Problems
Paloma wrote digits from \(0\) to \(9\) in each of the \(9\) dots below, using each digit at most once. Since there are \(9\) dots and \(10\) digits, she must have missed one digit.
In the triangles, Paloma started writing either the three digits at the corners added together (the sum), or the three digits at the corners multiplied together (the product). She gave up before finishing the final two triangles.
What numbers could Paloma have written in the interior of the red triangle? Demonstrate that you’ve found all of the possibilities.
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?
How many subsets are there of \(\{1,2,...,10\}\) (the integers from \(1\) to \(10\) inclusive) containing no consecutive
digits? That is, we do count \(\{1,3,6,8\}\) but do not count \(\{1,3,6,7\}\).
For example, when \(n=3\), we have
\(8\) subsets overall but only \(5\) contain no consecutive integers. The
\(8\) subsets are \(\varnothing\) (the empty set), \(\{1\}\), \(\{2\}\), \(\{3\}\), \(\{1,3\}\), \(\{1,2\}\), \(\{2,3\}\) and \(\{1,2,3\}\), but we exclude the final three
of these.
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.
