# TÀI LIỆU TOÁN - WWW.MOLYMPIAD.NET - ĐỀ THI TOÁN

## $show=home 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$E F$. ##$type=three$c=3$source=random$title=oot$p=1$h=1$m=hide$rm=hide ## Anniversary_$type=three$c=12$title=oot$h=1$m=hide\$rm=hide

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Mathematics & Youth: 2019 Issue 508
2019 Issue 508
Mathematics & Youth