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

## $show=home 1. Show that it is impossible to write$2^{9^{2018}}$as a sum of$n$consecutive positive integers for any$n \in \mathbb{N}$,$n \geq 2$. 2. Find the last twelve digits of the number$5^{1040}$. 3. Find integral solutions of the following systems of equations a)$\begin{cases}3 a^{2}+2 a b+3 b^{2}&=12 \\ a^{2}+b^{2}&=c^{2}\end{cases}.$b)$\begin{cases}(z-3)\left(x^{2}+y^{2}\right)-2 x y &=0 \\ x+y &=z\end{cases}.$4. Given a right triangle$A B C$with the right angle$A$. In the angle$\widehat{B A C}$draw the rays$A x$,$A y$such that$\widehat{C A x}=\dfrac{1}{2} \widehat{A B C}$,$\widehat{B A y}=\dfrac{1}{2} \widehat{A C B}$. The ray$A x$intersects the angle bisector of$\widehat{A C B}$at$Q$, the ray$A y$intersects$B C$at$K$. Compute the ratio$\dfrac{A K}{A Q}$. 5. Solve the equation $$3 \sqrt{\frac{x^{2}-2 x+2}{2 x-1}}+2 x=5$$ 6. Given real numbers$x$,$y$,$z$such that$x y z=1$. Show that $$\left(\frac{x}{1+x y}\right)^{2}+\left(\frac{y}{1+z y}\right)^{2}+\left(\frac{z}{1+x z}\right)^{2} \geq \frac{3}{4}$$ 7. Suppose that the polynomial$P(x)=x^{2018}-a x^{2016}+a$($a$is a real parameter) has$2018$real solutions. Show that there exists$\left|x_{0}\right| \leq \sqrt{2}$8. Given a pyramid$S . A B C D$with the base$A B C D$is a parallelogram, and$S A=S B=S C=a$,$A B=2 a$,$B C=3 a$. Let$S D=x(0<x<a \sqrt{14})$. Find$x$in terms of$a$so that the product$A C \cdot S D$obtains its maximum value. 9. Suppose that$x, y, z$are nonnegative numbers satisfying$x+y+z=1$. Find the maximum value of the expression $$P=x y+y z+z x+\frac{2}{9}(M-m)^{3}$$ where$M=\max \{x, y, z\}$and$m=\min \{x, y, z\}$10. Consider the sequence$a_{n}=n+[\sqrt{n}]$,$n$positive integers,$[\sqrt{n}]$is the integral part of$\sqrt{n}$. Suppose there exists a positive integer$k$such that the terms$a_{k} ; a_{k+1} ; \ldots ; a_{k+p}$are$p+1$consecutive natural numbers with$p=6015 \times 2006+1 .$Show that $$k>8.10^{9}+6.10^{7}.$$ 11. Find the least number$k$so that in any subset of$k$elements of$\{1,2, \ldots, 25\}$we can always find at least a Pythagorean triple. 12. Let$A B C$be a triangle inscribed in a circle ($O$). Suppose that$A H$is the altitude and the line$A O$intersects$B C$at$D .$Let$K$be the second intersection of the circumcircle of$A D C$and the circumcircle of$A H B$. Suppose that the circumcircle of$K H D$intersects$(O)$at$M$and$N .$Let$X$be the intersection of$M N$and$B C .$Show that$X A=X K$. ##$type=three$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: 2018 Issue 495
2018 Issue 495
Mathematics & Youth