MathDB

2

Part of 1989 Romania Team Selection Test

Problems(4)

set M = {P_n | n \ge 0} is dense in C

Source: Romania BMO TST 1989 p2

2/17/2020
Let PP be a point on a circle CC and let ϕ\phi be a given angle incommensurable with 2π2\pi. For each n∈N,Pnn \in N, P_n denotes the image of PP under the rotation about the center OO of CC by the angle αn=nϕ\alpha_n = n \phi. Prove that the set M={Pn∣n≥0}M = \{P_n | n \ge 0\} is dense in CC.
setdense setcirclerotationcombinatorics
monic integer polynomials Q(0) = 0 and P(Q(x)) = (x-1)(x-2)...(x-15).

Source: Romania IMO TST 1989 1.2

2/17/2020
Find all monic polynomials P(x),Q(x)P(x),Q(x) with integer coefficients such that Q(0)=0Q(0) =0 and P(Q(x))=(x−1)(x−2)...(x−15)P(Q(x)) = (x-1)(x-2)...(x-15).
Integer Polynomialpolynomialalgebra
au+bv+cw = 0, a = nw- pv, b = pu-mw, c = mv-nu

Source: Romania IMO TST 1989 2.2

2/17/2020
Let a,b,ca,b,c be coprime nonzero integers. Prove that for any coprime integers u,v,wu,v,w with au+bv+cw=0au+bv+cw = 0 there exist integers m,n,pm,n, p such that {a=nw−pvb=pu−mwc=mv−nu\begin{cases} a = nw- pv \\ b = pu-mw \\ c = mv-nu \end{cases}
system of equationsalgebranumber theorycoprime
a_{n+1} =(1989+a_na_{n-1})/a_{n-2}

Source: Romania IMO TST 1989 3.2

2/17/2020
The sequence (ana_n) is defined by a1=a2=1,a3=199a_1 = a_2 = 1, a_3 = 199 and an+1=1989+anan−1an−2a_{n+1} =\frac{1989+a_na_{n-1}}{a_{n-2}} for all n≥3n \ge 3. Prove that all terms of the sequence are positive integers
recurrence relationSequencenumber theoryalgebra