MathDB

4

Part of 1996 IMO Shortlist

Problems(4)

Very strange inequality

Source: 1996 IMO Shortlist

6/11/2006
Let a1,a2...an a_{1}, a_{2}...a_{n} be non-negative reals, not all zero. Show that that (a) The polynomial p(x) \equal{} x^{n} \minus{} a_{1}x^{n \minus{} 1} \plus{} ... \minus{} a_{n \minus{} 1}x \minus{} a_{n} has preceisely 1 positive real root R R. (b) let A \equal{} \sum_{i \equal{} 1}^n a_{i} and B \equal{} \sum_{i \equal{} 1}^n ia_{i}. Show that AA≤RB A^{A} \leq R^{B}.
inequalitiesalgebrapolynomialfunctioncalculusIMO Shortlist
Cool cevians problem

Source: 1996 IMO Shortlist

3/28/2006
Let ABCABC be an equilateral triangle and let PP be a point in its interior. Let the lines APAP, BPBP, CPCP meet the sides BCBC, CACA, ABAB at the points A1A_1, B1B_1, C1C_1, respectively. Prove that A1B1⋅B1C1⋅C1A1≥A1B⋅B1C⋅C1AA_1B_1 \cdot B_1C_1 \cdot C_1A_1 \ge A_1B \cdot B_1C \cdot C_1A.
trigonometrytrig identitiesLaw of CosinesIMO Shortlist
Two disjoint infinite sets A and B of points in the plane

Source: IMO Shortlist 1996, C4

8/9/2008
Determine whether or nor there exist two disjoint infinite sets A A and B B of points in the plane satisfying the following conditions: a.) No three points in A∪B A \cup B are collinear, and the distance between any two points in A∪B A \cup B is at least 1. b.) There is a point of A A in any triangle whose vertices are in B, B, and there is a point of B B in any triangle whose vertices are in A. A.
combinatorial geometrycombinatoricspoint setTriangleIMO Shortlist
Find all positive integers a and b

Source: IMO Shortlist 1996, N4

8/9/2008
Find all positive integers a a and b b for which \left \lfloor \frac{a^2}{b} \right \rfloor \plus{} \left \lfloor \frac{b^2}{a} \right \rfloor \equal{} \left \lfloor \frac{a^2 \plus{} b^2}{ab} \right \rfloor \plus{} ab.
floor functionnumber theoryequationalgebraIMO Shortlist