2022 Vietnam National Olympiad

Day 1

1

Let a be a non-negative real number and a sequence (un) defined as: u1=6,un+1=2n+an+n+anun+4,n1 a) With a=0, prove that there exist a finite limit of (un) and find that limit b) With a0, prove that there exist a finite limit of (un)

2

Find all function f:R+R+ such that: f(f(x)x+y)=1+f(y),x,yR+.

3

Let ABC be a triangle. Point E,F moves on the opposite ray of BA,CA such that BF=CE. Let M,N be the midpoint of BE,CF. BF cuts CE at D a) Suppost that I is the circumcenter of (DBE) and J is the circumcenter of (DCF), Prove that MNIJ b) Let K be the midpoint of MN and H be the orthocenter of triangle AEF. Prove that when E varies on the opposite ray of BA, HK go through a fixed point

4

For every pair of positive integers (n,m) with n<m, denote s(n,m) be the number of positive integers such that the number is in the range [n,m] and the number is coprime with m. Find all positive integers m2 such that m satisfy these condition: i) s(n,m)mns(1,m)m for all n=1,2,...,m1; ii) 2022m+1 is divisible by m2

Day 2

1

Consider 2 non-constant polynomials P(x),Q(x), with nonnegative coefficients. The coefficients of P(x) is not larger than 2021 and Q(x) has at least one coefficient larger than 2021. Assume that P(2022)=Q(2022) and P(x),Q(x) has a root pq0(p,qZ,(p,q)=1). Prove that |p|+n|q|Q(n)P(n) for all n=1,2,...,2021

2

We are given 4 similar dices. Denote xi(1xi6) be the number of dots on a face appearing on the i-th dice 1i4 a) Find the numbers of (x1,x2,x3,x4) b) Find the probability that there is a number xj such that xj is equal to the sum of the other 3 numbers c) Find the probability that we can divide x1,x2,x3,x4 into 2 groups has the same sum

3

Let ABC be an acute triangle, B,C fixed, A moves on the big arc BC of (ABC). Let O be the circumcenter of (ABC) (B,O,C are not collinear, ABAC), (I) is the incircle of triangle ABC. (I) tangents to BC at D. Let Ia be the A-excenter of triangle ABC. IaD cuts OI at L. Let E lies on (I) such that DEAI. a) LE cuts AI at F. Prove that AF=AI. b) Let M lies on the circle (J) go through Ia,B,C such that IaMAD. MD cuts (J) again at N. Prove that the midpoint T of MN lies on a fixed circle.