2013 Issue 437

  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.




Mathematics & Youth: 2013 Issue 437
2013 Issue 437
Mathematics & Youth
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