problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
|---|---|
Simplify: $\dfrac {\tan (2\pi-\theta)\sin (-2\pi-\theta)\cos (6\pi-\theta)}{\cos (\theta -\pi )\sin (5\pi +\theta )}$. | \tan \theta |
4. Given $y=|x-1|-2|x|+|x+2|$, and $-2 \leqslant x$ $\leqslant 1$. Then the sum of the maximum and minimum values of $y$ is ( ).
(A) -1
(B) 2
(C) 4
(D) 5 | B |
1. Given that $f(x)$ is an odd function defined on $\mathbf{R}$, and for any $x \in \mathbf{R}$, we have
$$
f(2+x)+f(2-x)=0 .
$$
If for $x \in[-1,0)$,
$$
f(x)=\log _{2}(1-x) \text {, }
$$
then $f(1)+f(2)+\cdots+f(2021)=$ $\qquad$ | -1 |
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 a positive review. A certain online store has received 60 angry and 20 positive reviews. Using this s... | 0.64 |
The graph of the power function $f(x)$ passes through the point $(2, \frac{1}{8})$. Find the explicit form of the function $f(x)$. | f(x) = x^{-3} |
XXIX OM - III - Problem 4
Let $ X $ be an $ n $-element set. Prove that the sum of the number of elements in the sets $ A \cap B $ extended over all ordered pairs $ (A, B) $ of subsets of $ X $ equals $ n \cdot 4^{n-1} $. | n\cdot4^{n-1} |
Question 206, Given the pair of positive integers $(a, b)$ with the greatest common divisor $(a, b)=2015$, find the minimum value of $a+b$ such that $(a+b) \mid\left[(a-b)^{2016}+b^{2016}\right]$. | 10075 |
6・74 Let $R$ be the triangular region in the plane with vertices $A(4,1), B(-1,-6), C(-3,2)$ (including the interior and boundary of the triangle, as shown in the figure). Try to find the maximum and minimum values of the function $4 x-3 y$ as $(x, y)$ varies over $R$ (you need to prove your conclusion). | 14 \text{ and } -18 |
Task No. 1.1
## Condition:
On the potion-making exam, each student at Hogwarts School had to brew 4 potions. Hermione managed to complete the task in half an hour, Harry in 40 minutes, and Ron took 1 hour. How many potions would Ron, Hermione, and Harry brew together in 2 hours if they continued working at the same s... | 36 |
Given the equation $x^2+2x-a=0$, where $a<0$, the correct conclusion about the roots of this equation in the complex number range is ( )
A: The equation definitely has a pair of conjugate imaginary roots.
B: The equation may have two positive real roots.
C: The sum of the real parts of the roots of the equation equ... | \text{C} |
5. Given the line $l_{1}: y=a x+3 a+2$ and $l_{2}: y$ $=-3 x+3$ intersect at a point in the first quadrant. Then the range of values for $a$ is $(\quad)$.
(A) $\left(-\frac{1}{2}, \frac{1}{3}\right)$
(B) $\left(-\infty, \frac{1}{3}\right)$
(C) $\left(-3, \frac{1}{3}\right)$
(D) $\left(-\frac{1}{2},+\infty\right)$ | A |
Vojta started writing the number of the current school year in his notebook: 2019202020192020... and continued doing so indefinitely. When he wrote 2020 digits, he got bored and stopped.
How many twos did he write?
(L. Růžičková)
Hint. How many twos would Vojta write if he only wrote 20 digits? | 757 |
7.38 From the right angle vertex of a right triangle, draw lines to the trisection points of the hypotenuse. These two line segments are of lengths $\sin x$ and $\cos x$, where $x$ is a real number such that $0 < x < \frac{\pi}{2}$. The length of the hypotenuse is
(A) $\frac{4}{3}$.
(B) $\frac{3}{2}$.
(C) $\frac{3 \sq... | \frac{3\sqrt{5}}{5} |
1. (a) What is $5^{a}$, if $a=\frac{\log _{7} 4\left(\log _{7} 5-\log _{7} 2\right)}{\log _{7} 25\left(\log _{7} 8-\log _{7} 4\right)}$?
