problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
|---|---|
8.3. In triangle $A B C$, the bisector $A M$ and the median $B N$ intersect at point $O$. It turns out that the areas of triangles $A B M$ and $M N C$ are equal. Find $\angle M O N$. | 90 |
Let Sn be the sum of the first n terms of a geometric sequence {an}. Given that a1, a2, and S2 form an arithmetic sequence, and S3 = 42.
(1) Find the general term formula an for the sequence {an};
(2) Let $$b_{n}= \frac {na_{n}}{2^{n}}$$. If $$c_{n}= \frac {1}{b_{n}b_{n+1}}$$, find the sum of the first n terms Tn for t... | \frac {n}{9(n+1)} |
Three, (16 points) Divide each side of an equilateral triangle with side length $a$ into 1993 equal parts, then connect the corresponding points on each side with parallel lines, resulting in several smaller equilateral triangles. Find the sum of the areas of the incircles of all the smaller equilateral triangles.
| \frac{\pi a^{2}}{12} |
Given that the function f(x) satisfies the conditions: f(2-x) + f(x) = 0 and f(x+2) + f(x) = 0 for any real number x, and f(x) = -(x-1)^2 when x ∈ [0, 1], find f(x) when x ∈ [2017, 2018]. | (2017 - x)^2 |
Given that the points (3, 1) and (-4, 6) are on opposite sides of the line $3x-2y+a=0$, the range of values for $a$ is ( )
A: $a24$
B: $a=7$ or $a=24$
C: $-7<a<24$
D: $-24<a<7$ | C |
Find the equation of the tangent line to the curve $y = x + \ln x$ at the point $(1, f(1))$. | y = 2x - 1 |
4. In the cube $A B C D-A_{1} B_{1} C_{1} D_{1}$, there is a moving point $M$ on the base $A_{1} B_{1} C_{1} D_{1}$, and $B M / /$ plane $A D_{1} C$. Then the maximum value of $\tan \angle D_{1} M D$ is $\qquad$ . | \sqrt{2} |
In the polar coordinate system, the curve $\rho=4\sin (\theta- \dfrac {\pi}{3})$ is symmetric about
A: The line $\theta= \dfrac {\pi}{3}$
B: The line $\theta= \dfrac {5\pi}{6}$
C: The point $(2, \dfrac {\pi}{3})$
D: The pole | B |
Define an odd function on $(-\infty, +\infty)$, such that when $x > 0$, $f(x) = x^2 + 1$. Find $f(-2) =$ ___. | -5 |
Given a geometric sequence $\{a_n\}$ with the sum of the first $n$ terms denoted as $S_n$, and given that $a_1 + a_3 = \frac{5}{2}$ and $a_2 + a_4 = \frac{5}{4}$, find the value of $\frac{S_5}{a_5}$. | 31 |
In a bag, there are 2 red balls and 3 black balls, all of the same size. Two balls are drawn from the bag without replacement. Let event $A$ be "the first ball drawn is red," and event $B$ be "the two balls drawn are of the same color." Then, the probability $P(B|A)$ is ( ).
A: $\dfrac{1}{4}$
B: $\dfrac{1}{2}$
C: $\... | A |
5. Form an $n$-digit number using the digits $1, 2, 3$, requiring that each of $1, 2, 3$ appears at least once in the $n$-digit number. The total number of such $n$-digit numbers is $\qquad$ . | 3^{n}-3\cdot2^{n}+3 |
What is the maximum area of a triangle that can be inscribed in an ellipse with semi-axes $a$ and $b$?
$\text{(A) }ab\frac{3\sqrt{3}}{4}\qquad\text{(B) }ab\qquad\text{(C) }ab\sqrt{2}\qquad\text{(D) }\left(a+b\right)\frac{3\sqrt{3}}{4}\qquad\text{(E) }\left(a+b\right)\sqrt{2}$ | ab \frac{3\sqrt{3}}{4} |
34. In a triangular prism $A B C A_{1} B_{1} C_{1}{ }^{*}$, two planes are drawn: one passes through the vertices $A, B$ and $C_{1}$, and the other through the vertices $A_{1}, B_{1}$ and $C$. These planes divide the prism into four parts. The volume of the smallest of these parts is $V$. Find the volume of the prism. | 12V |
10. Let the sequence $\left\{a_{n}\right\}$ have the sum of the first $n$ terms $S_{n}$ satisfying: $S_{n}+a_{n}=\frac{n-1}{n(n+1)}, n=1,2, \cdots$, then the general term $a_{n}=$ . $\qquad$ | a_{n}=(\frac{1}{2})^{n}-\frac{1}{n(n+1)} |
A quadrilateral $ABCD$ has a right angle at $\angle ABC$ and satisfies $AB = 12$, $BC = 9$, $CD = 20$, and $DA = 25$. Determine $BD^2$.
