problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
|---|---|
6. (10 points) As shown in the figure, divide a line segment $AB$ of length 9 into 9 equal parts. Then the total length of all line segments in the figure is ( )
A. 132
B. 144
C. 156
D. 165 | 165 |
7.200. $\frac{1+2 \log _{9} 2}{\log _{9} x}-1=2 \log _{x} 3 \cdot \log _{9}(12-x)$. | 6 |
$36 \cdot 48$ In one night, the number of times the hour, minute, and second hands overlap is
(A) 2.
(B) 4.
(C) 6.
(D) 24.
(China Hubei Province Huanggang Junior High School Mathematics Competition, 1993) | 2 |
XLVII OM - I - Problem 2
A palindromic number is defined as a natural number whose decimal representation read from left to right is the same as when read from right to left. Let $ (x_n) $ be the increasing sequence of all palindromic numbers. Determine all prime numbers that are divisors of at least one of the differ... | 2,5,11 |
11. (25 points) Let positive numbers $a, b$ satisfy $a+b=1$. Find
$$
M=\sqrt{1+2 a^{2}}+2 \sqrt{\left(\frac{5}{12}\right)^{2}+b^{2}}
$$
the minimum value. | \frac{5\sqrt{34}}{12} |
2. (3 points) Solve the equation:
$$
\left[\frac{9 x-4}{6}\right]=\frac{12 x+7}{4}
$$ | -\frac{9}{4};-\frac{23}{12} |
Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfy $2\overrightarrow{a} + \overrightarrow{b} = 0$ and $\overrightarrow{a} \cdot \overrightarrow{b} = -2$, then $(3\overrightarrow{a} + \overrightarrow{b}) \cdot (\overrightarrow{a} - \overrightarrow{b}) =$ ?
A: $1$
B: $3$
C: $4$
D: $5$ | B |
Example 2 As shown in Figure 2, in $\triangle A B C$, it is known that $\angle A B C=50^{\circ}$. Rotate $\triangle A B C$ counterclockwise around point $B$ to $\triangle A^{\prime} B C^{\prime}$, such that $A A^{\prime} / / B C$. Find the size of $\angle C B C^{\prime}$ at this time. | 80^{\circ} |
3. Let the sequence $\left(x_{n}\right)_{n \geq 0}$ be given by $x_{0}=1, x_{1}=\frac{1}{2}$ and $x_{n+2}=x_{n+1} \cdot x_{n}^{2}, \forall n \geq 0$. Find the general term $x_{n}$ of the sequence and then calculate $\lim _{n \rightarrow \infty} x_{n}$. | x_{n}=2^{\frac{(-1)^{n}-2^{n}}{3}},\foralln\geq0\lim_{narrow\infty}x_{n}=0 |
Let $f_0(x) = \cos x$, $f_1(x) = f'_0(x)$, $f_2(x) = f'_1(x)$, $f_{n+1}(x) = f'_n(x)$ ($n \in \mathbb{N}$). Then $f_{2012}(x) =$ （ ）
A: $\sin x$
B: $-\sin x$
C: $\cos x$
D: $-\cos x$ | C |
6. Find the area of triangle $ABC$, the vertices of which have coordinates $A(0,0), B(1424233,2848467), C(1424234,2848469)$. Round your answer to the hundredths. | 0.50 |
## Problem Statement
Calculate the indefinite integral:
$$
\int \frac{x^{3}-3 x^{2}-12}{(x-4)(x-3)(x-2)} d x
$$ | x+2\cdot\ln|x-4|+12\cdot\ln|x-3|-8\cdot\ln|x-2|+C |
You have exactly $\$ 4.40$ ( 440 ) in quarters ( $25 \phi$ coins), dimes ( $10 \phi$ coins), and nickels ( 5 coins). You have the same number of each type of coin. How many dimes do you have?
