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

## $show=home 1. Find the last two digits of the difference$2^{9^{2016}}-2^{9^{1945}}$. 2. Find natural numbers$n$such that $$n^{6}+8n^{3}+19n^{2}-33n-90=0.$$ 3. Find positive integers$x$and$y$so that $$A=x^{2}+y^{2}+\frac{x^{2}y^{2}}{\left(x+y\right)^{2}}$$ is a perfect square. 4. Given an acute triangle$ABC$with two altitudes$BE$and$CK$. Let$R$and$S$respectively be the perpendicular projections of$K$on$BC$and$BE$, and$P$and$Q$respectively the perpendicular projections of$E$on$BC$and$CK$. Prove that the lines$RS,PQ$and$EK$are concurrent. 5. Solve the system of equations $$\begin{cases} 5x^{2}+2y^{2}+z^{2} & =2\\ xy+yz+zx & =1 \end{cases}$$ 6. Find integers$m$so that the equation $$x^{3}+\left(m+1\right)x^{2}-\left(2m-1\right)x-\left(2m^{2}+m+4\right)=0$$ has an integer solution. 7. Let$x,y,x$be real numbers satisfying $$\begin{cases} x+z-yz & =1\\ y-3z+xz & =1 \end{cases}.$$ Find the maximum and minimum values of the expression$T=x^{2}+y^{2}$. 8. Given a triangle$ABC$. Let$\left(O\right)$be its circumcircle. The excircle$\left(I\right)$relative to the vertex$A$is tangent to$AB$and$AC$respectively at$D$and$E$. Let$A'$be the intersection between$OI$and$DE$. Similarly, we construct the points$B'$and$C'$. Prove that$AA',BB'$and$CC'$are concurrent. 9. Let$a,b,c$be positive numbers such that$a+b+c=1$. Prove that $$a^{b}b^{c}c^{a}\leq ab+bc+ca.$$ 10. Show that there exist infinitely many positive integers$n$so that$n^{2}+1$has a prime divisor$p$which is greater than$2n+\sqrt{10n}$. 11. Given two arbitrary positive integers$n$and$p$. Find the number of functions $$f:\left\{ 1,2,3,\ldots,n\right\} \to\left\{ -p,-p+1,-p+2,\ldots,p\right\}$$ which satisfy the property$\left|f\left(i\right)-f\left(j\right)\right|\leq p$for any$i,j\in\left\{ 1,2,3,\ldots,n\right\} $. 12. Given a non-right triangle$ABC$. Let$O$and$H$respectively be the circumcenter and the orthocenter of$ABC$. Choose an arbitrary point$M$on$\left(O\right)$so that$M$is different from$A,B$and$C$. Let$N$denote the symmetric point of$M$through$BC$. Let$P$be the second intersection between$AM$and the circumcircle of$OMN$. Prove that$HN$goes through the orthocenter of$AOP$. ##$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 473
2016 Issue 473
Mathematics & Youth