2020 Issue 521

  1. Find the smallest positive integer $n$ so that for cach set of $n$ numbers chosen from $1 ; 2 ; 3 ; \ldots ; 100,$ there always exist two numbers $a, b$ so that $a+b$ is a prime number.
  2. Given an odd prime number $p$. Find all pairs of positive integers $(x ; y)$ so that both $x+y$ and $x y+1$ are powers of $p$
  3. Find all integers $x, y$ satisfying $$x^{3}(3 y+1)+y^{2}(3 x+1)+(x+y)\left(x^{2}-x y+y^{2}+1\right)+2 x y=343$$
  4. Two circles $(O)$ and $\left(O^{\prime}\right)$ intersect at $A$ and $B$. An exterior common tangent touchs $(O)$ and $\left(O^{'}\right)$ at $C$ and $D .$ Show that $$\frac{A C}{A D}+\frac{B D}{B C} \geq 2$$
  5. Solve the system of equations $$\begin{cases} 3 x^{2} y-x y-y&=1 \\-x y^{2}-y+y^{2} &=3\end{cases}.$$
  6. Given real numbers $x, y \in(0 ; 1)$. Find the maximum value of the expression $$P=\sqrt{x}+\sqrt{y}+\sqrt[4]{12} \sqrt{x \sqrt{1-y^{2}}+y \sqrt{1-x^{2}}}$$
  7. Find the real roots of the equation $$(x+1)\left(x^{2}+1\right)\left(x^{3}+1\right)=30 x^{3}$$
  8. Given a triangle $A B C$ with the centroid $G$, the nine-point center $E$ and the circumradius $R$. Sbow that
    a) $E A+E B+E C \leq 3 R ;$
    b) $E A^{2}+E B^{2}+E C^{4} \geq G A^{2}+G B^{2}+G C^{2}$.
  9. Solve the following trigonometric equation $$\sqrt{4^{n} \cos ^{4 n} x+3}+\sqrt{4^{n} \sin ^{4n} x+3}=4$$ where $n$ is an arbitrary natural number.
  10. Let $a, b, c$ be positive integers. Show that there exists an natural number $k$ so that the three integers $a^{t}+b c$, $b^{4}+c a$, $c^{2}+a b$ have at least one common prime divisor.
  11. Find all functions $f: Z \rightarrow Z$ satisfying $$f\left(f(x)+y f\left(x^{2}\right)\right)=x+x^{2} f(y)$$ for all $x, y \in \mathbb{Z}$
  12. Given a triangle $A B C$ and $M$ is an arbitrary point on the side $B C$. The incircle $(I)$ of the triangle $A B M$ is tangent to the sides $B M$, $M A$, $A B$ respectively at $D$, $E$, $F,$ The incircle $(J)$ of the triangle $A C M$ is tangent to the sides $C M$, $M A$, $A C$ respectively at $X$, $Y$, $Z$ Let $H$ be the intersection between $D F$ and $X Z$. Show that the lines $A H$, $D E$, $X Y$ are concurrent.




Mathematics & Youth: 2020 Issue 521
2020 Issue 521
Mathematics & Youth
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