problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
|---|---|
Problem 5.1. Dasha calls a natural number special if four different digits are used to write it. For example, the number 3429 is special, while the number 3430 is not special.
What is the smallest special number greater than 3429? | 3450 |
Given $f(x)$ and $g(x)$ are respectively even and odd functions defined on $\mathbb{R}$, and $f(x) - g(x) = x^3 + x^2 + 1$, then $f(1) + g(1) = \ $
A: $-3$
B: $-1$
C: $1$
D: $3$ | C |
If $4^{a}=6^{b}=24$, then the value of $\frac{1}{a}+\frac{1}{b}$ is:
A: $0$
B: $1$
C: $2$
D: $3$ | B |
Given the function $f(x) = 2^{x-1}$, its inverse function is denoted as $y = f^{-1}(x)$. If $f^{-1}(a) + f^{-1}(b) = 4$, then $ab =$ ? | 4 |
Given the sets $A=\{x\in\mathbb{R}|2x^{2}-x-1 < 0\}$ and $B=\{x\in\mathbb{R}|\log_{\frac{1}{2}}x < 3\}$, find $A\cap B$. | (\frac{1}{8},1) |
1. If $M=\left\{z \left\lvert\, z=\frac{t}{1+t}+i \frac{1+t}{t}\right., t \in \mathrm{R}, t \neq-1, t \neq 0\right\}, N=\{z \mid z$ $=\sqrt{2}[\cos (\arcsin t)+i \cos (\arccos t)], t \in \mathrm{R},|t| \leqslant 1\}$, then the number of elements in $M \cap N$ is
(A) 0
(B) 1
(C) 2
(D) 4 | 0 |
66. Given a triangle $A B C$, where $|A B|=|A C|$, $\widehat{B A C}=80^{\circ}$. Inside the triangle, a point $M$ is taken such that $\widehat{M} \overrightarrow{B C}=30^{\circ}, \widehat{M C B}=10^{\circ}$. Find $\widehat{A M C}$. | 70 |
For a sale, a store owner reduces the price of a $$10$ scarf by $20\%$. Later the price is lowered again, this time by one-half the reduced price. The price is now
$\text{(A)}\ 2.00\text{ dollars} \qquad \text{(B)}\ 3.75\text{ dollars} \qquad \text{(C)}\ 4.00\text{ dollars} \qquad \text{(D)}\ 4.90\text{ dollars} \qqu... | 4 |
10. Person A and Person B start from points $A$ and $B$ respectively (Person A starts from $A$), walking towards each other and continuously moving back and forth between the two points. Person A's speed is 4 times that of Person B. It is known that the distance between $A$ and $B$ is $S$ kilometers, where $S$ is a pos... | 105 |
# Variant 4.
1) Let's draw an arbitrary line $y=k-b$, intersecting the parabola at points $A$ and $B$. Then the abscissas of the endpoints of the chord $A$ and $B$ must satisfy the conditions: $k x - b = x^2$, i.e., $x^2 - k x + b = 0$, $x_A = \frac{k - \sqrt{k^2 - 4b}}{2}$, $x_B = \frac{k + \sqrt{k^2 - 4b}}{2}$. Then... | f(x)=\frac{5}{8x^{2}}-\frac{x^{3}}{8} |
7. Given a sequence of positive numbers $a_{1}, a_{2}, \ldots, a_{10}$, satisfying the relation $a_{n}\left(a_{n-1}+a_{n+1}\right)=2 a_{n-1} a_{n+1}\left(a_{n}+1\right)$ for $n=2,3, \ldots, 9$. Find $a_{5}$ if it is known that $a_{1}=1$ and $a_{10}=0.01$ | 0.04 |
## Task 2 - 190812
From the digits $0,1, \ldots, 9$, exactly seven are to be selected, none of which are the same.
Determine the number of (in the decimal system) seven-digit numbers that contain each of the selected digits in their (decimal) digit representation!
