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

## $show=home 1. Suppose that$a,b$and$c$are positive integers such that $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}<1.$ Find the maximum value of the expression $S=\frac{1}{a}+\frac{1}{b}+\frac{1}{c}.$ 2. Assume that the triangle$ABC$is isosceles at$A$. Draw the height$AH$. Let$D$be the midpoint of$AH$. Choose$E$on$CD$such that$HE$is perpendicular to$CD$. Prove that$\widehat{AEB}=90^{0}$. 3. Solve the system of equations $\begin{cases} \dfrac{x^{2}-1}{y}+\dfrac{y^{2}-1}{x} & =3\\ x^{2}-y^{2}+\dfrac{12}{x} & =9 \end{cases}.$ 4. Two circles$(O,R)$and$(O',R')$intersect at$A$and$B$. A point$M$varies arbitrary on the opposite ray of the ray$AB$. From$M$, draw two tangent lines to the circle$(O',R')$at$C$and$D$with$D$is inside the circle$(O,R)$. The lines$AD$and$AC$intersect$(O,R)$respectively at$P$and$Q$($P$,$Q$are different from$A$). Prove that the line$PQ$always goes through a fixed point when$M$varies. 5. Find the minimum value of the expression $A=\frac{2}{3xy}+\sqrt{\frac{3}{y+1}}$ where$x,y$are positive real numbers satisfying$x+y\leq3$. 6. Consider the quadratic equation$ax^{2}+bx+c=0$($a\ne0$), where$a,b$and$c$are real numbers satisfying$145a+144b+144c=0$. Prove that this equation cannot have two solutions which are inverses of two nonzero consecutive perfect squares. 7. Given a tetrahedron$ABCD$inscribed in a sphere$(S)$. Let$A_{1},B_{1},C_{1},D_{1}$resprectively the centroids of the triangles$BCD$,$ACD$,$ABD$,$ABC$. The lines$AA_{1}$,$BB_{1}$,$CC_{1}$,$DD_{1}$intersect$(S)$respectively at$A_{2},B_{2},C_{2},D_{2}$. Find the minimum value of the expression $P=\frac{AA_{2}^{2}+BB_{2}^{2}+CC_{2}^{2}+DD_{2}^{2}}{AB\cdot CD+AC\cdot BD+AD\cdot BC}.$ 8. Let$a,b,c$be three positive real numbers. Prove that $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\geq\frac{a^{2}+nb^{2}}{a^{3}+nb^{3}}+\frac{b^{2}+nc^{2}}{b^{3}+nc^{3}}+\frac{c^{2}+na^{2}}{c^{3}+na^{3}}$ for all$n\in\mathbb{N}$,$n\geq2$. 9. Find all quadruples of posotive integers$(x,y,z,p)$,$p$is a prime, such that $p^{x}+(p-1)^{2y}=(2p-1)^{z}.$ 10. Find the maximal$K$such that the following inequality $\sqrt{a+K|b-c|^{\alpha}}+\sqrt{b+K|c-a|^{\alpha}}+\sqrt{c+K|a-b|^{\alpha}}\leq2$ always holds for all$\alpha\geq1$and$a,b,c$are arbitrary nonnegative real numbers satisfying$a+b+c=1$. 11. Find all functions$f:\mathbb{R}\to\mathbb{R}$such that for all$x,y\in\mathbb{R}$$f(x^{2}+f(y))=4y+\frac{1}{2}f^{2}(x).$ 12. Given a circle$(O)$,$ABC$is an acute triangle inscribed in$(O)$. The tangent line to$(O)$at$A$intersects$BC$at$S$;$K$is the perpendicular projection of$A$on$OS$;$BK$and$CK$intersect$CA$,$AB$respectively at$E$and$F$. Prove that the line which contains$A$and is perpendicular to$EF$always goes through a fixed point when$A$varies on$(O)$($B$and$C$are fixed). ##$type=three$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

Name

2006,1,2007,12,2008,12,2009,12,2010,12,2011,12,2012,12,2013,12,2014,12,2015,12,2016,12,2017,12,2018,11,2019,12,2020,12,2021,6,Anniversary,4,
ltr
item
Mathematics & Youth: 2015 Issue 451
2015 Issue 451
Mathematics & Youth