MathDB

3

Part of 2006 Estonia National Olympiad

Problems(4)

Estonian Math Competitions 2005/2006

Source: Final Round Grade 9 Pro 2

7/30/2008
Let there be n≥2 n \ge 2 real numbers such that none of them is greater than the arithmetic mean of the other numbers. Prove that all the numbers are equal.
algebra unsolvedalgebra
angle chasing, given angle bisectors and circumcircles

Source: 2006 Estonia National Olympiad Final Round grade 10 p3

3/12/2020
Let AG,CHAG, CH be the angle bisectors of a triangle ABCABC. It is known that one of the intersections of the circles of triangles ABGABG and ACHACH lies on the side BCBC. Prove that the angle BACBAC is 60o60 ^o
geometryangle bisectorAngle Chasingcircumcircle
Estonian Math Competitions 2005/2006

Source: Final Round Grade 11 Pro 3

7/30/2008
The sequence (Fn) (F_n) of Fibonacci numbers satisfies F_1 \equal{} 1, F_2 \equal{} 1 and F_n \equal{} F_{n\minus{}1} \plus{}F_{n\minus{}2} for all n≥3 n \ge 3. Find all pairs of positive integers (m,n) (m, n), such that F_m . F_n \equal{} mn.
inductionnumber theory unsolvednumber theory
Estonian Math Competitions 2005/2006

Source: Final Round Grade 12 Pro 3

7/30/2008
Prove or disprove the following statements. a) For every integer n≥3 n \ge 3, there exist n n pairwise distinct positive integers such that the product of any two of them is divisible by the sum of the remaining n \minus{} 2 numbers. b) For some integer n≥3 n \ge 3, there exist n n pairwise distinct positive integers, such that the sum of any n \minus{} 2 of them is divisible by the product of the remaining two numbers.
modular arithmeticnumber theory unsolvednumber theory