My Problems

Some of My Problems

M. Jamaali 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 e1 a1 + e2 a2 + · · · + en an 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 follo followin wingg condit conditions ions:: (1) the set of prime prime diviso divisors rs of the element elementss 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 funct function ion 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 Prove that that f (n) = n for each n ∈ N.

(N3 in Shortlist Problems for IMO 2004, Greece) 4. We call a positive positive integer integer alternate if its decimal digits are alternately alternately odd and even. 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 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 divide any of the numbers numbers 3 a1 , 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 Find all function functionss f  from the set of positive integers m, n ∈

N

N

into itself such that for all

the number m + n is divisible by f (m) + f (n).

(Second Round of the Iranian Mathematical Olympiad, 2004) 10. We call a positive positive integer 3-partite 3-partite if the set of it’s divisors divisors can be partitioned partitioned into three subsets subsets whose sum of elements elements are equal. 1) Find a 3-partite 3-partite number. number. 2) Show that there exist infinitely many 3-partite numbers. (Second Round of the Iranian Mathematical Olympiad, 2003) 11. Show Show that for each positive 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 ≤ ja i . (Second Round of the Iranian Mathematical Olympiad, 2009)

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13. Find all integer integer polynomials polynomials f (x) such that for each positive integers m, n, we have m|n if and only if  f (m)|f (n).

(Summe (Summerr Camp Camp of Mathem Mathemati atical cal Olympiad Olympiad,, 2003 2003.. Cited Cited in Book ,

Problem Problems s From rom The

by Titu Andreescu)

14. We are given positive integers integers a1 , a2 , . . . , an , mutually relatively prime, such that for each positive integer k, with 1 ≤ k ≤ n we have k a1 + a2 + · · · + an |ak1 + ak2 + · · · + an

Find them. (Iran Team Selection Test 2006) 15. Show Show that that there there dose not exist an infinit infinitee subset subset A of  N such for each x, y ∈ A, x2 − xy + y 2 |(xy )2 .

(Summer Camp of Mathematical Olympiad, 2002) 16. Positive Positive integers integers a,b,c are given such that an + bn + cn is a prime number, show that a = b = c = 1.

17. Find all polynomial polynomialss 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 + a22 + · · · + a2φ(n) .

19. Find all integer integer solutions solutions of  p3 = p2 + q 2 + r 2 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 positive integers integers such that the number of positive divisors of  na. (Summer Camp of Mathematical Olympiad, 2002) 3

p −1  p−1 a

= 2n . Find

21. A positiv positivee integ integer er k is given. given. Find Find all function functionss f  : m, n ∈

N,

N

−→

N

such that for each

we have f (m) + f (n)|(m + n)k .

(Iran Team Selection Test 2008 ) 22. Find all polynomials polynomials P (x) with integer coefficients such that if  a and b are natural numbers 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 Find all monic monic polynomia polynomials ls 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 bm cm 1 m−n + + (a + bm−n + cm−n ) ≥ n n n n (b + c) (a + c) (a + b) 2

(Iran Team Selection Test 2001) 25. Does there exist a strictly increasing 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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