problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
|---|---|
Given that the graph of the function $f(x)=\frac{1}{x-a-1}$ is symmetrical about the point $(3,0)$, find the value of the real number $a$. | a=2 |
Given the function $f(x) = x^{2}(\ln x - a)$, where $a$ is a real number.
$(1)$ Find the intervals where the function $f(x)$ is monotonic.
$(2)$ If the function $f(x)$ has an extremum at $x=e$, where ${f'}(x)$ is the derivative of $f(x)$, and ${f'}(x_{1})={f'}(x_{2})$, with $x_{1} \lt x_{2}$, prove that $2 \lt x_{1... | 2 < x_{1}+x_{2} < e |
9. As shown in the figure, in $\triangle A B C$, $A B$ $=425, B C=450, C A=510, P$ is inside the triangle, $D E 、 F G 、 H I$ all pass through $P$, have the same length $d$, and are parallel to $A B 、 B C 、 C A$ respectively. Find $d$. | 306 |
Given that the tangent line of the curve $y=f\left(x\right)$ at point $\left(0,0\right)$ coincides with the tangent line of the curve $y=\frac{f(x)}{x}$ at point $\left(2,1\right)$, find ${f'}\left(2\right)$. | 2 |
Given sets $A=\{x|log_{2}x＜\;1\}$ and $B=\{x|x^{2}+x-2≤0\}$, then $A\cap B=\left(\ \ \right)$
A: $\{x\left|\right.-2 \lt x \lt 2\}$
B: $\{x\left|\right.-2\leqslant x\leqslant 1\}$
C: $\{x\left|\right.0 \lt x\leqslant 1\}$
D: $\{x\left|\right.0 \lt x \lt 2\}$ | C |
78. Find the slope of the line and the segment it intercepts on the y-axis, given that the line passes through the points $M(2, -1)$ and $P(-1, 8)$. | k=-3,b=5 |
Given a sequence $\{a_n\}$ satisfying $a_1=1$ and $a_{n+1}= \frac{2a_{n}}{a_{n}+2}$, ($n\in\mathbb{N}^*$), let $b_{n}= \frac{1}{a_{n}}$.
(1) Prove that the sequence $\{b_n\}$ is an arithmetic sequence;
(2) Find the general formula for the sequence $\{a_n\}$. | a_{n}= \frac{2}{n+1} |
6. Let the set $I=\{1,2, \cdots, n\}(n \geqslant 3)$. If two non-empty proper subsets $A$ and $B$ of $I$ satisfy $A \cap B=\varnothing, A \cup$ $B=I$, then $A$ and $B$ are called a partition of $I$. If for any partition $A, B$ of the set $I$, there exist two numbers in $A$ or $B$ such that their sum is a perfect square... | 15 |
131. Find the coordinates of the foci, the lengths of the axes, the eccentricity, and the equations of the asymptotes of the hyperbola given by the equation $16 x^{2}-$ $-25 y^{2}=400$. | F_{1}(-\sqrt{41};0);F_{2}(\sqrt{41};0),2a=10,2b=8,\varepsilon=\sqrt{41}/5,\(4/5)x |
Let the universal set \\(U=\mathbb{Z}\\), \\(A=\{-1,1,3,5,7,9\}\\), \\(B=\{-1,5,7\}\\). Then, \\(A∩(\complement_{U}B)=\\) \\(\\)
A: \\(\{1,3,9\}\\)
B: \\(\{-1,5,7\}\\)
C: \\(\{-1,1,3,9\}\\)
D: \\(\{-1,1,3,5,9\}\\) | A |
To commemorate the contribution of the ancient Chinese mathematician Zu Chongzhi to the calculation of pi, the 40th session of the United Nations Educational, Scientific and Cultural Organization (UNESCO) designated March 14th of each year as "International Mathematics Day." On March 14, 2023, a school held a math cult... | \frac{3}{10} |
Zuzka wrote a five-digit number. When she appended a one to the end of this number, she got a number that is three times larger than the number she would get if she wrote a one before the original number.
