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

## $show=home 1. Find all the$4$-digit perfect squares such that when we reverse their digits we also get perfect squares. 2. Given an isosceles triangle$ABC$with the vertex angle$A$. On the half plane determined by$BC$which does not contain$A$choose a point$D$such that$\widehat{BAD}=2\widehat{ADC}$and$\widehat{CAD}=2\widehat{ADB}$. Prove that$CBD$is an isosceles triangle with the vertex angle$D$. 3. Prove thta$3^{n+2}|10^{3^{n}}-1$for any natural number$n$. 4. Given a trapezoid$ABCD$($AB\parallel CD$) with$AB<CD$. Let$P$and$Q$respectively be on the diagonals$AC$and$BD$such that$PQ$is not parallel to$AB$. The ray$QP$intersects$BC$at$M$and the ray$PQ$intersects$AD$at$N$. Let$O$be the intersect of$AC$and$BD$. Suppose futhermore that$MP=PQ=QN$. Prove that $\frac{OP}{OA}+\frac{OQ}{OB}=1.$ 5. Let$a,b$and$c$be positive numbers such that$\sqrt{a}+\sqrt{b}+\sqrt{c}=1$. Find the maximum value of the expression $P=\sqrt{abc}\left(\frac{1}{\sqrt{(a+b)(a+c)}}+\frac{1}{\sqrt{(b+c)(b+a)}}+\frac{1}{\sqrt{(c+b)(c+a)}}\right).$ 6. Solve the equation$f(f(x))=x$in terms of the parameter$m$where $f(x)=x^{2}+2x+m.$ 7. Suppose that$a,b$and$c$are the lengths of three sides of a triangle. Let $F(a,b,c)=a^{3}+b^{3}+c^{3}+3abc-ab(a+b)-bc(b+c)-ca(c+a).$ Prove that $F(a,b,c)\leq\min\left\{ F(a+b,b+c,c+a),4a^{2}b^{2}c^{2}F\left(\frac{1}{a},\frac{1}{b},\frac{1}{c}\right)\right\}.$ 8. Given an acute triangle$ABC$. Let$a,b$and$c$respectively be the lengths of$BC$,$CA$and$AB$. Let$R$and$r$respectively be the circumradius and the inradius of$ABC$. Prove that $\frac{r}{2R}\leq\frac{abc}{\sqrt{2(a^{2}+b^{2})(b^{2}+c^{2})(c^{2}+a^{2})}}.$ 9. Suppose that$a$and$b$are real numbers such that$0<a\ne1$and that the equation$a^{x}-\dfrac{1}{a^{x}}=2\cos(bx)$has exactly$2017$real roots and they are all different real numbers. How many distinct real roots does the equation $a^{x}+\frac{1}{a^{x}}=2\cos(bx)+4$ have?. 10. In a country, the length of any direct road between two cities (if any) is smaller than$100$km and we can travel from a city to any other one by roads which have total length is smaller than$100$km. When a road is closed under construction, we still can travel from one city to another by other roads. Prove that we can choose a route which has the total length is smaller than$300$km. 11. Prove that$\gcd(1,2,\ldots,2n)$is divisible by$C_{2n}^{n}$for any positive interger$n$. 12. Given a triangle$ABC$with$AB<AC$. Let$M$be the midpoint of$BC$. Let$H$be the perpendicular projection of$B$on$AM$. Suppose that$Q$is the point of the opposite ray of$AM$such that$AQ=4MH$. Assume that$AC$intersects$BQ$at$D$. Prove that the circumcenter of$ADQ$lies on the circumcircle of$DBC$. ##$type=three$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 478
2017 Issue 478
Mathematics & Youth