.
| 769 |
Given the function $f(x) = \begin{cases} x-\sin(x), & x<0 \\ x^{3}+1, & x\geq0\end{cases}$, which of the following statements is correct? ( )
A: $f(x)$ has an extremum.
B: $f(x)$ has a zero point.
C: $f(x)$ is an odd function.
D: $f(x)$ is an increasing function. | \text{D} |
Svyatov M.
On a plane, grasshopper Kolya and 2020 of his friends are sitting. Kolya is going to jump over each of the other grasshoppers (in any order) such that the starting and ending points of each jump are symmetric relative to the grasshopper being jumped over. We will call a point a finishing point if Kolya can ... | C_{2020}^{1010} |
Given vectors $\vec{a}=(\cos x,\sin x)$, $\vec{b}=(-\cos x,\cos x)$, $\vec{c}=(-1,0)$
$(1)$ If $x= \frac{\pi}{6}$, find the angle between vectors $\vec{a}$ and $\vec{c}$;
$(2)$ When $x\in\left[ \frac{\pi}{2}, \frac{9\pi}{8}\right]$, find the minimum value of the function $f(x)=2\vec{a}\cdot \vec{b}+1$. | - \sqrt{2} |
4. Let $m$ be a natural number that has 7 positive divisors, and $n$ be a natural number that has 9 positive divisors. How many divisors can the product $m \cdot n$ have? | 15,27,63 |
B2. On the circle $\mathcal{K}$, points $A$ and $B$ are such that the length of the arc $\widehat{A B}$ is equal to $\frac{1}{3}$ of the circumference of the circle $\mathcal{K}$. Point $C$ lies on the arc $\widehat{A B}$, and point $D$ lies on the chord $A B$. Calculate the area of triangle $A C B$, given that $|A D|=... | \frac{3\sqrt{2}}{4} |
The pages of a notebook are numbered consecutively such that the first sheet has the numbers 1 and 2, the second sheet has the numbers 3 and 4, and so on. A sheet is torn out of this notebook. The page numbers of the remaining sheets are added up, resulting in a sum of 2021.
(a) How many pages could the notebook origi... | Thenotebookhad64pagesthesheetwiththepage2930wastornout |
Among the following expressions, the one that is a fraction is:
A: $3x+\frac{1}{2}$
B: $-\frac{{m+n}}{3}$
C: $\frac{3}{{x+3}}$
D: $x-1$ | C |
Given that the edge length of a cube is $2$, and its $8$ vertices are all on the surface of a sphere, then the volume of the sphere is ______________. | 4 \sqrt{3}\pi |
4.40 Find $\operatorname{ctg} 2 \alpha$, if it is known that $\sin \left(\alpha-90^{\circ}\right)=-\frac{2}{3}$ and $270^{\circ}<\alpha<360^{\circ}$. | \frac{\sqrt{5}}{20} |
Which of the following conditions cannot determine that $\triangle ABC$ is a right triangle?
A: $\angle A:\angle B:\angle C=3:4:5$
B: $AB:BC:AC=3:4:5$
C: $\angle A+\angle B=\angle C$
D: $AB^{2}=BC^{2}+AC^{2}$ | A |
Given a sequence $\{a_n\}$, the sum of the first $n$ terms is $S_n$, $a_1=1$, $a_{n+1}=3S_n+2$, then $a_4=$ （ ）
A: 64
B: 80
C: 256
D: 320 | B |
The number of intersection points between the graph of the function $y=f(x)$ and the y-axis is ( )
A: One
B: At least one
C: At most two
D: At most one | \text{D} |
57 Let $f(x)=2^{x}, g(x)=\log _{\sqrt{2}} 8 x$, then the value of $x$ that satisfies $f[g(x)]=g[f(x)]$ is
Let's translate the problem and solution step by step.
