Problems

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Prove that \(\angle ABC < \angle BAC\) if and only if \(AC < BC\), that is, the larger side lies opposite the larger angle of the triangle, and opposite the larger side lies the larger angle.

The point \(D\) lies on the base \(AC\) of the isosceles triangle \(ABC\). Prove that the radii of the circumscribed circles of the triangles \(ABD\) and \(CBD\) are equal.

Prove that the point \(X\) lies on the line \(AB\) if and only if \(\overrightarrow{OX} = t \overrightarrow{OA} + (1 - t) \overrightarrow{OB}\) for some \(t\) and any point \(O\).

Several points are given and for some pairs \((A, B)\) of these points the vectors \(\overrightarrow{AB}\) are taken, and at each point the same number of vectors begin and end. Prove that the sum of all the chosen vectors is \(\vec{0}\).

Prove that the medians of the triangle \(ABC\) intersect at one point and that point divides the medians in a ratio of \(2: 1\), counting from the vertex.