Assume that
a) the 0 does not occur among the selec... | 5040 |
Given the function $f(x) = x + b\cos x$, where $b$ is a constant. Then "b=0" is the condition for "$f(x)$ is an odd function" is ( )
A: Sufficient but not necessary condition
B: Necessary but not sufficient condition
C: Sufficient and necessary condition
D: Neither sufficient nor necessary condition | \text{C} |
2. Given the line $l: 2 x+y=10$, draw a line $l^{\prime} \perp l$ through the point $(-10,0)$, then the coordinates of the intersection point of $l^{\prime}$ and $l$ are $\qquad$ | (2,6) |
5. If $a b c=4, a+b+c=2$, and $a \geqslant b \geqslant c$, then the minimum value of $|a|+|b|+|c|$ is ( ).
(A) 2
(B) 4
(C) 6
(D) 8 | C |
Given the sets $M=\{0,x\}$ and $N=\{1,2\}$, if $M \cap N = \{2\}$, then $M \cup N =$ ( )
A: $\{0,x,1,2\}$
B: $\{0,1,2\}$
C: $\{2,0,1,2\}$
D: Cannot be determined | B |
A rectangular grazing area is to be fenced off on three sides using part of a $100$ meter rock wall as the fourth side. Fence posts are to be placed every $12$ meters along the fence including the two posts where the fence meets the rock wall. What is the fewest number of posts required to fence an area $36$ m by $60... | 12 |
61. In a car factory, 8 workers produce 48 car parts every day. At this rate, if 300 parts are to be produced in 5 days, $\qquad$ workers are needed. | 10 |
Given a geometric sequence $\{a_n\}$ with the sum of the first $n$ terms denoted as $S_n$, if $\frac{S_4}{S_2} = 3$, then the value of $\frac{S_6}{S_4}$ is ( )
A: 2
B: $\frac{7}{3}$
C: $\frac{8}{3}$
D: 3 | B |
Let set $A=\{(x_1, x_2, x_3, x_4, x_5) | x_i \in \{-1, 0, 1\}, i=1, 2, 3, 4, 5\}$. Find the number of elements in set $A$ that satisfy the condition $1 \leq |x_1| + |x_2| + |x_3| + |x_4| + |x_5| \leq 3$. | 130 |
3. Given that a triangle can be divided into $N$ similar quadrilaterals. Find all possible values of the positive integer $N$.
| N\geqslant3 |
4. In all other cases - o points.
## Task 2
Maximum 15 points
Solve the equation $2 \sqrt{2} \sin ^{3}\left(\frac{\pi x}{4}\right)=\cos \left(\frac{\pi}{4}(1-x)\right)$.
How many solutions of this equation satisfy the condition:
$0 \leq x \leq 2020 ?$ | 505 |
The coordinates of the point symmetric to point M(2,1) with respect to the line $x+y+1=0$ are \_\_\_\_\_\_. | (-2, -3) |
9.061. $25^{x}<6 \cdot 5^{x}-5$. | x\in(0;1) |
Suppose $f$ is a function that satisfies $f(2)=20$ and $f(2 n)+n f(2)=f(2 n+2)$ for all positive integers $n$. What is the value of $f(10)$ ? | 220 |
In the plane, $n$ lines have been drawn in such a way that no two lines are parallel and no three lines are concurrent. How many regions are delimited by these lines? | u_{n}=\frac{n(n+1)}{2}+1 |
Given 8 dimes ( $10 \phi$ coins) and 3 quarters ( $25 \phi$ coins), how many different amounts of money can be created using one or more of the 11 coins?
