MathDB

3

Part of 1995 South africa National Olympiad

Problems(2)

Sum of squares of difference between permutations is even

Source: South African MO 1995 Q3

5/27/2012
Suppose that a1,a2,…,ana_1,a_2,\dots,a_n are the numbers 1,2,3,…,n1,2,3,\dots,n but written in any order. Prove that (a1−1)2+(a2−2)2+⋯+(an−n)2(a_1-1)^2+(a_2-2)^2+\cdots+(a_n-n)^2 is always even.
algebra unsolvedalgebra
AP BP CP < (1+x)^2 (1-x)

Source: South African MO 1995 Q7

5/27/2012
The circumcircle of △ABC\triangle ABC has radius 11 and centre OO and PP is a point inside the triangle such that OP=xOP=x. Prove that AP⋅BP⋅CP≤(1+x)2(1−x),AP\cdot BP\cdot CP\le(1+x)^2(1-x), with equality if and only if P=OP=O.
geometrycircumcirclegeometry unsolved