Which five-digit number did Zuzka write? | 42857 |
3. In triangle $ABC (AB < AC)$, the interior bisector of angle $A$ intersects side $BC$ at point $D$ and the line parallel to $AB$ through $C$ at point $E$, and the median corresponding to side $BC$ intersects the same parallel line at point $N$. Let $M$ be the reflection of $A$ over $B$, $O$ be the midpoint of $[BC]$,... | \frac{2}{3} |
The random variables $X, Y$ can each take a finite number of integer values. They are not necessarily independent. Express $P(\min(X,Y)=k)$ in terms of $p_1=P(X=k)$, $p_2=P(Y=k)$ and $p_3=P(\max(X,Y)=k)$. | p_1 + p_2 - p_3 |
Determine the smallest positive integer, $n$, which satisfies the equation $n^3+2n^2 = b$, where $b$ is the square of an odd integer. | 7 |
9. The ages of Jo, her daughter and her grandson are all even numbers. The product of their three ages is 2024 . How old is Jo?
A 42
B 44
C 46
D 48
E 50 | 46 |
5. Put 10 identical balls into 3 boxes numbered $1, 2, 3$ (all 10 balls must be placed each time), with the requirement that the number of balls in each box is no less than the box's number. The number of such ways to place the balls is ( .
(A) 9
(B) 12
(C) 15
(D) 18 | C |
Given the function $f(x)=\frac{lnx}{x}$, if $x_{2} \gt x_{1} \gt 1$, which of the following options is correct?
A: $(x_{1}-x_{2})[f(x_{1})-f(x_{2})] \lt 0$
B: $|f({x}_{1})-f({x}_{2})|＜\frac{1}{e}$
C: ${x}_{1}^{2}f({x}_{1})＞{x}_{2}^{2}f({x}_{2})$
D: $f(x_{1})-f(x_{2}) \lt x_{2}-x_{1}$ | D |
Given that $\{a_n\}$ is an arithmetic sequence, let $b_n = a_na_{n+1}a_{n+2}$ (where $n$ is a positive integer), and let $S_n$ denote the sum of the first $n$ terms of $\{b_n\}$. Given that $3a_5 = 8a_{12} > 0$, find the value of $n$ for which $S_n$ is maximized. | n = 16 |
Given the parametric equation of line $l$ as $\begin{cases} x= \frac{1}{2}t \
y= \frac{ \sqrt{3}}{2}t-1\end{cases} (t \text{ is the parameter })$, and the parametric equation of curve $C$ as $\begin{cases}x=\cos \theta \
y=2+\sin \theta\end{cases} (\theta \text{ is the parameter })$
(I) In the polar coordinate system w... | \frac{5}{2} |
8. One of these options gives the value of $17^{2}+19^{2}+23^{2}+29^{2}$. Which is it?
A 2004
B 2008
C 2012
D 2016
E 2020 | 2020 |
A math interest group has 3 male students and 2 female students. If 2 students are randomly selected to participate in a math competition, the two mutually exclusive events are ( )
A: Exactly 1 female student and exactly 2 female students
B: At least 1 male student and all male students
C: At least 1 male student an... | D |
The sum of all integers whose absolute value is less than $5$ is ____. | 0 |
Given \\(M=\{1,2,4\}\\) and \\(N=\{1,3,4\}\\), the union of \\(M\\) and \\(N\\), \\(M\cup N\\), is equal to ( ).
A: \\(\{1,4\}\\)
B: \\(M\\)
C: \\(N\\)
D: \\(\{1,2,3,4\}\\) | D |
3. Given the function $f(x)=x^{2}-4 x+3$, the sets are
$$
\begin{array}{l}
M=\{(x, y) \mid f(x)+f(y) \leqslant 0\}, \\
N=\{(x, y) \mid f(x)-f(y) \geqslant 0\} .
\end{array}
$$
Then in the Cartesian coordinate system, the area of the region represented by the set $M \cap N$ is ( ).
(A) $\frac{\pi}{4}$
(B) $\frac{\pi}{2... | C |
39. 15 volleyball teams played a round-robin tournament, where each team won exactly seven matches. How many triplets of teams are there in this tournament where each team has exactly one win against the other two? | 140 |
In the positive integer sequence, starting from 1, the items are selected according to the following rule: the first time, take 1; the second time, take 2 consecutive even numbers 2 and 4; the third time, take 3 consecutive odd numbers 5, 7, and 9; the fourth time, take 4 consecutive even numbers 10, 12, 14, and 16; th... | 3965 |
Problem 10.8. Rectangle $ABCD$ is such that $AD = 2AB$. Point $M$ is the midpoint of side $AD$. Inside the rectangle, there is a point $K$ such that $\angle AMK = 80^{\circ}$ and ray $KD$ is the bisector of angle $MKC$. How many degrees does angle $KDA$ measure?
