[Booklet] Vietnamese Mathematical Competitions 2018

Vietnam National Mathematical Olympiad 2018

1. The sequence $(x_n)$ is defined as follows $$x_1=2,\quad x_{n+1}=\sqrt{x_n+8}-\sqrt{x_n+3},\, \forall n\geq 1.$$ a) Prove that $(x_n)$ has a finite limit and find that limit.
   b) For every $n\geq 1$, prove that $$n\leq x_1+x_2+\dots +x_n\leq n+1.$$
2. We have a scalene acute triangle $ABC$ (triangle with no two equal sides) and a point $D$ on side $BC$. Pick a point $E$ on side $AB$ and a point $F$ on side $AC$ such that $\angle DEB=\angle DFC$. Lines $DF$, $DE$ intersect $AB$, $AC$ at points $M$, $N$, respectively. Denote $(I_1)$, $(I_2)$ by the circumcircles of triangles $DEM$, $DFN$ in that order. The circle $(J_1)$ touches $(I_1)$ internally at $D$ and touches $AB$ at $K$, circle $(J_2)$ touches $(I_2)$ internally at $D$ and touches $AC$ at $H$. $P$ is the intersection of $(I_1)$, $(I_2)$ different from $D$. $Q$ is the intersection of $(J_1)$, $(J_2)$ different from $D$.
   a) Prove that all points $D$, $P$, $Q$ lie on the same line.
   b) The circumcircles of triangles $AEF$, $AHK$ intersect at $A$, $G$. $(AEF)$ also cut $AQ$ at $A$, $L$. Prove that the tangent at $D$ of $(DQG)$ cuts $EF$ at a point on $(DLG)$. 
3. An investor has two rectangular pieces of land of size $120\times 100$.
   a) On the first land, she want to build a house with a rectangular base of size $25\times 35$ and nines circular flower pots with diameter $5$ outside the house. Prove that even the flower pots positions are chosen arbitrary on the land, the remaining land is still sufficient to build the desired house.
   b) On the second land, she want to construct a polygonal fish pond such that the distance from an arbitrary point on the land, outside the pond, to the nearest pond edge is not over $5$. Prove that the perimeter of the pond is not smaller than $440-20\sqrt{2}$. 
4. On the Cartesian plane the curve $(C)$ has equation $x^2=y^3$. A line $d$ varies on the plane such that $d$ always cut $(C)$ at three distinct points with $x$-coordinates $x_1$, $x_2$, $x_3$.
   a) Prove that the following quantity is a constant $$\sqrt[3]{\frac{x_1x_2}{x_3^2}}+\sqrt[3]{\frac{x_2x_3}{x_1^2}}+\sqrt[3]{\frac{x_3x_1}{x_2^2}}.$$b. Prove the following inequality $$\sqrt[3]{\frac{x_1^2}{x_2x_3}}+\sqrt[3]{\frac{x_2^2}{x_3x_1}}+\sqrt[3]{\frac{x_3^2}{x_3x_1}}<-\frac{15}{4}.$$
5. For two positive integers $n$ and $d$, let $S_n(d)$ be the set of all ordered $d$-tuples $(x_1,x_2,\dots ,x_d)$ that satisfy all of the following conditions
   
   • $x_i\in \{1,2,\dots ,n\}$ for every $i\in\{1,2,\dots ,d\}$;
   • $x_i\ne x_{i+1}$ for every $i\in\{1,2,\dots ,d-1\}$;
   • There does not exist $i,j,k,l\in\{1,2,\dots ,d\}$ such that $i<j<k<l$ and $x_i=x_k$, $x_j=x_l$;
   a) Compute $|S_3(5)|$.
   b) Prove that $|S_n(d)|>0$ if and only if $d\leq 2n-1$. 
6. The sequence $(x_n)$ is defined as follows $$x_0=2,\, x_1=1,\quad x_{n+2}=x_{n+1}+x_n$$ for every non-negative integer $n$.
   a) For each $n\geq 1$, prove that $x_n$ is a prime number only if $n$ is a prime number or $n$ has no odd prime divisors.
   b) Find all non-negative pairs of integers $(m,n)$ such that $x_m|x_n$. 
7. Acute scalene triangle $ABC$ has $G$ as its centroid and $O$ as its circumcenter. Let $H_a$, $H_b$, $H_c$ be the projections of $A$, $B$, $C$ on respective opposite sides and $D$, $E$, $F$ be the midpoints of $BC$, $CA$, $AB$ in that order. $\overrightarrow{GH_a}$, $\overrightarrow{GH_b}$, $\overrightarrow{GH_c}$ intersect $(O)$ at $X$, $Y$, $Z$ respectively.
   a) Prove that the circle $(XCE)$ pass through the midpoint of $BH_a$.
   b) Let $M$, $N$, $P$ be the midpoints of $AX$, $BY$, $CZ$ respectively. Prove that $\overleftrightarrow{DM}$, $\overleftrightarrow{EN}$, $\overleftrightarrow{FP}$ are concurrent.

