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2014 Issue 450

  1. Find all positive integers $a$ and $b$ such that $b|a+2$ and $a|b+3$.
  2. Given a right triangle $ABC$ with the right angle $A$. Choose $E$ on the side $BC$ such that $EC=2EB$. Prove that $AC^{2}=3(EC^{2}-EA^{2})$.
  3. Solve the following equation \[\frac{1}{x+\sqrt{x^{2}-1}}=\frac{1}{4x}+\frac{3x}{2x^{2}+2}.\]
  4. Let $BC$ be a chord of a circle with center $O$ and radius $R$. Assume that $BC=R$. Let $A$ be apoint on the major arc $BC$ ($A\ne B$, $A\ne C$), and $M,N$ points on the chord $AC$ such that $AC=2AN=\frac{3}{2}AM$. Choose $P$ on $AB$ such that $MP$ is perpendicular to $AB$. Prove that three points $P,O$ and $N$ are collinear.
  5. Assume that equation \[ax^{3}-x^{2}+bx-1=0,\quad(a\ne0)\] has three positive real solutions. Find the minimum value of the expression \[M=(1-2ab)\frac{b}{a^{2}}.\]
  6. Let $x$ and $y$ be two positive real numbers satisfying $32x^{6}+4y^{3}=1$. Find the maximum value of the expression \[P=\frac{(2x^{2}+y+3)^{3}}{3(x^{2}+y^{2})-3(x+y)+2}.\]
  7. Given an acute triangle $ABC$ ($AB>AC$). The heights $BB'$ and $CC'$ intersect at $H$. Let $M,N$ respectively be the midpoints of the sides $AB,AC$ and $O$ the circumcenter. $AH$ intersects $B'C'$ at $E$, and $AO$ intersects $MN$ at $F$. Prove that $EF\parallel OH$.
  8. Given three positive numbers $a,b,c$. Find the maximum value of $k$ so that the following inequality holds \[\frac{a}{b}+\frac{b}{c}+\frac{c}{a}-3\geq3\left(\frac{a^{2}+b^{2}+c^{2}}{ab+bc+ca}-1\right).\]
  9. Find all positive integers $x,y,z$ which form an airthmetic progression and satisfy the following equation \[\frac{x^{2}(x+y)(x+z)}{(x-y)(x-z)}+\frac{y^{2}(y+z)(y+x)}{(y-z)(y-x)}+\frac{z^{2}(z+x)(z+y)}{(z-x)(z-y)}=2016+(x+y-z)^{2}.\]
  10. Given a $999\times999$ table of squares. Each square is colored by white or red. Consider a set of triples of squares $(C_{1},C_{2},C_{3})$ which satisfy the following properties: the first two squares $C_{1},C_{2}$ are in the same row, the last two squares $C_{2},C_{3}$ are in the same column, $C_{1},C_{3}$ are white, and $C_{2}$ is red. Find the maximum number of elements in such a set.
  11. Find all positive integers $n>1$ and all primes $p$ such that the polynomial $f(x)=x^{n}-px+p^{2}$ ca be factorized as a product of two non-constant polynomials with integer coefficients. 
  12. Assume that $ABC$ is an equilateral triangle and $M$ is a point which is not on the lines through $BC$, $CA$ and $AB$. Prove that the Euler lines of the triangles $MBC$, $MCA$, and $MAB$ are either concurrent or parallel.

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Mathematics & Youth: 2014 Issue 450
2014 Issue 450
Mathematics & Youth
https://www.molympiad.org/2017/08/mathematics-and-youth-magazine-problems_54.html
https://www.molympiad.org/
https://www.molympiad.org/
https://www.molympiad.org/2017/08/mathematics-and-youth-magazine-problems_54.html
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