SMO 2022 Junior Round 2

Let $n$ be a 6-digit number. The first 3 digits and the last 3 digits of $n$ form two 3-digit numbers. It turns out that the product of these two 3-digit numbers is one-seventh of the value of $n$. Find the value of $n$.

In triangle $ABC$, $M$ is the midpoint of $AC$, the bisector of $\angle A$ meets the side $BC$ at $P$ and $MB$ at $N$. Suppose $\angle ANM=90^\circ$. Prove that $CP=2PM$.

Let $n=p_1p_2\cdots p_k$ be the product of distinct primes $p_1,p_2,\ldots,p_k$, $k>1$. Find all possible values of $n$ if it is a multiple of $p_i-1$ for each $i=1,2,\ldots,k$.

Let $n$ be a positive integer. Prove that $$n\sqrt{19}\{n\sqrt{19}\}>1,$$ where $\{x\}$ is the fractional part of the number $x$. (For example, $\{4.3\}=0.3$.)

Let $n$ be an integer greater than $6$ such that if the integers $6,7,8,\ldots,n$ are divided into two groups, one of the groups contains the numbers $a,b,c$ with $ab=c$, where $a=b$ is allowed. Determine the smallest possible value of $n$.