SMO 2009 Open Round 1

The expression $1000\sin10^\circ\cos20^\circ\cos30^\circ\cos40^\circ$ can be simplified as $a\sin b^\circ$, where $a,b$ are positive integers with $0\lt b\lt 90$. Find $100a+b$.

Let $A_1,A_2,A_3,A_4,A_5,A_6$ be six points on a circle in this order, with arc lengths $\widehat{A_1A_2}=\widehat{A_2A_3}$, $\widehat{A_3A_4}=\widehat{A_4A_5}$, and $\widehat{A_5A_6}=\widehat{A_6A_1}$. Here each hat denotes the length of the corresponding consecutive arc. Given $\angle A_1A_3A_5=72^\circ$, find $\angle A_4A_6A_2$ in degrees.

Let $P_1,P_2,\ldots,P_{41}$ be $41$ distinct points on the segment $BC$ of triangle $ABC$, where $AB=AC=7$. Evaluate $\displaystyle\sum_{i=1}^{41}(AP_i^2+P_iB\cdot P_iC)$.

Determine the largest real $x$ for which $|x^2-11x+24|+|2x^2+6x-56|=|x^2+17x-80|$.

Let $f(x)=a_0+a_1x+a_2x^2+a_3x^3+a_4x^4+a_5x^5$ be a polynomial, with constant coefficients and $a_5=7$. When divided by $x-2004,x-2005,x-2006,x-2007,x-2008$, respectively, it leaves remainders $72,-30,32,-24,24$. Find $f(2009)$.

Find $\displaystyle\frac{\sin80^\circ}{\sin20^\circ}-\frac{\sqrt3}{2\sin80^\circ}$.

Determine the number of eight-digit positive integers such that, after deleting any one digit, the remaining seven-digit number is divisible by $7$.

It is given that $\sqrt a-\sqrt b=20$, where $a,b$ are real numbers. Find the maximum possible value of $a-5b$.

Let $ABC$ be a triangle with $AB=7$, $BC=8$, $AC=9$. A unique circle touches the side $AC$ and the lines $BA$ produced and $BC$ produced. Let $D$ be its centre. Find $BD^2$.

If $x=\dfrac12\left(\sqrt[3]{2009}-\dfrac1{\sqrt[3]{2009}}\right)$, find $\left(x+\sqrt{1+x^2}\right)^3$.

Let $S=\{1,2,3,\ldots,30\}$. Determine the number of vectors $(x,y,z,w)$ with $x,y,z,w\in S$ such that $x\lt w$ and $y\lt z\lt w$.

Let $f(n)$ be the number of zeros in the decimal representation of the positive integer $n$. For example, $f(10001123)=3$ and $f(1234567)=0$. Find $f(1)+f(2)+\cdots+f(99999)$.

Let $k$ be a positive integer not exceeding $99$ such that there are no natural numbers $x,y$ satisfying $x^2-ky^2=8$. Find the difference between the maximum and minimum possible values of $k$.

Let $S=\{1,2,3,4,\ldots,16\}$. Each of the subsets $\{6\}$, $\{1,2,3\}$, $\{5,7,9,10,11,12\}$, $\{1,2,3,4,5,6,7,8,9\}$ has sum divisible by $3$. Find the total number of nonempty subsets $A\subseteq S$ whose element sum is divisible by $3$.

A function $f:\mathbb R\to\mathbb R$ satisfies $f(x)f(y)=f(2xy+3)+3f(x+y)-3f(x)+6x$ for all real $x,y$. Find $f(2009)$.

Let $(a_n)$ be positive integers with $a_1=1$, $a_2=2009$, and $a_{n+2}a_n-a_{n+1}^2-a_{n+1}a_n=0$ for $n\ge1$. Determine $\dfrac{a_{993}}{100a_{991}}$.

Determine the number of ways of tiling a $4\times9$ rectangle with $1\times2$ tiles.

Find the number of seven-digit positive integers whose digits from left to right are non-increasing. Examples include $9998766$ and $5555555$; $7776556$ is not non-increasing.

Determine the largest prime less than $5000$ of the form $a^n-1$, where $a,n$ are positive integers and $n\gt 1$.

Determine the least constant $M$ such that $\displaystyle\frac{x_1}{x_1+x_2}+\frac{x_2}{x_2+x_3}+\cdots+\frac{x_{2009}}{x_{2009}+x_1}\lt M$ for all positive real $x_1,\ldots,x_{2009}$.