(A) 20
(B) 11
(C) 10
(D) 12
(E) 4 | 11 |
The circumference of one base of a frustum is three times that of the other base, the slant height is 3, and the lateral area of the frustum is $84\pi$. The radius of the smaller base of the frustum is ( )
A: 7
B: 6
C: 5
D: 3 | A |
Given in the expansion of $\left( \sqrt{x}+ \frac{2}{x^{2}} \right)^{n}$, the ratio of the coefficient of the fifth term to the coefficient of the third term is $56:3$, find the constant term in the expansion. | 180 |
Let $\Gamma_1$ be a circle with radius $\frac{5}{2}$. $A$, $B$, and $C$ are points on $\Gamma_1$ such that $\overline{AB} = 3$ and $\overline{AC} = 5$. Let $\Gamma_2$ be a circle such that $\Gamma_2$ is tangent to $AB$ and $BC$ at $Q$ and $R$, and $\Gamma_2$ is also internally tangent to $\Gamma_1$ at $P$. $\Gamma_2$ i... | 19 |
Let $x, y$ be positive integers such that
$$
x^4=(x-1)\left(y^3-23\right)-1 .
$$
Find the maximum possible value of $x+y$. | 7 |
45. Given a triangle $A B C, \overline{A B}=20$. On the sides $A B$ and $A C$ of triangle $A B C$, points $X$ and $Y$ are chosen such that $\overline{A X}=\overline{A Y}=15$ and the area of triangle $A B C$ is twice the area of triangle $A X Y$. Find the ratio of the areas of triangles $B X Y$ and $C X Y$. | \frac{2}{3} |
Given the set $A=\{x|y=\ln x\}$, and the set $B=\{-2, -1, 1, 2\}$, then $A \cap B=$ ( )
A: $(1, 2)$
B: $\{1, 2\}$
C: $\{-1, -2\}$
D: $(0, +\infty)$ | \text{B} |
## Task 2
Subtract from 3182100 the product of the numbers 56823 and 56. By how much is the difference greater than 10? | 2 |
The root of the equation $49^{x} - 6 \cdot 7^{x} - 7 = 0$ is. | 1 |
10.120. Two circles of radius $R$ with centers at points $O_{1}$ and $O_{2}$ touch each other. They are intersected by a line at points $A, B, C$ and $D$ such that $A B=B C=C D$. Find the area of the quadrilateral $O_{1} A D O_{2}$. | \frac{5R^{2}\sqrt{3}}{4} |
Given $f(x)=\frac{x}{x-a}$.$(1)$ If $a \gt 0$ and $f\left(x\right)$ is monotonically decreasing in $\left(1,+\infty \right)$, find the range of values for $a$;$(2)$ The necessary and sufficient condition for the graph of the function $y=g\left(x\right)$ to be symmetric with respect to the point $P\left(m,n\right)$ is t... | (1,-1) |
3.22 *If the sequence $\left\{a_{n}\right\}$ satisfies $a_{1}=c, a_{n+1}=p a_{n}+q n(p \neq 0)$, find the general term $a_{n}$.
| a_{n}=p^{n-1}\frac{q}{1-p}(n-1)+\frac{q}{(1-p)^{2}}(p^{n}-p) |
Given two non-zero vectors $\overrightarrow{a}, \overrightarrow{b}$ with an angle of $60^{\circ}$ between them, and $|\overrightarrow{a} - \overrightarrow{b}| = 1$, determine the maximum value of $|\overrightarrow{a} + \overrightarrow{b}|$. | \sqrt{3} |
Exercise 4. We are given five numbers in ascending order, which are the lengths of the sides of a quadrilateral (not crossed, but not necessarily convex, meaning a diagonal is not necessarily inside the polygon) and one of its diagonals $D$. These numbers are 3, 5, 7, 13, and 19. What can be the length of the diagonal ... | 7 |
Given $12$ points in a plane no three of which are collinear, the number of lines they determine is:
$\textbf{(A)}\ 24 \qquad \textbf{(B)}\ 54 \qquad \textbf{(C)}\ 120 \qquad \textbf{(D)}\ 66 \qquad \textbf{(E)}\ \text{none of these}$ | 66 |
66. $P$ is a point inside triangle $ABC$, it is known that $\angle ABC=20^{\circ}, \angle ACB=30^{\circ}, \angle PBC=10^{\circ}$, $\angle PCB=20^{\circ}$, find the degree measure of $\angle PAB$. | 100 |
13.335. If a two-digit number is divided by the product of its digits, the quotient is 3 and the remainder is 8. If the number, formed by the same digits but in reverse order, is divided by the product of the digits, the quotient is 2 and the remainder is 5. Find this number. | 53 |
10. (14 points) Given the set $A$ consisting of the positive integers $1,2, \cdots, 1000$, first randomly select a number $a$ from set $A$, and after selecting, return $a$ to set $A$; then randomly select another number $b$ from set $A$. Find the probability that $\frac{a}{b}>\frac{1}{3}$. | \frac{1667}{2000} |
8.3. On the table, 28 coins of the same size but possibly different masses are arranged in a triangular shape (see figure). It is known that the total mass of any triplet of coins that touch each other pairwise is 10 g. Find the total mass of all 18 coins on the boundary of the triangle. | 60 |
Given an odd function $f(x)$ defined on $R$ such that $f(x) = f(2-x)$. When $1 \leq x \leq 2$, $f(x) = \log_{2}(x+7)$. Find $f(2021)$.
