problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
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## Task Condition
Find the derivative.
$$
y=\frac{1}{18 \sqrt{2}} \ln \frac{1+\sqrt{2} \operatorname{coth} x}{1-\sqrt{2} \operatorname{coth} x}
$$ | \frac{1}{9\cdot(1+\operatorname{ch}^{2}x)} |
3. Given a point on the plane that does not coincide with the origin. How many different points can be obtained from it by sequentially applying symmetries relative to the $O y$ axis and the line $y=-x$ (in any order and any number of times)? If the point itself can also be obtained, it should be counted. If different ... | 4;8 |
1. (10 points) Chess clubs from Moscow, Saint Petersburg, and Kazan agreed to hold a tournament. Each Muscovite played exactly 9 Saint Petersburg residents and $n$ Kazan residents. Each Saint Petersburg resident played exactly 6 Muscovites and 2 Kazan residents. Each Kazan resident played exactly 8 Muscovites and 6 Sai... | 4 |
3. Point $O$ is the center of the circumcircle of triangle $A B C$. Points $Q$ and $R$ are chosen on sides $A B$ and $B C$ respectively. Line $Q R$ intersects the circumcircle of triangle $A B R$ again at point $P$ and intersects the circumcircle of triangle $B C Q$ again at point $S$. Lines $A P$ and $C S$ intersect a... | 90 |
7.3 Toss a coin, each time heads appears you get 1 point, and tails appears you get 2 points. Prove that the probability of getting exactly $n$ points is $\frac{1}{3}\left[2+\left(-\frac{1}{2}\right)^{n}\right]$. | p_{n}=\frac{1}{3}[2+(-\frac{1}{2})^{n}] |
10. [15] Suppose we have an $n$-gon such that each interior angle, measured in degrees, is a positive integer. Suppose further that all angles are less than $180^{\circ}$, and that all angles are different sizes. What is the maximum possible value of $n$ ? Prove your answer. | 26 |
Given $a, b \in \mathbb{R}$, if $1+i$ is a root of the real-coefficient equation $x^{2}+ax+b=0$, where $i$ is the imaginary unit, then $a+b=$____. | 0 |
1. In a regular quadrilateral pyramid $P-A B C D$, $\angle A P C=60^{\circ}$, then the cosine value of the dihedral angle $A-P B-C$ is
A. $\frac{1}{7}$
B. $-\frac{1}{7}$
C. $\frac{1}{2}$
D. $-\frac{1}{2}$ | -\frac{1}{7} |
The $2007 \text{ AMC }10$ will be scored by awarding $6$ points for each correct response, $0$ points for each incorrect response, and $1.5$ points for each problem left unanswered. After looking over the $25$ problems, Sarah has decided to attempt the first $22$ and leave only the last $3$ unanswered. How many of the ... | \textbf{(D)}16 |
In the triangle $ABC$, point $D$ is given on the extension of side $CA$ beyond $A$, and point $E$ is given on the extension of side $CB$ beyond $B$, such that $AB = AD = BE$. The angle bisectors from $A$ and $B$ of triangle $ABC$ intersect the opposite sides at points $A_1$ and $B_1$, respectively. What is the area of ... | 6 |
If circle $C: x^2 + y^2 - 4x + 2y + m = 0$ intersects the y-axis at points $A$ and $B$, and $\angle ACB = 90^\circ$, then the value of the real number $m$ is ___. | -3 |
21.3.5 ** In a triangle with integer side lengths, given one side length as $n$, and the other two side lengths do not exceed $n$, congruent triangles are counted as one. Find the number of such triangles. | [\frac{(n+1)^{2}}{4}] |
The convex pentagon $ABCDE$ has $\angle{A}=\angle{B}=120^\circ,EA=AB=BC=2$ and $CD=DE=4$. What is the area of ABCDE?