(A) 27
(B) 29
(C) 35
(D) 26
(E) 28 | 27 |
The complex number $z= \frac {-1+i}{1+i}-1$ corresponds to a point in the complex plane, which is located in ( )
A: the first quadrant
B: the second quadrant
C: the third quadrant
D: the fourth quadrant | \text{B} |
a) Show that the product of all differences of possible couples of six given positive integers is divisible by $960$
b) Show that the product of all differences of possible couples of six given positive integers is divisible by $34560$
PS. a) original from Albania
b) modified by problem selecting committee | 34560 |
Example 4 As shown in Figure 2, there is a rectangular plot of land $A B C D$, and a rectangular flower bed $E F G H$ is to be built in the center, with its area being half of the area of this plot, and the width of the paths around the flower bed being equal. Now, without any measuring tools, only a sufficiently long ... | x=\frac{a+b-\sqrt{a^{2}+b^{2}}}{4} |
8. Let there be $n$ real numbers, satisfying $\left|x_{i}\right|<1(i=1, \cdots, n),\left|x_{1}\right|+\left|x_{2}\right|+\cdots+\left|x_{n}\right| \geqslant 19+\mid x_{1}+\cdots$ $+x_{n}$ |. Then the minimum value of $n$ is $\qquad$. | 20 |
Point $P$ lies on the diagonal $AC$ of [square](https://artofproblemsolving.com/wiki/index.php/Square) $ABCD$ with $AP > CP$. Let $O_{1}$ and $O_{2}$ be the [circumcenters](https://artofproblemsolving.com/wiki/index.php/Circumcenter) of triangles $ABP$ and $CDP$ respectively. Given that $AB = 12$ and $\angle O_{1}PO_{2... | 96 |
30.9. Find the largest three-digit number such that the number minus the sum of its digits is a perfect square. | 919 |
For any real numbers $a$ and $b$, prove that $\frac{{a}^{4}+{b}^{4}}{2}\geqslant {\left(\frac{a+b}{2}\right)}^{4}$. | \frac{{a}^{4}+{b}^{4}}{2}\geqslant {\left(\frac{a+b}{2}\right)}^{4} |
Distribute 5 volunteers to 3 different Olympic venues for ushering duties, with at least one volunteer assigned to each venue. Determine the number of distribution schemes. | 150 |
Find all real numbers$ x$ and $y$ such that $$x^2 + y^2 = 2$$
$$\frac{x^2}{2 - y}+\frac{y^2}{2 - x}= 2.$$ | (1, 1) |
There are 5 identical balls labeled with numbers 1, 2, 3, 4, 5. Two balls are randomly drawn with replacement, one ball each time. Let event A represent "the number on the ball drawn the first time is 1," event B represent "the number on the ball drawn the second time is 2," event C represent "the sum of the numbers on... | \text{C} |
When two triangular boards with sides of the same shape and size containing a $45^{\circ}$ angle are spliced together, the resulting shape cannot be ( ).
A: square
B: equilateral triangle
C: isosceles right triangle
D: parallelogram (non-rectangle, rhombus, square) | B |
130. Indicate the necessary and sufficient condition for a complex number $a+b i$ to be 1) equal to its conjugate, 2) equal to the reciprocal of its conjugate. | )b=0,b)^{2}+b^{2}=1 |
Which of the following expressions is incorrect when transformed from left to right?
A: $\frac{-2}{3a}=-\frac{2}{3a}$
B: $\frac{-x}{-3y}=\frac{x}{3y}$
C: $\frac{5y}{5x}=\frac{y}{x}$
D: $\frac{a}{b}=\frac{a+1}{b+1}$ | D |
## Problem A1
Find all positive integers $n<1000$ such that the cube of the sum of the digits of $n$ equals $n^{2}$.
| 1,27 |
One, (20 points) A research study group from Class 1, Grade 3 of Yuhong Middle School conducted a survey on students' lunch time at the school canteen. It was found that within a unit of time, the number of people buying lunch at each window and the number of people choosing to eat outside due to unwillingness to wait ... | 5 |
5. In the Cartesian coordinate system, points with both coordinates being integers are called "integer points". Let $k$ be a non-zero real number. If the intersection point of the two distinct lines $y=k x-\frac{1}{k}$ and $y=\frac{1}{k} x-1$ is an integer point, then the number of possible values of $k$ is ( ).
