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

## $show=home 1. Compare the following two numbers $$A=\frac{2^{2009}+1}{2^{2010}+1} \text { and } B=\frac{2^{2010}+1}{2^{2011}+1}.$$ 2. Let$A B C$be a triangle with$\widehat{B A C}=45^{\circ}$,$A M$is its median,$A D$is the angle bisector of the triangle$M A C$, draw$D K$perpendicular to$A B$($K$lies on$A B$).$A M$cuts$D K$at$I$. Prove that if$A M$is the angle bisector of$\widehat{B A D}$then$B I$is the angle bisector of$\widehat{A B D}$. 3. Find all positive numbers$a$and$b$such that$\dfrac{a^{2}+b}{b^{2}-a}$and$\dfrac{b^{2}+a}{a^{2}-b}$are both integers. 4. Find the minimum value of the expression$P=a+b+c$given that$3 \leq a, b,c \leq 5$and$a^{2}+b^{2}+c^{2}=50$. 5. Let$A B C$be a triangle. The angle bisector of$\widehat{B A C}$cuts the angle bisector of$\widehat{A B C}$at$I$and meets$B C$at$E$. The line perpendicular to$A E$at$E$meets the arc$\widehat{B I C}$of the circumcircle of the triangle$B I C$at$H$. Prove that$A H$touches the arc$\widehat{B I C}$. 6. Let$I$be the incenter of the triangle$A B C$with$B C=a$,$C A=b$and$A B=c$. The lines$A I$,$B I$,$C I$cut the circumcircle of$A B C$at$A^{\prime}$,$B^{\prime}$,$C^{\prime}$respectively. Prove that $$(p-a) I A^{\prime 2}+(p-b) I B^{\prime 2}+(p-c) I C^{\prime 2}=\frac{1}{2} a b c$$ where$p=\dfrac{1}{2}(a+b+c)$. 7. Let$A B C$($B C=a$,$A C=b$,$A B=c$) be a triangle where$A$,$B$,$C$satisfying the condition $$\cos A+\cos B=2 \cos C.$$ Prove the inequality $$c \geq \frac{8}{9} \max \{a, b\}.$$ When does the equality occur? 8. Solve the equation $$x^{\log _{7} 11}+3^{\log _{7} x}=2 x.$$ 9. Solve the equation $$\sqrt{3 x+4}=x^{3}+3 x^{2}+x-2$$ 10. Let$X$be the subset of the set$\{1,2,\ldots, 2010\}$satisfying conditions$|X|=62$and for every$x \in X$, there exist$a, b \in X \cup\{0 ; 2011\}$($a$and$b$differ from$x$) such that$x=\dfrac{a+b}{2}$. Prove that there exist two elements$x$,$y$in$X$such that$|x-y| \geq 11$and$\dfrac{x+y}{2}$is not in$X$. 11. Make a torus-shaped chessboard by first identifying a pair of opposite edges of an$n \times n$chessboard to get a cylinder and then identifying the opposite bases of the resulting cylinder. Prove that it is possible to place$n$queens on this torus chessboard so that none of them are able to capture any other using the standard chess queen's moves if and only if$(n, 6)=1$. (A queen can capture another if they share the same row, column or diagonal.) 12. Let ABCDEF be an inscribed hexagon,$A C$is parallel to$D F$and$B E$is the circumdiameter.$A B$cuts$E F$at$M$and$B C$cuts$D E$at$N$;$I$is the intersection point of$A N$and$CM$. Prove that$E I$is perpendicular to$A C$. ##$type=three$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: 2010 Issue 395
2010 Issue 395
Mathematics & Youth