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2021 Issue 526

  1. Given positive integers $m, n$ which are coprime and satisfy $m+n \neq 90$. Find the maximal value of the product $mn$.
  2. Find all positive integers $n$ so that $n^{2021}+n+1$ is a prime.
  3. Solve the equation $$x^{4}-6 x-1=2(x+4) \sqrt{2 x^{3}+8 x^{2}+6 x+1}.$$
  4. Given acute triangle $A B C$ with $A B+A C=2 B C$ which is inscribed in the circle $(O)$ and is circumscribed the circle $(I)$. Let $M$ be the midpoint of the major arc $B C$. The line $M I$ intersects the circumcircle of $B I C$ at $N$. The line $OI$ intersects $B C$ at $P$. Show that the line $P N$ is tangent to the circumcircle of $B I C$.
  5. Find all pairs of integers $(a ; b)$ so that the following system of equations $$\begin{cases}x^{2}+2 a x-3 a-1 &=0 \\ y^{2}-2 b y+x &=0\end{cases}$$ has exactly $3$ different real roots $(x ; y)$.
  6. Given positive numbers $a, b, c$. Show that $$\dfrac{(a-b)^{2}}{(b+c)(c+a)}+\dfrac{(b-c)^{2}}{(c+a)(a+b)}+\dfrac{(c-a)^{2}}{(a+b)(b+c)} \geq \frac{3\left(a^{2}+b^{2}+c^{2}\right)}{(a+b+c)^{2}}-1.$$
  7. Given a triangle $A B C$ with the lengths of the altitudes $h_{a}$, $h_{b}$, $h_{c}$ and its half perimeter $p$ satisfying $h_{a}^{2}+h_{b}^{2}+h_{c}^{2}=p^{2}$. Show that the triangle $A B C$ is equilateral.
  8. Given a tetrahedron $S.ABC$ whose base is a right triangle with the right angle $A$ and $A C > A B$. Let $A M$ be the median and $I$ the incenter of the base. Suppose that $I M$ is perpendicular to $B I$. Compute the value of the expression $$T = \frac{V_{S . AI B}}{V_{S . A I C}}+\frac{V_{S . A I C}}{V_{S . B I C}}+\frac{V_{S . B I C}}{V_{S . A B}}.$$
  9. Suppose that $a, b, c$ are the lengths of three sides of a triangle and $n \in \mathbb{N}^{*}$. Prove that $$\frac{a^{n}}{(b+c)^{n}-a^{n}}+\frac{b^{n}}{(c+a)^{n}-b^{n}}+\frac{c^{n}}{(a+b)^{n}-c^{n}} \geq \frac{3}{2^{n}-1}.$$
  10. For each positive integer $n$, let $S_{n}=\sum_{k=1}^{n}\left[\sqrt{k^{2}+4 \sqrt{k^{2}+2 k+2}}\right]$, where $[x]$ is the greatest integer which does not exceed $x .$ Find all the positive integers $n$ so that $S_{n}$ is a power of a prime.
  11. Find all functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$ satisfying $$f(x+f(y))=f(x),\, \forall x, y \in \mathbb{Z}.$$
  12. Given an acute triangle $A B C$ and $(O ; R)$ is its circumcircle. The altitude $A D=\sqrt{2} R$. Let $M, N$ respectively be the intersections between $A B$, $A C$ and the circle with the diameter $A D$. Let $P$ be the second intersection between $(O ; R)$ the circle with the diameter $A D$ and $Q$ the reflection point of $P$ in $M N$. Show that
    a) $2 S_{A M N}=S_{A B C}$.
    b) $\widehat{B Q N}=\widehat{C Q M}=\dfrac{\pi}{2}$.

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Mathematics & Youth: 2021 Issue 526
2021 Issue 526
Mathematics & Youth
https://www.molympiad.org/2021/04/2021-issue-526.html
https://www.molympiad.org/
https://www.molympiad.org/
https://www.molympiad.org/2021/04/2021-issue-526.html
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