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2017 Issue 485

  1. Let $a=n^{3}+2n$ and $b=n^{4}+3n^{2}+1$. For each $n\in\mathbb{N}$, find the greatest common divisor ($\gcd$) of $a$ and $b$.
  2. Given an isoscesles triangle $ABC$ with $\widehat{A}=100^{0}$, $BC=a$, $AC=AB=b$. Outside $ABC$, we construct the isosceles triangle $ABD$ with $\widehat{ADB}=140^{0}$. Compute the perimeter of $ABD$ in terms of $a$ and $b$.
  3. Find all pairs of intergers $x,y$ such that \[x^{3}y+xy^{3}+2x^{2}y^{2}-4x-4y+4=0.\]
  4. Given an isosceles trapezoid $ABCD$ with $AB//CD$ and $DA=AB=BC$. Let $(K)$ be the circle which goes through $A$, $B$ and tangent to $AD$, $BC$. Let $P$ be a point on $(K)$ and inside $ABCD$. Assume that $PA$ and $PB$ respectively intersect $CD$ at $E$ and $F$. Assume that $BE$ and $BF$ respectively intersect $AD$ and $BC$ at $M$ and $N$. Prove that $PM=PN$.
  5. Solve the system of equations \[\begin{cases}x^{2}+y^{2} & =4y+1\\x^{3}+(y-2)^{3} & =7\end{cases}.\]
  6. Let $a,b,c$ be he length of three sides of a triangle. Prove that \[\frac{a^{3}(a+b)}{a^{2}+b^{2}}+\frac{b^{3}(b+c)}{b^{2}+c^{2}}+\frac{c^{3}(c+a)}{c^{2}+a^{2}}\geq a^{2}+b^{2}+c^{2}.\]
  7. Given a function $f(x)$ which is continuous on $[a,b]$ and differentiable on $(a,b)$, where $0<a<b$. Prove that there exists $c\in(a,b)$ such that \[f'(c)=\frac{1}{a-c}+\frac{1}{b-c}+\frac{1}{a+b}.\]
  8. Given a triangle $ABC$ inscribed the circle $(O)$. Construct the altitude $AH$. Let $M$ be the midpoint of $BC$. Assume that $AM$ intersects $OH$ at $G$. Prove that $G$ belongs to the radial axis of the circumcircle of $BOC$ and the Euler circle of $ABC$.
  9. Given three non-negative real numbers $a,b,c$ satisfying $ab+bc+ca=1$. Find the minimum value of the expression \[P=\frac{(a^{3}+b^{3}+c^{3}+3abc)^{2}}{a^{2}b^{2}+b^{2}c^{2}+c^{2}a^{2}}.\]
  10. Find positive integers $n\leq40$ and positive numbers $a,b,c$ satisfying \[\begin{cases} \sqrt[3]{a}+\sqrt[3]{b}+\sqrt[3]{c} & =1\\ \sqrt[3]{a+n}+\sqrt[3]{b+n}+\sqrt[3]{c+n} & \in\mathbb{Z} \end{cases}.\]
  11. Given three positive integers $a,b,c$. Each time, we tranform the triple $(a,b,c)$ into the triple \[\left(\left[\frac{a+b}{2}\right],\left[\frac{b+c}{2}\right],\left[\frac{c+a}{2}\right]\right).\] Prove that after a finite number of such transformations, we will get a triple wit equal components. (The notation $[x]$ denotes the biggest integer which does not exceed $x$).
  12. Given a triangle $ABC$. Let $(D)$ be he circle which is tangent to the rays $AB$, $AC$ and is internally tangent to the circumcircle of $ABC$ at $X$. Let $J$, $K$ respectively be the incenters of the triangles $XAB$, $XAC$. Let $P$ be the midpoint of the arc $BAC$. Prove that $P(AXJK)$ is a harmonic range.

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Mathematics & Youth: 2017 Issue 485
2017 Issue 485
Mathematics & Youth
https://www.molympiad.org/2017/11/mathematics-and-youth-magazine-problems_25.html
https://www.molympiad.org/
https://www.molympiad.org/
https://www.molympiad.org/2017/11/mathematics-and-youth-magazine-problems_25.html
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