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

## $show=home 1. Find all integer$x$such that $$(x-2018)^{3}+(x-2019)^{2}=2020-x.$$ 2. Let$x_{n}=2^{2^{n}}+1$,$n=1,2, \ldots, 2019$. Show that $$\frac{1}{x_{1}}+\frac{2}{x_{2}}+\frac{2^{2}}{x_{3}}+\ldots+\frac{2^{2018}}{x_{2019}}<\frac{1}{3}.$$ 3. Given$0<x<y \leq z \leq 2$and$3 x+2 y+z=9$. Find the maximum value of the expression $$A=3 x^{2}+2 y^{2}+z^{2}.$$ 4. Given triangle$A B C$with$A B<B C$. On the side$B C$choose$D$so that$C D=A B$. Through$D$draw the line which is parallel to$A C$. This line intersects the angle bisector of$\widehat{A B C}$at$I$. From$I$draw$IH$perpendicular to$B C$($H$is on$B C$). From$H$draw$H E$perpendicular to$A B$($E$is on$A B$) and draw$H F$perpendicular to$A D$($F$is on$A D$). Prove that$\widehat{A E I}=\widehat{A F I}$. 5. Solve the system of equations $$\begin{cases} x y(x+y)-x(y-1)-\dfrac{3}{8}\left(y^{2}+1\right) &=0 \\ x y^{2}+x-y &=0 \end{cases}.$$ 6. Find$x \in[-1 ; 0]$such that $$\sqrt{x+1}+\sqrt{x^{2}+1}=2.$$ 7. Solve the equation $$(2+\sqrt{3})^{|\sin x|}+\left(\frac{\sqrt{2}+\sqrt{6}}{2}\right)^{|\cos x|}=\frac{2+\sqrt{2}+\sqrt{6}}{2}.$$ 8. Given a triangle$A B C$with$h_{\alpha}$,$h_{b}$,$h_{c}$and$a$,$b$,$c$are the lengths of altitudes and the corresponding sides. Prove that $$\frac{a}{h_{c}}+\frac{b}{h_{b}}+\frac{c}{h_{c}} \geq 2\left(\tan \frac{A}{2}+\tan \frac{B}{2}+\tan \frac{C}{2}\right).$$ 9. Find all positive integers$n$such that $$3 n=2 S^{3}(n)+7 S^{2}(n)+16$$ where$S(n)$is the sum of all the digits of$n$10. Given$0<\lambda<1$and$2019$positive numbers$x_{1}, x_{2}, \ldots, x_{2019}$such that$\displaystyle \sum_{i=1}^{200} x=2019$. Prove that $$673 \leq \sum_{i=1}^{2019} f\left(\frac{2019-\lambda x_{i}}{2019-\lambda}\right) \leq \sum_{i=1}^{2019} f\left(x_{i}\right)$$ where$f(x)=\dfrac{1}{x^{2}+x+1}$. 11. Two sequences$\left(x_{n}\right)$,$\left(y_{n}\right)$are determined by$x_{1} \in\left[\dfrac{1}{2},\dfrac{\sqrt{3}}{2}\right]$,$y_{1} \in\left[\dfrac{1}{2}, \dfrac{\sqrt{3}}{2}\right]$and $$x_{n+1}=\sqrt{1-y_{2}^{2}},\,y_{m+}=\frac{1}{\sqrt{21}} \sqrt{9-5 x_{n}^{2}},\,\forall n \in \mathbb{N}.$$ Show that the sequences$\left(x_{n}\right)$,$\left(y_{n}\right)$converse and then find$\displaystyle \lim_{n\to\infty} x_{2}$and$\displaystyle \lim_{n\to\infty} y_{n}$. 12. Given a triangular pyramid$S . A B C$with$\widehat{A S B}=60^{\circ}$,$\widehat{B S C}=90^{\circ}$,$\widehat{C S A}=120^{\circ}$;$S A=2$,$S B=9$,$S C=4$. Let$X$be the set of all values$d$such that there exists a point$M$from which the distances to the planes$(S B C)$,$(S C A)$,$(S A B)$,$(A B C)$are respectively$\sqrt{3} d$,$\dfrac{d}{2}$,$d$,$d$. Find the cardinality of$X$and also find the largest possible value for$d$. ##$type=three$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: 2019 Issue 500
2019 Issue 500
Mathematics & Youth