![](https://cdn.mathpix.com/cropped/2024_05_06_d7ec6e60d... | 35 |
$2 \cdot 63$ Inserting "+" or "-" between $1^{2}, 2^{2}, 3^{2}, \cdots, 1989^{2}$, what is the smallest non-negative number that can be obtained from the resulting sum expression? | 1 |
Given the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{5} = 1$ has its right focus coinciding with the focus of the parabola $y^2 = 12x$, then the distance from the focus of the hyperbola to its asymptote is ( ).
A: $\sqrt{5}$
B: $3$
C: $5$
D: $4\sqrt{2}$ | A |
A dealer bought $n$ radios for $d$ dollars, $d$ a positive integer. He contributed two radios to a community bazaar at half their cost. The rest he sold at a profit of $8 on each radio sold. If the overall profit was $72, then the least possible value of $n$ for the given information is:
$\textbf{(A)}\ 18\qquad \tex... | 12 |
9. (12 points) Solve the inequality
$$
\sqrt{x+\frac{1}{x^{2}}}-\sqrt{x-\frac{1}{x^{2}}}<\frac{1}{x} .
$$ | x>\frac{\sqrt[3]{10}}{2} |
Task 2. (10 points) A circle touches the extensions of two sides $AB$ and $AD$ of square $ABCD$, and the point of tangency cuts off a segment of length $2+\sqrt{5-\sqrt{5}}$ cm from vertex $A$. Two tangents are drawn from point $C$ to this circle. Find the side of the square if the angle between the tangents is $72^{\c... | \frac{\sqrt{\sqrt{5}-1}\cdot\sqrt[4]{125}}{5} |
10. (5 points) If 2 cows can be exchanged for 42 sheep, 3 sheep can be exchanged for 26 rabbits, 2 rabbits can be exchanged for 3 chickens, then how many chickens can 3 cows be exchanged for? | 819 |
4. The volume of a rectangular prism is 8 cubic centimeters, and the total surface area is 32 square centimeters. If the length, width, and height form a geometric sequence, then the sum of all the edges of this rectangular prism is $\qquad$ | 32 |
Xinhua Bookstore is preparing to order a batch of books. There are two suppliers, A and B, both priced at $40$ yuan per book. To promote sales, A said: "All books purchased at my place will be discounted by 10%." B said: "If you purchase more than $100$ books, the excess will be discounted by 20%."
$(1)$ If Xinhua Bo... | 200 |
Given the function $f(x)=
\begin{cases}
\frac{1}{3}x+1 & \text{if } x\leqslant 1 \\
\ln x & \text{if } x > 1
\end{cases}$, if the equation $f(x) - ax = 0$ has exactly two different roots, then the range of the real number $a$ is $(\quad)$.
A: $(0, \frac{1}{3})$
B: $[\frac{1}{3}, \frac{1}{e})$
C: $(\frac{1}{e}, \... | B |
18. (12 points) As shown in Figure 4, in the equilateral $\triangle ABC$, points $D$ and $E$ are on sides $AC$ and $AB$ respectively, and $AD = \frac{1}{3} AC$, $AE = \frac{2}{3} AB$, $BD$ intersects $CE$ at point $F$.
(1) Prove that points $A$, $E$, $F$, and $D$ are concyclic;
(2) If the side length of the equilateral... | \frac{2}{3} |
In the quadrilateral $ABCD$, $AB = BC = CD$ and $\angle BMC = 90^\circ$, where $M$ is the midpoint of $AD$. Determine the acute angle between the lines $AC$ and $BD$. | 30^\circ |
In the diagram, each of the circles with centres $X, Y$ and $Z$ is tangent to the two other circles. Also, the circle with centre $X$ touches three sides of rectangle $P Q R S$ and the circle with centre $Z$ touches two sides of rectangle $P Q R S$, as shown.
