problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
|---|---|
Given the equation in terms of $x$: $ax^{2}-2x+1=0$, in order for the equation to have two distinct real roots, the possible values for $a$ are:
A: $0$
B: $-1$
C: $1$
D: $2$ | B |
6. The brakes of a car allow it to stand on an inclined asphalt surface with an angle at the base of no more than $30^{\circ}$. Determine the minimum braking distance of this car when moving at a speed of $30 \, \text{m/s}$ on a flat horizontal road with the same surface. The acceleration due to gravity $g=10 \, \text{... | 78\, |
A group consists of 3 boys and 2 girls. If 2 students are randomly selected to participate in a speech contest, what is the probability that at most one girl participates?
A: $ \dfrac {1}{10}$
B: $ \dfrac {3}{10}$
C: $ \dfrac {3}{5}$
D: $ \dfrac {9}{10}$ | \text{D} |
Given the set $A=\{x\in \mathbb{R} | x^2-x=0\}$, which of the following statements is correct? ( )
A: $1 \subseteq A$
B: $\{0\} \in A$
C: $\emptyset \subseteq A$
D: $\phi \in A$ | C |
In the expansion of ${({x-\frac{2}{x}})^8}$, the coefficient of $x^{2}$ is:
A: $-C_8^3×{2^3}$
B: $C_8^3×{2^3}$
C: $-C_8^3$
D: $C_8^3$ | \text{A} |
Given a geometric sequence $\{a_n\}$ with a common ratio $q > 1$, and it satisfies $a_2+a_3+a_4=28$, and $a_3+2$ is the arithmetic mean of $a_2$ and $a_4$.
$(1)$ Find the general formula for the sequence $\{a_n\}$;
$(2)$ If $b_n=\log_{\sqrt{2}}{a_n}-3$, $S_n=b_1+b_2+\ldots+b_n$, find the maximum positive integer $n$ ... | 12 |
## Problem Statement
Derive the equations of the tangent and normal to the curve at the point corresponding to the parameter value $t=t_{0}$.
\[
\left\{\begin{array}{l}
x=\frac{3 a t}{1+t^{2}} \\
y=\frac{3 a t^{2}}{1+t^{2}}
\end{array}\right.
\]
$t_{0}=2$ | -\frac{4}{3}\cdotx-4 |
In right triangle $ABC$, $\angle C=90^\circ$, and $a$, $b$, $c$ are the sides opposite to $\angle A$, $\angle B$, $\angle C$ respectively. $a$ and $b$ are the roots of the equation $x^2 - 7x + c + 7 = 0$ related to $x$. The length of the median to the side $AB$ is ( )
A: $\frac{3}{2}$
B: $\frac{5}{2}$
C: $5$
D: $2... | B |
\section*{Problem 3 - 191013}
Determine all real numbers \(x\) with \(-1 \leq x \leq 1\) for which the expression \(x^{2}+3 x+4\) is the square of a natural number! | 0 |
Three, (18 points) Solve the system of equations:
$$
\left\{\begin{array}{l}
2(x+y+z)-5 \sqrt{x+y+z+5}=2, \\
\frac{x}{3}=\frac{y}{4}=\frac{z}{5} .
\end{array}\right.
$$ | x=\frac{11}{4}, y=\frac{11}{3}, z=\frac{55}{12} |
Proizvolov V.V.
