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2009 Anniversary 45th

Junior

  1. Let $a,b,c$ be postitive real numbers such that \[5a^{2}+4b^{2}+3c^{2}+2abc=60.\] Prove the inequality $a+b+c\leq60$.
  2. A point $P$ is chosen inside a tangential quadrilateral $ABCD$ such that $AP$ and $DP$ meet $BC$ at $S$ and $R$ respectively. Prove that the incenters of the triangles $ABS$, $DCR$, $PAD$, $PSR$ lie on the same circle.
  3. Solve for $x$ \[\sqrt{\frac{x^{3}}{3-4x}}-\frac{1}{2\sqrt{x}}=\sqrt{x}.\]
  4. Two circles $(O_{1})$ and $(O_{2})$ intersect at $M$ and $N$. $AB$ is the common tangent to $(O_{1})$ and $(O_{2})$ which is closer to $M$ than $N$, and with $A\in(O_{1})$, $B\in(O_{2})$. Choose a point $C$ on $(O_{1})$, $B\in(O_{2})$. Choose $D$ on $(O_{2})$ which lies inside $(O_{1})$. $AC$ and $BD$ intersects at $K$. Prove that if $CD//AB$ then $\widehat{KMC}=\widehat{KMD}$.
  5. Solve the following system of equations \[ \begin{cases}(x+\sqrt{x^{2}+1})(y+\sqrt{y^{2}+1}) & =1\\ y+\frac{y}{\sqrt{x^{2}-1}}+\frac{35}{12} & =0 \end{cases}.\]
  6. Let $ABC$ be a triangle with centroid $G$ and let $O$ be a point inside the triangle. The lines $AO$, $BO$ and $CO$ meets $BC$, $CA$ and $AB$ at $A_{1},B_{1},C_{1}$ respectively. Let $A_{2},B_{2},C_{2}$ be the points of symmetry through $O$ of the midpoints of $B_{1}C_{1}$, $C_{1}A_{1}$ and $A_{1}B_{1}$. Prove that $AA_{2}$, $BB_{2}$, $CC_{2}$ meet at a common point which lies on $OG$.
  7. Solve the equation \[\sqrt{x^{2}+9x-1}+x\sqrt{11-3x}=2x+3.\]
  8. The points $X,Y$ and $Z$ are chosen on the sides $BC$, $CA$, $AB$ of a given triangle $ABC$, respectively such that $BX=CY=AZ$. Prove that the triangle $XYZ$ is equilateral if and only if so the triangle $ABC$.

Senior

  1. Let $(u_{n})$ be a sequence given recursively as follows: $u_{1}=-1$, $u_{2}=-2$ and \[nu_{n+2}-(3n-1)u_{n+1}+2(n+1)u_{n}=3,\quad n\in\mathbb{N}^{*}.\] Let $S={\displaystyle \sum_{n=1}^{2009}u_{n}+2(2^{2009}-1)}.$ Prove that $S$ is divisible by $2009$.
  2. Let $ABCD$ be a quadrilateral inscribed in a circle $(O)$ and let $a,b$ be the perpendicular bisectors of the line segments $OC$, $OD$ respectively. $M$ is a point on the circle $(O)$. The line $MA$ meets $b$ at $E$, the line $MB$ meets $a$ at $F$. Prove that the line $EF$ is always tangent to a fixed circle when $M$ moves on the circle $(O)$.
  3. In a triangle $ABC$, let $BC=a$, $AC=b$, $AB=c$. Let $G$ be its centroid, $R$ is its circumradius and $G_{1}$, $G_{2}$ and $G_{3}$ are the feet of the altitudes from $G$ onto $BC$, $CA$ and $AB$ respectively. Prove the identity \[\frac{S_{G_{1}G_{2}G_{3}}}{S_{ABC}}=\frac{a^{2}+b^{2}+c^{2}}{36R^{2}}.\]
  4. Determine all functions $f:\mathbb{R}\to\mathbb{R}$, such that \[f(xy)f(yz)f(zx)f(x+y)f(y+z)f(z+x)=2009\] for all positive numbers $x,y,z$.
  5. Let $f$ be a funtion of natural numbers $\mathbb{N}\to\mathbb{N}$ with the following properties \[(f(2n)+f(2n+1)+1)(f(2n+1)-f(2n)-1)=3(1+2f(n))\] and $f(2n)\geq f(n)$ for all natural numbers $n$. Determine all values of $n$ such that $f(n)\leq2009$.
  6. Fixes two distinct points $A$ and $B$ on a given straight line $\Delta$ in a plane. For each point $M$ on $\Delta$, let $N$ be on $\Delta$ such that $\overrightarrow{BN}=k\overrightarrow{BA}$, where $k=\dfrac{\overline{MA}}{\overline{MB}}$. Let $(\alpha)$ be the half-circle that lies on a chosen half-plane given by $\Delta$ whose diameter is $MN$. Prove that $(\alpha)$ is always tangent to a fixed circle when $M$ moves on $\Delta$.
  7. Given a collection of real numbers $A=(a_{1},a_{2},\ldots,a_{n})$, denote by $A^{(2)}$ the 2-sums set of $A$, which is the set of all sum $a_{i}+a_{j}$ for $1\leq i<j\leq n$. Given that \[A^{(2)}=(2,2,3,3,3,4,4,4,4,4,5,5,5,6,6),\] determine the sum of squares of all elements of the original set $A$.
  8. Let $M$ be a point on a given line segment $AB$. Draw three semicircles whose diameters are $AM$, $BM$, $AB$ respectively, such that they are on the same side with respect to $AB$. Let $I$ be the incenter and $r$ be the inradius of the curvilinear triangle $ABM$ (whose sides are the three semicircles just constructed). Prove that when $M$ moves on the line segment $AB$, the locus of $I$ is an arc of an ellipse whose spanning chord passes through one of its foci.

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Mathematics & Youth: 2009 Anniversary 45th
2009 Anniversary 45th
Mathematics & Youth
https://www.molympiad.org/2017/09/mathematics-and-youth-magazine-problems_33.html
https://www.molympiad.org/
https://www.molympiad.org/
https://www.molympiad.org/2017/09/mathematics-and-youth-magazine-problems_33.html
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