2013 Issue 435

  1. The number $s=\overline{3\ldots3}^{2}+\overline{5\ldots54\ldots4}^{2}$, written in decimal system, consist of $n+1$ digits $3$, $n-1$ digits $5$ and $n$ digits $4$. Given that $s=r^{2}$, find the value of $r$.
  2. Let $AM$ be the median of triangle $ABC$. On the half-plane containing $C$ created by the side $AB$, draw line segment $AE$ perpendicular to $AB$ such that $AB=AE$. On the half-plane containing $B$ created by $AC$, draw $AF$ perpendiclar to $AC$ such that $AF=AC$. Prove that $EF=2AM$ and $EF\perp AM$.
  3. Consider $n$ positive integers $a_{1},a_{2},\ldots a_{n}$ ($n>1$) satisfying \[a_{1}+a_{2}+\ldots+a_{n}=a_{1}a_{2}\ldots a_{n}.\] a) Prove that for any given value of $n$, the above equation always has solution.
    b) Determine all values of $n$ such that the equation $a_{1}<a_{2}<\ldots<a_{n}$.
  4. The numbers $a,b,c$ satisfy \[ab+bc+ca=2013abc,\quad2013(a+b+c)=1.\] Find the sum $A=a^{2013}+b^{2013}+c^{2013}$. 
  5. Prove that if in a trapezium $ABCD$ ($AB||CD$), $AC+CB=AD+DB$, then $ABCD$ is an isosceles trapezium.
  6. Solve the inequality on $\mathbb{R}$ \[(\sqrt{13}-\sqrt{2x^{2}-2x+5}-\sqrt{2x^{2}-4x+4})(x^{6}-x^{3}+x^{2}-x+1)\geq0.\]
  7. The three angles of an acute triangle $ABC$ are such that $A>\dfrac{\pi}{4}$, $B>\dfrac{\pi}{4}$, $C>\dfrac{\pi}{4}$. Determine the smallese value of the expression \[\frac{\tan A-2}{\tan^{2}C}+\frac{\tan B-2}{\tan^{2}A}+\frac{\tan C-2}{\tan^{2}B}.\]
  8. Let $S.ABCD$ be a pyramid inscribed in a sphere centred at $O$, and $AB=a$, $CD=b$. Draw parallellograms $ADKB$ and $SDHC$. Determine the ratio $\dfrac{HK}{EF}$ in terms of $a$ and $b$, where $E$ is the point of intersection of $AD$ and $BC$, and $F$ is the point of intersecion of $AC$ and $BD$.
  9. How many positive integers $n$ are there such that $n$ has $2013$ digits in decimal number system and $\dfrac{n}{7}$ is a positive integer with $2013$ odd digits?.
  10. Find all funtions $f:\mathbb{R}\to\mathbb{R}$, $g:\mathbb{R}\to\mathbb{R}$ such that the following two conditions are satisfied
    a) $f(x)-2g(x)=g(y)+4y$, for all $x,y\in\mathbb{R}$;
    b) $f(x)g(x)\leq33x^{2}$, for all $x\in\mathbb{R}$.
  11. Find all polynomials $T(x,y)$ such that \[T(x,y)T(z,t)=T(xz+yt,xt+yz)\] for all $x,y,z,t\in\mathbb{R}$. 
  12. Let $ABCD$ be a cyclic quadrilateral. The circle whose diameter is $AB$ meets $CA$, $CB$, $DA$ and $DB$ at $E,F,I$ and $J$ respectively (all differ from $A$ and $B$). Prove that the angle-bisector of an angle between $EF$ and $IJ$ is perpendicular to the line $CD.$




Mathematics & Youth: 2013 Issue 435
2013 Issue 435
Mathematics & Youth
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