Problems

Age
Difficulty
Found: 500

Let C1 and C2 be two concentric circles with C1 inside C2 and the center A. Let B and D be two points on C1 that are not diametrically opposite. Extend the segment BD past D until it meets the circle C2 in C. The tangent to C2 at C and the tangent to C1 at B meet in a point E. Draw from E the second tangent to C2 which meets C2 at the point F. Show that BE bisects angle FBC.

image

Michael made a cube with edge 1 out of eight bars as in the picture. All 8 bars have the same volume. The dimensions of the grey bars are the same as each other. Similarly, the dimensions of the white bars are the same as each other. Find the lengths of the edges of the white bars.

image

Peter went to the Museum of Modern Art and saw a square painting in a frame of an unusual shape. The frame consisted of 21 congruent triangles. Peter was interested in what the angles of these triangles were equal to. Help him find these angles.

image

Let A, B, C and D be four points labelled clockwise on the circumference of a circle. The diagonals AC and BD intersect at the centre O of the circle. What can be deduced about the quadrilateral ABCD?

Consider the 7 different tetrominoes. Is it possible to cover a 4×7 rectangle with exactly one copy of each of the tetrominoes? If it is possible, then provide an example layout. If it is not possible, then prove that it’s impossible.

We allow rotation of the tetrominoes, but not reflection. This means that we consider S and Z as different, as well as L and J.

image

Let ABCDE be a regular pentagon. The point G is the midpoint of CD, the point F is the midpoint of AE. The lines EG and BF intersect at the point H. Find the angle EHF.

image

A paper band of constant width is tied into a simple knot and tightened. Prove that the knot has the shape of a regular polygon.

image

ABC is a triangle. The circumscribed circle is the circle that touches all three vertices of the triangle ABC. It is also the smallest circle lying entirely outside the triangle. The center of the circumscribed circle is D.

The inscribed circle is the circle which touches all three sides of the triangle ABC. It is also the largest circle lying entirely inside the triangle. The center of the inscribed circle is E.

The points D and E are symmetric with respect to the segment AC. Find the angles of the triangle ABC.

image

Between two mirrors AB and AC, forming a sharp angle two points D and E are located. In what direction should one shine a ray of light from the point D in such a way that it would reflect off both mirrors and hit the point E?
If a ray of light comes towards a surface under a certain angle, it is reflected with the same angle as on the picture.

image

Let a, b and c be the three side lengths of a triangle. Does there exist a triangle with side lengths a+1, b+1 and c+1? Does it depend on what a, b and c are?