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 $X A=X K$.




Mathematics & Youth: 2018 Issue 495
2018 Issue 495
Mathematics & Youth
Loaded All Posts Not found any posts VIEW ALL Readmore Reply Cancel reply Delete By Home PAGES POSTS View All RECOMMENDED FOR YOU LABEL ARCHIVE SEARCH ALL POSTS Not found any post match with your request Back Home Sunday Monday Tuesday Wednesday Thursday Friday Saturday Sun Mon Tue Wed Thu Fri Sat January February March April May June July August September October November December Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec just now 1 minute ago $$1$$ minutes ago 1 hour ago $$1$$ hours ago Yesterday $$1$$ days ago $$1$$ weeks ago more than 5 weeks ago Followers Follow THIS CONTENT IS PREMIUM Please share to unlock Copy All Code Select All Code All codes were copied to your clipboard Can not copy the codes / texts, please press [CTRL]+[C] (or CMD+C with Mac) to copy