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Part of 2015 Poland - Second Round

Problems(2)

6 double lengths of segments given, parallelogram wanted

Source: Polish National Olympiad 2015 2nd round, p1

8/28/2019
Points E,F,GE, F, G lie, and on the sides BC,CA,ABBC, CA, AB, respectively of a triangle ABCABC, with 2AG=GB,2BE=EC2AG=GB, 2BE=EC and 2CF=FA2CF=FA. Points PP and QQ lie on segments EGEG and FGFG, respectively such that 2EP=PG2EP = PG and 2GQ=QF2GQ=QF. Prove that the quadrilateral AGPQAGPQ is a parallelogram.
geometryparallelogram
x_3 + x_4 \in Q, x_3 x_4 \notin Q => x_1 + x_2 = x_3 + x_4

Source: Polish National Olympiad 2015 2nd round, p4

8/28/2019
Real numbers x1,x2,x3,x4x_1, x_2, x_3, x_4 are roots of the fourth degree polynomial W(x)W (x) with integer coefficients. Prove that if x3+x4x_3 + x_4 is a rational number and x3x4x_3x_4 is a irrational number, then x1+x2=x3+x4x_1 + x_2 = x_3 + x_4.
polynomialalgebraSumrationalirrational number