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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 \[S_{MNPQRS}\leq\frac{2}{3}S_{A_{1}B_{1}C_{1}}.\]

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Mathematics & Youth: 2013 Issue 429
2013 Issue 429
Mathematics & Youth
https://www.molympiad.org/2017/08/mathematics-and-youth-magazine-problems_71.html
https://www.molympiad.org/
https://www.molympiad.org/
https://www.molympiad.org/2017/08/mathematics-and-youth-magazine-problems_71.html
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