MathDB
geometric miniatures from V All-Ukrainian TYM

Source: V All-Ukrainian Tournament of Young Mathematicians, Qualifying p3

May 20, 2021
geometrygeometric inequalityUkrainian TYM

Problem Statement

Fix the triangle ABCABC on the plane. 1. Denote by SL,SMS_L,S_M and SKS_K the areas of triangles whose vertices are, respectively, the bases of bisectors, medians and points of tangency of the inscribed circle of a given triangle ABCABC. Prove that SK≤SL≤SMS_K\le S_L\le S_M.
2. For the point XX, which is inside the triangle ABCABC, consider the triangle TXT_X, the vertices of which are the points of intersection of the lines AX,BX,CXAX, BX, CX with the lines BC,AC,ABBC, AC, AB, respectively. 2.1. Find the position of the point XX for which the area of ​​the triangle TxT_x is the largest possible. 2.2. Suggest an effective criterion for comparing the areas of triangles TxT_x for different positions of the point XX. 2.3. Find the positions of the point XX for which the perimeter of the triangle TxT_x is the smallest possible and the largest possible. 2.4. Propose an effective criterion for comparing the perimeters of triangles TxT_x for different positions of point XX. 2.5. Suggest and solve similar problems with respect to the extreme values ​​of other parameters (for example, the radius of the circumscribed circle, the length of the greatest height) of triangles TxT_x.
3. For the point YY, which is inside the circle ω\omega, circumscribed around the triangle ABCABC, consider the triangle ΔY\Delta_Y, the vertices of which are the points of intersection AY,BX,CXAY, BX, CX with the circle ω\omega. Suggest and solve similar problems for triangles ΔY\Delta_Y for different positions of point YY.
4. Suggest and solve similar problems for convex polygons.
5. For the point ZZ, which is inside the circle ω\omega, circumscribed around the triangle ABCABC, consider the triangle FZF_Z, the vertices of which are orthogonal projections of the point ZZ on the lines BCBC, ACAC and ABAB. Suggest and solve similar problems for triangles FZF_Z for different positions of the point ZZ.
Back to ProblemsView on AoPS