MathDB

3

Part of 2009 Romania Team Selection Test

Problems(6)

Switching the lights on and off

Source: Romania TST 1 2009, Problem 3

5/4/2012
Some n>2n>2 lamps are cyclically connected: lamp 11 with lamp 22, ..., lamp kk with lamp k+1k+1,..., lamp n−1n-1 with lamp nn, lamp nn with lamp 11. At the beginning all lamps are off. When one pushes the switch of a lamp, that lamp and the two ones connected to it change status (from off to on, or vice versa). Determine the number of configurations of lamps reachable from the initial one, through some set of switches being pushed.
inductionlinear algebramatrixcombinatorics
Cyclic pentagon and product of distances

Source: Romania TST 2009. Day 2. Problem 3.

4/21/2009
Prove that pentagon ABCDE ABCDE is cyclic if and only if \mathrm{d(}E,AB\mathrm{)}\cdot \mathrm{d(}E,CD\mathrm{)} \equal{} \mathrm{d(}E,AC\mathrm{)}\cdot \mathrm{d(}E,BD\mathrm{)} \equal{} \mathrm{d(}E,AD\mathrm{)}\cdot \mathrm{d(}E,BC\mathrm{)} where d(X,YZ) \mathrm{d(}X,YZ\mathrm{)} denotes the distance from point X X ot the line YZ YZ.
conicsanalytic geometrygeometrycircumcircletrigonometry
Perpendicular bisector

Source: Romania TST 2009, Day 3, Problem 3

7/26/2009
Let ABC ABC be a non-isosceles triangle, in which X,Y, X,Y, and Z Z are the tangency points of the incircle of center I I with sides BC,CA BC,CA and AB AB respectively. Denoting by O O the circumcircle of △ABC \triangle{ABC}, line OI OI meets BC BC at a point D. D. The perpendicular dropped from X X to YZ YZ intersects AD AD at E E. Prove that YZ YZ is the perpendicular bisector of [EX] [EX].
geometrycircumcirclegeometric transformationreflectionEulerincentertrigonometry
Cardinality of a special set of n-vectors

Source: Romania TST 5 2009, Problem 3

5/5/2012
Given two integers n≥1n\geq 1 and q≥2q\geq 2, let A={(a1,…,an):ai∈{0,…,q−1},i=1,…,n}A=\{(a_1,\ldots ,a_n):a_i\in\{0,\ldots ,q-1\}, i=1,\ldots ,n\}. If a=(a1,…,an)a=(a_1,\ldots ,a_n) and b=(b1,…,bn)b=(b_1,\ldots ,b_n) are two elements of AA, let δ(a,b)=#{i:ai≠bi}\delta(a,b)=\#\{i:a_i\neq b_i\}. Let further tt be a non-negative integer and BB a non-empty subset of AA such that δ(a,b)≥2t+1\delta(a,b)\geq 2t+1, whenever aa and bb are distinct elements of BB. Prove that the two statements below are equivalent: a) For any a∈Aa\in A, there is a unique b∈Bb\in B, such that δ(a,b)≤t\delta (a,b)\leq t; b) ∣B∣⋅∑k=0t(nk)(q−1)k=qn\displaystyle|B|\cdot \sum_{k=0}^t \binom{n}{k}(q-1)^k=q^n
vectorcombinatorics proposedcombinatorics
Maximum of products of segments determined by n points

Source: Romania TST 6 2009, Problem 3

5/5/2012
Given an integer n≥2n\geq 2 and a closed unit disc, evaluate the maximum of the product of the lengths of all n(n−1)2\frac{n(n-1)}{2} segments determined by nn points in that disc.
functioncomplex analysistrigonometryinequalitiesfloor functioncomplex numbersalgebra proposed
p divides 2^{q-1}-1 and q divides 2^{p-1}-1

Source: Romania TST 7 2009, Problem 3

5/5/2012
Show that there are infinitely many pairs of prime numbers (p,q)(p,q) such that p∣2q−1−1p\mid 2^{q-1}-1 and q∣2p−1−1q\mid 2^{p-1}-1.
quadraticsalgebrapolynomialsearchnumber theory proposednumber theory