$\text{(A) } 10\quad \text{(B) } 7\sqrt{3}\quad \text{(C) } 15\quad \text{(D) } 9\sqrt{3}\quad \text{(E) } 12\sqrt{5}$ | 7\sqrt{3} |
Example 4 Plant all 3 types of crops in the 5 plots of experimental fields as shown in Figure 4, with each plot planting one type of crop and no two adjacent plots can plant the same type of crop. How many different planting methods are there? | 42 |
1. Nезнayka jumped from his house to Znayka's house. For the first three-quarters of the path, he jumped with jumps the length of two of his usual steps, and for the remaining quarter of the path, he jumped with jumps the length of three of his usual steps. It turned out that the number of jumps of two steps was 350 mo... | 1200 |
How many of the numbers $-5,-4,-3,-2,-1,0$, $1,2,3$ satisfy the inequality $-3 x^{2}<-14$?
(a) 1
(b) 2
(c) 3
(d) 4
(e) 5 | 4 |
$$
\text { II. (40 points) Let } a_{i} \in(0,1](1 \leqslant i \leqslant 45) \text {. }
$$
Find the maximum value of $\lambda$ such that the inequality
$$
\sqrt{\frac{45}{\sum_{i=1}^{45} a_{i}}} \geqslant 1+\lambda \prod_{i=1}^{45}\left(1-a_{i}\right)
$$
holds. | 8(\frac{81}{80})^{45} |
4 In a regular $\triangle A B C$, $D$ and $E$ are the midpoints of $A B$ and $A C$ respectively. The eccentricity of the hyperbola with foci at $B$ and $C$ and passing through points $D$ and $E$ is $\qquad$. | \sqrt{3}+1 |
12th VMO 1974 Problem A2 Find all positive integers n and b with 0 < b < 10 such that if a n is the positive integer with n digits, all of them 1, then a 2n - b a n is a square. | 2foranyn,7forn=1 |
6. (USS) ${ }^{1 \mathrm{MO4}}$ Let $A$ be the sum of the digits of the number $16^{16}$ and $B$ the sum of the digits of the number $A$. Find the sum of the digits of the number $B$ without calculating $16^{16}$. | 7 |
Elective 4-4: Coordinate System and Parametric Equations
In the plane Cartesian coordinate system $xOy$, the parametric equation of line $l$ is $\begin{cases} x=-8+t, \\ y=\frac{t}{2} \end{cases}$ (where $t$ is the parameter), and the parametric equation of curve $C$ is $\begin{cases} x=2s^{2}, \\ y=2\sqrt{2}s \end{ca... | \frac{4\sqrt{5}}{5} |
For the one-variable quadratic equation $x^{2}+4x+k=0$ with two real roots for $x$, the range of values for $k$ is ( )
A: $k\leqslant -4$
B: $k \lt -4$
C: $k\leqslant 4$
D: $k \lt 4$ | C |
A child lines up $2020^2$ pieces of bricks in a row, and then remove bricks whose positions are square numbers (i.e. the 1st, 4th, 9th, 16th, ... bricks). Then he lines up the remaining bricks again and remove those that are in a 'square position'. This process is repeated until the number of bricks remaining drops bel... | 240 |
Given an odd function $f(x)$ with domain $\mathbb{R}$ and its derivative $f′(x)$, when $x \neq 0$, $f′(x)+ \dfrac {f(x)}{x} > 0$. If $a= \dfrac {1}{2}f( \dfrac {1}{2})$, $b=-2f(-2)$, and $c=(\ln \dfrac {1}{2})f(\ln \dfrac {1}{2})$, then the correct relationship among $a$, $b$, and $c$ is ( ).
