2021 Issue 527

  1. Find natural numbers $a$ and $b$ given that the sum of four numbers $a+b$, $a-b$, $ab$, $a: b$ is equal to $1575$.
  2. Given triangle $A B C$ with $\widehat{A}=120^{\circ}$, $\widehat{B}=40^{\circ}$. On the side $A C$ choose the point $M$ such that $A B=A M$. On the opposite ray of $A B$ choose the point $N$ such that $\widehat{A M N}=40^{\circ}$. Find the measurement of the angle $\widehat{B N C}$.
  3. Find all pairs of integers $(x ; y)$ satisfying $0 \leq x+y \leq 6$ and $$x-\frac{1}{x^{3}}=y-\frac{1}{y^{3}}.$$
  4. Given an acute triangle $A B C$ inscribed in a circle $(O ; R)$. The altitudes $A D$, $B E$, $C F$ meet at $H$. Let $M$ be the midpoint of $A H$. Draw $M N$ perpendicular to $B M$, $N$ is on $A C$. Show that $O N || B C$ and $E M=R \cos A$.
  5. Solve the equation $$x^{2} \sqrt[4]{2-x^{4}}-x^{4}+x^{3}-1=0.$$
  6. Given real numbers $x, y, z$ such that $x^{2}+y^{2}+z^{2}=3$. Prove that $$8(2-x)(2-y)(2-z) \geq(x+y z)(y+x z)(z+x y)$$
  7. Find all triangles $A B C$ such that the lengths of all sides are positive integers and the length of $A C$ is equal to the length of the interior angle bisector of the angle $A$.
  8. Given an acute triangle $A B C$ inscribed in a circle $(O)$. Let $I_{a}$, $I_{b}$, $I_{c}$  respectively be the centers of the excircles of the angles $A$, $B$, $C$. The line $A I_{a}$ intersects $(O)$ at $D$ which is different from $A$. On $I_{b} D$, $I_{c} D$ respectively choose the points $E$, $F$ such that $\widehat{A B C}=2 \widehat{I_{a} B E}$, $\widehat{A C B}=2 \widehat{I_{a} C F}$, $E$, $F$ are inside the trangle $I_{a} B C$. Show that $E F$ intersects $I_{b} I_{c}$ at some point on $(O)$.
  9. Given a function $$f(x)=\frac{x^{2}+a x+b}{x^{2}+1}$$ with $a, b$ are integers. Suppose the range of $f(x)$ is the set of $11$ integers. Find the maximum and minimum values of the expression $M=a^{2}+b^{2}$.
  10. For each positive integer $n$, let $f(n)=\left(n^{2}+n+1\right)^{2}+1$. Find the smallest positive integer $k$ such that$$f(n) \cdot f(n+1) \ldots f(n+k-1)$$ is a perfect square for some positive integer $n$.
  11. Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying $$f(f(x)+y)=f(f(x))-y f(x)+f(-y)-2,\, \forall x, y \in \mathbb{R}.$$
  12. Given an isosceles triangle $A B C$ with the vertex angle $A$ inscribed in a circle $(O)$. Suppose that $A D$ is a diameter of $(O)$. The points $E$, $F$ respectively on $D C$, $D B$. Let $G$ be on $E F$ such that $\dfrac{G F}{G E}=\dfrac{F B}{C E}$. Show that $C G$ and $A F$ meets each other at some point on $(O)$.




Mathematics & Youth: 2021 Issue 527
2021 Issue 527
Mathematics & Youth
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