There are 19 weights of $1, 2, 3, \ldots, 19$ grams: nine iron, nine bronze, and one gold. It is known that the total weight of all iron weights is 90 grams more than the total weight of the bronze weights. Find the weight of the gold weight. | 10 |
(6) Given the universal set $U=\{x \mid x$ is a prime number not greater than 20 $\}$. If $A \cap \complement_{U} B=\{3,5\}, \complement_{U} A \cap$ $B=\{7,19\}, \complement_{U}(A \cup B)=\{2,17\}$, then $A=$ $\qquad$ , $B=$ $\qquad$ . | {3,5,11,13},{7,19,11,13} |
8.25 Three people, A, B, and C, are playing a game: There are three cards, each with a positive integer $p$, $q$, $r$ written on it, and $p<q<r$. After shuffling, the cards are distributed to the three people, one each, and marbles are given according to the number on each person's card. Then the cards are collected an... | C |
8. Let $a_{n}$ be the coefficient of the $x$ term in the expansion of $(3-\sqrt{x})^{n}$ $(n=2,3,4, \cdots)$. Then $\lim _{n \rightarrow \infty}\left(\frac{3^{2}}{a_{2}}+\frac{3^{3}}{a_{3}}+\cdots+\frac{3^{n}}{a_{n}}\right)=$ $\qquad$ . | 18 |
9.7. A green chameleon always tells the truth, while a brown chameleon lies and immediately turns green after lying. In a company of 2019 chameleons (green and brown), each in turn answered the question of how many of them are currently green. The answers were the numbers $1,2,3, \ldots, 2019$ (in some order, not neces... | 1010 |
A certain three-digit number, when the first and third digits are swapped, increases by 198; when the second and third digits are swapped, it increases by 9; and the sum of the squares of the digits is 2 more than four times their sum. Which number is this $?$ | 345 |
Calculate:$(1)3\sqrt{3}-\sqrt{8}+\sqrt{2}-\sqrt{27}$;$(2)({\sqrt{5}-\sqrt{3}})({\sqrt{5}+\sqrt{3}})$. | 2 |
6. The inequality $(k-1) x+2021 \geq 0$ about $x$ has exactly three positive integer solutions. Then the maximum value of the integer $k$ is ( ).
A. -672
B. -505
C. -504
D. 504
E. 505 | -505 |
If the function $f\left(x\right)=x^{2}+\left(m-1\right)x$ is an even function, then $m=\_\_\_\_\_\_.$ | 1 |
Given $a = \log_{2}3.6$, $b = \log_{4}3.2$, $c = \log_{4}3.6$, then ( )
(A) $a > b > c$ (B) $a > c > b$ (C) $b > a > c$ (D) $c > a > b$ | \text{B} |
Let $S=\{1,2,3,\ldots,280\}$. Find the smallest integer $n$ such that each $n$-element subset of $S$ contains five numbers which are pairwise relatively prime. | 217 |
Starting with some gold coins and some empty treasure chests, I tried to put $9$ gold coins in each treasure chest, but that left $2$ treasure chests empty. So instead I put $6$ gold coins in each treasure chest, but then I had $3$ gold coins left over. How many gold coins did I have?
$\textbf{(A) }9\qquad\textbf{(B)... | 45 |
3. As shown in Figure 1, in Tamara's garden, there are 3 rows and 2 columns of rectangular flower beds, totaling six beds, each 6 feet long and 2 feet wide. These flower beds are separated by 1-foot-wide walkways, and there is also a 1-foot-wide walkway around the perimeter of the garden. What is the total area $S$ of ... | B |
Let $A=\frac{2^2+3\cdot 2 + 1}{3! \cdot 4!} + \frac{3^2+3\cdot 3 + 1}{4! \cdot 5!} + \frac{4^2+3\cdot 4 + 1}{5! \cdot 6!} + \dots + \frac{10^2+3\cdot 10 + 1}{11! \cdot 12!}$. What is the remainder when $11!\cdot 12! \cdot A$ is divided by $11$?
