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

## $show=home 1. Find all pairs of prime numbers$(p,q)$satisfying $p^{q}-q^{p}=79.$ 2. Find integers$a,b,c,d$satisfying $\sqrt{a^{2}+b^{2}+c^{2}}=\sqrt{a+b+c}=d.$ 3. Let$a,b,c$be positive numbers such that$a+b+c=1$. Prove that $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\geq\frac{21}{1+36abc}.$ 4. Given a triangle$ABC$circumscribing a circle$(O)$. The sides$AB$,$BC$and$CA$are tangent to$(O)$at$D$,$E$and$F$respectively and assume furthermore that$EC=2EB$. Suppose that$EI$is a diameter of$(O)$. Through$D$draw a line which is parallel to$BC$. This line intersects the line segment$EF$at$K$. Prove that$A$,$I$,$K$are collinear. 5. Find the maximal number$M$such that the inequality$x^{2}\geq M[x]\{x\}$holds for every$x$(where$[x]$,$\{x\}$respectively are the integral part and the fractional part of$x$). 6. Solve the equation $x^{3}+x+6=2(x+1)\sqrt{3+2x-x^{2}}.$ 7. Solve the system of equations $\begin{cases}\dfrac{1}{\sqrt{x}}+\dfrac{1}{\sqrt{y}}+\dfrac{1}{\sqrt{z}} & =3\\ \dfrac{x^{2}}{y}+\dfrac{y^{2}}{z}+\dfrac{z^{2}}{x} & =\sqrt{\dfrac{x^{4}+y^{4}}{2}}+\sqrt{\dfrac{y^{4}+z^{4}}{2}}+\sqrt{\dfrac{z^{4}+x^{4}}{2}}\end{cases}.$ 8. Given a triangle$ABC$with the exradii$r_{a}$,$r_{b}$,$r_{c}$, the medians$m_{a}$,$m_{b}$,$m_{c}$and the area$S$. Prove that $r_{a}^{2}+r_{b}^{2}+r_{c}^{2}\geq3\sqrt{3}S+(m_{a}-m_{b})^{2}+(m_{b}-m_{c})^{2}+(m_{c}-m_{a})^{2}.$ 9. Given a positive integer$n$and positive numbers$a_{1},a_{2},\ldots a_{n}$. Find a real number$\lambda$such that $a_{1}^{x}+a_{2}^{x}+\ldots+a_{2}^{x}\geq n+\lambda x,\quad\forall x\in\mathbb{R}.$ 10. a) Prove that for every positive integer$n$the equation$2012^{x}(x^{2}-n^{2})=1$has a unique solution (denoted by$x_{n}$). b) Find${\displaystyle \lim_{n\to\infty}(x_{n+1}-x_{n})}$. 11. Let$T$be the set of all positive factors of$n=2004^{2010}$. Suppose that$S$be an arbitrary nonempty subset of$T$satisfying the fact that for all$a$,$b$belong to$S$and$a>b$then$a$is not divisible by$b$. Find the maximal number of elements of such subset$S$. 12. Given a triangle$ABC$whose the incircle$(I)$is tangent to$BC$at$D$. Let$H$be the perpendicular projection of$A$on$BC$. Let$N$be the midpoint pf$AH$. The line through$D$and$N$intersects$CA$,$AB$respectively at$J$and$S$. Assume that$BJ$intersects$CS$at$P$. Suppose that$DA$,$DP$intersect$(I)$respectively at$G$,$L$. Prove that$B$,$C$,$G$,$L$lie on some circle. ##$type=three$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 487
2018 Issue 487
Mathematics & Youth