problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
|---|---|
How many positive integer solutions does the equation have $$\left\lfloor\frac{x}{10}\right\rfloor= \left\lfloor\frac{x}{11}\right\rfloor + 1?$$
($\lfloor x \rfloor$ denotes the integer part of $x$, for example $\lfloor 2\rfloor = 2$, $\lfloor \pi\rfloor = 3$, $\lfloor \sqrt2 \rfloor =1$) | 110 |
Given four non-collinear points $O$, $A$, $B$, $C$ on a plane, if $\overrightarrow{OA}-4\overrightarrow{OB}+3\overrightarrow{OC}=\overrightarrow{0}$, then $\frac{|\overrightarrow{AB}|}{|\overrightarrow{CA}|}$ is ____. | \frac{3}{4} |
5. Express $(A-B)^{-1}$ in terms of $x$, if $A=\frac{3 x^{-\frac{1}{3}}}{x^{\frac{2}{3}}-2 x^{-\frac{1}{3}}}$ and $B=\frac{x^{\frac{1}{3}}}{x^{\frac{4}{3}}-x^{\frac{1}{3}}}$. Simplify the expression.
## You have 120 minutes to solve.
Solve each problem independently on the provided paper, with each problem on a separ... | \frac{(x-2)(x-1)}{2x-1} |
Given a random variable $ξ \sim B(10,0.6)$, then $E(ξ)$ and $D(ξ)$ are respectively ( ).
A: $6$ and $2.4$
B: $4$ and $2.4$
C: $4$ and $3.6$
D: $6$ and $1.6$ | A |
Given $f(x) = x^3 - 2x - 5$, and $f(2.5) > 0$, use the "bisection method" to find the real root of the equation $x^3 - 2x - 5 = 0$ within the interval $(2, 3)$. Taking the midpoint of the interval as $x = 2.5$, the next interval containing the root is ____. | (2, 2.5) |
[color=darkred]Let $n$ and $m$ be two natural numbers, $m\ge n\ge 2$ . Find the number of injective functions
\[f\colon\{1,2,\ldots,n\}\to\{1,2,\ldots,m\}\]
such that there exists a unique number $i\in\{1,2,\ldots,n-1\}$ for which $f(i)>f(i+1)\, .$[/color] | \binom{m}{n} (2^n - (n+1)) |
13. Where is the deviation greater? We will toss a symmetric coin $n$ times. Suppose that heads appeared $m$ times. The number $m / n$ is called the frequency of heads. The number $\frac{m}{n}-0.5$ is called the deviation of the frequency from the probability, and the number $\left|\frac{m}{n}-0.5\right|$ is called the... | Theexpectedabsolutedeviationinseriesof10tossesisgreaterthaninseriesof100tosses |
Given $\cos \left( \frac {\pi}{4}-\theta \right)= \frac { \sqrt {2}}{10}$, and $\theta \in (0,\pi)$.
$(1)$ Find the value of $\sin \left( \frac {\pi}{4}+\theta \right)$;
$(2)$ Find the value of $\sin ^{4}\theta-\cos ^{4}\theta$. | \frac {7}{25} |
8,9, |
In triangle $A B C \angle B=110^{\circ}, \angle C=50^{\circ}$. On side $A B$ a point $P$ is chosen such that $\angle P C B=30^{\circ}$, and on side $A C$ a point $Q$ is chosen such that
$\angle A B Q=40^{\circ}$. Find the angle $Q P C$. | 40 |
## Task 4 - 320824
On the left scale of a balance, there are three candles, and on the right, there is one candle. The four candles are such that each of them loses the same amount of mass during each hour of burning as any other. Each of the three left candles would take 9 hours to burn completely, while the right ca... | 7\frac{1}{2} |
Let the sequence $\{a_n\}$ have a sum of the first $n$ terms denoted as $S_n$, and it satisfies the equation $S_n^2 - 2S_n - a_n \cdot S_n + 1 = 0$, where $n \in \mathbb{N}^*$.
(Ⅰ) Find the relationship between $S_n$ and $S_{n-1}$ (for $n \geq 2$), and prove that the sequence $\left\{ \frac{1}{S_n - 1} \right\}$ is a... | \frac{n}{2(n+2)} < T_n < \frac{2}{3} |
A set of points $M$ in the plane satisfies the following conditions:
(a) No three points are collinear;
(b) Each point in $M$ can be colored either red or blue such that every triangle with vertices of the same color has at least one point of the other color inside it.
