a_{5}+a_{8}, \\
a_{7}=a_{6}+a_{9}, a_{9}=a_{8}+a_{10} .
\end{array}
$$
then the minimum possible value of $a_{4}$ is | 20 |
23. The solution set of the inequality $\left|\frac{x}{x-2}\right|>\frac{x}{2-x}$ is | (-\infty, 0) \cup (2,+\infty) |
Given points $A(2,0)$, $B(3,2)$, and vector $\overrightarrow{a}=(2,\lambda)$, if $\overrightarrow{a} \perp \overrightarrow{AB}$, then the magnitude of $\overrightarrow{a}$ is ( ).
A: $\sqrt{5}$
B: $\sqrt{3}$
C: $2\sqrt{6}$
D: $4$ | A |
Let $\{x_1, x_2, x_3, ..., x_n\}$ be a set of $n$ distinct positive integers, such that the sum of any $3$ of them is a prime number. What is the maximum value of $n$? | 4 |
Given the function $f(x)=a(x+\ln x)$, where $a \neq 0$, and $g(x)=x^2$.
(1) If the tangent line of the graph of $f(x)$ at $x=1$ is also the tangent line of the graph of $g(x)$, find the value of the real number $a$;
(2) When $0 < a < 1$, prove that for any two distinct real numbers $x_1$, $x_2$ in the interval $[1,2... | a=1 |
Find the smallest positive value taken by $a^3 + b^3 + c^3 - 3abc$ for positive integers $a$, $b$, $c$ .
Find all $a$, $b$, $c$ which give the smallest value | 4 |
1.10. (CSSR, 79). Find all natural numbers $n>2$, not exceeding the number 10000000 and possessing the following property: any number $m$, coprime with $n$ and satisfying the inequalities $1<m<n$, is prime. | {3;4;6;8;12;18;24;30} |
8.1. Let $n=34000$. Among the vertices of a regular $n$-gon $A_{1} A_{2} \ldots A_{n}$, the vertices $A_{i}$ are painted red for which the number $i$ is a power of two, i.e., $i=1,2,4,8,16, \ldots$ In how many ways can 400 vertices of this $n$-gon be chosen so that they are the vertices of a regular 400-gon and none of... | 77 |
4. (6 points) A number when divided by 3 leaves a remainder of 2, when divided by 4 leaves a remainder of 3, and when divided by 5 leaves a remainder of 4. This number is $\qquad$ | 59 |
Given set A={1, 3, 5} and set B={2, 3}, find the number of elements in set A∪B. | 4 |
75. Among the cards that differ only in color, there are 5 red, 3 blue, 2 green, and one yellow. In how many ways can they be laid out in a row to form a colored strip? | 27720 |
10. (20 points) Given $f(x)=\frac{2 x}{x+1}$. The sequence $\left\{a_{n}\right\}$ satisfies $a_{1}=2$, and
$$
a_{n}=\frac{1}{2} f\left(a_{n-1}\right)\left(n \in \mathbf{N}_{+}, n \geqslant 2\right) \text {, }
$$
(1) Find the general term formula for the sequence $\left\{a_{n}\right\}$;
(2) Prove: For all positive integ... | a_{n}=\frac{2}{2 n-1}\left(n \in \mathbf{N}_{+}\right) |
For the polynomial $f$, it holds that $f\left(x^{2}+1\right)-f\left(x^{2}-1\right)=4 x^{2}+6$. Determine the polynomial $f\left(x^{2}+1\right)-f\left(x^{2}\right)$. | 2x^{2}+4 |
## Problem 1.
Let $ABC$ be a triangle and points $M, N, P$ such that $\overrightarrow{BM}=\overrightarrow{MC} ; \overrightarrow{AN}=2 \overrightarrow{NC} ; \overrightarrow{AP}=3 \overrightarrow{PB}$ and $Q$ is the midpoint of segment $(PM)$.
a) Show that $\overrightarrow{BN}=\frac{2}{3} \overrightarrow{BC}+\frac{1}{3... | \frac{BQ}{QN}=\frac{3}{5} |
9. (5 points) Among the students participating in the gymnastics, martial arts, piano, and calligraphy interest groups, each student can participate in at most two interest groups. To ensure that there are no fewer than 6 students with exactly the same selection of interest groups, the number of students participating ... | 51 |
5. For which integers $a$ and $b$ does the system of equations
$$
\left\{\begin{array}{c}
|a x-2 y+b|=3 \\
(3 x-4 y-1)(4 x-y-10)(x+5 y+6)=0
\end{array}\right.
$$
have an infinite number of solutions? | =8,b=-17;=8,b=-23 |
Let $a, b, c$ be the three sides of $\triangle ABC$, with $a \leqslant b \leqslant c$, and $R$ and $r$ be the circumradius and inradius of $\triangle ABC$, respectively. Let $f=a+b-2R-2r$. Try to determine the sign of $f$ using the size of angle $C$.
