problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
|---|---|
6. Let $\mathbb{R}^{+}$ denote the set of all positive real numbers. Determine all functions $f$ : $\mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$ such that for all $x, y \in \mathbb{R}^{+}$,
$$
f(x) f(y)=f(y) f(x f(y))+\frac{1}{x y} .
$$
(Pavel Calábek) | f(x)=1+\frac{1}{x} |
7. Arrange $\{1,2, \cdots, 7\}$ randomly into $\left\{a_{1}, a_{2}, \cdots\right.$, $\left.a_{7}\right\}$. The probability that $a_{1}+a_{2}+a_{3} \geqslant a_{5}+a_{6}+a_{7}$ is $\qquad$ | \frac{73}{140} |
Given that the line \\((a-4)x+y+1=0\\) is perpendicular to the line \\(2x+3y-5=0\\), then \\(a=\\) .
A: \\( \dfrac {14}{3}\\)
B: \\( \dfrac {5}{2}\\)
C: \\( \dfrac {11}{2}\\)
D: \\(3\\) | B |
4. If the inequality $a x^{2}+7 x-1>2 x+5$ holds for -1 $\leqslant a \leqslant 1$, then the range of $x$ is ( ).
(A) $2 \leqslant x \leqslant 3$
(B) $2<x<3$
(C) $-1 \leqslant x \leqslant 1$
(D) $-1<x<1$ | B |
Exercise 2. Let $x$ be a real number such that $x^{3}=x+4$. Let $a, b, c, d$ be non-zero integers such that $a+b x^{2}+c x^{3}+d x^{4}=0$. Determine the value of $\frac{a c}{b d}$. | 64 |
3. Let $a=256$. Find the unique real number $x>a^{2}$ such that
$$
\log _{a} \log _{a} \log _{a} x=\log _{a^{2}} \log _{a^{2}} \log _{a^{2}} x .
$$ | 2^{32} |
How many numbers can be formed using the digits from 1 to 4, where all the digits are different? | 64 |
A [circle](https://artofproblemsolving.com/wiki/index.php/Circle) is [inscribed](https://artofproblemsolving.com/wiki/index.php/Inscribe) in [quadrilateral](https://artofproblemsolving.com/wiki/index.php/Quadrilateral) $ABCD$, [tangent](https://artofproblemsolving.com/wiki/index.php/Tangent) to $\overline{AB}$ at $P$ a... | 647 |
$12 \cdot 60$ In a right-angled triangle with side lengths all being positive integers, one of the legs is 1575 less than the hypotenuse, and the other leg is less than 1991. Find the length of the hypotenuse of this right-angled triangle.
(China Beijing High School Grade 1 Mathematics Competition, 1991) | 1799 |
If distinct digits $D,E,L,M,Q$ (between $0$ and $9$ inclusive) satisfy
\begin{tabular}{c@{\,}c@{\,}c@{\,}c}
& & $E$ & $L$ \\
+ & $M$ & $E$ & $M$ \\\hline
& $Q$ & $E$ & $D$ \\
\end{tabular}
what is the maximum possible value of the three digit integer $QED$?
[i]2019 CCA Math Bonanza Individual... | 893 |
11. Find the greatest integer less than $(2+\sqrt{3})^{4}$. | 193 |
Bakayev E.V.
In triangle $\mathrm{ABC}$, medians $\mathrm{BK}$ and $\mathrm{CN}$ are drawn, intersecting at point M. What is the maximum number of sides of quadrilateral ANMK that can have a length of 1? | 2 |
Problem 4. A rectangular piece of paper $20 \times 19$, divided into unit squares, is cut into several square pieces, the cuts being made along the sides of the unit squares. A square piece of this kind is called an odd square if the length of its side is an odd number.
a) What is the minimum possible number of odd sq... | 4 |
Given the function $f(x) = x^2 + 2ax + 2$, where $x \in [-5, 5]$.
(Ⅰ) When $a = -1$, find the maximum and minimum values of the function $f(x)$.
