MathDB

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Part of 2021 239 Open Mathematical Olympiad

Problems(2)

integer root of x^n+a_1x^{n -1}+..+a_n=0 where a_i=product of first i primes

Source: 239 MO 2021 10-11 p1 , 8-9 p2

5/2/2021
You are given nn different primes p1,p2,...,pnp_1, p_2,..., p_n. Consider the polynomial xn+a1xn−1+a2xn−2+...+an−1x+anx^n + a_1x^{n -1} + a_2x^{n - 2} + ...+ a_{n - 1}x + a_n, where aia_i is the product of the first ii given prime numbers. For what nn can it have an integer root?
primesIntegerpolynomialInteger Polynomialalgebra
angle wanted, arc midpoints of circumcircle, angle bisectors

Source: 239 MO 2021 8-9 p1

5/2/2021
Points XX and YY are the midpoints of arcs ABAB and BCBC of the circumscribed circle of triangle ABCABC. Point TT lies on side ACAC. It turned out that the bisectors of the angles ATBATB and BTCBTC pass through points XX and YY respectively. What angle BB can be in triangle ABCABC?
geometrycircumcircleanglesarc midpointangle bisector