Determine the maximum possible number of elements... | 8 |
$\sin 36^\circ \cos 36^\circ - \cos 36^\circ \sin 36^\circ =$ | 0 |
Determine the functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that:
$$
f(f(f(x)))+f(f(y))=f(y)+x
$$ | f(x)=x |
On August 20, 2021, the country passed a decision to amend the Population and Family Planning Law, which advocates for timely marriage and childbearing, eugenics, and nurturing superior offspring. According to the amended population and family planning law, a couple can have up to three children, known as the "three-ch... | \text{C} |
The eccentricity of the hyperbola $C: \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 (a > 0, b > 0)$ is $2$, and its asymptotes are tangent to the circle $(x-a)^2 + y^2 = \frac{3}{4}$. Find the equation of this hyperbola. | x^2 - \frac{y^2}{3} = 1 |
Given the function $f(x) = \begin{cases} 2^{x}, & x > 0 \\ x + 1, & x \leqslant 0 \end{cases}$, then $f(-2)=$ ()
A: $-1$
B: $0$
C: $\dfrac{1}{4}$
D: $4$ | \text{A} |
5. Let $x_{1}, x_{2}, \cdots, x_{7}$ be natural numbers, and $x_{1}<x_{2}$ $<\cdots<x_{6}<x_{7}$, also $x_{1}+x_{2}+\cdots+x_{7}=159$. Then the maximum value of $x_{1}+x_{2}+x_{3}$ is $\qquad$ | 61 |
G2 Let $A D, B F$ and $C E$ be the altitudes of $\triangle A B C$. A line passing through $D$ and parallel to $A B$ intersects the line $E F$ at the point $G$. If $H$ is the orthocenter of $\triangle A B C$, find the angle $\widehat{C G H}$.
| 90 |
2. Given $|x| \leqslant 1,|y| \leqslant 1$, and
$$
k=|x+y|+|y+1|+|2 y-x-4| \text {. }
$$
Then the sum of the maximum and minimum values of $k$ is $\qquad$ | 10 |
## Task 2 - 030922
In a shooting competition, a shooter scored 40 points with 5 shots on a ten-ring target. With each shot, he hit at least 7 rings.
How many possibilities are there for the rings scored with each individual shot?
Note: The order must be taken into account. For example, 7, 7, 7, 9, 10 and 7, 7, 7, 10... | 101 |
【Question 14】
Given a rectangular piece of paper $A B C D$ with a perimeter of 10 (as shown in the figure), now cut it twice along directions parallel to the length and width, respectively, to form 9 rectangles of unequal sizes. The total perimeter of these 9 rectangles is $\qquad$.
| 30 |
4. A natural number $a$ has 107 different divisors, including 1 and $a$. Find the sum and product of these divisors. | \Sigma_{}=\frac{\sqrt[106]{}-1}{\sqrt[106]{}-1};\Pi_{}=\sqrt{} |
Given $z=3-i$, then $|z|=\left(\ \ \right)$
A: $3$
B: $4$
C: $\sqrt{10}$
D: $10$ | C |
Example 5 Find the range of the function $y=\left|x^{2}-4\right|-3 x(-2 \leqslant x \leqslant 5)$. | [-6,\frac{25}{4}] |
Which of the following equations is correct?
A: $(1+\tan 1^{\circ})\left(1+\tan 44^{\circ}\right)=2$
B: $\frac{1}{\sin{10}°}-\frac{\sqrt{3}}{\cos{10}°}=2$
C: $\frac{3-\sin{70}°}{2-\cos^{2}{10}°}=2$
D: $\tan{70}°\cdot\cos{10}°(\sqrt{3}\tan{20}°-1)=2$ | A \text{ and } C |
A circle and a square are drawn on the plane so that they overlap. Together, the two shapes cover an area of $329$ square units. The area common to both shapes is $101$ square units. The area of the circle is $234$ square units. What is the perimeter of the square in units?
$\mathrm{(A) \ } 14 \qquad \mathrm{(B) \ } 4... | 56 |
A lucky number is a number whose digits are only $4$ or $7.$ What is the $17$th smallest lucky number?
