MathDB

2

Part of 2003 IMO Shortlist

Problems(4)

Simple triangle geometry [a fixed point]

Source: German TST 2004, IMO ShortList 2003, geometry problem 2

5/18/2004
Three distinct points AA, BB, and CC are fixed on a line in this order. Let Γ\Gamma be a circle passing through AA and CC whose center does not lie on the line ACAC. Denote by PP the intersection of the tangents to Γ\Gamma at AA and CC. Suppose Γ\Gamma meets the segment PBPB at QQ. Prove that the intersection of the bisector of ∠AQC\angle AQC and the line ACAC does not depend on the choice of Γ\Gamma.
geometryIMO ShortlistFixed point
A perverse one

Source: German TST 2004, IMO ShortList 2003, number problem 2

5/18/2004
Each positive integer aa undergoes the following procedure in order to obtain the number d=d(a)d = d\left(a\right):
(i) move the last digit of aa to the first position to obtain the numb er bb; (ii) square bb to obtain the number cc; (iii) move the first digit of cc to the end to obtain the number dd.
(All the numbers in the problem are considered to be represented in base 1010.) For example, for a=2003a=2003, we get b=3200b=3200, c=10240000c=10240000, and d=02400001=2400001=d(2003)d = 02400001 = 2400001 = d(2003).)
Find all numbers aa for which d(a)=a2d\left( a\right) =a^2.
Proposed by Zoran Sunic, USA
number theorydecimal representationalgorithmcombinatoricsIMO Shortlist
Functional equation on R

Source: IMO ShortList 2003, algebra problem 2

9/30/2004
Find all nondecreasing functions f:R→Rf: \mathbb{R}\rightarrow\mathbb{R} such that (i) f(0)=0,f(1)=1;f(0) = 0, f(1) = 1; (ii) f(a)+f(b)=f(a)f(b)+f(a+b−ab)f(a) + f(b) = f(a)f(b) + f(a + b - ab) for all real numbers a,ba, b such that a<1<ba < 1 < b.
Proposed by A. Di Pisquale & D. Matthews, Australia
functionalgebrafunctional equationIMO Shortlist
IMO ShortList 2003, combinatorics problem 2

Source:

5/17/2004
Let D1D_1, D2D_2, ..., DnD_n be closed discs in the plane. (A closed disc is the region limited by a circle, taken jointly with this circle.) Suppose that every point in the plane is contained in at most 20032003 discs DiD_i. Prove that there exists a disc DkD_k which intersects at most 7⋅2003−1=140207\cdot 2003 - 1 = 14020 other discs DiD_i.
geometrycirclesIntersectionIMO Shortlist