# TÀI LIỆU TOÁN - WWW.MOLYMPIAD.NET - ĐỀ THI TOÁN

## $show=home 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 $$2(a+b+c+d)-4 \geq \sqrt{\frac{a^{3}+b^{3}}{2}}+\sqrt{\frac{b^{3}+c^{3}}{2}}+\sqrt{\frac{c^{3}+d^{3}}{2}}+\sqrt{\frac{d^{3}+a^{3}}{2}}$$ 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. ##$type=three$c=3$source=random$title=oot$p=1$h=1$m=hide$rm=hide ## Anniversary_$type=three$c=12$title=oot$h=1$m=hide\$rm=hide

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Mathematics & Youth: 2020 Issue 516
2020 Issue 516
Mathematics & Youth