MathDB

Problem 7

Part of 2000 Moldova National Olympiad

Problems(6)

no fibonacci number is the sum of 2000 others

Source: Moldova 2000 Grade 7 P7

4/23/2021
The Fibonacci sequence is defined by F0=F1=1F_0=F_1=1 and Fn+2=Fn+1+FnF_{n+2}=F_{n+1}+F_n for n≥0n\ge0. Prove that the sum of 20002000 consecutive terms of the Fibonacci sequence is never a term of the sequence.
number theoryFibonacci NumbersMoldovaSequence
(a^3+a^2+3)^2>4a^3(a-1)^2 over R

Source: Moldova 2000 Grade 8 P7

4/25/2021
For any real number aa, prove the inequality: (a3+a2+3)2>4a3(a−1)2.\left(a^3+a^2+3\right)^2>4a^3(a-1)^2.
algebrainequalities
collinearity of centers of hexagon in trapezoid

Source: Moldova 2000 Grade 9 P7

4/26/2021
In a trapezoid ABCDABCD with AB∥CDAB\parallel CD, the diagonals ACAC and BDBD meet at OO. Let MM and NN be the centers of the regular hexagons constructed on the sides ABAB and CDCD in the exterior of the trapezoid. Prove that M,OM,O and NN are collinear.
geometrytrapezoid
perpendicular sides in triangle

Source: Moldova 2000 Grade 10 P7

4/26/2021
In an isosceles triangle ABCABC with BC=ACBC=AC, II is the incenter and OO the circumcenter. The line through II parallel to ACAC meets BCBC at DD. Prove that the lines DODO and BIBI are perpendicular.
Trianglegeometry
if triangle is in square then center of square is in triangle

Source: Moldova 2000 Grade 11 P7

4/27/2021
A triangle whose all sides have lengths greater than 11 is contained in a unit square. Show that the center of the square lies inside the triangle.
geometry
existence of matrix so that entries of An are squares

Source: Moldova 2000 Grade 12 P7

4/28/2021
Prove that for any positive integer nn there exists a matrix of the form A=(1abc01ab001a0001),A=\begin{pmatrix}1&a&b&c\\0&1&a&b\\0&0&1&a\\0&0&0&1\end{pmatrix}, (a) with nonzero entries, (b) with positive entries,
such that the entries of AnA^n are all perfect squares.
Matriceslinear algebra