Olympiad

Few Problems - 3

May 11, 2017 • 2 min read

Let $ABC$ be an acute angled triangle such that $\angle BAC = 45 ^o$ . Let $D$ be a point on $AB$ such that $CD \perp AB$ . Let $P$ be an internal point of the segment $CD$ . Prove that $AP\perp BC$ if and only if $|AP| = |BC|$ .
Show that there are infinitely many positive integers $n$ such that $n|3^{n}-1$ .
Show that $n^5+n^4+1$ is always composite for all natural numbers $n > 1$ . (Hint : Factorise $x^5+x^4+1$ )
Show that the number of natural numbers divisible by $k\in \mathbb{N}$ less than or equal to $n\in\mathbb{N}$ is $\lfloor n/k \rfloor$ .
For real numbers $a,b,c>0$ prove that
$3(a^2+b^2+c^2)\geq (a+b+c)^2\geq 3(ab+bc+ca)$ .

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