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IMO Shortlist 2007 problem N6

Let $k$ be a positive integer. Prove that the number $(4 \cdot k^2 - 1)^2$ has a positive divisor of the form $8kn - 1$ if and only if $k$ is even.

Actual IMO 2007 Problem, posed as question 5 in the contest, which was used as a lemma in the official solutions for problem N6 as shown above.

Author: Kevin Buzzard and Edward Crane, United Kingdom

Slični zadaci

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Zadaci

IMO Shortlist 2000 problem N3

Does there exist a positive integer $n$ such that $n$ has exactly 2000 prime divisors and $n$ divides $2^n + 1$ ?

IMO Shortlist 2002 problem N6

Find all pairs of positive integers $m,n\geq3$ for which there exist infinitely many positive integers $a$ such that $\frac{a^m+a-1}{a^n+a^2-1}$ is itself an integer.

Laurentiu Panaitopol, Romania

IMO Shortlist 2003 problem N6

Let $p$ be a prime number. Prove that there exists a prime number $q$ such that for every integer $n$ , the number $n^p-p$ is not divisible by $q$ .

IMO Shortlist 2004 problem N5

We call a positive integer alternating if every two consecutive digits in its decimal representation are of different parity.

Find all positive integers $n$ such that $n$ has a multiple which is alternating.

IMO Shortlist 2006 problem N4

Let $P(x)$ be a polynomial of degree $n > 1$ with integer coefficients and let $k$ be a positive integer. Consider the polynomial $Q(x) = P(P(\ldots P(P(x)) \ldots ))$ , where $P$ occurs $k$ times. Prove that there are at most $n$ integers $t$ such that $Q(t) = t$ .

IMO Shortlist 2008 problem N6

Prove that there are infinitely many positive integers $n$ such that $n^{2} + 1$ has a prime divisor greater than $2n + \sqrt {2n}$ .

Author: Kestutis Cesnavicius, Lithuania