(a) Determine a nonzero polynomial in $x$ of least degree with integer coefficients having $\sqrt2+\sqrt[3]3$ as a root.
(b) Solve
$$1+\sqrt{1+x^2}\left(\sqrt{(1+x)^3}-\sqrt{(1-x)^3}\right)=2+\sqrt{1-x^2}.$$
Let $u_1=1$, $u_2=2$, $u_{n+1}=3u_n-u_{n-1}$ for $n=2,3,\ldots$. Define $v_n=\sum_{k=1}^{n}\operatorname{arccot}u_i$ as printed. Determine $\lim_{n\to+\infty}v_n$.
In a plane $P$, $ABCD$ is a square of side $a$. On the ray $Az$ perpendicular to $P$, choose $S$ with $SA=2a$.
(a) Points $M,N$ move on $BC,DC$, respectively.
(i) Find their positions such that $BM+DN\ge3a/2$, the planes $SAM$ and $SMN$ are perpendicular, and $BM\cdot DN$ is least.
(ii) Find their positions such that $\angle NAM=45^\circ$ and the volume of tetrahedron $SAMN$ is greatest or least; calculate these volumes.
(b) A moving point $Q$ sees both $AB$ and $AD$ at right angles. Let $\pi$ be the plane perpendicular to $P$ along $AB$, and let $DQ$ meet $\pi$ at $Q'$.
(i) Find the locus of $Q'$.
(ii) The points $Q$ trace a circle $(K)$. Let $R\ne Q$ be the second intersection of $CQ$ and $(K)$. Prove that $\sin^2\angle Q'DB+\sin^2\angle R'DB$ is constant, where $R'=DR\cap\pi$.
(a) For integers $x,y$, not both zero, find the least value of $A=|5x^2+11xy-5y^2|$.
(b) Find all positive numbers $t$ such that $9t=10[t]/(t-[t])$, where $[t]$ denotes the integer part of $t$.
Let $Sxyz$ be a trihedral angle with vertex $S$, where $\angle xSz=\alpha$, $\angle xSy=\beta$, $\angle ySz=\gamma$. Let $A,B,C$ be its dihedral angles at edges $Sy,Sz,Sx$, respectively.
(a) Prove that (i) $\sin\alpha/\sin A=\sin\beta/\sin B=\sin\gamma/\sin C$; (ii) $\alpha+\beta=90^\circ$ if and only if $A+B=180^\circ$.
(b) Suppose $\alpha=\beta=\gamma=90^\circ$. Fix $O$ on $Sz$ with $SO=a$. Points $M,N$ move on $Sx,Sy$ with $SM+SN=a$. Prove that $\angle SOM+\angle SON+\angle MON$ is constant. Find the locus of the center $I$ of the sphere circumscribed about the quadrilateral $OSMN$ (source wording).