(Ⅱ) Find the range of real numbers $a$ such that $y = f(x)$ is a monotonic function on the interval $[-5, 5]$. | (-\infty, -5] \cup [5, +\infty) |
In the diagram, $P T$ and $P S$ are tangents to a circle with centre $O$. The point $Y$ lies on the circumference of the circle; and the point $Z$ is where the line $P Y$ meets the radius $O S$.
Also, $\angle S P Z=10^{\circ}$ and $\angle T O S=150^{\circ}$.
How many degrees are there in the sum of $\angle P T Y$ and... | 160 |
Each entry in the list below is a positive integer:
$$
a, 8, b, c, d, e, f, g, 2
$$
If the sum of any four consecutive terms in the list is 17 , what is the value of $c+f$ ? | 7 |
Problem 6.7. Anya places pebbles on the sand. First, she placed one stone, then added pebbles to form a pentagon, then made a larger outer pentagon with pebbles, then another outer pentagon, and so on, as shown in the picture. The number of stones she had arranged on the first four pictures: 1, 5, 12, and 22. If she co... | 145 |
Given that $\alpha$ and $\beta$ are acute angles, $\tan \alpha = 2$, and $\sin(\alpha-\beta) = -\frac{{\sqrt{10}}}{{10}}$.
$(1)$ Find the value of $\sin 2\alpha$.
$(2)$ Find the value of $\tan (\alpha + \beta)$. | \tan(\alpha + \beta) = -\frac{9}{13} |
15. A piece fell out of the book, the first page of which is numbered 143, and the number of the last page consists of the same digits but written in a different order. How many pages fell out of the book? | 172 |
Given that the center of the circle C lies on the positive x-axis and has a radius of 2, and the chord of circle C cut by the line y=$\sqrt{3}$x has a length of $\sqrt{13}$. (1) Find the equation of circle C; (2) Let the moving line y=k(x-2) intersect circle C at points A and B. Is there a fixed point N on the positi... | (5,0) |
2. Given numbers $x, y \in\left(0, \frac{\pi}{2}\right)$. Find the maximum value of the expression
$$
A=\frac{\sqrt[4]{\sin x \sin y}}{\sqrt[4]{\operatorname{tg} x}+\sqrt[4]{\operatorname{tg} y}}
$$ | \frac{\sqrt[4]{8}}{4} |
Given that the point $P(3-m, 2m+6)$ lies in the fourth quadrant of the Cartesian coordinate system and is equidistant from the two coordinate axes, the value of $m$ is ______. | -9 |
Example 2 Let the odd function $f(x)$ have the domain $\mathbf{R}$. It satisfies $f(x)+f(x+2)=a, f(1)=0$, where $a$ is a constant. Try to determine how many roots the equation $f(x)=0$ has at least in the interval $(-3,7)$.
Translate the above text into English, please keep the original text's line breaks and format, ... | 9 |
Given $\overrightarrow{a}=(2-k,3)$, $\overrightarrow{b}=(2,-6)$, and $\overrightarrow{a} \parallel \overrightarrow{b}$, find the real number $k$. | 3 |
4. In rectangle $A B C D$, $A B=2, A D=1$, point $P$ on side $D C$ (including points $D, C$) and point $Q$ on the extension of $C B$ (including point $B$) satisfy $|\overrightarrow{D P}|=|\overrightarrow{B Q}|$, then the dot product $\overrightarrow{P A} \cdot \overrightarrow{P Q}$ of vector $\overrightarrow{P A}$ and ... | \frac{3}{4} |
Exercise 2. In the figure below, the triangles ABO, BCO, CDO, DEO, and FEO are right isosceles triangles. We assume that $\mathrm{OA}=8 \mathrm{~cm}$. Determine the area of $A O F$ in $\mathrm{cm}^{2}$.
^{(1.4)^{(1.4)}}$ is well-defined, but Bob thinks it diverges. Who is right? | Aerith |
1. (Optional question on coordinate systems and parametric equations) If the polar equation of a curve is $p=2\sin\theta+4\cos\theta$, and a Cartesian coordinate system is established with the pole as the origin and the polar axis as the positive x-axis, then the Cartesian equation of the curve is.
