MathDB

6

Part of 2024 Saint Petersburg Mathematical Olympiad

Problems(3)

If P(0),P(1),...,P(2^n+1) are divides by 2^{2^n}, then all values are also

Source: Saint Petersburg olympiad 2024, 11.6

9/22/2024
Polynomial P(x)P(x) with integer coefficients is given. For some positive integer nn numbers P(0),P(1),…,P(2n+1)P(0),P(1),\dots,P(2^n+1) are all divisible by 22n2^{2^n}. Prove that values of P(x)P(x) in all integer points are divisible by 22n2^{2^n}.
number theoryPolynomialsalgebrapolynomial
Triangles ABC and A_1B_1C_1 with same incircle and circumcircle

Source: Saint Petersburg olympiad 2024, 10.6

9/22/2024
Inscribed hexagon AB1CA1BC1AB_1CA_1BC_1 is given. Circle ω\omega is inscribed in both triangles ABCABC and A1B1C1A_1B_1C_1 and touches segments ABAB and A1B1A_1B_1 at points DD and D1D_1 respectively. Prove that if ∠ACD=∠BCD1\angle ACD = \angle BCD_1, then ∠A1C1D1=∠B1C1D\angle A_1C_1D_1 = \angle B_1C_1D.
geometrycircumcircle
Poor number with 100 000 distinct prime divisors

Source: Saint Petersburg olympiad 2024, 9.6

9/22/2024
Call a positive integer number nn poor if equation x1x2…x101=(n−x1)(n−x2)…(n−x101)x_1x_2 \dots x_{101}=(n-x_1)(n-x_2)\dots (n-x_{101}) has no solutions in positive integers 1<xi<n1<x_i<n. Does there exist poor number, which has more than 100 000100 \ 000 distinct prime divisors?
number theory