Let $A$ be the closed region bounded by the following three lines in the $xy$ plane: $x=1, y=0$ and $y=t(2x-t)$, where $0<t<1$. Prove that the area of any triangle inside the region $A$, with two vertices $P(t,t^2)$ and $Q(1,0)$, does not exceed $\frac{1}{4}.$

As shown in the figure below, $ABCD$ is a trapezoid, $AB \parallel CD$. The sides $DA$, $AB$, $BC$ are tangent to $\odot O_1$ and $AB$ touches $\odot O_1$ at $P$. The sides $BC$, $CD$, $DA$ are tangent to $\odot O_2$, and $CD$ touches $\odot O_2$ at $Q$. Prove that the lines $AC$, $BD$, $PQ$ meet at the same point. [asy][asy] size(200); defaultpen(linewidth(0.8)+fontsize(10pt)); pair A=origin,B=(1,-7),C=(30,-15),D=(26,6); pair bisA=bisectorpoint(B,A,D),bisB=bisectorpoint(A,B,C),bisC=bisectorpoint(B,C,D),bisD=bisectorpoint(C,D,A); path bA=A--(bisA+100*(bisA-A)),bB=B--(bisB+100*(bisB-B)),bC=C--(bisC+100*(bisC-C)),bD=D--(bisD+100*(bisD-D)); pair O1=intersectionpoint(bA,bB),O2=intersectionpoint(bC,bD); dot(O1^^O2,linewidth(2)); pair h1=foot(O1,A,B),h2=foot(O2,C,D); real r1=abs(O1-h1),r2=abs(O2-h2); draw(circle(O1,r1)^^circle(O2,r2)); draw(A--B--C--D--cycle); draw(A--C^^B--D^^h1--h2); label("$A$",A,NW); label("$B$",B,SW); label("$C$",C,dir(350)); label("$D$",D,dir(350)); label("$P$",h1,dir(200)); label("$Q$",h2,dir(350)); label("$O_1$",O1,dir(150)); label("$O_2$",O2,dir(300)); [/asy][/asy]

In a group of $m$ girls and $n$ boys, any two persons either know each other or do not know each other. For any two boys and any two girls, there are at least one boy and one girl among them,who do not know each other. Prove that the number of unordered pairs of (boy, girl) who know each other does not exceed $m+\frac{n(n-1)}{2}$.

Find the number of polynomials $f(x)=ax^3+bx$ satisfying both following conditions: (i) $a,b\in\{1,2,\ldots,2013\}$; (ii) the difference between any two of $f(1),f(2),\ldots,f(2013)$ is not a multiple of $2013$.

For any given positive numbers $a_1,a_2,\ldots,a_n$, prove that there exist positive numbers $x_1,x_2,\ldots,x_n$ satisfying $\sum_{i=1}^n x_i=1$, such that for any positive numbers $y_1,y_2,\ldots,y_n$ with $\sum_{i=1}^n y_i=1$, the inequality $\sum_{i=1}^n \frac{a_ix_i}{x_i+y_i}\ge \frac{1}{2}\sum_{i=1}^n a_i$ holds.

Let $S$ be a subset of $\{0,1,2,\ldots,98 \}$ with exactly $m\geq 3$ (distinct) elements, such that for any $x,y\in S$ there exists $z\in S$ satisfying $x+y \equiv 2z \pmod{99}$. Determine all possible values of $m$.

As shown in the figure, $\odot O_1$ and $\odot O_2$ touches each other externally at a point $T$, quadrilateral $ABCD$ is inscribed in $\odot O_1$, and the lines $DA$, $CB$ are tangent to $\odot O_2$ at points $E$ and $F$ respectively. Line $BN$ bisects $\angle ABF$ and meets segment $EF$ at $N$. Line $FT$ meets the arc $\widehat{AT}$ (not passing through the point $B$) at another point $M$ different from $A$. Prove that $M$ is the circumcenter of $\triangle BCN$.

Let $n$ ($\ge 4$) be an even integer. We label $n$ pairwise distinct real numbers arbitrarily on the $n$ vertices of a regular $n$-gon, and label the $n$ sides clockwise as $e_1, e_2, \ldots, e_n$. A side is called positive if the numbers on both endpoints are increasing in clockwise direction. An unordered pair of distinct sides $\left\{ e_i,e_j \right\}$ is called alternating if it satisfies both conditions: (i) $2 \mid (i+j)$; and (ii) if one rearranges the four numbers on the vertices of these two sides $e_i$ and $e_j$ in increasing order $a < b < c < d$, then $a$ and $c$ are the numbers on the two endpoints of one of sides $e_i$ or $e_j$. Prove that the number of alternating pairs of sides and the number of positive sides are of different parity.