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2020 Issue 516

  1. Let $$A=\frac{1}{1 ! 3}+\frac{1}{2 ! 3}+\frac{1}{3 ! 5}+\ldots+\frac{1}{(n-2) ! n}$$ for $n \in \mathbb{N}, n \geq 3,$ where $n !=1.2 .3 . . n .$ Show that $A<\dfrac{1}{2}$.
  2. For any natural number $n$ which is divisible by $4,$ show that the number $a=27.7^{n}+2021$ cannot be a product of $m$ consecutive natural numbers $(m \in \mathbb{N}, m \geq 2)$
  3. Given positive numbers $a, b, c, d$ satisfying $\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}+\dfrac{1}{d}=4$. Show that $$2(a+b+c+d)-4 \geq \sqrt[3]{\frac{a^{3}+b^{3}}{2}}+\sqrt[3]{\frac{b^{3}+c^{3}}{2}}+\sqrt[3]{\frac{c^{3}+d^{3}}{2}}+\sqrt[3]{\frac{d^{3}+a^{3}}{2}}$$
  4. Given an acute triangle $\triangle A B C$. Outside the triangle, draw the equilateral triangle $\Delta A C E$ and the isosceles triangle $\Delta A B D$ with $\widehat{A B D}=120^{\circ}$. Let $I$ be the midpoint of $D E$ and $F(F \neq E)$ the other intersection between $D E$ and the circumcircle of $\Delta A C E .$ Show that $B$, $C$, $I$, $F$ lie on a same circle.
  5. Let $f(x)=x^{2}+b x+c .$ Show that if the equation $f(x)=x$ has two distinct roots and $b^{2}-2 b-3 \geq 4 c$ then the equation $f[f(x)]=x$ has four distinct roots.
  6. Let $f(x)=3 x^{2}+8 x+4 .$ Find the coefficient of $x^{4}$ in the polynomial $g(x)=f(f(f(x))))$
  7. Given positive numbers $a, b, c .$ Prove that $$ \frac{5 a+c}{b+c}+\frac{6 b}{c+a}+\frac{5 c+a}{a+b} \geq 9.$$
  8. A triangle $ABC$ inscribed in a circle O. Let $G$ be the centroid of the triangle. The medians $A A_{1}, B B_{1}, C C_{1}$ of the triangle respectively intersect (O) at $A_{2}, B_{2}, C_{2}$. Show that $$\frac{A_{1} A_{2}}{G A_{1}}+\frac{B_{1} B_{2}}{G B_{1}}+\frac{C_{1} C_{2}}{G C_{1}} \geq 3.$$
  9. Solve the system of equations $$\begin{cases}\sqrt{x^{2}+x+1}-\sqrt{y^{2}-y+1} &=\sqrt{x^{2}+y^{2}-\dfrac{1}{2}} \\ 2 x^{3} y-x^{2} &=\sqrt{x^{4}+x^{2}}-2 x^{3} y \sqrt{4 y^{2}+1}\end{cases}.$$
  10. Find all the prime numbers $p$ and $q$ so that $p^{2}+1 \mid 3^{4}+1$ and $q^{2}+1 \mid 3^{\prime}+1$
  11. Find all continuous functions $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying $$f(4 x y)=f\left(2 x^{2}+2 y^{2}\right)+4(x-y)^{2},\, \forall x, y \in \mathbb R$$
  12. Suppose that $A B C D$ is a parallelogram with the angle $\widehat{B A D}$ is acute. $A$ moving line $d$, which always passes through $B$, intersects $C D$ at $M .$ The line $A M$ intersects $B C$ at $N$. Two lines $B M, D N$ respectively intersect the circumcircle of the triangle $C M N$ at $K$, $L$ (besides $M$, $N$). Choose $P$ and $Q$ so that $P N=P K$,  $P N \perp A D$, $Q M=Q L$ and $Q M \perp A B$. Show that $P Q$ always passes through a fixed point when $d$ varies.

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Mathematics & Youth: 2020 Issue 516
2020 Issue 516
Mathematics & Youth
https://www.molympiad.org/2020/09/2020-issue-516.html
https://www.molympiad.org/
https://www.molympiad.org/
https://www.molympiad.org/2020/09/2020-issue-516.html
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