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2017 Issue 478

  1. Find all the $4$-digit perfect squares such that when we reverse their digits we also get perfect squares.
  2. Given an isosceles triangle $ABC$ with the vertex angle $A$. On the half plane determined by $BC$ which does not contain $A$ choose a point $D$ such that $\widehat{BAD}=2\widehat{ADC}$ and $\widehat{CAD}=2\widehat{ADB}$. Prove that $CBD$ is an isosceles triangle with the vertex angle $D$.
  3. Prove thta $3^{n+2}|10^{3^{n}}-1$ for any natural number $n$.
  4. Given a trapezoid $ABCD$ ($AB\parallel CD$) with $AB<CD$. Let $P$ and $Q$ respectively be on the diagonals $AC$ and $BD$ such that $PQ$ is not parallel to $AB$. The ray $QP$ intersects $BC$ at $M$ and the ray $PQ$ intersects $AD$ at $N$. Let $O$ be the intersect of $AC$ and $BD$. Suppose futhermore that $MP=PQ=QN$. Prove that \[ \frac{OP}{OA}+\frac{OQ}{OB}=1.\]
  5. Let $a,b$ and $c$ be positive numbers such that $\sqrt{a}+\sqrt{b}+\sqrt{c}=1$. Find the maximum value of the expression \[ P=\sqrt{abc}\left(\frac{1}{\sqrt{(a+b)(a+c)}}+\frac{1}{\sqrt{(b+c)(b+a)}}+\frac{1}{\sqrt{(c+b)(c+a)}}\right).\]
  6. Solve the equation $f(f(x))=x$ in terms of the parameter $m$ where \[ f(x)=x^{2}+2x+m. \]
  7. Suppose that $a,b$ and $c$ are the lengths of three sides of a triangle. Let \[ F(a,b,c)=a^{3}+b^{3}+c^{3}+3abc-ab(a+b)-bc(b+c)-ca(c+a). \] Prove that \[F(a,b,c)\leq\min\left\{ F(a+b,b+c,c+a),4a^{2}b^{2}c^{2}F\left(\frac{1}{a},\frac{1}{b},\frac{1}{c}\right)\right\}. \]
  8. Given an acute triangle $ABC$. Let $a,b$ and $c$ respectively be the lengths of $BC$, $CA$ and $AB$. Let $R$ and $r$ respectively be the circumradius and the inradius of $ABC$. Prove that \[\frac{r}{2R}\leq\frac{abc}{\sqrt{2(a^{2}+b^{2})(b^{2}+c^{2})(c^{2}+a^{2})}}.\]
  9. Suppose that $a$ and $b$ are real numbers such that $0<a\ne1$ and that the equation $a^{x}-\dfrac{1}{a^{x}}=2\cos(bx)$ has exactly $2017$ real roots and they are all different real numbers. How many distinct real roots does the equation \[a^{x}+\frac{1}{a^{x}}=2\cos(bx)+4\] have?.
  10. In a country, the length of any direct road between two cities (if any) is smaller than $100$ km and we can travel from a city to any other one by roads which have total length is smaller than $100$ km. When a road is closed under construction, we still can travel from one city to another by other roads. Prove that we can choose a route which has the total length is smaller than $300$ km.
  11. Prove that $\gcd(1,2,\ldots,2n)$ is divisible by $C_{2n}^{n}$ for any positive interger $n$.
  12. Given a triangle $ABC$ with $AB<AC$. Let $M$ be the midpoint of $BC$. Let $H$ be the perpendicular projection of $B$ on $AM$. Suppose that $Q$ is the point of the opposite ray of $AM$ such that $AQ=4MH$. Assume that $AC$ intersects $BQ$ at $D$. Prove that the circumcenter of $ADQ$ lies on the circumcircle of $DBC$.

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Mathematics & Youth: 2017 Issue 478
2017 Issue 478
Mathematics & Youth
https://www.molympiad.org/2017/06/mathematics-and-youth-magazine-problems_2.html
https://www.molympiad.org/
https://www.molympiad.org/
https://www.molympiad.org/2017/06/mathematics-and-youth-magazine-problems_2.html
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