Find all the real number triples $(x, y, z)$ satisfying the following system of inequalities under the condition $0 < x, y, z < \sqrt{2}$: $$y\sqrt{4-x^2y^2}\ge \frac{2}{\sqrt{xz}}$$ $$x\sqrt{4-x^2z^2}\ge \frac{2}{\sqrt{yz}}$$ $$z\sqrt{4-y^2z^2}\ge \frac{2}{\sqrt{xy}}$$.

Let $a_1>a_2>a_3>\cdots$ be a sequence of real numbers which converges to 0. We put circles of radius $a_1$ into a unit square until no more can fit. (A previously laid circle must not be moved.) Then we put circles of radius $a_2$ in the remaining space until no more can fit, continuing the process for $a_3$,... What can the area covered by the circles be? a similar problem involving circles in a square: [url]https://artofproblemsolving.com/community/c7h1979044[/url]

Let $ABC$ be a triangle with $AB < BC$. The bisector of angle $C$ meets the line parallel to $AC$ and passing through $B$, at point $P$. The tangent at $B$ to the circumcircle of $ABC$ meets this bisector at point $R$. Let $R'$ be the reflection of $R$ with respect to $AB$. Prove that $\angle R'P B = \angle RPA$.

Consider the sequence: $x_1=19,x_2=95,x_{n+2}=\text{lcm} (x_{n+1},x_n)+x_n$, for $n>1$, where $\text{lcm} (a,b)$ means the least common multiple of $a$ and $b$. Find the greatest common divisor of $x_{1995}$ and $x_{1996}$.

Evaluate \[\int_{-\pi}^{\pi} (\cos2x)(\cos 2^2x)\cdots (\cos 2^{2006}x)dx\]

Evaluate the integrals as follows. (1) $\int \frac{x^2}{2-x}\ dx$ (2) $\int \sqrt[3]{x^5+x^3}\ dx$ (3) $\int_0^1 (1-x)\cos \pi x\ dx$

Let $ m,n$ be two integers such that $ \varphi(m) \equal{}\varphi(n) \equal{} c$. Prove that there exist natural numbers $ b_{1},b_{2},\dots,b_{c}$ such that $ \{b_{1},b_{2},\dots,b_{c}\}$ is a reduced residue system with both $ m$ and $ n$.

