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

## $show=home 1. Find all possible ways of inserting three distinct digits into the positions represented by a star in$\overline{155*710*4*16}$so that the resulting number is divisible by 396. 2. The triangles$XBC$,$YCA$and$ZAB$are constructed externally on the sides of a triangle$ABC$such that triangle$XBC$is isosceles with angle$BXC$equals$120^{0}$and$YCA$,$ZAB$are both equilateral. Prove that$XA$is perpendicular to$YZ$. 3. Find all positive integer solutions$x,y$of the equation $(x^{2}-9y^{2})^{2}=33y+16.$ 4. Solve the following system of equations $\begin{cases}6(1-x)^{2} & =\frac{1}{y}\\ 6(1-y)^{2} & =\frac{1}{x} \end{cases}.$ 5. Point$C$lies on a half-circle$(O)$with diameter$AB=2R$,$CH$is the altitude from$C$to$AB$($H$differs from$O$). The points$E,F$move on the half-circle such that$\widehat{CHE}=\widehat{CHF}$. Prove that the line$EF$always passes through a fixed point. 6. Solve for$x$$\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+x}}}}=\sqrt{6+\sqrt{6+\sqrt{6+\sqrt{6+}x}}}.$ 7. Solve the following system of equations $\begin{cases} 4x^{2} & =(\sqrt{x^{2}+1}+1)(x^{2}-y^{3}+3y-2)\\ (x^{2}+y^{2})^{2}+1 & =x^{2}+2y \end{cases}.$ 8. Let$BC=a$,$CA=b$,$AB=c$be the side lengths of a triangle$ABC$;$R$and$r$denote its circumradius and inradius respectively. If$S$is the area of triangle$ABC$, prove that $\frac{R}{r}\geq\max\left\{ \frac{1}{2};\sqrt{\frac{ab^{3}+bc^{3}+ca^{3}}{3S^{2}}};\sqrt{\frac{ab^{3}+bc^{3}+ca^{3}}{3S^{2}}}\right\} .$ 9. Find all odd positive integers$n$such that$15^{n}+1$is divisible by$n$. 10. Determine all possible pairs of functions$f:\mathbb{R}\to\mathbb{R}$;$g:\mathbb{R}\to\mathbb{R}$such that for any$x,y\in\mathbb{R}$, the following identity holds $f(x+g(y))=xf(y)-yg(y)+g(x).$ 11.$n$students ($n\geq2$) are standing in a straigh line. Each time the teacher blow a whistle, exactly two students exchange their positions. Can it be possible that after an odd number of such whistles, all students returned to their original positions?. 12. Let$AH$($H\in BC$) be the altitude of an acute triangle$ABC$. Point$P$moves on the segment$AH$. Let$E,F$denote the feet of the perpendicular from$P$to$AB,AC$respectively. a) Prove that the points$B,R,F,C$are concyclic. b) Let$O'$denote the center of the circle containing$B,E,F,C$. Prove that$PO'$always passes through a fix point, independent of the position of point$P$chosen on$AH$. ##$type=three$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 439
2014 Issue 439
Mathematics & Youth