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2015 Issue 455

  1. Find a natural number with more than $3$ digits knowing that if we delete its the last $3$ digits, we will get a new number whose cube is exactly equal to that wanted number.
  2. Given two positive real numbers $a$ and $b$ satisfying the following conditions $a^{2015}-a-1=0$ and $b^{4030}-b-3a=0$. Compare $a$ and $b$.
  3. Solve the equation \[x+y+x\sqrt{1-y^{2}}+y\sqrt{1-x^{2}}=\frac{3\sqrt{3}}{2}.\]
  4. On a circle centered at the point $I$, fix two points $B$ and $C$. A point $A$ varies on the circle such that the triangle $ABC$ is always acute. On the side $AC$, choose $M$ so that $MA=3MC$. Let $H$ be the perpendicular projection of $M$ on the side $AB$. Prove that $H$ always lies on a fixed circle.
  5. Find all prime numbers $x$ and $y$ such that \[(x^{2}+2)^{2}=2y^{4}+11y^{2}+x^{2}y^{2}+9.\]
  6. Solve the following system of equations \[\begin{cases} x^{3}+y^{3} & =4y^{2}-5y+4x+4\\ 2y^{3}+z^{3} & =4z^{2}-5z+6y+6\\ 3z^{3}+x^{3} & =4y^{2}-5x+9z+8 \end{cases}.\]
  7. Given a triangle $ABC$. Suppose that the length of the sides are given by $BC=a$, $CA=b$, $AB=c$ and $m_{a}$, $m_{b}$ and $m_{c}$ are the length of the corresponding medians. Prove that \[(a^{2}+b^{2}+c^{2})\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)\geq2\sqrt{3}(m_{a}+m_{b}+m_{c}).\] When does the equality happen?.
  8. Consider an acute triangle $ABC$ with the angles $A,B$ and $C$. Find the maximum value of the expression \[M=\frac{\tan^{2}A+\tan^{2}B}{\tan^{4}A+\tan^{4}B}+\frac{\tan^{2}B+\tan^{2}C}{\tan^{4}B+\tan^{4}C}+\frac{\tan^{2}C+\tan^{2}A}{\tan^{4}C+\tan^{4}A}.\]
  9. Find that coefficient of $x^{2}$ in the expansion of the following expression \[(1+x)(1+2x)(1+4x)\ldots(1+2^{2013}x).\]
  10. Let $a_{1},a_{2},\ldots,a_{n}$ be positive numbers such that \[a_{1}+a_{2}+\ldots+a_{n}=\frac{1}{a_{1}}+\frac{1}{a_{2}}+\ldots+\frac{1}{a_{n}}.\]Find the minimum value of the expression \[A=a_{1}+\frac{a_{2}^{2}}{2}+\ldots+\frac{a_{n}^{2}}{n}.\]
  11. Find the biggest real number $k$ satisfying the condition: for any 3 real numbers $a,b,c$ such that $|a|+|b|+|c|<k$, the following system of inequalities has no solution \[\begin{cases} x^{16}+ax^{9}+bx^{4}+cx+15 & \leq0\\ |x^{16}-x^{9}+1|+|x^{4}-x+1| & \leq2 \end{cases}.\]
  12. Suppose that $ABC$ is an acute triangle inscribed in the circle $(O)$ and $AD$ is an altitude. The tangent lines at $B$, $C$ of $(O)$ intersect at $T$. On the line segment $AD$, choose $K$ such that $\widehat{BKC}=90^{0}$. Let $G$ be the centroid of $ABC$. Suppose that $KG$ intersects $OT$ at $L$. Choose the point $P$, $Q$ on the side $BC$ so that $LP\parallel OB$, $LQ\parallel OC$. Choose the points $E$ and $F$ respectively on the sides $CA$ and $AB$ such that $QE$, $PF$ are both perpendicular to $BC$. Let $(T)$ be the circle centerd at $T$ and containing $B$ and $C$. Prove that the circle circumscribing the triangle $AEF$ is tangent to $(T)$.

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Mathematics & Youth: 2015 Issue 455
2015 Issue 455
Mathematics & Youth
https://www.molympiad.org/2017/08/mathematics-and-youth-magazine-problems_46.html
https://www.molympiad.org/
https://www.molympiad.org/
https://www.molympiad.org/2017/08/mathematics-and-youth-magazine-problems_46.html
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