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2019 Issue 508

  1. Let $$A=11.13 .15+13.15 .17+\ldots+91.93 .95+93.95 .97.$$ Is $A$ divisible by $5 ?$
  2. Find $2019$ numbers so that the absolute values of these numbers do not exceed 0,5 and the sum of any $3$ arbitrary numbers among these is an integer.
  3. Let $x, y, z$ be positive numbers. Find the minimum value of the expression $$P=x^{2}+y^{2}+z^{2}+\frac{x^{3}}{x^{2}+y^{2}}+\frac{y^{3}}{y^{2}+z^{2}}+\frac{z^{3}}{z^{2}+x^{2}}-\frac{7}{6}(x+y+z).$$
  4. Given a triangle $A B C$ with $\widehat{A B C}$ and $\widehat{A C B}$ are acute. Let $M$ be the midpoint of $A B .$ On the opposite ray of the ray $B C$ choose the point $D$ such that $\widehat{D A B}=\widehat{B C M}$. Through $B$ draw a line perpendicular to $C D$. This line intersects the perpendicular bisector of $A B$ at $E$. Show that $D E$ is perpendicular to $A C$.
  5. Solve the equation $$x^{2010}-2011 x^{670}+\sqrt{2010}=0.$$
  6. Given non-negative numbers $a$, $b$, $c$ with at most one of them is equal to $0$. Show that for every positive integer $n$ we have $$\frac{a^{2^{2}}+b^{2^{n}}}{a^{2^{2}}+c^{2^{n}}}+\frac{b^{2^{n}}+c^{2^{n}}}{b^{2^{n}}+a^{2^{n}}}+\frac{c^{2^{n}}+a^{2^{n}}}{c^{2^{n}}+b^{2^{n}}} \geq \frac{a+b}{a+c}+\frac{b+c}{b+a}+\frac{c+a}{c+b}.$$
  7. Find the value of the expression $$f(A, B, C)=\sin A+\sin B+\sin C-\sin A \sin B \sin C$$ where $A$, $B$, $C$ are the angles of a triangle.
  8. Given a tetrahedron $O A B C$ with $O A$, $O B$, $O C$ are pairwise perpendicular and $O A=a$, $O B=b$, $O C=c$. Let $r$ be the radius of the inscribed sphere of $O A B C$. Show that $$\frac{1}{r} \geq \frac{\sqrt{3}+1}{\sqrt{3}}\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right).$$
  9. Suppose that the positive numbers $a, b, c, d$ form a pregression (in that order) with the common difference $m$. Show that $$e^{a^2}\left(4 m^{2}+2 m a+1\right)+e^{b^{2}} \cdot 2 m a+e^{c^{2}}\left(2 m^{2}+2 m a\right)<e^{d^{2}}$$
  10. Show that, for any integer $n \geq 1$, the equation $x^{2 n+1}=x+1$ has exactly one real solution which is denoted by $x_{n}$. Then find $\displaystyle \lim _{n \rightarrow+\infty} x_{n}$
  11. Find all the functions $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying $$f(x) f(y)+f(x+y)=x f(y)+y f(x)+f(x y)+x+y+1,\,\forall x, y \in \mathbb{R}.$$
  12. Given a harmonic quadrilateral $A B C D$ (a cyclic quadrilateral in which the products of two opposite sides are equal) inscribed the circle $(O)$. Let $M$ be the midpoint of $A C$. Let $X$, $Y$, $Z$, $T$ respectively be the perpendicular projection of $M$ on $A B$, $B C$, $C D$, $D A$. Let $E=A B \cap C D$, $F=A D \cap C B$, $P=A C \cap B D$, $Q=X Z \cap Y T$. Show that $P Q$ passes through the midpoint of $E F$.

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Mathematics & Youth: 2019 Issue 508
2019 Issue 508
Mathematics & Youth
https://www.molympiad.org/2020/09/2019-issue-508.html
https://www.molympiad.org/
https://www.molympiad.org/
https://www.molympiad.org/2020/09/2019-issue-508.html
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