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

## $show=home 1. Find all pairs of integers$(x,y)$satisfying $x^{2}+x=3^{2018y}+1.$ 2. Given a triangle$ABC$with$\angle B=45^{0}$,$\angle C=30^{0}$. Let$BM$be one of the medians of$ABC$. Find the angle$\widehat{AMB}$. 3. Given real numbers$x,y$satisfying$0<x,y<1$. Find the minimum value of the expression $F=x^{2}+y^{2}+\frac{2xy-x-y+1}{4xy}.$ 4. Given a circle$(O)$with a diameter$AB$. On$(O)$pick a point$C$($C$is different from$A$and$B$). Draw$CH$perpendicular to$AB$at$H$. Choose$M$and$N$on the line segments$CH$and$BC$respectively such that$MN$is parallel to$AB$. Through$N$draw a line perpendicular to$BC$. This line intersects the ray$AM$at$D$. On the line$DO$choose two points$F$and$K$such that$O$is the midpoint of$FK$. The lines$AF$and$AK$respectively intersect$(O)$at$P$and$Q$. Prove that$D$,$P$,$Q$are colinear. 5. Suppose that the polynomial $f(x)=x^{3}+ax^{2}+bx+c$ has$3$non-negative real solutions. Find the maximal real number$\alpha$so that $f(x)\geq\alpha(x-a)^{3},\,\forall x\geq0.$ 6. Solve the equation $(1-\sqrt{2}\sin x)(\cos2x+\sin2x)=\frac{1}{2}.$ 7. Given the following system of equations $\begin{cases}\dfrac{yz(y+z-x)}{x+y+z} & =a\\ \dfrac{zx(z+x-y)}{x+y+z} & =b\\ \dfrac{xy(x+y-z)}{x+y+z} & =c\end{cases}$ where$a,b,c$are positive parameters. a) Show that the system always has a positive solution. b) Solve the system when$a=2$,$b=5$,$c=10$. 8. Suppose that$a,b,c$are the lengths of three sides of a triangle. Prove that $\frac{ab}{a^{2}+b^{2}}+\frac{bc}{b^{2}+c^{2}}+\frac{ca}{c^{2}+a^{2}}\geq\frac{1}{2}+\frac{2r}{R}$ with$R$,$r$respectively are the inradius and the circumradius of the triangle. 9. For any integer$n$which is greater than$3$, let $P=\sqrt{3}\cdot\sqrt{4}\ldots\sqrt[n^{3}-n]{n-1}.$ Show that $\sqrt[24n^{2}+24n]{3^{n^{2}+n-12}}\leq P<\sqrt{3}.$ 10. Find natural numbers$n$so that$4^{m}+2^{n}+29$cannot be a perfect square for any natural number$m$. 11. The sequence$(a_{n})$is given as follows $a_{1}=\frac{1}{2},\quad a_{n+1}=\frac{(a_{n}-1)^{2}}{2-a_{n}},\, n\in\mathbb{N^{*}.}$ a) Find${\displaystyle \lim_{n\to\infty}a_{n}}$. b) Show that${\displaystyle \frac{a_{1}+a_{2}+\ldots+a_{n}}{n}\geq1-\frac{\sqrt{2}}{2}}$for all$n\in\mathbb{N^{*}}$. 12. Given a triangle$ABC$inscribed in a circle$(O)$. A point$P$varies on$(O)$but is different from$A$,$B$and$C$. Choose$M$,$N$respectively on$PB$,$PC$so that$AMPN$is a parallelogram. a) Prove that there exists a fixed point which is equidistant from$M$and$N$. b) Prove that the Euler line of$AMN$always goes through a fixed point. ##$type=three$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: 2018 Issue 489
2018 Issue 489
Mathematics & Youth