MathDB

5

Part of 2003 IMO Shortlist

Problems(4)

ISL 2003 G5 page doesnot exist on AoPS?

Source: ISL 2003 G5

10/25/2019
Let ABCABC be an isosceles triangle with AC=BCAC=BC, whose incentre is II. Let PP be a point on the circumcircle of the triangle AIBAIB lying inside the triangle ABCABC. The lines through PP parallel to CACA and CBCB meet ABAB at DD and EE, respectively. The line through PP parallel to ABAB meets CACA and CBCB at FF and GG, respectively. Prove that the lines DFDF and EGEG intersect on the circumcircle of the triangle ABCABC.
Proposed by Hojoo Lee
geometryIMO Shortlist
classical number theory

Source: IMO ShortList 2003, number theory problem 5

5/12/2004
An integer nn is said to be good if ∣n∣|n| is not the square of an integer. Determine all integers mm with the following property: mm can be represented, in infinitely many ways, as a sum of three distinct good integers whose product is the square of an odd integer.
Proposed by Hojoo Lee, Korea
number theoryPerfect SquaresIMO Shortlist
IMO ShortList 2003, algebra problem 5

Source: IMO ShortList 2003, algebra problem 5

10/4/2004
Let R+\mathbb{R}^+ be the set of all positive real numbers. Find all functions f:R+→R+f: \mathbb{R}^+ \to \mathbb{R}^+ that satisfy the following conditions:
- f(xyz)+f(x)+f(y)+f(z)=f(xy)f(yz)f(zx)f(xyz)+f(x)+f(y)+f(z)=f(\sqrt{xy})f(\sqrt{yz})f(\sqrt{zx}) for all x,y,z∈R+x,y,z\in\mathbb{R}^+;
- f(x)<f(y)f(x)<f(y) for all 1≤x<y1\le x<y.
Proposed by Hojoo Lee, Korea
functionalgebrafunctional equationIMO Shortlist
Nice property of &quot;almost&quot; lattice points

Source: German TST 2004, IMO ShortList 2003, combinatorics problem 5

5/18/2004
Every point with integer coordinates in the plane is the center of a disk with radius 1/10001/1000.
(1) Prove that there exists an equilateral triangle whose vertices lie in different discs.
(2) Prove that every equilateral triangle with vertices in different discs has side-length greater than 9696.
Radu Gologan, Romania [hide="Remark"] The "> 96" in (b) can be strengthened to "> 124". By the way, part (a) of this problem is the place where I used [url=http://mathlinks.ro/viewtopic.php?t=5537]the well-known "Dedekind" theorem.
geometrycoordinate geometrycirclescoordinatesTrianglecombinatoricsIMO Shortlist