Which of the following acceleration [i]vs.[/i] time graphs most closely represents the acceleration of the toy car? [asy] size(300); Label f; f.p=fontsize(8); xaxis(0,3); yaxis(-2,2); label("Time (s)",(2.8,0),0.5*N); label(rotate(90)*"Acceleration",(-0.2,0),W); label("$0$",(-0,0),SW,fontsize(9)); label("1",(1,0),2*S,fontsize(9)); label("2",(2,0),2*S,fontsize(9)); label("3",(3,0),2*S,fontsize(9)); draw((0.5,0)--(0.5,-0.1)); draw((1,0)--(1,-0.1)); draw((1.5,0)--(1.5,-0.1)); draw((2,0)--(2,-0.1)); draw((2.5,0)--(2.5,-0.1)); draw((3,0)--(3,-0.1)); label("(a)",(1.5,-2),N); pair A, B, C, D, E, F; A = (0,1); B = (1,1); C = (1,0); D = (1.5,0); E = (1.5, 0.5); F = (3, 0.5); draw(A--B--C--D--E--F); real x=6; Label f; f.p=fontsize(8); draw((x+3,0)--(x+0,0)); draw((x,-2)--(x,2)); label("Time (s)",(x+2.8,0.03),0.5*N); label(rotate(90)*"Acceleration",(x-0.2,0),W); label("$0$",(x+0,0),SW,fontsize(9)); label("1",(x+1,0),2*S,fontsize(9)); label("2",(x+2,0),2*S,fontsize(9)); label("3",(x+3,0),2*S,fontsize(9)); draw((x+0.5,0)--(x+0.5,-0.1)); draw((x+1,0)--(x+1,-0.1)); draw((x+1.5,0)--(x+1.5,-0.1)); draw((x+2,0)--(x+2,-0.1)); draw((x+2.5,0)--(x+2.5,-0.1)); draw((x+3,0)--(x+3,-0.1)); label("(b)",(x+1.5,-2),N); /*The lines*/ pair G, H, I, J, K, L; G = (x+0,1); H = (x+1,1); I = (x+1,0); J = (x+1.5,0); K = (x+1.5, -0.5); L = (x+3, -0.5); draw(G--H--I--J--K--L);[/asy][asy] size(300); Label f; f.p=fontsize(8); xaxis(0,3); yaxis(-2,2); label("Time (s)",(2.8,0),0.5*N); label(rotate(90)*"Acceleration",(-0.1,0),W); label("$0$",(-0,0),SW,fontsize(9)); label("1",(1,0),2*S,fontsize(9)); label("2",(2,0),2*S,fontsize(9)); label("3",(3,0),2*S,fontsize(9)); draw((0.5,0)--(0.5,-0.1)); draw((1,0)--(1,-0.1)); draw((1.5,0)--(1.5,-0.1)); draw((2,0)--(2,-0.1)); draw((2.5,0)--(2.5,-0.1)); draw((3,0)--(3,-0.1)); label("(c)",(1.5,-2),N); pair A, B, C, D, E, F; A = (0,0.5); B = (1,0.5); C = (1,0); D = (1.5,0); E = (1.5, -1); F = (3, -1); draw(A--B--C--D--E--F); real x = 6; Label f; f.p=fontsize(8); draw((x+3,0)--(x+0,0)); draw((x,-2)--(x,2)); label("Time (s)",(x+3.4,0),0.5*N); label(rotate(90)*"Acceleration",(x-0.2,0),W); label("$0$",(x+0,0),SW,fontsize(9)); label("1",(x+1,0),2*S,fontsize(9)); label("2",(x+2,0),2*S,fontsize(9)); label("3",(x+3,0),2*S,fontsize(9)); draw((x+0.5,0)--(x+0.5,-0.1)); draw((x+1,0)--(x+1,-0.1)); draw((x+1.5,0)--(x+1.5,-0.1)); draw((x+2,0)--(x+2,-0.1)); draw((x+2.5,0)--(x+2.5,-0.1)); draw((x+3,0)--(x+3,-0.1)); label("(d)",(x+1.5,-2),N); /*The lines*/ pair K, L, M, N, O, P, Q, R; K = (x+0,1); L = (x+1,1); M = (x+1,0.5); N= (x+1.5,0.5); O= (x+1.5, -0.5); P = (x+2.5, -0.5); Q = (x+2.5, 0.5); R = (x+3, 0.5); draw(K--L--M--N--O--P--Q--R);[/asy][asy] size(150); Label f; f.p=fontsize(8); xaxis(0,3); yaxis(-2,2); label("Time (s)",(3.2,0.03),N); label(rotate(90)*"Acceleration",(-0.1,0),W); label("$0$",(-0,0),SW,fontsize(9)); label("1",(1,0),2*S,fontsize(9)); label("2",(2,0),2*S,fontsize(9)); label("3",(3,0),2*S,fontsize(9)); draw((0.5,0)--(0.5,-0.1)); draw((1,0)--(1,-0.1)); draw((1.5,0)--(1.5,-0.1)); draw((2,0)--(2,-0.1)); draw((2.5,0)--(2.5,-0.1)); draw((3,0)--(3,-0.1)); label(rotate(90)*"Acceleration",(-0.1,0),W); label("(e)",(1.5,-2),N); /*The lines*/ pair A, B, C, D, E, F, G, H; A = (0,1); B = (1,1); C = (1,0.5); D = (1.5,0.5); E = (1.5, -0.5); F = (2.5, -0.5); G = (2.5, 0.5); H = (3, 0.5); draw(A--B--C--D--E--F--G--H); [/asy]

Define a sequence $a_n$ satisfying : \[a_1=1,\ \ a_{n+1}=\frac{na_n}{2+n(a_n+1)}\ (n=1,\ 2,\ 3,\ \cdots).\] Find $\lim_{m\to\infty} m\sum_{n=m+1}^{2m} a_n.$

There are $m \geq 3$ positive integers, not necessarily distinct, that are arranged in a circle so that any positive integer divides the sum of its neighbours. Show that if there is exactly one $1$, then for any positive integer $n$, there are at most $\phi(n)$ copies of $n$. [i]Proposed By- (usjl, adapted from 2014 Taiwan TST)[/i]

Let $f(x) = x^3 + ax^2 + bx + c$ have solutions that are distinct negative integers. If $a+b+c = 2014$, find $c$.

There are $ 63$ points arbitrarily on the circle $ \mathcal{C}$ with its diameter being $ 20$. Let $ S$ denote the number of triangles whose vertices are three of the $ 63$ points and the length of its sides is no less than $ 9$. Fine the maximum of $ S$.

