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

## $show=home 1. Give a pentagon$ABCDE$. Assume that$BC$is parallel to$AD$,$CD$is parallel to$BE$,$DE$is parallel to$AC$, and$AE$is parallel to$BD$. Show that$AB$is parallel to$CE$. 2. Prove that $\cfrac{1+\frac{1}{3}+\frac{1}{5}+\ldots+\frac{1}{4035}}{1+\frac{1}{2}+\frac{1}{3}+\ldots+\frac{1}{2018}}>\frac{2019}{4036}.$ 3. Given two triples$(a,b,c)$;$(x,y,z)$none of them contains all$0$'s, such that $a+b+c=x+y+z=ax+by+cz=0.$ Prove that the expression$P=\dfrac{(b+c)^{2}}{ab+bc+ca}+\dfrac{(y+z)}{xy+yz+zx}^{2}$is a constant. 4. Outside a triangle$ABC$, draw triangles$ABD$,$BCE$,$CAF$such that$\widehat{ADB}=\widehat{BEC}=\widehat{CFA}=90^{0}$,$\widehat{ABD}=\widehat{CBE}=\widehat{CAF}=\alpha$. Prove that$DF=AE. 5. Show that the following sum is a positive integer \begin{align} S=&1+\frac{1}{2}+\frac{1}{3}+\ldots+\frac{1}{2017}+\left(1+\frac{1}{2}+\frac{1}{3}+\ldots+\frac{1}{2017}\right)^{2}+\\&+\left(\frac{1}{2}+\frac{1}{3}+\ldots+\frac{1}{2017}\right)^{2}+\ldots+\left(\frac{1}{2017}\right)^{2}.\end{align} 6. Solve the system of equations $\begin{cases}2x^{5}-2x^{3}y-x^{2}y+10x^{3}+y^{2}-5y & =0\\ (x+1)\sqrt{y-5}-y+3x^{2}-x+2 & =0\end{cases}.$ 7. Prove thatx_{0}=\cos\dfrac{\pi}{21}+\cos\dfrac{8\pi}{21}+\cos\dfrac{10\pi}{21}$is a solution of the equation $4x^{3}+2x^{2}-7x-5=0.$ 8. In any triangle$ABC$, show that $\cos(A-B)+\cos(B-C)+\cos(C-A)\leq\frac{1}{2}\left(\dfrac{a+b}{c}+\frac{b+c}{a}+\frac{c+a}{b}\right).$ 9. Given$6$positive numbers$a,b,c$,$x,y,z$and assume that$x+y+z=1$. Show that $ax+by+cz\geq a^{x}b^{y}c^{z}.$ 10. Given an infinite sequence of positive integers$a_{1}<a_{2}<\ldots a_{n}<\ldots$such that$a_{i+1}-a_{i}\geq8$for all$i=1,2,3,\ldots$. For each$n$, let$s_{n}=a_{1}+a_{2}+\ldots a_{n}$. Show that for each$n$, there are at lease two square numbers inside the half-open interval$[s_{n},s_{n+1})$. 11. Given two positive sequences$(a_{n})_{n\geq0}$and$(b_{n})_{n\geq0}$which are determined as follows $a_{0}=\sqrt{3},\,b_{0}=2,\quad a_{n}^{2}+1=b_{n}^{2},\,a_{n}+b_{n}=\frac{1+a_{n+1}}{1-a_{n+1}},\,\forall n\in\mathbb{N}.$ Show that they converges and find the limits. 12. Given a triangle$ABC$with$AB\ne AC$. A circle$(O)$passing through$B$,$C$intersects the line segments$AB$and$AC$at$M$and$N$respectively. Let$P$be the intersection of$BN$and$CM$. Let$Q$be the midpoint of the arc$BC$which does not contian$M$,$N$. Let$K$be the incenter of$PBC$. Show that$KQ$always goes through a fixed point when$(O)$varies. ##$type=three$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 488
2018 Issue 488
Mathematics & Youth