$
\textbf{(A)}\ 0
\qquad\textbf{(B)}\ 1
\qquad\textbf{(C)}\ 5
\qquad\text... | 10 |
Problem 2. Consider an acute triangle $A B C$ with area S. Let $C D \perp A B \quad(D \in A B)$, $D M \perp A C \quad(M \in A C)$ and $\quad D N \perp B C \quad(N \in B C)$. Denote by $H_{1}$ and $H_{2}$ the orthocentres of the triangles $M N C$ and $M N D$ respectively. Find the area of the quadrilateral $\mathrm{AH}... | S |
1. Given real numbers $x, y$ satisfy $3 x^{2}+4 y^{2}=1$. If $|3 x+4 y-\lambda|+|\lambda+7-3 x-4 y|$ is independent of $x, y$, then the range of values for $\lambda$ is | [\sqrt{7}-7,-\sqrt{7}] |
6. In a Cartesian coordinate system, a point whose both vertical and horizontal coordinates are integers is called an integer point. The number of integer points $(x, y)$ that satisfy the inequality $(|x|-1)^{2}+(|y|-1)^{2}<2$ is $\qquad$ . | 16 |
Given the sets $\{a_1\}$, $\{a_1, a_2\}$, $\{a_1, a_2, a_3\}$, ..., the number of subsets for the set $\{a_1, a_2, a_3, \ldots, a_n\}$ is ( )
A: $n$
B: $n+1$
C: $2^n$
D: $2^n - 1$ | C |
3. The triangle $A B C$ is isosceles with $A B=B C$. The point $D$ is a point on $B C$, between $B$ and $C$, so that $A C=A D=B D$.
What is the size of angle $A B C$ ? | 36 |
[ [Decimal numeral system ] [ Problems with constraints ]
Find the number of five-digit numbers in decimal notation that contain at least one digit 8. | 37512 |
Find all pairs $(a, b)$ of positive integers such that $a \le b$ and
$$ \gcd(x, a) \gcd(x, b) = \gcd(x, 20) \gcd(x, 22) $$
holds for every positive integer $x$. | (2, 220), (4, 110), (10, 44), (20, 22) |
Given that for all $x \in \mathbb{R}$, it holds that $f(x) = f(2-x)$, and the equation $f(x) = 0$ has five distinct real roots, find the sum of these five roots. | 5 |
15. $y=\frac{1}{x}$. | (-\infty,0)\cup(0,\infty) |
Example 10 Find the integer solution of $1215 x \equiv 560(\bmod 2755)$. | x \equiv 200,751,1302,1853,2404 \quad(\bmod 2755) |
## Task B-3.5.
A regular right quadrilateral pyramid with apex $V$ and base $ABCD$ is inscribed in a sphere of radius $R$. Let $\cos \angle AVB = \frac{3}{4}$. What is the height of the pyramid expressed in terms of $R$? | \frac{3}{2}R |
A high school has a total of 1000 students, among which there are 380 students in the first year, and 180 male students in the second year. If one student is randomly selected from all the students, the probability of selecting a female student from the second year is 0.19. Now, using stratified sampling (by grade) to ... | 25 |
6. In $\triangle A B C$, it is known that $\angle C=90^{\circ}, \angle B=30^{\circ}$, $A C=2$, $M$ is the midpoint of $A B$, and $\triangle A C M$ is folded along $C M$ such that the distance between points $A$ and $B$ is $2 \sqrt{2}$. Then the volume $V$ of the tetrahedron $A-B C M$ is ( ).
(A) $\frac{\sqrt{2}}{3}$
(B... | D |
3. Let $F_{1}$ and $F_{2}$ be the left and right foci of the ellipse $C$:
$$
\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)
$$
If there exists a point $P$ on the ellipse $C$ such that $P F_{1} \perp P F_{2}$, then the range of the eccentricity $e$ of the ellipse is $\qquad$ | \left[\frac{\sqrt{2}}{2}, 1\right) |
Which of the following options does not reflect the same type of correlation between two variables as the other three?