The positive integers $v, w, x, y$, and $z$ satisfy the equation \[v + \frac{1}{w + \frac{1}{x + \frac{1}{y + \frac{1}{z}}}} = \frac{222}{155}.\] Compute $10^4 v + 10^3 w + 10^2 x + 10 y + z$.

We draw $n$ convex quadrilaterals in the plane. They divide the plane into regions (one of the regions is infinite). Determine the maximal possible number of these regions.

Let $ABCDEF$ be a convex equilateral hexagon such that lines $BC$, $AD$, and $EF$ are parallel. Let $H$ be the orthocenter of triangle $ABD$. If the smallest interior angle of the hexagon is $4$ degrees, determine the smallest angle of the triangle $HAD$ in degrees.

There are $n$ cities in Qingqiu Country. The distance between any two cities are different. The king of the country plans to number the cities and set up two-way air lines in such ways: The first time, set up a two-way air line between city 1 and the city nearest to it. The second time, set up a two-way air line between city 2 and the city second nearest to it. ... The $n-1$th time, set up a two-way air line between city $n-1$ and the city farthest to it. Prove: The king can number the cities in a proper way so that he can go to any other city from any city by plane.

Let $A$, $B$, $C$, $D$, $E$, and $F$ be $6$ points around a circle, listed in clockwise order. We have $AB = 3\sqrt{2}$, $BC = 3\sqrt{3}$, $CD = 6\sqrt{6}$, $DE = 4\sqrt{2}$, and $EF = 5\sqrt{2}$. Given that $\overline{AD}$, $\overline{BE}$, and $\overline{CF}$ are concurrent, determine the square of $AF$.

The parallelepiped contains a sphere of radius $r$ and is contained within a sphere of radius $R$. Prove that $ \frac{R}{r} \geq \sqrt{3} $.

Suppose that $p(n)$ is the number of partitions of a natural number $n$. Prove that there exists $c>0$ such that $P(n)\ge n^{c \cdot \log n}$. [i]proposed by Mohammad Mansouri[/i]

In rectangle $ABCD$, points $E$ and $F$ are on sides $\overline{BC}$ and $\overline{AD}$, respectively. Two congruent semicircles are drawn with centers $E$ and $F$ such that they both lie entirely on or inside the rectangle, the semicircle with center $E$ passes through $C$, and the semicircle with center $F$ passes through $A$. Given that $AB=8$, $CE=5$, and the semicircles are tangent, find the length $BC$. [i]Proposed by Ada Tsui[/i]

[color=darkred] Let $(R,+,\cdot)$ be a ring and let $f$ be a surjective endomorphism of $R$ such that $[x,f(x)]=0$ for any $x\in R$ , where $[a,b]=ab-ba$ , $a,b\in R$ . Prove that: [list] [b]a)[/b] $[x,f(y)]=[f(x),y]$ and $x[x,y]=f(x)[x,y]$ , for any $x,y\in R\ ;$ [b]b)[/b] If $R$ is a division ring and $f$ is different from the identity function, then $R$ is commutative. [/list] [/color]

An integer number $n \geq 2$ is called [i]feirense[/i] if it is possible to write on a sheet of paper some integers such that every positive divisor of $n$ less than $n$ is the difference between two numbers on the sheet, and no other positive number is. Find all the feirense numbers.

prove the third sylow theorem: suppose that $G$ is a group and $|G|=p^em$ which $p$ is a prime number and $(p,m)=1$. suppose that $a$ is the number of $p$-sylow subgroups of $G$ ($H<G$ that $|H|=p^e$). prove that $a|m$ and $p|a-1$.(Hint: you can use this: every two $p$-sylow subgroups are conjugate.)(20 points)

Prove that for any sequence $a_1, a_2, \ldots , a_{2002}$ of non-negative integers written in the usual decimal notation with $a_1 > 0$ there exists an integer $n$ such that $n^2$ starts with digits $a_1, a_2, \ldots , a_{2002}$ (in this order).

A pair of positive integers $(a,b)$ is called an [b]average couple[/b] if there exist positive integers $k$ and $c_1, \dots, c_k$ for which \[\frac{c_1+c_2+\cdots+c_k}{k}=a\qquad \text{and} \qquad \frac{s(c_1)+s(c_2)+\cdots+s(c_k)}{k}=b\] where $s(n)$ denotes the sum of digits of $n$ in decimal representation. Find the number of average couples $(a,b)$ for which $a,b<10^{10}$.