^2 + (x^2 + 1)^2 + \ldots + (x^n + 1)^2$. | \frac{x^2 - x^{2n+2}}{1 - x^2} + \frac{2x(1 - x^n)}{1 - x} + n |
The function $y=\log _{2}(ax+1)+1$ passes through the fixed point ______. | (0,1) |
Adam, Bendeguz, Cathy, and Dennis all see a positive integer $n$. Adam says, "$n$ leaves a remainder of $2$ when divided by $3$." Bendeguz says, "For some $k$, $n$ is the sum of the first $k$ positive integers." Cathy says, "Let $s$ be the largest perfect square that is less than $2n$. Then $2n - s = 20$." Dennis says,... | 210 |
If the line passing through points $K(1-a, 1+a)$ and $Q(3, 2a)$ has an obtuse angle $\alpha$ with the x-axis, then the range of real numbers $a$ is __________. | (-2, 1) |
Given $k=1, 2, 3, 4, 5$, the coefficient of $x^k$ in the expansion of $(x+2)^5$ cannot be ( )
A: 10
B: 40
C: 50
D: 80 | C |
Given $x \gt 0$, the minimum value of the function $y=x+\frac{2}{2x+1}-1$ is ______. | \frac{1}{2} |
11. (2002 National Competition Question) Real numbers $a, b, c$ and a positive number $\lambda$ make $f(x)=x^{3}+a x^{2}+b x+c$ have three real roots $x_{1}, x_{2}, x_{3}$, and satisfy
(1) $x_{2}-x_{1}=\lambda$; (2) $x_{3}>\frac{1}{2}\left(x_{1}+x_{2}\right)$.
Find the maximum value of $\frac{2 a^{3}+27 c-9 a b}{\lambd... | \frac{3\sqrt{3}}{2} |
3. (24 points) Let the equation of the ellipse be $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ $(a>b>0)$, and let segment $P Q$ be a focal chord passing through the left focus $F$ and not perpendicular to the $x$-axis. If there exists a point $R$ on the left directrix such that $\triangle P Q R$ is an equilateral triang... | \left(\frac{\sqrt{3}}{3}, 1\right), \pm \frac{1}{\sqrt{3 e^{2}-1}} |
Given $$a=2^{- \frac {1}{3}}$$, $$b=(2^{\log_{2}3})^{- \frac {1}{2}}$$, $$c= \frac {1}{4} \int_{ 0 }^{ \pi }\sin xdx$$, then the relationship among the real numbers $a$, $b$, $c$ is ( )
A: $a>c>b$
B: $b>a>c$
C: $a>b>c$
D: $c>b>a$ | \text{C} |
4・131 If
$$
\left\{\begin{array}{l}
a^{2}+b^{2}=1, \\
c^{2}+d^{2}=1, \\
a c+b d=0 .
\end{array}\right.
$$
Try to find the value of $a b+c d$. | 0 |
4. 199 A car departs from point $O$ along a straight highway, maintaining a constant speed $v$. At the same time the car starts, a person on a bicycle departs from a point that is a distance $a$ from $O$ and a distance $b$ from the highway, aiming to deliver a letter to the car's driver. What is the minimum speed the c... | x_{\min }=v \sin \alpha=\frac{v b}{a} |
5. Given the sequence $\left\{a_{n}\right\}$ satisfies:
$$
a_{1}=\frac{3}{2}, a_{n+1}=\frac{1}{2}\left(a_{n}+\frac{1}{a_{n}}\right)\left(n \in \mathbf{Z}_{+}\right) \text {. }
$$
Then the number of consecutive 0s after the decimal point in the decimal representation of $a_{6}$ is . $\qquad$ | 22 |
Given $$\overrightarrow {AC}=(\cos \frac {x}{2}+\sin \frac {x}{2}, -\sin \frac {x}{2})$$, $$\overrightarrow {BC}=(\cos \frac {x}{2}-\sin \frac {x}{2}, 2\cos \frac {x}{2})$$, let $f(x) = \overrightarrow {AC}\cdot \overrightarrow {BC}$
(1) Find the smallest positive period and the intervals of monotonic decrease for $f(... | [1, \sqrt {2}) |
1. Calculate the expression with mixed numbers
$895231755 \frac{234}{357118} \cdot 895231754 \frac{234}{357118}-895231756 \frac{234}{357118} \cdot 895231753 \frac{234}{357118}$ | 2 |
### 6.244. Solve the equation
$$
\begin{aligned}
& x(x+1)+(x+1)(x+2)+(x+2)(x+3)+(x+3)(x+4)+\ldots \\
& \ldots+(x+8)(x+9)+(x+9)(x+10)=1 \cdot 2+2 \cdot 3+\ldots+8 \cdot 9+9 \cdot 10
\end{aligned}
$$ | x_{1}=0,x_{2}=-10 |
# Task 7.2
There are 30 logs, the lengths of which are 3 or 4 meters, and their total length is one hundred meters. How many cuts are needed to saw all these logs into pieces 1 meter long? (Each cut saws exactly one log).
