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

## $show=home 1. Let$x,y$be positive natural numbers. Find the minimum value of the expression$A=|36^{x}-5^{y}|$. 2. Let$O$be the midpoint of the interval$AB$. On a half plane determined by the lines through$AB$, draw two rays$Ox$,$Oy$which are perpendicular to each other. On the$Ox$,$Oy$, respectively choose two points$M,N$which are different from$O$. Prove that$AM+BN\geq MN$. 3. Solve the system of inequalities $\begin{cases} 2\sqrt{x^{2}-xy+y^{2}} & \leq(x+y)^{2}\\ \sqrt{1-(x+y)^{2}} & =1-x \end{cases}.$ 4. Given a triangle$ABC$which is isosceles at$A$and is inscribed in a circle$(O)$. Let$AK$be a diameter. Let$I$be any point on the minor are$AB$($I$is different from$A$and$B$).$KI$intersects$BC$at$M$. The perpendicular bisector of$MI$intersects the sides$AB$,$AC$respectively at$D,E$. Let$N$be the midpoint of$DE$. Prove that$A,M$and$N$are colinear. 5. Find integers$m$so that the following equation $x^{3}+(m+1)x^{2}-(2m-1)x-(2m^{2}+m+4)=0$ has integer solutions. 6. Suppose that$a,b,c$are three nonnegative numbers. Prove that $(a+bc)^{2}+(b+ca)^{2}+(c+ab)^{2}\geq\sqrt{2}(a+b)(b+c)(c+a).$ 7. Given a triangle$ABC$which is isosceles at$A$and is inscribed in a circle$(O)$. Let$D$be the midpoint of$AB$. The ray$CD$intersects$(O)$at$E$. Let$F$be the point on$(O)$so that$CF\parallel AE$. The ray$EF$intersects$AC$at$G$. Prove that$BG$is tangent to the circle$(O)$. 8. Determine the minimum value of the funtion $f(x)=\sqrt{\sin x+\tan x}+\sqrt{\cos x+\cot x}.$ 9. On the plane$Oxy$, consider the set$M$consisting of the points$(x,y)$such that$x,y\in\mathbb{N}^{*}$and$x\leq12,y\leq12$. Each point in$M$is colored be red, white or blue. Prove that there exists a rectangle satisfying the following properties: its sides are parallel to coordinate axes and its vertives are in$M$and are colored by the same color. 10. Find the minimum positive integer$t$so that there exist$t$integers$x_{1},x_{2},\ldots,x_{t}$satisfying $x_{1}^{3}-x_{2}^{3}+x_{3}^{3}-\ldots+(-1)^{t+1}x_{t}^{3}=2065^{2014}.$ 11. Let$a$be a positive integer and$(x_{n})$a sequence given by$x_{1}=1$and $x_{n+1}=\sqrt{x_{n}^{2}+2ax_{n}+2a+1}-\sqrt{x_{n}^{2}-2ax_{n}+2a+1},\forall n\in\mathbb{N}^{*}.$ Find$a$so that the sequence$(x_{n})$has a finite limit. 12. Given a triangle$ABC$. A line$\Delta$which does not contain$A,B,C$intersects$BC,CA,AB$respectively at$A_{1},B_{1},C_{1}$. Let$A_{b},A_{c}$respectively be the symmetric points of$A_{1}$through$AB,AC$. Let$A_{a}$be the midpoint of$A_{b}A_{c}$. The points$B_{b},C_{c}$are determined similarly to the way we construct the point$A_{a}$. Prove that the points$A_{a},B_{b},C_{c}$are collinear. ##$type=three$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: 2015 Issue 452
2015 Issue 452
Mathematics & Youth