2021 Issue 523

  1. Find all prime numbers $p$, $q$ and positive integers $n$ so that $$p(p+1)+q(q+1)=n(n+1).$$
  2. Find all prime numbers $a, b, c, d$ satisfying $$\begin{cases}a+b^{2} &=c d \\ c+d^{2} &=17 b \end{cases}.$$
  3. Find all solutions which are prime numbers of the following equation $$x^{y}+y^{x}+(x+y+1)^{3}=x^{3}+y^{3}+z+1.$$
  4. Let $M$ be a point inside a square $A B C D$. The rays $A M$, $B M$, $C M$, $D M$ respectively intersect the circle circumscribing the square at $E$, $F$, $G$, $H$. The tangents of the circle at $F$ and $H$ meet at $K$ Show that three points $K$, $G$, $E$ are collinear.
  5. Given positive numbers $a, b, c$ satisfying $a+b+c=3$. Show that $$\frac{a^{3}+b^{2}+c^{2}}{a^{2}+1}+\frac{b^{3}+c^{2}+a^{2}}{b^{2}+1}+\frac{c^{3}+a^{2}+b^{2}}{c^{2}+1} \geq \frac{9}{2}.$$
  6. Solve the system of equations $$\begin{cases} x^{z}+y^{z}-2 &=z^{3}-z \\ y^{x}+z^{x}-2 &=x^{3}-x \\ z^{y}+x^{y}-2 &=y^{3}-y\end{cases}$$ where $x, y, z$ are integers.
  7. Find real solutions of the equation $$x 2^{x^{2}}=2^{2 x+1}.$$
  8. Given a triangle $A B C$ inscribed in a circle $(O)$. Let $M$ be a point on the arc $B C$ which does not contain $A$ ($M$ is different from $B$ and $C$). Draw $B E$ perpendicular to $A M$ ($E$ is on $A M$). Let $N$ be the intersection hetween $A M$ and $B C$, and $H$ the orthocenter of the triangle $C M N$. Show that the line $H E$ always passes through a fixed point when $M$ varies.
  9. Given $n$ positive numbers $x_{1}, x_{2}, \ldots, x_{n}$. Find the minimum value of the expression $$S=\frac{\left(x_{1}+x_{2}+\ldots+x_{n}\right)^{1+2+\ldots+n}}{x_{1} x_{2}^{2} \ldots x_{n}^{n}}.$$
  10. Find all functions $f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$ satisfying $$f(x f(y))+f\left(y^{2021} f(x)\right)=x y+x y^{2021},\, \forall x, y \in \mathbb{R}^{+}.$$
  11. Consider the sequence $\left\{a_{n}\right\}$ with $$a_{1}=1,\, a_{2}=\frac{1}{2},\quad n a_{n}=(n-1) a_{n-1}+(n-2) a_{n-2} .$$ Find $\displaystyle\lim_{n\to\infty} \frac{a_{n}}{a_{n-1}}$.
  12. Given a triangle $A B C$ and a point $M$ on the line passes through $B$, $C$ ($M$ is different from $B$, $C$). Let $K$, $L$ respectively be the second intersections between the circumcircles of the triangles $A M B$, $A M C$ and another line which passes through $M$ and different from $M A$ and $B C$. Let $P$, $Q$ respectively be the perpendicular projections of $A$ on $B K$, $C L$. $B K$ and $C L$ meet at $R$, $P O$ and $M K$ meet at $N$. Show that
    a) $\dfrac{N P}{N Q}=\dfrac{M B}{M C}$.
    b) If $M$ is the midpoint of $B C$ then $A K R L$ is a harmonic quadrilateral (a harmonic quadrilateral is a cyclic quadrilateral in which the products of the lengths of opposite sides are equal).




Mathematics & Youth: 2021 Issue 523
2021 Issue 523
Mathematics & Youth
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