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

## $show=home 1. Find all finite sets of primes such that for each set, the product of its elements is 10 times the sum of its element. 2. Let$ABC$be an isosceles triangle with the vertex angle$\widehat{BAC}=80^{0}$. Choose$D$and$E$on the sides$BC$and$CA$respectively such that$\widehat{BAD}=\widehat{ABE}=30^{0}$. Find the angle$\widehat{BED}$. 3. Solve the equation $\frac{1}{\sqrt{x}}+\frac{1}{\sqrt{2x-1}}=\sqrt{5}\left(\frac{1}{\sqrt{6x-1}}+\frac{1}{\sqrt{9x-4}}\right).$ 4. Let$ABCD$be a square and let$a$be the length each side. On the sides$AB$and$BC$, choose$M$and$N$respectively such that$\widehat{MDN}=45^{0}$. Find the positions of$M$and$N$so that the length$MN$is minimal. 5. Find all positive integers$x$and$y$such that $x^{4}+y^{2}+13y+1\leq(y-2)x^{2}+8xy.$ 6. Solve the following system of equations $\begin{cases} x+y+z & =3\\ x^{2}y+y^{2}z+z^{2}x & =4\\ x^{2}+y^{2}+z^{2} & =5 \end{cases}.$ 7. Given a quadrilateral pyramid$S.ABCD$with the following properties: the base$ABCD$is a rectangle and$SA$is perpendicular to the plane$(ABCD)$. Suppose that$G$is the centroid of the triangle$SBC$and let$d$be the distance from$G$to the plane$(SBD)$. Let$SB=a$,$BD=b$and$SD=c$. Prove that $a^{2}+b^{2}+c^{2}\geq162d^{2}.$ 8. Prove that the following equation $(x+1)^{\frac{1}{x+1}}=x^{\frac{1}{x}}$ has a unique solution. 9. Given positive integers$a_{1},a_{2},\ldots a_{15}$satisfying a)$a_{1}<a_{2}<\ldots<a_{15}$, b) for each$k$($k=1,\ldots,15$), if we denote$b_{k}$the largest divisor of$a_{k}$such that$b_{k}<a_{k}$, then$b_{1}>b_{2}>\ldots>b_{15}$. Prove that$a_{15}>2015$. 10. Given the following polynomial $f(x)=x^{3}+3x^{2}+6x+1975.$ In the interval$[1,3^{2015}]$, how many are there integers$a$such that$f(a)$is divisible by$3^{2015}$?. 11. Find all injections$f:\mathbb{R\to\mathbb{R}}$satisfying $$f(x^{5})+f(y^{5}) = (x+y)[f^{4}(x)-f^{3}(x)f(y)+f^{2}(x)f^{2}(y)-f(x)f^{3}(y)+f^{4}(y)]$$ for all$x,y\in\mathbb{R}$. 12. Given a triangle$ABC$and let$G$be its centroid. Choose a point$M$, which is different from$G$, inside the triangle. Suppose that$AM$,$BM$, and$CM$intersect$BC$,$CA$and$AB$at$A_{0},B_{0},C_{0}$respectively. Choose$A_{1},A_{2}$on$B_{0}C_{0}$such that$A_{0}A_{1}\parallel CA$and$A_{0}A_{2}\parallel AB$. We choose four points$B_{1},B_{2},C_{1},C_{2}$similarly. Let$G_{1},G_{2}$be the centroids of the triangles$A_{1}B_{1}C_{1}$,$A_{2}B_{2}C_{2}$respectively. Prove that a)$A_{1}B_{2}\parallel B_{1}C_{2}\parallel C_{1}A_{2}$, b)$MG$goes through the midpoint of$G_{1}G_{2}$. ##$type=three$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: 2015 Issue 456
2015 Issue 456
Mathematics & Youth