Practice library
Problems
1The squares \(ABCD\) and \(EFGH\) both have side \(1\). The first is cut into nine congruent small squares and the second into sixteen congruent small squares; in every small square the inscribed circle …Open2Every edge of a certain pyramid has length \(5\) cm, and the pyramid has five vertices in all, so its base is a square. All five vertices are sliced off, each by a single plane, and the cuts are made so …3The figure shows the route a hare ran while a wolf was chasing it through the fog. The hare first ran east; then it turned right, after a while it turned left, and shortly afterwards it turned left once …4At most how many interior angles of a polygon with \(n\) vertices can be greater than \(180^\circ\)? (The polygon is simple: its sides meet only at the shared endpoints of neighbouring sides.) A \(n - 1\) …5Janez drew a pattern on a sheet of paper, made up of congruent squares and congruent hexagons. On top of the pattern he then drew two dashed lines perpendicular to each other, as in the figure. What is …6A regular octagon is inscribed in a square of side length \(a\) so that four sides of the octagon lie on the four sides of the square. Express the side length of the octagon in terms of \(a\).7In a triangle \(ABC\), the bisector of the angle \(\angle BAC\) meets the side \(BC\) at the point \(D\). The triangle \(ADC\) is isosceles with apex \(D\), that is, \(|DA| = |DC|\). Given \(|CD| = 36\) …8Three straight cuts divide a rectangle into four pieces, as shown in the figure; the cut meeting the top edge is perpendicular to it. The four pieces are then rearranged, without gaps or overlaps, into …9In a triangle \(ABC\), the bisector of the angle \(\angle BAC\) meets the side \(BC\) at \(D\), and the bisector of the angle \(\angle CBA\) meets the side \(AC\) at \(E\). Suppose that \(|CD| = |CE|\). …10Let \(ABC\) be a right triangle with its right angle at \(C\), and write \(|BC| = a\), \(|AC| = b\). Let \(D\) be a point on the opposite side of the line \(AC\) from \(B\) for which the triangle \(ACD\) …
Showing 10 of 651 - problem statements are free for everyone.