Points 7
# | 70 |
What is the relationship between "$m \gt 0$" and the statement "$\forall x\in R$, $x^{2}+2x+m \gt 0$ is a true proposition"?
A: Sufficient but not necessary condition
B: Necessary but not sufficient condition
C: Necessary and sufficient condition
D: Neither necessary nor sufficient condition | B |
10. Given that the circumcenter of $\square A B C$ is $\mathrm{O}$, and $2 \overrightarrow{O A}+3 \overrightarrow{O B}+4 \overrightarrow{O C}=0$, then the value of $\cos \angle B A C$ is $\qquad$ | \frac{1}{4} |
In a class with $23$ students, each pair of students have watched a movie together. Let the set of movies watched by a student be his [i]movie collection[/i]. If every student has watched every movie at most once, at least how many different movie collections can these students have? | 23 |
12. As shown in Figure 3, $\triangle A B C$ is an equilateral triangle, point $C$ is on the side $D E$ of rectangle $A B D E$, the inradius of $\triangle A B C$ is 1. Then the diameter of the circumcircle of rectangle $A B D E$ is $\qquad$ | \sqrt{21} |
The statement "a=1" is a condition for the function $f(x) = ax^2 - 2x + 1$ to have only one zero (root) is ( )
A: Necessary and sufficient condition
B: Necessary but not sufficient condition
C: Sufficient but not necessary condition
D: Neither sufficient nor necessary condition | C |
22. The center of the regular octagon $A B C D E F G H$ is $J$. Each vertex and the center, these nine points, correspond to one of the numbers $1 \sim 9$, and each point corresponds to a different number. The sums of the numbers corresponding to the diagonals $A J E$, $B J F$, $C J G$, and $D J H$ are equal. How many ... | C |
## Task A-2.3.
Let $A B C$ be a right triangle and $\overline{C N}$ its altitude. If $|A C|=|B N|=1$, what is the length of the hypotenuse $\overline{A B}$? | \frac{1+\sqrt{5}}{2} |
Given that $P$ is any point on the ellipse $\frac{x^2}{4}+y^2=1$ ($y\neq-1$), two tangents $PM$ and $PN$ are drawn from $P$ to the circle $C: x^{2}+(y+2)^{2}=1$, with tangents intersecting at $M$ and $N$. Find the minimum value of $\overrightarrow{CM} \cdot \overrightarrow{CN}$.
A. $0$
B. $-\frac{3}{4}$
C. $-\frac{7... | D |
Determine the integer $k$ such that the zero $x_0$ of the function $f(x) = \ln x + 2x - 6$ lies in the interval $[k, k+1)$. | k=2 |
1. The maximum value of $y=2 x+\sqrt{6-x}$ is | 12 \frac{1}{8} |
Lines $a_1, b_1, c_1$ pass through the vertices $A, B, C$ respectively of a triange $ABC$; $a_2, b_2, c_2$ are the reflections of $a_1, b_1, c_1$ about the corresponding bisectors of $ABC$; $A_1 = b_1 \cap c_1, B_1 = a_1 \cap c_1, C_1 = a_1 \cap b_1$, and $A_2, B_2, C_2$ are defined similarly. Prove that the triangles ... | \frac{S_1}{R_1} = \frac{S_2}{R_2} |
Problem 11.2. Solve the system
$$
\left\lvert\, \begin{aligned}
& \left(4^{\sqrt{x^{2}+x}}+7.2^{\sqrt{x^{2}+x}}-1\right) \sin (\pi y)=7|\sin (\pi y)| \\
& x^{2}+4 x+y^{2}=0
\end{aligned}\right.