### Problem:
Let \( f(x) = 2^x \) and \( g(x) = \log_{\sqrt{2}} (8x) \). Find the value of \( x \) that satisfies \( f[g(x)] = g[f(x)] \).
### Solution:
1. ... | \frac{1+\sqrt{385}}{64} |
Poker is played with 52 cards. In poker, a full house consists of 3 cards of one rank and 2 cards of another rank. How many possible full houses are there? | 3744 |
The area of a rectangle is $12a^{2}-6ab$. If one side is $3a$, then the other side is ______. | 4a-2b |
Let $n \geq 2$ be an integer. Find the maximal cardinality of a set $M$ of pairs $(j, k)$ of integers, $1 \leq j < k \leq n$, with the following property: If $(j, k) \in M$, then $(k,m) \not \in M$ for any $m.$ | \left\lfloor \frac{n^2}{4} \right\rfloor |
Example 3 Let $m$ be an integer, and $4<m<40$, and the equation $x^{2}-2(2 m-3) x+4 m^{2}-14 m+8=0$ has two integer roots. Find the value of $m$ and the roots of the equation. | m=12, x_{1}=26, x_{2}=16 \text{ or } m=24, x_{1}=52, x_{2}=38 |
Lines $L_1,L_2,\dots,L_{100}$ are distinct. All lines $L_{4n}, n$ a positive integer, are parallel to each other.
All lines $L_{4n-3}, n$ a positive integer, pass through a given point $A.$ The maximum number of points of intersection of pairs of lines from the complete set $\{L_1,L_2,\dots,L_{100}\}$ is
$\textbf{(A) ... | 4351 |
Let $N = 24 \times 25 \times 26 \times 27 + 1$. Then, $N$ is the square of ____. | 24^2 + 3 \times 24 + 1 |
Consider two fair dice being thrown consecutively. If the upward-facing numbers after the dice land are denoted by $x$ and $y$ respectively, find the probability that $\log_{2x}(y) = 1$. | \frac{1}{12} |
4. As shown in Figure 1, in trapezoid $A B C D$, $A D / / B C$, $A D \perp C D, B C=C D=2 A D, E$ is a point on $C D$, $\angle A B E=45^{\circ}$. Then the value of $\tan \angle A E B$ is ( ).
(A) $\frac{3}{2}$
(B) 2
(C) $\frac{5}{2}$
(D) 3 | D |
1. Solve the inequality $\operatorname{tg} \arccos x \leqslant \sin \operatorname{arctg} x$. Answer: $\left[-\frac{1}{\sqrt[4]{2}} ; 0\right) \cup\left[\frac{1}{\sqrt[4]{2}} ; 1\right]$. | [-\frac{1}{\sqrt[4]{2}};0)\cup[\frac{1}{\sqrt[4]{2}};1] |
222 Find the smallest natural number $n$, such that for any $n$ points taken on a circle, among the central angles of the arcs connecting any two points, at least 2005 angles are not greater than $120^{\circ}$. | 91 |
4. Given a tetrahedron $ABCD$ satisfying: $AB=2, CD=2\sqrt{5}, AC=BD=3, AD=BC=\sqrt{5}$, then the volume of the tetrahedron is $\qquad$. | \frac{4}{3} |
10. In $\triangle A B C$, $A B=\sqrt{30}, A C=\sqrt{6}, B C$ $=\sqrt{15}$, there is a point $D$ such that $A D$ bisects $B C$ and $\angle A D B$ is a right angle, the ratio $\frac{S_{\triangle A D B}}{S_{\triangle A B C}}$ can be written as $\frac{m}{n}$. ($m, n$ are coprime positive integers). Then $m+n=$ | 65 |
11.4. a) Given a rectangular parallelepiped with a volume of 2017 and integer coordinates of vertices in a Cartesian coordinate system in space. Find the diagonal of the parallelepiped, given that its edges are parallel to the coordinate axes. b) Does there exist a rectangular parallelepiped with a volume of 2017 and i... | \sqrt{4035} |
## Problem Statement
Calculate the area of the parallelogram constructed on vectors $a$ and $b$.
