SMO 2010 Open Round 1

Let $S$ be the set of all integers $n$ such that $\dfrac{8n^3-96n^2+360n-400}{2n-7}$ is an integer. Find $\displaystyle\sum_{n\in S}|n|$.

Determine the largest real $x$ such that $|x^2-4x-39601|\ge|x^2+4x-39601|$.

Given $x=\lfloor1^{1/3}\rfloor+\lfloor2^{1/3}\rfloor+\cdots+\lfloor7999^{1/3}\rfloor$, find $\lfloor x/100\rfloor$. Here $\lfloor y\rfloor$ denotes the greatest integer not exceeding $y$; for example, $\lfloor2.1\rfloor=2$, $\lfloor30\rfloor=30$, $\lfloor-10.5\rfloor=-11$.

Determine the smallest positive integer $C$ such that $6^n/n!\le C$ for every positive integer $n$.

Let $CD$ be a chord of circle $\Gamma_1$ and $AB$ a diameter of $\Gamma_1$ perpendicular to $CD$ at $N$, with $AN\gt NB$. A circle $\Gamma_2$ centred at $C$ with radius $CN$ intersects $\Gamma_1$ at $P,Q$, and segments $PQ,CD$ intersect at $M$. The radii of $\Gamma_1,\Gamma_2$ are $61,60$, respectively. Find $AM$.

Determine the minimum of $\displaystyle\sum_{k=1}^{50}x_k$ over all positive numbers $x_1,\ldots,x_{50}$ satisfying $\displaystyle\sum_{k=1}^{50}1/x_k=1$.

Find the sum of all positive integers $p$ for which $(x-p)(x-13)+4$ can be expressed as $(x+q)(x+r)$ for distinct integers $q,r$.

Let $p_k=1+1/k-1/k^2-1/k^3$ for positive integer $k$. Find the least positive integer $n$ such that $p_2p_3\cdots p_n\gt 2010$.

Let $B$ lie on the circle centred at $O$ with diameter $AC$, and let $D,E$ be the circumcentres of triangles $OAB,OBC$, respectively. Given $\sin\angle BOC=4/5$ and $AC=24$, find the area of triangle $BDE$.

Let $f(x)=x^3+3x^2+6x+14$ for all real $x$. Real numbers $a,b$ satisfy $f(a)=1$, $f(b)=19$. Find $(a+b)^2$.

If $\cot\alpha+\cot\beta+\cot\gamma=-4/5$, $\tan\alpha+\tan\beta+\tan\gamma=17/6$, and $\cot\alpha\cot\beta+\cot\beta\cot\gamma+\cot\gamma\cot\alpha=-17/5$, find $\tan(\alpha+\beta+\gamma)$.

The figure shows a road map connecting shopping malls $A,B$ in a city. Each side of a smallest square is a $1\text{ km}$ road. Regions $C,D$ are two large residential estates. Find the number of shortest routes from $A$ to $B$ along the roads shown.

Let $a_1=1$, $a_2=2$, and $a_{n+1}=\dfrac{2n}{n+1}a_n-\dfrac{n-1}{n+1}a_{n-1}$ for $n\ge2$. It is known that $a_n\gt 2+2009/2010$ for every $n\ge m$, where $m$ is a positive integer. Find the least $m$.

It is known that $\sqrt{9-8\sin50^\circ}=a+b\sin c^\circ$ for exactly one triple of positive integers $(a,b,c)$ with $0\lt c\lt 90$. Find $(b+c)/a$.

If $\alpha$ is a real root of $x^5-x^3+x-2=0$, find $[\alpha^6]$, where $[x]$ is the least positive integer not exceeding $x$.

A positive integer is called “cute” if it cannot be written as the difference of two square numbers. The first three cute integers are $1,2,4$. Find the $2010$th cute integer. A square number means the square of a positive integer; for example, $1,4,9,16$ are the first four square numbers.

Let $f(x)$ be a polynomial of degree $5$. When divided by $x-1,x-2,x-3,x-4,x^2-x-1$, respectively, it leaves remainders $3,1,7,36,x-1$. Find the square of the remainder when $f(x)$ is divided by $x+1$.

Find the number of ordered pairs of positive integers $(a,b)$ satisfying $100(a+b)=ab-100$. The pairs $(1,2)$ and $(2,1)$ are considered distinct.

Let $p=a^b+b^a$. If $a,b,p$ are all prime, what is $p$?

Evaluate $\displaystyle\sum_{k=1}^{2009}\left\lfloor\frac{11k}{2010}\right\rfloor$, where $\lfloor y\rfloor$ is the greatest integer not exceeding $y$.