(b) What is $5^{b}$, if $b=\frac{\log _{77} 4\left(\log _{77} 5-\log _{77} 2\right)}{\log _{77} 25\left(\log _{77} 8-\log _{77} 4\right)}$? | \frac{5}{2} |
Given that the real numbers $4$, $m$, $1$ form a geometric sequence, the eccentricity of the conic section $\frac{x^2}{m} + y^2 = 1$ is ( )
A: $\frac{\sqrt{2}}{2}$
B: $\sqrt{3}$
C: $\frac{\sqrt{2}}{2}$ or $\sqrt{3}$
D: $\frac{1}{2}$ or $3$ | C |
14. (6 points) As shown in the figure, even numbers starting from 2 are arranged in a clockwise right-angle spiral, with 4, 6, 10, 14, 20, 26, 34, ... appearing at the corners of the spiral. Then 2010 $\qquad$ (fill in “will” or “will not”) appear at the corner of the spiral. | willnot |
5. Let a regular n-gon be denoted as $A_{1} A_{2} \ldots A_{n}$. The point $A_{3}$ is reflected over the axis $A_{2} A_{4}$, resulting in the point $A_{3}^{\prime}$. Then, the point $A_{3}^{\prime}$ is reflected over the axis $A_{1} A_{3}$, resulting in the point $A_{3}^{\prime \prime}$. For which $n \geqq 4$ is the po... | 10 |
A train station starts to form a queue several minutes before checking tickets. The number of people in the queue increases at a certain rate. If one ticket window is opened, it takes 40 minutes for the queue at the ticket window to disappear. If two ticket windows are opened simultaneously, the queue disappears in 16 ... | 10 |
Given the function $f(x) = 4^x + m \cdot 2^x + 1$ has exactly one zero, then the value of the real number $m$ is. | -2 |
Four normal students, A, B, C, and D, are to be assigned to work at three schools, School A, School B, and School C, with at least one person at each school. It is known that A is assigned to School A. What is the probability that B is assigned to School B? | \dfrac{5}{12} |
8. The image set interval of the interval $[0, m]$ under the mapping $f: x \rightarrow 2 x+m$ is $[a, b]$. If the length of the interval $[a, b]$ is 5 units greater than the length of the interval $[0, m]$, then $m=(\quad)$.
(A) 5
(B) 10
(C) 2.5
(D) 1 | A |
12. (18 points) The inverse function of $f(x)$ is $y=\frac{x}{1+x}$, $g_{n}(x)+\frac{1}{f_{n}(x)}=0$, let $f_{1}(x)=f(x)$, and for $n>1\left(n \in \mathbf{N}_{+}\right)$, $f_{n}(x)=f_{n-1}\left(f_{n-1}(x)\right)$. Find the analytical expression for $g_{n}(x)\left(n \in \mathbf{N}_{+}\right)$. | g_{n}(x)=2^{n-1}-\frac{1}{x} |
The steps of an algorithm are as follows: First, input the value of $x$. Second, calculate the largest integer $y$ that does not exceed $x$. Third, calculate $z = 2^{y} - y$. Fourth, output the value of $z$. If the output value of $z$ is $27$, then the possible value of input $x$ could be ( )
A: $3.3$
B: $4.4$ ... | \text{C} |
Factor the expression first, then evaluate it: $5a(b-2) + 2a(2-b)$, where $a=2$ and $b=-2$. | -24 |
In how many ways can the integers from $1$ to $n$ be ordered subject to the condition that, except for the first integer on the left, every integer differs by $1$ from some integer to the left of it? | 2^{n-1} |
Andile and Zandre play a game on a $2017 \times 2017$ board. At the beginning, Andile declares some of the squares [i]forbidden[/i], meaning the nothing may be placed on such a square. After that, they take turns to place coins on the board, with Zandre placing the first coin. It is not allowed to place a coin on a for... | 2017 |
7.4. A rectangular grid with a cell side of 1 cm and an area of $2021 \mathrm{~cm}^{2}$ is cut into four rectangular pieces by two perpendicular cuts along the grid lines. Prove that at least one of the pieces has an area of at least $528 \mathrm{~cm}^{2}$. | 528 |
In the arithmetic sequence $\{a_n\}$, $a_2 + a_6 = 8$, $a_3 + a_4 = 3$. What is the common difference?
