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

## $show=home 1. Which number is greater?$P$or$Q, given that \begin{align*} P & =\frac{20}{30}+\frac{20}{70}+\frac{20}{126}+\ldots+\frac{20}{798};\\ Q & =\left(\frac{31}{2}\cdot\frac{32}{2}\cdot\frac{33}{2}\ldots\frac{60}{2}\right):(1.3.5\ldots59). \end{align*} 2. Given2014$points$A_{1},A_{2},\ldots,A_{2014}$and a circle with radius$1$on the plane, prove that there always exists a point$M$on the circle such that $MA_{1}+MA_{2}+\ldots+MA_{2014}\geq2014.$ 3. Prove that for any natural number$n$, $n^{4}-5n^{3}-2n^{2}-10n+4$ is not divisible by$49$. 4. Let$R$denote the radius of the circumcircle of a given triangle$ABC$. The internal and external angle bisector of angle$\widehat{ACB}$meet$AB$at$E$and$F$respectively. Prove that if$CE=CF$, then$AC^{2}+BC^{2}=4R^{2}$. 5. Solve the following system of equations $\begin{cases} 2x\left(1+\frac{1}{x^{2}-y^{2}}\right) & =5\\ 2(x^{2}+y^{2})\left(1+\frac{1}{(x^{2}-y^{2})^{2}}\right) & =\frac{17}{2}\end{cases}.$ 6. Determine the funtion $f(x)=ax^{2}+bx+c$ where$a,b,c$are integers such that$f(0)=2014$,$f(2014)=0$and$f(2^{n})$is a multiple of$3$for any natural number$n$. 7. The positive real numbers$x,y,z$satisfy the equation$xy=1+z(x+y)$. Find the greatest value of $P=\frac{2xy(xy+1)}{(1+x^{2})(1+y^{2})}+\frac{z}{1+z^{2}}.$ 8. In an acute triangle$ABC$, the three altitudes$AA_{1},BB_{1},CC_{1}$meet at$H$. Prove that$ABC$is an equilateral triangle if and only if $HA^{2}+HB^{2}+HC^{2}=4(HA_{1}^{2}+HB_{1}^{2}+HC_{1}^{2}).$ ##$type=three$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: 2014 Issue 440
2014 Issue 440
Mathematics & Youth