Mathematics and Youth Magazine Problems - Feb 2015, Issue 452

  1. Let $x,y$ be positive natural numbers. Find the minimum value of the expression $A=|36^{x}-5^{y}|$.
  2. Let $O$ be the midpoint of the interval $AB$. On a half plane determined by the lines through $AB$, draw two rays $Ox$, $Oy$ which are perpendicular to each other. On the $Ox$, $Oy$, respectively choose two points $M,N$ which are different from $O$. Prove that $AM+BN\geq MN$. 
  3. Solve the system of inequalities \[\begin{cases} 2\sqrt{x^{2}-xy+y^{2}} & \leq(x+y)^{2}\\ \sqrt{1-(x+y)^{2}} & =1-x \end{cases}. \]
  4. Given a triangle $ABC$ which is isosceles at $A$ and is inscribed in a circle $(O)$. Let $AK$ be a diameter. Let $I$ be any point on the minor are $AB$ ($I$ is different from $A$ and $B$). $KI$ intersects $BC$ at $M$. The perpendicular bisector of $MI$ intersects the sides $AB$,$AC$ respectively at $D,E$. Let $N$ be the midpoint of $DE$. Prove that $A,M$ and $N$ are colinear.
  5. Find integers $m$ so that the following equation \[x^{3}+(m+1)x^{2}-(2m-1)x-(2m^{2}+m+4)=0\] has integer solutions.
  6. Suppose that $a,b,c$ are three nonnegative numbers. Prove that \[(a+bc)^{2}+(b+ca)^{2}+(c+ab)^{2}\geq\sqrt{2}(a+b)(b+c)(c+a).\]
  7. Given a triangle $ABC$ which is isosceles at $A$ and is inscribed in a circle $(O)$. Let $D$ be the midpoint of $AB$. The ray $CD$ intersects $(O)$ at $E$. Let $F$ be the point on $(O)$ so that $CF\parallel AE$. The ray $EF$ intersects $AC$ at $G$. Prove that $BG$ is tangent to the circle $(O)$.
  8. Determine the minimum value of the funtion \[f(x)=\sqrt{\sin x+\tan x}+\sqrt{\cos x+\cot x}.\]
  9. On the plane $Oxy$, consider the set $M$ consisting of the points $(x,y)$ such that $x,y\in\mathbb{N}^{*}$ and $x\leq12,y\leq12$. Each point in $M$ is colored be red, white or blue. Prove that there exists a rectangle satisfying the following properties: its sides are parallel to coordinate axes and its vertives are in $M$ and are colored by the same color.
  10. Find the minimum positive integer $t$ so that there exist $t$ integers $x_{1},x_{2},\ldots,x_{t}$ satisfying \[x_{1}^{3}-x_{2}^{3}+x_{3}^{3}-\ldots+(-1)^{t+1}x_{t}^{3}=2065^{2014}.\]
  11. Let $a$ be a positive integer and $(x_{n})$ a sequence given by $x_{1}=1$ and \[x_{n+1}=\sqrt{x_{n}^{2}+2ax_{n}+2a+1}-\sqrt{x_{n}^{2}-2ax_{n}+2a+1},\forall n\in\mathbb{N}^{*}.\] Find $a$ so that the sequence $(x_{n})$ has a finite limit.
  12. Given a triangle $ABC$. A line $\Delta$ which does not contain $A,B,C$ intersects $BC,CA,AB$ respectively at $A_{1},B_{1},C_{1}$. Let $A_{b},A_{c}$ respectively be the symmetric points of $A_{1}$ through $AB,AC$. Let $A_{a}$ be the midpoint of $A_{b}A_{c}$. The points $B_{b},C_{c}$ are determined similarly to the way we construct the point $A_{a}$. Prove that the points $A_{a},B_{b},C_{c}$ are collinear.

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