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2015 Issue 451

  1. Suppose that $a,b$ and $c$ are positive integers such that \[\frac{1}{a}+\frac{1}{b}+\frac{1}{c}<1.\] Find the maximum value of the expression \[S=\frac{1}{a}+\frac{1}{b}+\frac{1}{c}.\]
  2. Assume that the triangle $ABC$ is isosceles at $A$. Draw the height $AH$. Let $D$ be the midpoint of $AH$. Choose $E$ on $CD$ such that $HE$ is perpendicular to $CD$. Prove that $\widehat{AEB}=90^{0}$.
  3. Solve the system of equations \[\begin{cases} \dfrac{x^{2}-1}{y}+\dfrac{y^{2}-1}{x} & =3\\ x^{2}-y^{2}+\dfrac{12}{x} & =9 \end{cases}.\]
  4. Two circles $(O,R)$ and $(O',R')$ intersect at $A$ and $B$. A point $M$ varies arbitrary on the opposite ray of the ray $AB$. From $M$, draw two tangent lines to the circle $(O',R')$ at $C$ and $D$ with $D$ is inside the circle $(O,R)$. The lines $AD$ and $AC$ intersect $(O,R)$ respectively at $P$ and $Q$ ($P$, $Q$ are different from $A$). Prove that the line $PQ$ always goes through a fixed point when $M$ varies. 
  5. Find the minimum value of the expression \[A=\frac{2}{3xy}+\sqrt{\frac{3}{y+1}}\] where $x,y$ are positive real numbers satisfying $x+y\leq3$.
  6. Consider the quadratic equation $ax^{2}+bx+c=0$ ($a\ne0$), where $a,b$ and $c$ are real numbers satisfying $145a+144b+144c=0$. Prove that this equation cannot have two solutions which are inverses of two nonzero consecutive perfect squares.
  7. Given a tetrahedron $ABCD$ inscribed in a sphere $(S)$. Let $A_{1},B_{1},C_{1},D_{1}$ resprectively the centroids of the triangles $BCD$, $ACD$, $ABD$, $ABC$. The lines $AA_{1}$, $BB_{1}$, $CC_{1}$, $DD_{1}$ intersect $(S)$ respectively at $A_{2},B_{2},C_{2},D_{2}$. Find the minimum value of the expression \[P=\frac{AA_{2}^{2}+BB_{2}^{2}+CC_{2}^{2}+DD_{2}^{2}}{AB\cdot CD+AC\cdot BD+AD\cdot BC}.\]
  8. Let $a,b,c$ be three positive real numbers. Prove that \[\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\geq\frac{a^{2}+nb^{2}}{a^{3}+nb^{3}}+\frac{b^{2}+nc^{2}}{b^{3}+nc^{3}}+\frac{c^{2}+na^{2}}{c^{3}+na^{3}}\] for all $n\in\mathbb{N}$, $n\geq2$.
  9. Find all quadruples of posotive integers $(x,y,z,p)$, $p$ is a prime, such that \[p^{x}+(p-1)^{2y}=(2p-1)^{z}.\]
  10. Find the maximal $K$ such that the following inequality \[\sqrt{a+K|b-c|^{\alpha}}+\sqrt{b+K|c-a|^{\alpha}}+\sqrt{c+K|a-b|^{\alpha}}\leq2\] always holds for all $\alpha\geq1$ and $a,b,c$ are arbitrary nonnegative real numbers satisfying $a+b+c=1$.
  11. Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that for all $x,y\in\mathbb{R}$ \[f(x^{2}+f(y))=4y+\frac{1}{2}f^{2}(x).\]
  12. Given a circle $(O)$, $ABC$ is an acute triangle inscribed in $(O)$. The tangent line to $(O)$ at $A$ intersects $BC$ at $S$; $K$ is the perpendicular projection of $A$ on $OS$; $BK$ and $CK$ intersect $CA$, $AB$ respectively at $E$ and $F$. Prove that the line which contains $A$ and is perpendicular to $EF$ always goes through a fixed point when $A$ varies on $(O)$ ($B$ and $C$ are fixed).

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Mathematics & Youth: 2015 Issue 451
2015 Issue 451
Mathematics & Youth
https://www.molympiad.org/2017/07/mathematics-and-youth-magazine-problems_37.html
https://www.molympiad.org/
https://www.molympiad.org/
https://www.molympiad.org/2017/07/mathematics-and-youth-magazine-problems_37.html
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