(A) 1
... | A |
If the function $f(x)=\sin (\omega x+\varphi)$ where $\omega > 0$ is monotonically increasing on the interval $\left(0,\frac{\pi}{3}\right)$, and $f\left(\frac{\pi}{6}\right)+f\left(\frac{\pi}{3}\right)=0$, $f(0)=-1$, then $\omega=\boxed{2}$. | 2 |
Find the constant term in the expansion of $$(x^{2}- \frac {1}{x})^{6}$$. | 15 |
3. Let $a$ be a non-zero real number, in the expansion of the binomial $(a x+1)^{5}$, the coefficient of the $x^{4}$ term is equal to the coefficient of the $x^{3}$ term, then the value of $a$ is $\qquad$ . | 2 |
To understand the scores of 1252 students participating in a knowledge competition, it was decided to use systematic sampling to draw a sample of 50. The number of individuals that should be randomly removed from the population is \_\_\_\_\_\_. | 2 |
The domain of the function $f(x)=\frac{1}{\sqrt{1-x}}$ is $M$, and the domain of the function $g(x)=\ln(1+x)$ is $N$. Determine $M \cap N$ ___. | (-1, 1) |
We consider the sequence $u_{n}$ as follows: $u_{0}=2010^{2010}$, and $u_{n+1}=u_{n}+7$ if $u_{n}$ is odd, and $\frac{u_{n}}{2}$ if it is even. What is the smallest integer that this sequence will reach? | 1 |
Calculate: $\dfrac {12^{0}-3^{2}\times6^{-1}\times2^{2}}{-3^{-2}}\times5^{-1}=$ \_\_\_\_\_\_ . | 9 |
Given that $F_1$ and $F_2$ are the left and right foci of the hyperbola C: $$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$$ (a > 0, b > 0) with an eccentricity of $e = \frac{\sqrt{7}}{2}$, and P is a point on the right branch of the hyperbola C. If $PF_2 \perp F_1F_2$ and $PF_2 = \frac{9}{2}$, find the length of the conjugate... | 6\sqrt{3} |
Given that for any $x \in \mathbb{R}$, $2x^{2}+(a-1)x+ \frac{1}{2} > 0$ always holds true, find the range of values that the real number $a$ can take.
A: $(-\infty,-1)$
B: $(-1,3)$
C: $(-3,+\infty)$
D: $(-3,1)$ | \text{B} |
From a group of 2012 students, 50 students are selected to participate in a math competition. The selection method is as follows: first, 12 people are randomly eliminated from the 2012 individuals using simple random sampling, and then 50 people are selected from the remaining 2000 individuals using systematic sampling... | D |
The slope of the tangent line to the curve $y=(ax+1)e^{x}$ at the point $(0,1)$ is $-2$. Find the value of $a$. | -3 |
There are 10 steps, and one can take 1, 2, or 3 steps at a time to complete them in 7 moves. The total number of different ways to do this is ( )
A: 175
B: 42
C: 77
D: 35 | C |
A circle with a radius of $2m$ has an arc length corresponding to a central angle of $\frac{\pi}{3}$. Determine the length of the arc. | \frac{2\pi}{3} |
Given $\tan (α- \dfrac {π}{6})= \dfrac {3}{7}$ and $\tan ( \dfrac {π}{6}+β)= \dfrac {2}{5}$, find $\tan (α+β)=$ _____ | 1 |
3. Given that $n$ is a positive integer, $S=1+2+\cdots+n$. Then the units digit of $S$ cannot be ( ).
(A) $0,1,2,3$
(B) $3,4,5,6$
(C) $3,5,6,8$
(D) $2,4,7,9$ | D |
Given a random variable $X \sim N(2,\sigma^{2})$, if $P(X \leqslant 1-a) + P(X \leqslant 1+2a) = 1$, then the real number $a$ is
A: $0$
B: $1$
C: $2$
D: $4$ | C |
The sum of ten consecutive integers is $S$. Ten times the smallest of these integers is $T$. What is the value of $S-T$ ?