A: $3$
B: $-3$
C: $-5$
D: $5$ | A |
Given the function $y = -x^2 - 2(a-1)x + 5$ is decreasing on the interval $[-1, +\infty)$, then the range of the real number $a$ is ( ).
A: $a \geq 2$
B: $a \leq 2$
C: $a \geq -2$
D: $a \leq -2$ | A |
8.394. $\left\{\begin{array}{l}\sin x+\cos y=0, \\ \sin ^{2} x+\cos ^{2} y=\frac{1}{2} .\end{array}\right.$ | x_{1}=(-1)^{k+1}\frac{\pi}{6}+\pik,y_{1}=\\frac{\pi}{3}+2\pin,\quadx_{2}=(-1)^{k}\frac{\pi}{6}+\pik,\quady_{2}=\\frac{2}{3}\pi+2\pin,\text{where}k\text{} |
12. In the sequence $\left\{a_{n}\right\}$, if $a_{n}^{2}-a_{n-1}^{2}=p(n \geqslant 2, n \in \mathbf{N}, p$ is a constant $)$, then $\left\{a_{n}\right\}$ is called a "sequence of equal variances". The following are judgments on sequences of equal variances:
(1) The sequence $\left\{(-1)^{n}\right\}$ is a sequence of e... | (1)(2)(3)(4) |
Given vectors $\overrightarrow{a}=(-3,2)$ and $\overrightarrow{b}=(-1,0)$, and the vector $λ \overrightarrow{a}+ \overrightarrow{b}$ is perpendicular to $\overrightarrow{a}-2 \overrightarrow{b}$, find the value of the real number $λ$. | λ = -\frac{1}{7} |
6・159 Functions $f, g: R \rightarrow R$ are not constant and satisfy
$$\begin{array}{l}
f(x+y) \equiv f(x) g(y)+g(x) f(y), \\
g(x+y) \equiv g(x) g(y)-f(x) f(y), \\
x, y \in R \text {. }
\end{array}$$
Find all possible values of $f(0)$ and $g(0)$. | f(0)=0, g(0)=1 |
If $m$ and $n$ are opposite numbers, and $p$ and $q$ are reciprocals, then the value of $-2023m+\frac{3}{{pq}}-2023n$ is ______. | 3 |
7. The set
$\left\{x \left\lvert\,-1 \leqslant \log _{\frac{1}{x}} 10<-\frac{1}{2}\right., x \in \mathbf{N}\right\}$ has $\qquad$ proper subsets. | 2^{90}-1 |
The math scores of grade 12 in our city from the second quality inspection, X, approximately follow a normal distribution N(82, σ²). P(74 < X < 82) = 0.42. Given that 800 students from a certain school participated in this test, estimate the number of students with a math score no less than 90.
