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

## $show=home 1. Find the natural number$n$given that$n^{5}+n+1$has only one prime factor. 2. Given a right isosceles triangle$A B C$with the vertex angle$A$. Let$M$,$N$,$I$respectively be the midpoints of$A B$,$A C$and$N C$. Assume that$K$is the perpendicular projection of$N$on$B C$. Show that$A K$,$B I$and$C M$are concurrent. 3. Given positive numbers$a$,$b$,$c$,$d$such that$a+b=c+d=2019$and$a b \geq c d$. Find the minimum value of the expression $$P=\frac{a+3 \sqrt{b}+2}{\sqrt{c}+\sqrt{d}}.$$ 4. Given a triangle$A B C$with the angle$\hat{A} > 90^{\circ}$. Choose a point$I$inside the line segment$B C$so that$B A$intersects the circumcircle$\Delta A C I$at$D(D \neq A)$and$C A$intersects the circumcircle$\Delta A B I$at$E(E \neq A)$,$B E$intersects$CD$at$N$. Let$M$be the midpoint of$B C$,$M A$intersects the circumcircle$\triangle B N C$at$F$. Show that$A$,$D$,$E$,$F$,$N$both belong to a circle. 5. Solve the system of equations $$\begin{cases} x^{4}+3 x &=y^{4}+y \\ x^{2}-y^{2} &=2\end{cases}.$$ 6. Find the sum of the squares of all real roots of the equation $$x^{5}+2018 x^{2}+2019=x^{4}+2019 x^{3}+2020 x.$$ 7. Given$3$positive numbers$x, y, z$such that$x y z=1$. Prove that $$\frac{1}{x^{k+1}(y+z)}+\frac{1}{y^{k+1}(z+x)}+\frac{1}{z^{k+1}(x+y)} \geq \frac{3}{2} \quad \left(k \in \mathbb{N}^{*}\right)$$ 8. Given two equilateral triangles$A B C$and$A B^{\prime} C^{\prime}$with the same orientation. Let$K$be the second intersection between the circumcircles of$A B C$and$A B^{\prime} C^{\prime}$. Let$M$be the intersection between$B C^{\prime}$and$C B^{\prime}$. Show that$M A=M K$. 9. Determine the coefficient of$x$in the polynomial expansion of $$(1+x)(1+2 x)^{2} \ldots(1+n x)^{n}.$$ 10. Given the real sequence$\left\{x_{n}\right\}$determined as follows $$x_{1}=1,\quad x_{n+1}=x_{n}+\frac{1}{2 x_{n}},\, \forall n \geq 1.$$ Show that$\left[9 x_{81}\right]=81$(where$[x]$denotes the integral part of$x$). 11. Given an integer$k \geq 2$and aninteger$n \geq \dfrac{k(k+1)}{2}$. Find the maximal positive integer$m$so that among$n$arbitrary distinct positive integers which do not exceed$m$there always exist$k+1$numbers of which some number is equal to the sum of the remaining ones. 12. Given a non-right triangle$A B C$. The altitudes$B B^{\prime}$and$C C^{\prime}$intersect at$H$. Let$M$be the midpoint of$A H$. Let$K$be an arbitrary point on$B^{\prime} C^{\prime}$($K$is different from$B^{\prime}$,$C^{\prime}$). The line$A K$intersects$M B^{\prime}$,$M C^{\prime}$respectively at$E$,$F$. Let$N$be the intersection between$B E$and$C F$. Show that$K$is the orthocenter of the triangle$N B C$. ##$type=three$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: 2019 Issue 504
2019 Issue 504
Mathematics & Youth