2005 JBMO Problems

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Problem 1

Find all positive integers $x,y$ satisfying the equation \[9(x^2+y^2+1) + 2(3xy+2) = 2005 .\]

Solution

Problem 2

Let $ABC$ be an acute-angled triangle inscribed in a circle $k$. It is given that the tangent from $A$ to the circle meets the line $BC$ at point $P$. Let $M$ be the midpoint of the line segment $AP$ and $R$ be the second intersection point of the circle $k$ with the line $BM$. The line $PR$ meets again the circle $k$ at point $S$ different from $R$.

Prove that the lines $AP$ and $CS$ are parallel.

Solution

Problem 3

Prove that there exist

(a) 5 points in the plane so that among all the triangles with vertices among these points there are 8 right-angled ones;

(b) 64 points in the plane so that among all the triangles with vertices among these points there are at least 2005 right-angled ones.

Solution

Problem 4

Find all 3-digit positive integers $\overline{abc}$ such that \[\overline{abc} = abc(a+b+c) ,\] where $\overline{abc}$ is the decimal representation of the number.

Solution

See Also

2005 JBMO (ProblemsResources)
Preceded by
2004 JBMO
Followed by
2006 JBMO
1 2 3 4
All JBMO Problems and Solutions


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