MathDB

1

Part of 2010 Romania National Olympiad

Problems(6)

673 element subset has a,b such that 6|a+b

Source: Romanian MO 2010 Grade 7

8/6/2012
Let SS be a subset with 673673 elements of the set {1,2,…,2010}\{1,2,\ldots ,2010\}. Prove that one can find two distinct elements of SS, say aa and bb, such that 66 divides a+ba+b.
combinatorics proposedcombinatorics
a, b, c are integers greater than 1

Source: Romanian MO 2010 Grade 8

8/6/2012
Let a,b,ca,b,c be integers larger than 11. Prove that a(a−1)+b(b−1)+c(c−1)≤(a+b+c−4)(a+b+c−5)+4.a(a-1)+b(b-1)+c(c-1)\le (a+b+c-4)(a+b+c-5)+4.
inequalities proposedinequalities
Orthocentre of DEF is incentre of ABC

Source: Romanian MO 2010 Grade 9

8/6/2012
In a triangle ABCABC denote by D,E,FD,E,F the points where the angle bisectors of ∠CAB,∠ABC,∠BCA\angle CAB,\angle ABC,\angle BCA respectively meet it's circumcircle. a) Prove that the orthocenter of triangle DEFDEF coincides with the incentre of triangle ABCABC. b) Prove that if AD→+BE→+CF→=0\overrightarrow{AD}+\overrightarrow{BE}+\overrightarrow{CF}=0, then the triangle ABCABC is equilateral.
Marin Ionescu
geometryincentergeometry proposed
Prove that it is a geometric sequence

Source: Romanian MO 2010 Grade 10

8/6/2012
Let (an)n≥0(a_n)_{n\ge0} be a sequence of positive real numbers such that ∑k=0nCnkakan−k=an2, for any n≥0.\sum_{k=0}^nC_n^ka_ka_{n-k}=a_n^2,\ \text{for any }n\ge 0. Prove that (an)n≥0(a_n)_{n\ge0} is a geometric sequence.
Lucian Dragomir
quadraticsVietafunctionalgebraalgebra unsolved
Romania National Olympiad 2010 - Grade XI

Source:

4/10/2011
Let a,b∈Ra,b\in \mathbb{R} such that b>a2b>a^2. Find all the matrices A∈M2(R)A\in \mathcal{M}_2(\mathbb{R}) such that det⁡(A2−2aA+bI2)=0\det(A^2-2aA+bI_2)=0.
linear algebramatrixlinear algebra unsolved
f is continuous if F has a finite derivative

Source: Romanian MO 2010 Grade 12

8/6/2012
Let f:R→Rf:\mathbb{R}\to\mathbb{R} be a monotonic function and F:R→RF:\mathbb{R}\to\mathbb{R} given by F(x)=∫0xf(t) dt.F(x)=\int_0^xf(t)\ \text{d}t. Prove that if FF has a finite derivative, then ff is continuous.
Dorin Andrica & Mihai Piticari
calculusderivativefunctionintegrationlimitreal analysisreal analysis unsolved