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2021 Issue 524

  1. Find all prime numbers $x, y$ satisfying $x^{2}-2 y^{2}-1=0$.
  2. Given a triangle $A B C$ $(A B < B C)$, the bisector of the angles $B A C$, $A B C$ intersect at $I$. Draw $ID$ perpendicular to $A B$ at $D$, $I E$ perpendicular to $A C$ at $E$. Let $M$, $N$ respectively be the midpoints of $B C$, $A C$. Denote $K$ the intersection between $D E$ and $M N$. Show that the points $B$, $I$, $K$ are collinear.
  3. Find all pairs of integers $(x ; y)$ satisfying $$2025^{x}=y^{3}+3 y^{2}+2 y+6.$$
  4. Given a triangle $A B C$ with the altitude $A H$. The incircle $I$ of the triangle is tangent to $B C$, $C A$, $A B$ respectively at $D$, $E$, $F$. Let $G$ be the intersection between $E F$ and $B I$, and $P$ the intersection between $A I$ and $F D$. Show that $G D$ is perpendicular to $H P$.
  5. Suppose that $a, b, c, d$ are positive integers which satisfy $a b=cd$ and $c>a$, $d>a$. Show that $$\sqrt{b}-\sqrt{a} \geq 1.$$
  6. Solve the system of equations $$\begin{cases} x^{3}+y^{3}+3 y &=x^{2}+2 y^{2}+x+8 \\ y^{3}+z^{3}+3 z &=y^{2}+2 z^{2}+y+8 \\ z^{3}+x^{3}+3 x &=z^{2}+2 x^{2}+z+8 \end{cases}$$
  7. Given real numbers $a, b, c>\dfrac{1}{2}$. Show that $$\frac{1}{2 a-1}+\frac{1}{2 b-1}+\frac{1}{2 c-1}+\frac{4 a b}{1+a b}+\frac{4 b c}{1+b c}+\frac{4 c a}{1+c a} \geq 9.$$
  8. Given a triangle $A B C$. Show that $$\cos ^{2} \frac{A}{2}+\cos ^{2} \frac{B}{2}+\cos ^{2} \frac{C}{2} \geq 18 \sin \frac{A}{2} \sin \frac{B}{2} \sin \frac{C}{2}.$$
  9. Given positive numbers $x, y$ with $x<y$. Prove the following inequalities $$\sqrt{x^{2}-2 \sqrt{2} x+4} \cdot \sqrt{y^{2}-2 \sqrt{2} y+4} + \sqrt{x^{2}+2 \sqrt{2} x+4} \cdot \sqrt{y^{2}+2 \sqrt{2} y+4} \geq 4(x+y).$$
  10. Show that $$\left(\frac{3+\sqrt{5}}{2}\right)^{3^{n}}+\left(\frac{3-\sqrt{5}}{2}\right)^{3^{n}}$$ is an integer which is greater or equal to $3^{n+1}$ and is divisible by 3 for every $n \in \mathbb{N}$.
  11. Given positive numbers $\alpha$ and $\beta$. Consider the following sequences $\left(x_{n}\right)$, $\left(y_{n}\right)$ $$x_{1}=\alpha,\, y_{1}=\beta,\quad x_{n+1}=\frac{5 x_{n} y_{n}}{2 y_{n}+3 x_{n}},\, y_{n+1}=\frac{5 x_{n+1} y_{n}}{2 y_{n}+3 x_{n+1}},\, \forall n=1,2, \ldots$$ Find $\lim _{n \rightarrow \infty} x_{n}$ and $\lim _{n \rightarrow \infty} y_{n}$.
  12. Given an acute triangle $A B C$ inscribed in a circle $(O)$ and suppose that $A D$ is the altitude. The tangent lines of $(O)$ at $B$ $C$ intersect at $T .$ On the line segment $A D$, choose $K$ so that $\widehat{B K C}=90^{\circ}$. Let $G$ be the centroid of $A B C . K G$ intersects $O T$ at $L .$ The points $P$, $Q$ are on the line segments $B C$ so that $L P || O B$, $L Q || O C$. The points $E$, $F$ respectively on the line segments $C A$, $A B$ so that $Q E$, $P F$ are both perpendicular to $B C$. Let $(T)$ be the circle with center $T$ which passes through $B$, $C$. Show that the circumcircle of $A E F$ is tangent to $(T)$.

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Mathematics & Youth: 2021 Issue 524
2021 Issue 524
Mathematics & Youth
https://www.molympiad.org/2021/04/2021-issue-524.html
https://www.molympiad.org/
https://www.molympiad.org/
https://www.molympiad.org/2021/04/2021-issue-524.html
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