2. (Optional questio... | 5 |
685. For what base of the numeral system does the following rebus have a solution
$$
\begin{aligned}
& \text { KITO } \\
& + \text { KIOTO } \\
& \hline \text { TOKIO, }
\end{aligned}
$$
where identical letters represent identical digits, and different letters represent different digits? Find all solutions to the reb... | 4350_{7}+43050_{7}=50430_{7} |
Given the sets $A=\{5\}$, $B=\{1,2\}$, and $C=\{1,3,4\}$. If we take one element from each of these three sets to form the coordinates of a point in the Cartesian coordinate system, then the number of distinct points determined is ( )
A: 33
B: 34
C: 35
D: 36 | \text{A} |
10.061. A perpendicular line drawn from the vertex of a parallelogram to its diagonal divides this diagonal into segments of length 6 and $15 \mathrm{~cm}$. The difference in the lengths of the sides of the parallelogram is 7 cm (Fig. 10.60). Find the lengths of the sides of the parallelogram and its diagonals. | 10;17;21;\sqrt{337}\mathrm{} |
Given that in the binomial expansion of ${(\sqrt[3]{x}-\frac{1}{2\sqrt[3]{x}})}^{n}$, the $6$th term is a constant term, then ( )
A: $n=10$
B: The total number of terms in the expansion is $13$
C: The coefficient of the term containing $x^{2}$ is $\frac{45}{4}$
D: The number of rational terms in the expansion is $3... | ACD |
Xiao Mei, based on her experience in learning "Numbers and Expressions," wants to explore the rules of operation of the following quadratic radicals through the method of "from specific to general." Here is her exploration process, please complete it:The first equation: $\sqrt{1+\frac{1}{3}}=\sqrt{\frac{3+1}{3}}=\sqrt{... | 2023\sqrt{2} |
Example 1 Given that $a$ and $b$ are real numbers, and $a^{2} + ab + b^{2} = 3$. If the maximum value of $a^{2} - ab + b^{2}$ is $m$, and the minimum value is $n$, find the value of $m + n$. ${ }^{\text {[2] }}$ | 10 |
[u]level C[/u]
p10. Construct the following figure, where $ABCD$ is a rectangle, $PQC$ is equilateral and $2 PD = PA$.
[img]https://cdn.artofproblemsolving.com/attachments/3/d/6fd25309c373e91b8d837df7378725b6b6fd7d.gif[/img]
p11. Let $ABC$ be an isosceles triangle, with base $AB = 10$ cm. Let $M$ and $N$ be the mid... | 75 \text{ cm}^2 |
A point is "bouncing" inside a unit equilateral triangle with vertices $(0,0)$, $(1,0)$, and $(1/2,\sqrt{3}/2)$. The point moves in straight lines inside the triangle and bounces elastically off an edge at an angle equal to the angle of incidence. Suppose that the point starts at the origin and begins motion in the dir... | 30 - 17\sqrt{3} |
10. From the set $\mid 1,2, \cdots, 20$ \}, select 5 numbers”, such that the difference between any two numbers is not less than 4, then there are a total of different ways to select. | 56 |
What is the coefficient of the $x^3$ term in the expansion of $(2x-1)^4$? | -32 |
The equation of a curve in polar coordinates is given: $r=p \sin 5 \vartheta$ (where $p$ is a given length, $r$ is the radius vector to individual points on the curve, and $\vartheta$ is the corresponding direction angle). Construct the curve and express its equation in Cartesian coordinates. | x^{6}-5px^{4}y+3x^{4}y^{2}+10px^{2}y^{3}+3x^{2}y^{4}-py^{5}+y^{6}=0 |
10.009. In an isosceles triangle, the base is 16 cm, and the lateral side is $10 \mathrm{~cm}$. Find the radii of the inscribed and circumscribed circles and the distance between their centers. | \frac{8}{3},\frac{25}{3},5 |
6. Among the following statements, the correct option is ( ).
(A) A sample has only one mode.
