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

## $show=home 1. Compare$\dfrac{5}{24}$ưith the sum $$\frac{1}{1.2 .4}+\frac{1}{2.5 .7}+\frac{1}{3.8 .10}+\ldots+\frac{1}{1000.2999 .3001}$$ 2. Let$A B C$be an isosceles right triangle, with right angle at vertex$A .$Let$D$be a point inside the triangle such that$\Delta A B D$is an isosceles triangle and$\widehat{A D B}=150^{\circ} .$Let$A C E$be an equilateral triangle so that points$D$and$E$are on the different side of the half-plane$A C .$Prove that$B$,$D$,$E$are collinear. 3. Find all positive integer numbers$m$such that the following equation $$x^{2}-m x y+y^{2}+1=0$$ has positive integer roots. 4. Let$A B C$be an acute triangle and let$A H$,$B K$,$C L$be its three altitudes. Prove that $$A K \cdot B L \cdot C H=A L \cdot B H \cdot C K=H K \cdot K L \cdot L H.$$ 5. Solve the equation. $$\frac{8 x\left(1-x^{2}\right)}{\left(1+x^{2}\right)^{2}}-\frac{2 \sqrt{2} x(x+3)}{1+x^{2}}=5-\sqrt{2}$$ 6. Let$a$,$b$,$c$be three positive real numbers such that$a b c=1$. Prove that $$\dfrac{1}{\sqrt{a^{5}-a^{2}+3 a b+6}}+\dfrac{1}{\sqrt{b^{5}-b^{2}+3 b c+6}} +\frac{1}{\sqrt{c^{5}-c^{2}+3 c a+6}} \leq 1.$$ 7. Solve the equation $$2 \sin \left(x+\frac{\pi}{3}\right) +2^{2} \sin \left(x+\frac{2 \pi}{3}\right)+\ldots +2^{2010} \sin \left(x+\frac{2010 \pi}{3}\right)=0.$$ 8. Let$O A B C$be a tetrahedron where$O A$,$O B$,$O C$are pairwise orthogonal. Let$M$be a point on the plane containing the base$A B C$. Let$G_{1}$,$G_{2}$,$G_{3}$be respectively the centroids of triangles$O A B$,$O B C$and$O C A$. Put$O A=a$,$O B=b$,$O C=c$. Prove the inequality $$M G_{1}^{2}+M G_{2}^{2}+M G_{3}^{2} \geq \frac{a^{2} b^{2} c^{2}}{a^{2} b^{2}+b^{2} c^{2}+c^{2} a^{2}}.$$ When does equality occur? 9. Let$A B C$be an acute triangle with altitude$A D$.$M$is a point on$A D$. The lines$B M$,$C M$meets$A C$,$A B$at$E$,$F$respectively.$D E$,$D F$intersects with the circles whose diameters$A B$,$A C$at$K$,$L$respectively. Prove that the line connecting the midpoints of$E F$,$K L$goes through$A$. 10. Prove that there exist infinitely many triples of positive integers$(a, b, c)$such that$a b+1$,$b c+1$,$c a+1$are all square numbers. 11. Find all positive integers$a$,$b$,$c$so that the equation $$x+3 \sqrt[4]{x}+\sqrt{4-x}+3 \sqrt[4]{4-x}=a+b+c$$ has solution and the expression$P=a b+2 a c+3 b c$is greatest possible. 12. Find all positive real numbers$a$such that there exists a positive real number$k$and a function$f: \mathbb{R} \rightarrow \mathbb{R}$such that $$\frac{f(x)+f(y)}{2} \geq f\left(\frac{x+y}{2}\right)+k|x-y|^{a}$$ for all real numbers$x$,$y$. ##$type=three$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: 2010 Isue 402
2010 Isue 402
Mathematics & Youth