MathDB

2008 Moldova Team Selection Test

Part of Moldova Team Selection Test

Subcontests

(4)
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Probably Known Property of homogeneous Polynomials in R[X,Y]

A non-zero polynomial S∈R[X,Y] S\in\mathbb{R}[X,Y] is called homogeneous of degree d d if there is a positive integer d d so that S(\lambda x,\lambda y)\equal{}\lambda^dS(x,y) for any λ∈R \lambda\in\mathbb{R}. Let P,Q∈R[X,Y] P,Q\in\mathbb{R}[X,Y] so that Q Q is homogeneous and P P divides Q Q (that is, P∣Q P|Q). Prove that P P is homogeneous too.

Nice: find number of even permutations with no fixed points

Find the number of even permutations of {1,2,…,n} \{1,2,\ldots,n\} with no fixed points.

Good sets and coloring of all positive integers.

A non-empty set S S of positive integers is said to be good if there is a coloring with 2008 2008 colors of all positive integers so that no number in S S is the sum of two different positive integers (not necessarily in S S) of the same color. Find the largest value t t can take so that the set S\equal{}\{a\plus{}1,a\plus{}2,a\plus{}3,\ldots,a\plus{}t\} is good, for any positive integer a a. [hide="P.S."]I have the feeling that I've seen this problem before, so if I'm right, maybe someone can post some links...
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Centroids of triangles inscribed/circumscribed to 2 circles.

Let Γ(I,r) \Gamma(I,r) and Γ(O,R) \Gamma(O,R) denote the incircle and circumcircle, respectively, of a triangle ABC ABC. Consider all the triangels AiBiCi A_iB_iC_i which are simultaneously inscribed in Γ(O,R) \Gamma(O,R) and circumscribed to Γ(I,r) \Gamma(I,r). Prove that the centroids of these triangles are concyclic.

Prove that some circles are concurrent.

Let ω \omega be the circumcircle of ABC ABC and let D D be a fixed point on BC BC, D≠B D\neq B, D≠C D\neq C. Let X X be a variable point on (BC) (BC), X≠D X\neq D. Let Y Y be the second intersection point of AX AX and ω \omega. Prove that the circumcircle of XYD XYD passes through a fixed point.

Some circles and a constant segment.

In triangle ABC ABC the bisector of ∠ACB \angle ACB intersects AB AB at D D. Consider an arbitrary circle O O passing through C C and D D, so that it is not tangent to BC BC or CA CA. Let O\cap BC \equal{} \{M\} and O\cap CA \equal{} \{N\}. a) Prove that there is a circle S S so that DM DM and DN DN are tangent to S S in M M and N N, respectively. b) Circle S S intersects lines BC BC and CA CA in P P and Q Q respectively. Prove that the lengths of MP MP and NQ NQ do not depend on the choice of circle O O.
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Minimum Value of a Non-symmetric Expression

Let a1,…,an a_1,\ldots,a_n be positive reals so that a_1\plus{}a_2\plus{}\ldots\plus{}a_n\le\frac n2. Find the minimal value of \sqrt{a_1^2\plus{}\frac1{a_2^2}}\plus{}\sqrt{a_2^2\plus{}\frac1{a_3^2}}\plus{}\ldots\plus{}\sqrt{a_n^2\plus{}\frac1{a_1^2}}.

Again Warm-Up Prob for a TST-Partition of a set into triples

We say the set {1,2,…,3k} \{1,2,\ldots,3k\} has property D D if it can be partitioned into disjoint triples so that in each of them a number equals the sum of the other two. (a) Prove that {1,2,…,3324} \{1,2,\ldots,3324\} has property D D. (b) Prove that {1,2,…,3309} \{1,2,\ldots,3309\} hasn't property D D.

Prove that a binomial coefficient is not divisible by p.

Let p p be a prime number and k,n k,n positive integers so that \gcd(p,n)\equal{}1. Prove that (n⋅pkpk) \binom{n\cdot p^k}{p^k} and p p are coprime.
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