(B) The median of a sample is always one of the numbers in the sample.
(C) The variance of a sample is always a positive number.
(D) Fans A, B, and C decide who gets the only ticket by drawing lots. They prepare three pieces o... | D |
3. In the coordinate plane, the area enclosed by the curve ||$x|+| x-1||+|y|=2$ is equal to $\qquad$
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly. | 3 |
Ant Amelia starts on the number line at $0$ and crawls in the following manner. For $n=1,2,3,$ Amelia chooses a time duration $t_n$ and an increment $x_n$ independently and uniformly at random from the interval $(0,1).$ During the $n$th step of the process, Amelia moves $x_n$ units in the positive direction, using up $... | \textbf{(C)}\frac{2}{3} |
How many different ways can 6 different books be distributed according to the following requirements?
(1) Among three people, A, B, and C, one person gets 1 book, another gets 2 books, and the last one gets 3 books;
(2) The books are evenly distributed to A, B, and C, with each person getting 2 books;
(3) The boo... | 30 |
3. Let $\varepsilon$ be a 7th root of 1, compute: $\frac{\varepsilon}{1+\varepsilon^{2}}+\frac{\varepsilon^{2}}{1+\varepsilon^{4}}+\frac{\varepsilon^{3}}{1+\varepsilon^{6}}$. | \frac{3}{2}or-2 |
feira 13 - What is the maximum number of Friday the 13ths that can occur in a non-leap year? In this case, what is the $10^{\text{th}}$ day of the year? | 3 |
The general term of the sequence $\{a_n\}$ is $a_n = \frac{n}{n^2 + 90}$. What is the largest term in the sequence $\{a_n\}$?
A: The 9th term
B: Both the 9th and 10th terms
C: The 10th term
D: Both the 9th and 8th terms | B |
30. If a die is tossed 500 times at random, how many times is the probability of getting a 1 (the face with 1 dot facing up) the greatest? | 83 |
$ABCDEF$ is a regular hexagon. Let $R$ be the overlap between $\vartriangle ACE$ and $\vartriangle BDF$. What is the area of $R$ divided by the area of $ABCDEF$? | \frac{1}{3} |
Given the function $y=|x-a|+|x+19|+|x-a-96|$, where $a$ is a constant, and it satisfies $19 < a < 96$. When the range of the independent variable $x$ is $a \leq x \leq 96$, find the maximum value of $y$. | 211 |
12. Given four integers $a, b, c, d$ are all even, and $0<a<b<c<d, d-a=90$, if $a, b, c$ form an arithmetic sequence, and $b, c, d$ form a geometric sequence, then $a+b+c+d=$ $\qquad$ . | 194 |
Let the sum of the first $n$ terms of an arithmetic sequence $\{a_n\}$ be $S_n$. If $S_3 = 9$ and $S_5 = 20$, then $a_7 + a_8 + a_9 =$ ( )
\[
A: 63
\]
\[
B: 45
\]
\[
C: 27
\]
\[
D: 36
\] | C |
Determine all positive integer numbers $k$ for which the numbers $k+9$ are perfect squares and the only prime factors of $k$ are 2 and 3. | \{16,27,72,216,432,2592\} |
The interval of decrease for the function $f(x) = \ln(-x^2+2x+3)$ is ______. | (1, 3) |
Let the complex number $z = (a^2 + a - 2) + (a^2 - 7a + 6)i$, where $a \in \mathbb{R}$. For what value(s) of $a$ does:
(1) $z \in \mathbb{R}$?
(2) $z$ is a pure imaginary number?
