MathDB

4

Part of 2006 IMO Shortlist

Problems(3)

n variable inequality

Source: IMO Shortlist 2006, Algebra 4, AIMO 2007, TST 7, P1

6/28/2007
Prove the inequality: \sum_{i < j}{\frac {a_{i}a_{j}}{a_{i} \plus{} a_{j}}}\leq \frac {n}{2(a_{1} \plus{} a_{2} \plus{}\cdots \plus{} a_{n})}\cdot \sum_{i < j}{a_{i}a_{j}} for positive reals a1,a2,…,an a_{1},a_{2},\ldots,a_{n}.
Proposed by Dusan Dukic, Serbia
inequalitiesfunctionalgebraIMO Shortlist
n x n square and strawberries

Source: IMO Shortlist 2006, Combinatorics 4, AIMO 2007, TST 4, P2

6/28/2007
A cake has the form of an n n x n n square composed of n2 n^{2} unit squares. Strawberries lie on some of the unit squares so that each row or column contains exactly one strawberry; call this arrangement A\mathcal{A}.
Let B\mathcal{B} be another such arrangement. Suppose that every grid rectangle with one vertex at the top left corner of the cake contains no fewer strawberries of arrangement B\mathcal{B} than of arrangement A\mathcal{A}. Prove that arrangement B\mathcal{B} can be obtained from A \mathcal{A} by performing a number of switches, defined as follows:
A switch consists in selecting a grid rectangle with only two strawberries, situated at its top right corner and bottom left corner, and moving these two strawberries to the other two corners of that rectangle.
matrixcombinatoricsPartial OrdersIMO Shortlist
France TST 2007

Source: ISL 2006, G4, France TST 2007/6 1st Brazilian TST 2007, AIMO 2007, TST 4, P1

5/16/2007
A point DD is chosen on the side ACAC of a triangle ABCABC with ∠C<∠A<90∘\angle C < \angle A < 90^\circ in such a way that BD=BABD=BA. The incircle of ABCABC is tangent to ABAB and ACAC at points KK and LL, respectively. Let JJ be the incenter of triangle BCDBCD. Prove that the line KLKL intersects the line segment AJAJ at its midpoint.
geometryincenterTriangleIMO Shortlistgeometry solvedmidpointcongruent triangles