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2019 Issue 510

  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

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Mathematics & Youth: 2019 Issue 510
2019 Issue 510
Mathematics & Youth
https://www.molympiad.org/2020/09/2019-issue-510.html
https://www.molympiad.org/
https://www.molympiad.org/
https://www.molympiad.org/2020/09/2019-issue-510.html
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