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

## $show=home 1. Find positive integers$x$,$y$such that $$x^{y}+y^{x}=23-x y$$ 2. Given a triangle$A B C$with right angle at$A .$Let$A H$be the altitude. If$\dfrac{A H}{B C}=\dfrac{12}{25},$find$\dfrac{A B}{A C}$. 3. Suppose that$a$,$b$are positive and$9 a^{2}+9 b^{2}+82 a b+10 a+10 b \geq 1 .$Show that $$41 a^{2}+41 b^{2}+18 a b \geq 3-2 \sqrt{2}.$$ 4. Given a rhombus$A B C D$with$\widehat{A B C}=60^{\circ} .$The diagonals$A C$and$B D$intersect at$O$. A line$d$through$D$intersects the opposite rays of the rays$A B$,$C B$respectively at$E$,$F(E \neq A, F \neq C) .$Let$M$be the intersection between$C E$and$A F,$and then$H$the intersection between$O M$and$E F .$Show that$A$,$C$,$D$,$H$lie on some circle. 5. Solve the equation $$\frac{2020 x^{4}+x^{4} \sqrt{x^{2}+2020}+x^{2}}{2019}=2020.$$ 6. Solve the system of equations $$\begin{cases}\sqrt{x^{2}+1}+x-8 y^{2}+8 \sqrt{2 y}-8 & = \dfrac{1}{x-\sqrt{x^{2}+1}}-8 y^{2}+8 \sqrt{2 y}-4 \\ 2 y^{2}-2 \sqrt{2 y}-\sqrt{x^{2}+1}+x+2 &=0\end{cases}$$ 7. Given the function$f(x)=x^{2}-x+1$. Let $$f_{1}(x)=f(x),\quad f_{n}(x)=f\left(f_{n-1}(x)\right),\,n=2,3,4, \ldots$$ Find the coefficient of$x^{2}$in the polynomial expansion of$f_{2021}(x)$. 8. Find the point$M$inside a given tetrahedron$A B C D$such that the product of the distances from$M$to the faces of$A B C D$is maximal. 9. Does there exist a triangle$A B C$with$\sin A=\cos B=\tan C ?$10. The sequence determined as follows $$a_{0}=\frac{1}{2},\quad a_{n+1}=\frac{4 n+5}{4 n+6} a_{n},\, \forall n \geq 0.$$ Let$\displaystyle b_{n}=\sum_{i=0}^{n} a_{i}$,$n \in \mathbb{N}$. Find$\displaystyle \lim _{n \rightarrow+\infty} \dfrac{b_{n}}{n}$. 11. Given real numbers$a, b, c \in[1 ; 5]$so that$a+b+c=9 .$Find the minimum and maximum values of the expressions a)$P=a b c$. b)$F=a b+b c+c a$. c)$S_{\lambda}=a^{\lambda}+b^{\lambda}+c^{\lambda}$, where$\lambda$is a constant$\lambda \in\{0\} \cup[1 ;+\infty)$. 12. Given a triangle$A B C$which is not equilateral and$G$is its centroid. The lines$A G$,$B G$,$C G$respectively intersect the circumcircles of$G B C$,$G C A$,$G A B$at$D$,$E$,$F$. Show that the Euler lines of the triangles$D B C$,$E C A$,$F A B$are concurrent. ##$type=three$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 519
2020 Issue 519
Mathematics & Youth