A: 64
B: 81
C: 100
D: 1... | A |
5. Nathaniel and Obediah play a game in which they take turns rolling a fair six-sided die and keep a running tally of the sum of the results of all rolls made. A player wins if, after he rolls, the number on the running tally is a multiple of 7 . Play continues until either player wins, or else indefinitely. If Nathan... | \frac{5}{11} |
1. (2 points) Solve the equation $5 p=q^{3}-r^{3}$, where $p, q, r$ are prime numbers. | q=7,r=2,p=67 |
39. $\tan ^{-1}\left(\frac{1}{2}\right)+\tan ^{-1}\left(\frac{1}{3}\right)$ The value is ( ).
(A) $\frac{\pi}{6}$
(B) $\frac{\pi}{4}$
(C) $\frac{\pi}{3}$
(D) $\frac{\pi}{2}$
$(\mathrm{E}) *$ | B |
12. Sarah and Hagar play a game of darts. Let $O_{0}$ be a circle of radius 1. On the $n$th turn, the player whose turn it is throws a dart and hits a point $p_{n}$ randomly selected from the points of $O_{n-1}$. The player then draws the largest circle that is centered at $p_{n}$ and contained in $O_{n-1}$, and calls ... | \frac{6\pi}{7} |
(1) In the expansion of ${(1+x)^{n}}$, if the coefficient of the third term is equal to that of the sixth term, what is the value of $n$?
(2) In the expansion of ${(x\sqrt{x}-\frac{2}{\sqrt{x}})}^{n}$, the sum of the binomial coefficients of the odd terms is 128. Find the maximum binomial coefficient term in the expans... | T_{5}=C_{8}^{4}{(x\sqrt{x})}^{4}{(\frac{-2}{\sqrt{x}})}^{4}=1120x^{4} |
Task B-1.6. Lovro writes down natural numbers one after another in a sequence:
$$
12345678910111213141516 \ldots
$$
and so on, without spaces or punctuation marks. He has written a total of 2013 digits. How many times did he write the digit 7? Justify your answer. | 149 |
13. As shown in Figure $3, \triangle A B C$ is divided into three equal areas (i.e., $S_{1}=S_{2}=S_{3}$) by $D E, F G$, and $D E / / F G$ $/ / B C, B C=\sqrt{6}$. Then $F G-D E=(\quad)$.
(A) $\sqrt{3}-1$
(B) $\sqrt{6}-\sqrt{3}$
(C) $\sqrt{6}-\sqrt{2}$
(D) $2-\sqrt{2}$ | D |
A high-speed train, weighing $P=150$ tons, travels at a maximum speed of $v=72 \frac{\mathrm{km}}{\text { hour }}$ on a horizontal track where $\rho=0,005$. At what speed can it travel on a track that has the same friction conditions but an elevation of $e=0,030$? [^0]
[^0]: ${ }^{1}$ In this problem, $\rho$ is th... | 10.3\frac{\mathrm{}}{\text{}} |
Given points $A(x_{1}, y_{1})$, $B(x_{2}, y_{2})$, $C(x_{3}, y_{3})$ are all on the graph of the inverse proportion function $y=\frac{k}{x}$ where $k>0$, with $x_{1} < 0 < x_{2} < x_{3}$, then the relationship between $y_{1}$, $y_{2}$, $y_{3}$ is ______. | y_{1} < y_{3} < y_{2} |
1. Consider all the subsets of $\{1,2,3, \ldots, 2018,2019\}$ having exactly 100 elements. For each subset, take the greatest element. Find the average of all these greatest elements. | 2000 |
Given the ellipse $E$: $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1(a > b > 0)$, its minor axis length is $2$, and the eccentricity is $\frac{\sqrt{6}}{3}$. The line $l$ passes through the point $(-1,0)$ and intersects the ellipse $E$ at points $A$ and $B$. $O$ is the coordinate origin.
(1) Find the equation of the ell... | \frac{\sqrt{6}}{3} |
Given $\frac{2\sin\alpha+3\cos\alpha}{\sin\alpha-2\cos\alpha}=\frac{1}{4}$.
