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

## $show=home 1. Find all pairs of coprime positive integers$x, y$so that $$\frac{x+y}{x^{2}+y^{2}}=\frac{7}{25}$$ 2. Let$A B C$be an equilateral triangle whose altitudes$A H$,$B K$intersect at$G .$The angle-bisector of angle$B K H$meets$C G$,$A H$,$B C$at$M$,$N$,$P$respectively. Prove that$K M=N P$. 3. Find the minimum value of the expression$S=2011 c a-a b-b c$where$a, b, c$satisfy$a^{2}+b^{2}+c^{2} \leq 2$. 4. Let$A B C$be an isosceles right triangle with right angle at$A .$Let$M$,$N$,$O$be respectively the midpoints of$A B$,$A C$,$B C$. The line perpendicular to$C M$from$O$cuts$M N$at$G .$Compare the lengths of the two segments$G M$and$G N$. 5. Solve the equation $$\sqrt{7 x^{2}+25 x+19}-\sqrt{x^{2}-2 x-35}=7 \sqrt{x+2}$$ 6. Let$A B C$be a triangle. Let$A M$,$B N$,$C P$be its internal angle-bisectors ($M \in B C$,$N \in C A$,$P \in A B$). Find the measure of angle$B A C$so that$P M$is perpendicular to$N M$7. Solve the equation $$(\sin x-2)\left(\sin ^{2} x-\sin x+1\right)=3 \sqrt{3 \sin x-1}+1.$$ 8. Find all values of$a, b$so that the equation $$x^{4}+a x^{3}+b x^{2}+a x+1=0$$ has at least one solution and the sum$a^{2}+b^{2}$is smallest possible. 9. Let$a, b, c$be positive numbers. Prove that $$\left(a^{2012}-a^{2010}+3\right)\left(b^{2012}-b^{2010}+3\right)\left(c^{2012}-c^{2010}+3\right) \geq 9(a b+b c+c a).$$ When does equality occur? 10. Let$A B C$be a triangle. An arbitrary line cuts the lines$B C$,$C A$,$A B$at$M$,$N$,$P$respectively. Let$X$,$Y$,$Z$be respectively the centroids of triangles$A N P$,$B P M$,$C M N$. Prove that $$S_{X Y Z}=\frac{2}{9} S_{A B C}$$ 11. Let$\left(a_{n}\right)\left(n \in \mathbb{N}^{*}\right)$be a sequence given by $$a_{1}=0 ,\, a_{2}=38,\, a_{3}=-90,\quad a_{n+1}=19 a_{n-1}-30 a_{n-2},\, \forall n \geq 3.$$ Prove that$a_{2011}$is divisible by 2011 . 12. For all positive integers$n$greater than$2$. Find the number of functions $$f:\{1,2,3, \ldots, n\} \rightarrow\{1,2,3,4,5\}$$ satisfying$|f(k+1)-f(k)| \geq 3$where$k \in\{1,2, \ldots, n-1\}$. ##$type=three$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: 2011 Issue 410
2011 Issue 410
Mathematics & Youth