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2013 Issue 434

  1. Find the largest possible perfect square of the form $4^{27}+4^{1020}+4^{x}$, where $x$ is a natural number.
  2. Let $ABC$ be a right triangle with right angle at vertex $A$ and $\widehat{ABC}=54^{0}$. The median $AM$ meets the internal angle-bisector $CD$ at $E$. Prove that $CE=AB$.
  3. Find all prime numbers $p$ such that $p-1$ and $p+1$ each has exactly $6$ divisors.
  4. Solve the system of equations \[ \begin{cases} 3\sqrt{x}+2\sqrt{y}+\sqrt{z} & =\dfrac{1}{6}\sqrt{xyz}\\ 6\sqrt{xy}+2\sqrt{yz}+3\sqrt{zx} & =108+18\sqrt{x+4}+12\sqrt{y+9}+6\sqrt{z+36} \end{cases}.\]
  5. Let $AB$ and $AC$ be the tangent lines to a circle $(O)$ through an external point $A$ ($B$ and $C$ are the poins of tangency). The median $BM$ of triangle $ABC$ intersects $(O)$ at $D$, the ray $AD$ meets $(O)$ at $E$. Prove that $BE||AC$.
  6. Given that $a,b,c$ are the sides of a triangle, prove the inequality \begin{align*} & \sqrt{a^{2}-(b-c)^{2}}+\sqrt{b^{2}-(c-a)^{2}}+\sqrt{c^{2}-(a-b)^{2}}\\ \leq & \sqrt{ab}+\sqrt{bc}+\sqrt{ca}\leq a+b+c \end{align*}
  7. The circle $(O)$ is inscribed in a triangle $ABC$. The tangents to $(O)$ which are parallel to the sides of the triangle are drawn, they intersect these sides at points $M,N,P,Q,R$ and $S$ ($M,S\in AB$, $N,P\in AC$, $Q,R\in BC$). Let $l_{1},l_{2},l_{3}$ be the lengths of the internal angle-bisectors from $A,B,C$ of triangles $AMN$, $BSR$ and $CPQ$ respectively. Prove that \[\frac{1}{l_{1}^{2}}+\frac{1}{l_{2}^{2}}+\frac{1}{l_{3}^{2}}\geq\frac{81}{p^{2}},\] where $p$ denotes the semiperimeter of triangle $ABC$.
  8. It is given that in a triangle $ABC$, $\tan\dfrac{B}{2}\tan\dfrac{C}{2}=\dfrac{1}{3}$. Solve for $x$ \[x^{2}+x-\cos A-\frac{1}{4}\cos(B-C)=0.\]
  9. Find all real numbers $x$ such that \[\left\{ \frac{x^{2}+1}{x^{2}+x+1}\right\} =\frac{1}{2}\] where $\{a\}$ denote te fractional part of $a$, that is $\{a\}=a-[a]$.
  10. Consider the sequence $(x_{n})$, \[x_{n+1}=x_{n}+\frac{1}{x_{n}}+\frac{2}{x_{n}^{2}}+\ldots+\frac{2012}{x_{n}^{2012}}\quad(n\in\mathbb{N}^{*})\] where $x_{1}>0$ is given. Determine all values of $\alpha$ such that the sequence $(nx_{n}^{\alpha})$ has a non-zero limit.
  11. Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that \[f(xf(y)+3y^{2})+f(3xy+y)=f(3y^{2}+x)+4xy-x+y\] for all $x,y\in\mathbb{R}$.
  12. Let $G$ be the centroid of a tetrahedron $ABCD$. Points $X,Y<Z,T$ are chosen on the faces $(BCD)$, $(CDA)$, $(DAB)$, and $(ABC)$ respectively such that $XY$, $YZ$, $ZT$, $TX$ are parallel to $GA$, $GB$, $GC$ and $GD$. Determine the volume ratio of the two tetrahedrons $ABCD$ and $XYZT$.

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Mathematics & Youth: 2013 Issue 434
2013 Issue 434
Mathematics & Youth
https://www.molympiad.org/2017/10/mathematics-and-youth-magazine-problems_75.html
https://www.molympiad.org/
https://www.molympiad.org/
https://www.molympiad.org/2017/10/mathematics-and-youth-magazine-problems_75.html
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