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 $(T)$.




Mathematics & Youth: 2021 Issue 524
2021 Issue 524
Mathematics & Youth
Loaded All Posts Not found any posts VIEW ALL Readmore Reply Cancel reply Delete By Home PAGES POSTS View All RECOMMENDED FOR YOU LABEL ARCHIVE SEARCH ALL POSTS Not found any post match with your request Back Home Sunday Monday Tuesday Wednesday Thursday Friday Saturday Sun Mon Tue Wed Thu Fri Sat January February March April May June July August September October November December Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec just now 1 minute ago $$1$$ minutes ago 1 hour ago $$1$$ hours ago Yesterday $$1$$ days ago $$1$$ weeks ago more than 5 weeks ago Followers Follow THIS CONTENT IS PREMIUM Please share to unlock Copy All Code Select All Code All codes were copied to your clipboard Can not copy the codes / texts, please press [CTRL]+[C] (or CMD+C with Mac) to copy