(Xu Yichao, problem contributor) | f>0 \Leftrightarrow C<\frac{\pi}{2} |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$, respectively. Given vectors $\overrightarrow{m}=(\sin B+\sin C, \sin A-\sin B)$ and $\overrightarrow{n}=(\sin B-\sin C, \sin(B+C))$, and $\overrightarrow{m} \perp \overrightarrow{n}$.
(1) Find the magnitude of angle $C$... | \frac{4\sqrt{3}-3}{10} |
The complex number $\dfrac{2}{i(3-i)}=$
A: $\dfrac{1-3i}{5}$
B: $\dfrac{1+3i}{5}$
C: $\dfrac{3+i}{5}$
D: $\dfrac{3-i}{5}$ | \text{A} |
8. Reviews (from 7th grade. 1 point). Angry reviews about the work of an online store are left by $80 \%$ of dissatisfied customers (those who were poorly served in the store). Only $15 \%$ of satisfied customers leave positive reviews. A certain online store has received 60 angry and 20 positive reviews. Using this st... | 0.64 |
Is an 8 kg block of wood heavier than an 8000 g brick? (Judge true or false) | \text{False} |
C1. Consider a regular $2 n+1$-gon $P$ in the plane, where $n$ is a positive integer. We say that a point $S$ on one of the sides of $P$ can be seen from a point $E$ that is external to $P$, if the line segment $S E$ contains no other points that lie on the sides of $P$ except $S$. We want to color the sides of $P$ in... | 1 |
11.4. 2011 Warehouses are connected by roads in such a way that from any warehouse you can drive to any other, possibly by driving along several roads. On the warehouses, there are $x_{1}, \ldots, x_{2011}$ kg of cement respectively. In one trip, you can transport an arbitrary amount of cement from any warehouse to ano... | 2010 |
Given lines $a$ and $b$, and planes $\alpha$ and $\beta$, with $a \perp \alpha$ and $b \subset \beta$, then "$a \perp b$" is a condition for "$\alpha \parallel \beta$" to be ( )
A: A sufficient but not necessary condition
B: A necessary but not sufficient condition
C: A necessary and sufficient condition
D: Ne... | \text{B} |
A certain company has two research and development teams, A and B, whose probabilities of successfully developing a new product are $\frac {2}{3}$ and $\frac {3}{5}$, respectively. Team A is tasked with developing new product A, and team B with developing new product B. Assuming the research and development efforts of ... | \frac {13}{15} |
Given the function $f(x) = mx^{2} - (m+1)x + 1$.
$(1)$ If $m > 0$, find the solution set of the inequality $f(x) < 0$.
$(2)$ If $f(x) \leq 2$ holds for all $x \in [1,2]$, find the range of real numbers for $m$. | (-\infty, \frac{3}{2}] |
For the power function $f(x) = (m^2 - m - 1)x^{m^2 + m - 3}$, if it is an increasing function on the interval $(0, +\infty)$, then $m = \_\_\_\_\_\_$. | 2 |
10. [5] Let
$$
\begin{array}{l}
A=(1+2 \sqrt{2}+3 \sqrt{3}+6 \sqrt{6})(2+6 \sqrt{2}+\sqrt{3}+3 \sqrt{6})(3+\sqrt{2}+6 \sqrt{3}+2 \sqrt{6})(6+3 \sqrt{2}+2 \sqrt{3}+\sqrt{6}), \\
B=(1+3 \sqrt{2}+2 \sqrt{3}+6 \sqrt{6})(2+\sqrt{2}+6 \sqrt{3}+3 \sqrt{6})(3+6 \sqrt{2}+\sqrt{3}+2 \sqrt{6})(6+2 \sqrt{2}+3 \sqrt{3}+\sqrt{6})
\e... | 1 |
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