2014 Issue 447

  1. Find all the integer solutions of the following equation \[1+x+x^{2}+x^{3}=y^{2}.\]
  2. Let $x,y,z$ be three coprime positive integers satisfying \[(x-z)(y-z)=z^{2}.\] Prove that $xyz$ is a perfect square. 
  3. Solve the following equation \[\frac{1}{\sqrt{3x}}+\frac{1}{\sqrt{9x-3}}=\frac{1}{\sqrt{5x-1}}+\frac{1}{\sqrt{7x-2}}.\]
  4. Given a circle $(O,R)$ and a chord $AB$ with the distance from $O$ is $d$ ($0<d<R$). Two circles $(I)$, $(K)$ are externally tangent at $C$, are both tangent to $AB$ and are internally tangent with $(O)$ ($I$ and $K$ are in the same half-plane determined by the line through $AB$). Find the locus of the points $C$ which vary when $(I)$ and $(K)$ vary.
  5. Find all positive integers $a$ and $b$ so that both equations $x^{2}-2ax-3b=0$ and $x^{2}-2bx-3a=0$ have positive integer solution.
  6. Find all positive real numbers $x,y,z$ satisfying system of equations \[\begin{cases} \dfrac{1}{x+1}+\dfrac{1}{y+1}+\dfrac{1}{z+1} & =1\\ xyz(x+y+z)(x+1)(y+1)(z+1 & =1296 \end{cases}.\]
  7. Given a tetrahedron $ABCD$ and the lengths of its sides $AB=BD=DC=x$, $BC=CA=AD=y$. Prove that \[ \frac{3}{5}<\frac{x}{y}<\frac{5}{3}.\]
  8. Find the maximum value of the expression \[P=|(a^{2}-b^{2})(b^{2}-c^{2})(c^{2}-a^{2})|,\] in which $a,b,c$ are nonnegative numbers satisfying $a+b+c=\sqrt{5}$.
  9. Solve equation \[x^{4}+ax^{3}+bx^{2}+2ax+4\] given $9(a^{2}+b^{2})=16$.
  10. Find all pairs of positive integers $(a,b)$ satisfying the following properties: $4a+1$ and $4b-1$ are coprime and $a+b$ is a divisor of $16ab+1$.
  11. Given two sequences \[a_{1}=0,\,a_{2}=16,\,a_{3}=18,\,a_{n+2}=8a_{n}+6a_{n-1}\] and \[b_{1}=3,\,b_{2}=19,\,b_{3}=69,\,b_{n+2}=3b_{n+1}+5b_{n}-b_{n-1}\] for $n\geq2$. Prove that \begin{align*} b_{n} & =C_{n}^{0}a_{n}+C_{n}^{1}a_{n-1}+\ldots+C_{n}^{n-1}a_{1}+3C_{n}^{n},\\ a_{n} & =C_{n}^{0}b_{n}-C_{n}^{1}b_{n-1}+C_{n}^{2}b_{n-2}-\ldots+(-1)^{n-1}C_{n}^{n-1}b_{1}+(-1)^{n}3C_{n}^{n}. \end{align*}
  12. Given a triangle $ABC$ and its circumscribed circle $(O)$. The points $A_{1},B_{1}$and $C_{1}$ are on the sides $BC,CA$ and $AB$ respectively. The circumscribed circles $(AB_{1}C_{1})$, $(BC_{1}A_{1})$, and $(CA_{1}B_{1})$ intersect $(O)$ at $A_{2},B_{2}$ and $C_{2}$ respectively. Find the positions of $A_{1},B_{1}$ and $C_{1}$ so that $\dfrac{S_{A_{1}B_{1}C_{1}}}{S_{A_{2}B_{2}C_{2}}}$ is minimal.




Mathematics & Youth: 2014 Issue 447
2014 Issue 447
Mathematics & Youth
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