A: $a < c < b$
B: $b < c ... | A |
Given that the arithmetic sequence $\left\{ {{a}\_{n}} \right\}$ has a common difference of $d$, and the geometric sequence $\left\{ {{b}\_{n}} \right\}$ has a common ratio of $q$. Let the sum of the first $n$ terms of $\left\{ {{a}\_{n}} \right\}$ and $\left\{ {{b}\_{n}} \right\}$ be ${{S}\_{n}}$ and ${{T}\_{n}}$ resp... | q = 2 |
10. In the tetrahedron $P-ABC$, the three edges $PA$, $PB$, and $PC$ are pairwise perpendicular, and $PA=1$, $PB=PC=2$. If $Q$ is any point on the surface of the circumscribed sphere of the tetrahedron $P-ABC$, then the maximum distance from $Q$ to the plane $ABC$ is $\qquad$ | \frac{3}{2}+\frac{\sqrt{6}}{6} |
Given that the coordinates satisfying the equation $f(x,y)=0$ are all on the curve $C$, then
A: The coordinates of points on the curve $C$ all satisfy the equation $f(x,y)=0$
B: Any point whose coordinates do not satisfy $f(x,y)=0$ are not on $C$
C: The coordinates of points not on $C$ must not satisfy $f(x,y)=0$
D: So... | C |
## 166. Math Puzzle $3 / 79$
Frank needs four minutes to cover the $300 \mathrm{~m}$ to the street corner, then runs $50 \mathrm{~m}$ up the stairs in five minutes, and completes the remaining $600 \mathrm{~m}$ of the journey in ten minutes.
What is his average speed?
Can he be considered a "fast walker" if we allow... | 3\mathrm{~}/\mathrm{} |
9. Positive integers $x, y$ satisfy $\frac{2}{5}<\frac{x}{y}<\frac{3}{7}$, then the minimum value of $y$ is | 12 |
Given that the sum of the first $n$ terms of a sequence $\left\{a_{n}\right\}$ is $S_{n}=\dfrac{3n^{2}-n}{2}$.
(1) Find the general formula for $\left\{a_{n}\right\}$;
(2) Let $b_{n}=\dfrac{1}{a_{n}a_{n+1}}$, and the sum of the first $n$ terms of the sequence $\left\{b_{n}\right\}$ is $T_{n}$. If $t\leqslant 4T_{n}$ ho... | 1 |
If the sum of $3x^2 - 2x + b$ and $x^2 + bx - 1$ does not contain any terms with $x$, find the value of $b$, write out their sum, and prove that no matter what value $x$ takes, the sum is always positive. | b = 2 |
Given that the midline of an isosceles trapezoid is 15 units long and its height is 3 units, the area of this isosceles trapezoid is $\boxed{45}$. | 45 |
4. Solve the equation $x^{3}-9[x]=8$. Here $[x]$ is the integer part of the number $x$ - the greatest integer not exceeding $x$. | x_{1}=-\sqrt[3]{28},\quadx_{2}=-\sqrt[3]{19},\quadx_{3}=-1,\quadx_{4}=\sqrt[3]{26},\quadx_{5}=\sqrt[3]{35} |
Given the point $P(2,2)$ on the curve $y=ax^{3}+bx$, if the slope of the tangent line at point $P$ is $9$, find:
(i) The value of $ab=$ \_\_\_\_\_\_;
(ii) The range of the function $f(x)=ax^{3}+bx$, $x\in\left[- \frac {3}{2},3\right]$ is \_\_\_\_\_\_. | [-2,18] |
15. As shown in the figure, $O$ is the origin, point $F$ is the focus of the parabola $C_{1}: x^{2}=2 p y$ $(p>0)$, and the tangent line at point $P$ on the parabola $C_{1}$ is tangent to the circle $C_{2}: x^{2}+y^{2}=1$ at point $Q$.
