IMO Shortlist 1982 problem 14
Kvaliteta:
Avg: 0,0Težina:
Avg: 0,0 Let
be a convex plane quadrilateral and let
denote the circumcenter of
. Define
in a corresponding way.
(a) Prove that either all of
coincide in one point, or they are all distinct. Assuming the latter case, show that
, C1 are on opposite sides of the line
, and similarly,
are on opposite sides of the line
. (This establishes the convexity of the quadrilateral
.)
(b) Denote by
the circumcenter of
, and define
in an analogous way. Show that the quadrilateral
is similar to the quadrilateral
![ABCD](/media/m/9/c/e/9ce25711ba18d9663b73c3580de4bf5a.png)
![A_1](/media/m/5/a/6/5a6ce1347567551c02239ff8d4ebee67.png)
![\triangle BCD](/media/m/3/b/1/3b147a09016f2df03b75355ec5030ba8.png)
![B_1, C_1,D_1](/media/m/6/4/8/6481b404346b869a1e772295276fa10a.png)
(a) Prove that either all of
![A_1,B_1, C_1,D_1](/media/m/5/5/b/55bf1def0f50c199671f4e7348c5c4cc.png)
![A_1](/media/m/5/a/6/5a6ce1347567551c02239ff8d4ebee67.png)
![B_1D_1](/media/m/7/b/d/7bdc96821214a01a74a5c673ad1c9f60.png)
![B_1,D_1](/media/m/4/2/9/42978b9f1373ba6be613137e689a7a56.png)
![A_1C_1](/media/m/9/3/2/932752a76a906e245c34e4943fa92183.png)
![A_1B_1C_1D_1](/media/m/8/e/a/8ea8991888072519f65b0a7e2f45de2d.png)
(b) Denote by
![A_2](/media/m/a/2/5/a25c6dade4a684fc874981a7d65625f5.png)
![B_1C_1D_1](/media/m/b/b/b/bbb680be8195ad070334c9c80677bb1c.png)
![B_2, C_2,D_2](/media/m/0/1/2/01274b6137b61cc8d147630de76910b8.png)
![A_2B_2C_2D_2](/media/m/6/2/9/629cf266bd45fcf773bea434616f461c.png)
![ABCD.](/media/m/2/f/c/2fc42d859a3093b5385a997cb44fcf63.png)
Izvor: Međunarodna matematička olimpijada, shortlist 1982