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

## $show=home 1. Find all the integer solutions of the following equation $1+x+x^{2}+x^{3}=y^{2}.$ 2. Let$x,y,z$be three coprime positive integers satisfying $(x-z)(y-z)=z^{2}.$ Prove that$xyz$is a perfect square. 3. Solve the following equation $\frac{1}{\sqrt{3x}}+\frac{1}{\sqrt{9x-3}}=\frac{1}{\sqrt{5x-1}}+\frac{1}{\sqrt{7x-2}}.$ 4. Given a circle$(O,R)$and a chord$AB$with the distance from$O$is$d$($0<d<R$). Two circles$(I)$,$(K)$are externally tangent at$C$, are both tangent to$AB$and are internally tangent with$(O)$($I$and$K$are in the same half-plane determined by the line through$AB$). Find the locus of the points$C$which vary when$(I)$and$(K)$vary. 5. Find all positive integers$a$and$b$so that both equations$x^{2}-2ax-3b=0$and$x^{2}-2bx-3a=0$have positive integer solution. 6. Find all positive real numbers$x,y,z$satisfying system of equations $\begin{cases} \dfrac{1}{x+1}+\dfrac{1}{y+1}+\dfrac{1}{z+1} & =1\\ xyz(x+y+z)(x+1)(y+1)(z+1 & =1296 \end{cases}.$ 7. Given a tetrahedron$ABCD$and the lengths of its sides$AB=BD=DC=x$,$BC=CA=AD=y$. Prove that $\frac{3}{5}<\frac{x}{y}<\frac{5}{3}.$ 8. Find the maximum value of the expression $P=|(a^{2}-b^{2})(b^{2}-c^{2})(c^{2}-a^{2})|,$ in which$a,b,c$are nonnegative numbers satisfying$a+b+c=\sqrt{5}$. 9. Solve equation $x^{4}+ax^{3}+bx^{2}+2ax+4$ given$9(a^{2}+b^{2})=16$. 10. Find all pairs of positive integers$(a,b)$satisfying the following properties:$4a+1$and$4b-1$are coprime and$a+b$is a divisor of$16ab+1$. 11. Given two sequences $a_{1}=0,\,a_{2}=16,\,a_{3}=18,\,a_{n+2}=8a_{n}+6a_{n-1}$ and $b_{1}=3,\,b_{2}=19,\,b_{3}=69,\,b_{n+2}=3b_{n+1}+5b_{n}-b_{n-1}$ for$n\geq2. Prove that \begin{align*} b_{n} & =C_{n}^{0}a_{n}+C_{n}^{1}a_{n-1}+\ldots+C_{n}^{n-1}a_{1}+3C_{n}^{n},\\ a_{n} & =C_{n}^{0}b_{n}-C_{n}^{1}b_{n-1}+C_{n}^{2}b_{n-2}-\ldots+(-1)^{n-1}C_{n}^{n-1}b_{1}+(-1)^{n}3C_{n}^{n}. \end{align*} 12. Given a triangleABC$and its circumscribed circle$(O)$. The points$A_{1},B_{1}$and$C_{1}$are on the sides$BC,CA$and$AB$respectively. The circumscribed circles$(AB_{1}C_{1})$,$(BC_{1}A_{1})$, and$(CA_{1}B_{1})$intersect$(O)$at$A_{2},B_{2}$and$C_{2}$respectively. Find the positions of$A_{1},B_{1}$and$C_{1}$so that$\dfrac{S_{A_{1}B_{1}C_{1}}}{S_{A_{2}B_{2}C_{2}}}$is minimal. ##$type=three$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 Issue 447
2014 Issue 447
Mathematics & Youth