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

## $show=home 1. Consider all pairs of integers$x$,$y$which are greater than$1$and satisfy$x^{2017}=y^{2018}$. Find the pair with the smallest possible value for$y$. 2. Given an isosceles triangle$ABC$with the apex$A$. Suppose that$\hat{A}=180^{0}$,$BC=a$,$AC=b$. Outside$ABC$, construct the isosceles triangle$ABD$with the apex$A$and$\widehat{BAD}=36^{0}$. Find the perimeter of the triangle$ABD$in terms of$a$and$b$. 3. Find positive integers$n$such that if the positive integer$a$is a divisor of$n$then$a+2$is also a divisor of$n+2$. 4. Given a triangle$ABC$inscribed the circle$(O)$with diameter$AC$. Draw a line which is perpendicular to$AC$at$A$and intersects$BC$at$K$. Choose a point$T$on the minor$AB$($T$is different from$A$and$B$). The line$KT$intersects$(O)$at the second point$P$. On the tangent line to the circle$(O)$at the point$T$choose two points$I$and$J$such that$KIA$and$KAJ$are isosceles triangles with the apex$K$. Show that a)$\widehat{TIP}=\widehat{TKJ}$. b) The circle$(O)$and the circumcircle of the triangle$KPJ$are tangent to each other. 5. Find integral solutions of the equation $y^{2}+2y=4x^{2}y+8x+7.$ 6. Solve the system of equations $\begin{cases}\log_{2}x+\log_{2}y+\log_{2}z & =3\\ \sqrt{x^{2}+4}+\sqrt{y^{2}+4}+\sqrt{z^{2}+4} & =\sqrt{2}(x+y+z)\end{cases}$ 7. Show that $\lim_{n\to\infty}\sqrt{1+2\sqrt{1+3\sqrt{1+\ldots\sqrt{1+(n-1)\sqrt{1+n}}}}}=3.$ 8. Let$m_{a}$,$m_{b}$,$m_{c}$be the lengths of the medians of the triangle with the perimeter$2$. Show that $\max\{1;3\sqrt{r}\}\leq m_{a}+m_{b}+m_{c}<\frac{3}{\sqrt{2}}$ where$r$is the inradius of the triangle. 9. Assume that$a,b,c$are three non-negative numbers such that$a+b+c=3$. Find the maximum value of the expression $P=a\sqrt{b^{3}+1}+b\sqrt{c^{3}+1}+c\sqrt{a^{3}+1}.$ 10. For every$n\in\mathbb N$, let$F_{n}=2^{2^{n}}+1$. For each$n\in\mathbb N$, let$q$be a prime divisor of$F_{n}$. Show that$2^{n+1}\mid q-1$. Futhermore, if$n\geq 2$, show even more that$2^{n+2}\mid q-1$. 11. Find all funtions$f:\mathbb{R\to\mathbb{R}}$satisfying $f(x)f(y)+f(xy)+f(x)+f(y)=f(x+y)+2xy$ for all$x,y\in\mathbb{R}.$12. Given a convex hexagon$ABCDEF$circumscribing a circle$(O)$. Assume that$O$is the circumcenter of the triangle$ACE$. Prove that the circumcenter of the triangles$OAD$,$OBE$and$OCF$has another common point besides$O$. ##$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: 2018 Issue 491
2018 Issue 491
Mathematics & Youth