There are several triangles. From them a new triangle is obtained according to the following rule. The largest side of the new triangle is equal to the sum of the large sides of the data, the middle one is equal to the sum of the middle sides, and the smallest one is the sum of the smaller ones. Prove that if all the angles of these triangles were less than $a$, and $\phi$, where $\phi$ is the largest angle of the resulting triangle, then $\cos \phi \ge 1-\sin (a/2)$.

The expression $ \frac{2x^2-x}{(x+1)(x-2)}-\frac{4+x}{(x+1)(x-2)}$ cannot be evaluated for $ x=-1$ or $ x=2$, since division by zero is not allowed. For other values of $ x$: $\textbf{(A)}\ \text{The expression takes on many different values.} \\ \textbf{(B)}\ \text{The expression has only the value 2.} \\ \textbf{(C)}\ \text{The expression has only the value 1.} \\ \textbf{(D)}\ \text{The expression always has a value between } -1 \text{ and } +2. \\ \textbf{(E)}\ \text{The expression has a value greater than 2 or less than } -1.$

$\alpha,\beta$ are two different solutions to the equation $4x^2-4tx+1=0(t\in\mathbb{R})$, the domain of definition of the function $f(x)=\frac{2x-t}{x^2+1}$ is $[\alpha,\beta](\alpha<\beta)$. [b](a)[/b] Find $g(t)=\max f(x)-\min f(x)$. [b](b)[/b] Prove: for $u_i\in\left(0,\frac{\pi}{2}\right)(i=1,2,3)$, if $\sin u_1+\sin u_2+\sin u_3=1$, then $\frac{1}{g(\tan u_1)}+\frac{1}{g(\tan u_2)}+\frac{1}{g(\tan u_3)}<\frac{3}{4}\sqrt6$.

Fix positive integers $n$ and $k$, and $2n$ positive (not neccesarily distinct) real numbers $a_1,\cdots, a_n$, $b_1, \cdots, b_n$. An equation is written on a whiteboard: $$t=*\times*\times\cdots\times*$$ where $t$ is a fixed positive real number, with exactly $k$ asterisks. Ebi fills each asterisk with a number from $a_1, a_2,\cdots, a_n$, while Rubi fills each asterisk with a number from $b_1, b_2,\cdots, b_n$, so that the equation on the whiteboard is correct. Suppose for every positive real number $t$, the number of ways for Ebi and Rubi to do so are equal. Prove that the sequences $a_1,\cdots, a_n$ and $b_1, \cdots, b_n$ are permutations of each other. [i](Note: $t=a_1a_2a_3$ and $t=a_2a_3a_1$ are considered different ways to fill the asterisks, and the chosen terms need not be distinct, for example $t=a_1a_1a_2$.)[/i] [i]Proposed by Wong Jer Ren[/i]

Given a sequence $\{x_k\}$ such that $x_1 = 1$, $x_{n+1} = n \sin x_n+ 1$. Prove that the sequence is non-periodic.

Points on a spherical surface with radius $4$ are colored in $4$ different colors. Prove that there exist two points with the same color such that the distance between them is either $4\sqrt{3}$ or $2\sqrt{6}$. (Distance is Euclidean, that is, the length of the straight segment between the points)

The side length of the largest square below is $8\sqrt{2}$, as shown. Find the area of the shaded region. [asy] size(10cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-18.99425911800572,xmax=23.81538435842469,ymin=-15.51769962526155,ymax=6.464951807764648; pen zzttqq=rgb(0.6,0.2,0.); pair A=(0.,1.), B=(0.,0.), C=(1.,0.), D=(1.,1.), F=(1.,2.), G=(2.,3.), H=(0.,3.), I=(0.,5.), J=(-2.,3.), K=(-4.,5.), L=(-4.,1.), M=(-8.,1.), O=(-8.,-7.), P=(0.,-7.); draw(B--A--D--C--cycle); draw(A--C--(2.,1.)--F--cycle); draw(A--(2.,1.)--G--H--cycle); draw(A--G--I--J--cycle); draw(A--I--K--L--cycle); draw(A--K--M--(-4.,-3.)--cycle); draw(A--M--O--P--cycle); draw(A--O--(0.,-15.)--(8.,-7.)--cycle); filldraw(A--B--C--D--cycle,opacity(0.2)+black); filldraw(A--(2.,1.)--F--cycle,opacity(0.2)+black); filldraw(A--G--H--cycle,opacity(0.2)+black); filldraw(A--I--J--cycle,opacity(0.2)+black); filldraw(A--K--L--cycle,opacity(0.2)+black); filldraw(A--M--(-4.,-3.)--cycle, opacity(0.2)+black); filldraw(A--O--P--cycle,opacity(0.2)+black); draw(B--A); draw(A--D); draw(D--C); draw(C--B); draw(A--C); draw(C--(2.,1.)); draw((2.,1.)--F); draw(F--A); draw(A--(2.,1.)); draw((2.,1.)--G); draw(G--H); draw(H--A); draw(A--G); draw(G--I); draw(I--J); draw(J--A); draw(A--I); draw(I--K); draw(K--L); draw(L--A); draw(A--K); draw(K--M); draw(M--(-4.,-3.)); draw((-4.,-3.)--A); draw(A--M); draw(M--O); draw(O--P); draw(P--A); draw(A--O); draw(O--(0.,-15.)); draw((0.,-15.)--(8.,-7.)); draw((8.,-7.)--A); draw(A--B,black); draw(B--C,black); draw(C--D,black); draw(D--A,black); draw(A--(2.,1.),black); draw((2.,1.)--F,black); draw(F--A,black); draw(A--G,black); draw(G--H,black); draw(H--A,black); draw(A--I,black); draw(I--J,black); draw(J--A,black); draw(A--K,black); draw(K--L,black); draw(L--A,black); draw(A--M,black); draw(M--(-4.,-3.),black); draw((-4.,-3.)--A,black); draw(A--O,black); draw(O--P,black); draw(P--A,black); label("$8\sqrt{2}$",(-8,-7)--(0,-15)); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); [/asy] [i]Lightning 2.4[/i]

A square has vertices $(0,10)$, $(0, 0)$, $(10, 0)$, and $(10,10)$ on the $x-y$ coordinate plane. A second quadrilateral is constructed with vertices $(0,10)$, $(0, 0)$, $(10, 0)$, and $(15,15)$. Find the positive difference between the areas of the original square and the second quadrilateral. [i]Proposed byWilliam Hua[/i]

A quadrilateral with sides $a,b,c,d$ is inscribed in a circle of radius $R$. Prove that if $a^2+b^2+c^2+d^2=8R^2$, then either one of the angles of the quadrilateral is right or the diagonals of the quadrilateral are perpendicular.

Determine all the triples $(a, b, c)$ of positive real numbers such that the system \[ax + by -cz = 0,\]\[a \sqrt{1-x^2}+b \sqrt{1-y^2}-c \sqrt{1-z^2}=0,\] is compatible in the set of real numbers, and then find all its real solutions.