MathDB

7

Part of 2023 All-Russian Olympiad

Problems(2)

Two exsimilicenters in a trapezoid

Source: All-Russian MO 2023 Final stage 9.7/10.7

4/23/2023
Given a trapezoid ABCDABCD, in which AD∥BCAD \parallel BC, and rays ABAB and DCDC intersect at point GG. The common external tangents to the circles (ABC),(ACD)(ABC), (ACD) intersect at point EE. The common external tangents to circles (ABD),(CBD)(ABD), (CBD) meet at FF. Prove that the points E,FE, F and GG are collinear.
geometrytrapezoid
Characterisation of special polynomials

Source: All-Russian MO 2023 Final stage 11.7

4/23/2023
We call a polynomial P(x)P(x) good if the numbers P(k)P(k) and P′(k)P'(k) are integers for all integers kk. Let P(x)P(x) be a good polynomial of degree dd, and let NdN_d be the product of all composite numbers not exceeding dd. Prove that the leading coefficient of the polynomial Nd⋅P(x)N_d \cdot P(x) is integer.
algebrapolynomial