Wednesday, September 21, 2011

Mathematics and Youth Magazine, Problem in 411 Issue

Problem in this Issue (Vol 411/09/2011)

FOR LOWER SECONDARY SCHOOLS

T1/411 (For 6 grade). The natural numbers $1,2,3,...,2011^2$ are arranged in some order in a $2011\times 2011$ square table, each square contians one number. Prove that there exits two adjacent square (that is two square having a common edge or common vertex) such that the difference between the corresponding assigned numbers is not smaller than 2012.

T2/411 (For 7 grade). Find the value of the following 2009-terms sum

$S=(1+\dfrac{1}{1.3})(1+\dfrac{1}{2.4})(1+\dfrac{1}{3.5})...(1+\dfrac{1}{2009.2011}).$

T3/411. Find the integers $x,y$ are positive real numbers expression

$x^3+x^2y+xy^2+y^3=4(x^2+y^2+xy+3).$

T4/411. $M$ is a point in the interior of a triangle $ABC$ . Let $P, Q, R, H, G$ be respectively the centroid of triangles $MBC, MCA, MAB, PQR, ABC$ . Prove that points $M, H, G$ are colinear.

T5/411. Let $a,b,c$ are positive real numbers whose sum is 3. Prove the inequality

$\dfrac{4}{(a+b)^3}+\dfrac{4}{(b+c)^3}+\dfrac{4}{(c+a)^3}\geq \dfrac{a}{b+c}+\dfrac{b}{c+a}+\dfrac{c}{a+b}.$

FOR UPPER SECONDARY SCHOOLS

T6/411. The incircle $(I)$ of a triangle $ABC$ touches $BC, CA, AB$ at $D, E, F$ respectively. The line passing through $A$ and parallel to $BC$ meets $EF$ at $K,M$ is the midpoint of $BC$ . Prove that $IM$ is perpendicular to $DK$ .

T7/411. Slove the systerm of the equations

$\begin{cases}\sqrt{\dfrac{x^2+y^2}{4}}+\sqrt{\dfrac{x^2+xy+y^2}{3}}=x+y // x\sqrt{2xy+5x+3}=4xy-5x-3. \end{cases}$

T8/411. Let $a,b,c$ be real numbers such that the equation $ax^2+bx+c=0$ has two solutions, both are in the closed interval $[0,1]$ . Find the maximum and minimum values of the expression

$M=\dfrac{(a-b)(2a-c)}{a(a-b+c)}$ .

TOWARD MATHEMATICAL OLYMPIAD

T9/411. Let $P(x)$ and $Q(x)$ be two polynomials with real coefficients, each has at least one real solution, so that

$P(1+x+Q(x)+Q^2(x))=Q(1+x+P(x)+P^2(x)),\forall x\in\mathbb{R}.$

Prove that $P(x)\equiv Q(x)$ .

T10/411. Let $a,b,c,d$ be positive numbers such that $a\geq b\geq c\geq d$ and $abcd=1$ . Find smallest constant k such that the following inequality holds

$\dfrac{1}{a+1}+\dfrac{1}{b+1}+\dfrac{1}{c+1}+\dfrac{k}{d+1}\geq \dfrac{3+k}{2}$ .

T11/411. Find all continuous functions $f:\mathbb{R}\to\mathbb{R}$ satisfying

$\{f(x+y)\}=\{f(x)+f(y)\},\forall x,y\in\mathbb{R}.$

Where $[t]$ is the largest integer not exceed $t$ and $\{t\}=t-[t]$ .

T12/411. Let $ABC$ be a triangle, $P$ ia an arbitrary point inside the triangle. Let $d_a,d_b,d_c$ be respectively the distances from $P$ to $BC, CA, AB$ . $R_a, R_b,R_c$ are the circumradii of triangle $PBC, PCA,PAB$ respectively. Prove that

$\dfrac{(d_a+d_b+d_c)^2}{PA.PB.PC}\geq \dfrac{\sqrt{3}}{2}(\dfrac{\sin A}{R_a}+\dfrac{\sin B}{R_b}+\dfrac{\sin C}{R_c})$ .

Mathematical Olympiad

Wednesday, September 21, 2011

Mathematics and Youth Magazine, Problem in 411 Issue

Problem in this Issue (Vol 411/09/2011)

FOR LOWER SECONDARY SCHOOLS

FOR UPPER SECONDARY SCHOOLS

TOWARD MATHEMATICAL OLYMPIAD

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