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

## $show=home ### Junior 1. Solve the system of equations $\begin{cases} x^{3}+y^{3} & =x^{2}+xy+y^{2}\\ \sqrt{6x^{2}y^{2}-x^{4}-y^{4}} & =\frac{13}{4}(x+y)-2xy-\frac{3}{4}\end{cases}.$ 2. In a triangle$ABC$, let$(O)$,$(I)$and$I_{a}$denote the circumcircle, incircle and the center of its$A$-excircle. The incircle$(I)$touches segment$BC$at$D$,$P$is the midpoint of arc$BAC$of the circle$(O)$, and$PI_{a}$intersects$(O)$again at$K$. Prove that$\widehat{DAI}=\widehat{KAI}$. 3. Find all triples of positive integers$(a,b,c)$such that$a,b,c$are the side lengths of a triangle and $\sqrt{\frac{19}{a+b-c}}+\sqrt{\frac{5}{b+c-a}}+\sqrt{\frac{79}{c+a-b}}$ is an odd integer, differ from 1. 4. In a triangle$ABC$, segment$CD$is the altitude from$C$. Points$E,F$chosen on the side$AB$such that$\widehat{ACE}=\widehat{BCF}=90^{0}$.$X$is a point on segment$CD$; point$K$on segment$FX$such that$BK=BC$and point$L$on segment$EX$such that$AL=AC$;$AL$meets$BK$at$M$. Prove that$ML=MK$. 5. Find all triple of positive integers$(x,y,p)$where$p$is a prime number and $8x^{3}+y^{3}-6xy=p-1.$ 6. In a triangle$ABC$, let$I$denote a point on the internal angle-bisector of the angle$BAC$. The line through$B$and parallel to$CI$intersects$AC$at$D$; the line through$C$and parallel to$BI$meets$AB$at$E$. Let$M,N$denote the midpoints of$BD,CE$respectively. Prove that$AI$is perpendicular to$MN$. ### Senior 1. The sequence of positive real numbers$\{a_{n}\}_{n=0}^{\infty}$satisfies $a_{n+1}=\frac{2}{a_{n}+a_{n-1}},\quad n=1,2,3,\ldots.$ Prove that there exists a pair of real numbers$s,t$such that$s\leq a_{n}\leq t$for all$n=0,1,2,\ldots$. 2. In a triangle$ABC$($AC>AB$), the altitudes$BB'$and$CC'$intersects at point$H$. Let$M,N$be the midpoints of$BC'$,$CB'$respectively.$MH$meets the circumcircle of triangle$CHB'$at$I$;$NH$meets the circumcircle of triangle$BHC'$at point$J$. If$P$is the midpoint of segment$BC$, prove that$AP\perp IJ$. 3. Find all triples of positive integers$(a,b,p)$such that$p$is a prime number,$a$and$b$have no common divisor, and the set of prime divisors of$a+b$is the same as that of$a^{p}+b^{p}$. 4. Let$ABC$be a non-isosceles triangle. The incircle$(I)$touches$BC$,$CA$and$AB$at$A_{0}$,$B_{0}$and$C_{0}$respectively;$AI$,$BI$,$CI$intersect$BC$,$CA$,$AB$at$A_{1}$,$B_{1}$,$C_{1}$;$B_{0}C_{0}$,$C_{0}A_{0}$,$A_{0}B_{0}$meet$B_{1}C_{1}$,$C_{1}A_{1}$,$A_{1}B_{1}$at$A_{2}$,$B_{2}$,$C_{2}$respectively. Prove that the lines$A_{0}A_{2}$,$B_{0}B_{2}$,$C_{0}C_{2}$intersect at a point on the circle$(I)$. 5. Find all nonempty subsets$A,B$of the set of positive integers$\mathbb{Z}^{+}$such that the following conditions are satisfied •$A\cap B=\emptyset,$• If$a\in A$,$b\in B$then$a+b\in A$and$2a+b\in B$. 6. Let$ABC$be a triangle, non-isosceles at vertex$A$. Let$O,H$denote its circumcenter and orthocenter respectively. The line through$A$and perpendicular to$OH$intersects$BC$at$K$. Prove that$H$and the centers of the Euler circles of triangles$ABC$,$ABK$and$ACK$lie on a 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: 2014 Anniversary 50th
2014 Anniversary 50th
Mathematics & Youth