$(1)$ Find the value of $\tan \alpha$;
$(2)$ Find the value of $\frac{{\sin2\alpha+1}}{{1+\sin2\alpha+\cos2\alpha}}$. | -\frac{1}{2} |
3. Calculate: $0.1+0.3+0.5+0.7+0.9+0.11+0.13+\cdots+0.97+0.99=$ | 27.25 |
Example 3 Given the real-coefficient equation $x^{3}+2(k-1) x^{2}+9 x+5(k-1)=0$ has a complex root with modulus $\sqrt{5}$, find the value of $k$, and solve this equation. | k=-1ork=3 |
We cut a circular piece of paper into $n$ identical sectors, and then shape each of the resulting sectors into a cone. For what value of $n$ will the total volume of the cones determined by the cone-shaped sectors be maximized? | 2 |
5. Let $a, b$ be real numbers, and the function $f(x)=a x+b$ satisfies: for any $x \in[0,1]$, we have $|f(x)| \leqslant 1$. Then the maximum value of $a b$ is $\qquad$ . | \frac{1}{4} |
286. Which of the following statements are consequences of the judgment: "Only some $A$ are $B$ "?
1) Not every $B$ is $A$. 2) Not every $A$ is $B$. 3) Some $B$ are not $A$. 4) Some $A$ are not $B$. 5) Some $B$ are $A$. 6) Some $A$ are $B$. 7) Some non-$B$ are $A$. 8) Some non-$A$ are $B$. 9) All $B$ are $A$. | 2)4)5)6)7) |
4. The sequence $\left\{x_{n}\right\}$ is defined as follows: $x_{1}=\frac{1}{2}, x_{k+1}=x_{k}^{2}+x_{k}, k=1,2, \cdots$. Find the integer part of the following sum:
$$
\frac{1}{1+x_{1}}+\frac{1}{1+x_{2}}+\cdots+\frac{1}{1+x_{2005}}
$$ | 1 |
6. What is the smallest area that a right triangle can have, with its hypotenuse containing the point \( M(1 ; 0) \), and its legs lying on the lines \( y = -2 \) and \( x = 0 \)?
(12 points) | 4 |
Example 4 Convert the pure periodic continued fraction
$$
2+\frac{1}{3}+\ldots=2+\frac{1}{3}+\frac{1}{2}+\frac{1}{3}+\cdots=[\overline{2,3}]
$$
into a quadratic irrational number. | x=\frac{3+\sqrt{15}}{3} |
## Task A-1.7.
How many five-digit natural numbers are there whose product of digits is equal to 900? | 210 |
72. There are 6 students, each shaking hands with every other student once, in total they shake hands $\qquad$ times. | 15 |
Six The terms of the sequence $x_{1}, x_{2}, x_{3}, \cdots$ are non-zero real numbers, and satisfy $x_{\mathrm{a}+2}=\frac{x_{\square} \cdot x_{\mathrm{n}+1}}{2 x_{\mathrm{a}}-x_{\mathrm{a}+1}} \quad(n=1,2, \cdots)$,
(1) Find the necessary and sufficient conditions for $x_{1}$ and $x_{2}$ such that all $x_{0}$ are inte... | 27 |
$20 \%$ of 20 is equal to
(A) 400
(B) 100
(C) 5
(D) 2
(E) 4 | 4 |
14. The range of the projection area of a regular tetrahedron with edge length 1 on a plane is what? | [\frac{\sqrt{2}}{4},\frac{1}{2}] |
Simplify first, then evaluate: $3{x^2}y-[{6x{y^2}-2({xy+\frac{3}{2}{x^2}y})}]+2({3x{y^2}-xy})$, where $x=\frac{1}{3}$ and $y=-6$. | -4 |
On an island live 34 chameleons, which can take on 3 colors: yellow, red, and green. At the beginning, 7 are yellow, 10 are red, and 17 are green. When two chameleons of different colors meet, they simultaneously take on the third color. It turns out that after a certain amount of time, all the chameleons on the island... | green |
Given space vectors $\overrightarrow{a}=(2,1,-3)$, $\overrightarrow{b}=(-1,2,3)$, $\overrightarrow{c}=(7,6,z)$, if the three vectors $\overrightarrow{a}$, $\overrightarrow{b}$, $\overrightarrow{c}$ lie in the same plane, then the real number $z=\_\_\_\_\_\_$. | -9 |
The standard equation of circle C, whose center is on the line $y= \frac {1}{3}x$ and is tangent to the positive half of the y-axis, and the chord intercepted on the x-axis is $4 \sqrt {2}$ in length, is ( )
A: $(x-3)^{2}+(y-1)^{2}=9$
B: $(x+3)^{2}+(y+1)^{2}=9$
C: $(x-4)^{2}+(y- \frac {4}{3})^{2}=16$
D: $(x-6)^{2}... | \text{A} |
2. Honza has three cards, each with a different non-zero digit. The sum of all three-digit numbers that can be formed from these cards is a number 6 greater than three times one of them. What digits are on the cards? | 8,2,1 |
I2.1 It is given that $m, n>0$ and $m+n=1$. If the minimum value of $\left(1+\frac{1}{m}\right)\left(1+\frac{1}{n}\right)$ is $a$, find the value of $a$.