(3) $z$ is zero? | a = 1 |
In the rectangular coordinate system $xOy$, the parametric equations of line $l$ are given by $$\begin{cases} x=1+ \frac {3}{5}t \\ y=1+ \frac {4}{5}t\end{cases}$$ ($t$ is the parameter). Establish a polar coordinate system with the coordinate origin as the pole and the positive semi-axis of the $x$-axis as the polar a... | \frac{55}{41} |
648. Find all natural $n$, for which the number $2^{n}+65$ is a perfect square. | 10,4 |
2. What is the maximum possible number of consecutive terms of an increasing geometric progression that can be three-digit natural numbers? Provide an example of such a sequence. (16 points) | 6 |
Given that $a$, $b$, $c$ are the sides opposite to angles $A$, $B$, $C$ in $\triangle ABC$, and vectors $\overrightarrow{m} = (\sqrt{3}, -1)$, $\overrightarrow{n} = (\cos A, \sin A)$. If $\overrightarrow{m} \perp \overrightarrow{n}$, and $a\cos B + b\cos A = c\sin C$, then angle $B = \boxed{\frac{\pi}{6}}$. | \frac{\pi}{6} |
2.2. Find all possible values of
$$
\left\lfloor\frac{x-p}{p}\right\rfloor+\left\lfloor\frac{-x-1}{p}\right\rfloor,
$$
where $x$ is a real number and $p$ is a nonzero integer.
Here $\lfloor z\rfloor$ denotes the greatest integer less than or equal to $z$. | -3,-2,-1,0 |
3. In the number $2016 * * * * 02 *$, each of the 5 asterisks needs to be replaced with any of the digits $0,2,4,5,7,9$ (digits can be repeated) so that the resulting 11-digit number is divisible by 15. In how many ways can this be done? | 864 |
44. Given that $a$ and $b$ are prime numbers $(a>b)$, and $ab$ represents the product of $a$ and $b$. If $a + ab + b = 55$, then the value of $a - b$ is _. $\qquad$ | 10 |
Problem 11.6. One chooses a point in the interior of $\triangle A B C$ with area 1 and connects it with the vertices of the triangle. Then one chooses a point in the interior of one of the three new triangles and connects it with its vertices, etc. At any step one chooses a point in the interior of one of the triangles... | \frac{2}{2n+1} |
Given the function $f(x)=2\sin (\pi x+ \frac {\pi}{6})$,
(1) Find the maximum and minimum values of $f(x)$ when $x\in[-\frac {1}{2}, \frac {1}{2}]$;
(2) If $f(\frac {\alpha}{2\pi })= \frac {1}{4}$, find the value of $\cos ( \frac {2\pi}{3}-\alpha)$. | -\frac {31}{32} |
An ordinary domino tile can be identifi ed as a pair $(k,m)$ where numbers $k$ and $m$ can get values $0, 1, 2, 3, 4, 5$ and $6.$
Pairs $(k,m)$ and $(m, k)$ determine the same tile. In particular, the pair $(k, k)$ determines one tile.
We say that two domino tiles [i]match[/i], if they have a common component.
[i]Ge... | \frac{4(n+1)}{(n+2)(n+3)} |
3. Given $a^{2}+b c=14, b^{2}-2 b c=-6$. Then $3 a^{2}+4 b^{2}-5 b c=$ $\qquad$ | 18 |
Class $\mathbf{M}$.
In the class, there are 15 boys and 15 girls. On March 8, some boys called some girls to congratulate them on the holiday (no boy called the same girl twice). It turned out that the children could be uniquely divided into 15 pairs so that in each pair there was a boy who called the girl. What is th... | 120 |
The sequence $\{c_{n}\}$ is determined by the following equation. \[c_{n}=(n+1)\int_{0}^{1}x^{n}\cos \pi x\ dx\ (n=1,\ 2,\ \cdots).\] Let $\lambda$ be the limit value $\lim_{n\to\infty}c_{n}.$ Find $\lim_{n\to\infty}\frac{c_{n+1}-\lambda}{c_{n}-\lambda}.$ | 1 |
Example 3. Find the image of the circle
$$
z=R \cos t+i R \sin t \quad(0 \leqslant t<2 \pi)
$$
under the mapping $\boldsymbol{w}=\frac{\boldsymbol{z}}{\bar{z}}$. | u^2+v^2=1 |
Given an arithmetic sequence $\{a_n\}$, where $a_2=7$ and $a_4=15$, find the sum of the first $10$ terms, $S_{10}=$ .
