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2015 Issue 460

  1. Let $a$ be a natural number with all different digits and $b$ is another number obtained by using the all the digits of $a$ but in a different order. Given \[a-b=\underset{n}{\underbrace{111\ldots1}}\] ($n$ digit $1$ where $n$ is a positive integer). Find the maximum value of $n$.
  2. Find positive integers $x,y,z$ such that \[(x-y)^{3}+(y-z)^{3}+4|z-x|=27.\]
  3. Let $a,b,c$ be the lengths of three sides of a triangle. Prove that \[2\left(\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\right)\geq\frac{a}{c}+\frac{b}{a}+\frac{c}{b}+3.\]
  4. Let $ABC$ be an isosceles triangle ($AB=AC$) inscribed in a given circle $(O,R)$. Draw $BH$ perpendicular to $AC$ at $H$. Find the maximum length of $BH$.
  5. Solve the system of equations \[\begin{cases} \dfrac{7}{2}+\dfrac{3y}{x+y} & =\sqrt{x}+4\sqrt{y}\\ (x^{2}+y^{2})(x+1) & =4+2xy(x-1) \end{cases}.\]
  6. For any $m>1$, prove that the following equation has a unique solution \[x^{3}-3\sqrt[3]{3x+2m}=2m.\]
  7. Find all values of the parameters $p$ and $q$ such that the corresponding system of equations \[ \begin{cases} x^{2}+y^{2}+5 & =q^{2}+2x-4y\\ x^{2}+(12-2p)x+y^{2} & =2py+12p-2p^{2}-27 \end{cases}\] has two solutions $(x_{1},y_{1})$ and $(x_{2},y_{2})$ satisfying \[ x_{1}^{2}+y_{1}^{2}=x_{2}^{2}+y_{2}^{2}. \]
  8. Given a triangle $ABC$ with $BC=a$, $CA=b$ and $AB=c$. Let $R,r$ and $p$ respectively be the circumradius, the inradius, and the semiperimeter of $ABC$. Prove that \[\frac{ab+bc+ca}{p^{2}+9Rr}\geq\frac{4}{5}.\] When does the equality occur?.
  9. Given three positive real numbers $a,b,c$ such that $abc\geq1$. Prove that \[\frac{a^{4}b^{2}c^{2}}{bc+1}+\frac{b^{4}c^{2}a^{2}}{ca+1}+\frac{c^{4}a^{2}b^{2}}{ab+1}\geq\frac{3}{2}.\]
  10. Find all positive integers $k$ such that there exist 2015 different positive integers whose sum is divisible by the sum of any $k$ numbers among them.
  11. Find the maximum $k$ such that the inequality \[\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+1+2k\geq(k+2)\sqrt{a+b+c+1}\] holds for any positive real numbers $a,b,c$ satisfying $abc=1$.
  12. Given a triangle $ABC$ and $D$ varies on the side $BC$. Let $(I_{1})$, $(I_{2})$ respectively be the incircles of the triangles $ABD$, $ACD$. Suppose that $(I_{1})$ is tangent to $AB$, $BD$ at $E,X$ and $(I_{2})$ is tangent to $AC$, $CD$ at $F,Y$. Assume that $AI_{1}$, $AI_{2}$ respectively intersects $EX$, $FY$ at $Z,T$. Show that
    a) $X,Y,Z,T$ both belong to a circle with center $K$.
    b) $K$ belongs to a fixed line.

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Mathematics & Youth: 2015 Issue 460
2015 Issue 460
Mathematics & Youth
https://www.molympiad.org/2017/08/mathematics-and-youth-magazine-problems_99.html
https://www.molympiad.org/
https://www.molympiad.org/
https://www.molympiad.org/2017/08/mathematics-and-youth-magazine-problems_99.html
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