2005 JBMO Problems

Problem 1

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

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

2005 JBMO (Problems • Resources)
Preceded by 2004 JBMO	Followed by 2006 JBMO
1 • 2 • 3 • 4
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2005 JBMO Problems

Contents

Problem 1

Problem 2

Problem 3

Problem 4

See Also