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2017 Issue 480

  1. Find positive integers $x,y,z$ such that $$x^{y}+y^{z}+z^{x}=105.$$
  2. Given a right triangle $ABC$ with the right angle $A$, $AB=AC$ and $BC=a\sqrt{2}$. Let $M,D$ and $E$ be arbitrary points on the sides $BC,AB$ and $AC$ respectively. Find the minimum value of $MD+ME$
  3. Solve the system of equation \begin{align*} \sqrt{x}-\frac{2}{\sqrt{y}}+\frac{3}{\sqrt{z}} & =1\\ \sqrt{y}-\frac{2}{\sqrt{z}}+\frac{3}{\sqrt{x}} & =2\\ \sqrt{z}-\frac{2}{\sqrt{x}}+\frac{3}{\sqrt{y}} & =3 \end{align*}
  4. Two circles $(O_{1},R_{1})$ and $(O_{2,}R_{2})$ intersect at $A$ and $B$. A line $d$ is tangent to $(O_{1},R_{1})$ and $(O_{2},R_{2})$ at $C$ and $D$ respectively. Let $R_{3},R_{4}$ respectively be the circumradii of $ACD$ and $BCD$. Prove that $R_{1}\cdot R_{2}=R_{3}\cdot R_{4}$
  5. Suppose that $a,b,c$ are three positive numbers and given $\alpha,\beta,\lambda\geq2$. Prove that \[\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}\] \[\geq\frac{3}{2}+\frac{(a-b)^{2}}{\alpha(b+c)(c+a)}+\frac{(b-c)^{2}}{\beta(c+a)(a+b)}+\frac{(c-a)^{2}}{\lambda(a+b)(b+c)}.\]
  6. Solve the equation \[ 4x^{3}-24x^{2}+45x-26=\sqrt{-x^{2}+4x-3}.\]
  7. Let $x_{1},x_{2}$ be two solutions of the equation $x^{2}-2ax-1=0$ where a is an integer. Prove that for every natural number $n$, $\frac{1}{8}(x_{1}^{2n}-x_{2}^{2n})(x_{1}^{4n}-x_{2}^{4n})$ is always a product of three consecutive natural numbers.
  8. The incircle $I$ of the triangle $ABC$ is tangent to $BC$, $CA$ and $AB$ at $D,E$ and $F$ respectivly. Suppose that $X$ is the intersection of the lines $DE$ and $AB$, and $Y$ is the intersection of the lines $CF$ and $DE$. Prove that $IY\perp CX$.
  9. Given three positive numbers $a,b,c$ such that $24ab+44bc+33ca\leq1$. Find the minimum value of the expression $P=\frac{1}{a}+\frac{1}{b}+\frac{1}{c}$.
  10. Find all sets of $19$ distinct positive integers such that for each set the sum of the numbers is 2017 and the sums of all the digits of the numbers are all equal.
  11. The sequence $(a_{n})$, $n=0,1,2,\ldots$, is determined as follows: $a_{0}=610$, $a_{1}=89$ and $a_{n+2}=7a_{n+1}-a_{n}$. Find all $n$ such that $2a_{n+1}a_{n}-3$ is a fourth power of an integer.
  12. Given a triangle $ABC$ where $AB\ne AC$ and $\widehat{A}=90^{0}$. Let $BE$ and $CF$ be the altitudes from $B$ and $C$. Let $(O_{1})$, $(O_{2})$ be the circles which pass through $A$ and are tangent to $BC$ at $B,C$ respectively. Assume that $D$ is the second intersection of $(O_{1})$ and $(O_{2})$. Let $M,N$ (resp. $P,Q$) respectively be the second intersections of $BA,BD$ (resp. $CA,CD$) and $(O_{2})$ (resp. $(O_{1})$). Let $R$ be the intersection of $AD$ and $EF$, $S$ the intersection of $MP$ and $NQ$. Show that $RS\perp BC$.

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Mathematics & Youth: 2017 Issue 480
2017 Issue 480
Mathematics & Youth
https://www.molympiad.org/2017/07/mathematics-and-youth-magazine-problems_75.html
https://www.molympiad.org/
https://www.molympiad.org/
https://www.molympiad.org/2017/07/mathematics-and-youth-magazine-problems_75.html
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