[i]Author: Ray Li[/i]
[hide="Clarifications"]
[list=1][*]Lucky numbers are positive.
[*]"only 4 or 7" includes combinations of 4 and 7, as well as only 4 and only 7. That is, 4 and 47 are both lucky numbers.[/list][... | 4474 |
7.1. The sides of a rectangle are in the ratio $3: 4$, and its area is numerically equal to its perimeter. Find the sides of the rectangle. | \frac{7}{2},\frac{14}{3} |
Which of the following idioms describes a random event?
A: Fishing for the moon in the water
B: Castles in the air
C: Waiting by a stump for a hare
D: Catching a turtle in a jar | C |
Example 8. Find the surface integral
$$
I=\int_{\Sigma}\left(\varphi \frac{\partial \psi}{\partial n}-\psi \frac{\partial \varphi}{\partial n}\right) d \sigma
$$
taken over the surface $\Sigma: x^{2}+y^{2}=R^{2}, z=0, z=H(H>0)$, if $\varphi=x^{2}+y^{2}+x+z, \psi=x^{2}+y^{2}+2 z+x$. | -2\piR^{2}H^{2} |
2. Solve the equation $\cos 2 x+\cos 6 x+2 \sin ^{2} x=1$.
In the answer, specify the number equal to the sum of the roots of the equation that belong to the interval $A$, rounding this number to two decimal places if necessary.
$$
A=\left[\frac{m \pi}{6} ; \frac{(m+1) \pi}{6}\right], m=5
$$ | 2.88 |
Example 3. Find the real roots of the equation
$$
\mathrm{f}(\mathrm{x})=\mathrm{x}^{41}+\mathrm{x}^{3}+1=0
$$ | x=-0.9524838 |
Ященно I.v.
Brothers Petya and Vasya decided to shoot a funny video and post it on the internet. First, they filmed each of them walking from home to school - Vasya walked for 8 minutes, while Petya walked for 5 minutes. Then they came home and sat down at the computer to edit the video: they started Vasya's video fro... | 5 |
5. Find all pairs of positive numbers $(x, y)$ that satisfy the system of equations $\left\{\begin{array}{l}x-3 \sqrt{x y}-2 \sqrt{\frac{x}{y}}+6=0, \\ x^{2} y^{2}+x^{4}=82 .\end{array}\right.$ Answer: $\left(3 ; \frac{1}{3}\right),\left(\sqrt[4]{66} ; \frac{4}{\sqrt[4]{66}}\right)$ | (3;\frac{1}{3}),(\sqrt[4]{66};\frac{4}{\sqrt[4]{66}}) |
Example 5 (IMO-31 Shortlist) The integer 9 can be expressed as the sum of two consecutive integers: $9=4+5$. At the same time, 9 can be written as the sum of consecutive natural numbers in two different ways: $9=4+5=2+3+4$.
Is there a natural number $N$ such that:
(1) It is the sum of 1990 consecutive integers;
(2) It ... | 5^{10}\cdot199^{180} |
Insert $n$ numbers between $-9$ and $3$ such that these $n+2$ numbers form an arithmetic sequence with a sum of $-21$. Find the value of $n$. | 5 |
6. Solve the equation $2^{x}+15^{y}=z^{2}$ in natural numbers. | (6,2,17) |
Given the set $A=\{x|2x+a>0\}$ ($a\in \mathbb{R}$), and $1 \notin A$, $2 \in A$, then ( )
A: $a>-4$
B: $a\leq-2$
C: $-4<a<-2$
D: $-4<a\leq-2$ | \text{D} |
Find the number of integers $n$ with $1 \le n \le 2017$ so that $(n-2)(n-0)(n-1)(n-7)$ is an integer
multiple of $1001$.
| 99 |
Example 1.3.1 (Shanghai 2000) As shown in the figure, $ABCD$ is a square with a side length of 1, $U, V$ are points on $AB, CD$ respectively, $AV \cap DU=P, BV \cap CU=Q$, find the maximum value of the area $S_{PUQV}$. | \frac{1}{4} |
3. (10 points) There are several nine-digit numbers composed of 8 threes and 1 zero. Among them, the numbers that leave a remainder of 1 when divided by 4 are ( ) in number.
