MathDB

6

Part of 2006 Iran MO (3rd Round)

Problems(4)

Beautiful polynomial problem

Source: Iranian National Olympiad (3rd Round) 2006

8/26/2006
a) P(x),R(x)P(x),R(x) are polynomials with rational coefficients and P(x)P(x) is not the zero polynomial. Prove that there exist a non-zero polynomial Q(x)∈Q[x]Q(x)\in\mathbb Q[x] that P(x)∣Q(R(x)).P(x)\mid Q(R(x)). b) P,RP,R are polynomial with integer coefficients and PP is monic. Prove that there exist a monic polynomial Q(x)∈Z[x]Q(x)\in\mathbb Z[x] that P(x)∣Q(R(x)).P(x)\mid Q(R(x)).
algebrapolynomialnumber theory proposednumber theory
Polynomials

Source: Iranian National Olympiad (3rd Round) 2006

9/19/2006
P,Q,RP,Q,R are non-zero polynomials that for each z∈Cz\in\mathbb C, P(z)Q(zˉ)=R(z)P(z)Q(\bar z)=R(z). a) If P,Q,R∈R[x]P,Q,R\in\mathbb R[x], prove that QQ is constant polynomial. b) Is the above statement correct for P,Q,R∈C[x]P,Q,R\in\mathbb C[x]?
algebrapolynomialfunctioncomplex analysisdomainalgebra proposed
The NFoH

Source: Iranian National Olympiad (3rd Round) 2006

9/11/2006
The National Foundation of Happiness (NFoH) wants to estimate the happiness of people of country. NFoH selected nn random persons, and on every morning asked from each of them whether she is happy or not. On any two distinct days, exactly half of the persons gave the same answer. Show that after kk days, there were at most n−nkn-\frac{n}{k} persons whose “yes” answers equals their “no” answers.
floor functionvectorcombinatorics proposedcombinatorics
Translations [Combinatorial Geometry]

Source: Iranian National Math Olympiad (Final exam) 2006

9/14/2006
Assume that CC is a convex subset of Rd\mathbb R^{d}. Suppose that C1,C2,…,CnC_{1},C_{2},\dots,C_{n} are translations of CC that Ci∩C≠∅C_{i}\cap C\neq\emptyset but Ci∩Cj=∅C_{i}\cap C_{j}=\emptyset. Prove that n≤3d−1n\leq 3^{d}-1 Prove that 3d−13^{d}-1 is the best bound. P.S. In the exam problem was given for n=3n=3.
geometrygeometric transformationvectortrigonometrydilationratioreal analysis