## 10 May Few Problems – 3

1. 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 $|AP| = |BC|$.
2. Show that there are infinitely many positive integers $n$ such that $n|3^{n}-1$.
3. Show that $n^5+n^4+1$ is always composite for all natural numbers $n > 1$. (Hint : Factorise $x^5+x^4+1$)
4. Show that the number of natural numbers divisible by $k\in \mathbb{N}$ less than or equal to $n\in\mathbb{N}$ is $\lfloor n/k \rfloor$.
5. 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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• ##### R gogoi
Posted at 08:16h, 17 May Reply

Please update Higher Secondary final year Science Question paper 2017