A. 5
B. 6
C. 7
D. 8 | B |
98. A small boat travels back and forth between two docks, A and B, which are 144 kilometers apart. It takes 6 hours to travel from A to B downstream, and 8 hours to travel from B to A upstream. How many hours would it take for a bottle to drift from dock A to dock B downstream? $\qquad$ | 48 |
There are four outstanding students, A, B, C, and D, who are all guaranteed admission to one of three schools: School Alpha, School Beta, and School Gamma. Each school must admit at least one student. The total number of different admission plans is ( )
A: 18
B: 36
C: 72
D: 108 | B |
Example 5. $n$ circles on a plane, any two of which intersect at two points, and no three circles pass through the same point, divide the plane into how many regions?
---
The translation maintains the original text's format and line breaks as requested. | n^{2}-n+2 |
6.1. A three-digit number is 56 times greater than its last digit. How many times greater is it than its first digit? Justify your answer.
# | 112 |
In a domino game with 45 tiles, we draw 3 tiles one after another. What is the probability that the second tile can be placed next to the first tile, and the third tile can also be attached to either of the first two? How many attempts do we need to make at least so that the probability of the occurrence is at least 0.... | 19 |
2. [6 points] Solve the equation $\sqrt{x+1}-\sqrt{4-x}+3=2 \sqrt{4+3 x-x^{2}}$. | 3 |
Given positive numbers $a, b (a < b)$ and a positive integer $n$, and a sequence of numbers $x_{1}, x_{2}, \cdots, x_{n}$ satisfying $x_{i} \in [a, b], i=1,2, \cdots, n$. Find
$$
f=\frac{x_{1} x_{2} \cdots x_{n}}{\left(a+x_{1}\right)\left(x_{1}+x_{2}\right) \cdots\left(x_{n-1}+x_{n}\right)\left(x_{n}+b\right)}
$$
the ... | f \leqslant\left(a^{\frac{1}{n+1}}+b^{\frac{1}{n+1}}\right)^{-(n+1)} |
1. Given a triangular pyramid $S A B C$, the base of which is an equilateral triangle $A B C$, and all plane angles at vertex $S$ are equal to $\alpha$. For what least $\alpha$ can we assert that this pyramid is regular?
M. Malkin | 60 |
To commemorate the 100th anniversary of the Xinhai Revolution, a TV drama production team plans to create a cover poster featuring 7 of its main actors, who are all of different heights, arranged in an umbrella shape (taller in the middle, shorter on the sides). The number of different posters that can be created is ( ... | \text{A} |
Let \\(P(x,y)\\) be a moving point on the ellipse \\(2x^{2}+3y^{2}=12\\). Find the range of values for \\(x+2y\\). | \left[- \sqrt{22}, \sqrt{22}\right] |
Given a line $l$ with a direction vector $\vec{m}=(2,m,1)$, and a plane $\alpha$ with a normal vector $\vec{n}=(1, \frac{1}{2},2)$, and $l$ is parallel to $\alpha$, find the value of $m$. | -8 |
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 |
Exercise 2. A train is rolling at a constant speed. It takes 7 seconds to pass a fixed point by the track. On the other hand, from the moment the front of the train approaches an 870-meter bridge until the rear of the train leaves the bridge, 37 seconds elapse. Determine the length of the train, expressed in meters. | 203 |
Given the function $f(x)=\begin{cases} \log_{2}(-x), & x\leqslant 0 \\ 2x-1, & x > 0 \end{cases}$, find the range of $x$ such that $f(x)\geqslant 1$.
A) $(-\infty,-2]$
B) $(-\infty,-2]\cup[1,+\infty)$
C) $(-\infty,0]\cup[1,+\infty)$
D) $[1,+\infty)$ | \text{B} |
Calculate
(1) $$27^{ \frac {1}{3}}-(- \frac {1}{2})^{-2}+( \frac {1}{16})^{- \frac {1}{4}}+(\sqrt {2}-1)^{0}$$
(2) $\text{lg}8+\text{lg}125-\text{lg}2-\text{lg}5$． | 2 |
Example 7: From 5 cards each written with a number $1,2,3,4,5$, any two cards are taken out. The number on the first card is used as the tens digit, and the number on the second card is used as the units digit to form a two-digit number. Then the probability that the number formed is a multiple of 3 is ( ).