(1) When the equation of line $P Q$ is $x-y-\sqrt{2}=0$, find the equation of the pa... | 2\sqrt{2}+3 |
1. There are 100 nuts, among which 96 are genuine and 4 are defective. If 2 are randomly selected, what is the probability that both are defective? | \frac{1}{825} |
Given the geometric sequence $\{a_n\}$ with a positive common ratio $q$, and $a_2 \cdot a_6 = 9a_4$, $a_2=1$, find the value of $q$ and the expression for $a_n$. | 3^{n-2} |
9. (This question is worth 14 points) Find the smallest positive integer $n$, such that for any positive integer $k \geq n$, in the set $M=\{1,2, \cdots, k\}$ of the first $k$ positive integers, for any $x \in M$, there always exists another number $y \in M(y \neq x)$, such that $x+y$ is a perfect square. | 7 |
12.17. Find all solutions to the equation $x^{y}=y^{x}$:
a) in natural numbers;
b) in rational numbers.
## 12.3. Finding some solutions | 4,2 |
The approximate interval where the zero of the function $f(x) = 2^x + 2x - 3$ is located is ( )
A: $(0, \frac{1}{2})$
B: $(\frac{1}{2}, 1)$
C: $(1, 2)$
D: $(2, 3)$ | B |
Draw a tangent line from point $P(3,5)$ to the circle $(x-1)^{2}+(y-1)^{2}=4$. Then, the length of the tangent line is ______. | 4 |
The sum of the coefficients of the terms in the expansion of ${{(4-3x+2y)}^{n}}$ $(n\in \mathbb{N}^*)$ that do not contain $y$ is ______. | 1 |
If the direction vector of line $l$ is $\overrightarrow{u}$ and the normal vector of plane $\alpha$ is $\overrightarrow{n}$, which of the following options can make $l \perp \alpha$ hold true?
A: $\overrightarrow{u} = \left(2,1,1\right)$, $\overrightarrow{n} = \left(-1,1,1\right)$
B: $\overrightarrow{u} = \left(1,-2,... | B |
Maya lists all the positive divisors of $2010^2$. She then randomly selects two distinct divisors from this list. Let $p$ be the [probability](https://artofproblemsolving.com/wiki/index.php/Probability) that exactly one of the selected divisors is a [perfect square](https://artofproblemsolving.com/wiki/index.php/Perfec... | 107 |
Given points $P(2\cos 10^{\circ}$,$2\sin 10^{\circ})$, $Q(2\cos 50^{\circ}$,$-2\sin 50^{\circ})$, the inclination angle of the line $PQ$ is ______. | 70^{\circ} |
Solve the equation
$$
\sqrt{x^{2}+5 x-2}-\sqrt{x^{2}+5 x-5}=1 . \text { (1) }
$$ | x_{1}=1, x_{2}=-6 |
If $\alpha$ is any real number, then $sin({\frac{{5π}}{2}+α})=\left(\ \ \right)$
A: $\sin \alpha$
B: $-\sin \alpha$
C: $\cos \alpha$
D: $-\cos \alpha$ | C |
25. Thomas has constant speeds for both running and walking. When a down-escalator is moving, Thomas can run down it in 15 seconds or walk down it in 30 seconds. One day, when the escalator was broken (and stationary), it took Thomas 20 seconds to run down it.
How long, in seconds, would it take Thomas to walk down the... | 60 |
11.1. While walking in the park, Seryozha and Misha stumbled upon a meadow surrounded by lindens. Seryozha walked around the meadow, counting the trees. Misha did the same but started from a different tree (although he went in the same direction). The tree that was the $20-\mathrm{th}$ for Seryozha was the $7-\mathrm{t... | 100 |
Given a complex number $z$ satisfying $zi = 1 - i$, find the modulus $|z|$. | \sqrt{2} |
The reciprocal of $5$ is ( ).
