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

## $show=home 1. Find natural numbers$a,b,c$satisfying$9^{a}+952=(b+41)^{2}$and$a=2^{b}\cdot c$. 2. Prove that $A=\frac{1}{1000}+\frac{1}{1002}+\ldots+\frac{1}{2000}<\frac{1}{2}.$ 3. Find positive integral solutions of the equation $x^{5}+y^{5}+2016=(x+2017)^{5}+(y-2018)^{5}.$ 4. Given a quadrilateral$ABCD$inscribed in a circle$(O)$with$AB=BD$. The lines$AB$and$DC$intersect at$N$. The line$CB$and the tangent line of the circle$(O)$at the point$A$intersect at$M$. Prove that$\widehat{AMN}=\widehat{ABD}$. 5. Solve the equation $(1-2x)\sqrt{2-x^{2}}=x-1.$ 6. Solve the system of equations $\begin{cases} \dfrac{x+\sqrt{1+x^{2}}}{1+\sqrt{1+y^{2}}} & =x^{2}y\\ 1+y^{4}-\dfrac{4y}{x}+\dfrac{3}{x^{4}} & =0 \end{cases}.$ 7. Given an integer$n>1$. Prove that $\frac{2^{2n}}{2\sqrt{n}}<C_{2n}^{n}<\frac{2^{2n}}{\sqrt{2n+1}}.$ 8. Given a quadrilateral$ABCD$circumsribing a circle$(O)$. Let$E,F$and$G$respectively be the intersection of three pairs of line$(AB,CD)$,$(AD,CB)$and$(AC,BD)$. Let$K$be the intersection between$EF$and$OG$. Prove that$\widehat{AKG}=\widehat{CKG}$and$\widehat{BKG}=\widehat{DKG}$. 9. Given three nonnegative numbers$x,y,z$such that$2^{x}+4^{y}+8^{z}=4$. Find the maximum and minimum values of the expression $S=\frac{x}{6}+\frac{y}{3}+\frac{z}{2}.$ 10. Find all pairs of natural number$(m,n)$so that$2^{m}\cdot3^{n}-1$is a perfect square. 11. Let$(u_{n})$be a sequence defined as follows $u_{1}=0,\quad\log_{\frac{1}{4}}u_{n+1}=\left(\frac{1}{4}\right)^{u_{n}}\quad\forall n\in\mathbb{N}^{*}.$ Prove that the sequence has a finite limit and find that limit. 12. Given a triangle$ABC$and its circumcircle$(O)$. Let$(O_{B})$and$(O_{C})$respectively be the mixtilinear incircles corresponding to$B$and$C$(recall that a mixtilinear incircle corresponding to$B$is the circle which is internally tangent to$BA$,$BC$and the circumcircle$(O)$). Let$M$(resp.$N$) be the tangent point between$(O)$and$(O_{B})$(resp.$(O_{C})$). Let$(M)$and$(N)$be the circles with centers at$M$and$N$and are tangent to$AC$and$AB$respectively. Prove that the center of dilation of$(M)$and$(N)$belongs to$BC$. ##$type=three$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 481
2017 Issue 481
Mathematics & Youth