MathDB

1

Part of 1999 Romania National Olympiad

Problems(3)

a triangle

Source:

7/1/2006
Source: Romania 1999 7.1 Determine the side lengths of a right trianlge if they are intgers and the product of the leg lengths is equal to three times the perimeter.
natural sets related to P(x) = 2x^3-3x^2+2 1999 Romania NMO VIII p2

Source:

8/14/2024
Let P(x)=2x3−3x2+2P(x) = 2x^3-3x^2+2, and the sets: A={P(n)∣n∈N,n≤1999},B={p2+1∣p∈N},C={q2+2∣q∈N}A =\{ P(n) | n \in N, n \le 1999\}, B=\{p^2+1 |p \in N\}, C=\{ q^2+2 | q \in N\} Prove that the sets A∩BA \cap B and A∩CA\cap C have the same number of elements
number theory
AD^3 = AB x AC x AP, equal sngles

Source: 1999 Romania NMO IX p1

8/14/2024
Let ADAD be the bisector of angle AA of the triangle ABCABC. One considers the points M, N on the half-lines (AB(AB and (AC(AC, respectively, such that ∠MDA=∠B\angle MDA = \angle B and ∠NDA=∠C\angle NDA = \angle C. Let AD∩MN={P}AD \cap MN=\{P\}. Prove that: AD3=AB⋅AC⋅APAD^3 = AB \cdot AC\cdot AP
equal anglesgeometry