( )
A: A great teacher produces outstanding students
B: A rising tide lifts all boats
C: The brighter the moon, the fewer the stars
D: Climbing high to see far | \text{C} |
Find the difference between the maximum and minimum distance from a point on the circle $x^2 + y^2 - 4x - 4y + 5 = 0$ to the line $x + y - 9 = 0$. | 2\sqrt{3} |
4. The directrix of the parabola $y=-\frac{1}{8} x^{2}$ intersects the $y$-axis at point $A$. A line through $A$ intersects the parabola at points $M$ and $N$. Point $B$ is on the axis of symmetry of the parabola, and $\left(\overrightarrow{B M}+\frac{\overrightarrow{M N}}{2}\right) \perp \overrightarrow{M N}$. Then th... | D |
In the arithmetic sequence $\{a_n\}$, $a_3=4$, and $d=-2$, then $a_n=$ ___. | 10-2n |
Given that $f(x)$ is an odd function, and when $x > 0$, $f(x) = x + 1$, then the value of $f(-1)$ is. | -2 |
17. Let $x_{0}=5, x_{n+1}=x_{n}+\frac{1}{x_{n}}(n=0,1,2, \cdots)$, prove: $45<x_{1000}<45.1$. | 45<x_{1000}<45.1 |
7.1. Halfway between Vasya's house and school stands a traffic light. On Monday, Vasya caught the green light. On Tuesday, he walked at the same speed but waited at the traffic light for 5 minutes, and then doubled his speed. On both Monday and Tuesday, he spent the same amount of time on the journey from home to schoo... | 20 |
5. (2004 National College Entrance Examination - Tianjin Paper) The interval of the function $y=2 \sin \left(\frac{\pi}{6}-2 x\right)(x \in[0, \pi]$ where it is monotonically increasing is
A. $\left[0, \frac{\pi}{3}\right]$
B. $\left[\frac{\pi}{12}, \frac{5 \pi}{12}\right]$
C. $\left[\frac{\pi}{3}, \frac{5 \pi}{6}\righ... | C |
## [ The larger side is opposite the larger angle ]
[Sum of the angles of a triangle. Theorem about the exterior angle.] In triangle $ABC$, it is known that $AB < BC < AC$, and one of the angles is twice as small as another and three times as small as the third. Find the angle at vertex $A$. | 60 |
16. As shown in Figure 2, two semicircles with centers at $A$ and $B$ and radii of 2 and 1, respectively, are internally tangent to the semicircle with diameter $JK$, and the two smaller semicircles are also tangent to each other. $\odot P$ is externally tangent to both smaller semicircles and internally tangent to the... | B |
In the ancient Chinese book "The Mathematical Classic of Sunzi", there is a problem about "multiple people sharing a carriage", which is described as follows: "There are three people sharing a carriage, with two carriages empty; when two people share a carriage, nine people have to walk. The question is to find out the... | D |
1. 2. A car with a load traveled from one city to another at a speed of 60 km/h, and returned empty at a speed of 90 km/h. Find the average speed of the car for the entire route. Give your answer in kilometers per hour, rounding to the nearest whole number if necessary.
$\{72\}$ | 72 |
Given a sequence $\{a_n\}$ with the sum of the first n terms being $S_n$, it satisfies the equation $S_n = 2a_n - n$.
(1) Find the general formula for the $n$-th term of the sequence $\{a_n\}$.
(2) Let $b_n = (2n+1)(a_n + 1)$, find the sum of the first n terms, $T_n$, of the sequence $\{b_n\}$. | T_n = 2 + (2n-1) \cdot 2^{n+1} |
9. When $\theta$ varies, the equation of the locus of the vertex of the parabola $y=x^{2}+2 x \sin \theta+1$ is | -x^{2}+1,-1\leqslantx\leqslant1 |
27. There are 30 arithmetic expressions:
$$
134+7,134+14,134+21, \cdots \cdots, 134+210 \text {. }
$$
Each expression results in a three-digit number, and the sum of the hundreds digits of these three-digit numbers is $\qquad$ | 58 |
Given that $x=2$ is a root of the quadratic equation $x^{2}+mx+2=0$, the value of $m$ is ____. | -3 |
Calculate the result of $(\sqrt{5}-2)^{2023}(\sqrt{5}+2)^{2023}$. | 1 |
The calculation formula for the earthquake magnitude $M$ (Richter magnitude) is $M= \lg A- \lg A_{0}$ (where $A$ is the maximum amplitude of the measured earthquake, and the constant $A_{0}$ is the amplitude of the "standard earthquake"). The seismic feeling of a 5-magnitude earthquake is already quite noticeable. On M... | 1000 |
11.2. How much greater is one of two positive numbers than the other if their arithmetic mean is $2 \sqrt{3}$ and the geometric mean is $\sqrt{3}$?