A: $-5$
B: $-\frac{1}{5}$
C: $\frac{1}{5}$
D: $\pm 5$ | C |
34. Find the number of pairs of positive integers $(x, y)$ are there which satisfy the equation $2 x+3 y=2007$. | 334 |
Solve the inequality system $\left\{\begin{array}{l}5x+1>3(x-1)\\ \frac{{x-1}}{2}\geq 2x-4\end{array}\right.$, and find its integer solutions. | -1, 0, 1, 2 |
Find all polynomials $p(x)\in\mathbb{R}[x]$ such that for all $x\in \mathbb{R}$:
$p(5x)^2-3=p(5x^2+1)$ such that:
$a) p(0)\neq 0$
$b) p(0)=0$ | p(x) = \frac{1 + \sqrt{13}}{2} |
In the two-dimensional rectangular coordinate system, determine the graph of the following equations after undergoing the stretching transformation $\begin{cases} x'=\frac{1}{2}x \\ y'=\frac{1}{3}y \end{cases}$.
$(1) \quad 5x+2y=0$
$(2) \quad x^2+y^2=1$. | 4x'^2+9y'^2=1 |
For $0\leq a\leq 2$, find the minimum value of $\int_0^2 \left|\frac{1}{1+e^x}-\frac{1}{1+e^a}\right|\ dx.$
[i]2010 Kyoto Institute of Technology entrance exam/Textile e.t.c.[/i] | \ln \left( \frac{2+2e^2}{1+2e+e^2} \right) |
## Task 2 - 270522
A tourist who lives in Magdeburg (M) wants to visit each of the cities Schwerin (S), Neubrandenburg (N), and Berlin (B) exactly once on a round trip and then return to his place of residence.
One possible route would be from Magdeburg via Berlin, Schwerin, and Neubrandenburg back to Magdeburg (see ... | 6 |
The maximum and minimum values of the function $y=2x^3-3x^2-12x+5$ on the interval $[0,3]$ are respectively ( )
A: 5, -4
B: 5, -15
C: -4, -15
D: 5, -16 | \text{B} |
Example 25 (2007 Bulgarian National Team Selection Test) Find all positive integers $x, y$ such that $\left(x^{2}+\right.$ $y)\left(y^{2}+x\right)$ is a fifth power of a prime. | (2,5)(5,2) |
When $x^{13}+1$ is divided by $x-1$, the remainder is:
$\textbf{(A)}\ 1\qquad\textbf{(B)}\ -1\qquad\textbf{(C)}\ 0\qquad\textbf{(D)}\ 2\qquad\textbf{(E)}\ \text{None of these answers}$ | 2 |
For how many ordered pairs $(b,c)$ of positive integers does neither $x^2+bx+c=0$ nor $x^2+cx+b=0$ have two distinct real solutions?
$\textbf{(A) } 4 \qquad \textbf{(B) } 6 \qquad \textbf{(C) } 8 \qquad \textbf{(D) } 12 \qquad \textbf{(E) } 16 \qquad$ | 6 |
## 45. Thirty-two Cards
Alfred, Bruno, Christoph, and Damien are playing cards using a deck of 32 cards. Damien dealt them all to the players, and when it was pointed out that the number of cards each player had was different, he said: "If you want us all to have the same number of cards, do as I say. You, Alfred, div... | 4,7,13,8 |
A complex number $z$ satisfies $z+1$ is a pure imaginary number, then the real part of $z$ is ______. | -1 |
Given circle $C$: $x^{2}+y^{2}-4x-14y+45=0$ and point $Q(-2,3)$.
(1) If $M$ is any point on circle $C$, find the maximum and minimum values of $|MQ|$;
(2) If real numbers $m$, $n$ satisfy $m^{2}+n^{2}-4m-14n+45=0$, find the maximum and minimum values of $k= \frac {n-3}{m+2}$. | k_{max}=2+ \sqrt {3}, k_{min}=2- \sqrt {3} |
Consider a sequence $\{a_n\}$ with the sum of the first $n$ terms denoted by $S_n$, satisfying the condition $2S_n = a_n^2 + n$ with $a_n > 0$ ($n \in \mathbb{N}^*$).
(I) Find $a_1$, $a_2$, $a_3$.
