2011 Issue 410

  1. Find all pairs of coprime positive integers $x, y$ so that $$\frac{x+y}{x^{2}+y^{2}}=\frac{7}{25}$$
  2. Let $A B C$ be an equilateral triangle whose altitudes $A H$, $B K$ intersect at $G .$ The angle-bisector of angle $B K H$ meets $C G$, $A H$, $B C$ at $M$, $N$, $P$ respectively. Prove that $K M=N P$.
  3. Find the minimum value of the expression $S=2011 c a-a b-b c$ where $a, b, c$ satisfy $a^{2}+b^{2}+c^{2} \leq 2$.
  4. Let $A B C$ be an isosceles right triangle with right angle at $A .$ Let $M$, $N$, $O$ be respectively the midpoints of $A B$, $A C$, $B C$. The line perpendicular to $C M$ from $O$ cuts $M N$ at $G .$ Compare the lengths of the two segments $G M$ and $G N$.
  5. Solve the equation $$\sqrt{7 x^{2}+25 x+19}-\sqrt{x^{2}-2 x-35}=7 \sqrt{x+2}$$
  6. Let $A B C$ be a triangle. Let $A M$, $B N$, $C P$ be its internal angle-bisectors ($M \in B C$, $N \in C A$, $P \in A B$). Find the measure of angle $B A C$ so that $P M$ is perpendicular to $N M$
  7. Solve the equation $$(\sin x-2)\left(\sin ^{2} x-\sin x+1\right)=3 \sqrt[3]{3 \sin x-1}+1.$$
  8. Find all values of $a, b$ so that the equation $$x^{4}+a x^{3}+b x^{2}+a x+1=0$$ has at least one solution and the sum $a^{2}+b^{2}$ is smallest possible.
  9. Let $a, b, c$ be positive numbers. Prove that $$\left(a^{2012}-a^{2010}+3\right)\left(b^{2012}-b^{2010}+3\right)\left(c^{2012}-c^{2010}+3\right) \geq 9(a b+b c+c a).$$ When does equality occur?
  10. Let $A B C$ be a triangle. An arbitrary line cuts the lines $B C$, $C A$, $A B$ at $M$, $N$, $P$ respectively. Let $X$, $Y$, $Z$ be respectively the centroids of triangles $A N P$, $B P M$, $C M N$. Prove that $$S_{X Y Z}=\frac{2}{9} S_{A B C}$$
  11. Let $\left(a_{n}\right)$ $\left(n \in \mathbb{N}^{*}\right)$ be a sequence given by $$a_{1}=0 ,\, a_{2}=38,\, a_{3}=-90,\quad a_{n+1}=19 a_{n-1}-30 a_{n-2},\, \forall n \geq 3.$$ Prove that $a_{2011}$ is divisible by 2011 .
  12. For all positive integers $n$ greater than $2$. Find the number of functions $$f:\{1,2,3, \ldots, n\} \rightarrow\{1,2,3,4,5\}$$ satisfying $|f(k+1)-f(k)| \geq 3$ where $k \in\{1,2, \ldots, n-1\}$.




Mathematics & Youth: 2011 Issue 410
2011 Issue 410
Mathematics & Youth
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