SMO 2012 Open Round 1

The sum of the squares of $50$ consecutive odd integers is $300850$. Find the largest odd integer whose square is the last term of this sum.

Find the value of $\displaystyle\sum_{k=3}^{1000}\lfloor\log_2 k\rfloor$.

Here $\lfloor x\rfloor$ denotes the greatest integer less than or equal to $x$, as defined in the source paper.

Given that $f(x)$ is a polynomial of degree $2012$, and that $f(k)=2/k$ for $k=1,2,3,\ldots,2013$, find $2014f(2014)$.

Find the total number of sets of positive integers $(x,y,z)$ with $x\lt y\lt z$ such that $x+y+z=203$.

There are a few integers $n$ such that $n^2+n+1$ divides $n^{2013}+61$. Find the sum of the squares of these integers.

The sequence $(a_n)_{n=1}^{\infty}$, with $a_1=a_2=2$, satisfies

$$\frac{2a_{n-1}a_n}{a_{n-1}a_{n+1}-a_n^2}=n^3-n\qquad(n=2,3,4,\ldots).$$

Find the integer closest to $\displaystyle\sum_{n=2}^{2011}\frac{a_{n+1}}{a_n}$.

Determine the largest even positive integer which cannot be expressed as the sum of two composite odd positive integers.

The lengths of the sides of a triangle are successive terms of a geometric progression. Let $A$ and $C$ be its smallest and largest interior angles respectively. The shortest side is $16$ cm and

$$\frac{\sin A-2\sin B+3\sin C}{\sin C-2\sin B+3\sin A}=\frac{19}{9}.$$

Find the perimeter in centimetres.

Find the least positive integer $n$ for which

$$x_1^3+x_2^3+\cdots+x_n^3=2002^{2002}$$

has integer solutions $(x_1,x_2,\ldots,x_n)$.

Let $\alpha_n$ be a real root of $nx^3+2x-n=0$, where $n$ is a positive integer. If $\beta_n=\lfloor(n+1)\alpha_n\rfloor$ for $n=2,3,4,\ldots$, find $\displaystyle\frac1{1006}\sum_{n=2}^{2013}\beta_n$.

Here $\lfloor x\rfloor$ denotes the greatest integer less than or equal to $x$, as defined in the source paper.

In the diagram, $D$ lies inside triangle $ABC$ such that $\angle BAD=\angle BCD$ and $\angle BDC=90^\circ$. Given $AB=5$, $BC=6$, and that $M$ is the midpoint of $AC$, find $8DM^2$.

Suppose the real numbers $x,y$ satisfy $x^3-3x^2+5x=1$ and $y^3-3y^2+5y=5$. Find $x+y$.

The product of two of the four roots of $x^4-18x^3+kx^2+200x-1984=0$ is $-32$. Determine $k$.

Determine the smallest integer $n\ge2$ such that $\sqrt{(n+1)(2n+1)/6}$ is an integer.

Let $f$ be a real-valued function on all real numbers such that, for any real $a,b$, $f(af(b))=ab$. Find $f(2011)$.

The real solutions of $x^3-4\lfloor x\rfloor=5$ are denoted by $x_1,x_2,\ldots,x_k$ for some positive integer $k$. Find $\displaystyle\sum_{i=1}^{k}x_i^3$.

Here $\lfloor x\rfloor$ denotes the greatest integer less than or equal to $x$, as defined in the source paper.

Determine the maximum integer solution of

$$\left\lfloor\frac{x}{1!}\right\rfloor+\left\lfloor\frac{x}{2!}\right\rfloor+\cdots+\left\lfloor\frac{x}{10!}\right\rfloor=1001.$$

Here $\lfloor x\rfloor$ denotes the greatest integer less than or equal to $x$, as defined in the source paper.

Let $A,B,C$ be the three angles of a triangle, and let $L$ be the maximum of $\sin3A+\sin3B+\sin3C$. Determine $\lfloor10L\rfloor$.

Here $\lfloor x\rfloor$ denotes the greatest integer less than or equal to $x$, as defined in the source paper.

Determine the number of integer triples $(x,y,z)$ satisfying $x^2+y^2+z^2=x^2y^2$.

Sets of $13$ distinct positive integers can have sum $2142$. Find the largest possible greatest common divisor of these $13$ integers.