Vietnam Team Selection Test 2018

1. Let $ABC$ be a acute, non-isosceles triangle. $D$, $E$, $F$ are the midpoints of sides $AB$, $BC$, $AC$, resp. Denote by $(O)$, $(O')$ the circumcircle and Euler circle of $ABC$. An arbitrary point $P$ lies inside triangle $DEF$ and $DP$, $EP$, $FP$ intersect $(O')$ at $D'$, $E'$, $F'$, resp. Point $A'$ is the point such that $D'$ is the midpoint of $AA'$. Points $B'$, $C'$ are defined similarly.
   a) Prove that if $PO=PO'$ then $O\in(A'B'C')$;
   b) Point $A'$ is mirrored by $OD$, its image is $X$. $Y$, $Z$ are created in the same manner. $H$ is the orthocenter of $ABC$ and $XH$, $YH$, $ZH$ intersect $BC, AC, AB$ at $M$, $N$, $L$ resp. Prove that $M$, $N$, $L$ are collinear. 
2. For every positive integer $m$, a $m\times 2018$ rectangle consists of unit squares (called "cell") is called complete if the following conditions are met
   
   1. In each cell is written either a "$0$", a "$1$" or nothing;
   2. For any binary string $S$ with length $2018$, one may choose a row and complete the empty cells so that the numbers in that row, if read from left to right, produce $S$ (In particular, if a row is already full and it produces $S$ in the same manner then this condition ii. is satisfied).
   A complete rectangle is called minimal, if we remove any of its rows and then making it no longer complete.
   a) Prove that for any positive integer $k\le 2018$ there exists a minimal $2^k\times 2018$ rectangle with exactly $k$ columns containing both $0$ and $1$.
   b) A minimal $m\times 2018$ rectangle has exactly $k$ columns containing at least some $0$ or $1$ and the rest of columns are empty. Prove that $m\le 2^k$. 
3. For every positive integer $n\ge 3$, let $\phi_n$ be the set of all positive integers less than and coprime to $n$. Consider the polynomial $$P_n(x)=\sum_{k\in\phi_n} {x^{k-1}}.$$ a) Prove that $P_n(x)=(x^{r_n}+1)Q_n(x)$ for some positive integer $r_n$ and polynomial $Q_n(x)\in\mathbb{Z}[x]$ (not necessary non-constant polynomial).
   b) Find all $n$ such that $P_n(x)$ is irreducible over $\mathbb{Z}[x]$. 
4. Let $a\in\left[ \tfrac{1}{2}, \tfrac{3}{2}\right]$ be a real number. Sequences $(u_n)$, $(v_n)$ are defined as follows $$u_n=\frac{3}{2^{n+1}}\cdot (-1)^{\lfloor2^{n+1}a\rfloor},\quad v_n=\frac{3}{2^{n+1}}\cdot (-1)^{n+\lfloor 2^{n+1}a\rfloor}.$$ a) Prove that $${{({{u}_{0}}+{{u}_{1}}+\cdots +{{u}_{2018}})}^{2}}+{{({{v}_{0}}+{{v}_{1}}+\cdots +{{v}_{2018}})}^{2}}\le 72{{a}^{2}}-48a+10+\frac{2}{{{4}^{2019}}}.$$ b) Find all values of $a$ in the equality case. 
5. In a $m\times n$ square grid, with top-left corner is $A$, there is route along the edges of the grid starting from $A$ and visits all lattice points (called "nodes") exactly once and ending also at $A$. A turning point is a node that is different from $A$ and if two edges on the route intersect at the node are perpendicular.
   a) Prove that this route exists if and only if at least one of $m$, $n$ is odd.
   b) If such a route exists, then what is the least possible of turning points?
6. Triangle $ABC$ circumscribed $(O)$ has $A$-excircle $(J)$ that touches $AB$, $BC$, $AC$ at $F$, $D$, $E$, resp.
   a) $L$ is the midpoint of $BC$. Circle with diameter $LJ$ cuts $DE$, $DF$ at $K$, $H$. Prove that $(BDK)$, $(CDH)$ has an intersecting point on $(J)$.
   b) Let $EF\cap BC =\{G\}$ and $GJ$ cuts $AB$, $AC$ at $M$, $N$, resp. $P\in JB$ and $Q\in JC$ such that $$\angle PAB=\angle QAC=90{}^\circ .$$$PM\cap QN=\{T\}$ and $S$ is the midpoint of the larger $BC$-arc of $(O)$. $(I)$ is the incircle of $ABC$. Prove that $SI\cap AT\in (O)$.

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