Let \(m_a, m_b, m_c\) be the medians of the triangle \(ABC\) and \(r_a, r_b, r_c\) be its exradii. Then prove that \(m_a+m_b+m_c\le r_a+r_b+r_c\).

**Tags:**
triangle
calculation
inequality
geometric inequality

The points \(D\) and \(E\) on the side \(AC\) of the triangle \(ABC\) are chosen so that the incircles of \(ABD\) and \(CBE\) with respective centers \(J_a\) and \(K_c\) have equal radii. Also \(J_c\) and \(K_a\) are the incenters of \(CBD\) and \(ABE\) respectively. Prove that the circles \(BJ_aJ_c\) and \(BK_aK_c\) concur second time on the bisector of angle \(B\).

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XXIV Kolmogorov Cup, individual olympiad, senior group, conclusion, problem 7
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**Note.** In fact this problem occurred in the olympiad in generalized form for circumscribed
quadrilaterals by I. Bogdanov and I. Frolov.

**Tags:**
triangle
circles
incircle

In triangle \(ABC\) \(I\) is the incenter and \(J\) is the \(B\)-excenter. \(BH\) is height. Reconstruct \(ABC\) given the points \(I,J,H\).

**Tags:**
triangle
circles
incircle

Let the side \(AC\) of triangle \(ABC\) touch the incircle and the corresponding excircle at points \(K\) and \(L\) respectively. Let \(P\) be the projection of the incenter onto the perpendicular bisector of \(AC\). It is known that the tangents to the circumcircle of triangle \(BKL\) at \(K\) and \(L\) meet on the circumcircle of \(ABC\). Prove that the lines \(AB\) and \(BC\) touch the circumcircle of triangle \(PKL\).

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XV Geometrical Olympiad in Honour of I.F. Sharygin, correspondence round, problem 14
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**Tags:**
triangle
circles
incircle
circumcircle

Let us be given a convex quadrilateral \(ABCD\) none of whose diagonals is its axis of symmetry and \(DA+AB=DC+CB\). Let \(N_a\) and \(N_c\) be the Nagel points of \(ABD\) and \(CBD\) respectively. Prove that \(N_aN_c\perp BD\) iff the length of \(BD\) is the quarter of \(ABCD\)'s perimeter.

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taken from "The Nagel point and the difference triangle",
journal “Quantum”, 2019 №9, pages 32-33
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**Tags:**
quadrilateral
triangle
circles
incircle
Nagel point

Let \(I\) be the incenter of the triangle \(ABC\). Let the respective excircle touch the segment \(AC\) at the point \(K\). Let \(W\) be the second point of intersection of \(BI\) and the circumcircle of \(ABC\), and let \(P\) be the reflection of \(I\) in the \(B\)-height of \(ABC\). It is known that \(\angle IKW=90^\circ\). Then find the measure of \(\angle BPK\).

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XXIV Kolmogorov Cup, first tour, senior league, problem 6
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**Tags:**
triangle
circles
incircle
circumcircle
symmedian

Let us be given an isosceles tetrahedron \(ABCD\), i.e. it has equal opposite edges. Choose a point \(X\) on its circumsphere \(\Omega\) different from the points \(A, B, C, D\). Prove that the center of the sphere passing through the reflections of \(X\) in the faces of \(ABCD\) lies on \(\Omega\).

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XXIII Kolmogorov Cup, second tour, senior league, problem 1
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**Tags:**
3D
tetrahedron
isosceles tetrahedron
spheres
circumsphere
isogonals

Let the sphere \(\omega\) with center \(I\) be inscribed in a

- hexahedron which has only quadriangular faces and in each vertex of which exactly three faces meet;
- dodecahedron which has only pentagonal faces and in each vertex of which exactly three faces meet.

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XXIV Kolmogorov Cup, third tour, senior league, problem 8
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**Note.** In fact this problem occurred in the olympiad in simplified form by I. Frolov.

**Tags:**
3D
dodecahedron
hexahedron
spheres
insphere
isogonals
conics