A: $100$
B: $210$
C: $380$
D: $400$ | B |
Question 207, Find all positive integers that can be written in the form $\frac{[x, y]+[y, z]}{[x, z]}$, where $x, y, z$ are three positive integers, and $[a, b]$ denotes the least common multiple of the positive integers $a, b$. | 2m |
For the hyperbola $\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$ ($a > 0, b > 0$), a chord $AB$ passing through focus $F_1$ ($A$ and $B$ are on the same branch of the hyperbola) has a length of $m$. If the other focus is $F_2$, find the perimeter of $\triangle ABF_2$. | 4a + 2m |
Given $$\alpha \in \mathbb{R}, \sin \alpha + 2\cos \alpha = \frac{\sqrt{10}}{2}$$, then $\tan 2\alpha =$ （ ）
A: $$\frac{4}{3}$$
B: $$\frac{3}{4}$$
C: $$- \frac{3}{4}$$
D: $$- \frac{4}{3}$$ | C |
If the general term of the sequence $\{a_n\}$ is $a_n = n(n + 2)$, then which of the following numbers is an item of this sequence?
A: 18
B: 20
C: 24
D: 30 | C |
“$t \geq 0$” is a condition for “the function $f(x) = x^2 + tx - t$ to have a root in $(-\infty, +\infty)$” is ( )
A: A sufficient but not necessary condition
B: A necessary but not sufficient condition
C: A necessary and sufficient condition
D: Neither sufficient nor necessary condition | A |
7. Plot the figure $\Phi$ on the plane, consisting of points $(x ; y)$ of the coordinate plane such that the system of inequalities is satisfied
$$
\left\{\begin{array}{l}
\sqrt{x^{2}-3 y^{2}+4 x+4} \leqslant 2 x+1 \\
x^{2}+y^{2} \leqslant 4
\end{array}\right.
$$
Determine how many parts the figure $\Phi$ consists of... | 1 |
3. In a tournament among six teams, every team plays against each other team exactly once. When a team wins, it receives 3 points and the losing team receives 0 points. If the game is a draw, the two teams receive 1 point each.
Can the final scores of the six teams be six consecutive numbers $a, a+1, \ldots, a+5$ ? I... | 4 |
10. From a point $S$ outside a circle $\gamma$ of radius 1, draw a tangent line to $\gamma$ and let $T$ be the point of tangency. As a point $P$ varies on $\gamma$, the centroid of the triangle $P S T$ describes a curve $\gamma^{\prime}$. What is the ratio between the length of $\gamma^{\prime}$ and that of $\gamma$?
(... | \frac{1}{3} |
If the function $f(x)$ is increasing in some interval $I$ of its domain $D$, and $\frac{f(x)}{x}$ is decreasing in $I$, then $y=f(x)$ is said to be a "weakly increasing function" in $I$. It is known that the function $h(x)=x^{2}-(b-1)x+b$ is a "weakly increasing function" in $(0,1]$, then the value of the real number $... | 1 |
Given $a=3^{0.3}$, $b=\log_{\pi}3$, $c=\log_{0.3}e$, then
A: $a > b > c$
B: $b > a > c$
C: $c > a > b$
D: $b > c > a$ | \text{A} |
7. The minimum value of the function $f(x)=\sum_{k=1}^{2013}|x-k|$ is | 1013042 |
5. Draw the tangent line to the curve $y=3x-x^{3}$ through the point $A(2,-2)$. Then the equation of the tangent line is $\qquad$ . | y=-2 \text{ or } 9x+y-16=0 |
Given two sequences $\{a_n\}$ and $\{b_n\}$ ($b_n \neq 0$, $n \in \mathbb{N}^*$), satisfying $a_1 = b_1 = 1$, $a_n b_{n+1} - a_{n+1} b_n + b_{n+1} b_n = 0$
(1) Let $c_n = \frac{a_n}{b_n}$, prove that the sequence $\{c_n\}$ is an arithmetic sequence, and find the general formula for $\{c_n\}$.
