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2019 Issue 507

  1. Does it exist a natural number $n$ so that the last digit of the sum $1+2+3+\ldots+n$ is $2$, $4$, $7$ or $9$?.
  2. Given a right triangle $A B C$ with the right angle $A$ and $A B<A C$. Let $E$ and $F$ be the points on the sides $A C$ and $B C$ respectively such that $E F \perp B C$ and $E F=F B$. Let $D$ be the point on the side $A C$ such that $A D=A B$. Prove that $E F D$ is an isosceles triangle.
  3. Find positive integral solutions of the equation $$1+5^{x}=2^{y}+5.2^{2}.$$
  4. Given an acute triangle $A B C$. Outside the triangle, draw two equilateral triangles $A B D$ and $A C E$. On the line segments $A D$, $CE$, $CB$ choose the points $M$, $N$, $F$ respectively so that $$\frac{A M}{A D}=\frac{C N}{C E}=\frac{C F}{C B}=\frac{1}{3} .$$ Compare the lengths of two length segments $M N$ and $E F$.
  5. Given real numbers $x,y, z \geq 0$ such that $\max \{x ; y ; z\} \geq 1$. Show that $$x^{3}+y^{3}+z^{3}+(x+y+z-1)^{2} \geq 1+3 x y z.$$
  6. Solve the system of equations $$\begin{cases}x^{3}+x+2 &=8 y^{3}-6 x y+2 y \\ \sqrt{x^{2}-2 y+2}+2 \sqrt[4]{x^{3}(5-4 y)} &=2 y^{2}-x+2\end{cases}.$$
  7. Suppose that $$P(x)=x^{n}+x^{n-1}+a_{n-2} x^{n-2}+\ldots+a_{1} x+a_{0}$$ has $n$ distinct real roots $x_{1}, x_{2}, \ldots, x_{n}$. Show that $$\frac{x_{1}^{n}}{P^{\prime}\left(x_{1}\right)}+\frac{x_{2}^{n}}{P^{\prime}\left(x_{2}\right)}+\ldots+\frac{x_{n}^{n}}{P^{\prime}\left(x_{n}\right)}=-1$$ where $P^{\prime}(x)$ is the derivative of $P(x)$.
  8. Suppose that the inscribed sphere of the tetrahedron $A_{1} A_{2} A_{3} A_{4}$ is tangent to the face which is opposite to $A_{i}$ at $B_{I}$ $(i=1,2,3,4)$. Prove that if $B_{1} B_{2} B_{3} B_{4}$ is almost-regular (opposite sides have the same length) if and only if $A_{1} A_{2} A_{3} A_{4}$ is almost-regular.
  9. Find the minimum and maximum values of the expression $$P=\frac{\left(2 x^{2}+5 x+5\right)^{2}}{(x+1)^{4}+1}$$
  10. Find all prime numbers $p$ and positive integers $a, b$ so that $p^{a}+p^{b}$ is a perfect square.
  11. Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying $$f((x+z)(y+z))=(f(x)+f(z))(f(y)+f(z)),\,\forall x, y, z \in \mathbb{R}.$$
  12. Given a triangle $A B C$ and a point $M$ on the side $B C$. The symmedians through $M$ of the triangles $M A B$, $M A C$ intersect the circles $(M A B)$, $(M A C)$ respectively at $Q$, $R$ which are different from $M$. Let $P$ be the point on $B C$ so that $AP \perp AM$. Denote by $l$ the external common tangent, which closer to $A$, of two circles $(M A B)$, $(MAC)$. Suppose that $l$ is parallel to $B C$. Show that $l$ is tangent to $(P Q R)$. (The notion $(X Y Z)$ is for the circumcircle of the triangle $X Y Z$).

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Mathematics & Youth: 2019 Issue 507
2019 Issue 507
Mathematics & Youth
https://www.molympiad.org/2020/09/2019-issue-507.html
https://www.molympiad.org/
https://www.molympiad.org/
https://www.molympiad.org/2020/09/2019-issue-507.html
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