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2010 Issue 398

  1. Let $n$ be a positive integer so that the first digit of $2^{n}$ and $5^{n}$ are the same. Prove that the number obtained by writing $2^{n}$ and $5^{n}$ consecutively has $n+1$ digits, where the digit 3 appears at least twice.
  2. Let $A B C$ be an isosceles triangle with $A B=A C$. Point $E$ on the median $B D$ is chosen so that $\widehat{D A E}=\widehat{A B D}$. Prove that $\widehat{D A E}=\widehat{E C B}$.
  3. Find all positive integer solutions of the equation $$x(x+2 y)^{3}-y(y+2 x)^{3}=27$$
  4. Find the value of the following expression $$P=\sqrt{12 \sqrt[3]{2}-15}+2 \sqrt{3 \sqrt[3]{4}-3}$$
  5. Let $ABC$ be a triangle with $\widehat{A C B}=70^{\circ}$, $\widehat{A B C}=50^{\circ}$. Let $D$, $E$ be respectively the midpoints of $B C$ and $A D$. Draw $EF$ perpendicular to $B C$ ($F$ is on $B C$). Let $M$ be a point on $E F$; let $N$, $P$ be respectively the orthogonal projections of $M$ onto $A C$, $A B$. Given that the three points $N$, $E$, $P$ are colinear, find the measure of angle $M A B$.
  6. Find the least value of the expression $$T=3 \sqrt{1+2 x^{2}}+2 \sqrt{40+9 y^{2}}$$ where $x, y$ are non-negative real numbers such that $x+y=1$.
  7. Let $A B C$ be an acute triangle. Prove that $$\frac{\cos A}{\cos \frac{B}{2} \cos \frac{C}{2}}+\frac{\cos B}{\cos \frac{C}{2} \cos \frac{A}{2}}+\frac{\cos C}{\cos \frac{A}{2} \cos \frac{B}{2}} \geq 2.$$
  8. Let $A B C$ be a triangle. A straight line cut the lines $B C$, $C A$ and $A B$ at $A^{\prime}$, $B^{\prime}$ and $C^{\prime}$ respectively. Let $A^{\prime \prime}$, $B^{\prime \prime}$ and $C^{\prime \prime}$ be the points reflection of $A^{\prime}$, $B^{\prime}$ and $C^{\prime}$ with centers at $A$, $B$ and $C$ respectively. Prove that the area of the triangle $A B C$.
  9. The positive integers are colored with either black or white such that the sum of two numbers with different color is painted black, and there are infinitely many numbers with white color. Let $q$ $(q>1)$ be the smallest positive integer with black color. Prove that $q$ is prime.
  10. Find all functions $f: \mathbb{N}^{*} \rightarrow \mathbb{N}^{*}$ such that $$f\left(f^{2}(m)+2 f^{2}(n)\right)=m^{2}+2 n^{2},\,\forall m, n \in \mathbb{N}^{*}$$
  11. The sequence $\left(x_{n}\right)(n \geq 1)$ of real numbers is defined inductively as follows $$x_{1}=a \in \mathbb{R},\quad x_{n+1}=2 x_{n}^{3}-5 x_{n}^{2}+4 x_{n},\,\forall n \geq 1 .$$ Find all possible values of $a$ such that the sequence $\left(x_{n}\right)$ has finite limit. Determine the limit of $\left(x_{n}\right)$ with respect to each such value of $a$.
  12. Let $A B C D$ be a tetrahedron. Find all points $P$ inside the tetrahedron such that $$x d_{A}+y d_{B}+z d_{C}+t d_{D}=c$$ where $x, y, z, t, c$ are given positive constants and $d_{A}$, $d_{B}$, $d_{C}$, $d_{D}$ are respectively the distances from $P$ to the four faces $B C D$, $C D A$, $D A B$, $A B C$ of the tetrahedron.

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Mathematics & Youth: 2010 Issue 398
2010 Issue 398
Mathematics & Youth
https://www.molympiad.org/2020/09/2010-issu-398.html
https://www.molympiad.org/
https://www.molympiad.org/
https://www.molympiad.org/2020/09/2010-issu-398.html
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