2006 Issue 354

  1. a) Find all natural number, each of which can be written as the sum of two relatively prime  relatively prime integers greater than $1$.
    b) Find all natural numbers, each of which can be written as the sum of three pairwise
  2. Let $A B C$ be a triangle with acute angle $\widehat{A B C}$. integers greater than $1 .$ Let $K$ be a point on the side $A B$, and $H$ be its orthogonal projection on the line $B C .$ A ray $B x$ cuts the segment $K H$ at $E$ and cuts the line passing through $K$ parallel to $B C$ at $F .$ Prove that $\widehat{A B C}=3 \widehat{C B F}$ when and only when $E F=$ $2 B K$.
  3. Find all natural numbers n such that the product of the digits of $n$ is equal to $$(n-86)^{2}\left(n^{2}-85 n+40\right)$$
  4. Prove that $a b+b c+c a<\sqrt{3} d^{2}$ where $a, b, c, d$ are real numbers satisfying the following conditions
    • $0<a, b, c<d$
    • $\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}-\sqrt{\dfrac{1}{a^{2}}+\dfrac{1}{b^{2}}+\dfrac{1}{c^{2}}}=\dfrac{2}{d}$.
  5. Solve the equation $$x^{4}+2 x^{3}+2 x^{2}-2 x+1=\left(x^{3}+x\right) \sqrt{\frac{1-x^{2}}{x}}$$
  6. Let $A B C D$ be a square with sides equal to a. On the side $A D,$ take the point $M$ such that $A M=3 M D .$ Draw the ray $B x$ cutting the side $C D$ at $I$ such that $\widehat{A B M}=\widehat{M B I}$. The angle bisector of $\widehat{C B I}$ cuts the side $C D$ at $N$. Calculate the area of triangle $B M N$.
  7. Let $B C$ be a fixed chord (which is not a diameter of a circle. On the major arc $B C$ of the circle, take a point $A$ not coinciding with $B, C .$ Let $H$ be the orthocenter of triangle $A B C .$ The second points of intersection of the line $B C$ with the circumcircles of triangles $A B H$ and $A C H$ are $E$ and $F$ respectively. The line $E H$ cuts the side $A C$ at $M$ and the line $F H$ cuts the side $A B$ at $N$. Determine the position of $A$ so that the measure of the segment $M N$ attains its least value.
  8. How many are there natural 9-digit numbers with 3 distinct odd digits, 3 distinct even digits and every even digit in each number appears exactly two times (in this number
  9. For every positive integer $n$, consider the function $f_{n}$ defined on $\mathbb{R}$ by $$f_{n}(x)=x^{2 n}+x^{2 n-1}+\ldots+x^{2}+x+1.$$ a) Prove that the function $f_{n}$ attains its least value at a unique value $x_{n}$ of $x$.
    b) Let $S_{n}$ be the least value of $f_{n}$. Prove that 
    • $S_{n}>\dfrac{1}{2}$ for all $n$ and there does not exist a real number $a>\dfrac{1}{2}$ such that $S_{n}>a$ for all $n$
    • $\left(S_{n}\right)(n=1,2, \ldots)$ is a decreasing sequence and $\lim S_{n}=\dfrac{1}{2}$
    • $\lim x_{n}=-1$
  10. Let $$A=\sqrt{x^{2}+\sqrt{4 x^{2}+\sqrt{16 x^{2}+\sqrt{100 x^{2}+39 x+\sqrt{3}}}}}.$$ Find the greatest integer not exceeding $A$ when $x=20062007$
  11. Let $A B C$ be a triangle with $B C=a$, $C A=b$, $A B=c,$ with inradius $r$ and with incenter $I .$ Let $A_{1}, B_{1}, C_{1}$ be respectively the touching points of the sides $B C$, $C A$, $A B$ with the incircle. The rays $I A$, $I B$, $I C$ cut the incircle respectively at $A_{2}$, $B_{2}$, $C_{2}$. Let $B_{i}C_{i}=a_{i}$, $C_{i} A_{i}=b_{i}$ $A_{i} B_{i}=c_{i}$ $(i=1,2)$. Prove that $$\frac{a_{2}^{3} b_{2}^{3} c_{2}^{3}}{a_{1}^{2} b_{1}^{2} c_{1}^{2}} \geq \frac{216 r^{6}}{a b c}.$$ When does equality occur?
  12. Let $O A B C$ be a tetrahedron with $$\widehat{A O B}+\widehat{B O C}+\widehat{C O A}=180^{\circ},$$ $O A_{1}$, $O B_{1}$, $O C_{1}$ are internal angle bisectors respectively of the triangles $O B C$, $O C A$, $O A B$, $O A_{2}$, $O B_{2}$, $O C_{2}$ are internal angle bisectors respectively of the triangles $O A A_{1}$, $O B B_{1}$, $O C C_{1}$. Prove that $$\left(\frac{A A_{1}}{A_{2} A_{1}}\right)^{2}+\left(\frac{B B_{1}}{B_{2} B_{1}}\right)^{2}+\left(\frac{C C_{1}}{C_{2} C_{1}}\right)^{2} \geq(2+\sqrt{3})^{2}.$$ When does equality occur?




Mathematics & Youth: 2006 Issue 354
2006 Issue 354
Mathematics & Youth
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