MathDB

2

Part of 1993 IMO Shortlist

Problems(4)

1993 points in each quadrant of the circle

Source: IMO Shortlist 1993, Brasil 1

3/14/2006
Show that there exists a finite set A⊂R2A \subset \mathbb{R}^2 such that for every X∈AX \in A there are points Y1,Y2,…,Y1993Y_1, Y_2, \ldots, Y_{1993} in AA such that the distance between XX and YiY_i is equal to 1, for every i.i.
algebrageometrypoint setdistancecombinatorial geometryIMO Shortlist
T contains at least k(n-k)+1 distinct elements

Source: IMO Shortlist 1993, Ireland 2

3/15/2006
Let n,k∈Z+n,k \in \mathbb{Z}^{+} with k≤nk \leq n and let SS be a set containing nn distinct real numbers. Let TT be a set of all real numbers of the form x1+x2+…+xkx_1 + x_2 + \ldots + x_k where x1,x2,…,xkx_1, x_2, \ldots, x_k are distinct elements of S.S. Prove that TT contains at least k(n−k)+1k(n-k)+1 distinct elements.
combinatoricsIMO ShortlistAdditive combinatoricsAdditive Number Theory
All such circles pass through two fixed points

Source: IMO Shortlist 1993, United Kingdom 1

10/24/2005
A circle SS bisects a circle S′S' if it cuts S′S' at opposite ends of a diameter. SAS_A, SBS_B,SCS_C are circles with distinct centers A,B,CA, B, C (respectively). Show that A,B,CA, B, C are collinear iff there is no unique circle SS which bisects each of SAS_A, SBS_B,SCS_C . Show that if there is more than one circle SS which bisects each of SAS_A, SBS_B,SCS_C , then all such circles pass through two fixed points. Find these points. Original Statement: A circle SS is said to cut a circle Σ\Sigma diametrically if and only if their common chord is a diameter of Σ.\Sigma. Let SA,SB,SCS_A, S_B, S_C be three circles with distinct centres A,B,CA,B,C respectively. Prove that A,B,CA,B,C are collinear if and only if there is no unique circle SS which cuts each of SA,SB,SCS_A, S_B, S_C diametrically. Prove further that if there exists more than one circle SS which cuts each SA,SB,SCS_A, S_B, S_C diametrically, then all such circles SS pass through two fixed points. Locate these points in relation to the circles SA,SB,SC.S_A, S_B, S_C.
geometrypower of a pointperpendicular bisectorcirclesIMO Shortlist
Show that every prime number n has property P

Source: IMO Shortlist 1993, India 5

3/15/2006
A natural number nn is said to have the property P,P, if, for all a,n2a, n^2 divides an−1a^n - 1 whenever nn divides an−1.a^n - 1. a.) Show that every prime number nn has property P.P. b.) Show that there are infinitely many composite numbers nn that possess property P.P.
modular arithmeticnumber theoryDivisibilityprime numberscomposite numbersIMO Shortlist