Problems

Certain geometric objects nicely blend when they happen to be together in a problem. One possible example of such a pair of objects is a circle and an inscribed angle.
We will be using the following statements in the examples and problems:
1. The supplementary angles (angles “hugging" a straight line) add up to \(180^{\circ}\).
2. The sum of all internal angles of a triangle is also \(180^{\circ}\).

3. Two triangles are said to be “congruent" if ALL of their corresponding sides and angles are equal.
The following terminology will also be quite helpful. In the picture below, the points \(B\) and \(C\) lie on the circumference of the circle while the vertex \(A\) lies at the centre of the circle. We say that the angle \(\angle BAC\) is a central angle. The angle \(\angle DFE\) is called an inscribed angle because the vertices \(D\), \(F\) and \(E\) all lie on the circumference of the circle.

Prove that the product of five consecutive integers is divisible by \(120\).

Prove that the vertices of a planar graph can be coloured in (at most) six different colours such that every pair of vertices joined by an edge are of different colours.

Note: a graph is planar if it can be drawn in the plane with no edges crossing. For example, three houses, each of which is connected to three utilities, is not a planar graph.
You may find it useful to use the Euler characteristic: a planar graph with \(v\) vertices, \(e\) edges and \(f\) faces satisfies \(v-e+f=2\).

Norman painted the plane using two colours: red and yellow. Both colours are used at least once. Show that no matter how Norman does this, there is a red point and a yellow point exactly \(1\)cm apart.

Two players are playing a game. The first player is thinking of a finite sequence of positive integers \(a_1\), \(a_2\), ..., \(a_n\). The second player can try to find the first player’s sequence by naming their own sequence \(b_1\), \(b_2\), ..., \(b_n\). After this, the first player will give the result \(a_1b_1 + a_2b_2 + ...+a_nb_n\). Then the second player can say another sequence \(c_1\), \(c_2\), ..., \(c_n\) to get another answer \(a_1c_1+ a_2c_2 + ... +a_nc_n\) from the first player. Find the smallest number of sequences the second player has to name to find out the sequence \(a_1\), \(a_2\), ..., \(a_n\).

The letters \(A\), \(R\), \(S\) and \(T\) represent different digits from \(1\) to \(9\). The same letters correspond to the same digits, while different letters correspond to different digits.
Find \(ART\), given that \(ARTS+STAR=10,T31\).

Let \(ABC\) be a non-isosceles triangle. The point \(G\) is the point of intersection of the medians \(AE\), \(BF\), \(CD\). The point \(H\) is the point of intersection of all heights. The point \(I\) is the center of the circumscribed circle of \(ABC\), or the point of intersection of all perpendicular bisectors to the segments \(AB\), \(BC\), \(AC\).
Prove that points \(I,G,H\) lie on one line and that the ratio \(IG:GH = 1:2\). The line that all of \(I\), \(G\) and \(H\) lie on is called the Euler line of triangle \(ABC\).

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.