MathDB

Problem 3

Part of 2022 Kyiv City MO Round 2

Problems(3)

Very cute algebra

Source: Kyiv City MO 2022 Round 2, Problem 8.3, 9.3

1/30/2022
Nonzero real numbers x1,x2,…,xnx_1, x_2, \ldots, x_n satisfy the following condition:
x1−1x2=x2−1x3=…=xn−1−1xn=xn−1x1x_1 - \frac{1}{x_2} = x_2 - \frac{1}{x_3} = \ldots = x_{n-1} - \frac{1}{x_n} = x_n - \frac{1}{x_1}
Determine all nn for which x1,x2,…,xnx_1, x_2, \ldots, x_n have to be equal.
(Proposed by Oleksii Masalitin, Anton Trygub)
algebra
Equal ratio implies tangency

Source: Kyiv City MO 2022 Round 2, Problem 10.3

1/30/2022
Let AHA,BHB,CHCAH_A, BH_B, CH_C be the altitudes of triangle ABCABC. Prove that if HBCAC=HCAAB\frac{H_BC}{AC} = \frac{H_CA}{AB}, then the line symmetric to BCBC with respect to line HBHCH_BH_C is tangent to the circumscribed circle of triangle HBHCAH_BH_CA.
(Proposed by Mykhailo Bondarenko)
geometryratio
Prefix sums of permutation

Source: Kyiv City MO 2022 Round 2, Problem 11.3

1/30/2022
Find the largest kk for which there exists a permutation (a1,a2,…,a2022)(a_1, a_2, \ldots, a_{2022}) of integers from 11 to 20222022 such that for at least kk distinct ii with 1≤i≤20221 \le i \le 2022 the number a1+a2+…+ai1+2+…+i\frac{a_1 + a_2 + \ldots + a_i}{1 + 2 + \ldots + i} is an integer larger than 11.
(Proposed by Oleksii Masalitin)
number theorypermutationsalgebra