A: 4
B: 5
C: 6
D: 7 | \text{B} |
Suppose $(a_n)$, $(b_n)$, $(c_n)$ are arithmetic progressions. Given that $a_1+b_1+c_1 = 0$ and $a_2+b_2+c_2 = 1$, compute $a_{2014}+b_{2014}+c_{2014}$.
[i]Proposed by Evan Chen[/i] | 2013 |
3. A circular cone has a height of 12 and a base radius of 5. Let the radius of the inscribed sphere be $r$, and the radius of the circumscribed sphere be $R$. Then $\frac{r}{R}=(\quad)$.
(A) $\frac{89}{169}$
(B) $\frac{83}{169}$
(C) $\frac{80}{169}$
(D) $\frac{60}{169}$ | \frac{80}{169} |
7.055. $\log _{3}\left(3^{x}-8\right)=2-x$.
Translate the above text into English, please keep the original text's line breaks and format, and output the translation result directly.
7.055. $\log _{3}\left(3^{x}-8\right)=2-x$. | 2 |
Task 12. (16 points)
The Vasilievs' family budget consists of the following income items:
- parents' salary after income tax deduction - 71000 rubles;
- income from renting out property - 11000 rubles;
- daughter's scholarship - 2600 rubles
The average monthly expenses of the family include:
- utility payments - 84... | 31320 |
Maria and Joe are jogging towards each other on a long straight path. Joe is running at $10$ mph and Maria at $8$ mph. When they are $3$ miles apart, a fly begins to fly back and forth between them at a constant rate of $15$ mph, turning around instantaneously whenever it reachers one of the runners. How far, in miles,... | \frac{5}{2} |
4. The sequence $\left\{x_{n}\right\}$ is defined as follows:
$$
x_{1}=\frac{2}{3}, x_{n+1}=\frac{x_{n}}{2(2 n+1) x_{n}+1}\left(n \in \mathbf{Z}_{+}\right) \text {. }
$$
Then $x_{1}+x_{2}+\cdots+x_{2014}=$ | \frac{4028}{4029} |
Let us have $6050$ points in the plane, no three collinear. Find the maximum number $k$ of non-overlapping triangles without common vertices in this plane. | 2016 |
Given that the sequence $\{a\_n\}$ satisfies $a\_p + a\_q = a\_{p+q}$ for any $p, q \in \mathbb{N}^*$, and $a\_1 = 2$, find the expression for $a\_n$. | 2n |
Calculate the area of the triangle formed by the tangent line to the curve $y = \frac{1}{3}x^3 + x$ at the point $(1, \frac{4}{3})$ and the coordinate axes. | \frac{1}{9} |
5-33 If $a, b$ are integers, and $x^{2}-x-1$ is a factor of $a x^{17}+b x^{16}+1$, try to find the value of $a$.
| 987 |
In $\triangle ABC$, $a=7$, $b=8$, $A= \frac {\pi}{3}$.
(I) Find the value of $\sin B$;
(II) If $\triangle ABC$ is an acute triangle, find the area of $\triangle ABC$. | 10 \sqrt {3} |
Given the function $f(x)=|x+ \frac {4}{x}-m|+m$.
- (I) When $m=0$, find the minimum value of the function $f(x)$;
- (II) If the function $f(x)\leqslant 5$ holds true for all $x\in[1,4]$, find the range of the real number $m$. | (-\infty, \frac {9}{2}] |
4. For $x \in \mathbf{R}$, the function $f(x+2)+f(x-2)$ $=f(x)$, then it is a periodic function. The smallest positive period of such a function is ( ).
(A) 4
(B) 6
(C) 8
(D) 12 | D |
Problem 10.1. Consider the equations
$$
3^{2 x+3}-2^{x+2}=2^{x+5}-9^{x+1}
$$
and
$$
a .5^{2 x}+|a-1| 5^{x}=1
$$
where $a$ is a real number.
a) Solve the equation (1).
b) Find the values of $a$ such that the equations (1) and (2) are equivalent.
