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## $show=home 1. Let $$S=\frac{1}{1.2.3.4}+\frac{1}{2.3.4.5}+\ldots+\frac{1}{20.21.22.23}.$$ Compare$S$and$\frac{1}{18}$. 2. Given an isosceles triangle$ABC$with the base$BC$. Let$BD$be the angle bisector. On the ray$BA$, choose$E$such that$BE=2CD$. Show that$\widehat{EDB}=90^{0}$. 3. Let$x,y,z$be positive numbers such that$x+y+z=xyz$. Find the minimum value of the expression $$S=\frac{x}{y^{2}}+\frac{y}{z^{2}}+\frac{z}{x^{2}}.$$ 4. Given an isosceles triangle$ABC$with base$BC$and let$M$be a point inside the triangle such that$\widehat{ABM}=\widehat{BCM}$. Let$H$,$I$and$K$respectively be the perpendicular projections of$M$on$AB$,$BC$and$CA$. Suppose that$E$is the intersection between$MB$and$IH$, and$F$is the intersection between$MC$and$IK$. Assume that the circumcircles of the triangles$MEH$and$MFK$intersect at$N\ne M$. Prove that the line$NM$contains the midpoint of$BC$. 5. Solve the equation $$(x^{2}+x+1)(\sqrt{(3x-2)^{2}}+\sqrt{3x-2}+1)=9.$$ 6. Suppose that the equation $$x^{3}-ax^{2}+bx-c=0$$ has$3$positive solutions. Prove that if $$2a^{3}+3a^{2}-7ab+9c-6b-3a+2=0$$ then$1\leq a\leq2$. 7. Let$a,b,c,d$and$e$be the real numbers such that $$\sin a+\sin b+\sin c+\sin d+\sin e = 0$$ $$\cos2a+\cos2b+\cos2c+\cos2d+\cos2e =-3.$$ Find the maximum and minimum values of the expression $$\cos a+2\cos2b+3\cos3c.$$ 8. Assume that$H$is a point inside the triangle$ABC$. Let$K$be the orthocenter of$ABH$. The line which goes through$H$and is perpendicular to$BC$intersects$AK$at$E$. The line which goes through$H$and is perpendicular to$AC$intersects$HK$at$F$. Prove that$CH\bot EF$. 9. For$n\in\mathbb{N}$, let$S_{n}=1+\frac{1}{2}+\ldots+\frac{1}{n}$. Let$\{S_{n}\}$denote the fractional part of$S_{n}$. Prove that for every$\epsilon\in(0,1)$, there always exists$n\in\mathbb{N}^{+}$such that$\{S_{n}\}\leq\epsilon$. 10. Find the smallest$k$such that we can use$k$colors to color the numbers$1,2,\ldots,20$in the following way. For each number, we use exactly one color, we can use one color for more than one number, and no$3$numbers with the same color forms an arithmetic sequence. 11. Consider the sequence$\{x_{n}\}$determined as follows $$x_{1}=-\pi,\,x_{2}=-1,$$ $$x_{n+2}=x_{n+1}+\log_{2}\frac{9+3(\cos x_{n+1}-\cos x_{n})-\cos x_{n+1}\cos x_{n}}{8+\sin^{2}x_{n}},\quad n=1,2,3,\ldots.$$ Prove that the sequence$(x_{n})$has a finite limit and find that limit. 12. Given a quadrilateral$ABCD$inscribed a circle$(O)$. The external angle bisectors of the angles$\widehat{DAB}$,$\widehat{ABC}$,$\widehat{BCD}$,$\widehat{CDA}$respectively intersects the external angle bisector of the angle$\widehat{ABC}$,$\widehat{BCD}$,$\widehat{CDA}$,$\widehat{DAB}$at$X,Y,Z,T$. Let$E$and$F$respectively be the midpoints of$XZ$and$YT$. Prove that a)$XYZT$is a cyclic quadrilateral and$XZ\bot YT$. b)$O$,$E$and$F$are collinear. ##$type=three$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: 2017 Issue 477
2017 Issue 477
Mathematics & Youth