2013 Issue 429

  1. Find an integer size square whose area is a 4-digit number such that the last rightmost three digits are idnentical.
  2. Determine the values of $a_{1},a_{2},a_{3},a_{4},a_{5},a_{6},a_{7}$. Given that $2a_{1}=3a_{2}$, $2a_{3}=4a_{4}$, $5a_{4}=2a_{5}$, $2a_{5}=5a_{6}$, $2a_{6}=3a_{7}$, $2a_{7}=3a_{1}$ and $$a_{1}+a_{2}+a_{3}+a_{4}+a_{5}+a_{6}+a_{7}=400.$$
  3. Find all pairs of integers $(x,y)$ such that $x^{2}(x^{2}+y^{2})=y^{p+1}$, where $p$ is a prime number.
  4. Find the maximum and minimum values of the expression \[P=xy+yz+zx-xyz\] where $x,y,z$ are non-negative real numbers satisfying \[x^{2}+y^{2}+z^{2}=3.\]
  5. Triangle $ABC$ inscribed in circle $(O)$ with $\widehat{BAC}=70^{0}$, $\widehat{ACB}=50^{0}$. The points $M,N,P,Q$ and $R$ on circle $(O)$ are such that $PA=AB=BR$, $QB=BC=CM$ and $NC=CA=AN$. Let $S$ be the intersection of arc $NQ$ and the diameter $PP'$ of $(O)$. Prove that $\Delta NRS\backsim\Delta NQR$.
  6. Solve the equation \[x^{3}-3x=\sqrt{x+2}.\]
  7. Find the measure of the angles of a triangle $ABC$ such that the expression \[T=-3\tan\frac{C}{2}+4(\sin^{2}A-\sin^{2}B)\] is greatest possible.
  8. Let $S.ABC$ be a triangular pyramid where the sides $SA,SB,SC$ are pairwise orthogonal, $SA=a$, $SB=b$, $Sc=c$. $H$ is the foot of the perpendicular from $S$ onto $ABC$. Prove the inequality \[aS_{HBC}+bS_{HAC}+cS_{HAB}\leq\frac{abc\sqrt{3}}{2}.\]
  9. Let $p$ be an odd prime number, and $x,y$ are two positive integers such that $\sqrt{x}+\sqrt{y}\leq\sqrt{2p}$. Find the minimum value of the following expression \[A=\sqrt{2p}-\sqrt{x}-\sqrt{y}.\]
  10. Does there exist a funtion $f:\mathbb{N}^{*}\to\mathbb{N}^{*}$ such that \[f(mf(n))=n+f(2013m),\quad\forall m,n\in\mathbb{N}^{*}?.\]
  11. The non-negative real numbers $a,b,c$ are such that $$\max\{a,b,c\}\leq4\min\{a,b,c\}.$$ Prove the inequality \[2(a+b+c)(ab+bc+ca)^{2}\geq9abc(a^{2}+b^{2}+c^{2}+ab+bc+ca).\]
  12. Let $ABC$ be a traingle inscribed in circle centered at $O$, and let $I$ be its incenter. $AI,BI,CI$ intersect $(O)$ at $A_{1},B_{1},C_{1}$; $A_{1}C_{1},A_{1B_{1}}$ meet $BC$ at $M,N$; $B_{1}A_{1},B_{1}C_{1}$ meet $CA$ at $P,Q$; $C_{1}B_{1},C_{1}A_{1}$ meet $AB$ at $R,S$ respectively. Prove that \[S_{MNPQRS}\leq\frac{2}{3}S_{A_{1}B_{1}C_{1}}.\]




Mathematics & Youth: 2013 Issue 429
2013 Issue 429
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