$$
a = p - 2q
$$
$$
\begin{aligned}
& b = 2p + q \\
& |p| = 2 \\
& |q| = 3 \\
& (\widehat{p, q}) = \frac{3 \pi}{4}
\end{aligned}
$$ | 15\sqrt{2} |
B2. Write the equation of a line with a positive slope that forms a right-angled triangle with the coordinate axes, having an area of 9 square units, and intersects the abscissa at $x=-3$, in explicit, implicit, and segment form. | 2x+6 |
Given a sequence $\{a_n\}$ satisfies $a_1 = -1$, and $S_n = 2a_n + n$, (where $S_n$ is the sum of the first $n$ terms of $\{a_n\}$), then $a_6$ = ( )
A: -31
B: -32
C: -62
D: -63 | \text{D} |
2. A. If the graph of the direct proportion function $y=a x(a \neq 0)$ and the graph of the inverse proportion function $y=\frac{b}{x}(b \neq 0)$ intersect at two points, and one of the intersection points has coordinates $(-3,-2)$, then the coordinates of the other intersection point are ( ).
(A) $(2,3)$
(B) $(3,-2)$
... | D |
4. A net of a cube is shown with one integer on each face. A larger cube is constructed using 27 copies of this cube. What is the minimum possible sum of all of the integers showing on the six faces of the larger cube? | 90 |
[ Divisibility of numbers. General properties ]
On a bus rode Andrey
To the circle and back home,
Paying 115 rubles,
He bought himself a pass.
In January he didn't get it,
And therefore for several days
He bought a ticket from the driver
For 15 rubles for himself.
And on another day the conductor
Took only 11... | 9 |
Which of the following sequences of statements is correctly arranged according to the "syllogism" pattern?
1. $y=\cos x (x \in \mathbb{R})$ is a trigonometric function;
2. Trigonometric functions are periodic functions;
3. $y=\cos x (x \in \mathbb{R})$ is a periodic function.
A: 1 2 3
B: 3 2 1
C: 2 3 1
D: 2... | D |
Given that the random variable $X$ follows a normal distribution $N(1, \sigma^2)$, if $P(0 < X \leq 1) = 0.3$, then find the value of $P(X \geq 2)$. | 0.2 |
In an independence test, assume $H_0$: Variable $X$ is not associated with variable $Y$, then under the condition that $H_0$ is true, $P(K^2 \geq 6.635) \approx 0.010$ implies which of the following?
A: With a probability of making an error not exceeding 0.1%, we consider "Variable $X$ is related to variable $Y$"
B: W... | D |
1. Determine the number of all infinite arithmetic sequences of integers that have both numbers 1 and 2005 among their first ten terms. | 68 |
Suppose that $f_{1}(x)=\frac{1}{2-x}$. For each positive integer $n \geq 2$, define $f_{n}(x)=f_{1}\left(f_{n-1}(x)\right)$ for all real numbers $x$ in the domain of $f_{1}\left(f_{n-1}(x)\right)$. The value of $f_{2019}(4)$ can be written as $\frac{a}{b}$ where $a$ and $b$ are positive integers with no common divisor ... | (6053,6056) |
The edge lengths of the top and bottom bases of a frustum of a regular triangular pyramid are 3cm and 6cm, respectively, and its height is $\frac{3}{2}$cm. The lateral surface area of the frustum is \_\_\_\_\_\_ cm2. | \frac{27 \sqrt{3}}{2} |
To obtain the graph of the function $$y=\cos(2x+ \frac {\pi}{3})$$, it is only necessary to shift the graph of the function $y=\sin 2x$ ( )
A: to the right by $$\frac {5\pi}{6}$$ units
B: to the right by $$\frac {5\pi}{12}$$ units
C: to the left by $$\frac {5\pi}{6}$$ units
D: to the left by $$\frac {5\pi}{12}$$ units | D |
II. Given that $AB$ and $CD$ are the diameter and chord of a unit circle, respectively, and $AB \perp CD, \overparen{BD}=2 \overparen{AD}, E$ is the midpoint of $BD$, connect $AE$, intersecting $CD$ at point $P$, extend $AE$ to intersect the circle at point $F$; then connect $CF$, intersecting $AB$ at point $Q$.
