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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)

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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)

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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)

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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)

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