I2.2 If the roots of the equation $x^{2}-(10+a) x+25=0$ are the square of the roots of the equation $x^{2}+b x=5$, find the positive value of $b$. (Reference: 2001 F... | 15 |
Calculate: $|-\sqrt{3}|+(\frac{1}{2})^{-1}+\left(\pi +1\right)^{0}-\tan 60^{\circ}$. | 3 |
The constant term in the expansion of the binomial $(4^x - 2^{-x})^6$ ($x \in \mathbb{R}$) is ______. | 15 |
Simplify:$(1)7x+3(x^2-2)-3(\frac{1}{2}x^2-x+3)$;$(2)3(2x^{2}y-xy^{2})-4(-xy^{2}+3x^{2}y)$. | -6x^{2}y + xy^{2} |
7. $\arcsin \left(\sin 2000^{\circ}\right)=$
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly.
7. $\arcsin \left(\sin 2000^{\circ}\right)=$ | -20 |
3. If the volume of a certain cylinder is numerically equal to its surface area, then the minimum possible value of the volume of the cylinder is ( ).
(A) $48 \pi$
(B) $50 \pi$
(C) $54 \pi$
(D) $66 \pi$ | C |
Given $$f(x)= \begin{cases} \overset{2^{x},x\leq 0 , }{log_{2}x,x>0}\end{cases}$$ find $f(f(-1))$. | -1 |
A polygon has each of its exterior angles measuring $36^{\circ}$. Determine the number of sides of this polygon. | 10 |
[Parallel lines, properties and signs. Transversals]
Points $M$ and $N$ are located on the lateral sides $A B$ and $C D$ of trapezoid $A B C D$, respectively, such that $M N \| A D$. It is known that the area of trapezoid $M B C N$ is to the area of trapezoid $A M N D$ as $2: 3$. Find $M N$, if $B C = a, A D = b$. | \sqrt{\frac{3a^2+2b^2}{5}} |
2. Find the number of roots of the equation: $2^{\lg \left(x^{2}-2023\right)}-\lg 2^{x^{2}-2022}=0$. | 4 |
Find all natural numbers a, b such that $ a^{n}\plus{} b^{n} \equal{} c^{n\plus{}1}$ where c and n are naturals. | a = b = c = 2 |
If $f(2x-1)=3x^{2}+1$, find the expression for $f(x)$. | f(x)= \frac {3}{4}x^{2}+ \frac {3}{2}x+ \frac {7}{4} |
8. (4 points) Pleasant Goat and Lazy Goat have a total of 70 stamps. Pleasant Goat has 5 more than 4 times the number of stamps Lazy Goat has. Pleasant Goat has $\qquad$ stamps, and Lazy Goat has $\qquad$ stamps. | 57;13 |
Let $\alpha$ and $\beta$ be two different planes, $l$ and $m$ be two different lines, and $l \subset \alpha$, $m \subset \beta$. There are two propositions as follows: ① If $\alpha \parallel \beta$, then $l \parallel m$; ② If $l \perp \beta$, then $\alpha \perp \beta$. Then, which of the following is correct?
