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 $E F$.




Mathematics & Youth: 2019 Issue 508
2019 Issue 508
Mathematics & Youth
Loaded All Posts Not found any posts VIEW ALL Readmore Reply Cancel reply Delete By Home PAGES POSTS View All RECOMMENDED FOR YOU LABEL ARCHIVE SEARCH ALL POSTS Not found any post match with your request Back Home Sunday Monday Tuesday Wednesday Thursday Friday Saturday Sun Mon Tue Wed Thu Fri Sat January February March April May June July August September October November December Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec just now 1 minute ago $$1$$ minutes ago 1 hour ago $$1$$ hours ago Yesterday $$1$$ days ago $$1$$ weeks ago more than 5 weeks ago Followers Follow THIS CONTENT IS PREMIUM Please share to unlock Copy All Code Select All Code All codes were copied to your clipboard Can not copy the codes / texts, please press [CTRL]+[C] (or CMD+C with Mac) to copy