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

## $show=home 1. Find the smallest positive integer$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 $$P=\sqrt{x}+\sqrt{y}+\sqrt{12} \sqrt{x \sqrt{1-y^{2}}+y \sqrt{1-x^{2}}}$$ 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$are concurrent. ##$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 521
2020 Issue 521
Mathematics & Youth