MathDB

3

Part of 1988 China Team Selection Test

Problems(2)

OI is perpendicular to DE

Source: China TST 1988, problem 3

6/27/2005
In triangle ABCABC, ∠C=30∘\angle C = 30^{\circ}, OO and II are the circumcenter and incenter respectively, Points D∈ACD \in AC and E∈BCE \in BC, such that AD=BE=ABAD = BE = AB. Prove that OI=DEOI = DE and OI⊥DEOI \bot DE.
geometrycircumcircleincentergeometry solved
n+1 points in the plane with integer differences

Source: China TST 1988, problem 7

6/27/2005
A polygon ∏\prod is given in the OXYOXY plane and its area exceeds n.n. Prove that there exist n+1n+1 points P1(x1,y1),P2(x2,y2),…,Pn+1(xn+1,yn+1)P_{1}(x_1, y_1), P_{2}(x_2, y_2), \ldots, P_{n+1}(x_{n+1}, y_{n+1}) in ∏\prod such that ∀i,j∈{1,2,…,n+1}\forall i,j \in \{1, 2, \ldots, n+1\}, xj−xix_j - x_i and yj−yiy_j - y_i are all integers.
geometrygeometric transformationvectoranalytic geometrycombinatorics unsolvedcombinatorics