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

## show=home 1. Let \begin{align*} A & =\frac{1}{1.2^{2}}+\frac{2}{2.3^{2}}+\frac{1}{3.4^{2}}+\ldots+\frac{1}{49.50^{2}},\\ B & =\frac{1}{2^{2}}+\frac{1}{3^{2}}+\frac{1}{4^{2}}+\ldots+\frac{1}{50^{2}}.\end{align*} CompareA$and$B$with$\dfrac{1}{2}$. 2. For all pairs of real numbers$(a,b)$such that the polynomial $A(x)=x^{2}-2ax+2a^{2}+b^{2}-5$ has solutions. Find the minimum value of the expression $P=(a+1)(b+1).$ 3. Suppose that$a_{1},a_{2},\ldots,a_{n}$are different positive integers such that $\frac{1}{a_{1}}+\frac{1}{a_{2}}+\ldots+\frac{1}{a_{n}}=1$ and also assume that the biggest number among them is equal to$2p$, where$p$is some prime number. Determine$a_{i}$($i=\overline{1,n}$). 4. Let$ABC$be a isosceles right triangle ($AB=BC$) and let$O$be the midpoint of$AC$. Through$C$draw the line$d$perpendicular to$BC$. Let$Cx$be the opposite ray of the ray$CB$and$M$be an arbitrary point on$Cx$. Assume that$E$is the intersection between$BE$and$OM$. Prove that when$M$varies on$Cx$,$I$always belongs to a fixed curve. 5. Solve the following equation $x^{2}-2x=2\sqrt{2x-1}.$ 6. Solve the following inequation $\frac{2x^{3}+3x}{7-2x}>\sqrt{2-x}.$ 7. Given an acute and non-isosceles triangle$ABC$with the altitudes$AH$,$BE$,$CF$. Let$I$be the incenter of$ABC$(the center of the inscribed circle) and$R$be the circumradius of$ABC$(the radius of the circumscribed circle). Let$M,N$and$P$respectively be the midpoint of$BC,CA$and$AB$. Assume that$K,J$and$L$respectively are the intersections between$MI$and$AH$,$NI$and$BE$, and$PI$and$CF$. Prove that $\frac{1}{HK}+\frac{1}{EJ}+\frac{1}{FL}>\frac{3}{R}.$ 8. Let$a,b,c$be the lengths of three sides of a triangle whose perimeter is equal to$3$. Find the minimum value of the expression $T=a^{3}+b^{3}+c^{3}+\sqrt{5}abc.$ 9. For every positive integer$n$prove that the following numbers are perfect square $10([(4+\sqrt{15})^{n}+1])([(4+\sqrt{15})^{n+1}]+1)-60,$ $6([(4+\sqrt{15})^{n}+1])([(4+\sqrt{15})^{n+1}]+1)-60,$ $15([(4+\sqrt{15})^{n}+1])^{2}-60,$ where$[x]$is the integerpart of$x$. 10. Let$f(x)=x^{3}-3x^{2}+1$. a) Find the number of different real solutions of the equation$f(f(x))=0$. b) Let$\alpha be$the maximal positive solution of$f(x)$. Prove that$[\alpha^{2020}]$is divisible by$17$(notice that$[x]$is the integer part of$x$). 11. Give the sequence$(u_{n})$where$u_{1}=2$,$u_{2}=20$,$u_{3}=56$and $u_{n+3}=7u_{n+2}-11u_{n+1}+5u_{n}-3.2^{n},\quad\forall n\in\mathbb{N}^{*}.$ Find the remainder of the division$u_{2011}$by$2011$. 12. Consider any triangle$ABC$and any line$d$. Let$A_{1},B_{1},C_{1}$are the projections of$A,B,C$onto$d$. It is a fact that the line through$A_{1}$and perpendicular to$BC$, the line through$B_{1}$and perpendicular to$AC$, and the line through$C_{1}$and perpendicular to$AB$are concurrent ar a point which is called the orthogonal pole of$d$with respect to$ABC$. Prove that for any triangle and any point$P$on its circumcircle, the Simon line corresponding to$P$and the orthogonal pole of the Simon line with respect to the given triangle. ##$type=three$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 459
2015 Issue 459
Mathematics & Youth