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

## $show=home 1. Find a natural number with more than$3$digits knowing that if we delete its the last$3$digits, we will get a new number whose cube is exactly equal to that wanted number. 2. Given two positive real numbers$a$and$b$satisfying the following conditions$a^{2015}-a-1=0$and$b^{4030}-b-3a=0$. Compare$a$and$b$. 3. Solve the equation $x+y+x\sqrt{1-y^{2}}+y\sqrt{1-x^{2}}=\frac{3\sqrt{3}}{2}.$ 4. On a circle centered at the point$I$, fix two points$B$and$C$. A point$A$varies on the circle such that the triangle$ABC$is always acute. On the side$AC$, choose$M$so that$MA=3MC$. Let$H$be the perpendicular projection of$M$on the side$AB$. Prove that$H$always lies on a fixed circle. 5. Find all prime numbers$x$and$y$such that $(x^{2}+2)^{2}=2y^{4}+11y^{2}+x^{2}y^{2}+9.$ 6. Solve the following system of equations $\begin{cases} x^{3}+y^{3} & =4y^{2}-5y+4x+4\\ 2y^{3}+z^{3} & =4z^{2}-5z+6y+6\\ 3z^{3}+x^{3} & =4y^{2}-5x+9z+8 \end{cases}.$ 7. Given a triangle$ABC$. Suppose that the length of the sides are given by$BC=a$,$CA=b$,$AB=c$and$m_{a}$,$m_{b}$and$m_{c}$are the length of the corresponding medians. Prove that $(a^{2}+b^{2}+c^{2})\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)\geq2\sqrt{3}(m_{a}+m_{b}+m_{c}).$ When does the equality happen?. 8. Consider an acute triangle$ABC$with the angles$A,B$and$C$. Find the maximum value of the expression $M=\frac{\tan^{2}A+\tan^{2}B}{\tan^{4}A+\tan^{4}B}+\frac{\tan^{2}B+\tan^{2}C}{\tan^{4}B+\tan^{4}C}+\frac{\tan^{2}C+\tan^{2}A}{\tan^{4}C+\tan^{4}A}.$ 9. Find that coefficient of$x^{2}$in the expansion of the following expression $(1+x)(1+2x)(1+4x)\ldots(1+2^{2013}x).$ 10. Let$a_{1},a_{2},\ldots,a_{n}$be positive numbers such that $a_{1}+a_{2}+\ldots+a_{n}=\frac{1}{a_{1}}+\frac{1}{a_{2}}+\ldots+\frac{1}{a_{n}}.$Find the minimum value of the expression $A=a_{1}+\frac{a_{2}^{2}}{2}+\ldots+\frac{a_{n}^{2}}{n}.$ 11. Find the biggest real number$k$satisfying the condition: for any 3 real numbers$a,b,c$such that$|a|+|b|+|c|<k$, the following system of inequalities has no solution $\begin{cases} x^{16}+ax^{9}+bx^{4}+cx+15 & \leq0\\ |x^{16}-x^{9}+1|+|x^{4}-x+1| & \leq2 \end{cases}.$ 12. Suppose that$ABC$is an acute triangle inscribed in the circle$(O)$and$AD$is an altitude. The tangent lines at$B$,$C$of$(O)$intersect at$T$. On the line segment$AD$, choose$K$such that$\widehat{BKC}=90^{0}$. Let$G$be the centroid of$ABC$. Suppose that$KG$intersects$OT$at$L$. Choose the point$P$,$Q$on the side$BC$so that$LP\parallel OB$,$LQ\parallel OC$. Choose the points$E$and$F$respectively on the sides$CA$and$AB$such that$QE$,$PF$are both perpendicular to$BC$. Let$(T)$be the circle centerd at$T$and containing$B$and$C$. Prove that the circle circumscribing the triangle$AEF$is tangent to$(T)$. ##$type=three$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 455
2015 Issue 455
Mathematics & Youth