MathDB

5

Part of 2015 IFYM, Sozopol

Problems(5)

Problem 5 of First round

Source: VI International Festival of Young Mathematicians Sozopol, Theme for 10-12 grade

1/13/2020
Does there exist a natural number nn with exactly 3 different prime divisors pp, qq, and rr, so that p−1∣np-1\mid n, qr−1∣nqr-1\mid n, q−1∤nq-1\nmid n, r−1∤nr-1\nmid n, and 3∤q+r3\nmid q+r?
number theoryprime divisorsprime numbersDivisibility
Problem 5 of Third round - Divisibility of a^n+b^n+c^n+d^n

Source: VI International Festival of Young Mathematicians Sozopol, Theme for 10-12 grade

12/24/2019
Let p>3p>3 be a prime number. The natural numbers a,b,c,da,b,c, d are such that a+b+c+da+b+c+d and a3+b3+c3+d3a^3+b^3+c^3+d^3 are divisible by pp. Prove that for all odd nn, an+bn+cn+dna^n+b^n+c^n+d^n is divisible by pp.
number theoryDivisibilityprimes
An indeterminate equation

Source: 2013 Taiwan TST

7/12/2015
If x,y,zx,y,z are positive integers and z(xz+1)2=(5z+2y)(2z+y)z(xz+1)^2=(5z+2y)(2z+y), prove that zz is an odd perfect square.
number theoryTaiwanTaiwan TST 2013
VietNam TST 2015, day 1, problem 3

Source:

3/26/2015
A positive interger number kk is called “t−mt-m”-property if forall positive interger number aa, there exists a positive integer number nn such that 1k+2k+3k+...+nk≡a( mod m).{{1}^{k}}+{{2}^{k}}+{{3}^{k}}+...+{{n}^{k}} \equiv a (\bmod m). a) Find all positive integer numbers kk which has t−20t-20-property. b) Find smallest positive integer number kk which has t−2015t-{{20}^{15}}-property.
number theory
Problem 5 of Finals - Product of primitive roots

Source: VI International Festival of Young Mathematicians Sozopol, Theme for 10-12 grade

1/11/2020
Let p>3p>3 be a prime number. Prove that the product of all primitive roots between 1 and p−1p-1 is congruent 1 modulo pp.
number theoryPrimitive Roots