MathDB

Day 1

Part of 2019 Vietnam National Olympiad

Problems(4)

Integer sequence

Source: VMO 2019

4/15/2019
Let (xn)({{x}_{n}}) be an integer sequence such that 0≤x0<x1≤1000\le {{x}_{0}}<{{x}_{1}}\le 100 and xn+2=7xn+1−xn+280, ∀n≥0.{{x}_{n+2}}=7{{x}_{n+1}}-{{x}_{n}}+280,\text{ }\forall n\ge 0. a) Prove that if x0=2,x1=3{{x}_{0}}=2,{{x}_{1}}=3 then for each positive integer n,n, the sum of divisors of the following number is divisible by 2424 xnxn+1+xn+1xn+2+xn+2xn+3+2018.{{x}_{n}}{{x}_{n+1}}+{{x}_{n+1}}{{x}_{n+2}}+{{x}_{n+2}}{{x}_{n+3}}+2018. b) Find all pairs of numbers (x0,x1)({{x}_{0}},{{x}_{1}}) such that xnxn+1+2019{{x}_{n}}{{x}_{n+1}}+2019 is a perfect square for infinitely many nonnegative integer numbers n.n.
Integer sequenceperfect numbersum of divisors
Continuous function

Source: VMO 2019

4/15/2019
Let f:R→(0;+∞)f:\mathbb{R}\to (0;+\infty ) be a continuous function such that lim⁡x→−∞ f(x)=lim⁡x→+∞ f(x)=0.\underset{x\to -\infty }{\mathop{\lim }}\,f(x)=\underset{x\to +\infty }{\mathop{\lim }}\,f(x)=0. a) Prove that f(x)f(x) has the maximum value on R.\mathbb{R}. b) Prove that there exist two sequeneces (xn),(yn)({{x}_{n}}),({{y}_{n}}) with xn<yn,∀n=1,2,3,...{{x}_{n}}<{{y}_{n}},\forall n=1,2,3,... such that they have the same limit when nn tends to infinity and f(xn)=f(yn)f({{x}_{n}})=f({{y}_{n}}) for all n.n.
function
Points on the sides of triangle

Source: VMO 2019

4/15/2019
Let ABCABC be triangle with HH is the orthocenter and II is incenter. Denote A1,A2,B1,B2,C1,C2A_{1}, A_{2}, B_{1}, B_{2}, C_{1}, C_{2} be the points on the rays AB,AC,BC,CA,CBAB, AC, BC, CA, CB, respectively such that AA1=AA2=BC,BB1=BB2=CA,CC1=CC2=AB.AA_{1} = AA_{2} = BC, BB_{1} = BB_{2} = CA, CC_{1} = CC_{2} = AB. Suppose that B1B2B_{1}B_{2} cuts C1C2C_{1}C_{2} at A′A', C1C2C_{1}C_{2} cuts A1A2A_{1}A_{2} at B′B' and A1A2A_{1}A_{2} cuts B1B2B_{1}B_{2} at C′C'. a) Prove that area of triangle A′B′C′A'B'C' is smaller than or equal to the area of triangle ABCABC. b) Let JJ be circumcenter of triangle A′B′C′A'B'C'. AJAJ cuts BCBC at RR, BJBJ cuts CACA at SS and CJCJ cuts ABAB at TT. Suppose that (AST),(BTR),(CRS)(AST), (BTR), (CRS) intersect at KK. Prove that if triangle ABCABC is not isosceles then HIJKHIJK is a parallelogram.
geometryincenterparallelogramorthocenterarea of a triangle
Sum of square of coefficients

Source: VMO 2019

4/15/2019
For each real coefficient polynomial f(x)=a0+a1x+⋯+anxnf(x)={{a}_{0}}+{{a}_{1}}x+\cdots +{{a}_{n}}{{x}^{n}}, let Γ(f(x))=a02+a12+⋯+am2.\Gamma (f(x))=a_{0}^{2}+a_{1}^{2}+\cdots +a_{m}^{2}. Let be given polynomial P(x)=(x+1)(x+2)…(x+2020).P(x)=(x+1)(x+2)\ldots (x+2020). Prove that there exists at least 20192019 pairwise distinct polynomials Qk(x){{Q}_{k}}(x) with 1≤k≤220191\le k\le {{2}^{2019}} and each of it satisfies two following conditions: i) deg⁡Qk(x)=2020.\deg {{Q}_{k}}(x)=2020. ii) Γ(Qk(x)n)=Γ(P(x)n)\Gamma \left( {{Q}_{k}}{{(x)}^{n}} \right)=\Gamma \left( P{{(x)}^{n}} \right) for all positive initeger nn.
polynomial