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

## $show=home 1. Let$a$be a natural number with all different digits and$b$is another number obtained by using the all the digits of$a$but in a different order. Given $a-b=\underset{n}{\underbrace{111\ldots1}}$ ($n$digit$1$where$n$is a positive integer). Find the maximum value of$n$. 2. Find positive integers$x,y,z$such that $(x-y)^{3}+(y-z)^{3}+4|z-x|=27.$ 3. Let$a,b,c$be the lengths of three sides of a triangle. Prove that $2\left(\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\right)\geq\frac{a}{c}+\frac{b}{a}+\frac{c}{b}+3.$ 4. Let$ABC$be an isosceles triangle ($AB=AC$) inscribed in a given circle$(O,R)$. Draw$BH$perpendicular to$AC$at$H$. Find the maximum length of$BH$. 5. Solve the system of equations $\begin{cases} \dfrac{7}{2}+\dfrac{3y}{x+y} & =\sqrt{x}+4\sqrt{y}\\ (x^{2}+y^{2})(x+1) & =4+2xy(x-1) \end{cases}.$ 6. For any$m>1$, prove that the following equation has a unique solution $x^{3}-3\sqrt{3x+2m}=2m.$ 7. Find all values of the parameters$p$and$q$such that the corresponding system of equations $\begin{cases} x^{2}+y^{2}+5 & =q^{2}+2x-4y\\ x^{2}+(12-2p)x+y^{2} & =2py+12p-2p^{2}-27 \end{cases}$ has two solutions$(x_{1},y_{1})$and$(x_{2},y_{2})$satisfying $x_{1}^{2}+y_{1}^{2}=x_{2}^{2}+y_{2}^{2}.$ 8. Given a triangle$ABC$with$BC=a$,$CA=b$and$AB=c$. Let$R,r$and$p$respectively be the circumradius, the inradius, and the semiperimeter of$ABC$. Prove that $\frac{ab+bc+ca}{p^{2}+9Rr}\geq\frac{4}{5}.$ When does the equality occur?. 9. Given three positive real numbers$a,b,c$such that$abc\geq1$. Prove that $\frac{a^{4}b^{2}c^{2}}{bc+1}+\frac{b^{4}c^{2}a^{2}}{ca+1}+\frac{c^{4}a^{2}b^{2}}{ab+1}\geq\frac{3}{2}.$ 10. Find all positive integers$k$such that there exist 2015 different positive integers whose sum is divisible by the sum of any$k$numbers among them. 11. Find the maximum$k$such that the inequality $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+1+2k\geq(k+2)\sqrt{a+b+c+1}$ holds for any positive real numbers$a,b,c$satisfying$abc=1$. 12. Given a triangle$ABC$and$D$varies on the side$BC$. Let$(I_{1})$,$(I_{2})$respectively be the incircles of the triangles$ABD$,$ACD$. Suppose that$(I_{1})$is tangent to$AB$,$BD$at$E,X$and$(I_{2})$is tangent to$AC$,$CD$at$F,Y$. Assume that$AI_{1}$,$AI_{2}$respectively intersects$EX$,$FY$at$Z,T$. Show that a)$X,Y,Z,T$both belong to a circle with center$K$. b)$K$belongs to a fixed line. ##$type=three$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 460
2015 Issue 460
Mathematics & Youth