Found problems: 9
2020 Vietnam National Olympiad, 2
a)Let$a,b,c\in\mathbb{R}$ and $a^2+b^2+c^2=1$.Prove that:
$|a-b|+|b-c|+|c-a|\le2\sqrt{2}$
b) Let $a_1,a_2,..a_{2019}\in\mathbb{R}$ and $\sum_{i=1}^{2019}a_i^2=1$.Find the maximum of:
$S=|a_1-a_2|+|a_2-a_3|+...+|a_{2019}-a_1|$
2023 Vietnam National Olympiad, 7
Let $\triangle{ABC}$ be a scalene triangle with orthocenter $H$ and circumcenter $O$. Incircle $(I)$ of the $\triangle{ABC}$ is tangent to the sides $BC,CA,AB$ at $M,N,P$ respectively. Denote $\Omega_A$ to be the circle passing through point $A$, external tangent to $(I)$ at $A'$ and cut again $AB,AC$ at $A_b,A_c$ respectively. The circles $\Omega_B,\Omega_C$ and points $B',B_a,B_c,C',C_a,C_b$ are defined similarly.
$a)$ Prove $B_cC_b+C_aA_c+A_bB_a \ge NP+PM+MN$.
$b)$ Suppose $A',B',C'$ lie on $AM,BN,CP$ respectively. Denote $K$ as the circumcenter of the triangle formed by lines $A_bA_c,B_cB_a,C_aC_b.$ Prove $OH//IK$.
2022 Vietnam National Olympiad, 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 $MN \parallel IJ$
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
2022 Vietnam National Olympiad, 2
Find all function $f:\mathbb R^+ \rightarrow \mathbb R^+$ such that:
\[f\left(\frac{f(x)}{x}+y\right)=1+f(y), \quad \forall x,y \in \mathbb R^+.\]
2022 Vietnam National Olympiad, 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 $\frac p q \ne 0 (p,q\in \mathbb Z,(p,q)=1)$. Prove that $|p|+n|q|\le Q(n)-P(n)$ for all $n=1,2,...,2021$
2022 Vietnam National Olympiad, 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, $AB \ne AC)$, $(I)$ is the incircle of triangle $ABC$. $(I)$ tangents to $BC$ at $D$. Let $I_a$ be the $A$-excenter of triangle $ABC$. $I_aD$ cuts $OI$ at $L$. Let $E$ lies on $(I)$ such that $DE \parallel AI$.
a) $LE$ cuts $AI$ at $F$. Prove that $AF=AI$.
b) Let $M$ lies on the circle $(J)$ go through $I_a,B,C$ such that $I_aM \parallel AD$. $MD$ cuts $(J)$ again at $N$. Prove that the midpoint $T$ of $MN$ lies on a fixed circle.
2022 Vietnam National Olympiad, 1
Let $a$ be a non-negative real number and a sequence $(u_n)$ defined as:
$u_1=6,u_{n+1} = \frac{2n+a}{n} + \sqrt{\frac{n+a}{n}u_n+4}, \forall n \ge 1$
a) With $a=0$, prove that there exist a finite limit of $(u_n)$ and find that limit
b) With $a \ge 0$, prove that there exist a finite limit of $(u_n)$
2022 Vietnam National Olympiad, 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 $m\ge 2$ such that $m$ satisfy these condition:
i) $\frac{s(n,m)}{m-n} \ge \frac{s(1,m)}{m}$ for all $n=1,2,...,m-1$;
ii) $2022^m+1$ is divisible by $m^2$
2022 Vietnam National Olympiad, 2
We are given 4 similar dices. Denote $x_i (1\le x_i \le 6)$ be the number of dots on a face appearing on the $i$-th dice $1\le i \le 4$
a) Find the numbers of $(x_1,x_2,x_3,x_4)$
b) Find the probability that there is a number $x_j$ such that $x_j$ is equal to the sum of the other 3 numbers
c) Find the probability that we can divide $x_1,x_2,x_3,x_4$ into 2 groups has the same sum