(II) Conjecture a general formula for $\{a_n\}$ and prove it using mathematical induction.
(III) Let $b_n = 2a_n - 1$... | \frac{2\sqrt{3}}{3} |
Triangle $ABC$ has $AB=BC=10$ and $CA=16$. The circle $\Omega$ is drawn with diameter $BC$. $\Omega$ meets $AC$ at points $C$ and $D$. Find the area of triangle $ABD$. | 24 |
$23 \cdot 34$ is the base area of a rectangular box, the areas of the side, and the front are all known, the product of these areas equals
(A) the volume of the box.
(B) the square root of the volume.
(C) twice the volume.
(D) the square of the volume.
(E) the cube of the volume. | D |
Let's simplify the following fraction:
$$
\frac{x y^{2}+2 y z^{2}+y z u+2 x y z+2 x z u+y^{2} z+2 z^{2} u+x y u}{x u^{2}+y z^{2}+y z u+x u z+x y u+u z^{2}+z u^{2}+x y z}
$$ | \frac{y+2z}{u+z} |
A fair six-faced cubic die, labeled with numbers 0, 1, 2, 3, 4, 5, and a fair four-faced tetrahedral die, labeled with numbers 1, 2, 3, 4, are rolled once simultaneously. Let $z = a + bi$ be a complex number, where $a$ is the number on the face-up side of the cube, and $b$ is the sum of the numbers on the three visible... | \dfrac{11}{24} |
1. Let $\alpha, \beta$ be a pair of conjugate complex numbers. If $|\alpha-\beta|=2 \sqrt{3}$, and $\frac{\alpha}{\beta^{2}}$ is a real number, then $|\alpha|=$ | 2 |
3. As shown in the figure, in the vertical coordinate plane $x o y$, line $l$ passes through the origin $O$, and $l$ is in the first and the I quadrant, with the angle between $l$ and the $x$-axis being $\alpha\left(0^{\circ}<\alpha<90^{\circ}\right)$. There is a point mass (neglecting the size of the point mass) at po... | C |
1. Given point $P(1,2)$ is inside (including the boundary) the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$, and outside (including the boundary) the circle $x^{2}+y^{2}=\frac{a^{2}+2 b^{2}}{3}$. If $a, b \in \mathbf{R}_{+}$. Then the minimum value of $a+b$ is ( ).
(A) $\sqrt{5}+2$
(B) $\sqrt{6}+\sqrt{3}$
(C) $2... | B |
The determinant $\begin{vmatrix}1 & 4 & 7 \\ 2 & 5 & 8 \\ 3 & 6 & 9\end{vmatrix}$, the value of the algebraic cofactor of the element $7$ is ( ).
A: $-15$
B: $-3$
C: $3$
D: $12$ | B |
2. (I. Akulich) Find the largest natural $n$ with the following property: for any odd prime $p$ less than $n$, the difference $n-p$ is also a prime number. | 10 |
Given the ellipse $C$: $\frac{x^2}{a^2} + y^2 = 1 (a > 1)$, $A(a,0)$, $B(0,1)$, and the center of the circle $O$: $x^2 + y^2 = 1$ is at a distance of $\frac{\sqrt{3}}{2}$ from the line $AB$.
(I) Find the equation of the ellipse $C$;
(II) If the line $l$ is tangent to the circle $O$ and intersects the ellipse $C$ at... | \sqrt{3} |
(15) Let $\left\{a_{n}\right\}$ be a geometric sequence, and each term is greater than 1, then $\lg a_{1} \lg a_{2012} \sum_{i=1}^{20111} \frac{1}{\lg a_{i} \lg a_{i+1}}$ has the value $\qquad$. | 2011 |
On a line segment \\(AB\\) with a length of \\(12 cm\\), a point \\(M\\) is randomly chosen, and a square is constructed with the line segment \\(AM\\) as its side length. The probability that the area of this square is between \\(36 cm^2\\) and \\(81 cm^2\\) is ( )
A: \\( \dfrac{36}{81} \\)
B: \\( \dfrac{12}{36} \\... | \text{D} |
17. In the equation "中环杯是 + 最棒的 $=2013$", different Chinese characters represent different digits. What are the possible values of "中 + 环 + 杯 + 是 + 最 + 棒 + 的"? (If there are multiple solutions, please list them all.)
