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1

Part of 1998 Iran MO (2nd round)

Problems(2)

a_1<a_2<...<a_n - Iran NMO 1998 (Second Round) Problem1

Source:

10/4/2010
If a1<a2<⋯<ana_1<a_2<\cdots<a_n be real numbers, prove that: a1a24+a2a34+⋯+an−1an4+ana14≥a2a14+a3a24+⋯+anan−14+a1an4. a_1a_2^4+a_2a_3^4+\cdots+a_{n-1}a_n^4+a_na_1^4\geq a_2a_1^4+a_3a_2^4+\cdots+a_na_{n-1}^4+a_1a_n^4.
inequalitiesinductionrearrangement inequalityinequalities proposed
n = d_1^2+... - Iran NMO 1998 (Second Round) Problem4

Source:

10/4/2010
Let the positive integer nn have at least for positive divisors and 0<d1<d2<d3<d40<d_1<d_2<d_3<d_4 be its least positive divisors. Find all positive integers nn such that: n=d12+d22+d32+d42. n=d_1^2+d_2^2+d_3^2+d_4^2.
number theory proposednumber theory