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

## $show=home 1. Let$(2n-1)!!$and$(2n)!!$donote the products$1.3.5\ldots(2n-1)$and$2.4.6\ldots(2n)$respectively. Prove that the number $A=(2013)!!+(2014)!!$ is divisible by$2015.$2. Given an isolates triangle$ABC$with$AB=AC$and$\widehat{A}=3\widehat{B}$. On the half-plane determined by$BC$that contians$A$, draw the array$Cy$such that$\widehat{BCy}=132^{0}$. The array$Cy$intersects the bisector$Bx$of the angle$B$at$D$. Calculate$\widehat{ADB}$. 3. Solve the equation $\frac{1}{\sqrt{x^{2}+3}}+\frac{1}{\sqrt{1+3x^{2}}}=\frac{2}{x+1}.$ 4. Given an equilateral triangle$ABC$and a point$)$inside the triangle. Let$M,N,P$respectively be the intersections between$AO,BO,CO$and the sides of the triangle. Prove that a)${\displaystyle \frac{1}{AM}+\frac{1}{BN}+\frac{1}{CP}\leq\frac{1}{3}\left(\frac{1}{OM}+\frac{1}{ON}+\frac{1}{OP}\right)}$, b)${\displaystyle \frac{1}{AM}+\frac{1}{BN}+\frac{1}{CP}\leq\frac{2}{3}\left(\frac{1}{OA}+\frac{1}{OB}+\frac{1}{OC}\right)}$. 5. Solve the system of equations $\begin{cases} \sqrt{9(x\sqrt{x}+y^{3}+z^{3})} & =x+y+z\\ x^{2}+\sqrt{y} & =2z\\ \sqrt{y}+z^{2} & =\sqrt{1-x}+2 \end{cases}.$ 6. Solve the equation $(x+1)(x+2)(x+3)=\frac{720}{(x+4)(x+5)(x+6)}.$ 7. Determine the number of solutions of each following equation a)${\displaystyle \sin x=\frac{x}{1964}}$, b)$\sin x=\log_{100}x$, 8. Given a triangle$ABC$inscribed in a circle$(O)$. The bisectors of the angles$A,B,C$respectively intersects the circle at$D$,$E$,$F$. Denote respectively by$h_{a},h_{b},h_{c},S$the heights from$A,B,C$and the area of$ABC$. Prove that $AD.h_{1}+BE.h_{b}+CF.h_{c}\geq4\sqrt{3}S.$ 9. Given real numbers$a,b,c,d$satisfying $a^{2}+b^{2}+c^{2}+d^{2}=1.$ Find the maximum and minimum values of the expression $P=ab+ac+ad+bc+cd+3cd.$ 10. Given$k\geq1$and positive numbers$x,y$. For any positive integer$n\geq2$, show the following inequalities $\sqrt{xy}\leq\sqrt[n]{\frac{x^{n}+y^{n}+k[(x+y)^{n}-x^{n}-y^{n}]}{2+k(2^{n}-2)}}\leq\frac{x+y}{2}.$ 11. A pair of positive integers is called a good pair if their quotient if either$2$or$3$. What is the most number of good pairs we can get among$2015^{2016}$arbitrary different positive integers?. 12. Given a triangle$ABC$. Let$E,F$respectively be the perpendicular projections of$B,C$on$AC,AB$; and then let$T$be the perpendicular projection of$A$on$EF$. Denote the midpoints of$BE$and$CF$by$M$and$N$respectively. Suppose that$TM,TN$intersects$AB,AC$respectively at$P,Q$. Prove that$EF$goes through the midpoint of$PQ$, ##$type=three$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 461
2015 Issue 461
Mathematics & Youth