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

## $show=home 1. Find all$2$-digit numbers such that when multiplied by$2,3,4,5,6,7,8,9,$the sum of the digits of the resulting numbers are equal. 2. Let $$S=\frac{2}{2013+1}+\frac{2^{2}}{2012^{2}+1}+\frac{2^{3}}{2013^{2^{2}}+1}+\ldots+\frac{2^{2014}}{2013^{2^{2013}}+1}.$$ Which number is greater?$S$or$\dfrac{1}{1006}$?. 3. Find all integer solutions of the equation $$(y-2) x^{2}+\left(y^{2}-6 y+8\right) x=y^{2}-5 y+62$$ 4. Let$x$,$y$be two rational numbers such that $$x^{2}+y^{2}+\left(\frac{x y+1}{x+y}\right)^{2}=2 .$$ Prove that$\sqrt{1+x y}$. is also a rational number. 5. Let$O$denote the point of intersection of the two diagonals$A C$and$B D$of a convex quadrilateral$A B C D$. Let$E$,$F$,$H$be the feet of the altitudes from$B$,$C$and$O$respectively onto$A D$. Prove that $$A D \cdot B E \cdot C F \leq A C \cdot B D \cdot O H.$$ When does equality holds? 6.$a, b, c$are positive real numbers satisfying$a b c=1$. Prove that $$\frac{a^{3}+5}{a^{3}(b+c)}+\frac{b^{3}+5}{b^{3}(c+a)}+\frac{c^{3}+5}{c^{3}(a+b)} \geq 9$$ 7. Solve the equation $$\left(x^{3}+\frac{1}{x^{3}}+1\right)^{4}=3\left(x^{4}+\frac{1}{x^{4}}+1\right)^{3}$$ 8. Let$A B C$be a triangle with acute angle$A$. Point$P$inside the triangle$A B C$such that$\widehat{B A P}=\widehat{A C P}$and$\widehat{C A P}=\widehat{A B P}$. Let$M$and$N$be the incenters of triangles$A B P$and$A C P$respectively,$R$is the circumradius of triangle$A M N$. Prove that $$\frac{1}{R}=\frac{1}{A B}+\frac{1}{A C}+\frac{1}{A P}.$$ 9. Solve the equation $$[x]^{3}+2 x^{2}=x^{3}+2[x]^{2}$$ where$[t]$denotes the largest integer not exceeding$t$. 10. In the interior of a unit square, there are$n\left(n \in \mathbb{N}^{*}\right)$circles whose sum of areas is greater than$n-1$. Prove that the circles has at least a common point of intersection. 11. Given that the following equation $$a_{0} x^{n}+a_{1} x^{n-1}+\ldots+a_{n-1} x+a_{n}=0$$ has$n$distinct roots. Prove that $$\frac{n-1}{n}>\frac{2 a_{0} a_{2}}{a_{1}^{2}}.$$ 12. Let$O$,$I$and$I_{a}$denote the circumcenter, incenter and excenter in the angle$A$of a triangle$A B C$.$A I$meets$B C$at$D$. BI meets$C A$at$E$. The line through$I$and perpendicular to$O I_{a}$intersects$A C$at$M$. Prove that$D E$passes through the midpoint of line segment$I M$. ##$type=three$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: 2012 Issue 424
2012 Issue 424
Mathematics & Youth