MathDB

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Part of 2005 All-Russian Olympiad

Problems(5)

a parallelogram and a common point of some circles

Source: ARO 2005 - 9.1

4/30/2005
Given a parallelogram ABCDABCD with AB<BCAB<BC, show that the circumcircles of the triangles APQAPQ share a second common point (apart from AA) as P,QP,Q move on the sides BC,CDBC,CD respectively s.t. CP=CQCP=CQ.
geometryparallelogramcircumcirclegeometric transformationreflectionperpendicular bisectorangle bisector
squares of all these numbers are rational

Source: ARO 2005 - problem 9.5

4/30/2005
Ten mutually distinct non-zero reals are given such that for any two, either their sum or their product is rational. Prove that squares of all these numbers are rational.
symmetrynumber theory unsolvednumber theory
selected cells on a chessboard

Source: ARO 2005 - 10.5

4/30/2005
We select 1616 cells on an 8×88\times 8 chessboard. What is the minimal number of pairs of selected cells in the same row or column?
linear algebramatrixinequalitiescombinatorics proposedcombinatorics
f^2(x+y) &gt;= f^2(x)+2f(xy)+f^2(y)

Source: ARO 2005 - problem 11.5

4/30/2005
Do there exist a bounded function f:R→Rf: \mathbb{R}\to\mathbb{R} such that f(1)>0f(1)>0 and f(x)f(x) satisfies an inequality f2(x+y)≥f2(x)+2f(xy)+f2(y)f^2(x+y)\ge f^2(x)+2f(xy)+f^2(y)?
functioninequalitiesalgebra unsolvedalgebra
|x-a_1|+ ... +|x-a_{50}|=|x-b_1|+...+|x-b_50|

Source: ARO 2005 - problem 11.1

4/30/2005
Find the maximal possible finite number of roots of the equation ∣x−a1∣+⋯+∣x−a50∣=∣x−b1∣+⋯+∣x−b50∣|x-a_1|+\dots+|x-a_{50}|=|x-b_1|+\dots+|x-b_{50}|, where a1, a2, …,a50, b1,…, b50a_1,\,a_2,\,\dots,a_{50},\,b_1,\dots,\,b_{50} are distinct reals.
functionabstract algebramodular arithmeticalgebraabsolute value