# TÀI LIỆU TOÁN - WWW.MOLYMPIAD.NET - ĐỀ THI TOÁN

## $show=home 1. Find all prime numbers$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$BC$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$(T)$. ##$type=three$c=3$source=random$title=oot$p=1$h=1$m=hide$rm=hide ## Anniversary_$type=three$c=12$title=oot$h=1$m=hide\$rm=hide

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Mathematics & Youth: 2021 Issue 524
2021 Issue 524
Mathematics & Youth