$$
Aleksandar Ivanov | -1,-\sqrt{3} |
1. Compute the area of the figure bounded on the plane Oxy by the lines $x=2, x=3, y=0, y=x^{2 / \ln x}$. | e^{2} |
Given the vector $\overrightarrow{a}=(3,4)$, if $λ\overrightarrow{a}=(3λ,2μ)(λ,μ∈R)$, and $|λ\overrightarrow{a}|=5$, then $λ+μ=(\ $ $\)$
A: $3$
B: $-3$
C: $±3$
D: $-1$ | C |
Alice and Bob play a game. Alice writes an equation of the form $ax^2 + bx + c =0$, choosing $a$, $b$, $c$ to be real numbers (possibly zero). Bob can choose to add (or subtract) any real number to each of $a$, $b$, $c$, resulting in a new equation. Bob wins if the resulting equation is quadratic and has two distinct r... | (a, b, c) = (0, 0, 0) |
Given that the real part and the imaginary part of $(1+2i)(a+i)$ are equal, where $a$ is a real number, find the value of $a$. | a = -3 |
Example 10 The real number $a$ makes the inequality
$$x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}+x_{5}^{2} \geqslant a\left(x_{1} x_{2}+x_{2} x_{3}+x_{3} x_{4}+x_{4} x_{5}\right)$$
hold for any real numbers $x_{1}, x_{2}, x_{3}, x_{4}, x_{5}$. Find the maximum value of $a$. (2010 Shanghai High School Mathematics Competiti... | \frac{2 \sqrt{3}}{3} |
Starting from an equilateral triangle like the one in figure a, trisect each side of the triangle, and then construct new equilateral triangles outward using the middle segments as sides, as shown in figure b, to get a "snowflake hexagon". Next, trisect each of the 12 sides of the "snowflake hexagon" and construct new ... | 40:27 |
4. In the rectangular prism $A^{\prime} C$, $A B=5, B C=4, B^{\prime} B=6$, and $E$ is the midpoint of $A A^{\prime}$. Find the distance between the skew lines $B E$ and $A^{\prime} C^{\prime}$. | \frac{60}{\sqrt{769}} |
$1 \cdot 35$ A four-digit number is composed of four consecutive digits in sequence. If the first two digits from the left are swapped, the resulting number is a perfect square. Find the four-digit number.
The text above is translated into English, keeping the original text's line breaks and format. | 3456 |
12.153. A circle of radius $r$ is inscribed in a sector of radius $R$. Find the perimeter of the sector. | 2R(1+\arcsin\frac{r}{R-r}) |
If the proposition "$\forall x\in R$, $mx^{2}-2mx+1 \gt 0$" is false, then the range of real number $m$ is ( ).
A: $0\leqslant m \lt 1$
B: $m \lt 0$ or $m\geqslant 1$
C: $m\leqslant 0$ or $m\geqslant 1$
D: $0 \lt m \lt 1$ | \text{B} |
Let the complex number $z$ satisfy the relation: $z + | \bar{z} | = 2 + i$. Then $z$ equals to ( )
A: $-\frac{3}{4} + i$
B: $\frac{3}{4} + i$
C: $-\frac{3}{4} - i$
D: $\frac{3}{4} - i$ | \text{B} |
Given that the terms of the geometric sequence {an} are all positive, a6+a5=4, a4+a3-a2-a1=1, find the value of a1. | a_1 = \sqrt {2} - 1 |
The coefficient of $x^2$ in the expansion of $(1+2x)^3(1-x)^4$ is ______. | -6 |
## Problem Statement
Calculate the definite integral:
$$
\int_{0}^{3} \frac{d x}{\left(9+x^{2}\right)^{3 / 2}}
$$ | \frac{\sqrt{2}}{18} |
After moving terms in the equation $x-5=3x+7$, which of the following is correct?