翻译结果：
17. In the equation "Zhong Huan Bei Shi + Zui Bang De $=2013$", different Chinese characters rep... | 24,33 |
It takes $A$ algebra books (all the same thickness) and $H$ geometry books (all the same thickness,
which is greater than that of an algebra book) to completely fill a certain shelf.
Also, $S$ of the algebra books and $M$ of the geometry books would fill the same shelf.
Finally, $E$ of the algebra books alone would ... | \frac{AM-HS}{M-H} |
3. Given point $D$ on side $A B$ of $\triangle A B C$, $\angle A$ $=40^{\circ}, \angle B=50^{\circ}, \angle B C D=30^{\circ}, B C=a, A C$ $=b$. Then $C D=(\quad$.
(A) $\frac{\sqrt{2}}{2} \sqrt{a b}$
(B) $\frac{1}{2} \sqrt{a^{2}+b^{2}}$
(C) $\frac{1}{3}(a+b)$
(D) $\frac{1}{4}(2 a+b)$ | B |
10. (12 points) 1 kilogram of soybeans can be made into 3 kilograms of tofu, and 1 kilogram of soybean oil requires 6 kilograms of soybeans. Soybeans cost 2 yuan per kilogram, tofu costs 3 yuan per kilogram, and soybean oil costs 15 yuan per kilogram. A batch of soybeans was purchased for 920 yuan, and after being made... | 360 |
25.42. Prove that $\lim _{n \rightarrow \infty} \frac{\sqrt[n]{n!}}{n}=\frac{1}{e}$.
## 25.5. Conjugate numbers | \frac{1}{e} |
Divide a sample data with a capacity of 100 into 10 groups and complete the frequency distribution table. If the cumulative frequency of the first seven groups is 0.79, and the frequencies of the remaining three groups form a geometric sequence with a common ratio greater than 2, then the highest frequency among the re... | 16 |
Given $a_1 > 0$, $a_1 \neq 1$, and $a_{n+1} = \frac{2a_n}{1+a_n}$ (for $n=1,2,\ldots$),
(1) Prove that $a_{n+1} \neq a_n$;
(2) Let $a_1 = \frac{1}{2}$, write down the values of $a_2$, $a_3$, $a_4$, $a_5$, observe and deduce the general formula for $a_n$;
(3) Prove that there exists a non-zero constant $p$, such that... | q = \frac{1}{2} |
33. To answer the question: "Is the difference $9^{1986}-8^{1980}$ divisible by 5 and by 10?" - one could call upon a computer. But perhaps you would like to compete with it? Give it a try. | 5 |
3. We will factorize the differences of cubes in the numerator and the sums of cubes in the denominator:
$$
A=\frac{(2-1)\left(2^{2}+2+1\right)}{(2+1)\left(2^{2}-2+1\right)} \cdot \frac{(3-1)\left(3^{2}+3+1\right)}{(3+1)\left(3^{2}-3+1\right)} \cdot \cdots \cdot \frac{(99-1)\left(99^{2}+99+1\right)}{(99+1)\left(99^{2}... | \frac{3367}{5050} |
## 216. Math Puzzle $5 / 83$
Three brothers aged 7, 12, and 15 have a total of 10 marks pocket money per week. The youngest receives 10 percent. The 12-year-old gets half of what the oldest receives.
How much money does each of the three boys have? | 1,3,6 |
## 3. Benches
In the park, there is a group of students. If 7 students sit on each bench in the park, 8 students will be left without a seat. If the students were to sit 9 to a bench, four benches would remain empty. How many students are in the park?
