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2013 Issue 430

  1. Do there exist natural numbers $x,y,z$ such that \[5x^{2}+2016^{y+1}=2017^{z}?.\]
  2. Point $O$ is chosen in a right triangke $ABC$, right angle at $A$, such that $\widehat{ABO}=30^{0}$ and $OA=OC$. Point $E$ on side $BC$ such that $\widehat{EOB}=60^{0}.$ Determine the three angles of traingle $ABC$ given that the line $CO$ passes through the midpoint $I$ of the line segment $AE$.
  3. Find all pair of natural numbers $x,y$ such that $5^{x}=y^{4}+4y+1$.
  4. Solve the system of equations \[\begin{cases} x+\sqrt{y-2}+\sqrt{4-z} & =y^{2}-5z+11\\ y+\sqrt{z-2}+\sqrt{4-x} & =z^{2}-5x+11\\ z+\sqrt{x-2}+\sqrt{4-y} & =x^{2}-5y+11 \end{cases}.\]
  5. Let $AB=2a$ be a line segment with midpoint $O$. Two half-circles, one with center $O$ and diagonal $AB$, another with center $O'$ and diagonal $AO$ are drawn on the same half-plane divided by $AB$. Point $M$, different from $A$ and $O$, moves on the half-circle $(O')$. $OM$ meets the half-circle $(O)$ at $C$. Let $D$ be the second intersection point of $CA$ and half-circle $(O')$. The tangent line at $C$ of half-circle $(O)$ meets $OD$ at $E$. Find the position of point $M$ on $(O')$ such that $ME$ is parallel to $AB$.
  6. Let $ABCD$ be a quadrilateral where the diagonals $AC,BD$ are equal and perpendicular. The triangles $AMB$, $BNC$, $CPD$, $DQA$, similar in order, are constructed outside the given quadrilateral. $O_{1}$, $O_{2}$, $O_{3}$, $O_{4}$ are the midpoints of $MN$, $NP$, $PQ$, $QM$ respectively. Prove that the quadrilateral $O_{1}O_{2}O_{3}O_{4}$ is s square.
  7. Find a formula counting the number of all $2013$-digits natural numbers which are multiple of $3$ and all digits are taken from the set $X=\{3,5,7,9\}$.
  8. Solve for $x$, \[\log_{2}x=\log_{5-x}3.\]
  9. The positive integers $a_{1},a_{2},\ldots,a_{2013}$, $b_{1},b_{2},\ldots b_{2013}$ where $b_{k}>1$ for all $k$ are chosen from the set $X=\{1,2,\ldots,2013\}$. Prove that there exists a positive integer $n$ satisfying the following two conditions
    • ${\displaystyle n\leq\left(\prod_{i=1}^{2013}a_{i}\right)\left(\prod_{i=1}^{2013}b_{i}\right)+1}$.
    • $a_{k}b_{k}^{n}+1$ is a composite number for every $k\in X$.
  10. Let $a,b,c\in\left[0,\frac{1}{2}\right]$ be such that $a+b+c=1$. Prove the inequality \[a^{3}+b^{3}+c^{3}+4abc\leq\frac{9}{32}.\]
  11. Let $ap$ be a prime number, $p\equiv1\,(\text{mod }4)$. Determine the sum \[\sum_{k=1}^{p-1}\left[\frac{2k^{2}}{p}-2\left[\frac{k^{2}}{p}\right]\right],\] where $[a]$ denotes the largest integer not exceeding $a$.
  12. Let $ABC$ be a triangle inscribed inside circle $(O)$. Point $M$ not on lines $BC$, $CA$, $AB$ as weel as circle $(O)$; $AM$, $BM$, $CM$ intersect $(O)$ at $A_{1}$, $B_{1}$, $C_{1}$; $A_{2}$, $B_{2}$, $C_{2}$ are the circumcenters of triangles $MBC$, $MCA$, $MAB$ respectively. Prove that the lines $A_{1}A_{2}$, $B_{1}B_{2}$, $C_{1}C_{2}$ meet at a point on the circle $(O)$.

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Mathematics & Youth: 2013 Issue 430
2013 Issue 430
Mathematics & Youth
https://www.molympiad.org/2017/08/mathematics-and-youth-magazine-problems_9.html
https://www.molympiad.org/
https://www.molympiad.org/
https://www.molympiad.org/2017/08/mathematics-and-youth-magazine-problems_9.html
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