MathDB

6

Part of 2023 All-Russian Olympiad

Problems(2)

NT bijection

Source: All-Russian MO 2023 Final stage 9.6

4/23/2023
Consider all 100100-digit numbers divisible by 1919. Prove that the number of such numbers not containing the digits 4,54, 5, and 66 is the number of such numbers that do not contain the digits 1,41, 4 and 77.
number theory
3 midpoints of edges in tetrahedron and tangent's intersection, coplanar

Source: All-Russian MO 2023 Final stage 11.6

8/26/2024
The plane α\alpha intersects the edges ABAB, BCBC, CDCD and DADA of the tetrahedron ABCDABCD at points X,Y,ZX, Y, Z and TT, respectively. It turned out, that points YY and TT lie on a circle ω\omega constructed with segment XZXZ as the diameter. Point PP is marked in the plane α\alpha so that the lines PYP Y and PTP T are tangent to the circle ω\omega.Prove that the midpoints of the edges are ABAB, BCBC, CD,CD, DADA and the point PP lie in the same plane.
geometry3D geometrytetrahedron