(1) Pr... | \frac{3}{8} \sqrt{3} |
Example 7 Let $n$ be a given positive integer, $X=\{1,2,3, \cdots, n\}, A$ be a subset of $X$, and for any $x<y<z, x, y, z \in A$, there exists a triangle with side lengths $x, y, z$. Let $|A|$ denote the number of elements in the set $A$. Find the maximum value of $|A|$. (Original problem) | \left[\frac{n+2}{2}\right] |
Problem 9.6. In triangle $A B C$, the angles $\angle B=30^{\circ}$ and $\angle A=90^{\circ}$ are known. On side $A C$, point $K$ is marked, and on side $B C$, points $L$ and $M$ are marked such that $K L=K M$ (point $L$ lies on segment $B M$).
Find the length of segment $L M$, if it is known that $A K=4, B L=31, M C=3... | 14 |
If the complex number \\(z\\) satisfies \\(i \cdot z = -\dfrac{1}{2}(1 + i)\\), what is the imaginary part of \\(z\\)? | \dfrac{1}{2} |
Yan Yan's home is near People's Square, where she often sees many people parking their bicycles. Once, she asked the old man who watches over the bicycles and learned that the parking volume for that day was 6,882 times. The parking fee for regular bicycles is 0.2 yuan per time, and for electric bicycles, it is 0.5 yua... | \text{B} |
In the Cartesian coordinate system $xOy$, given circle $O: x^{2}+y^{2}=4$ and point $P(-1,1)$, the line $l$ passing through point $P$ intersects circle $O$ at points $A$ and $B$.
$(1)$ If $|AB|=2\sqrt{3}$, find the equation of line $l$;
$(2)$ Let the midpoint of chord $AB$ be $M$, find the equation of the trajector... | (x+\dfrac{1}{2})^{2}+(y-\dfrac{1}{2})^{2}=\dfrac{1}{2} |
There are two equilateral triangles with a vertex at $(0, 1)$, with another vertex on the line $y = x + 1$ and with the final vertex on the parabola $y = x^2 + 1$. Find the area of the larger of the two triangles. | 26\sqrt{3} + 45 |
10. In $\triangle A B C$, $A B=\sqrt{2}, A C=\sqrt{3}, \angle B A C=30^{\circ}, P$ is any point on the plane of $\triangle A B C$, then $\mu=\overrightarrow{P A} \cdot \overrightarrow{P B}+\overrightarrow{P B} \cdot \overrightarrow{P C}+\overrightarrow{P C} \cdot \overrightarrow{P A}$ has the minimum value of $\qquad$. | \frac{\sqrt{2}}{2}-\frac{5}{3} |
Calculate: $1+\frac{1}{1+2}+\frac{1}{1+2+3}+\frac{1}{1+2+3+4}+\ldots+\frac{1}{1+2+3+\ldots+2015}=\_\_\_\_\_\_$. | 1\frac{1007}{1008} |
The equation of the line that passes through the point (1,2) and has the maximum distance from the origin is ( )
A: $x+2y-5=0$
B: $2x+y-4=0$
C: $x+3y-7=0$
D: $x-2y+3=0$ | \text{A} |
If $a$ and $b$ are positive numbers, and $ab=a+b+3$, then the minimum value of $ab$ is ______. | 9 |
Given a sequence $\{a_n\}$ that satisfies: $a_1=1$, $a_n=2a_{n-1}+1$ (for $n\geq2$), find the value of $a_4$. | 15 |
2.2 In a closed bag lie apples. Three friends tried to lift the bag and guess how many fruits were inside. The first said that, in his opinion, there were 19 apples, the second thought there were 22, and the third said 23. When the friends opened the bag, it turned out that one of them was off by 1, another by 2, and t... | 20 |
MLD
Let $x, y, z$ be real numbers, satisfying the relations
$$
\left\{\begin{array}{l}
x \geq 20 \\
y \geq 40 \\
z \geq 1675 \\
x+y+z=2015
\end{array}\right.
$$
Find the greatest value of the product $P=x \cdot y \cdot z$. | 48407500 |
If the focus of the parabola $y^{2}=-2px$ coincides with the left focus of the ellipse $\dfrac {x^{2}}{16}+ \dfrac {y^{2}}{12}=1$, then the value of $p$ is ( ).
