2017 Issue 479

  1. Place the numbers from 1 to 20 on a circle such that the sum of any two numbers which are next to each other is a prime number.
  2. Given a right triangle $ABC$ with the right angle $A$ and $\widehat{ABC}<45^{0}$. On the haft plane determined by $AB$, containing $C$, choose two points $D$ and $E$ such that $BD=BA$, $\widehat{DBA}=90^{0}$, $\widehat{EBC}=\widehat{CBA}$, and $ED$ is perpendicular to $BD$. Prove that $BE=AC+DE$.
  3. Given two real numbers $x$ and $y$ such that $x^{2}+2xy+2y^{2}=1$. Find the maximum and minimum values of the expression \[P=x^{4}+y^{4}+(x+y)^{4}.\]
  4. Given a triangle $ABC$ with $AB<AC$. Let $(O)$ be the incircle of $ABC$. The side $BC$ is tangent to $(O)$ at $D$. Choose $I$ on $AD$ such that $OI$ is perpendicular to $AD$. The ray $IO$ intersect the perpendicular bisector of $BC$ at $K$. Prove that $BIKC$ is an inscribed quadrilateral.
  5. Solve the equation \[\frac{\sqrt{x^{2}+28x+4}}{x+2}+8=\frac{x+4}{\sqrt{x-1}}+2x.\]
  6. Given non-negative numbers $a,b$ and $c$ such that $ab+bc+ca=1$. Prove that \[\sqrt{3}\leq\sqrt{1+a^{2}}+\sqrt{1+b^{2}}+\sqrt{1+c^{2}}-a-b-c\leq2.\]
  7. Solve the system of equations \[\begin{cases} x^{8} & =21y+13\\ \frac{(x+y)^{25}}{2^{18}} & =(x^{3}+y^{3})^{3}(x^{4}+y^{4})^{4}\end{cases}.\]
  8. Given a triangle $ABC$ and $P$ is a point lying inside $ABC$. Let $E$ (resp. $AC$ and $AB$). Let $K$ be the perpendicular projection of $K$ on $BC$. On the side $AF$, choose an arbitrary point $M$. Assume that $N$ is the intersection between $MK$ and $PE$. Prove that $\widehat{KHM}=\widehat{KHN}$.
  9. Given a bijection $f:\mathbb{N}^{*}\to\mathbb{N}^{*}$. Prove that there exist positive intergers $a,b,c$ and $d$ such that $a<b<c<d$ and $f(x)+f(d)=f(b)+f(c)$.
  10. Given a sequence $(u_{n})$ whose terms are positive integers satisfying \[0\leq u_{m+n}-u_{m}-u_{n}\leq2\quad\forall m,n\geq1.\] Prove that there exist two positive numbers $a_{1},a_{2}$ such that \[[a_{1}n]+[a_{n}n]-1\leq u_{n}\leq[a_{1}n]+[a_{2}n]+1\] for all $n\leq2017$. ($[x]$ is the greatest integer which does not exceed $x$).
  11. Find all continuous functions $f:(0,+\infty)\to(0,\infty)$ such that \[f(x+2016)=f\left(\frac{x+2017}{x+2018}\right)+\frac{x^{2}+4033x+4066271}{x+2018},\forall x>0.\]
  12. Given a circle $(O)$ and two fiexd points $B,C$ on $(O)$. A point $A$ is moving on $(O)$ such that $ABC$ is always an acute triangle and $AB<AC$. Choose $D$ on the side $AC$ such that $AB=AD$. The line $BD$ intersects $(O)$ at $E$ which is different from $B$. The perpendicular projection of $E$ (resp. $D$) on $AC$ (resp. $AE$) is $H$ (resp. $M$). The circumcircle of $AMH$ intersects $(O)$ at $K$. Let $N$ be perpendicular projection of $K$ on $AB$.
    a) Prove that $MN$ goes through the midpoint $I$ of $BD$.
    b) The circumcircles of $BCD$ and $AMH$ intersect each other at $P$ and $Q$. Prove that the midpoint of $PQ$ always lies on a fixed line when $A$ is moving on $(O)$ in the given way.




Mathematics & Youth: 2017 Issue 479
2017 Issue 479
Mathematics & Youth
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