Problems

Let \(ABC\) be a triangle. Prove that the heights \(AD\), \(BE\), \(CF\) intersect in one point.

Let \(ABC\) be a triangle. Prove that the medians \(AD\), \(BE\), \(CF\) intersect in one point.

Let \(ABC\) be a triangle with medians \(AD\), \(BE\), \(CF\). Prove that the triangles \(ABC\) and \(DEF\) are similar. What is their similarity coefficient?

Let \(ABC\) be a triangle with midpoints \(D\) on the side \(BC\), \(E\) on the side \(AC\), and \(F\) on the side \(AB\). Prove that the perpendicular bisectors to the sides \(AB\), \(BC\), \(AC\) intersect at one point.

Let \(ABC\) be a triangle with midpoints \(D\) on the side \(BC\), \(E\) on the side \(AC\), and \(F\) on the side \(AB\). Let \(M\) be the point of intersection of all medians of the triangle \(ABC\), let \(H\) be the point of intersection of the heights \(AJ\), \(BI\) and \(CK\). Prove that the points \(D\), \(J\), \(I\), \(E\), \(F\) and \(K\) all lie on one circle.

Let \(ABC\) be a triangle with midpoints \(D\) on the side \(BC\), \(E\) on the side \(AC\) and \(F\) on the side \(AB\). Let \(M\) be the point of intersection of all medians of the triangle \(ABC\) and let \(H\) be the point of intersection of the heights \(AJ\), \(BI\) and \(CK\). Consider the Euler circle of the triangle \(ABC\), which is the one that contains the points \(D,J,I,E,F,K\). This circle intersects the segments \(AH\), \(BH\) and \(CH\) at the points \(O\), \(P\) and \(Q\) respectively. Prove that \(O\), \(P\) and \(Q\) are the midpoints of the segments \(AH\), \(BH\) and \(CH\).

Consider the point \(H\) of intersection of the heights of the triangle \(ABC\). Prove that Euler lines of the triangles \(ABC\), \(ABH\), \(BCH\) and \(ACH\) intersect at one point. On the diagram below the points \(R,S,T\) are the points of intersection of medians in triangles \(ABH\), \(BCH\), and \(ACH\) respectively.

Show that there are infinitely many composite numbers \(n\) such that \(3^{n-1}-2^{n-1}\) is divisible by \(n\).

Consider the triangle \(BCD\), inscribed in a circle with center \(A\). The segments \(EF\), \(FG\) and \(EG\) are tangent to the circle at the points \(C\), \(D\) and \(B\) respectively. Prove that the Euler line of the triangle \(BCD\) passes through the center of the circle circumscribed around the triangle \(EFG\).

Show that there are infinitely many integers \(n\) such that \(2^n+1\) is divisible by \(n\). Find all prime numbers that satisfy this property.