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2015 Issue 461

  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 \[\begin{cases} \sqrt[3]{9(x\sqrt{x}+y^{3}+z^{3})} & =x+y+z\\ x^{2}+\sqrt{y} & =2z\\ \sqrt{y}+z^{2} & =\sqrt{1-x}+2 \end{cases}.\]
  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 $PQ$,

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Mathematics & Youth: 2015 Issue 461
2015 Issue 461
Mathematics & Youth
https://www.molympiad.org/2017/08/mathematics-and-youth-magazine-problems_2.html
https://www.molympiad.org/
https://www.molympiad.org/
https://www.molympiad.org/2017/08/mathematics-and-youth-magazine-problems_2.html
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