MathDB

2

Part of 2007 Romania Team Selection Test

Problems(6)

Prove that the inequality is false!

Source: Romanian TST 3 2007, Problem 2

5/23/2007
Prove that for n,pn, p integers, n≥4n \geq 4 and p≥4p \geq 4, the proposition P(n,p)\mathcal{P}(n, p) \sum_{i=1}^{n}\frac{1}{{x_{i}}^{p}}\geq \sum_{i=1}^{n}{x_{i}}^{p}  \textrm{for}  x_{i}\in \mathbb{R},   x_{i}> 0 ,   i=1,\ldots,n \ ,  \sum_{i=1}^{n}x_{i}= n, is false.
Dan Schwarz
[hide="Remark"]In the competition, the students were informed (fact that doesn't actually relate to the problem's solution) that the propositions P(4,3)\mathcal{P}(4, 3) are P(3,4)\mathcal{P}(3, 4) true.
inequalitiesinequalities proposed
functional inequality

Source: Romanian I TST 2007

4/13/2007
Let f:Q→Rf: \mathbb{Q}\rightarrow \mathbb{R} be a function such that ∣f(x)−f(y)∣≤(x−y)2|f(x)-f(y)|\leq (x-y)^{2} for all x,y∈Qx,y \in\mathbb{Q}. Prove that ff is constant.
inequalitiesfunctionalgebra proposedalgebra
a problem about ABD and ACD having equal in-radii

Source: romanian tst 2 - 2007 , problem 2

4/15/2007
Let ABCABC be a triangle, EE and FF the points where the incircle and AA-excircle touch ABAB, and DD the point on BCBC such that the triangles ABDABD and ACDACD have equal in-radii. The lines DBDB and DEDE intersect the circumcircle of triangle ADFADF again in the points XX and YY. Prove that XY∥ABXY\parallel AB if and only if AB=ACAB=AC.
geometrycircumcirclegeometry proposed
circles tangent to eachother and to the sides of triangle

Source: romania TST 5 / 2007 problem 2

6/13/2007
Let ABCABC be a triangle, and ωa\omega_{a}, ωb\omega_{b}, ωc\omega_{c} be circles inside ABCABC, that are tangent (externally) one to each other, such that ωa\omega_{a} is tangent to ABAB and ACAC, ωb\omega_{b} is tangent to BABA and BCBC, and ωc\omega_{c} is tangent to CACA and CBCB. Let DD be the common point of ωb\omega_{b} and ωc\omega_{c}, EE the common point of ωc\omega_{c} and ωa\omega_{a}, and FF the common point of ωa\omega_{a} and ωb\omega_{b}. Show that the lines ADAD, BEBE and CFCF have a common point.
geometrygeometric transformationprojective geometrygeometry proposedDesarguesMonge theorem
a probably new characterization of AC/AB

Source: Romanian TST 2007, Day 6, Problem 2

6/8/2007
Let ABC ABC be a triangle, let E,F E, F be the tangency points of the incircle Γ(I) \Gamma(I) to the sides AC AC, respectively AB AB, and let M M be the midpoint of the side BC BC. Let N \equal{} AM \cap EF, let γ(M) \gamma(M) be the circle of diameter BC BC, and let X,Y X, Y be the other (than B,C B, C) intersection points of BI BI, respectively CI CI, with γ \gamma. Prove that \frac {NX} {NY} \equal{} \frac {AC} {AB}. Cosmin Pohoata
geometrytrigonometryAMCUSA(J)MOUSAJMOperpendicular bisectorgeometry proposed
Joric and social egalitarianism

Source: Romanian TST 5 2007, Problem 2

6/7/2007
The world-renowned Marxist theorist Joric is obsessed with both mathematics and social egalitarianism. Therefore, for any decimal representation of a positive integer nn, he tries to partition its digits into two groups, such that the difference between the sums of the digits in each group be as small as possible. Joric calls this difference the defect of the number nn. Determine the average value of the defect (over all positive integers), that is, if we denote by δ(n)\delta(n) the defect of nn, compute lim⁡n→∞∑k=1nδ(k)n.\lim_{n \rightarrow \infty}\frac{\sum_{k = 1}^{n}\delta(k)}{n}. Iurie Boreico
limitprobabilitynumber theory proposednumber theory