(A) 45
(B) 55
(C) 10
(D) 9
(E) 66 | 45 |
The order of the numbers $0.3^2$, $\log_{2}0.3$, and $2^{0.3}$ is ( )
A: $0.3^2 < 2^{0.3} < \log_{2}0.3$
B: $0.3^2 < \log_{2}0.3 < 2^{0.3}$
C: $\log_{2}0.3 < 0.3^2 < 2^{0.3}$
D: $\log_{2}0.3 < 2^{0.3} < 0.3^2$ | \text{C} |
68. A large group of tourists set off on an international trip. Out of them, 28 people speak English, 13 speak French, 10 speak German, 8 speak both English and French, 6 speak both English and German, 5 speak both French and German, 2 speak all three languages, and 41 people do not speak any of the three languages. Ho... | 75 |
4. Let three real numbers $x_{1}, x_{2}, x_{3}$ be such that the sum of any one of them and the product of the other two is 1. Then the number of ordered triples $\left(x_{1}, x_{2}, x_{3}\right)$ that satisfy this condition is ( ).
(A)1
(B) 3
(C) 5
(D) more than 5
Translate the above text into English, please keep th... | C |
Given two non-zero vectors $a$ and $b$ that are not collinear, if $|a|=|b|=1$, and the angle between $a$ and $b$ is $120^{\circ}$, find the value of the real number $x$ that minimizes $|a-xb|$ ($x \in \mathbb{R}$). | -\frac{1}{2} |
Given $a=\log_{2}6$, $b=\log_{3}12$, $c=2^{0.6}$, then ()
A: $a \lt b \lt c$
B: $c \lt b \lt a$
C: $b \lt a \lt c$
D: $c \lt a \lt b$ | B |
4. Given the set $S=\{1,2, \cdots, 2005\}, A \subseteq S, A$ such that the sum of any two numbers in $A$ is not divisible by 117, find the maximum value of $|A|$.
untranslated text remains the same as requested. However, if you need any further assistance or a different translation, feel free to let me know! | 1003 |
1. Find the sum of the first thirty-three terms of an arithmetic progression, given that the sum of the seventh, eleventh, twentieth, and thirtieth terms of this progression is 28. | 231 |
Chinese New Year is a traditional festival in China. Every year after New Year's Day is the peak shopping period. In January 2023, a fruit merchant purchased two types of Red Fuji apples, $A$ and $B$, from farmers. The purchase price of type $A$ Red Fuji apples is $28$ yuan per piece, and the selling price is $42$ yuan... | 27 |
8 If the three lines from the line system $C: x \cos t+(y+1) \sin t=2$ form an equilateral triangle region $D$, then the area of region $D$ is $\qquad$ . | 12\sqrt{3} |
Given $x > 0$, $y > 0$, and $x+2y+2xy=8$, find the minimum value of $x+2y$.
A: $\dfrac {11}{2}$
B: $3$
C: $\dfrac {9}{2}$
D: $4$ | D |
3. In triangle $ABC$, the median $AM$ is drawn. Circle $\alpha$ passes through point $A$, touches line $BC$ at point $M$, and intersects sides $AB$ and $AC$ at points $D$ and $E$ respectively. On the arc $AD$ that does not contain point $E$, a point $F$ is chosen such that $\angle BFE = 72^{\circ}$. It turns out that $... | 36 |
The [difference](https://artofproblemsolving.com/wiki/index.php/Subtraction) between a $6.5\%$ sales tax and a $6\%$ sales tax on an item priced at $$20$ before tax is
$\text{(A)}$ $$.01$
$\text{(B)}$ $$.10$
$\text{(C)}$ $$ .50$
$\text{(D)}$ $$ 1$
$\text{(E)}$ $$10$ | 0.10 |
Statement A: If the random variable $\xi \sim N(3, \sigma^2)$ and $P(\xi \leq 2) = 0.3$, then $P(\xi \leq 4) = 0.7$. Statement B: The random variable $\eta$ follows a binomial distribution $B(n, p)$, and $E\eta = 300$, $D\eta = 200$. It is given that $p = \frac{1}{3}$. Which of the following is correct?
