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

## $show=home 1. Find two whole numbers of the form$\overline{ab}$and$\overline{ba}$($a\ne b$) such that $\frac{\overline{ab}}{\overline{ba}}=\frac{\underset{2014\text{ digits}}{\overline{a\underbrace{3\ldots3}b}}}{\underset{2014\text{ digits}}{\overline{b\underbrace{3\ldots3}a}}}.$ 2. The sum$A$below consists of 2014 summands $A=\frac{1}{19^{1}}+\frac{2}{19^{2}}+\frac{3}{19^{3}}+\ldots+\frac{2014}{19^{2014}}.$ Compare the number$A^{2013}$with$A^{2014}$. 3. Let$ABCD$be a quadriteral whose diagonals$AC$and$BD$are perpendicular.$M$and$N$are the midpoints of line segments$AB,AD$respectively. Points$E,F$are the feet of perpendicular lines from$M$and$N$onto$CD,BC$respectively. Prove that$MNEF$is a cyclic quadrilateral. 4. Solve for$x$$4x^{3}+4x^{2}-5x+9=4\sqrt[4]{16x+8}.$ 5. The real numbers$x,y,z$satisfy$x+y+z=1$. Prove the inequality $44(xy+yz+zx)\leq(3x+4y+5z)^{2}.$ 6. Prove that the following equation has no real solutions $9x^{4}+x(12x^{2}+6x-1)+(x+1)(9x^{2}+12x+5)+1=0.$ 7. Triangle$ABC$inscribed in a circle centerd at$O$and radius$R$, where$CA\ne CB$,$\widehat{ACB}=90^{0}$. The circumcircle centered at$S$of triangle$AOB$meets$CA,CB$at points$M,N$respectively. Let$K$be the reflection of$S$in the line$MN$. Prove that$SK=R$. 8. The real numbers$x,y,z$satisfy$x^{2}+y^{2}+z^{2}=8$. Determine the largest and smallest values of the following expression $P=(x-y)^{5}+(y-z)^{5}+(z-x)^{5}.$ ##$type=three$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 442
2014 Issue 442
Mathematics & Youth