Result: $\quad 162$ | 162 |
Solve the following equations using the direct square root method:
①$x^2=2$
②$4x^2-1=0$
③$(x-1)^2-4=0$
④$12(3-x)^2-48=0$. | x_1=1, x_2=5 |
A $2 a$-sided square, from each of its vertices as the center, we draw quarter-circle arcs within the square with radius $a$. In the quadrilateral thus formed by the curved lines, we construct a square whose sides are tangents to the aforementioned arcs. The area of this square depends on the direction of its sides. In... | 4a^2(3-2\sqrt{2}) |
8. Let $I$ be the incenter of $\triangle A B C$, and
$$
3 \overrightarrow{I A}+4 \overrightarrow{I B}+5 \overrightarrow{I C}=\mathbf{0} \text {. }
$$
Then the size of $\angle C$ is $\qquad$ | 90 |
A 2. Find the maximum positive integer $k$ such that for any positive integers $m, n$ such that $m^{3}+n^{3}>$ $(m+n)^{2}$, we have
$$
m^{3}+n^{3} \geq(m+n)^{2}+k
$$
| 10 |
6. As shown in Figure 3, there are 81 uniformly spaced grid points, including the boundary points, in a square. $P$ is the center of the square. If a point $Q$ is randomly chosen from the remaining 80 grid points, the probability that the line $P Q$ is a line of symmetry of the square is ( ).
(A) $\frac{1}{5}$
(B) $\fr... | \frac{2}{5} |
Three. (50 points) The sequence $\left\{x_{n}\right\}$ satisfies
$$
x_{1}=3, x_{n+1}=\left[\sqrt{2} x_{n}\right]\left(n \in \mathbf{N}_{+}\right) \text {. }
$$
Find all $n$ such that $x_{n} 、 x_{n+1} 、 x_{n+2}$ form an arithmetic sequence, where $[x]$ denotes the greatest integer not exceeding the real number $x$.
保留... | 1 \text{ or } 3 |
8. The polar equation of a circle is $\rho=\sqrt{2}(\cos \theta+\sin \theta)$, the coordinates of the center of the circle are ( ).
A. $\left(2, \frac{\pi}{4}\right)$
B. $\left(1, \frac{\pi}{4}\right)$
C. $\left(1, \frac{3 \pi}{4}\right)$
D. $\left(\sqrt{2},-\frac{\pi}{4}\right)$ | (1,\frac{\pi}{4}) |
2. We have a pencil, a ruler, and a certain special device that, for any angle drawn on a plane, constructs two rays that divide this angle into three equal angles. Using these tools, construct an angle of $10^{0}$ on the plane. (Let's recall that a pencil can mark a point on the plane, in particular, the intersection ... | 10 |
2. The general solution of the equation $\cos \frac{x}{4}=\cos x$ is ( ), and in the interval $(0,24 \pi)$, there are ( ) distinct solutions. | 20 |
Given the function $f(x) = \log_{a}(2x - a)$, the function satisfies $f(x) > 0$ in the interval $[\frac{1}{2}, \frac{2}{3}]$. Determine the range of values for the real number $a$. | a \in (\frac{1}{3}, 1) |
2. Find all integer solutions of the inequality
$$
x^{2} y^{2}+y^{2} z^{2}+x^{2}+z^{2}-38(x y+z)-40(y z+x)+4 x y z+761 \leq 0
$$ | (6,2,7),(20,0,19) |
Example 5. Solve the system of equations
$$
\left\{\begin{array}{l}
x+y+z=\sqrt{x+y+z+1}+5, \\
\frac{x}{2}=\frac{y}{3}=\frac{z}{4} .
\end{array}\right.
$$ | x=\frac{16}{9}, \quad y=\frac{8}{3}, \quad z=\frac{32}{9} |
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