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2018 Issue 492

  1. Let $$M=\frac{1}{10}+\frac{1}{20}+\frac{1}{35}+\frac{1}{56}+\frac{1}{84}+\frac{1}{120}+\ldots$$ a) Is the fraction $\dfrac{1}{15400}$ a term of $M$ ? Why?.
    b) Compute the sum of the $8$ first terms of $M$. 
  2. Given a triangle $A B C$ with $A B < A C$. The angle bisector of $\widehat{B A C}$ intersects the perpendicular bisector of $B C$ at $M$. Let $H$, $K$, and $I$ respectively be the perpendicular projections of $M$ on $A B$, $A C$ and $B C$. Prove that $H$, $I$, $K$ is collinear. 
  3. Find integer solutions of the equation $$x^{3}=4 y^{3}+x^{2} y+y+13$$
  4. Given a quadrilateral $A B C D$ inscribed in the circle with the diameter $A C$. Let $M$ be the point on $A B$ such that $A M=A D$. The lines $D M$ and $B C$ intersects at $N,$ and the $A N$ intersects the circle at $K$. Let $H$ be the perpendicular projection of $D$ on $A C$. Assume that $A B$ intersects $N H$ and $C K$ at $P$ and $Q .$ Show that $$\frac{1}{M P}=\frac{1}{M A}+\frac{1}{M Q}.$$
  5. Solve the system of equations $$\begin{cases} 3 x^{3}+6 x+2 &=2 y^{3} \\ 3 y^{3}+6 y+2 &=2 z^{3} \\ 3 z^{3}+6 z+2 &=2 x^{3}\end{cases}$$
  6. Solve the inequality $$x^{3}+6 x^{2}+9 x \leq \sqrt{x+4}-2$$
  7. Let $a, b, c$ be positive numbers with the product is equal to $1$. Find the minimum value of the expression $$P=\frac{1}{a^{2017}+a^{2015}+1}+\frac{1}{b^{2017}+b^{2015}+1}+\frac{1}{c^{2017}+c^{2015}+1}$$
  8. Given a triangle $A B C$ inscribed in the circle $(O)$. The tangent lines of $(O)$ at $B$ and $C$ intersect at $P .$ The line which goes through $A$ and is parallel to $B P$ intersects $B C$ at $M$. The line which goes through $A$ and is parallel to $B C$ intersects $B P$ at $N$. Suppose that $I$ is the intersections between $A P$ and $M N$. Prove that four points $B$, $I$, $O$, $C$ lie on a circle.
  9. Given the equation $x^{3}+m x^{2}+n=0$. Find $m$, $n$ so the the equation has three distinct non-zero real roots $u$, $v$, $t$ satisfying $$\frac{u^{4}}{u^{3}-2 n}+\frac{v^{4}}{v^{3}-2 n}+\frac{t^{4}}{t^{3}-2 n}=3$$
  10. Let $p$ be an odd prime and $a_{1}, a_{2}, \ldots, a_{p}$ is an arithmetic progression with the common difference $d$ which is not divisible by $p$. Prove that $\prod_{i=1}^{p}\left(a_{i}+a_{1} a_{2} \ldots a_{p}\right): p^{2}$
  11. Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $$f(x+f(y))=f\left(x+y^{2018}\right)+f\left(y^{2018}-f(y)\right),\, \forall x, y \in \mathbb{R}.$$
  12. Given an acute triangle $A B C$ with the altitudes $B E$, $C F$. Let $S T$ be a chord of the circumcircle of $A E F$. Two circles which go through $S$, $T$ is tangent to $B C$ respectively at $P$ and $Q$. Prove that the intersection between $P E$, $Q F$ lies on the circumcircle of $A E F$.

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Mathematics & Youth: 2018 Issue 492
2018 Issue 492
Mathematics & Youth
https://www.molympiad.org/2020/09/2018-issue-492.html
https://www.molympiad.org/
https://www.molympiad.org/
https://www.molympiad.org/2020/09/2018-issue-492.html
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