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

## $show=home 1. Find all positive integers$x,y,z$such that $x^{2}+y^{3}+z^{4}=90.$ 2. Let$ABC$be an equilateral triangle whose altitudes are$AD$,$BE$and$CF$. Suppose$M$is an arbitrary point inside triangle$ABC$.$I$,$K$,$L$are the perpendicualr feet from$M$to$AD$,$BE$,$CF$. Prove that the sum$AI+BK+CL$does not depend on the position of$M$. 3. The rational numbers$a$,$b$satisfy the identity $a^{2013}+b^{2013}=2a^{1006}b^{1006}.$ Prove that the equation$x^{2}+2x+ab=0$has two rational solutions. 4. Find the minimum value of the expression $P=(x^{4}+y^{4}+z^{4})\left(\frac{1}{x^{4}}+\frac{1}{y^{4}}+\frac{1}{z^{4}}\right),$ where$x,y,z$are positive real numbers that satisfy$x+y\leq z$. 5. Let$AH$be the altitude from$A$of right triangle$ABC$, right angle at$A$. Point$D$on the oppostite ray of$HA$such that$HA=2HD$. Point$E$is the reflection of$B$through$D$;$I$is the midpoint of$AC$;$DI$and$EI$meet$BC$at$M$and$K$respectively. Prove that$\widehat{BDK}=\widehat{MCD}$. 6. Solve the equation $\sqrt{x+\sqrt{x^{2}-1}}=\frac{27\sqrt{2}}{8}(x-1)^{2}\sqrt{x-1}.$ 7. A convex quadrilateral$ABCD$with area$S$is inscribed in a circle whose radius is$R$and$AB=a$,$BC=b$,$CD=c$,$DA=d$,$AC=e$. If there exists a circle touching all the opposite rays of the rays$BA$,$DA$,$CD$and$CB$. Prove that a)$R=\dfrac{S\cdot e}{p^{2}-e^{2}}$, b)$a^{2}+b^{2}+c^{2}+d^{2}+\dfrac{8SR}{e}=2p^{2}$, where$2p=a+b+c+d$. 8. Find the maximum value of the expression $\alpha(\sin^{2}A+\sin^{2}B+\sin^{2}C)-\beta(\cos^{3}A+\cos^{3}B+\cos^{3}C)$ where$A$,$B$,$C$are three angles of an acute triangle and$\alpha$,$\beta$are two given positive numbers. 9. Find the maximum area of a convex pectagon in the coordinate plane$Oxy$having the following properties: all interior angles are the same, all vertices have integer coordinates, there exists a side that is parallel to the axis$Ox$, there are exactly$16$points, including the vertices, with integer coordinates on its boundary. 10. Find all continuous functions$f$such that $(x+y)f(x+y)=xf(x)+yf(y)+2xy,\,\forall x,y\in\mathbb{R}.$ 11. Let$(a_{n})$be a sequence where$a_{1}\in\mathbb{R}$and$a_{n+1}=|a_{n}-2^{1-n}|$,$\forall n\in\mathbb{N}^{*}$. Find${\displaystyle \lim_{n\to\infty}a_{n}}$. 12. A right triangle$ABC$with right angle at$C$is inscribed in circle$(O)$.$M$is an arbitrary point moving on circle$(O)$, different from$A$,$B$,$C$. Point$N$is the reflection of$M$in$AB$,$P$is the perpendicular foot of$N$to$AC$,$MP$meets$(O)$at a second point$Q$. Prove that the circumcenter of triangle$APQ$lies on a fixed circle. ##$type=three$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: 2013 Issue 433
2013 Issue 433
Mathematics & Youth