A: Statement A ... | D |
Given that $\theta$ is in the fourth quadrant, and $\sin\theta = -\frac{\sqrt{6}}{3}$, then $\tan\theta = \_\_\_\_\_\_$. | -\sqrt{2} |
9.164. $\log _{3} \log _{4} \frac{4 x-1}{x+1}-\log _{\frac{1}{3}} \log _{\frac{1}{4}} \frac{x+1}{4 x-1}<0$. | x\in(\frac{2}{3};\infty) |
[ Perpendicularity of a line and a plane (other).]
In the pyramid $A B C D$, the edges $A D, B D$ and $C D$ are equal to 5, the distance from point $D$ to the plane $A B C$ is 4. Find the radius of the circle circumscribed around triangle $A B C$. | 3 |
9.2. Petya says to Kolya: «If you think of a quadratic trinomial that has roots and tell me only the leading coefficient and the distance between the roots, then I will guess the y-coordinate of the vertex on its graph». Kolya believes that Petya is wrong: after all, to define a quadratic trinomial, you need to know th... | -(\frac{}{2})^{2} |
Determine all possible values of $A^3+B^3+C^3-3ABC$ where $A$, $B$, and $C$ are nonnegative integers. | \text{all } n \in \mathbb{Z}_{\ge 0}, n \not\equiv \{3, 6\} \pmod{9} |
4. Find all values of the parameter $a$ for which the system
$$
\left\{\begin{array}{c}
\left(a\left|y^{3}\right|+a\left|x^{3}\right|-8\right)\left(x^{6}+y^{6}-3 a^{2}\right)=0 \\
\sqrt{x^{6} y^{6}}=a
\end{array}\right.
$$
has eight distinct solutions. (16 points) | \in(0;2/3)\cup{2}\cup(2\sqrt[3]{2};+\infty) |
4. "24 Points" is a familiar math game to many people, the game process is as follows: arbitrarily draw 4 cards from 52 cards (excluding the jokers). Use the numbers on these 4 cards (from 1 to 13, where $A=1, J=11, Q=12, K=13$) to derive 24 using the four basic arithmetic operations. The person who finds the algorithm... | 2\times(11-5)+12 |
6. Pete came up with all the numbers that can be formed using the digits 2, 0, 1, 8 (each digit can be used no more than once). Find their sum. | 78331 |
7. Given the function $f(x)=\frac{25^{x}}{25^{x}+5}$, then $\sum_{k=1}^{2016} f\left(\frac{k}{2017}\right)=$ $\qquad$ | 1008 |
19. Find the remainder when $(x-1)^{100}+(x-2)^{200}$ is divided by $x^{2}-3 x+2$. | 1 |
7. A certain high school offers its students the choice of two sports: football and basketball. One fifth of the footballers also play basketball, and one seventh of the basketball players also play football. There are 110 students who practice exactly one of the sports. How many of them practice both sports? | 11 |
The sequence $\{a_n\}$ has the sum of its first $n$ terms equal to $S_n$. Given that $a_n + S_n = n$ and $c_n = a_n - 1$. For the sequence $\{b_n\}$, we have $b_1 = a_1$, and $b_n = a_n - a_{n-1}$ for $n \geq 2$.
(1) Prove that the sequence $\{c_n\}$ is a geometric sequence.
(2) Find the general formula for the ter... | \frac{1}{2^n} |
2. (1992 Ukrainian Mathematical Olympiad) Find all prime numbers $p$ not exceeding 1000 such that $2p+1$ is a power of a natural number (i.e., there exist natural numbers $m$ and $n, n \geqslant 2$, such that $2p+1=m^n$). | 13 |
In a dice game, a player rolls two dice. His score is the larger of the two numbers on the dice. For example, if he rolls 3 and 5 , his score is 5 , and if he rolls 4 and 4 , his score is 4 . What is the probability that his score is 3 or less?
