2020 Issue 519

  1. Find positive integers $x$, $y$ such that $$x^{y}+y^{x}=23-x y$$
  2. Given a triangle $A B C$ with right angle at $A .$ Let $A H$ be the altitude. If $\dfrac{A H}{B C}=\dfrac{12}{25},$ find $\dfrac{A B}{A C}$.
  3. Suppose that $a$, $b$ are positive and $9 a^{2}+9 b^{2}+82 a b+10 a+10 b \geq 1 .$ Show that $$41 a^{2}+41 b^{2}+18 a b \geq 3-2 \sqrt{2}.$$
  4. Given a rhombus $A B C D$ with $\widehat{A B C}=60^{\circ} .$ The diagonals $A C$ and $B D$ intersect at $O$. A line $d$ through $D$ intersects the opposite rays of the rays $A B$, $C B$ respectively at $E$, $F$ $(E \neq A, F \neq C) .$ Let $M$ be the intersection between $C E$ and $A F,$ and then $H$ the intersection between $O M$ and $E F .$ Show that $A$, $C$, $D$, $H$ lie on some circle.
  5. Solve the equation $$\frac{2020 x^{4}+x^{4} \sqrt{x^{2}+2020}+x^{2}}{2019}=2020.$$
  6. Solve the system of equations $$\begin{cases}\sqrt{x^{2}+1}+x-8 y^{2}+8 \sqrt{2 y}-8 & = \dfrac{1}{x-\sqrt{x^{2}+1}}-8 y^{2}+8 \sqrt{2 y}-4 \\ 2 y^{2}-2 \sqrt{2 y}-\sqrt{x^{2}+1}+x+2 &=0\end{cases}$$
  7. Given the function $f(x)=x^{2}-x+1$. Let $$f_{1}(x)=f(x),\quad f_{n}(x)=f\left(f_{n-1}(x)\right),\,n=2,3,4, \ldots$$ Find the coefficient of $x^{2}$ in the polynomial expansion of $f_{2021}(x)$.
  8. Find the point $M$ inside a given tetrahedron $A B C D$ such that the product of the distances from $M$ to the faces of $A B C D$ is maximal.
  9. Does there exist a triangle $A B C$ with $\sin A=\cos B=\tan C ?$
  10. The sequence determined as follows $$a_{0}=\frac{1}{2},\quad a_{n+1}=\frac{4 n+5}{4 n+6} a_{n},\, \forall n \geq 0.$$ Let $\displaystyle b_{n}=\sum_{i=0}^{n} a_{i}$, $n \in \mathbb{N}$. Find $\displaystyle \lim _{n \rightarrow+\infty} \dfrac{b_{n}}{n}$.
  11. Given real numbers $a, b, c \in[1 ; 5]$ so that $a+b+c=9 .$ Find the minimum and maximum values of the expressions
    a) $P=a b c$.
    b) $F=a b+b c+c a$.
    c) $S_{\lambda}=a^{\lambda}+b^{\lambda}+c^{\lambda}$, where $\lambda$ is a constant $\lambda \in\{0\} \cup[1 ;+\infty)$.
  12. Given a triangle $A B C$ which is not equilateral and $G$ is its centroid. The lines $A G$, $B G$, $C G$ respectively intersect the circumcircles of $G B C$, $G C A$, $G A B$ at $D$, $E$, $F$. Show that the Euler lines of the triangles $D B C$, $E C A$, $F A B$ are concurrent.




Mathematics & Youth: 2020 Issue 519
2020 Issue 519
Mathematics & Youth
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