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

## $show=home 1. Does it exist a natural number$n$so that the last digit of the sum$1+2+3+\ldots+n$is$2$,$4$,$7$or$9$?. 2. Given a right triangle$A B C$with the right angle$A$and$A B<A C$. Let$E$and$F$be the points on the sides$A C$and$B C$respectively such that$E F \perp B C$and$E F=F B$. Let$D$be the point on the side$A C$such that$A D=A B$. Prove that$E F D$is an isosceles triangle. 3. Find positive integral solutions of the equation $$1+5^{x}=2^{y}+5.2^{2}.$$ 4. Given an acute triangle$A B C$. Outside the triangle, draw two equilateral triangles$A B D$and$A C E$. On the line segments$A D$,$CE$,$CB$choose the points$M$,$N$,$F$respectively so that $$\frac{A M}{A D}=\frac{C N}{C E}=\frac{C F}{C B}=\frac{1}{3} .$$ Compare the lengths of two length segments$M N$and$E F$. 5. Given real numbers$x,y, z \geq 0$such that$\max \{x ; y ; z\} \geq 1$. Show that $$x^{3}+y^{3}+z^{3}+(x+y+z-1)^{2} \geq 1+3 x y z.$$ 6. Solve the system of equations $$\begin{cases}x^{3}+x+2 &=8 y^{3}-6 x y+2 y \\ \sqrt{x^{2}-2 y+2}+2 \sqrt{x^{3}(5-4 y)} &=2 y^{2}-x+2\end{cases}.$$ 7. Suppose that $$P(x)=x^{n}+x^{n-1}+a_{n-2} x^{n-2}+\ldots+a_{1} x+a_{0}$$ has$n$distinct real roots$x_{1}, x_{2}, \ldots, x_{n}$. Show that $$\frac{x_{1}^{n}}{P^{\prime}\left(x_{1}\right)}+\frac{x_{2}^{n}}{P^{\prime}\left(x_{2}\right)}+\ldots+\frac{x_{n}^{n}}{P^{\prime}\left(x_{n}\right)}=-1$$ where$P^{\prime}(x)$is the derivative of$P(x)$. 8. Suppose that the inscribed sphere of the tetrahedron$A_{1} A_{2} A_{3} A_{4}$is tangent to the face which is opposite to$A_{i}$at$B_{I}(i=1,2,3,4)$. Prove that if$B_{1} B_{2} B_{3} B_{4}$is almost-regular (opposite sides have the same length) if and only if$A_{1} A_{2} A_{3} A_{4}$is almost-regular. 9. Find the minimum and maximum values of the expression $$P=\frac{\left(2 x^{2}+5 x+5\right)^{2}}{(x+1)^{4}+1}$$ 10. Find all prime numbers$p$and positive integers$a, b$so that$p^{a}+p^{b}$is a perfect square. 11. Find all functions$f: \mathbb{R} \rightarrow \mathbb{R}$satisfying $$f((x+z)(y+z))=(f(x)+f(z))(f(y)+f(z)),\,\forall x, y, z \in \mathbb{R}.$$ 12. Given a triangle$A B C$and a point$M$on the side$B C$. The symmedians through$M$of the triangles$M A B$,$M A C$intersect the circles$(M A B)$,$(M A C)$respectively at$Q$,$R$which are different from$M$. Let$P$be the point on$B C$so that$AP \perp AM$. Denote by$l$the external common tangent, which closer to$A$, of two circles$(M A B)$,$(MAC)$. Suppose that$l$is parallel to$B C$. Show that$l$is tangent to$(P Q R)$. (The notion$(X Y Z)$is for the circumcircle of the triangle$X Y Z$). ##$type=three$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: 2019 Issue 509
2019 Issue 509
Mathematics & Youth
https://www.molympiad.org/2020/09/2019-issue-509_4.html
https://www.molympiad.org/
https://www.molympiad.org/
https://www.molympiad.org/2020/09/2019-issue-509_4.html
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