## 21 Jun Few problems - 4

1.  Find all integers $a,b,c$ satisfying $1 such that $(a-1)(b-1)(c-1)$ is a divisor of $abc-1$.
2. Find the number of ordered pairs $(x,y)$ of positive integers which satisfy $xy=27027$.
3. 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 ?
4. 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 $|AP|=|BC|$.
5. 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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#### Debashish Sharma

Assistant Professor at Gurucharan College
Dr. Debashish Sharma is an Assistant Professor in the Department of Mathematics, Gurucharan College, Silchar, Assam.

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• ##### ramesh nath jogi
Posted at 21:27h, 09 July Reply

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