A: Propo... | \text{B} |
6. For the geometric sequence $\left\{a_{n}\right\}$ with the first term $a_{1}=1536$, and common ratio $q=-\frac{1}{2}$, let $\pi_{n}$ denote the product of its first $n$ terms. The maximum value of $\pi_{n}$ is $\qquad$ . | \pi_{12} |
If the sum of the interior angles of a polygon is 1440°, then this polygon is a ____-sided figure. | 10 |
Find all pairs of integers $(x, y)$ such that $x^{2}-y!=2001$. | (45,4) |
Problem 2. In an aquarium, there are three types of fish: gold, silver, and red. If the cat eats all the gold fish, the number of fish will be 1 less than $2 / 3$ of the original number. If the cat eats all the red fish, the number of fish will be 4 more than $2 / 3$ of the original number. Which fish - gold or silver ... | 2 |
14. The set $P$ consisting of certain integers has the following properties:
(1) $P$ contains both positive and negative numbers;
(2) $P$ contains both odd and even numbers;
(3) $1 \notin P$;
(4) If $x, y \in P$, then $x+y \in P$.
Determine the relationship between 0, 2 and the set $P$. | 0\inP,2\notinP |
Given a sequence $\{a_n\}$ that satisfies the condition $$a_1 + a_2 + \ldots + a_n = n^3$$, find the value of $a_6 + a_7 + a_8 + a_9$. | 604 |
Given the function $$f(x)= \sqrt {3} \sin(2x) + \cos(2x)$$.
(I) The smallest positive period of the function $f(x)$ is ________;
(II) Find the maximum and minimum values of the function $f(x)$ on the interval $[0, \frac {\pi}{2}]$. | -1 |
A number $x$ is $2$ more than the product of its [reciprocal](https://artofproblemsolving.com/wiki/index.php/Reciprocal) and its additive [inverse](https://artofproblemsolving.com/wiki/index.php/Inverse). In which [interval](https://artofproblemsolving.com/wiki/index.php/Interval) does the number lie?
$\textbf{(A) }\ -... | 0<x\le2 |
In triangle $\triangle ABC$, the sides $a, b, c$ are opposite to the angles $A, B, C$, respectively, and it is given that $\sin A + \sqrt{3} \cos A = 2$.
(1) Find the measure of angle $A$.
(2) Given that $a=2$ and $c= \sqrt{3}b$, find the area of $\triangle ABC$. | \sqrt{3} |
A game developer has set up an "anti-addiction system" for a new game with the following rules:
1. Within 3 hours (including 3 hours) is considered healthy time. The relationship between the cumulative experience value $E$ (unit: $\exp$) obtained by the player during this time and the playing time $t$ (hours) satisfie... | [\dfrac{1}{4}, +\infty) |
Among the following functions, the graph of which is symmetric about the origin is ( )
A: $y=\lg x$
B: $y=\cos x$
C: $y=|x|$
D: $y=\sin x$ | \text{D} |
4. Define $9!=9 \times 8 \times \cdots \times 2 \times 1$, then 9 ! has $\qquad$ factors. | 160 |
If all the interior angles of a decagon are equal, then one of its interior angles is ______. | 144^{\circ} |
Given the circles $x^{2}+y^{2}=4$ and $x^{2}+y^{2}+2x-4y+4=0$, then ( )
A: The slope of the line passing through the centers of the two circles is $-\frac{1}{2}$
B: The equation of the line passing through the common chord of the two circles is $x-2y+4=0$
C: The two circles are symmetric with respect to the line $x-... | BD |
7.2. In 7a grade, there are 33 students. At the beginning of the school year, two clubs were organized in the class. According to school rules, a club can be organized if at least $70 \%$ of all students in the class sign up for it. What is the smallest number of students who could have signed up for both clubs simulta... | 15 |
3. Let $ABC$ be an equilateral triangle inscribed in a circle of radius 1, and let $M$ be any point on the circle. Prove that $M A^{2}+M B^{2}+M C^{2}=6$.
www.viitoriolimpici.ro | 6 |
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