(Hint: the geometric mean of two numbers $m$ and $n$ is the number $p=\sqrt{\mathrm{mn}}$). | 6 |
$a=0$ is the condition for the complex number $z=a+bi$ ($a, b \in \mathbb{R}$) to be a purely imaginary number ( ).
A: Sufficient but not necessary condition
B: Necessary but not sufficient condition
C: Necessary and sufficient condition
D: Neither sufficient nor necessary condition | \text{B} |
15. (6 points) There are three numbers $a, b, c$, $a \times b=24, a \times c=36, b \times c=54$, then $a+b+c=$ | 19 |
## Task Condition
Find the derivative.
$$
y=\frac{2\left(\sqrt{2^{x}-1}-\operatorname{arctg} \sqrt{2^{x}-1}\right)}{\ln 2}
$$ | \sqrt{2^{x}-1} |
In triangle $ABC$, the sides opposite to angles $A$, $B$, $C$ are denoted as $a$, $b$, $c$ respectively. It is given that $b=a\cos C+\frac{\sqrt{3}}{3}a\sin C$.
(Ⅰ) Find angle $A$;
(Ⅱ) If $a=\sqrt{7}$ and $bc=6$, find the perimeter of $\triangle ABC$. | 5+\sqrt{7} |
In the plane rectangular coordinate system $Oxy$, $A$ is a point on the line $l: y=2x$ in the first quadrant, $B(5,0)$, and the circle $C$ with diameter $AB$ intersects the line $l$ at another point $D$. If $\overrightarrow{AB} \cdot \overrightarrow{CD} = 0$, then the abscissa of point $A$ is ( ).
A: $3$
B: $-1$
C: ... | A |
Given $a+b=7$ and $ab=6$, find:① the value of $a^{2}+b^{2}$;② the value of $a^{3}b-2a^{2}b^{2}+ab^{3}$. | 150 |
Given $a=0.3^3$, $b=3^3$, $c=\log_{3}0.3$, the relationship among them is ( )
A: $a > b > c$
B: $c > b > a$
C: $b > c > a$
D: $b > a > c$ | \text{D} |
Given that $a$ is a real number, $z\_1=2-i$, and $z\_2=a+i$ ($i$ is the imaginary unit),
(1) If $a=1$, indicate the quadrant in which the point corresponding to $z\_1+ \overline {z\_2}$ in the complex plane is located;
(2) If $z\_1 \cdot z\_2$ is a purely imaginary number, find the value of $a$. | a=-\frac{1}{2} |
Given the function $f(x) = |x + a| + |x - 2|.$
(1) When $a = -4$, find the solution set of the inequality $f(x) \geq 6$;
(2) If the solution set of $f(x) \leq |x - 3|$ contains $[0, 1]$, find the range of the real number $a$. | -1 \leq a \leq 0 |
## Task 4
Pair the following numbers so that they always add up to the same sum.
$\begin{array}{llllllll}43 & 202 & 100 & 175 & 98 & 257 & 125 & 200\end{array}$ | 300 |
2. If the three sides of $\triangle A B C$ are $\sqrt{2}, \sqrt{3}, \sqrt{5}$, then its inradius $r=$ | \frac{\sqrt{2}+\sqrt{3}-\sqrt{5}}{2} |
(Full score for this problem is 12 points) Given $f(x) = e^x - ax - 1$.
(1) Find the intervals where $f(x)$ is monotonically increasing.
(2) If $f(x)$ is monotonically increasing on the domain $\mathbb{R}$, find the range of possible values for $a$.
