MathDB

2

Part of 1997 Estonia National Olympiad

Problems(4)

x = 2 +\sqrt3 solution of ax^2 + bx + c = 0

Source: 1997 Estonia National Olympiad Final Round grade 9 p2

3/13/2020
Find the integers a≠0,ba \ne 0, b and cc such that x=2+3x = 2 +\sqrt3 would be a solution of the quadratic equation ax2+bx+c=0ax^2 + bx + c = 0.
trinomialalgebraquadratic polynomial
Simple inequality (383)

Source: Estonian Mathematical Olympiad 1996-1997

2/13/2015
Let xx and yy be real numbers. Show thatx2+y2+1>xy2+1+yx2+1.x^2+y^2+1>x\sqrt{y^2+1}+y\sqrt{x^2+1}.
inequalitiesinequalities proposed
find angle C when (a^3+b^3+c^3)/(a+b+c)= c^2

Source: 1997 Estonia National Olympiad Final Round grade 11 p2

3/11/2020
Side lengths a,b,ca,b,c of a triangle satisfy a3+b3+c3a+b+c=c2\frac{a^3+b^3+c^3}{a+b+c}= c^2. Find the measure of the angle opposite to side cc.
Triangleanglesgeometry
f (1)+ f (2)+...+ f (n) = n^2 f (n), f (1) = 999, f (1997) =?

Source: 1997 Estonia National Olympiad Final Round grade 12 p2

3/11/2020
A function ff satisfies the following condition for each n∈Nn\in N: f(1)+f(2)+...+f(n)=n2f(n)f (1)+ f (2)+...+ f (n) = n^2 f (n). Find f(1997)f (1997) if f(1)=999f (1) = 999.
functionalfunctional equationfunctionalgebra