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

## $show=home 1. Two stations$A$and$B$are$999km$away. The milestones along the railway from$A$to$B$show the distances from that point to$A$and$B$as follows $$0 / 999 ; 1 / 998 ; 2 / 997 ; \ldots ; 999 / 0.$$ Among these milestones, how many of them contains only two different digits? 2. Given a triangle$A B C$with the side$B C$is fixed and the vertex$A$can vary. Draw the perpendicular bisector$A D .$Through$C$draw a perpendicular line to$A D$at$N$. Let$M$be the midpoint of$A C$. Show that when$A$is moving,$M N$always passes through a fixed point. 3. Suppose that$a$and$b$are positive integers so that$(a, 6)=1$and$3 \mid a+b$. Assume that$p, q$are prime numbers so that both$p q+a$and$b p+q$are also prime numbers. Prove that$a+6$is a prime number. 4. Given a circle$(O, R)$. From a point$A$outside the circle we draw two tangents$A B$,$A C$($B$,$C$are touch points) and a secant$A D E(D$is in between$A$and$E)$. The line$B C$intersects$OA$at$H$. From$H$draw the line parallel to$B E$. That line intersects$A B$at$K$. Show that$B D$passes through the midpoint of$H K$. 5. Solve the system of equations $$\begin{cases}x^{3}+x(y+z)^{2} &=26 \\ y^{3}+y(z+x)^{2} &=40 \\ z^{3}+z(x+y)^{2} &=54\end{cases}.$$ 6. Solve the equation $$\sqrt{x^{3}+x+2}=x^{4}-x^{3}-7 x^{2}-x+10.$$ 7. Suppose that$x$,$y$are positive numbers so that$x+y \leq 6$. Find the minimum value of the expression $$P=x^{2}(6-x)+y^{2}(6-y)+(x+y)\left(\frac{1}{x y}-x y\right).$$ 8. Given a triangle$A B C$. Let$O$and$I$be the circumcenter and the incenter of the triangle respectively. Let$D$be the second intersection between$(O)$and$AI$. Let$P$be the intersection between$B C$and the line which passes though$I$and perpendicular to$AI$. Let$Q$be the reflection point of$I$in$O$. Show that $$\widehat{P A Q}=\widehat{P D Q}=90^{\circ}.$$ 9. Find the limit $$\lim_{n \rightarrow+\infty} \frac{\mid \sqrt{1}]+[\sqrt{2}]+\ldots+\left[\sqrt{n^{3}+3 n^{2}+3 n}\right]}{n^{4}}$$ where$n$is a positive integer and the notion$[x]$denote the integer which does not exceed$x$. 10. Given a positive integer$n$so that both$6 n+1$and$20 n+1$are perfect squares. Show that$58 n+11$is a composite number. 11. Suppose that$g:[a, b] \rightarrow R$is$a$continuous functions with$g(a) \leq g(b)$and$f:[a, b] \rightarrow[g(a), g(b)]$is an increasing function. Show that the equation$f(x)=g(x)$has at least one solution. 12. Given a triangle$A B C$. Let$O$and$H$be the circumcenter and the orthocenter of the triangle respectively. Let$S$be the circumcenter of the triangle$O B C$. Denote$K$and$L$the reflection points of$S$in$A B$and$A C$respectively. Show that$K L$passes through the the midpoint of$O H$. ##$type=three$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 515
2020 Issue 515
Mathematics & Youth