RMO 2008 | Regional Mathematics Olympiad Problem

In this post, there are problems from Regional Mathematics Olympiad, RMO 2008. Try out these problems.

Let $ABC$ be an acute-angled triangle, let $D$, $F$ be the mid-points of $BC, AB$ respectively. Let the perpendicular from $F$ to $AC$ and the perpendicular at $B$ to $BC$ meet in $N$. Prove that $ND$ is equal to circum-radius of $ABC$.
Discussion
Prove that there exists two infinite sequences $ {\left \langle a_n \right \rangle}_{n \ge 1} $ and $ {\left \langle b_n \right \rangle}_{n \ge 1} $ of positive integers such that the following conditions holds simultaneously:
1. $1<a_1<a_2<a_3<..... $;
2. $a_n < b_n < a_n^{2} $, for all $n \ge 1 $;
3. $a_{n}-1 $ divides $ b_{n}-1 $, for all $n \ge 1 $;
4. $a_{n}^2-1 $ divides $b_{n}^2-1 $, for all $n \ge 1 $.
Suppose a and b are real numbers such that the roots of the cubic equation $ax^3-x^2+bx+1=0 $ are all positive real numbers. Prove that i) $0<3ab<1 $ and ii) $ b \ge \sqrt{3} $.
Find the number of all $6$-digits natural number such that the sum of their digits is $10$ and each of the digits $0,1,2,3$ occurs at least once in them.
Three non-zero numbers a,b,c are said to be in harmonic progression if $ \frac{1}{a}+\frac{1}{c}=\frac{2}{b} $. Find all three term harmonic progressions $a,b,c$ of strictly increasing positive integers in which $a=20$ and $b$ divides $c$.
Find the number of integer-sided isosceles obtuse-angled triangles with perimeter $2008$.
Discussion

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In this post, there are problems from Regional Mathematics Olympiad, RMO 2008. Try out these problems.

Let $ABC$ be an acute-angled triangle, let $D$, $F$ be the mid-points of $BC, AB$ respectively. Let the perpendicular from $F$ to $AC$ and the perpendicular at $B$ to $BC$ meet in $N$. Prove that $ND$ is equal to circum-radius of $ABC$.
Discussion
Prove that there exists two infinite sequences $ {\left \langle a_n \right \rangle}_{n \ge 1} $ and $ {\left \langle b_n \right \rangle}_{n \ge 1} $ of positive integers such that the following conditions holds simultaneously:
1. $1<a_1<a_2<a_3<..... $;
2. $a_n < b_n < a_n^{2} $, for all $n \ge 1 $;
3. $a_{n}-1 $ divides $ b_{n}-1 $, for all $n \ge 1 $;
4. $a_{n}^2-1 $ divides $b_{n}^2-1 $, for all $n \ge 1 $.
Suppose a and b are real numbers such that the roots of the cubic equation $ax^3-x^2+bx+1=0 $ are all positive real numbers. Prove that i) $0<3ab<1 $ and ii) $ b \ge \sqrt{3} $.
Find the number of all $6$-digits natural number such that the sum of their digits is $10$ and each of the digits $0,1,2,3$ occurs at least once in them.
Three non-zero numbers a,b,c are said to be in harmonic progression if $ \frac{1}{a}+\frac{1}{c}=\frac{2}{b} $. Find all three term harmonic progressions $a,b,c$ of strictly increasing positive integers in which $a=20$ and $b$ divides $c$.
Find the number of integer-sided isosceles obtuse-angled triangles with perimeter $2008$.
Discussion