MathDB

Problem 3

Part of 2023 Kyiv City MO Round 1

Problems(5)

NT for beginners

Source: Kyiv City MO 2023 7.3

5/14/2023
Prove that there don't exist positive integer numbers kk and nn which satisfy equation nn+(n+1)n+1+(n+2)n+2=2023kn^n+(n+1)^{n+1}+(n+2)^{n+2} = 2023^k. Proposed by Mykhailo Shtandenko
number theory
Hedgehogs going WILD

Source: Kyiv City MO 2023 Round 1, Problem 8.3

12/16/2023
A hedgehog is a circle without its boundaries. The diameter of the hedgehog is the diameter of the corresponding circle. We say that the hedgehog sits at the at the point where the center of the circle is located.
We are given a triangle with sides a,b,ca, b, c, with hedgehogs sitting at its vertices. It is known that inside the triangle there is a point from which you can reach any side of the triangle by walking along a straight line without hitting any hedgehog. What is the largest possible sum of the diameters of these hedgehogs?
Proposed by Oleksiy Masalitin
geometry
Arc midpoints in the right triangle

Source: Kyiv City MO 2023 Round 1, Problem 9.3

12/16/2023
You are given a right triangle ABCABC with ∠ACB=90∘\angle ACB = 90^\circ. Let WA,WBW_A , W_B respectively be the midpoints of the smaller arcs BCBC and ACAC of the circumcircle of △ABC\triangle ABC, and NA,NBN_A , N_B respectively be the midpoints of the larger arcs BCBC and ACAC of this circle. Denote by PP and QQ the points of intersection of segment ABAB with the lines NAWBN_AW_B and NBWAN_BW_A, respectively. Prove that AP=BQAP = BQ.
Proposed by Oleksiy Masalitin
geometrycircumcircle
This one was misplaced

Source: Kyiv City MO 2023 Round 1, Problem 11.3

12/16/2023
Let II be the incenter of triangle ABCABC with AB<ACAB < AC. Point XX is chosen on the external bisector of ∠ABC\angle ABC such that IC=IXIC = IX. Let the tangent to the circumscribed circle of △BXC\triangle BXC at point XX intersect the line ABAB at point YY. Prove that AC=AYAC = AY.
Proposed by Oleksiy Masalitin
circumcirclegeometry
Grid geometry

Source: Kyiv City MO 2023 Round 1, Problem 10.3

12/16/2023
Consider all pairs of distinct points on the Cartesian plane (A,B)(A, B) with integer coordinates. Among these pairs of points, find all for which there exist two distinct points (X,Y)(X, Y) with integer coordinates, such that the quadrilateral AXBYAXBY is convex and inscribed.
Proposed by Anton Trygub
analytic geometrygridgeometrycyclic quadrilateral