MathDB

3

Part of 1996 Iran MO (3rd Round)

Problems(3)

Find all sets of real numbers

Source: Iran Third Round 1997, E1, P3

3/27/2011
Find all sets of real numbers {a1,a2,…,1375}\{a_1,a_2,\ldots, _{1375}\} such that 2 \left( \sqrt{a_n - (n-1)}\right) \geq a_{n+1} - (n-1),   \forall n \in \{1,2,\ldots,1374\}, and 2(a1375−1374)≥a1+1.2 \left( \sqrt{a_{1375}-1374} \right) \geq a_1 +1.
algebra unsolvedalgebra
p=a_1 and (x-a1)(x-a2)...(x-an)≤x^n-a_1^n

Source: Iran Third Round 1996, E2, P3

3/25/2011
Let a1≥a2≥⋯≥ana_1 \geq a_2 \geq \cdots \geq a_n be nn real numbers such that a1k+a2k+⋯+ank≥0a_1^k +a_2^k + \cdots + a_n^k \geq 0 for all positive integers kk. Suppose that p=max⁡{∣a1∣,∣a2∣,…,∣an∣}p=\max\{|a_1|,|a_2|, \ldots,|a_n|\}. Prove that p=a1p=a_1, and (x−a1)(x−a2)⋯(x−an)≤xn−a1n∀x>a1.(x-a_1)(x-a_2)\cdots(x-a_n)\leq x^n-a_1^n \qquad \forall x>a_1.
inequalities proposedinequalities
Cyclic

Source: 14-th iranian mathematical olympiad 1996/1997

10/17/2005
Suppose that 1010 points are given in the plane, such that among any five of them there are four lying on a circle. Find the minimum number of these points which must lie on a circle.
geometry proposedgeometry