MathDB

Day 2

Part of 2019 Vietnam National Olympiad

Problems(3)

Quotient of two polynomials

Source: VMO 2019

4/15/2019
Consider polynomial f(x)=x2−αx+1f(x)={{x}^{2}}-\alpha x+1 with α∈R.\alpha \in \mathbb{R}. a) For α=152\alpha =\frac{\sqrt{15}}{2}, let write f(x)f(x) as the quotient of two polynomials with nonnegative coefficients. b) Find all value of α\alpha such that f(x)f(x) can be written as the quotient of two polynomials with nonnegative coefficients.
factoring polynomialspolynomialalgebra
Figure of midpoints and feet of altitude triangle

Source: VMO 2019

4/15/2019
Let ABCABC be an acute, nonisosceles triangle with inscribe in a circle (O)(O) and has orthocenter HH. Denote M,N,PM,N,P as the midpoints of sides BC,CA,ABBC,CA,AB and D,E,FD,E,F as the feet of the altitudes from vertices A,B,CA,B,C of triangle ABCABC. Let KK as the reflection of HH through BCBC. Two lines DE,MPDE,MP meet at XX; two lines DF,MNDF,MN meet at YY. a) The line XYXY cut the minor arc BCBC of (O)(O) at ZZ. Prove that K,Z,E,FK,Z,E,F are concyclic. b) Two lines KE,KFKE,KF cuts (O)(O) second time at S,TS,T. Prove that BS,CT,XYBS,CT,XY are concurrent.
geometryconcurrencyConcyclic
Painting the paper

Source: VMO 2019

4/15/2019
There are some papers of the size 5×55\times 5 with two sides which are divided into unit squares for both sides. One uses nn colors to paint each cell on the paper, one cell by one color, such that two cells on the same positions for two sides are painted by the same color. Two painted papers are consider as the same if the color of two corresponding cells are the same. Prove that there are no more than 18(n25+4n15+n13+2n7)\frac{1}{8}\left( {{n}^{25}}+4{{n}^{15}}+{{n}^{13}}+2{{n}^{7}} \right) pairwise distinct papers that painted by this way.
combinatoricstable