Kerope Chakarian | \in[0,1]\cup{-1} |
Given that the line $ax - y + 2a = 0$ is perpendicular to $(2a - 1)x + ay + a = 0$, find the value of $a$. | a = 1 |
Given an odd function \\(f(x)\\) with the domain of \\(\mathbb{R}\\), if \\(f(x+2)\\) is an even function, and \\(f(-1)=-1\\), then \\(f(2017)+f(2016)=\\) \\(\\)
A: \\(-2\\)
B: \\(-1\\)
C: \\(0\\)
D: \\(1\\) | \text{D} |
4. (20 points) Before the geometry lesson, the teacher wrote on the board the values of all the angles (in degrees) of a certain convex polygon. However, the duty students erased one of the written numbers. When the lesson began, it turned out that the sum of the remaining numbers was 1703. What number did the duty stu... | 97 |
Example 20 (1998 National High School League Question) If the ellipse $x^{2}+4(y-a)^{2}=4$ intersects the parabola $x^{2}=2 y$, then the range of real number $a$ is $\qquad$ . | -1\leqslant\leqslant\frac{17}{8} |
If the distribution of the random variable $X$ is $P(X=i)= \frac {i}{10} (i=1,2,3,4)$, then $P(X > 2)=$ \_\_\_\_\_\_. | 0.7 |
"2＜x＜5" is ( ) of "3＜x＜4".
A: Sufficient and necessary condition
B: Necessary but not sufficient condition
C: Sufficient and necessary condition
D: Neither sufficient nor necessary condition | \text{B} |
In triangle $ABC$, let sides $a$, $b$, and $c$ be opposite to angles $A$, $B$, and $C$ respectively. Given that $\sin A = 2\sin B$ and $a + b = \sqrt{3}c$, find the magnitude of angle $C$. | 60^{\circ} |
Given the curve $C$ with the polar coordinate equation $ρ=2\cos θ$, establish a rectangular coordinate system with the pole as the origin and the positive semi-axis of the polar axis as the $x$-axis. The parameter equation of line $L$ is $\begin{cases} x=\frac{\sqrt{3}}{2}t+m \\ y=\frac{1}{2}t \end{cases}$ (where $t$ i... | 1-\sqrt{2} |
## SUBJECT 1
a) Let $\mathrm{x}, \mathrm{y} \in \mathrm{R},\left|x-\frac{1}{2}\right| \leq \frac{3}{2},\left|y-\frac{1}{2}\right| \leq \frac{3}{2}$. Given that $\sqrt{(2-x)(y+1)}+\sqrt{(2-y)(x+1)}=3$, show that $\mathrm{x}+\mathrm{y}=1$.
b) Consider the numbers $\mathrm{a}=1+\sqrt{2}+\sqrt{2^{2}}+\sqrt{2^{3}}+\ldots ... | 6 |
3. In a psychiatric hospital, there is a chief doctor and many lunatics. During the week, each lunatic bit someone (possibly even themselves) once a day. At the end of the week, it turned out that each of the patients had two bites, and the chief doctor had a hundred bites. How many lunatics are there in the hospital | 20 |
$2023$ players participated in a tennis tournament, and any two players played exactly one match. There was no draw in any match, and no player won all the other players. If a player $A$ satisfies the following condition, let $A$ be "skilled player".
[b](Condition)[/b] For each player $B$ who won $A$, there is a playe... | 3 |
Given the function $f(x)=4\sin \omega x\cos (\omega x+\dfrac{\pi }{3})+2\sqrt{3}(\omega > 0)$.
(1) If the smallest positive period of $f(x)$ is $\pi$, find the maximum and minimum values of $f(x)$ in the interval $[\dfrac{-\pi }{4},\dfrac{\pi }{6}]$ and the values of $x$ when these extreme values are attained.
