2013 Issue 433

  1. Find all positive integers $x,y,z$ such that \[x^{2}+y^{3}+z^{4}=90.\]
  2. Let $ABC$ be an equilateral triangle whose altitudes are $AD$, $BE$ and $CF$. Suppose $M$ is an arbitrary point inside triangle $ABC$. $I$, $K$, $L$ are the perpendicualr feet from $M$ to $AD$, $BE$, $CF$. Prove that the sum $AI+BK+CL$ does not depend on the position of $M$.
  3. The rational numbers $a$, $b$ satisfy the identity \[a^{2013}+b^{2013}=2a^{1006}b^{1006}.\] Prove that the equation $x^{2}+2x+ab=0$ has two rational solutions.
  4. Find the minimum value of the expression \[P=(x^{4}+y^{4}+z^{4})\left(\frac{1}{x^{4}}+\frac{1}{y^{4}}+\frac{1}{z^{4}}\right),\] where $x,y,z$ are positive real numbers that satisfy $x+y\leq z$.
  5. Let $AH$ be the altitude from $A$ of right triangle $ABC$, right angle at $A$. Point $D$ on the oppostite ray of $HA$ such that $HA=2HD$. Point $E$ is the reflection of $B$ through $D$; $I$ is the midpoint of $AC$; $DI$ and $EI$ meet $BC$ at $M$ and $K$ respectively. Prove that $\widehat{BDK}=\widehat{MCD}$.
  6. Solve the equation \[\sqrt{x+\sqrt{x^{2}-1}}=\frac{27\sqrt{2}}{8}(x-1)^{2}\sqrt{x-1}.\]
  7. A convex quadrilateral $ABCD$ with area $S$ is inscribed in a circle whose radius is $R$ and $AB=a$, $BC=b$, $CD=c$, $DA=d$, $AC=e$. If there exists a circle touching all the opposite rays of the rays $BA$, $DA$, $CD$ and $CB$. Prove that
    a) $R=\dfrac{S\cdot e}{p^{2}-e^{2}}$,
    b) $a^{2}+b^{2}+c^{2}+d^{2}+\dfrac{8SR}{e}=2p^{2}$,
    where $2p=a+b+c+d$.
  8. Find the maximum value of the expression \[\alpha(\sin^{2}A+\sin^{2}B+\sin^{2}C)-\beta(\cos^{3}A+\cos^{3}B+\cos^{3}C)\] where $A$, $B$, $C$ are three angles of an acute triangle and $\alpha$, $\beta$ are two given positive numbers.
  9. Find the maximum area of a convex pectagon in the coordinate plane $Oxy$ having the following properties: all interior angles are the same, all vertices have integer coordinates, there exists a side that is parallel to the axis $Ox$, there are exactly $16$ points, including the vertices, with integer coordinates on its boundary.
  10. Find all continuous functions $f$ such that \[(x+y)f(x+y)=xf(x)+yf(y)+2xy,\,\forall x,y\in\mathbb{R}.\]
  11. Let $(a_{n})$ be a sequence where $a_{1}\in\mathbb{R}$ and $a_{n+1}=|a_{n}-2^{1-n}|$, $\forall n\in\mathbb{N}^{*}$. Find ${\displaystyle \lim_{n\to\infty}a_{n}}$.
  12. A right triangle $ABC$ with right angle at $C$ is inscribed in circle $(O)$. $M$ is an arbitrary point moving on circle $(O)$, different from $A$, $B$, $C$. Point $N$ is the reflection of $M$ in $AB$, $P$ is the perpendicular foot of $N$ to $AC$, $MP$ meets $(O)$ at a second point $Q$. Prove that the circumcenter of triangle $APQ$ lies on a fixed circle.




Mathematics & Youth: 2013 Issue 433
2013 Issue 433
Mathematics & Youth
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