2011 Issue 411

  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 $$\frac{\left(d_{a}+d_{b}+d_{c}\right)^{2}}{P A \cdot P B \cdot P C} \geq \frac{\sqrt{3}}{2}\left(\frac{\sin A}{R_{a}}+\frac{\sin B}{R_{b}}+\frac{\sin C}{R_{c}}\right)$$




Mathematics & Youth: 2011 Issue 411
2011 Issue 411
Mathematics & Youth
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