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

## $show=home 1. Replace the distinct letters by distinct numbers such that the following expression becomes a true equality $$\mathrm{VE}+\mathrm{TRUONG}+\mathrm{SA}=22 \times 12 \times 2009.$$ 2. It is well-known that the two right triangles whose side lengths are positive integers$(5,12,13)$and$(6,8,10)$possess additional property that the area of each triangle equals its perimeter. Are there other triangles with similar properties? 3. Suppose given$1003$nonzero rational numbers in which any quadruple form a proportion. Prove that at least$1000$numbers are equal. 4. Let$x$,$y$,$z$be non-negative real numbers such that$x+y+z=1$. Find the maximum valuc of the following expression $$P=(x+2 y+3 z)(6 x+3 y+2 z).$$ 5. Let$A B C D$be a cyclic quadrilateral, inscribed in a circle$(O)$. The angle bisectors of$B A D$and$B C D$meet at a point$K$on the diagonal$B D$. Let$Q$be the second intersection point (different from$A$) of$A P$and the circle$(O)$;$M$and$N$be respectively the midpoints of$B D$and$C P$. The line through$C$and parallel to$A D$meets$A M$at$P$. Prove that a)$S_{A BQ}=S_{A D Q}$. b)$DN$is perpendicular to$C P$. 6. For each natural number$n,$let$p(n)$be its largest odd divisor. Determine the sum $$\sum_{n=2006}^{4012} p(n)$$ 7. Solve for$x$$$\sqrt{3 x-2}=-4 x^{2}+21 x-22$$ 8. Let$A B C$be a triangle whose circumcircle is$(O)$and such that$A C<A B$. The tangent lines to$(O)$at$B$,$C$intersect at$T$. The line through$A$and perpendicular to$A T$meet$B C$at$S$. Choose the points$B_{1}$and$C_{1}$on$S T$such that$T B_{1}=T C_{1}=T B$and that$C_{1}$lies between$S$and$T$. Prove that$A B C$and$A B_{1} C_{1}$are similar. ##$type=three$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: 2009 Issue 384
2009 Issue 384
Mathematics & Youth