(2) If... | \dfrac{1}{2} |
## Zadatak B-1.7.
Dva pravokutna trokuta $P S T$ i $R S T$ imaju zajedničku hipotenuzu $\overline{S T}$. Vrhovi $P$ i $R$ nalaze se s iste strane hipotenuze $\overline{S T}$. Katete $\overline{P T}$ i $\overline{R S}$ sijeku se u točki $Q$. Ako je $|P S|=6 \mathrm{~cm},|P T|=17 \mathrm{~cm}$ i $|R T|=1 \mathrm{~cm}$, ... | \frac{39}{5}\mathrm{~}^{2} |
2. The minimum value of the function $f(x)=(\sqrt{1+x}+\sqrt{1-x}-3)\left(\sqrt{1-x^{2}}+1\right)$ is $m$, and the maximum value is $M$, then $\frac{M}{m}=$
$\qquad$ . | \frac{3-\sqrt{2}}{2} |
Given \\( \overrightarrow{a}=(2,-3,5) \\) and \\( \overrightarrow{b}=(-3,1,2) \\), then \\( | \overrightarrow{a}-2 \overrightarrow{b}|= \)
A: \\(7 \sqrt {2}\\)
B: \\(5 \sqrt {2}\\)
C: \\(3 \sqrt {10}\\)
D: \\(6 \sqrt {3}\\) | C |
Define a function $f(x)$ on $\mathbb{R}$ satisfying $f(x) = f(x+2)$, and when $x \in [3, 5]$, $f(x) = 1 - (x-4)^2$. Then $f(x)$ ( )
A: is increasing on the interval $[-2, -1]$ and increasing on the interval $[5, 6]$
B: is increasing on the interval $[-2, -1]$ and decreasing on the interval $[5, 6]$
C: is decreasing on ... | \text{C} |
Given a triangle $ABC$ with internal angles $A$, $B$, $C$ and opposite sides $a$, $b$, $c$ satisfying $(a+b)^{2}=10+c^{2}$, and $\cos C= \frac {2}{3}$, find the minimum value of $a^{2}+b^{2}$. | 6 |
29. Let $\mathbf{R}_{+}$ denote the set of positive real numbers. Find the function $f: \mathbf{R}_{+} \rightarrow \mathbf{R}_{+}$, such that for all $x, y \in \mathbf{R}_{+}$, we have $f(x f(y))=f(x y)+x$.
(51st Czech and Slovak Mathematical Olympiad (Final)) | f(x)=x+1 |
【8】There are six wooden strips, each 50 cm long. Now they are to be connected end-to-end, with each joint between two strips being 10 cm long. After nailing them together, the total length of the wooden strips is ( ) cm. | 250 |
4. Let $n$ students be such that among any 3 of them, 2 know each other, and among any 4 of them, 2 do not know each other. Find the maximum value of $n$.
untranslated text remains the same as requested. | 8 |
For the parabola $y^2=4x$ ($p>0$), two chords AB and CD passing through the focus of the parabola are perpendicular to each other. Then the value of $\frac {1}{|AB|}+ \frac {1}{|CD|}$ is ( )
A: 2
B: 4
C: $\frac {1}{2}$
D: $\frac {1}{4}$ | D |
(1) Let the set $S=\left\{x \mid x^{2}-5 x-6<0\right\}, T=\{x|| x+2 \mid \leqslant 3\}$, then $S \cap T=(\quad)$.
(A) $\{x \mid-5 \leqslant x<-1\}$
(B) $\{x \mid-5 \leqslant x<5\}$
(C) $\{x \mid-1<x \leqslant 1\}$
(D) $\{x \mid 1 \leqslant x<5\}$ | {x\mid-1<x\leqslant1} |
[ [Ratio of areas of triangles with a common base or common height] Difficult
In a circle, a trapezoid $ABCD$ is inscribed, with its bases $AB=1$ and $DC=2$. Let the point of intersection of the diagonals of this trapezoid be denoted as $F$. Find the ratio of the sum of the areas of triangles $ABF$ and $CDF$ to the su... | \frac{5}{4} |
The sum of the (decimal) digits of a natural number $n$ equals $100$, and the sum of digits of $44n$ equals $800$. Determine the sum of digits of $3n$. | 300 |
The minimum value of the function $f(x) = \sqrt{2} \sin x (\cos x + \sin x) - \frac{\sqrt{2}}{2}$ in the interval $[0, \frac{\pi}{2}]$ is __________. | -\frac{\sqrt{2}}{2} |
Let $A,B,C,D$, be four different points on a line $\ell$, so that $AB=BC=CD$. In one of the semiplanes determined by the line $\ell$, the points $P$ and $Q$ are chosen in such a way that the triangle $CPQ$ is equilateral with its vertices named clockwise. Let $M$ and $N$ be two points of the plane be such that the tria... | 60^\circ |
In △ABC, there are 2008 arbitrary points that are not collinear, plus the three vertices A, B, and C, making a total of 2011 points. If these 2011 points are connected to form non-overlapping small triangles, the total number of small triangles that can be formed is ( )
A: 4015
B: 4017
C: 4019
D: 4020 | \text{B} |
Given: $\neg p$ and $q$ are true, then among the following propositions, the false propositions are ( )
①$p$；②$p$ or $q$；③$p$ and $q$；④$\neg q$．
A: ①④
B: ①②③
C: ①③④
D: ②③④ | C |
A3. For which value of the parameter $m$ will the lines with equations $y=(m+5) x+m-1$ and $y=(m-1)^{2} x-3 m$ be parallel?
