MathDB

2

Part of 2010 Postal Coaching

Problems(6)

Diophantine equation at romanian tst

Source: Romanian IMO TST 2006, day 4, problem 2

5/19/2006
Find all non-negative integers m,n,p,qm,n,p,q such that pmqn=(p+q)2+1. p^mq^n = (p+q)^2 +1 .
quadraticsreal analysiscomplex numbersalgebrapolynomialsum of rootsnumber theory proposed
Hard Problem Involving Modulus

Source:

12/9/2010
Let a1,a2,…,ana_1, a_2, \ldots, a_n be real numbers lying in [−1,1][-1, 1] such that a1+a2+⋯+an=0a_1 + a_2 + \cdots + a_n = 0. Prove that there is a k∈{1,2,…,n}k \in \{1, 2, \ldots, n\} such that ∣a1+2a2+3a3+⋯+kak∣≤2k+14|a_1 + 2a_2 + 3a_3 + \cdots + k a_k | \le \frac{2k+1}4 .
inductionalgebra unsolvedalgebra
Determine a length

Source:

12/9/2010
In a circle with centre at OO and diameter ABAB, two chords BDBD and ACAC intersect at EE. FF is a point on ABAB such that EF⊥ABEF \perp AB. FCFC intersects BDBD in GG. If DE=5DE = 5 and EG=3EG =3, determine BGBG.
geometryangle bisectorgeometry unsolved
Nice triples

Source:

10/22/2010
Call a triple (a,b,c)(a, b, c) of positive integers a nice triple if a,b,ca, b, c forms a non-decreasing arithmetic progression, gcd(b,a)=gcd(b,c)=1gcd(b, a) = gcd(b, c) = 1 and the product abcabc is a perfect square. Prove that given a nice triple, there exists some other nice triple having at least one element common with the given triple.
arithmetic sequencenumber theory unsolvednumber theory
Inequality of Areas of triangle formed by circumcentres

Source:

12/9/2010
Let MM be an interior point of a △ABC\triangle ABC such that ∠AMB=150∘,∠BMC=120∘\angle AM B = 150^{\circ} , \angle BM C = 120^{\circ}. Let P,Q,RP, Q, R be the circumcentres of the △AMB,△BMC,△CMA\triangle AM B, \triangle BM C, \triangle CM A respectively. Prove that [PQR]≥[ABC][P QR] \ge [ABC].
inequalitiesgeometryinequalities unsolved
Reflection about the sides

Source:

12/9/2010
Suppose △ABC\triangle ABC has circumcircle Γ\Gamma, circumcentre OO and orthocentre HH. Parallel lines α,β,γ\alpha, \beta, \gamma are drawn through the vertices A,B,CA, B, C, respectively. Let α′,β′,γ′\alpha ', \beta ', \gamma ' be the reflections of α,β,γ\alpha, \beta, \gamma in the sides BC,CA,ABBC, CA, AB, respectively.
(a)(a) Show that α′,β′,γ′\alpha ', \beta ', \gamma ' are concurrent if and only if α,β,γ\alpha, \beta, \gamma are parallel to the Euler line OHOH.
(b)(b) Suppose that α′,β′,γ′\alpha ', \beta ' , \gamma ' are concurrent at the point PP . Show that Γ\Gamma bisects OPOP .
geometrygeometric transformationreflectioncircumcircleEuleranalytic geometrygeometry unsolved