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2021 Issue 525

  1. Find prime numbers $a, b, c, d$ so that $a>3 b>6 c>12 d$ and $$a^{2}-b^{2}+c^{2}-d^{2}=1749.$$
  2. Given an isosceles triangle $A B C$ with the vertex angle $A$ $(\hat{A}=20^{\circ})$. Let $D$, $E$ be the points on $A C$ so that $D$ is in between $A$ and $E$, and $A D=C E=B C$. Find the measurement of the angle $\widehat{D B E}$.
  3. Given a polynomial $f(x)=x^{2}+a x+b$ where $a$, $b$ are integers. Prove that there always exist integers $m$ so that $f(m)=f(2021) \cdot f(2022)$.
  4. Let $M$ be a point lying inside the triangle $A B C$ so that $\widehat{M B A}=\widehat{M C A}$. Draw the parallelogram $B M C D$. Show that the angle bisector of $\widehat{B A C}$ and the angle bisector of $\widehat{M C D}$ are perpendicular to each other.
  5. Solve the system of equations $$\begin{cases} x y z+x+y+z &=x y+y z+z x+2 \\ \dfrac{1}{x^{2}-x+1}+\dfrac{1}{y^{2}-y+1}+\dfrac{1}{z^{2}-z+1} &=1\end{cases}$$
  6. Given numbers $x, y, z$ which are greater than 1 and satisfy $x+y+z+2=x y z$. Prove that $$\sqrt{x^{2}-1}+\sqrt{y^{2}-1}+\sqrt{z^{2}-1} \geq 3 \sqrt{3}.$$ When does the equality happen?
  7. Find all triangles $A B C$ whose lengths of the sides are positive integers and the length of $A C$ is equal to the length of the internal angle bisector of the angle $A$.
  8. Two circles $(O)$ and $\left(O^{\prime}\right)$ intersect at two points $A$ and $B$. Through a point $C$ lying on the opposite ray of the ray $B A$ draw the tangents $C D$ and $C E$ with $(O)$. The line segment $D E$ intersects $\left(O^{\prime}\right)$ at $F$. The tangent of $\left(O^{\prime}\right)$ at $F$ intersects $C D$ and $C E$ respectively at $M$ and $N$. Show that $A B M N$ is a cyclic quadrilateral.
  9. Show that, for every positive integer $n$, we have $$\frac{1}{1} C_{8 n-1}^{1}-\frac{1}{2} C_{8 n-1}^{3}+\frac{1}{3} C_{8 n-1}^{\delta}-\ldots-\frac{1}{2 n-1} C_{8 n-1}^{4 n-5}+\frac{1}{2 n-1} C_{8 n-1}^{4 n-3} \leq \frac{(8 n-1) !}{((4 n) !)^{2}}-\frac{7}{4}.$$
  10. The sequence $\left(a_{n}\right)$ is determined as follows $$a_{1}=1,\, a_{2}=3,\quad \log _{2} a_{n+2}=\log _{3}\left(a_{n}+1\right),\, \forall n=1,2, \ldots.$$ Show that the sequence $\left(a_{n}\right)$ converges and find its limit.
  11. Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ so that $$f\left(x+y^{4}\right)=f(x)+y f\left(y^{3}\right),\, \forall x, y \in \mathbb{R}$$
  12. Given a quadrilateral $A B C D$ and a point $E$ is on $A B$. A point $F$ varies on $C D$. The points $M$ and $N$ respectively are the perpendicular projections of $C$ and $D$ on $E F$. Assume that $P$ is the intersection between the line passing through $M$ and perpendicular to $A D$ and the line passing through $N$ and perpendicular to $B C$. Show that the incenter of the triangle $M N P$ belongs to a fixed circle.

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Mathematics & Youth: 2021 Issue 525
2021 Issue 525
Mathematics & Youth
https://www.molympiad.org/2021/04/2021-issue-525.html
https://www.molympiad.org/
https://www.molympiad.org/
https://www.molympiad.org/2021/04/2021-issue-525.html
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