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2018 Issue 495

  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 $$3 \sqrt[3]{\frac{x^{2}-2 x+2}{2 x-1}}+2 x=5$$
  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 $a_{n}=n+[\sqrt[3]{n}]$, $n$ positive integers, $[\sqrt[3]{n}]$ is the integral part of $\sqrt[3]{n}$. 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 $X A=X K$.

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Mathematics & Youth: 2018 Issue 495
2018 Issue 495
Mathematics & Youth
https://www.molympiad.org/2020/09/2018-issue-495.html
https://www.molympiad.org/
https://www.molympiad.org/
https://www.molympiad.org/2020/09/2018-issue-495.html
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