MathDB

1

Part of 2002 IMO Shortlist

Problems(3)

IMO ShortList 2002, geometry problem 1

Source: IMO ShortList 2002, geometry problem 1

9/28/2004
Let BB be a point on a circle S1S_1, and let AA be a point distinct from BB on the tangent at BB to S1S_1. Let CC be a point not on S1S_1 such that the line segment ACAC meets S1S_1 at two distinct points. Let S2S_2 be the circle touching ACAC at CC and touching S1S_1 at a point DD on the opposite side of ACAC from BB. Prove that the circumcentre of triangle BCDBCD lies on the circumcircle of triangle ABCABC.
geometrycircumcirclehomothetyincenterIMO Shortlistgeometry solvedtangent circles
IMO ShortList 2002, number theory problem 1

Source: IMO ShortList 2002, number theory problem 1

9/28/2004
What is the smallest positive integer tt such that there exist integers x1,x2,…,xtx_1,x_2,\ldots,x_t with x13+x23+ … +xt3=20022002 ?x^3_1+x^3_2+\,\ldots\,+x^3_t=2002^{2002}\,?
number theoryIMO Shortlistmodular arithmeticsum of cubesHi
IMO ShortList 2002, algebra problem 1

Source: IMO ShortList 2002, algebra problem 1

9/28/2004
Find all functions ff from the reals to the reals such that f(f(x)+y)=2x+f(f(y)−x)f\left(f(x)+y\right)=2x+f\left(f(y)-x\right) for all real x,yx,y.
algebrafunctional equationIMO Shortlist