A: $-2$
B: $2$
C: $4$
D: $6$ | C |
1. The inequality is equivalent to
$$
\begin{gathered}
\frac{\sqrt{x^{2}+14 x+40}-2 \sqrt{x^{2}-x-2}}{\left(\sqrt{x^{2}-x-2}-2\right)\left(\sqrt{x^{2}+14 x+40}-4\right)} \leqslant 0 \Leftrightarrow \\
\left\{\begin{array} { l }
{ \frac { x ^ { 2 } + 1 4 x + 4 0 - 4 ( x ^ { 2 } - x - 2 ) } { ( x ^ { 2 } - x - 6 ) ( x ... | (-\infty;-12)\cup(-2;-1]\cup[2;3)\cup[8;+\infty) |
29. As shown in the figure, in the isosceles right triangle $A B C$, $\angle A B C=90^{\circ}$. Inside triangle $A B C$ there is a point $D$, such that $A B=A D$, and $\angle B A D=30^{\circ}$. Then $\angle A C D=$ $\qquad$ . | 30 |
In a school there are $1200$ students. Each student is part of exactly $k$ clubs. For any $23$ students, they are part of a common club. Finally, there is no club to which all students belong. Find the smallest possible value of $k$. | 23 |
4. On the shores of a circular island (viewed from above), there are cities $A, B, C$, and $D$. A straight asphalt road $A C$ divides the island into two equal halves. A straight asphalt road $B D$ is shorter than road $A C$ and intersects it. The speed of a cyclist on any asphalt road is 15 km/h. The island also has s... | 450 |
Eve and Martti have a whole number of euros.
Martti said to Eve: ''If you give give me three euros, so I have $n$ times the money compared to you. '' Eve in turn said to Martti: ''If you give me $n$ euros then I have triple the amount of money compared to you'' . Suppose, that both claims are valid.
What values can a... | 1, 2, 3, 7 |
Example 3 Given $p^{3}+q^{3}=2$, where $p, q$ are real numbers. Then the maximum value of $p+q$ is $\qquad$
(1987, Jiangsu Province Junior High School Mathematics Competition) | 2 |
In triangle $\triangle ABC$, it is known that $2c=2b\cos A-a$. Find:
1. $B$;
2. If $a+c=8$, $b=7$, and $C > A$, find the height on side $AC$. | \frac{15\sqrt{3}}{14} |
The wheel shown below consists of two circles and five spokes, with a label at each point where a spoke meets a circle. A bug walks along the wheel, starting at point $A$. At every step of the process, the bug walks from one labeled point to an adjacent labeled point. Along the inner circle the bug only walks in a coun... | 4 |
The vertices of a cube ABCD-A1B1C1D1 with edge length $a$ are all on the surface of a sphere $O$. $E$ and $F$ are the midpoints of edges $AA1$ and $DD1$, respectively. The length of the line segment cut off by sphere $O$ from line $EF$ is \_\_\_\_\_\_\_\_\_. | a |
In the geometric sequence ${a_n}$ with a common ratio $q=2$, if $a_1+a_4+a_7+\ldots+a_{97}=11$, then the sum of the first 99 terms of the sequence, $S_{99}$, equals to _________. | 77 |
# Task No. 8.1
## Condition:
For the "Handy Hands" club, Anton needs to cut several identical pieces of wire (the length of each piece is a whole number of centimeters). Initially, Anton took a piece of wire 10 meters long and was able to cut only 9 needed pieces from it. Then Anton took a piece 11 meters long, but i... | 111 |
Given the function $g(x)= \frac {10^{x}-1}{10^{x}+1}$, where $x\in\mathbb{R}$, the function $y=f(x)$ is the inverse function of $y=g(x)$.
(1) Find the expression for the function $y=f(x)$ and state its domain $D$;
(2) Let $h(x)= \frac {1}{x}-f(x)$, if the graph of the function $y=h(x)$ is a continuous and smooth cu... | -1 < t < - \frac {1}{2} |
4. Four friends came back from fishing. Every two of them counted the sum of their catches. Six numbers were obtained: $7,9,14,14,19,21$. How many fish were caught in total? | 28 |
Given the function $f(x)=\frac{1}{3}x^3-\frac{1}{2}ax^2+(a-1)x+1$, find the range of $a$ such that the function is monotonically decreasing in the interval $(1,4)$ and monotonically increasing in the interval $(6,+\infty)$. | [5,7] |
In the Cartesian coordinate system $xOy$, the parametric equation of curve $C_{1}$ is given by
$$
\begin{cases}
x=2+t\cos \alpha, \\
y=1+t\sin \alpha
\end{cases}
\text{(with parameter t)},
$$
where $\alpha \neq \frac{\pi}{2}$. In the polar coordinate system with the origin as the pole and the positive half-axis of $x$... | 2\sqrt{3} |
Let $S_{n}$ be the sum of the first $n$ terms of the sequence ${a_{n}}$, with $a_{1}=1$ and $a_{n+1}=a_{n}+2$. Find $S_{n}=\underline{\qquad\qquad}$. | n^{2} |
## Task Condition
Find the $n$-th order derivative.
