The point \(A\) is fixed on a circle. Find the locus of the point \(X\) which divides the chords that end at point \(A\) in a \(1:2\) ratio, starting from the point \(A\).
Let \(O\) be the center of the rectangle \(ABCD\). Find the geometric points of \(M\) for which \(AM \geq OM, BM \geq OM\), \(CM \geq OM\), and \(DM \geq OM\).
Construct a right-angled triangle along the leg and the hypotenuse.
Construct a circle with a given centre, tangent to a given circle.
Construct a straight line passing through a given point and tangent to a given circle.
Three segments whose lengths are equal to \(a, b\) and \(c\) are given. Using only straightedge and compass construct a segment of length: a) \(ab/c\); b) \(\sqrt {ab}\).
Construct the triangle ABC by the medians \(m_a, m_b\) and \(m_c\).
Solve the equations \(x^2 = 14 + y^2\) in integers.
Many maths problems begin with the question “Is it possible…?”. In these kinds of problems, what you need to do depends on what you think is true.
If you believe it is possible, then you must give an example that really satisfies the conditions in the problem.
If you believe it is not possible, then you must explain clearly why it cannot be done.
When trying to build an example, it often helps to ask yourself extra questions to narrow things down: “How could it be possible?”, or “What properties must a correct example have?”.
On the other hand, if you have been trying to build an example for a while and nothing works, perhaps the answer is that it is impossible. In that case, look for a property that any example would need to have — and then show why that property cannot actually happen. Let’s see some examples!
Suppose that a rectangle can be divided into \(13\) equal smaller squares. What could be the side lengths of this rectangle?