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

## $show=home 1. Find the minimum value of the products of$5$different integers among which the sum of any$3$arbitrary numbers is always greater than the sum of the remains. 2. Let$ABC$be a triangle with$AB>AC$and$AB>BC$. On the side$AB$choose$D$and$E$such that$BC=BD$and$AC=AE$. Choose$K$on$CA$and$I$on$CB$such that$DK$is parallel to$BC$and$EI$is parallel to$CA$. Prove that$CK=CI$. 3. Solve the follwowing equation $\frac{1}{\sqrt{x+3}}+\frac{1}{\sqrt{3x+1}}=\frac{2}{1+\sqrt{x}}.$ 4. Given an acute triangle$ABC$with the orthocenter$H$. Let$M$be a point inside the triangle such that$\widehat{MAB}=\widehat{MCA}$. Let$E$and$F$respectively be the orthogonal projections of$M$on$AB$and$AC$. Let$I$and$J$respectively be the midpoints of$BC$and$MA$. Prove that 3 lines$MH$,$EF$and$IJ$are concurrent. 5. Find all pairs of integers$(x,y)$satisfying $x^{4}+y^{3}=xy^{3}+1.$ 6. Solve the following equation $8^{x}-9|x|=2-3^{x}.$ 7. Given a triangle$ABC$with the sides$AB=c$,$CA=b$,$BC=a$. Assume that the radius of the circumscribed circle is$R$and the radius of the inscribed circle is$r$. Show that $\frac{r}{R}\leq\frac{3(ab+bc+ca)}{2(a+b+c)^{2}}.$ 8. Let$x,y,z$be 3 positive real numbers with$x\geq z$. Find the minimum value of the expression $P=\frac{xz}{y^{2}+yz}+\frac{y^{2}}{xz+yz}+\frac{x+2z}{x+z}.$ 9. Find the integer part of the expression $B=\frac{1}{3}+\frac{5}{7}+\frac{9}{13}+\ldots+\frac{2013}{2015}.$ 10. Find all polynomials$f(x)$with integer coefficients such that$f(n)$is a divisor of$3^{n}-1$for every positive integer$n$. 11. Let$\{x_{n}\}$be a sequence satisfying $x_{0}=4,\,x_{1}=34,\,x_{n+2}\cdot x_{n}=x_{n+1}^{2}+18\cdot10^{n+1},\,\forall n\in\mathbb{N}.$ Let${\displaystyle S_{n}=\sum_{k=0}^{26}x_{n+k}}$,$n\in\mathbb{N}^{*}$. Prove that, for every odd natural number$n$,$66|S_{n}$. 12. Given a triangle$ABC$. The point$E$and$F$respectively vary on the sides$CA$and$AB$such that$BF=CE$. Let$D$be the intersection of$BE$and$CF$. Let$H$and$K$respectively be the orthocenters of$DEF$and$DBC$. Prove that, when$E$and$F$change, the line$HK$always passes 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: 2014 Issue 449
2014 Issue 449
Mathematics & Youth