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

## $show=home 1. Given positive integers$m, n$which are coprime and satisfy$m+n \neq 90$. Find the maximal value of the product$mn$. 2. Find all positive integers$n$so that$n^{2021}+n+1$is a prime. 3. Solve the equation $$x^{4}-6 x-1=2(x+4) \sqrt{2 x^{3}+8 x^{2}+6 x+1}.$$ 4. Given acute triangle$A B C$with$A B+A C=2 B C$which is inscribed in the circle$(O)$and is circumscribed the circle$(I)$. Let$M$be the midpoint of the major arc$B C$. The line$M I$intersects the circumcircle of$B I C$at$N$. The line$OI$intersects$B C$at$P$. Show that the line$P N$is tangent to the circumcircle of$B I C$. 5. Find all pairs of integers$(a ; b)$so that the following system of equations $$\begin{cases}x^{2}+2 a x-3 a-1 &=0 \\ y^{2}-2 b y+x &=0\end{cases}$$ has exactly$3$different real roots$(x ; y)$. 6. Given positive numbers$a, b, c$. Show that $$\dfrac{(a-b)^{2}}{(b+c)(c+a)}+\dfrac{(b-c)^{2}}{(c+a)(a+b)}+\dfrac{(c-a)^{2}}{(a+b)(b+c)} \geq \frac{3\left(a^{2}+b^{2}+c^{2}\right)}{(a+b+c)^{2}}-1.$$ 7. Given a triangle$A B C$with the lengths of the altitudes$h_{a}$,$h_{b}$,$h_{c}$and its half perimeter$p$satisfying$h_{a}^{2}+h_{b}^{2}+h_{c}^{2}=p^{2}$. Show that the triangle$A B C$is equilateral. 8. Given a tetrahedron$S.ABC$whose base is a right triangle with the right angle$A$and$A C > A B$. Let$A M$be the median and$I$the incenter of the base. Suppose that$I M$is perpendicular to$B I$. Compute the value of the expression $$T = \frac{V_{S . AI B}}{V_{S . A I C}}+\frac{V_{S . A I C}}{V_{S . B I C}}+\frac{V_{S . B I C}}{V_{S . A B}}.$$ 9. Suppose that$a, b, c$are the lengths of three sides of a triangle and$n \in \mathbb{N}^{*}$. Prove that $$\frac{a^{n}}{(b+c)^{n}-a^{n}}+\frac{b^{n}}{(c+a)^{n}-b^{n}}+\frac{c^{n}}{(a+b)^{n}-c^{n}} \geq \frac{3}{2^{n}-1}.$$ 10. For each positive integer$n$, let$S_{n}=\sum_{k=1}^{n}\left[\sqrt{k^{2}+4 \sqrt{k^{2}+2 k+2}}\right]$, where$[x]$is the greatest integer which does not exceed$x .$Find all the positive integers$n$so that$S_{n}$is a power of a prime. 11. Find all functions$f: \mathbb{Z} \rightarrow \mathbb{Z}$satisfying $$f(x+f(y))=f(x),\, \forall x, y \in \mathbb{Z}.$$ 12. Given an acute triangle$A B C$and$(O ; R)$is its circumcircle. The altitude$A D=\sqrt{2} R$. Let$M, N$respectively be the intersections between$A B$,$A C$and the circle with the diameter$A D$. Let$P$be the second intersection between$(O ; R)$the circle with the diameter$A D$and$Q$the reflection point of$P$in$M N$. Show that a)$2 S_{A M N}=S_{A B C}$. b)$\widehat{B Q N}=\widehat{C Q M}=\dfrac{\pi}{2}$. ##$type=three$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

Name

2006,1,2007,12,2008,12,2009,12,2010,12,2011,12,2012,12,2013,12,2014,12,2015,12,2016,12,2017,12,2018,11,2019,12,2020,12,2021,6,Anniversary,4,
ltr
item
Mathematics & Youth: 2021 Issue 526
2021 Issue 526
Mathematics & Youth