MathDB

2

Part of 2023 India Regional Mathematical Olympiad

Problems(2)

RMO 2023 P2

Source:

10/29/2023
Let ω\omega be a semicircle with ABA B as the bounding diameter and let CDC D be a variable chord of the semicircle of constant length such that C,DC, D lie in the interior of the arc ABA B. Let EE be a point on the diameter ABA B such that CEC E and DED E are equally inclined to the line ABA B. Prove that (a) the measure of ∠CED\angle C E D is a constant; (b) the circumcircle of triangle CEDC E D passes through a fixed point.
geometry
RMO KV 2023 P2

Source: RMO 2023 P2

10/29/2023
Given a prime number pp such that 2p2p is equal to the sum of the squares of some four consecutive positive integers. Prove that p−7p-7 is divisible by 36.
number theoryprime numbers