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

## $show=home 1. Find all prime numbers$p$so that$2^{p}+p^{2}$is a prime. 2. Given an acute triangle$A B C$. Let$D$be the point which is equidistant to$3$vertices of the triangle. Suppose that$E$,$F$respectively are the midpoints of$B C$,$A C$. Let$G$be the perpendicular projection of$E$on$A B,$then call$J$the midpoint of$D E .$Show that the triangle$B G J$is isosceles. 3. Solve the following system of equations $$\begin{cases}x^{3}+y &=2 \\ y^{3}+z &=2 \\ z^{3}+t&=2 \\ t^{3}+x&=2\end{cases}.$$ 4. Given a quadrilateral$A B C D$with$\widehat{A B D}=\widehat{A C D}=90^{\circ} .$Draw$B H \perp A D$at$H$. On the diangonal$A C$we choose the point$I$so that$A I=A B .$Let$O$be the midpoint of$A D .$The line perpendicular to$O I$at$I$intersects$B H$and$C D$at$E$and$F$respectively. Show that$I F=2 I E$. 5. Solve the equation $$x+\frac{2 x \sqrt{6}}{\sqrt{x^{2}+1}}=1$$ 6. Solve the following system of equations $$\begin{cases}\sqrt{x}+\sqrt{y}+\sqrt{z}+1&=4 \sqrt{x y z} \\ \dfrac{1}{2 \sqrt{x}+1}+\dfrac{1}{2 \sqrt{y}+1}+\dfrac{1}{2 \sqrt{z}+1}&=\dfrac{3 \sqrt{x y z}}{x+y+z}\end{cases}.$$ 7. Given non-zero numbers$a, b, cd$whose sum is 4 and each of them is greater or equal to$-2 .$Show that $$\frac{1}{a^{2}}+\frac{1}{b^{2}}+\frac{1}{c^{2}}+\frac{1}{d^{2}} \geq \frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}$$ 8. Given a triangle$A B C$with$B C=a$,$C A=b$,$A B=c .$Prove that $$\frac{b}{a^{2} c^{2}}\left[a^{2}(b+c)+b^{2}(c+a)+c^{2}(a+b)\right] \geq \frac{6 \sin ^{3} B}{\cos \frac{A}{2} \cos \frac{B}{2} \cos \frac{C}{2}}.$$ When does the equality happen? 9. Solve the equation $$\sum_{r=1}^{\infty}(-1)^{-1} \frac{x(x-1) \ldots(x-r+1)}{(x+1) \ldots(x+r)}=\frac{1}{2}$$ where$n$is a given positive integer. 10. The sequence$\left(u_{e}\right)$is determined as follows $$u_1=3,\quad u_n = 4u_{n-1}+3 n^{2}-12 n^{3}+12 n-4 ,\,\forall n=2,3, \ldots$$ Show that for any odd prime number$p$,$\displaystyle 2019 \sum_{i=1}^{n-1} u_{i}$is always divisble by$p$. 11. Find all functions$f: \mathbb{R} \rightarrow \mathbb{R}$satisfying $$f(f(x)-2 y)=6 x+f(f(y)+x),\, \forall x, y \in \mathbb{R}$$ 12. Given a triangle$A B C$and let$(O)$be it circumcircle. Let$A_{0}$,$B_{0}$,$C_{0}$respectively be the midpoints of$B C$,$C A$,$A B$. Assume that$A_{1}$,$B_{1}$,$C_{1}$respectively are the perpendicular projections of$A$,$B$,$C$. Let$(O_a)$be the circle passing through$B_0$,$C_{0}$and is internally tangent to$(O)$at$A_{2}$which is different from$A$;$\left(O_{b}\right)$the circle passing through$C_{0}$,$A_{0}$and is internally tangent to$(O)$at$B,$which is different from$B$; and$\left(O_{c}\right)$the circle passing through$A_{0}$,$B_{0}$and is internally tangent to$(O)$at$C_{2}$which is different from$C$. Let$A_{3}$be the intersection between$B_{1} C_{1}$and$B_{2} C_{2}$, and similarly we get the points$B_{3}$,$C_{3}$. Show that$A_{3}, B_{3}$,$C_{3}$belong to a line which is perpendicular to the Euler line of the triangle$A B C$. ##$type=three$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: 2020 Issue 520
2020 Issue 520
Mathematics & Youth