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

## $show=home 1. Let$a=n^{3}+2n$and$b=n^{4}+3n^{2}+1$. For each$n\in\mathbb{N}$, find the greatest common divisor ($\gcd$) of$a$and$b$. 2. Given an isoscesles triangle$ABC$with$\widehat{A}=100^{0}$,$BC=a$,$AC=AB=b$. Outside$ABC$, we construct the isosceles triangle$ABD$with$\widehat{ADB}=140^{0}$. Compute the perimeter of$ABD$in terms of$a$and$b$. 3. Find all pairs of intergers$x,y$such that $x^{3}y+xy^{3}+2x^{2}y^{2}-4x-4y+4=0.$ 4. Given an isosceles trapezoid$ABCD$with$AB//CD$and$DA=AB=BC$. Let$(K)$be the circle which goes through$A$,$B$and tangent to$AD$,$BC$. Let$P$be a point on$(K)$and inside$ABCD$. Assume that$PA$and$PB$respectively intersect$CD$at$E$and$F$. Assume that$BE$and$BF$respectively intersect$AD$and$BC$at$M$and$N$. Prove that$PM=PN$. 5. Solve the system of equations $\begin{cases}x^{2}+y^{2} & =4y+1\\x^{3}+(y-2)^{3} & =7\end{cases}.$ 6. Let$a,b,c$be he length of three sides of a triangle. Prove that $\frac{a^{3}(a+b)}{a^{2}+b^{2}}+\frac{b^{3}(b+c)}{b^{2}+c^{2}}+\frac{c^{3}(c+a)}{c^{2}+a^{2}}\geq a^{2}+b^{2}+c^{2}.$ 7. Given a function$f(x)$which is continuous on$[a,b]$and differentiable on$(a,b)$, where$0<a<b$. Prove that there exists$c\in(a,b)$such that $f'(c)=\frac{1}{a-c}+\frac{1}{b-c}+\frac{1}{a+b}.$ 8. Given a triangle$ABC$inscribed the circle$(O)$. Construct the altitude$AH$. Let$M$be the midpoint of$BC$. Assume that$AM$intersects$OH$at$G$. Prove that$G$belongs to the radial axis of the circumcircle of$BOC$and the Euler circle of$ABC$. 9. Given three non-negative real numbers$a,b,c$satisfying$ab+bc+ca=1$. Find the minimum value of the expression $P=\frac{(a^{3}+b^{3}+c^{3}+3abc)^{2}}{a^{2}b^{2}+b^{2}c^{2}+c^{2}a^{2}}.$ 10. Find positive integers$n\leq40$and positive numbers$a,b,c$satisfying $\begin{cases} \sqrt{a}+\sqrt{b}+\sqrt{c} & =1\\ \sqrt{a+n}+\sqrt{b+n}+\sqrt{c+n} & \in\mathbb{Z} \end{cases}.$ 11. Given three positive integers$a,b,c$. Each time, we tranform the triple$(a,b,c)$into the triple $\left(\left[\frac{a+b}{2}\right],\left[\frac{b+c}{2}\right],\left[\frac{c+a}{2}\right]\right).$ Prove that after a finite number of such transformations, we will get a triple wit equal components. (The notation$[x]$denotes the biggest integer which does not exceed$x$). 12. Given a triangle$ABC$. Let$(D)$be he circle which is tangent to the rays$AB$,$AC$and is internally tangent to the circumcircle of$ABC$at$X$. Let$J$,$K$respectively be the incenters of the triangles$XAB$,$XAC$. Let$P$be the midpoint of the arc$BAC$. Prove that$P(AXJK)$is a harmonic range. ##$type=three$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: 2017 Issue 485
2017 Issue 485
Mathematics & Youth