(A) $\frac{... | C |
Given the array: \\(( \frac {1}{1}),( \frac {1}{2}, \frac {2}{1}),( \frac {1}{3}, \frac {2}{2}, \frac {3}{1}),( \frac {1}{4}, \frac {2}{3}, \frac {3}{2}, \frac {4}{1}),…,( \frac {1}{n}, \frac {2}{n-1}, \frac {3}{n-2},… \frac {n-1}{2}, \frac {n}{1})\\)
Denote this array as: \\((a_{1})\\), \\((a_{2},a_{3})\\), \\((a_{3},... | 7 |
19. A, B, C, D, E stand in a row, satisfying that A is not adjacent to B and C, and D is not adjacent to E. The number of different arrangements is ( ) kinds.
(A) 12
(B) 16
(C) 28
(D) 32
(E) 40 | C |
Let \\(O\\) be the origin of coordinates, and a moving point \\(N\\) lies on the circle \\(C\\): \\(x^{2}+y^{2}=8\\). A perpendicular line to the \\(y\\)-axis is drawn through \\(N\\), with the foot of the perpendicular being \\(M\\). Point \\(P\\) satisfies \\( \overrightarrow{MP}= \dfrac {1}{2} \overrightarrow{MN}\\)... | \text{B} |
In the arithmetic sequence $\{a_n\}$, it is known that $a_1=2$, and $a_3+a_5=10$. Then, $a_7$ equals ( )
A．5 B．6 C．8 D．10 | C |
3. Solve the system of equations $\left\{\begin{array}{l}x-y+z-w=2, \\ x^{2}-y^{2}+z^{2}-w^{2}=6, \\ x^{3}-y^{3}+z^{3}-w^{3}=20, \\ x^{4}-y^{4}+z^{4}-w^{4}=66 .\end{array}\right.$ | (1,0,3,2),(1,2,3,0),(3,0,1,2),(3,2,1,0) |
Calculate:
(1) $\left( \frac {1}{3} \right)^{-1} - \log_{2}8 + \left(0.5^{-2} - 2\right) \times \left( \frac {27}{8} \right)^{ \frac {2}{3}}$
(2) Given $\tan\alpha = -2$, compute the value of $\frac {\sin(\pi + \alpha) + 2\sin\left( \frac {\pi}{2} - \alpha \right)}{\sin(-\alpha ) + \cos(\pi - \alpha)}$. | 4 |
Given the inverse proportion function $y=\frac{{k-1}}{x}$ whose graph lies in the second and fourth quadrants, the range of $k$ is ______. | k < 1 |
Define two functions $g(x),\ f(x)\ (x\geq 0)$ by $g(x)=\int_0^x e^{-t^2}dt,\ f(x)=\int_0^1 \frac{e^{-(1+s^2)x}}{1+s^2}ds.$
Now we know that $f'(x)=-\int_0^1 e^{-(1+s^2)x}ds.$
(1) Find $f(0).$
(2) Show that $f(x)\leq \frac{\pi}{4}e^{-x}\ (x\geq 0).$
(3) Let $h(x)=\{g(\sqrt{x})\}^2$. Show that $f'(x)=-h'(x).$
(4) F... | \frac{\sqrt{\pi}}{2} |
10. If for all positive numbers $x, y$, the inequality $\sqrt{x}+\sqrt{y}$ $\leqslant a \sqrt{x+y}$ holds, then the minimum value of $a$ is $\qquad$ | \sqrt{2} |
In each square of the table below, we must write a different integer from $1$ to $17$, such that the sum of the numbers in each of the eight columns is the same, and the sum of the numbers in the top row is twice the sum of the numbers in the bottom row. Which number from $1$ to $17$ can be omitted?
[img]https://wiki... | 9 |
Given that $f'(x)$ is the derivative of the function $f(x)$, and it satisfies $xf'(x)-xf(x)-e^{x}=0$ in the domain $(0,+\infty)$, and $f(1)=2e$. If $f(1-\frac{1}{2a})≤e^{\frac{1}{e}}$, then the range of real number $a$ is ____. | \left(\dfrac{1}{2}, \dfrac{e}{2(e-1)} \right] |
Given the function $f(x)=x^{2}+a(b+1)x+a+b \ (a,b \in \mathbb{R})$, the statement "$a=0$" is the $(\ \ \ \ \ \ )$ for "$f(x)$ is an even function".
