MathDB

3

Part of 2010 Indonesia TST

Problems(10)

Problem from Indonesian TST

Source:

12/8/2010
For every natural number n n , define s(n) s(n) as the smallest natural number so that for every natural number a a relatively prime to nn, this equation holds: as(n)≡1(modn) a^{s(n)} \equiv 1 (mod n) Find all natural numbers n n such that s(n)=2010 s(n) = 2010
number theoryleast common multiplefunctionrelatively primenumber theory unsolved
Unique points and collinearity

Source:

3/6/2015
Given acute triangle ABCABC with circumcenter OO and the center of nine-point circle NN. Point N1N_1 are given such that ∠NAB=∠N1AC\angle NAB = \angle N_1AC and ∠NBC=∠N1BA\angle NBC = \angle N_1BA. Perpendicular bisector of segment OAOA intersects the line BCBC at A1A_1. Analogously define B1B_1 and C1C_1. Show that all three points A1,B1,C1A_1,B_1,C_1 are collinear at a line that is perpendicular to ON1ON_1.
geometrygeometry unsolved
find the locus of T

Source: itamo 2004, p2

3/2/2012
Two parallel lines r,sr,s and two points P∈rP \in r and Q∈sQ \in s are given in a plane. Consider all pairs of circles (CP,CQ)(C_P, C_Q) in that plane such that CPC_P touches rr at PP and CQC_Q touches ss at QQ and which touch each other externally at some point TT. Find the locus of TT.
geometrygeometric transformationhomothetygeometry proposed
collinearity

Source:

3/6/2015
Given a non-isosceles triangle ABCABC with incircle kk with center SS. kk touches the side BC,CA,ABBC,CA,AB at P,Q,RP,Q,R respectively. The line QRQR and line BCBC intersect at MM. A circle which passes through BB and CC touches kk at NN. The circumcircle of triangle MNPMNP intersects APAP at LL. Prove that S,L,MS,L,M are collinear.
geometry unsolvedgeometry
it is bisect iff the other is parallel

Source: unknown

9/24/2014
Let ABCDABCD be a convex quadrilateral with ABAB is not parallel to CDCD. Circle ω1\omega_1 with center O1O_1 passes through AA and BB, and touches segment CDCD at PP. Circle ω2\omega_2 with center O2O_2 passes through CC and DD, and touches segment ABAB at QQ. Let EE and FF be the intersection of circles ω1\omega_1 and ω2\omega_2. Prove that EFEF bisects segment PQPQ if and only if BCBC is parallel to ADAD.
geometryparallelogrampower of a pointradical axis
party but graph

Source: Indonesia IMO 2010 TST, Stage 1, Test 1, Problem 3

11/12/2009
In a party, each person knew exactly 22 22 other persons. For each two persons X X and Y Y, if X X and Y Y knew each other, there is no other person who knew both of them, and if X X and Y Y did not know each other, there are exactly 6 6 persons who knew both of them. Assume that X X knew Y Y iff Y Y knew X X. How many people did attend the party? Yudi Satria, Jakarta
combinatorics proposedcombinatorics
maximum value of integer z

Source: Indonesia IMO 2010 TST, Stage 1, Test 2, Problem 3

11/12/2009
Let x x, y y, and z z be integers satisfying the equation \dfrac{2008}{41y^2}\equal{}\dfrac{2z}{2009}\plus{}\dfrac{2007}{2x^2}. Determine the greatest value that z z can take. Budi Surodjo, Jogjakarta
number theory proposednumber theory
sequence and inequality

Source: Indonesia IMO 2010 TST, Stage 1, Test 3, Problem 3

11/12/2009
Let a1,a2,… a_1,a_2,\dots be sequence of real numbers such that a_1\equal{}1, a_2\equal{}\dfrac{4}{3}, and a_{n\plus{}1}\equal{}\sqrt{1\plus{}a_na_{n\minus{}1}},   \forall n \ge 2. Prove that for all n≥2 n \ge 2, a_n^2>a_{n\minus{}1}^2\plus{}\dfrac{1}{2} and 1\plus{}\dfrac{1}{a_1}\plus{}\dfrac{1}{a_2}\plus{}\dots\plus{}\dfrac{1}{a_n}>2a_n. Fajar Yuliawan, Bandung
inequalitiesinductionalgebra proposedalgebra
the existence of function

Source: Indonesia IMO 2010 TST, Stage 1, Test 4, Problem 3

11/12/2009
Determine all real numbers a a such that there is a function f:R→R f: \mathbb{R} \rightarrow \mathbb{R} satisfying x\plus{}f(y)\equal{}af(y\plus{}f(x)) for all real numbers x x and y y. Hery Susanto, Malang
functionalgebra proposedalgebraFunction equations
existence of bijective function

Source: Indonesia IMO 2010 TST, Stage 1, Test 5, Problem 3

11/12/2009
Let Z \mathbb{Z} be the set of all integers. Define the set H \mathbb{H} as follows: (1). 12∈H \dfrac{1}{2} \in \mathbb{H}, (2). if x∈H x \in \mathbb{H}, then \dfrac{1}{1\plus{}x} \in \mathbb{H} and also \dfrac{x}{1\plus{}x} \in \mathbb{H}. Prove that there exists a bijective function f:Z→H f: \mathbb{Z} \rightarrow \mathbb{H}.
functioninductionnumber theory proposednumber theory