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

## $show=home 1. Find all prome numbers$p,q,r$satisfying $(p+1)(q+2)(r+3)=4pqr.$ 2. Given a triangle$ABC$with$\widehat{A}=75^{0}$,$\widehat{B}=45^{0}$. On the side$AB$, choose a point$D$such that$\widehat{ACD}=45^{0}$. Prove that$DA=2DB$. 3. Solve the following system of equations $\begin{cases} \sqrt{x+y+2}+x+y & =2(x^{2}+y^{2})\\ \frac{1}{x}+\frac{1}{y} & =\frac{1}{x^{2}}+\frac{1}{y^{2}} \end{cases}.$ 4. Given a triangle$ABC$. Let$(I)$be the inscribed circle and$(J)$the escribed circle corresponding to the angle$A$. Suppose that$(J)$is tangent to the lines$BC$,$CA$and$AB$at$D,E$and$F$respectively. The line$JD$meets the line$EF$at$N$. The line which contains$I$and is perpendicular to the line$BC$intersects the line$AN$at$P$. Let$M$be the midpoint of$BC$. Prove that$MN=MP$. 5. Find all the integer solutions of the following equation $x^{3}=4y^{3}+x^{2}y+y+13.$ 6. Let $$f(x)=\frac{4^{x+2}}{4^{x}+2}.$$ Find $f(0)+f\left(\frac{1}{2014}\right)+f\left(\frac{2}{2014}\right)+\ldots+f\left(\frac{2013}{2014}\right)+f(1).$ 7. Given a tetrahedron$ABCD$. Let$d_{1},d_{2},d_{3}$be the distances between the pairs of opposite sides$AB$and$CD$,$AC$and$BD$,$AD$and$BC$. Prove that $V_{ABCD}\geq\frac{1}{3}d_{1}d_{2}d_{3}.$ 8. Given an integer$n$which is greater than$1$. Let$a_{1},a_{2},\ldots,a_{n}$be arbitrary positive real numbers satisfying $\frac{1}{a_{1}}+\frac{1}{a_{2}}+\ldots+\frac{1}{a_{n}}=1.$ Prove that $a_{1}^{a_{2}}+a_{2}^{a_{3}}+\ldots+a_{n-1}^{a_{n}}+a_{1}+a_{2}+\ldots+a_{n}>n^{3}+n.$ 9. Let$T$be a set of$n$elements. What is the maximal number of subsets of$T$which can be picked so that each subset has exactly 3 elements and any two subsets has nonempty intersection?. 10. Let$p$be a prime number. Find all the polynomials$f(x)$with integer coefficients such that for every positive integer$n$,$f(n)$is a divisor of$p^{n}-1$. 11. Let$x,y$be the positive real numbers satisfying$[x]\cdot[y]=30^{4}$, where$[a]$is the greatest integer not wxceeding$a$. Find the minimum and maximum values of $P=[x[x]]+[y[y]].$ 12. Given a triangle$ABC$. Let$E,F$be points on$CA$,$AB$respectively such that$EF\parallel BC$. The perpendicular bisector of$BC$intersects$AC$at$M$and the perpendicular bisector of$EF$intersects$AB$at$N$. The circle circumscribing the triangle$BCM$meets$CF$at$P$which is different from$C$. The circle circumscribing the triangle$EFN$meets$CF$at$Q$which is different from$F$. Prove that the perpecdicular bisector of$PQ$contains the midpoint of$MN$. ##$type=three$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 446
2014 Issue 446
Mathematics & Youth