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

## $show=home 1. Find all natural number of the form$\overline{abba}$such that $\overline{abba}=\overline{ab}^{2}+\overline{ba}^{2}+a-b.$ 2. Given a right isosceles triangle$ABC$with the right angle$A$. Inside the triangle, choose a point$D$such that$\angle ABD=15^{0}$,$\angle BAD=30^{0}$. Prove that a)$BC=2BD$. b)$\angle BCD>\angle ACD$. 3. Find all integer solutions of the equation $\sqrt{3x+4}=\sqrt{y^{3}+5y^{2}+7y+4}.$ 4. On a semicircle$O$with the diameter$AB$choose two points$E$,$F$($E$is on the arc$BF$). A point O varies on the opposite ray of the ray$EB$. The circumcircle of$ABP$intersects the line through$BF$at the second point$Q$. Let$R$be the midpoint of$PQ$. Prove that the circle with the diameter$AR$always goes through a fixed point. 5. Solve the system of equations $\begin{cases} x^{3}-7x+\sqrt{x-2} & =y+4\\ y^{3}-7y+\sqrt{y-2} & =z+4\\ z^{3}-7z+\sqrt{z-2} & =x+4\end{cases}$ 6. Given three non-negative numbers$a,b,c$such that$a+b+c=3$,$a^{2}+b^{2}+c^{2}=5$. Prove that $a^{3}b+b^{3}c+c^{3}a\leq8.$ 7. Solve the following equation with$m,n,k\in\mathbb{N}$,$n\geq m$,$k\geq2$. $\frac{1}{4}(|\sin x|^{n}+|\cos x|^{n})=\frac{|\sin x|^{m}+|\cos x|^{m}}{|\sin2x|^{k}+|\cos2x|^{k}}$ 8. Given a triangle$OBC$with$\angle AOB=120^{0}$,$OA=a$, and$OB=b$. Let$H$be the perpendicular projection of$O$on$AB$. Prove that $aHA+bHB\leq\sqrt{3}ab.$ 9. Prove that there exists infinitely many positive integers$n$such that$2018^{n-2017}-1$is divisible by$n$. 10. Find all natural numbers$n$satisfying$4^{n}+15^{2n+1}+19^{2n}$is divisible by$18^{17}-1$. 11. Find all funtions$f:\mathbb{R}\to\mathbb{R}$such that$f(0)$is rational and $f(x+f^{2}(y))=f^{2}(x+y),\,\forall x,y\in\mathbb{R}.$ 12. Given a triangle$ABC$. Prove that $\cos\frac{A}{2}+\cos\frac{B}{2}+\cos\frac{C}{2}\geq\sqrt{\frac{3}{2}}\left(\sqrt{\sin\frac{A}{2}}+\sqrt{\sin\frac{B}{2}}+\sqrt{\sin\frac{C}{2}}\right).$ ##$type=three$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: 2017 Issue 486
2017 Issue 486
Mathematics & Youth