Few problems - 4

Jun 22, 2017 • 2 min read

Find all integers $a,b,c$ satisfying $1<a<b<c$ such that $(a-1)(b-1)(c-1)$ is a divisor of $abc-1$ .
Find the number of ordered pairs $(x,y)$ of positive integers which satisfy $xy=27027$ .
The integer $N$ consists of 2017 consecutive nines. A computer calculates $N^3 = (999 . . . 999)^3$ . How many nines does the number $N^3$ contain in total ?
Let $ABC$ be an acute angled triangle such that $\angle BAC=45^0$ . 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 $|AP|=|BC|$ .
Let $f(x)=x^n+5x^{n-1}+3$ where $n>1$ is an integer. Prove that $f(x)$ cannot be expressed as the product of two non-constant polynomials with integer coefficients.

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