MathDB

1

Part of 1998 Romania Team Selection Test

Problems(5)

a^2.AM.AN=b^2.BN.CM

Source:

8/18/2010
We are given an isosceles triangle ABCABC such that BC=aBC=a and AB=BC=bAB=BC=b. The variable points M∈(AC)M\in (AC) and N∈(AB)N\in (AB) satisfy a2⋅AM⋅AN=b2⋅BN⋅CMa^2\cdot AM \cdot AN = b^2 \cdot BN \cdot CM. The straight lines BMBM and CNCN intersect in PP. Find the locus of the variable point PP.
Dan Branzei
geometrytrapezoidtrigonometryincentermodular arithmeticratioparallelogram
Two types of words formed by the letters a,b,c

Source: Romanian TST 1998

4/23/2011
A word of length nn is an ordered sequence x1x2…xnx_1x_2\ldots x_n where xix_i is a letter from the set {a,b,c}\{ a,b,c \}. Denote by AnA_n the set of words of length nn which do not contain any block xixi+1,i=1,2,…,n−1,x_ix_{i+1}, i=1,2,\ldots ,n-1, of the form aaaa or bbbb and by BnB_n the set of words of length nn in which none of the subsequences xixi+1xi+2,i=1,2,…n−2,x_ix_{i+1}x_{i+2}, i=1,2,\ldots n-2, contains all the letters a,b,ca,b,c. Prove that ∣Bn+1∣=3∣An∣|B_{n+1}|=3|A_n|.
Vasile Pop
functionsymmetrycombinatorics proposedcombinatorics
Intersection of A and P is empty

Source: Romanian TST 1998

4/23/2011
Let ABCABC be an equilateral triangle and n≥2n\ge 2 be an integer. Denote by A\mathcal{A} the set of n−1n-1 straight lines which are parallel to BCBC and divide the surface [ABC][ABC] into nn polygons having the same area and denote by P\mathcal{P} the set of n−1n-1 straight lines parallel to BCBC which divide the surface [ABC][ABC] into nn polygons having the same perimeter. Prove that the intersection A∩P\mathcal{A} \cap \mathcal{P} is empty.
Laurentiu Panaitopol
geometrygeometry proposed
Subset A satisfies two given conditions

Source: Romanian TST 1998

4/23/2011
Let n≥2n\ge 2 be an integer. Show that there exists a subset A∈{1,2,…,n}A\in \{1,2,\ldots ,n\} such that:
i) The number of elements of AA is at most 2⌊n⌋+12\lfloor\sqrt{n}\rfloor+1
ii) {∣x−y∣∣x,y∈A,x≠y}={1,2,…n−1}\{ |x-y| \mid x,y\in A, x\not= y\} = \{ 1,2,\ldots n-1 \}
Radu Todor
floor functioncombinatorics proposedcombinatorics
The monotonic function u(x)

Source: Romanian TST 1998

4/23/2011
Find all monotonic functions u:R→Ru:\mathbb{R}\rightarrow\mathbb{R} which have the property that there exists a strictly monotonic function f:R→Rf:\mathbb{R}\rightarrow\mathbb{R} such that f(x+y)=f(x)u(x)+f(y)f(x+y)=f(x)u(x)+f(y) for all x,y∈Rx,y\in\mathbb{R}.
Vasile Pop
functionalgebra proposedalgebrafunctional equation