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2017 Issue 483

  1. Find all single digit numbers $a,b,c$ such that the numbers $\overline{abc}$, $\overline{acb}$, $\overline{bca}$, $\overline{cab}$, $\overline{cba}$ are prime numbers.
  2. Given a right triangle $XYZ$ with the right angle $X$ ($XY<XZ$). Draw $XU$ perpendicular to $YZ$. Let $P$ be the midoint of $YU$. Choose $K$ on the half plane determined by $YZ$ which does not contain $X$ such that $KZ$ is perpendicular to $XZ$ and $XY=2KZ$. The line $d$ which passes through $Y$ and is parallel to $XU$ intersects $KP$ at $V$. Let $T$ be the intersection of $XP$ and $d$. Prove that $\widehat{VTK}=\widehat{XKZ}+\widehat{VXY}$.
  3. There are 3 people wanting to buy sheeps from Mr. An. The first one wants to buy $\dfrac{1}{a}$ of the herd, the second one wants to buy $\dfrac{1}{b}$ of the herd, and the third one wants to buy $\dfrac{1}{c}$ of the herd and it happens that
    • $a,b,c\in\mathbb{N}^{*}$ and $a<b<c$,
    • the numbers of sheeps each person wants to buy are positive integers,
    • after shelling, Mr. An still has exactly one sheep left.
    What are the possible numbers of sheeps Mr. An has?.
  4. Given a circle $(O)$ with a diameter $AB$. On $(O)$ choose a point $C$ such that $CA<CB$. On the open line segment $OB$ choose $E$. $CE$ intersect $(O)$ at $D$. The line which goes through $A$ and is parallel to $BD$ intersects $BC$ at $I$. The lines $OI$ and $CE$ meet at $F$. Prove that $FA$ is a tangent to the circle $(O)$.
  5. Given real numbers $a,b,c$ satisfying $a+b+c=3$ and $abc\geq-4$. Prove that \[3(abc+4)\geq5(ab+bc+ca).\]
  6. Solve the following system of equations ($a$ is a parameter) \[\begin{cases}2x(y^{2}+a^{2}) & =y(y^{2}+9a^{2})\\ 2y(z^{2}+a^{2}) & =z(z^{2}+9a^{2})\\ 2z(x^{2}+a^{2}) & =x(x^{2}+9a^{2})\end{cases}.\]
  7. Suppose that $x=\dfrac{2013}{2015}$ is a solution of the polynomial \[f(x)=a_{0}+a_{1}x+a_{2}x^{2}+\ldots+a_{n}x^{n}\in\mathbb{Z}[x].\] Can the sum of the coefficients of $f(x)$ be 2017?.
  8. Given three circles $(O_{1}R_{1})$, $(O_{2}R_{2})$, $(O_{3}R_{3})$ which are pariwise externally tangent to each other at $A,B,C$. Let $r$ be the radius of the incircle of $ABC$. Prove that \[r\leq\frac{R_{1}+R_{2}+R_{3}}{6\sqrt{3}}.\]
  9. Given positive numbers $x,y,z$ satisfying \[x^{2}+y^{2}-2z^{2}+2xy+yz+zx\leq0.\] Find the minimum value of the expression \[P=\frac{x^{4}+y^{4}}{z^{4}}+\frac{y^{4}+z^{4}}{x^{4}}+\frac{z^{4}+x^{4}}{y^{4}}.\]
  10. Find the maximum value of the expression \[T=\frac{a+b}{c+d}\left(\frac{a+c}{a+d}+\frac{b+d}{b+c}\right)\] where $a,b,c,d$ belong to $[\frac{1}{2},\frac{2}{3}]$.
  11. Let $R(t)$ be a polynomial of degree 2017. Prove that there exist infinitely many polynomials $P(x)$ such that \[P((R^{2017}(t)+R(t)+1)^{2}-2)=P^{2}(R^{2017}(t)+R(t)+1)-2.\] Find a relation between those polynomials $P(x)$.
  12. Given a triangle $ABC$. The incircle $(I)$ of $ABC$ is rangent to $AB$, $BC$ and $CA$ at $K,L$ and $M$. The line $t$ which passes through $B$ and is different from $AB$ and $BC$ intersects $MK$ at $ML$ respectively at $R$ and $S$. Prove that $\widehat{RIS}$ is an acute angle.

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Mathematics & Youth: 2017 Issue 483
2017 Issue 483
Mathematics & Youth
https://www.molympiad.org/2017/09/mathematics-and-youth-magazine-problems_20.html
https://www.molympiad.org/
https://www.molympiad.org/
https://www.molympiad.org/2017/09/mathematics-and-youth-magazine-problems_20.html
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