Advanced book on Mathematics Olympiad

260 5 Number Theory

We left the better problems as exercises.

749.Prove that among any three distinct integers we can find two, sayaandb, such that

the numbera^3 b−ab^3 is a multiple of 10.

750.Show that the number 2002^2002 can be written as the sum of four perfect cubes, but

not as the sum of three perfect cubes.

751.The last four digits of a perfect square are equal. Prove that they are all equal to

zero.

752.Solve in positive integers the equation

2 x· 3 y= 1 + 5 z.

753.Define the sequence(an)nrecursively bya 1 =2,a 2 =5, and

an+ 1 =( 2 −n^2 )an+( 2 +n^2 )an− 1 forn≥ 2.

Do there exist indicesp, q, rsuch thatap·aq=ar?

754.For some integerk>0, assume that an arithmetic progressionan+b,n≥1, with

aandbpositive integers, contains thekth power of an integer. Prove that for any

integerm>0 there exist an infinite number of values ofnfor whichan+bis the

sum ofmkth powers of nonzero integers.

755.Given a positive integern>1000, add the residues of 2nmodulo each of the

numbers 1, 2 , 3 ,...,n. Prove that this sum is greater than 2n.

756.Prove that ifn≥3 prime numbers form an arithmetic progression, then the common

difference of the progression is divisible by any prime numberp<n.

757.LetP(x)=amxm+am− 1 xm−^1 +···+a 0 andQ(x)=bnxn+bn− 1 xn−^1 +···+b 0 be

two polynomials with each coefficientaiandbiequal to either 1 or 2002. Assuming

thatP(x)dividesQ(x), show thatm+1 is a divisor ofn+1.

758.Prove that ifnis a positive integer that is divisible by at least two primes, then there

exists ann-gon with all angles equal and with side lengths the numbers 1, 2 ,...,n

in some order.

759.Find all prime numbersphaving the property that when divided by every prime

numberq<pyield a remainder that is a square-free integer.

5.2.4 Fermat’s Little Theorem...................................

A useful tool for solving problems about prime numbers is a theorem due to P. Fermat.