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2015 Issue 462

  1. Find the last digit of the following number \[A=1^{2015}+2^{2015}+3^{2015}+\ldots+2014^{2015}+2015^{2015}.\]
  2. Given a right trapezoid $ABCD$ ($A$ and $B$ are right angles) with $AD<BC$ and $AC\perp BD$. Prove that \[AC^{2}+BD^{2}=3AB^{2}+CD^{2}.\]
  3. Does there exist a pair of positive integers $(a,b)$ such that both the equation $x^{2}+2ax-b-2=0$ and the equation $x^{2}+bx-a=0$ have integer solutions?.
  4. Given a right triangle $ABC$ with the right angle $A$ such that \[BC^{2}=2BC\cdot AC+4AC^{2}.\] Find the angle $\widehat{ABC}$.
  5. Solve the equation \[\sqrt[5]{3x-2}-\sqrt[5]{2x+1}=\sqrt[5]{x-3}.\]
  6. Solve the system of equations \[\begin{cases} x+\sqrt{x^{2}+9} & =\sqrt[4]{3^{y+4}}\\ y+\sqrt{y^{2}+9} & =\sqrt[4]{3^{z+4}}\\ z+\sqrt{z^{2}+9} & =\sqrt[4]{3^{x+4}} \end{cases}.\]
  7. Given an acute triangle $ABC$ with angles measured in radian. Prove that \[\sin A+\sin B+\sin C>\frac{5}{2}-\frac{A^{2}+B^{2}+C^{2}}{\pi^{2}}.\]
  8. Given a tetrahedron $ABCD$ inscribed in a sphere of radius $1$. Assume that the product of the lengths of its sides is equal to $\frac{512}{27}$. Compute the lengths of its sides.
  9. Let $a,b,c$ be the lengths of three sides of a triangle. Prove that \[\frac{3(a^{2}+b^{2}+c^{2})}{(a+b+c)^{2}}+\frac{ab+bc+ca}{a^{2}+b^{2}+c^{2}}\leq2.\]
  10. Find the number of the triples $(a,b,c)$ of integers in the interval $[1,2015]$ such that $a^{3}+b^{3}+c^{3}$ is divisible by $9$?. 
  11. Let $a,b,c$ be real numbers such that $a^{2}+b^{2}+c^{2}=1$. Find the minimum value of the expression \[P=|6a^{3}+bc|+|6b^{3}+ca|+|6c^{3}+ab|.\]
  12. Given a triangle $ABC$ with $(O)$ and $H$ respectively are the circumcircle and the orthocenter of the triangle. Let $P$ be an arbitrary point on $OH$. Let $A_{0},B_{0},C_{0}$ respectively be the intersections between $AH,BC,CH$ and $BC,CA,AB$. Suppose that $A_{1},B_{1},C_{1}$ respectively be the second intersections between $AP,BP,CP$ and $(O)$. Let $A_{2},B_{2},C_{2}$ respectively be the reflection points of $A_{1},B_{1},C_{1}$ through $A_{0},B_{0},C_{0}$. Prove that $H,A_{2},B_{2},C_{2}$ belong to a circle with center is on $OH$.

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Mathematics & Youth: 2015 Issue 462
2015 Issue 462
Mathematics & Youth
https://www.molympiad.org/2017/08/mathematics-and-youth-magazine-problems_96.html
https://www.molympiad.org/
https://www.molympiad.org/
https://www.molympiad.org/2017/08/mathematics-and-youth-magazine-problems_96.html
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