A: $x+3x=7+5$
B: $x-3x=-5+7$
C: $x-3x=7-5$
D: $x-3x=7+5$ | D |
1. Find for which values of the real number $a$ all the roots of the polynomial, in the variable $x$,
$$
x^{3}-2 x^{2}-25 x+a
$$
are integers. | 50 |
Example 2.69. Find the center of gravity of the arc of the circle $x^{2}+$ $+y^{2}=R^{2}$, located in the first quadrant, if the density $\rho=\rho_{0} x y$. | x_{}=y_{}=\frac{2R}{3} |
Given the function $f(x) = a - \frac{2}{2^{x} + 1} \ (a \in \mathbb{R})$,
(I) Determine the monotonicity of the function $f(x)$ on $\mathbb{R}$, and prove it using the definition of a monotonic function;
(II) Does there exist a real number $a$ such that the function $f(x)$ is an odd function? If so, find the value of $... | a = 1 |
Which of the following inequalities are satisfied for all real numbers $a, b, c, x, y, z$ which satisfy the conditions $x < a, y < b$, and $z < c$?
$\text{I}. \ xy + yz + zx < ab + bc + ca \\ \text{II}. \ x^2 + y^2 + z^2 < a^2 + b^2 + c^2 \\ \text{III}. \ xyz < abc$
$\textbf{(A)}\ \text{None are satisfied.} \qquad \t... | \textbf{(A)} |
Find the sum of all integers $n$ not less than $3$ such that the measure, in degrees, of an interior angle of a regular $n$-gon is an integer.
[i]2016 CCA Math Bonanza Team #3[/i] | 1167 |
Suppose the random variable $X\sim N(\mu ,{{\sigma }^{2}})$, and $P(X > 5)=P(X < -1)=0.2$, then $P(2 < X < 5)=$ ___. | 0.3 |
The $2023$ Chongqing Marathon started at Haitang Yanyu Park in Nan'an District, Chongqing. Xiao Nan and Xiao Kai participated in the $5$-kilometer mini-marathon race. The two agreed to run from point $A$ along the same route to point $B$, which is $5$ kilometers away from point $A$. It is known that Xiao Nan's running ... | C |
7. Given that the 6027-digit number $\frac{a b c a b c \cdots a b c}{2000 \uparrow a b c}$ is a multiple of 91. Find the sum of the minimum and maximum values of the three-digit number $\overline{a b c}$. | 1092 |
In triangle $ABC$, given $a=18$, $b=24$, and $A=30\degree$, the number of solutions for this triangle is \_\_\_\_\_\_. | 2 |
If the angle of inclination of a line is $120^{\circ}$, then the slope of the line is ( )
A: $\sqrt {3}$
B: $- \sqrt {3}$
C: $\dfrac { \sqrt {3}}{3}$
D: $- \dfrac { \sqrt {3}}{3}$ | \text{B} |
João has a deck of 52 cards numbered from 1 to 52. A set of three cards is called lucky if the sum of the digits on each card is the same. What is the minimum number of cards João has to take from the deck, without looking, so that among the cards he takes there are necessarily three cards that form a set of lucky card... | 26 |
2. Given real numbers $a, b, x, y$ satisfy the system of equations:
$$
\left\{\begin{array}{l}
a x+b y=3, \\
a x^{2}+b y^{2}=7, \\
a x^{3}+b y^{3}=16, \\
a x^{4}+b y^{4}=42 .
\end{array}\right.
$$
Find the value of $a x^{5}+b y^{5}$. | 20 |
In the arithmetic sequence {an}, it is known that a3+a10<0, and S11>0, then the maximum value of Sn is ( )
A: S5
B: S6
C: S7
D: S8 | \text{B} |
## Task 2 - 200522
When making a purchase, the price of 170 Marks was paid with exactly 12 banknotes. Each of these banknotes was either a 10-Mark note or a 20-Mark note.
Determine the number of 10-Mark notes and the number of 20-Mark notes used to pay the specified amount! | 7 |
14. Xiaoding, Xiaoming, and Xiaoqian are playing a game together and need to determine the order of play. They agree to use "scissors, paper, rock" to decide. What is the probability that all three of them will choose "paper" in one round? $\qquad$ . | \frac{1}{27} |
Given a sphere \\(O\\) with a radius of \\(13\\), and three points \\(A, B, C\\) on the surface of the sphere such that \\(AB=12\sqrt{3}\\), \\(AC=BC=12\\), find the volume of the tetrahedron \\(OABC\\). | 60\sqrt{3} |
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