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

## $show=home 1. Let$A$be the sum (where$n!$denote the product$1.2.3.\ldots n$) $A=\frac{1}{1.1!}+\frac{1}{2.2!}+\ldots+\frac{1}{n.n!}+\ldots+\frac{1}{2013.2013!}.$ Prove that$A<\dfrac{3}{2}$. 2. In a right triangle$ABC$with right angle at$A$, let$D$be the midpoint of$AC$,$E$be the point on side$BC$such that$BE=2CE$. Prove that$BD=3ED$. 3. Determine all possible triples of integers$x,y,z$satisfying the equation $6(y^{2}-1)+3(x^{2}+y^{2}z^{2})+2(z^{2}-9x)=0.$ 4. Let$a,b,c$be three real numbers satisfying the conditions$a\ne0$and$2a+3b+6c=0$. Find the smallest possible distance between the two roots of equation$ax^{2}+bc+c=0$. 5. Given that in a triangle$ABC$,$AB=2$,$AC=3$,$BC=4$. Prove that $\widehat{BAC}=\widehat{ABC}+2\widehat{ACB}.$ 6. Let$a,b,c$be positive real numbers satisfuing $a+b+c=\frac{1}{a}+\frac{1}{b}+\frac{1}{c}.$ Prove the inequality $3(a+b+c)\geq\sqrt{8a^{2}+1}+\sqrt{8b^{2}+1}+\sqrt{8c^{2}+1}.$ 7. A circle centered at$O$and radius$R$circumscribes about a triangle$ABC$whose sides are$AB=c$,$BC=a$,$CA=b$. Let$E$be the center of the Euler circle of triangle$ABC$. Prove that if$a^{2}+b^{2}+c^{2}=5R^{2}$then$E$lies on the circle$(O;R)$. 8. Solve the equation $3^{x}-x-1=\log_{3}\frac{(2x+1)\log_{3}(2x+1)}{x}.$ 9. For which positive integers$n$will the following equation has positive integer solutions $\frac{1}{x_{1}^{2}}+\frac{1}{x_{n}^{2}}+\ldots+\frac{1}{x_{n}^{2}}=4.$ 10. Prove that in any triangle$ABC$, the following inequality holds $\frac{\cos\left(\frac{B}{2}-\frac{C}{2}\right)}{\sin\frac{A}{2}}+\frac{\cos\left(\frac{C}{2}-\frac{A}{2}\right)}{\sin\frac{B}{2}}+\frac{\cos\left(\frac{A}{2}-\frac{B}{2}\right)}{\sin\frac{C}{2}}\\ \leq 2\left(\frac{\tan\frac{A}{2}}{\tan\frac{B}{2}}+\frac{\tan\frac{B}{2}}{\tan\frac{C}{2}}+\frac{\tan\frac{C}{2}}{\tan\frac{A}{2}}\right).$ 11. Let$(a_{n})$be the sequence of real numbers such that $a_{1}=34,\quad a_{n+1}=4a_{n}^{3}-104a_{n}^{2}-107a_{n}$ for all positive integers$n$. Find all prime numbers$p$satisfying the following two conditions$p\equiv3(\text{mod }4)$and$a_{2013}+1$is divisible by$p$. 12. Given five points$A,B,C,D,E$on the same circle. Let$M$be the midpoint of$DE$. The Euler circles of triangles$ADE$and$BDE$meet at$C'$(different from$M$); the Euler circles of triangles$BDE$and$CDE$meet at$A'$(different from$M$); the Euler circles of triangles$CDE$and$ADE$meet at$B'$(different from$M$). Prove that the lines$AA'$,$BB'$and$CC'$are concurrent. ##$type=three$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: 2013 Issue 437
2013 Issue 437
Mathematics & Youth