2018 Issue 488

  1. Give a pentagon $ABCDE$. Assume that $BC$ is parallel to $AD$, $CD$ is parallel to $BE$, $DE$ is parallel to $AC$, and $AE$ is parallel to $BD$. Show that $AB$ is parallel to $CE$. 
  2. Prove that \[\cfrac{1+\frac{1}{3}+\frac{1}{5}+\ldots+\frac{1}{4035}}{1+\frac{1}{2}+\frac{1}{3}+\ldots+\frac{1}{2018}}>\frac{2019}{4036}.\]
  3. Given two triples $(a,b,c)$; $(x,y,z)$ none of them contains all $0$'s, such that \[a+b+c=x+y+z=ax+by+cz=0.\] Prove that the expression $P=\dfrac{(b+c)^{2}}{ab+bc+ca}+\dfrac{(y+z)}{xy+yz+zx}^{2}$ is a constant.
  4. Outside a triangle $ABC$, draw triangles $ABD$, $BCE$, $CAF$ such that $\widehat{ADB}=\widehat{BEC}=\widehat{CFA}=90^{0}$, $\widehat{ABD}=\widehat{CBE}=\widehat{CAF}=\alpha$. Prove that $DF=AE$.
  5. Show that the following sum is a positive integer \[\begin{align} S=&1+\frac{1}{2}+\frac{1}{3}+\ldots+\frac{1}{2017}+\left(1+\frac{1}{2}+\frac{1}{3}+\ldots+\frac{1}{2017}\right)^{2}+\\&+\left(\frac{1}{2}+\frac{1}{3}+\ldots+\frac{1}{2017}\right)^{2}+\ldots+\left(\frac{1}{2017}\right)^{2}.\end{align}\]
  6. Solve the system of equations \[\begin{cases}2x^{5}-2x^{3}y-x^{2}y+10x^{3}+y^{2}-5y & =0\\ (x+1)\sqrt{y-5}-y+3x^{2}-x+2 & =0\end{cases}.\]
  7. Prove that $x_{0}=\cos\dfrac{\pi}{21}+\cos\dfrac{8\pi}{21}+\cos\dfrac{10\pi}{21}$ is a solution of the equation \[4x^{3}+2x^{2}-7x-5=0.\]
  8. In any triangle $ABC$, show that \[\cos(A-B)+\cos(B-C)+\cos(C-A)\leq\frac{1}{2}\left(\dfrac{a+b}{c}+\frac{b+c}{a}+\frac{c+a}{b}\right).\]
  9. Given $6$ positive numbers $a,b,c$, $x,y,z$ and assume that $x+y+z=1$. Show that \[ax+by+cz\geq a^{x}b^{y}c^{z}.\]
  10. Given an infinite sequence of positive integers $a_{1}<a_{2}<\ldots a_{n}<\ldots$ such that $a_{i+1}-a_{i}\geq8$ for all $i=1,2,3,\ldots$. For each $n$, let $s_{n}=a_{1}+a_{2}+\ldots a_{n}$. Show that for each $n$, there are at lease two square numbers inside the half-open interval $[s_{n},s_{n+1})$.
  11. Given two positive sequences $(a_{n})_{n\geq0}$ and $(b_{n})_{n\geq0}$ which are determined as follows \[a_{0}=\sqrt{3},\,b_{0}=2,\quad a_{n}^{2}+1=b_{n}^{2},\,a_{n}+b_{n}=\frac{1+a_{n+1}}{1-a_{n+1}},\,\forall n\in\mathbb{N}.\] Show that they converges and find the limits.
  12. Given a triangle $ABC$ with $AB\ne AC$. A circle $(O)$ passing through $B$, $C$ intersects the line segments $AB$ and $AC$ at $M$ and $N$ respectively. Let $P$ be the intersection of $BN$ and $CM$. Let $Q$ be the midpoint of the arc $BC$ which does not contian $M$, $N$. Let $K$ be the incenter of $PBC$. Show that $KQ$ always goes through a fixed point when $(O)$ varies.




Mathematics & Youth: 2018 Issue 488
2018 Issue 488
Mathematics & Youth
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