A: Sufficient but not necessary condition
B: Necessary but not sufficient condition
C: Necessary and sufficient condition
D: Neither necessary nor sufficient condition | A |
3.3. Calculate: a) $(1+i)^{11}$, b) $(1-i \sqrt{3})^{73}$. | 2^{72}-2^{72}\sqrt{3}i |
A polygon is cut to form another polygon, and the sum of the interior angles of the new polygon is 1620°. The number of sides of the original polygon is ( )
A: 10
B: 11
C: 12
D: All of the above are possible | \text{D} |
6. A man, standing on a lawn, is wearing a circular sombrero of radius 3 feet. Unfortunately, the hat blocks the sunlight so effectively that the grass directly under it dies instantly. If the man walks in a circle of radius 5 feet, what area of dead grass will result? | 60\pi\mathrm{}^{2} |
Triangle $ABC$ satisfies $AB=104$, $BC=112$, and $CA=120$. Let $\omega$ and $\omega_A$ denote the incircle and $A$-excircle of $\triangle ABC$, respectively. There exists a unique circle $\Omega$ passing through $A$ which is internally tangent to $\omega$ and externally tangent to $\omega_A$. Compute the radius of $... | 49 |
Problem 6. A row of new recruits stood facing the sergeant. On the command "left," some turned left, while the rest turned right. It turned out that six times more soldiers were looking at the back of their neighbor than in the face. Then, on the command "about face," everyone turned in the opposite direction. Now, sev... | 98 |
2.2. Find all values of $a$ for which the quadratic function $f(x)=a x^{2}-4 a x+1$ takes values, the modulus of which does not exceed 3, at all points of the interval $[0 ; 4]$. In your answer, specify the total length of the intervals to which the found values of $a$ belong. | 1.5 |
(4) If the acute angle $\alpha$ satisfies $\frac{1}{\sqrt{\tan \frac{\alpha}{2}}}=\sqrt{2 \sqrt{3}} \sqrt{\tan 10^{\circ}}+\sqrt{\tan \frac{\alpha}{2}}$, then the degree of angle $\alpha$ is $\qquad$ | 50 |
1. Insert “+” or “- -” between $1,2,3, \cdots, 1989$, what is the smallest non-negative number that the sum can achieve? | 1 |
Given vectors $\overrightarrow{a}=(x,1)$ and $\overrightarrow{b}=(9,x)$, if $\overrightarrow{a}$ and $\overrightarrow{b}$ are collinear and in opposite directions, then $x=$ ______. | -3 |
1. The speed of light is 300,000 kilometers per second, and the distance from the Sun to the Earth is 150 million kilometers. Question: How many minutes does it take for light to travel from the Sun to the Earth (round the answer to one decimal place)? | 8.3 |
Find all pairs of nonnegative integers $(x, y)$ for which $(xy + 2)^2 = x^2 + y^2 $. | (0, 2) \text{ and } (2, 0) |
Among candidates A, B, C, and D, (1) if one person is elected as the class president and another as the vice president, how many ways are there to elect them? List all possible election outcomes; (2) if three class committee members are to be elected, how many ways are there to elect them? List all possible election ou... | 4 |
3. Two circles with equal radii touch at point $C$ and touch externally a third circle with radius $r=5$ cm at points $A$ and $B$. Calculate the area of triangle $ABC$, if $\overline{AB}=6$ cm. | 18 |
50. There is a natural number, when it is divided by the 5 natural numbers $1,2,3,4,5$, the remainders are all different. The smallest natural number is $\qquad$ . | 59 |
Task 1. A group of boys and girls collected 1700 denars for a birthday gift. The girls collected 200 denars each, and the boys collected 300 denars each. Determine the number of boys and girls in the group, given that the group has an odd number of members. | 4 |
8.1. A natural number $n$ was multiplied by the sum of the digits of the number $3 n$, and the resulting number was then multiplied by 2. As a result, 2022 was obtained. Find $n$. | 337 |
4.018. The denominator of the geometric progression is $1 / 3$, the fourth term of this progression is $1 / 54$, and the sum of all its terms is 121/162. Find the number of terms in the progression. | 5 |
4. Given that $f$ is a mapping from the set $M=\{a, b, c\}$ to the set $N=\{-3,-2, \cdots, 3\}$. Then the number of mappings $f$ that satisfy
$$
f(a)+f(b)+f(c)=0
$$
is ( ).