(2) If $b_n = 2^{n-1}$... | 2^n(n - 1) + 1 |
Example 2. Approximate the larger negative root of the equation $x^{3}-12 x-8=0$ using the tangent method with an accuracy of five decimal places after the decimal point. | -0.694593 |
Given vectors $\overrightarrow{a} = (2, -3, 5)$ and $\overrightarrow{b} = (4, x, y)$ are parallel, then the values of $x$ and $y$ are ( )
A: 6 and -10
B: -6 and 10
C: -6 and -10
D: 6 and 10 | \text{B} |
For which natural numbers $n$ is it true that the fraction
$$
\frac{n^{2}+2}{n(n+1)}
$$
is not simplifiable? | n=6k+1n=6k+3 |
Given the function $y=\log_{2}(x^{2}-2kx+k)$ has the range $\mathbb{R}$, then the range of $k$ is ( )
A: $0<k<1$
B: $0\leq k<1$
C: $k\leq 0$ or $k\geq 1$
D: $k=0$ or $k\geq 1$ | C |
In how many ways can the nine digits from 1 to 9 be placed in a $3 \times 3$ grid so that the sums of the rows and the sums of the columns are all equal? | 72 |
Given the parabola $y^{2}=2px (p > 0)$, a line passing through point $T(p,0)$ with a slope of $1$ intersects the parabola at points $A$ and $B$. Find the product of the slopes of lines $OA$ and $OB$, where $O$ is the coordinate origin. | -2 |
Example 13 Let the sequence of positive real numbers $x_{0}, x_{1}, x_{2}, \cdots, x_{1995}$ satisfy the conditions:
(1) $x_{0}=x_{1995}$;
(2) For $i=1,2, \cdots, 1995$, $x_{i-1}+\frac{2}{x_{i-1}}=2 x_{i}+\frac{1}{x_{i}}$, find the maximum value of $x_{0}$. | 2^{997} |
The eccentricity of the hyperbola $\dfrac{x^2}{m} - \dfrac{y^2}{n} = 1$ is $2$, and one of its foci coincides with the focus of the parabola $y^2 = 4x$. Find the value of $mn$. | \dfrac{3}{16} |
A set of positive integers is called tyû-de-good if there are no two numbers among them whose difference is 2. How many tyû-de-good subsets does the set $\{1,2,3, \ldots, 10\}$ have? | 169 |
Given that $a$, $b$, and $c$ are all positive real numbers, and $ab+bc+ac=1$, find the maximum value of $abc$. | \frac{\sqrt{3}}{9} |
3. The equation is transformed into the form
$$
t^{\alpha}=\alpha(t+1), \quad \alpha=\lg 2 \in(0,1), \quad t=x^{2}-3>0
$$
where:
1) the left side of the equation is a concave up function (since $\alpha \in(0 ; 1)$), defined for $t \geqslant 0$
2) the right side of the equation is a linear function with a positive sl... | 4 |
Example 15 (Problem from the 29th Russian Mathematical Olympiad) Find all prime numbers $p$ such that $p^{x}=y^{3}+1$ holds, where $x, y$ are positive integers. | p=2,1,1p=3,2,2 |
Three. (25 points) Find all positive integer triples $(x, y, z)$ such that $1+2^{x} \times 3^{y}=5^{z}$ holds.
(Zhang Lei) | (3,1,2) |
$2018$ people (call them $A, B, C, \ldots$) stand in a line with each permutation equally likely. Given that $A$ stands before $B$, what is the probability that $C$ stands after $B$? | \frac{1}{3} |
The complex numbers $5+10i$ and $3-4i$ represent vectors $\overrightarrow{OA}$ and $\overrightarrow{OB}$ respectively. What is the complex number representing the vector $\overrightarrow{AB}$? | -2-14i |
In the "Guess the Song Title" game, participants guess the song title based on the ringtone of the main melody of the song. A guest participates in the game show where the production team has prepared ringtone melodies of songs from two groups, $A$ and $B$. Two ringtone melodies are randomly played from each of the $A$... | \frac{10}{3} |
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