MathDB

3

Part of 1997 Moscow Mathematical Olympiad

Problems(3)

1997 MMO Grade 9 #3

Source:

10/3/2016
Convex octagon AC1BA1CB1AC_1BA_1CB_1 satisfies: AB1=AC1AB_1=AC_1, BC1=BA1BC_1=BA_1, CA1=CB1CA_1=CB_1 and ∠A+∠B+∠C=∠A1+∠B1+∠C1\angle{A}+\angle{B}+\angle{C}=\angle{A_1}+\angle{B_1}+\angle{C_1}. Prove that the area of △ABC\triangle{ABC} is equal to half the area of the octagon.
Grade 91997
1997 MMO Grade 8 #3

Source:

10/3/2016
Inside acute ∠XOY,\angle{XOY}, points MM and NN are taken so that ∠XON=∠YOM\angle{XON}=\angle{YOM}. Point QQ is taken on segment OXOX such that ∠NQO=∠MQX.\angle{NQO}=\angle{MQX}. Point PP is taken such that ∠NPO=∠MPY.\angle{NPO}=\angle{MPY}. Prove the lengths of the broken lines MPNMPN and MQNMQN are equal.
Grade 81997
1997 MMO Grade 10 #3

Source:

10/3/2016
A quadrilateral is rotated clockwise, and the sides are extended its length in the direction of the movement. It turns out the endpoints of the segments constructed form a square. Prove the initial quadrilateral must also be a square.
Generalization: Prove that if the same process is applied to any nn-gon and the result is a regular nn-gon, then the intial nn-gon must also be regular.
Grade 101997