(3) Does there exist a value of $a$ such that $f(x)$ is monotonically... | a = 1 |
3. A project needs 100 days to complete. Initially, 10 people worked for 30 days and completed $\frac{1}{5}$ of the entire project. Then, 10 more people were added to complete the project. How many days earlier can the task be completed? | 10 |
Example 1.15. Find $\int \frac{d x}{(x+3)^{1 / 2}+(x+3)^{2 / 3}}$. | 3\sqrt[3]{x+3}-6\sqrt[6]{x+3}+6\ln|\sqrt[6]{x+3}+1|+C |
## Problem Statement
Calculate the indefinite integral:
$$
\int \frac{3 x^{3}+1}{x^{2}-1} d x
$$ | \frac{3x^{2}}{2}+2\ln|x-1|+\ln|x+1|+C |
4. Solve the equation $20 x^{2}+17 y=2017$ in natural numbers. | (10;1);(7;61) |
3. The graph of the function $y=-x^{2}+p x+q$ intersects the $x$-axis at points $(a, 0),(b, 0)$. If $a>1>b$, then ( ).
(A) $p+q>1$
(B) $p-r q=1$
(C) $p+q<1$
(D) $f q$ ) $=0$ | A |
Let $f(x)$ be a monotonically decreasing odd function defined on $\mathbb{R}$. If $x_1+x_2>0$, $x_2+x_3>0$, and $x_3+x_1>0$, then ( )
A: $f(x_1)+f(x_2)+f(x_3)>0$
B: $f(x_1)+f(x_2)+f(x_3)f(x_3)$ | \text{B} |
Example 7 As shown in Figure 6, in rectangle $ABCD$, it is known that the diagonal length is 2, and
$$
\begin{array}{l}
\angle 1=\angle 2= \\
\angle 3=\angle 4 .
\end{array}
$$
Then the perimeter of quadrilateral
$\mathrm{EFGH}$
is ( ).
(A) $2 \sqrt{2}$
(B) 4
(C) $4 \sqrt{2}$
(D) 6
(2010, Sichuan Province Junior High ... | B |
Let $(x_n)_{n\ge 0}$ a sequence of real numbers defined by $x_0>0$ and $x_{n+1}=x_n+\frac{1}{\sqrt{x_n}}$. Compute $\lim_{n\to \infty}x_n$ and $\lim_{n\to \infty} \frac{x_n^3}{n^2}$. | \frac{9}{4} |
Arrange the sequence $\{2n+1\} (n\in\mathbb{N}^{*})$ sequentially in brackets with one number in the first bracket, two numbers in the second bracket, three numbers in the third bracket, four numbers in the fourth bracket, one number in the fifth bracket, and so on in a cycle: $(3)$, $(5,7)$, $(9,11,13)$, $(15,17,19,21... | B |
Which of the following functions is increasing in $\left(0,+\infty \right)$?
A: $y=\cos x$
B: $y=x^{3}-x$
C: $y=x\ln x$
D: $y=xe^{x}$ | D |
Calculate: $(-2m^{3})^{2}$ is equal to ( ).
