2017 Issue 484

  1. Compute \[A=3+4+6+9+13+18+\ldots+4953\] where the terms are determined by the formular $a_{n+1}=a_{n}+n$, $n\in\mathbb{N}^{*}$.
  2. Find all natural numbers $x,y$ such that \[(1+x!)(1+y!)=(x+y)!\] where $n!=1.2...n$.
  3. In each square in a $8\times8$ chess board we place some small stones such that the sum of the stones in an row or any column is even. Prove that the sum of the stones in the black squares is even.
  4. Let $ABCD$ be a rectangle with $AB=BC\sqrt{2}$. Choose some point $M$ on the line segment $CD$ such that $M$ is different from $D$. Draw $BI\perp AM$ ($I\in AM$). Assume that $CI$ and $DI$ intersect $AB$ at $E$ and $F$ respectively. Prove that $AE$, $BF$ and $AB$ can be the lengths of the three sides of a right triangle.
  5. Solve the equation \[2x^{4}-8x^{3}+60x^{2}-104x-240.\]
  6. Find the real roots of the following equation $$ 3\sqrt{x^{2}+y^{2}-2x-4y+5}+2\sqrt{5x^{2}+5y^{2}+10x+50y+130}+ \\ +\sqrt{5x^{2}+5y^{2}-30x+45} = \sqrt{102x^{2}+102y^{2}-204x+204y+1360}.$$
  7. Prove that the following inequalities hold for every positive integer $n$ \[\ln\frac{n+1}{2}<\frac{1}{2}+\frac{1}{3}+\ldots+\frac{1}{n}<\log_{2}\frac{n+1}{2}.\] And hence deduce that \[\lim_{n\to+\infty}\left(\frac{1}{2}+\frac{1}{3}+\ldots+\frac{1}{n}\right)=+\infty.\]
  8. Assume that the incircle $I$ of the triangle $ABC$ is tangent to $BC$, $CA$ and $AB$ respectively at $D$, $E$ and $F$. Draw $DG$ perpendicular to $EF$ ($G$ belongs to $EF$). Let $J$ be the midpoint of $DG$. The line $EJ$ intersects the circle $(I)$ at $H$. Let $K$ be the circumcenter of the triangle $FGH$. Prove that $IK||BH$.
  9. Given positivenumbers $x,y,z$ satisfying \[\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=\frac{16}{x+y+z}.\] Find the minimum value of the expression \[P=\frac{x}{y}+\frac{y}{z}+\frac{z}{x}.\]
  10. For each positive integer $n$, let $f(n)$ be the sum of the squares of the positive divisors of $n$. Find all positive integers $n$ such that \[\sum_{k=1}^{n}f(k)\geq\frac{10n^{3}+15n^{2}+2n}{24}.\]
  11. Given the sequence $(x_{n})$ as follows \[x_{1}=1,\,x_{2}=\frac{1}{2},\quad x_{n+2}x_{n}=x_{n+1}^{2}+4^{-n},\,n\in\mathbb{N}^{*}.\] Find ${\displaystyle \lim_{n\to\infty}(3-\sqrt{5})^{n}x^{n}}$.
  12. Given a triangle $ABC$. Let $(O)$ and $I$ respectively be the circumcircle and the incenter of $ABC$. Assume that $AI$ intersects $BC$ at $A_{1}$ and intersects $(O)$ at another point $A_{2}$. Similarly we get the points $B_{1},B_{2}$ and $C_{1},C_{2}$. Suppose that $B_{1}C_{1}$ intersects $B_{2}C_{2}$ at $A_{3}$, $A_{1}C_{1}$ intersects $A_{2}C_{2}$ at $B_{3}$, $A_{1}B_{1}$ intersects $A_{2}B_{2}$ at $C_{3}$. Prove that $A_{3},B_{3},C_{3}$ both belong to a line which is perpendicular to $OI$.




Mathematics & Youth: 2017 Issue 484
2017 Issue 484
Mathematics & Youth
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