$y=\sqrt{e^{3 x+1}}$ | (\frac{3}{2})^{n}\cdot\sqrt{e^{3x+1}} |
The function $f(x) = x\ln{x}$ has its minimum value on the interval $[1, t+1]$ ($t > 0$) as ____. | 0 |
Subject 3. Determine the cardinality of the set $A=\left\{\overline{a b} \left\lvert\, \frac{16}{a^{2}+b} \in \mathbb{N}\right.\right\}$. | 8 |
$2 \cdot 28$ If $a-b=x \neq 0$, and $a^{3}-b^{3}=19 x^{3}$, then
(A) $a=2 x$ or $a=3 x$.
(B) $a=2 x$ or $a=-3 x$.
(C) $a=-2 x$ or $a=-3 x$.
(D) $a=-2 x$ or $a=3 x$.
(China Junior High School Mathematics Correspondence Competition, 1990) | =-2xor=3x |
Consider sequences that consist entirely of $A$'s and $B$'s and that have the property that every run of consecutive $A$'s has even length, and every run of consecutive $B$'s has odd length. Examples of such sequences are $AA$, $B$, and $AABAA$, while $BBAB$ is not such a sequence. How many such sequences have length 1... | 172 |
8.5. In an isosceles triangle, the bisector drawn to the base turned out to be twice as short as the bisector drawn to the lateral side. Find the angles of this triangle.
 | 36,36,108 |
Given $1$, $a_{1}$, $a_{2}$, $a_{3}$ form an arithmetic sequence $(a_{1}$, $a_{2}$, $a_{3}$ are all positive numbers). If these $3$ terms form a geometric sequence in a certain order, the number of such geometric sequences is ().
A: $3$
B: $4$
C: $5$
D: $6$ | A |
4. At the vertices of a cube, the numbers $1^{2}, 2^{2}, \ldots, 8^{2}$ (one number per vertex) are placed. For each edge, the product of the numbers at its ends is calculated. Find the maximum possible sum of all these products. | 9420 |
## Task 1 - 130621
A rectangular glass pane is $24 \mathrm{~cm}$ long and $22 \mathrm{~cm}$ wide. From this, rectangular panes of $8 \mathrm{~cm}$ length and $6 \mathrm{~cm}$ width are to be cut.
What is the maximum number of such panes that can be obtained?
Illustrate a way to achieve this maximum number in a drawi... | 11 |
Jirka had three identical rectangles. First, he placed them together as shown in the picture and got a rectangle with a perimeter of $20 \mathrm{~cm}$. Then he placed them together differently and got a rectangle with a different perimeter. What was the perimeter of this rectangle?
(When placing the rectangles togethe... | 28\mathrm{~} |
Given the function $f(x)=2\sin (\omega x+\varphi)$ ($\omega > 0$, $|\varphi| < \frac {\pi}{2}$), its smallest positive period is $4\pi$, and its graph is symmetric about the line $x= \frac {2}{3}\pi$. Consider the following four statements:
1. The function $f(x)$ increases first and then decreases in the interval $\l... | C |
4. Task: On an $8 \times 8$ chessboard, 64 checkers numbered from 1 to 64 are placed. 64 students take turns approaching the board and flipping only those checkers whose numbers are divisible by the ordinal number of the current student. A "Queen" is a checker that has been flipped an odd number of times. How many "Que... | 8 |
Find the angle between the lines $A C$ and $B D$, if the distance between the midpoints of segments $A D$ and $B C$ is equal to the distance between the midpoints of segments $A B$ and $C D$.
# | 90 |
5. Let the function $f_{0}(x)=|x|, f_{1}(x)=\left|f_{0}(x)-1\right|, f_{2}(x)=$ $\left|f_{1}(x)-2\right|$, then the area of the closed part of the figure enclosed by the graph of the function $y=f_{2}(x)$ and the $x$-axis is $\qquad$ | 7 |
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