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

## $show=home 1. Find all$3$digit perfecy squares such that if we pick any of those numbers, say$n^{2}$, and then interchange the tens digit and the units digit of that one, we will get$\left(n+1\right)^{2}$. 2. Consider the following sequence$1$;$3+5$;$7+9+11$;$13+15+17+19$;$\ldots$. Prove that each term of the sequence is a cube of some positive integer. 3. Find integers$x$and$y$such that$x^{2}-2x=27y^{3}$. 4. Given a triangle$ABC$with$AB^{2}-AC^{2}=\frac{BC^{2}}{2}$. Suppose futhermore that the angle$BAC$is obtuse. Let$D$be the point on the side$AB$such that$BC=2CD$. The line which goes through$D$and is perpendicular to$AB$intersects the line through$AC$at$E$. Let$K$be the intersection between the line through$CD$the line through$BE$. Prove that$K$is the midpoint of$BE$. 5. Given positive numbers$a$and$b$. Prove that $$\frac{a+b}{1+ab}+\left(\frac{1}{1+a}+\frac{1}{1+b}\right)ab+\frac{a+b+2ab}{\left(1+a\right)\left(1+b\right)ab}\geq3.$$ When does the equality happen?. 6. Solve the equation $$\sqrt{2x^{3}+6}=x+\sqrt{x^{2}-3x+3}.$$ 7. Find the integral solutions of the inequality $$x^{6}-2x^{3}-6x^{2}-6x-17<0.$$ 8. Given a scalene triangle$ABC$with the orthocenter$H$. The incircle$\left(I\right)$of the triangle if tangent to the sides$BC$,$CA$,$AB$respectively at$A_{1}$,$B_{1}$,$C_{1}$. The line$d_{1}$which goes through$I$and is paralled to$BC$intersects the line through$\left(B_{1}C_{1}\right)$at$A_{2}$. Similarly, we construct the points$B_{2}$,$C_{2}$. Prove that$A_{2}$,$B_{2}$,$C_{2}$are colinear and the line through these points is perpendicular to$IH. 9. Given the equation $$x^{n}+a_{1}x^{n-1}+\ldots+a_{n-1}x+a_{n}=0$$ where the coefficients satisfy \begin{align*} \sum_{i=1}^{n}a_{i} & =0,\\ \,\sum_{i=1}^{n-1}\left(n-1\right)a_{i} & =2-n,\\ \sum_{i=1}^{n-2}\left(n-i\right)\left(n-i-1\right)a_{i} & =3-n\left(n-1\right). \end{align*} Suppose thatx_{1},x_{2},\ldots,x_{n}$are$nsolutions of the above equation. Compute the value of the following expressions \begin{align*} E & =\sum_{1\leq i<j\leq n}\frac{1}{\left(1-x_{i}\right)\left(1-x_{i}\right)},\\ F & =\sum_{i=1}^{n}\frac{1}{\left(1-x_{i}\right)^{2}}. \end{align*} 10. There are20$stones and we divide them into$3$piles. Then we start to moving stones from one pile to another with the follow rule. Each time we can move half of a pile with even number of stones to any pile. Prove that nomatter how we divide the stones into$3$piles, after finite of moves, we will get a pile with$10$stones. 11. For each positive integer$t$, let$\phi\left(t\right)$denote the numbers of positive integers which are not greater than$t$and are relatively prime to$t$(Euler phi function). Now assume that$n,k$are positive integers and$p$is an odd prime number. Prove that there exists a positive integer$a$such that$p^{k}$is a divisor of all numbers$\phi\left(a\right),\phi\left(a+1\right),\ldots$and$\phi\left(a+n\right)$. 12. Given an acute triangle$ABC$with the altitudes$AD$,$BE$and$CF$. The circles with the diameters$AB$and$AC$respectively intersect the rays$DF$and$DE$at$Q$and$P$. Let$N$be the cicumcenter of the triangle$DEF$. Prove that a)$AN\perp PQ$. b)$AN$,$BP$and$CQ$are concurrent. ##$type=three$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: 2016 Issue 472
2016 Issue 472
Mathematics & Youth