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

## $show=home 1. Given a natural number$n$. Find all prime numbers$p$such that the following number $$A=1010n^{2}+2010(n+p)+10^{10^{1954}}$$ can be written as a difference of two perfect squares. 2. Given a triangle$ABC$and let$M$be the midpoint of$BC$. Suppose that$\widehat{ABC}+\widehat{MAC}=90^{0}$. Prove that$ABC$is either an isosceles triangle or a right triangle. 3. Solve the equation $$\frac{x}{2\sqrt{x}+1}+\frac{x^{2}}{2\sqrt{x}+3}=\frac{\sqrt{x^{3}}+x}{4}.$$ 4. Give a isoceles trapezoid$ABCD$inscribed in a circle$(O)$with$AB$is parallel to$CD$and$AB<CD$. Let$M$be the midpoint of$CD$and let$P$be any point on the side$MD$. Suppose that$AP$intersects$(O)$at the second point$Q$, and$BP$intersects$(O)$at the second point$R$. Assume that QR intersects$CD$at$E$. Let$F$be the symmetry point of$P$over the point$E$. Suppose that$EA$is tangent to$(O)$, prove that$AF$is perpendicular to$AQ$. 5. Let$x,y$belong to$(0,1)$. Find the maximum value of the expression $$P=x\sqrt{1-y^{2}}+y\sqrt{1-x^{2}}+\frac{1}{\sqrt{3}}(x+y).$$ 6. Given$f\left(x\right)=x^{2}-6x+12$. Solve the equation$f(f(f(f(x))))=65539$. 7. Find all real solutions of the following system of equations: $$\begin{cases} \sqrt{2^{2^{x}}} & =2(y+1)\\ 2^{2^{y}} & =2x \end{cases}.$$ 8. Given a uniform triangular prism$ABC,A'B'C'$(the base$ABC$is an equilateral triangle). Let$\alpha$be the angle between$A'B$and$(ACC'A')$and let$\beta$be the angle between$(A'B'C')$and$(ACC'A')$. Prove that$\alpha<60^{0}$and$\cot^{2}\alpha-\cot^{2}\beta=\frac{1}{3}$. 9. Let$p$,$q$be two coprime numbers. Let $$T=\left[\frac{p}{q}\right]+\left[\frac{2p}{q}\right]+\ldots+\left[\frac{(q-1)p}{q}\right],$$ where$[x]$is the gretest integral number, that isn't execeed$x$and is called the integral part of$x$. a) Find$T$. b) Find$p$,$q$such that$T$is a prime number. 10. Let$S$be the number of all the binary strings of length$n$with the property that the sum of any$3$consecutive numbers on any of these strings is always positive. Prove that $$2\left(\frac{3}{2}\right)^{n}\leq S\leq\frac{13}{2}\cdot\left(\frac{20}{9}\right)^{n},\quad\forall n\geq3.$$ 11. Find all functions$f:\mathbb{R}^{+}\to\mathbb{R}^{+}$such that $$f\left(\frac{x}{x-y}\right)+f\left(xf(y)\right)=f\left(xf(x)\right),$$ for every$x>y>0$. 12. Given a triangle$ABC$. The incircle$(I)$of$ABC$is tangent to$BC$,$CA$and$AB$at$D$,$E$and$F$respectively. Let$B_{1}$(resp.$C_{1}$) be the intersection between the lines which go through$AB$and$DE$(resp.$AC$and$DF$). Let$H$and$K$respectively be the orthocenter of$ABC$and$AB_{1}C_{1}$. Prove that the line which goes through$HK$contains the point$I$. ##$type=three$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: 2017 Issue 475
2017 Issue 475
Mathematics & Youth