A: $-2m^{6}$
B: $4m^{6}$
C: $4m^{5}$
D: $-4m^{6}$ | B |
4. On a machine that operates with tokens of 1, 10, and 25 kuna, you can play this game: if you insert a 1 kuna token, you get one 10 kuna token; if you insert a 10 kuna token, you get one 1 kuna token and one 25 kuna token; if you insert a 25 kuna token, you get two 10 kuna tokens. At the beginning, you had one 10 kun... | 1110 |
1. Given the complex number $z$ satisfies $\frac{z-\mathrm{i}}{z-1}$ is a pure imaginary number (where $\mathrm{i}$ is the imaginary unit), then the maximum value of $|z|$ is | \sqrt{2} |
A $1.4 \mathrm{~m}$ long rod has $3 \mathrm{~kg}$ masses at its ends. Where should the rod be pivoted so that, when released from a horizontal position, the mass on the left side passes under the pivot at a speed of $1.6 \mathrm{~m} / \mathrm{s}$? | 0.8 |
Find the equation of the circle with center at the point (2, -1) that is tangent to the line $x+y=7$. | (x-2)^2+(y+1)^2=18 |
The coefficient of the linear term in the quadratic equation $x^{2}-x=0$ is ____. | -1 |
6.90. What is the maximum number of acute angles a convex polygon can have
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly. | 3 |
Given the function $f(x) = 2\ln(3x) + 8x$, the value of $$\lim_{\Delta x \to 0} \frac{f(1 + \Delta x) - f(1)}{\Delta x}$$ is ( )
A: -20
B: -10
C: 10
D: 20 | C |
If three different numbers are selected from the numbers $1$, $2$, $3$, and $4$, the probability that the sum of the three selected numbers is divisible by $6$ is:
A: $\frac{1}{2}$
B: $\frac{1}{5}$
C: $\frac{1}{4}$
D: $\frac{2}{5}$ | C |
A4. The solution to the equation $2 \cdot 3^{x-3}+3^{x-2}+3^{x-1}=14 \cdot 3^{5}$ is
(A) $x=2$
(B) $x=8$
(C) $x=-8$
(D) $x=-2$
(E) $x=0$ | 8 |
A class has 50 students, and their scores $\xi$ ($\xi \in \mathbb{N}$) in a test follow a normal distribution $N(100, 10^2)$. It is known that $P(90 \leq \xi \leq 100) = 0.3$. Estimate the number of students in this class whose math scores are above 110. | 10 |
Given the function $f(x)=e^{x}-ax$ (where $e$ is the base of the natural logarithm).
(Ⅰ) Discuss the monotonicity of $f(x)$;
(Ⅱ) If for any real number $x$, it always holds that $f(x) \geqslant 0$, find the range of the real number $a$. | [0,e] |
1. (20 points) On the board, all two-digit numbers divisible by 5, where the number of tens is greater than the number of units, were written down. There turned out to be $A$ such numbers. Then, all two-digit numbers divisible by 5, where the number of tens is less than the number of units, were written down. There tur... | 413 |
2. Given that $H$ is the orthocenter of $\triangle A B C$ and $B H=\sqrt{3} A C$. Then the degree measure of $\angle A B C$ is $\qquad$ . | 30^{\circ} \text{ or } 150^{\circ} |
## Task 1 - 200621
An airplane flying from A to B at a constant speed was still $2100 \mathrm{~km}$ away from B at 10:05 AM, and only $600 \mathrm{~km}$ away at 11:20 AM.
At what time will it arrive in B if it continues to fly at the same speed? | 11:50 |
11. Given the function defined on the set of non-zero natural numbers
$$
f(n)=\left\{\begin{array}{ll}
n+2, & n \leqslant 2005 ; \\
f(f(n-4)), & n>2005 .
\end{array}\right.
$$
then when $n \leqslant 2005$, $n-f(n)=$ $\qquad$
when $2005<n \leqslant 2007$, $n-f(n)=$
$\qquad$ | -2,0 |
Example 5. Investigate the convergence of the series $\sum_{n=1}^{\infty} \frac{x^{2 n-1}}{\left(n^{2}+1\right) \cdot 3^{n}}$. | -\sqrt{3}\leqslantx\leqslant\sqrt{3} |
# 8. Variant 1.
101 natural numbers are written in a circle. It is known that among any 5 consecutive numbers, there will be at least two even numbers. What is the minimum number of even numbers that can be among the written numbers? | 41 |
9.027. $(x+1)(3-x)(x-2)^{2}>0$. | x\in(-1;2)\cup(2;3) |
9,10
In a chess tournament, 12 people participated. After the tournament, each participant made 12 lists. The first list includes only themselves, the second list includes them and those they defeated, the third list includes everyone from the second list and those they defeated, and so on. The 12th list includes ever... | 54 |
There are 6 different books to be divided among three individuals, A, B, and C, with each person receiving two books. The number of distinct ways to distribute the books is \_\_\_\_\_\_. | 90 |
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