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

## $show=home 1. The natural numbers$1,2 \ldots, 2011^{2}$are arranged in some order in a$2011 \times 2011$square table, each square contains one number. Prove that there exists two adjacent squares (that is two squares having a common edge or a common vertex) such that the difference between the corresponding assigned numbers is not smaller than$2012$. 2. Find the value of the following$2009$-terms sum $$S=\left(1+\frac{1}{1.3}\right)\left(1+\frac{1}{2.4}\right)\left(1+\frac{1}{3.5}\right) \ldots\left(1+\frac{1}{2009.2011}\right)$$ 3. Find the integers$x$,$y$satisfying the expression $$x^{3}+x^{2} y+x y^{2}+y^{3}=4\left(x^{2}+y^{2}+x y+3\right)$$ 4.$M$is a point in the interior of a triangle$A B C .$Let$P$,$Q$,$R$,$H$,$G$be respectively the centroid of triangles$M B C$,$M A C$,$M A B$,$P Q R$,$A B C$. Prove that points$M$,$H$and$G$are colinear. 5.$a$,$b$and$c$are positive real numbers whose sum is$3$. Prove the inequality $$\frac{4}{(a+b)^{3}}+\frac{4}{(b+c)^{3}}+\frac{4}{(c+a)^{3}} \geq \frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}$$ 6. The incircle$(I)$of a triangle$A B C$touches$B C$,$C A$,$A B$at$D$,$E$,$F$respectively. The line passing through$A$and parallel to$B C$meets$E F$at$K$.$M$is the midpoint of$B C$. Prove that$I M$is perpendicular to$D K$. 7. Solve the system of equations $$\begin{cases}\sqrt{\dfrac{x^{2}+y^{2}}{2}}+\sqrt{\dfrac{x^{2}+x y+y^{2}}{3}}&=x+y \\ x \sqrt{2 x y+5 x+3} &=4 x y-5 x-3\end{cases}$$ 8. Let$a, b, c$be real numbers such that the equation$a x^{2}+b x+c=0$has two real solutions, both are in the closed interval$[0 ; 1] .$Find the maximum and mininum values of the expression $$M=\frac{(a-b)(2 a-c)}{a(a-b+c)}.$$ 9. Let$P(x)$and$Q(x)$be two polynomials with real coefficients, each has at least one real solution, so that $$P\left(1+x+Q(x)+(Q(x))^{2}\right)=Q\left(1+x+P(x)+(P(x))^{2}\right).$$ For any$x \in \mathbb{R}$. Prove that$P(x) \equiv Q(x)$. 10. Let$a, b, c, d$be positive numbers such that$a \geq b \geq c \geq d$and$a b c d=1 .$Find the smallest constant$k$such that the following inequality holds $$\frac{1}{a+1}+\frac{1}{b+1}+\frac{1}{c+1}+\frac{k}{d+1} \geq \frac{3+k}{2}.$$ 11. Find all continuous functions$f: \mathbb{R} \rightarrow \mathbb{R}$satisfying $$\{f(x+y)\}=\{f(x)+f(y)\}$$ for every$x, y \in \mathbb{R}$($[t]$is the largest integer not exceed$t$and$\{t\}=t-[t]$.) 12. Let$A B C$be a triangle,$P$is an arbitrary point inside the triangle. Let$d_{a}$,$d_{b}$,$d_{c}$be respectively the distances from$P$to$B C$,$C A$,$A B$;$R_{a}$,$R_{b}$,$R_{c}$are the circumradii of triangles$P B C$,$P C A$,$P A B$respectively. Prove that $$\frac{\left(d_{a}+d_{b}+d_{c}\right)^{2}}{P A \cdot P B \cdot P C} \geq \frac{\sqrt{3}}{2}\left(\frac{\sin A}{R_{a}}+\frac{\sin B}{R_{b}}+\frac{\sin C}{R_{c}}\right)$$ ##$type=three$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 411
2011 Issue 411
Mathematics & Youth