(A) $\frac{1}{4}$
(B) $\frac{7}{36}$
(C) $\frac{5}{36}$
(D) $\frac{1}{3}$
(... | \frac{1}{4} |
6. Given the sequence $\left\{a_{n}\right\}$ is an arithmetic sequence, and $S_{5}=$ $28, S_{10}=36$. Then $S_{15}$ equals ( ).
(A) 80
(B) 40
(C) 24
(D) -48 | C |
In $\triangle ABC$, $\angle C= \frac {2\pi}{3}$, $a=6$.
(I) If $c=14$, find the value of $\sin A$;
(II) If the area of $\triangle ABC$ is $3 \sqrt {3}$, find the value of $c$. | 2 \sqrt {13} |
Given the function $f(x)= \dfrac {1}{ \sqrt {1-x}}$ with domain $M$, and the function $g(x)=2^{x}$ with range $N$, find $M∩N=$ _____ . | (0,1) |
12.177. Inside the given angle $\alpha$ there is a point at a distance $a$ from the vertex and at a distance $b$ from one side. Find the distance of this point from the other side. | \sin\alpha\sqrt{^{2}-b^{2}}-b\cos\alpha |
7.3. A natural number is called curious if, after subtracting the sum of its digits from it, the result is a number consisting of identical digits. How many three-digit curious numbers exist? | 30 |
7.2. One morning at 9:00, pedestrian Fedia left the village of Fedino for the village of Novoverandovo. At the same time, cyclist Vera set off towards him from Novoverandovo. It is known that by the time they met, Fedia had walked one-third of the distance between the villages. However, if Fedia had left an hour earlie... | 10:20 |
6. Given the equation $\sqrt{x}+3 \sqrt{y}=\sqrt{300}$. The number of positive integer solutions to this equation is ( ).
(A) 1
(B) 2
(C) 3
(D) 4 | C |
Given that $y$ is directly proportional to $x-2$, when $x=3$, $y=-4$, the analytical expression of $y$ with respect to $x$ is ______. | y = -4x + 8 |
Given $\sin ( \frac {π}{4}-α)= \frac {1}{2}$, find $\cos ( \frac {π}{4}+α)=(\ $ $)$
A: $\frac { \sqrt {3}}{2}$
B: $- \frac {1}{2}$
C: $- \frac { \sqrt {3}}{2}$
D: $\frac {1}{2}$ | D |
From the set $\{1,2,3,4,5\}$, two distinct numbers are randomly selected. The probability that the sum of these two numbers is odd is $\_\_\_\_\_\_\_\_$. | \frac{3}{5} |
5. For a convex $n$-sided polygon, if circles are constructed with each side as the diameter, the convex $n$-sided polygon must be covered by these $n$ circles. Then the maximum value of $n$ is: ( ).
(A) 3 .
(B) 4 .
(C) 5 .
(D) Greater than 5 . | B |
Given an arithmetic sequence $\{a_{n}\}$ with the sum of the first $n$ terms as $S_{n}$, and a positive geometric sequence $\{b_{n}\}$ with the sum of the first $n$ terms as $T_{n}$, where $a_{1}=2$, $b_{1}=1$, and $b_{3}=3+a_{2}$. $(1)$ If $b_{2}=-2a_{4}$, find the general formula for the sequence $\{b_{n}\}$; $(2)$ I... | 18 |
(4) The parabola $y^{2}=2 p x(p>0)$ has a focus $F$, and a directrix $l$. $A$ and $B$ are two moving points on the parabola, and they satisfy $\angle A F B=\frac{\pi}{3}$. Let the midpoint of segment $A B$ be $M$, and the projection of $M$ on $l$ be $N$. Then the maximum value of $\frac{|M N|}{|A B|}$ is $\qquad$. | 1 |
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