Number Theory Problems
Amir Hossein Parvardi ∗
June 16, 2011
I’ve written the source of the problems beside their numbers. If you need
solutions, visit AoPS Resources Page, select petition, select the year
and go to the link of the problem. All (except very few) of these problems have
been posted by Orlando Doehring (orl).
Contents
1 Problems 1
IMO Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
ISL and ILL Problems . . . . . . . . . . . . . . . . . . . . . . . . 4
petitions . . . . . . . . . . . . . . . . . . . . . . . . . 8
China IMO Team Selection Test Problems . . . . . . . . . 8
Other Problems . . . . . . . . . . . . . . . . . . . . . . . . 8
1 Problems
IMO Problems
1. (IMO 1974, Day 1, Problem 3) Prove that for any n natural, the number
n
2n + 1
23k
2k + 1
Xk=0
cannot be divided by 5.
2. (IMO 1974, Day 2, Problem 3) Let P (x) be a polynomial with integer
coefficients. We denote deg(P ) its degree which is 1. Let n(P ) be the number
of all the integers k for which we have (P (k))2 = 1.≥Prove that n(P ) deg(P )
2. −≤
3. (IMO 1975, Day 1, Problem 2) Let a1, . . . , an be an infinite sequence of
strictly positive integers, so that ak < ak+1 for any k. Prove that there exists
an infinity of terms am, which can be written like am = x ap + y aq with x, y
strictly positive integers and p = q. · ·
6
∗email: ahpwsog@, blog: http://math-olympiad..
1
4. (IMO 1976, Day 2, Problem 4) Determine the greatest number, who is
the product of some positive integers, and the sum of these numbers is 1976.
5. (IMO 1977, Day 1, Problem 3) Let n be a given number greater than 2.
We consider the set Vn of all the integers of the form 1 + kn with k = 1, 2, . . . A
number m from Vn is called posable in Vn if there are not two numbers
p and q from Vn so that m = pq. Prove that there exist a number r Vn that
∈
can be expressed as the product of elements posable in Vn in more than
one way.
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