8/14/2019 My Problems
http://slidepdf.com/reader/full/my-problems 1/4
Some of My Problems
M. Jamaali
Department of Mathematical Sciences, Sharif University of Technology,
Young Scholars Club,
mjamaali@sharif .edu
September 23, 2009
1. Let m, n ≥ 2 be positive integers, and let a1, a2, . . . , an be integers, none of which is
a multiple of mn−1. Show that there exist integers e1, e2, . . . , en , not all zero, with
|ei| < m for all i, such that e1a1 + e2a2 + · · · + enan is a multiple of mn.
(N5 in Shortlist Problems for IMO 2002, Britain)
2. Let p be a prime number and let A be a set of positive integers that satisfies the
following conditions: (1) the set of prime divisors of the elements in A consists of
p − 1 elements; (2) for any nonempty subset of A, the product of its elements is not
a perfect pth power. What is the largest possible number of elements in A?
(N8 in Shortlist Problems for IMO 2003, Japan)
3. A function f from the set of positive integers N into itself is such that for all m, n ∈ N
the number (m2 + n)2 is divisible by f 2(m) + f (n). Prove that f (n) = n for each
n ∈ N.
(N3 in Shortlist Problems for IMO 2004, Greece)
4. We call a positive integer alternate if its decimal digits are alternately odd and even.
Find all positive integers n such that n has an alternate multiple.
(Problem 6 in IMO 2004, Greece, with Armin Morabbi)
1
8/14/2019 My Problems
http://slidepdf.com/reader/full/my-problems 2/4
5. Let a and b be positive integers such that an + n divides bn + n for every positive
integer n. Show that a = b.(N6 in Shortlist Problems for IMO 2005, Mexico)
6. Find all surjective functions f : N −→ N such that for every m, n ∈ N and every
prime p, the number f (m + n) is divisible by p if and only if f (m) + f (n) is divisible
by p.
(N5 in Shortlist Problems for IMO 2007, Vietnam, with N. Ahmadipour)
7. Let a1, a2, . . . , an be distinct positive integers, n ≥ 3. Prove that there exist distinct
indices i and j such that ai+a j does not divide any of the numbers 3a1, 3a2, . . . , 3an.
(N2 in Shortlist Problems for IMO 2008, Spain)
8. Let a be a positive integer such that 4(an + 1) is a perfect cube for each positive
integer n. Show that a = 1.
(Second Round of the Iranian Mathematical Olympiad, 2008)
9. Find all functions f from the set of positive integers N into itself such that for all
m, n ∈ N the number m + n is divisible by f (m) + f (n).
(Second Round of the Iranian Mathematical Olympiad, 2004)
10. We call a positive integer 3-partite if the set of it’s divisors can be partitioned into
three subsets whose sum of elements are equal. 1) Find a 3-partite number. 2) Show
that there exist infinitely many 3-partite numbers.
(Second Round of the Iranian Mathematical Olympiad, 2003)
11. Show that for each positive integer n we can find n distinct positive integers such
that their sum is a perfect square and their product is a perfect cube.
(Second Round of the Iranian Mathematical Olympiad, 2007)
12. Positive integers a1 < a2 < .. . < an are given, and for each i, j, (i = j) ai is divisible
by ai − a j . Show that if i < j, then ia j ≤ jai.
(Second Round of the Iranian Mathematical Olympiad, 2009)
2
8/14/2019 My Problems
http://slidepdf.com/reader/full/my-problems 3/4
13. Find all integer polynomials f (x) such that for each positive integers m, n, we have
m|n if and only if f (m)|f (n).(Summer Camp of Mathematical Olympiad, 2003. Cited in Problems From The
Book , by Titu Andreescu)
14. We are given positive integers a1, a2, . . . , an, mutually relatively prime, such that for
each positive integer k, with 1 ≤ k ≤ n we have
a1 + a2 + · · · + an|ak1 + ak2 + · · · + ak
n
Find them.
(Iran Team Selection Test 2006)
15. Show that there dose not exist an infinite subset A of N such for each x, y ∈ A,
x2 − xy + y2|(xy)2.
(Summer Camp of Mathematical Olympiad, 2002)
16. Positive integers a,b,c are given such that an + bn + cn is a prime number, show that
a = b = c = 1.
17. Find all polynomials f ∈ Z[x] such that for each a,b,c ∈ N
a + b + c|f (a) + f (b) + f (c)
(Summer Camp of Mathematical Olympiad, 2008)
18. Let n be a positive integer such that (n, 6) = 1. Let {a1, a2, . . . , aφ(n)} be a reduced
residue system for n. Prove that
n|a21 + a2
2 + · · · + a2φ(n).
19. Find all integer solutions of p3 = p2 + q2 + r2 where p , q , r are prime numbers.
(Summer Camp of Mathematical Olympiad, 2004)
20. Let p be a prime integer and a and n be positive integers such that pa−1 p−1 = 2n. Find
the number of positive divisors of na.
(Summer Camp of Mathematical Olympiad, 2002)
3
8/14/2019 My Problems
http://slidepdf.com/reader/full/my-problems 4/4
21. A positive integer k is given. Find all functions f : N −→ N such that for each
m, n ∈ N, we have f (m) + f (n)|(m + n)k.(Iran Team Selection Test 2008 )
22. Find all polynomials P (x) with integer coefficients such that if a and b are natural
numbers whose sum a + b is a perfect square, then P (a) + P (b) is a perfect square.
(Iran Team Selection Test 2008)
23. Find all monic polynomials f (x) ∈ Z[x] such that the set f (Z) is closed under
multiplication.
(Iran Team Selection Test 2007)
24. Let m, n ∈ N and a,b,c be positive real numbers. Show that
am
(b + c)n+
bm
(a + c)n+
cm
(a + b)n≥
1
2n(am−n + bm−n + cm−n)
(Iran Team Selection Test 2001)
25. Does there exist a strictly increasing function f : N\{1} −→ N, such that f (n2) =
f (n)2 for each positive integer n, and f (k) + k is an odd integer for each positive
integer k > 1?
(Summer Camp of Mathematical Olympiad, 2002)
4
Top Related