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 $(O)$.




Mathematics & Youth: 2013 Issue 430
2013 Issue 430
Mathematics & Youth
Loaded All Posts Not found any posts VIEW ALL Readmore Reply Cancel reply Delete By Home PAGES POSTS View All RECOMMENDED FOR YOU LABEL ARCHIVE SEARCH ALL POSTS Not found any post match with your request Back Home Sunday Monday Tuesday Wednesday Thursday Friday Saturday Sun Mon Tue Wed Thu Fri Sat January February March April May June July August September October November December Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec just now 1 minute ago $$1$$ minutes ago 1 hour ago $$1$$ hours ago Yesterday $$1$$ days ago $$1$$ weeks ago more than 5 weeks ago Followers Follow THIS CONTENT IS PREMIUM Please share to unlock Copy All Code Select All Code All codes were copied to your clipboard Can not copy the codes / texts, please press [CTRL]+[C] (or CMD+C with Mac) to copy