Let $(x_n)$ be defined by $x_1=2$ and $x_{n+1}=\sqrt{x_n+8}-\sqrt{x_n+3}$ for $n\ge1$.
(a) Prove that $(x_n)$ has a finite limit and find it.
(b) Prove that $n\le x_1+x_2+\cdots+x_n\le n+1$ for every positive integer $n$.
Let $ABC$ be an acute scalene triangle and $D$ a point on side $BC$. Choose $E$ on side $AB$ and $F$ on side $AC$ so that $\angle DEB=\angle DFC$. The lines $DF,DE$ meet $AB,AC$ at $M,N$, respectively. Let $(I_1),(I_2)$ be the circumcircles of $DEM,DFN$, respectively. Let $(J_1)$ be internally tangent to $(I_1)$ at $D$ and tangent to $AB$ at $K$, and let $(J_2)$ be internally tangent to $(I_2)$ at $D$ and tangent to $AC$ at $H$. Let $P\ne D$ be the other intersection of $(I_1),(I_2)$, and $Q\ne D$ the other intersection of $(J_1),(J_2)$.
(a) Prove that $D,P,Q$ are collinear.
(b) The circumcircle of $AEF$ meets the circumcircle of $AHK$ and the line $AQ$ again at $G,L$, respectively, with $G,L\ne A$. Prove that the tangent at $D$ to the circumcircle of $DQG$ meets $EF$ at a point on the circumcircle of $DLG$.
An investor has two rectangular plots of land, each measuring $120\,\mathrm m\times100\,\mathrm m$.
(a) On the first plot, the investor wants to build a house with a rectangular footprint measuring $25\,\mathrm m\times35\,\mathrm m$, and nine circular flowerbeds outside the house, each of diameter $5\,\mathrm m$. Prove that wherever the nine flowerbeds are built first, enough space remains for the house.
(b) On the second plot, the investor wants a pond shaped as a convex polygon, such that from any point of the remaining land one can reach the pond's bank by walking no more than $5\,\mathrm m$. Prove that the pond's perimeter is at least $(440-20\sqrt2)\,\mathrm m$.
In the Cartesian plane $Oxy$, let $(C)$ be the graph of $y=\sqrt[3]{x^2}$. A variable line $d$ meets $(C)$ at three distinct points with abscissas $x_1,x_2,x_3$.
(a) Prove that $\sqrt[3]{\dfrac{x_1x_2}{x_3^2}}+\sqrt[3]{\dfrac{x_2x_3}{x_1^2}}+\sqrt[3]{\dfrac{x_3x_1}{x_2^2}}$ is constant.
(b) Prove that $\sqrt[3]{\dfrac{x_1^2}{x_2x_3}}+\sqrt[3]{\dfrac{x_2^2}{x_3x_1}}+\sqrt[3]{\dfrac{x_3^2}{x_1x_2}}\lt -\dfrac{15}4$.
Let $n,d$ be positive integers. Let $S_n(d)$ be the set of all ordered tuples $(x_1,\ldots,x_d)$ satisfying:
(i) $x_i\in\{1,2,\ldots,n\}$ for $1\le i\le d$;
(ii) $x_i\ne x_{i+1}$ for $1\le i\le d-1$;
(iii) there are no indices $1\le i\lt j\lt k\lt l\le d$ such that $x_i=x_k$ and $x_j=x_l$.
(a) Find $|S_3(5)|$.
(b) Prove that $S_n(d)$ is nonempty if and only if $d\le2n-1$.
Let $(x_n)$ be defined by $x_0=2$, $x_1=1$ and $x_{n+2}=x_{n+1}+x_n$ for all $n\ge0$.
(a) For $n\ge1$, prove that if $x_n$ is prime, then either $n$ is prime or $n$ has no odd prime divisor.
(b) Find all pairs of nonnegative integers $(m,n)$ such that $x_n$ is divisible by $x_m$.
Let $ABC$ be an acute scalene triangle with centroid $G$, inscribed in a circle $(O)$. Let $H_a,H_b,H_c$ be the feet of the altitudes from $A,B,C$, and let $D,E,F$ be the midpoints of $BC,CA,AB$, respectively. The rays $GH_a,GH_b,GH_c$ meet $(O)$ at $X,Y,Z$, respectively.
(a) Prove that the circumcircle of $XCE$ passes through the midpoint of $BH_a$.
(b) Let $M,N,P$ be the midpoints of $AX,BY,CZ$, respectively. Prove that $DM,EN,FP$ are concurrent.