MathDB

P1

Part of 2017 Romania Team Selection Test

Problems(3)

5-tuples with special property

Source: Romania 2017 IMO TST 3, problem 1

3/18/2018
a) Determine all 4-tuples (x0,x1,x2,x3)(x_0,x_1,x_2,x_3) of pairwise distinct intergers such that each xkx_k is coprime to xk+1x_{k+1}(indices reduces modulo 4) and the cyclic sum x0x1+x1x2+x2x3+x3x1\frac{x_0}{x_1}+\frac{x_1}{x_2}+\frac{x_2}{x_3}+\frac{x_3}{x_1} is an interger. b)Show that there are infinitely many 5-tuples (x0,x1,x2,x3,x4)(x_0,x_1,x_2,x_3,x_4) of pairwise distinct intergers such that each xkx_k is coprime to xk+1x_{k+1}(indices reduces modulo 5) and the cyclic sum x0x1+x1x2+x2x3+x3x4+x4x0\frac{x_0}{x_1}+\frac{x_1}{x_2}+\frac{x_2}{x_3}+\frac{x_3}{x_4}+\frac{x_4}{x_0} is an interger.
number theoryabstract algebra
2 lines concurrent on the circumcircle

Source: Romania 2017 IMO TST 2, problem 1

3/18/2018
Let ABCABC be a triangle with AB<ACAB<AC, let G,HG,H be its centroid and otrhocenter. Let DD be the otrhogonal projection of AA on the line BCBC, and let MM be the midpoint of the side BCBC. The circumcircle of ABCABC crosses the ray HMHM emanating from MM at PP and the ray DGDG emanating from DD at QQ, outside the segment DGDG. Show that the lines DPDP and MQMQ meet on the circumcircle of ABCABC.
geometrycircumcircle
Number Theory problem on sequence

Source: Romania 2017 IMO TST 4, problem 1

3/18/2018
Let m be a positive interger, let pp be a prime, let a1=8pma_1=8p^m, and let an=(n+1)an−1na_n=(n+1)^{\frac{a_{n-1}}{n}}, n=2,3...n=2,3.... Determine the primes pp for which the products an(1−1a1)(1−1a2)...(1−1an)a_n(1-\frac{1}{a_1})(1-\frac{1}{a_2})...(1-\frac{1}{a_n}), n=1,2,3...n=1,2,3... are all integral.
number theory