# TÀI LIỆU TOÁN - WWW.MOLYMPIAD.NET - ĐỀ THI TOÁN

## $show=home ### Junior 1. Let$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. ##$type=three$c=3$source=random$title=oot$p=1$h=1$m=hide$rm=hide ## Anniversary_$type=three$c=12$title=oot$h=1$m=hide\$rm=hide

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Mathematics & Youth: 2009 Anniversary 45th
2009 Anniversary 45th
Mathematics & Youth