(A) $m_{1}=1, m_{2}=-5$
(B) $m_{1}=-4, m_{2}=1$
(C) $m_{1}=1, m_{2}=\frac{1}{3}$
(D) $m=-\frac{1}{4}$
(E) none of the above | E |
In the figure, the symbols □, △, and ○ each represent a number, and they satisfy the following three equations: □+□+△+○=17, □+△+△+○=14, □+△+○+○=13. What number does □ represent? | 6 |
Let $S_{n}$ be the sum of the first $n$ terms of the sequence $\{a_{n}\}$. Given that for scenario A: $\{a_{n}\}$ is an arithmetic sequence, and for scenario B: $2S_{n}=(a_{1}+a_{n})n$ (where $n\in N^{*}$), which of the following statements is correct?
A: Scenario A is a sufficient but not necessary condition for scen... | C |
For the equation $x^2 - 2|x| + 2 = m$, if the number of real roots of the equation is exactly 3, then the value of $m$ is ( )
A: 1
B: 2
C: 3
D: 2.5 | B |
Task 4. (20 points) To get from the first to the second building of the university, Sasha took a car-sharing vehicle, while Valya rented a scooter. Sasha and Valya set off from the first building to the second at the same time, and at the same time, teacher Sergei Vladimirovich set off from the second building to the f... | 10,45,60 |
Find the value of the constant $a$ in the expansion of ${(x-\frac{\sqrt{a}}{x^{2}})}^{6}$ if the constant term is $60$. | 4 |
## Task Condition
Find the $n$-th order derivative.
$y=\sin 2x + \cos (x+1)$ | y^{(n)}=2^{n}\cdot\sin(\frac{\pi}{2}\cdotn+2x)+\cos(\frac{\pi}{2}\cdotn+x+1) |
If the function $f(x)=\log_a(ax^2-x+\frac{3}{2})$ and $0 < a < 1$, is increasing in the interval $[1,2]$, then the range of values for $a$ is ______. | \left( \frac{1}{8}, \frac{1}{4} \right] |
Given the following propositions, the correct ones are \_\_\_\_\_\_ (write down all the correct proposition numbers)
① The graph of the function $f(x)=\log_{a}(x-3)+2$ always passes through the fixed point $(4,2)$;
② Given the sets $P=\{a,b\}$, $Q=\{0,1\}$, then there are $1$ mappings $f$: $P\rightarrow Q$ that sati... | ①④ |
135. Solve the equation
$$
x(x+1)(x+2)(x+3)=5040
$$ | x_{1}=-10,x_{2}=7 |
(1) Calculate $(\lg 2)^{2}+\lg 5\cdot \lg 20+(\sqrt {2016})^{0}+0.027^{ \frac {2}{3}}\times(\frac {1}{3})^{-2}$;
(2) Given $\frac {3\tan \alpha}{\tan \alpha -2}=-1$, find the value of $\frac {7}{\sin ^{2}\alpha +\sin \alpha \cdot \cos \alpha +\cos ^{2}\alpha }$. | 5 |
What is the sum of the solutions to the equation $\sqrt[4]{x} = \frac{12}{7 - \sqrt[4]{x}}$? | 337 |
## Problem Statement
Calculate the limit of the function:
$\lim _{x \rightarrow \frac{\pi}{2}} \sqrt{3 \sin x+(2 x-\pi) \sin \frac{x}{2 x-\pi}}$ | \sqrt{3} |
What is the probability when rolling 3 dice that
a) exactly one die shows a 6?
b) at least one die shows a 6?
c) at most one die shows a 6?
(For the example, see the article “>> A” Elements of Probability Calculation <<” in this issue.) | \frac{25}{72},\frac{91}{216},\frac{25}{27} |
6.10 The formula $N=8 \cdot 10^{8} \cdot x^{-\frac{3}{2}}$ represents the number of people in a group whose income exceeds $x$ yuan. The minimum income of the highest $800^{\circ}$ earners is at least
(A) $10^{4}$.
(B) $10^{6}$.
(C) $10^{8}$.
