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

## $show=home 1. Given seven distinct prime numbers$a$,$b$,$c$,$a+b+c$,$a+b-c$,$a-b+c$,$-a+b+c$in which the sum of two of three numbers$a$,$b$,$c$equals$800$. Let$d$be the difference between the largest and the smallest number among these seven integers. What is the maximum value of$d ?$2. A triangle$A B C$has sides$A B=2cm$,$A C=4cm$and median$A M=\sqrt{3}cm$. Find the measure of the angle$B A C$, the length of side$B C$and the area of triangle$A B C$. 3. Find the largest natural number$k$so that$n^{5}-2011 n$is divisible by$k$for all natural number$n$. 4. Solve the equation $$\left(x^{4}-625\right)^{2}-100 x^{2}-1=0$$ 5. Let$A B C$be an acute triangle$(A B \neq A C)$inscribed in the circle$(O),$and$H$is its orthocenter. Let$d$be an arbitrary line passing through$H$. Draw the line$d^{\prime}$symmetric to$d$through$B C$. Find the position of the line$d$so that$d^{\prime}$touches the circumcircle$(O)$. 6. Find the largest constant$k$such that $$\frac{a^{2}}{b}+\frac{b^{2}}{c}+\frac{c^{2}}{a} \geq k\left(a^{2}+b^{2}+c^{2}\right)$$ for all positive real numbers$a$,$b$, and$c$whose sum equals$1 .$7. Let$A B C$be a triangle inscribed in the circle$(O)$, and let$G$be its centroid;$D$,$E$, and$F$are the circumcenters of triangles$G B C$,$G C A$,$G A B$respectively. Prove that$O$is the centroid of$D E F$. 8. Solve the equation $$\sqrt{\cos 5 x+2 \cos x}-\sqrt{2 \cos 5 x+\cos x}=2 \sqrt{\cos x}(\cos 4 x-\cos 2 x).$$ 9. Let$x$,$y$,$z$be real numbers such that$x \geq 1$,$y \geq 2$,$z \geq 3$and $$\frac{1}{x+\sqrt{x-1}}+\frac{2}{y+\sqrt{y-2}}+\frac{3}{z+\sqrt{z-3}}=12$$ Find the maximum and minimum value of the function$f(x, y, z)=x+y+z$. 10. Let$\left(x_{n}\right)$be a sequence given by $$x_{1}=\frac{5}{2},\quad x_{n+1}=\sqrt{x_{n}^{3}-12 x_{n}+\frac{20 n+21}{n+1}},\,\forall n \in \mathbb{N}^{*}.$$ Prove that the sequence$\left(x_{n}\right)$converges and find its limit. 11. Find all functions$f: \mathbb{R} \rightarrow(0 ; 2011]$such that $$f(x) \leq 2011\left(2-\frac{2011}{f(y)}\right),\,\forall x>y.$$ 12. Given four points$A_{I}(i=1,2,3,4),$no three of them are colinear and a point$M$so that$A_{i}(i=1,2,3,4)$and$M$do not lie on the same circle. Let$T_{i}$be a triangle having$A_{j}(j=1,2,3,4 ; j \neq i)$as its vertices,$C_{i}$is the circle (or the line) passing through the feet of the projections through$M$onto three sides (or extended sides) of triangle$T_{i}$. Prove that$C_{I}(i=1,2,3,4)$have a common point. ##$type=three$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 404
2011 Issue 404
Mathematics & Youth