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

## $show=home 1. Let$a,b$and$c$be three integers such that$abc=2015^{2016}$. Find the remainder when$19a^{2}+5b^{2}+1890c^{2}$is divided by$24$. 2. Let$a,b\in\mathbb{Z}$. Prove that if$10\mid\left(a^{2}+ab+b^{2}\right)$then$1000\mid\left(a^{3}-b^{3}\right)$. 3. Solve the equation $$\sqrt{2x-2}+\sqrt[3]{x-2}=\frac{9-x}{\sqrt[3]{8x-16}}.$$ 4. Given a triangle$ABC$. Outside the triangle, we draw two equilateral triangles$ABD$and$ACE$. Let$O$be the centroid of$ACE$. On the opposite ray of the ray$OE$choose$F$such that$OF=OE$. Prove that$DF=BO$. 5. Consider the quadratic function$f\left(x\right)=20x^{2}-11x+2016$. Show that there exists an integer$\alpha$such that$2^{20^{11^{1960}}}\mid f\left(\alpha\right)$6. Solve the equation $$\frac{\log_{2}x}{x^{4}-10x^{2}+26}=\frac{5-x^{2}}{\log_{2}^{2}x+1}.$$ 7. Let$x$and$y$be positive numbers such that$\left(\sqrt{x}+1\right)\left(2\sqrt{y}+4\right)+y\geq13$. Find the minimum value of the expression $$P=\frac{x^{4}}{y}+\frac{y^{3}}{x}+y.$$ 8. Given an acute triangle$ABC$. Prove that $$\sqrt[3]{\cot A+\cot B}\geq\frac{16}{3}\cot A\cot B\cot C.$$ 9. Let$a,b$and$c$be three positive numbers such that$a+b+c=abc. Prove that the following inequality \begin{align*} & \sqrt{\left(1+a^{2}\right)\left(1+b^{2}\right)}+\sqrt{\left(1+b^{2}\right)\left(1+c^{2}\right)}+\sqrt{\left(1+c^{2}\right)\left(1+c^{2}\right)}\\ \geq & \sqrt{\left(1+a^{2}\right)\left(1+b^{2}\right)\left(1+c^{2}\right)}+4. \end{align*} 10. A school organizes 7 summer classes. Each student in the school attend at lease one class and each class has exactly 40 students. Besides, for any two classes, there are no more 9 students attending both of them. Prove that the number of students in that school is at least 120. 11. Find all continuous functionsf:\mathbb{R}\to\mathbb{R}$such that $$f\left(xy\right)=f\left(\frac{x^{2}+y^{2}}{2}\right)+\left(x-y\right)^{2}$$ for all$x,y\in\mathbb{R}$. 12. Given a triangle$ABC$and let$I$be its incenter. Assume that$\Delta$is the line which goes through$I$and is perpendicular to$AI$. The points$E$and$F$belong to$\Delta$such that$\widehat{EBA}=\widehat{FCA}=90^{0}$. The points$M$and$N$belong to$BC$such that$ME\parallel NF\parallel AI$. Prove that the circumcircles of the triangles$ABC$and$AMN$are tangent to each other. ##$type=three$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: 2016 Issue 471
2016 Issue 471
Mathematics & Youth