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2019 Issue 505

  1. Find the smallest positive interger $a$ so that $2 a$ is a square and $3 a$ is a cube.
  2. Find different non-zero digits $a$, $b$, $c$, $d$ so that $\overline{a b c d a 1}-4 n=n^{2}$ for some postitive integer $n$ (the last digit of $\overline{abcdal}$ is $1$).
  3. Find all polynomials $P(x)$ whose the coefficients are integers between $0$ and $8$ and $P(9)=32078$
  4. Let $ABCD$ be a convex quadrilateral. Denote the midpoints of $A B$, $A C$, $C D$, $D B$ respectively $M$, $N$, $P$, $Q$. Let the lengths of the sides $A B$, $B C$, $C D$, $D A$ respectively be $a$, $b$, $c$, $d$. Let the area of $M N P Q$ be $S$. Assume that $A D$ and $B C$ are perpendicular. Show that $$\frac{(c-a)^{2}-(b-d)^{2}}{8} \leq S \leq \frac{(b+d)^{2}-(c-a)^{2}}{8}$$
  5. Let $x$, $y$, $z$ be positive numbers such that $x+y+z=3$. Find the minimum value of the expression $$P=x^{5}+y^{5}+z^{5}+\frac{2}{x+y}+\frac{2}{y+z}+\frac{2}{z+x}+\frac{10}{x y z}.$$
  6. Find all real solutions of the equation $$\sqrt[3]{\frac{x^{3}-3 x+\left(x^{2}-1\right) \sqrt{x^{2}-4}}{2}}+\sqrt[3]{\frac{x^{3}-3 x-\left(x^{2}-1\right) \sqrt{x^{2}-4}}{2}}=x^{2}-2.$$
  7. Given the equation $$\frac{1}{3} x^{5}+2 x^{4}-5 x^{3}-7 x^{2}+12 x-1=0.$$ a) Show that the equation has $5$ distinct roots.
    b) Let $x_{I}$ $(i=1,5)$ be the roots of the equation. Find the sum $$S=\sum_{i=1}^{5} \frac{x_{i}-1}{x_{i}^{5}+6 x_{i}^{4}-3}.$$
  8. Given any triangle $A B C$ show that $$ \left(1+\sin ^{2} \frac{A}{2}\right)\left(1+\sin ^{2} \frac{B}{2}\right)\left(1+\sin ^{2} \frac{C}{2}\right) \geq \frac{125}{64}.$$
  9. Given positive numbers $a$, $b$, $c$ and a number $-2<k<2$. Prove that $$27\left(a^{2}+k a b+b^{2}\right)\left(b^{2}+k b c+c^{2}\right)\left(c^{2}+k c a+a^{2}\right) \geq (k+2)^{3}(a b+b c+c a)^{3}.$$
  10. A man using a map on his phone walked from the point $A$ to the point $B$. He arrived $B$ after a few straight walks and correspondingly a few rotations of the phone (to find the right directions). Assume that each time he needed to rotate his phone clockwisely an acute angle from the previous direction. Given that the sum of all the angles is $\alpha$ which is less than $180^{\circ}$. Show that the total distance that he walked is less than or equal to $\dfrac{A B}{\cos \frac{\alpha}{2}}.$
  11. Given the real sequence $\left(a_{n}\right)$ determined as follows $$a_{1}=2020, \quad a_{n+1}=1+\frac{2}{a_{n}},\, \forall n \geq 1.$$ a) Prove that $2 n<a_{1}+a_{2}+\ldots+a_{n}<2 n+2018$ for any arbitrary $n=1,2, \ldots$.
    b) Find the maximal real number $a$ such that the inequality $$\sqrt{x^{2}+a_{1}^{2}}+\sqrt{x^{2}+a_{2}^{2}}+\ldots+\sqrt{x^{2}+a_{n}^{2}} \geq n \sqrt{x^{2}+a^{2}}$$ holds for any given $x \in \mathbb{R}$, $n=1,2, \ldots$.
  12. Given a triangle $A B C$ which is not an isosceles triangle with the vertex angle $A$. Let $M$ be on the side $B C$. Let $I_{1}$, $I_{2}$ respectively be the incenters of the triangles $A B M$, $A C M$. Assume that $N$, $P$, $Q$ respectively be the second intersections between $A M$, $A B$, $A C$ and the circumcircle of $A I_{1} I_{2}$. Let $J_{1}$, $J_{2}$ respectively be the incenters of the triangles $A P N$, $A Q N$. Prove that the radical center of the circumcircles of $A I_{1} I_{2}$, $A J_{1} J_{2}$, $M I_{1} I_{2}$ belongs to $B C$.

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Mathematics & Youth: 2019 Issue 505
2019 Issue 505
Mathematics & Youth
https://www.molympiad.org/2020/09/2019-issue-505.html
https://www.molympiad.org/
https://www.molympiad.org/
https://www.molympiad.org/2020/09/2019-issue-505.html
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