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

## $show=home 1. Find natural numbers$x, y, z$satisfying $$3^{x}+5^{y}-2^{z}=(2 z+3)^{3}.$$ 2. Given a right triangle$A B C$with the right angle$A$and$\hat{B}=75^{\circ}$. Let$H$be the point on the opposite ray of$A B$such that$B H=2 A C$. Find the angle$\widehat{B H C}$. 3. Given that$x y(x+y)+y z(y+z)+z x(z+x)+2 x y z=0$. Show that $$x^{2019}+y^{2019}+z^{2019}=(x+y+z)^{2019}.$$ 4. Given a triangle$A B C$with$\widehat{A B C}=30^{\circ}$. Outside the triangle$A B C$, construct the isosceles triangle$A C D$with the right angle$D$. Show that $$2 B D^{2}=B A^{2}+B C^{2}+B A \cdot B C.$$ 5. Find the minimum value of the expression $$T=\frac{5-3 x}{\sqrt{1-y^{2}}}+\frac{5-3 y}{\sqrt{1-z^{2}}}+\frac{5-3 z}{\sqrt{1-x^{2}}}.$$ 6. Find all possible values for the parameter$m$so that the equation $$4^{x}+2=m \cdot 2^{x}(1-x) x$$ has a unique solution. 7. Let$a, b, c$be non-negative numbers such that$(a+b)(b+c)(c+a)>0$. Find the minimum value of the expression $$P=\sqrt{\frac{a}{b+c}}+\sqrt{\frac{b}{c+a}}+\sqrt{\frac{c}{a+b}}+\frac{4 \sqrt{a b+b c+c a}}{a+b+c}.$$ 8. Given a triangle$A B C$with$\widehat{C}=45^{\circ}$. Let$G$be the centroid of$A B C .$Let$\widehat{A G B}=\alpha$. Prove that $$\frac{\sqrt{2}}{\sin A \sin B}+3 \cot \alpha=1.$$ 9. Let$a, b, c$be positive number such that$a^{2}+b^{2}+c^{2}=3$. Show that $$\frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a} \geq \frac{1}{2 \sqrt{2}}\left(\sqrt{a^{2}+b^{2}}+\sqrt{b^{2}+c^{2}}+\sqrt{c^{2}+a^{2}}\right).$$ 10. For any real numbers$a, b, c$we let $$T(a, b, c)=|a-b|+|b-c|+|c-a|.$$ Consider a sequence$(*)$of integers$x_{1}, x_{2}, \ldots, x_{12}$satisfying the conditions: there exists a polynomial$f(x)$with integral coefficients so that$f\left(x_{1}\right), f\left(x_{2}\right), \ldots, f\left(x_{12}\right)$are different and $$880<\sum_{k<j=k \leq 12} T\left(f\left(x_{i}\right), f\left(x_{j}\right), f\left(x_{k}\right)\right) \leq 3758.$$ Show that from the sequence$(*)$we can always extract an arithmetic progression with at least four terms. 11. Find all functions$h(x): \mathbb{R} \rightarrow \mathbb{R}$which satisfy all of the following conditions •$h(2019)=0$•$h(x+1)=h(x)$,$\forall x \in \mathbb{R}$. •$3^{x+y}[h(x) h(y)+h(x+y)]=3^{x}(y+1) h(x)+3^{y}(x+1) h(y)+3^{x y} h(x y)$,$\forall x, y \in \mathbb{R}$. 12. Given an acute triangle$A B C$inscribed in a circle$(\Omega)$. The points$E$,$F$are on the sides$CA$,$A B$respectively so that the quadrilateral$B C E F$is cyclic. The perpendicular bisector of$C E$intersects$B C$,$E F$at$N$,$R$respectively. The perpendicular bisector of$B F$intersects$B C$,$E F$at$M$,$Q$respectively. Let$K$be the reflection point of$E$over the line$R M$. Let$L$be the reflection point of$F$over the line$Q N$. Suppose that the intersection between$R K$and$Q^{B}$is$S$; the intersection between$Q L$and$R C$is$T$. a) Show that four points$Q, R, S, T$both belong to a circle, say$(\omega)$. b) Show that$(\omega)$and$(\Omega)$are tangent to each ##$type=three$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 510
2019 Issue 510
Mathematics & Youth