(A) 19
(B) 30
(C) 31
(D) 37 | 37 |
Example 16. Given as shown, the area of $\triangle ABC$ is $S$, a line $l \parallel BC$ intersects $AB, AC$ at points $D, E$ respectively, and the area of $\triangle BED$ is denoted as $K$. Prove that $K \leqslant \frac{1}{4} S$.
(2nd Zu Chongzhi Cup Junior High School Mathematics Invitational) | K \leqslant \frac{1}{4} S |
Given the function $f(x) = \sqrt{3}\sin x \cos x - \cos^2 x + \frac{1}{2}$.
(1) Find the smallest positive period of $f(x)$ and the equation of the axis of symmetry.
(2) In $\triangle ABC$, let $a$, $b$, and $c$ be the lengths of the sides opposite to angles $A$, $B$, and $C$, respectively. If $f\left(\frac{A}{2}\rig... | \sqrt{6} |
Given that $A(-3, y_{1})$, $B(3, y_{2})$, and $C(4, y_{3})$ are three points on the parabola $y=2\left(x-2\right)^{2}+1$, arrange $y_{1}$, $y_{2}$, $y_{3}$ in ascending order.
A: $y_{1} < y_{2} < y_{3}$
B: $y_{2} < y_{1} < y_{3}$
C: $y_{3} < y_{2} < y_{1}$
D: $y_{2} < y_{3} < y_{1}$ | D |
In the Cartesian coordinate system $xOy$, the line $C_{1}$: $y=- \sqrt {3}x$, and the parametric equation of the curve $C_{2}$ is $\begin{cases}x=- \sqrt{3}+\cos \phi \\ y=-2+\sin \phi\end{cases}$ (where $\phi$ is the parameter), with the origin as the pole and the positive half-axis of $x$ as the polar axis, a polar c... | \sqrt {3} |
8. Let real numbers $x, y$ satisfy the conditions $\left\{\begin{array}{l}x-y+1 \geqslant 0, \\ y+1 \geqslant 0, \\ x+y+1 \leqslant 0,\end{array}\right.$ then the maximum value of $2 x-y$ is | 1 |
In a triangle, two angles $\beta$ and $\gamma$ and the radius $R$ of the circumscribed circle are given. Find the radius of the inscribed circle.
# | 4R\sin\beta/2\sin\gamma/2\cos(\beta+\gamma/2) |
8. Egor wrote a number on the board and encrypted it according to the rules of letter puzzles (different letters correspond to different digits, the same letters correspond to the same digits). The result was the word "GUATEMALA". How many different numbers could Egor have initially written if his number was divisible ... | 21600 |
17. A bag contains 15 balls, marked with the 15 numbers $2^{0}, 2^{1}, 2^{2}, \ldots, 2^{14}$ respectively. Each ball is either red or blue, and there is at least one ball of each colour. Let $a$ be the sum of the numbers on all red balls, $b$ be the sum of the numbers on all blue balls and $d$ be the H.C.F. of $a$ and... | 4681 |
Given $i$ as the imaginary unit, if $\frac{(3+i)(a+2i)}{1+i}$ is a real number, then the real number $a$ is $\left(\ \ \right)$
A: $-2$
B: $4$
C: $2$
D: $-4$ | B |
1. Find a polynomial of degree less than 4, such that it satisfies $f(0)=1, f(1)=1, f(2)=5, f(3)=11$. | -\frac{1}{3}(x^{3}-9x^{2}+8x-3) |
Given that the variable $x$ follows a normal distribution $N(4,\sigma^{2})$, and $P(x > 2)=0.6$, then $P(x > 6)=$ ()
A: $0.4$
B: $0.3$
C: $0.2$
D: $0.1$ | A |
For the correlation coefficient $r$ between two variables, consider the following statements: (1) The larger $|r|$, the greater the degree of correlation; (2) The smaller $|r|$, the greater the degree of correlation; (3) When $|r|$ is close to 0, there is no non-linear correlation; (4) The closer $|r|$ is to 1, the str... | (1), (4) |
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