2016 Issue 468

  1. Find positive integers $x,y,z$ such that $2^{x}+3^{y}+5^{z}=136$.
  2. Given a right triangle $ABC$ with the right angle $A$. Let $AH$ be the altitude. Choose a point $D$ on the opposite ray of the ray $AH$ such that $AD=BC$ and choose $E$ on the opposite ray of the ray $CA$ such that $AB=CE$. The line which goes through $A$ and is perpendicular to $BD$ intersects $BD$ and $DE$ repectively at $I$ and $K$. Find the angle $\widehat{CKE}$.
  3. Find all pairs of primes $\left(p,q\right)$ such that $p^{2}-pq-q^{3}=27$.
  4. Given a rhombus $ABCD$ with $\widehat{BAD}=120^{0}$. The points $M$ and $N$ respectively vary on the sides $BC$ and $CD$ such that $\widehat{MAN}=30^{0}$. Find the locus of the circumcenter $O$ of the triangle $AMN$.
  5. Solve the system of equations $$\begin{cases}\left(x+y\right)^{2}+\sqrt{3\left(x+y\right)} & =\sqrt{2\left(x+y+1\right)}+4\\ \left(x^{2}+y-2\right)\sqrt{2x+1} & =x^{3}+2y-5.\end{cases}$$
  6. Solve the equation $$-3x^{2}+x+3+\left(\sqrt{3x+2}-4\right)\sqrt{3x-2x^{2}}+\left(x-1\right)\sqrt{3x+2}=0.$$
  7. Let $ABC$ be an acute triangle. Prove that $$\tan^{2}A+\tan^{2}B+\tan^{2}C>4\left(\cot^{2}A+\cot^{2}B+\cot^{2}C\right). $$
  8. Let $ABCD$be a tetrahedron inscribed in a unit sphere $O$. Assume futher more that $O$ lies inside the $ABCD$. Prove that at least one face of $ABCD$ has perimeter greater than $2\sqrt{3}$.
  9. Find the maximum and minimum values of the expression $$A=\frac{x^{4}+y^{4}+2xy+9}{x^{4}+y^{4}+3},\quad\forall x,y\in\mathbb{R}^{3}.$$
  10. A worker has to tile floors of sizes $2^{n}\times2^{n}$ units square either by domino shaped tiles or by set square shaped tiles which are both 3 unites square. For every floor, he has to cover all except one $1\times1$ unit square spot which is finised later for the decoration purpose. Since ser square shaped tiles are much more expensive than the other tiles are much more expensive than the other ones, he wants to use them the fewer the better. He observes that no matter where the special spot is, he never needs more than $n$ set square shaped tiles. Show that his observation is mathematically correct for any positive integer $n$.
  11. Let $a,,b,c$ and $S$ respectively the sides and the area of a given triangle. Let $x,y$ and $z$ be positive real numbers. Prove that
    a) $xa^{2}+yb^{2}+zc^{2}\geq4\sqrt{xy+yz+zx}S$
    b) $\left(y+z\right)bc+\left(z+x\right)ca+\left(x+y\right)ab-xa^{2}-yb^{2}-zc^{2}\geq4\sqrt{xy+yz+zx}S.$
    When do the equalities happen?.
  12. Let $ABCD$ be a quadrilateral circumscribed about a circle $\left(I\right)$. The rays $AB$ and $CD$ intersect at $E$, the rays $DA$ and $CB$ intersect at $F$. Let $\left(I_{1}\right),\left(I_{2}\right)$ respectively be the incircles of the triangles $EFB$ and $EFD$. Prove that $\widehat{I_{1}IB}=\widehat{I_{2}ID}$.




Mathematics & Youth: 2016 Issue 468
2016 Issue 468
Mathematics & Youth
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