MathDB

7

Part of 2023 Assara - South Russian Girl's MO

Problems(2)

increasing sequence , a_n = 2023

Source: 2023 South Russian Girls MO - Assara Juniors p7

2/28/2024
Given an increasing sequence of different natural numbers a1<a2<a3<...<ana_1 < a_2 < a_3 < ... < a_n such that for any two distinct numbers in this sequence their sum is not divisible by 1010. It is known that an=2023a_n = 2023. a) Can nn be greater than 800800? b) What is the largest possible value of nn? c) For the value nn found in question b), find the number of such sequences with an=2023a_n = 2023.
algebraSequence
midpoint of median drawn from vertex A lies on axis of parabola

Source: 2023 South Russian Girls MO - Assara Seniors p7

3/1/2024
A parabola is drawn on the coordinate plane - the graph of a square trinomial. The vertices of triangle ABCABC lie on this parabola so that the bisector of angle ∠BAC\angle BAC is parallel to the axis OxOx . Prove that the midpoint of the median drawn from vertex AA lies on the axis of the parabola.
analytic geometryconicsparabolageometry