MathDB

15

Part of 2019 Durer Math Competition Finals

Problems(2)

<DAC=? AB = AC , <BAC = 96^o, <ACD = 48^o, AD=BC

Source: 2019 Dürer Math Competition Finals Day2 E15 https://artofproblemsolving.com/community/c1621835_2019_

1/5/2022
ABCABC is an isosceles triangle such that AB=ACAB = AC and ∠BAC=96o\angle BAC = 96^o. DD is the point for which ∠ACD=48o\angle ACD = 48^o, AD=BCAD = BC and triangle DACDAC is obtuse-angled. Find ∠DAC\angle DAC.
geometryanglesisosceles
m^{x_0} =\sum_{i=1}^{1001}m^{x_i}

Source: 2019 Dürer Math Competition Finals Day2 E+15 https://artofproblemsolving.com/community/c1621835_2019_

1/5/2022
The positive integer mm and non-negative integers x0,x1,...,x1001x_0, x_1,..., x_{1001} satisfy the following equation: mx0=∑i=11001mxi.m^{x_0} =\sum_{i=1}^{1001}m^{x_i}. How many possibilities are there for the value of mm?
diophantineDiophantine equationnumber theorySum