(D) $10^{12}$.
(E) $10^{16}$.
(11th American High School Mathematics Examinat... | 10^{4} |
3. The sum of positive numbers $a, b, c$ and $d$ does not exceed 4. Find the maximum value of the expression
$$
\sqrt[4]{a^{2}+3 a b}+\sqrt[4]{b^{2}+3 b c}+\sqrt[4]{c^{2}+3 c d}+\sqrt[4]{d^{2}+3 d a}
$$ | 4\sqrt{2} |
Let $l, m$ be two skew lines, and on $l$ there are three points $A, B, C$, with $A B = B C$. Draw perpendiculars from $A, B, C$ to $m$, denoted as $A D, B E, C F$, with the feet of the perpendiculars being $D$, $E, F$ respectively. Given that $A D = \sqrt{15}, B E = \frac{7}{2}, C F = \sqrt{10}$. Find the distance betw... | \sqrt{6} |
Among the following square roots, which one can be combined with $\sqrt{3}$?
A: $\sqrt{\frac{3}{2}}$
B: $\sqrt{8}$
C: $\sqrt{0.5}$
D: $\sqrt{12}$ | D |
Condition of the problem
Calculate the limit of the numerical sequence:
$\lim _{n \rightarrow \infty} \frac{e^{1 / n}+\sin \left(\frac{n}{n^{2}+1}\right) \cdot \cos n}{1+\cos \left(\frac{1}{n}\right)}$ | \frac{1}{2} |
# Task 3. Maximum 20 points
At the "Economics and Law" congress, a "Tournament of the Best" was held, in which more than 220 but fewer than 254 delegates—economists and lawyers—participated. Within one match, participants had to ask each other questions within a limited time and record the correct answers. Each partic... | 105 |
The value of $2 \frac{1}{10}+3 \frac{11}{100}+4 \frac{111}{1000}$ is
(A) 9.321
(B) 9.111
(C) 9.123
(D) 9.111111
(E) 9.11081081 | 9.321 |
Prove that
$$
\operatorname{tg} 9^{\circ}-\operatorname{tg} 27^{\circ}-\operatorname{tg} 63^{\circ}+\operatorname{tg} 81^{\circ}=4
$$ | 4 |
17. In a strange animal village, only cats and dogs live. 20% of the dogs think they are cats, and 20% of the cats think they are dogs, with the rest of the cats and dogs being normal. Among all the cats and dogs, 36% think they are cats. If there are 140 more dogs than cats in this strange animal village, then the num... | 220 |
198. Perform the division:
a) $\frac{2+3 i}{5-7 i}$;
b) $\frac{3+5 i}{2+6 i}$. | \frac{9}{10}-\frac{1}{5}i |
If $\sin\theta=1-\log_{2}x$, then the range of $x$ is \_\_\_\_\_\_ . | [1, 4] |
Example 10 Find $A^{2}$, where $A$ is the sum of the absolute values of all roots of the following equation:
$$
x=\sqrt{19}+\frac{91}{\sqrt{19}+\frac{91}{\sqrt{19}+\frac{91}{\sqrt{19}+\frac{91}{\sqrt{19}+\frac{91}{x}}}}}
$$
(9th American Invitational Mathematics Examination) | 383 |
At the beginning of the school year, a certain school conducted a mathematics quiz (out of $100$ points) to understand the mathematics learning situation of eighth-grade students during the winter vacation. They randomly selected $10$ students from each of Class A and Class B for statistical analysis. The collected dat... | 15 |
A Vinczlér. A poor Vinczlér was just milking, when the landowner approached and asked for 4 messzely of must to drink. The poor Vinczlér only had two measuring vessels, one 3 messzely, the other 5 messzely, the question is, how did he still manage to give 4 messzely of drink to the lord? 1[^0]
[^0]: 1* We are happ... | 4 |
The imaginary part of the complex number $1 - \sqrt {3}i$ is ( )
A: $\sqrt {3}i$
B: 1
C: $\sqrt {3}$
D: $-\sqrt {3}$ | D |
Example 11 Four spheres, each with a radius of 1, are mutually tangent, three on the floor, and the fourth above them. A tetrahedron with side length $S$ has faces that are tangent to all these spheres. Find $S$.
(IMO - 29 Shortlist) | 2+2\sqrt{6} |
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