We solve problems

Problems Geometry Plane geometry Geometrical inequalities Triangle inequality

The distance between London and Warsaw equals \(1450\) km, between Warsaw and Kyiv is \(680\) km. The distance from London to New Delhi, is \(6700\) km and the distance from Kyiv to New Delhi is \(4570\) km. What is the distance from London to Kyiv?

Problem #PRU-100679

Problems Geometry Plane geometry Geometrical inequalities Triangle inequality

Show that if all sides of a triangle have integer lengths and one of them is equal to \(1\), then the other two have lengths equal to each other.

Problem #PRU-100682

Problems Geometry Plane geometry Geometrical inequalities Triangle inequality

A billiard ball lies on a table in the shape of an acute angle. How should you hit the ball so that it returns to its starting location after hitting each of the two banks once? Is it always possible to do so?
(When the ball hits the bank, it bounces. The way it bounces is determined by the shortest path rule – if it begins at some point \(D\) and ends at some point \(D'\) after bouncing, the path it takes is the shortest possible path that includes the bounce.)

Problem #PRU-103818

Problems Geometry Plane geometry Quadrilateral Parallelogram Special cases Rectangles and squares. Criteria and properties

11–12 2.0

Which rectangles with whole sides are there more of: with perimeter of 1996 or with perimeter of 1998? (The rectangles \(a \times b\) and \(b \times a\) are assumed to be the same).

Problem #PRU-103947

Problems Geometry Plane geometry Area

10–12 2.0

Find the area of the figures shown below.

Problem #PRU-104117

Problems Geometry Plane geometry Visual geometry

12–13 2.0

In a board, 20 pins are placed (see the picture). The distance between any adjacent pins is 1 inch. Pull a string of length 19 inches from the first pin to the second one, so that it goes through all the pins.

Problem #PRU-107992

Problems Geometry Plane geometry Circles Central angle. Arc length and circumference Calculus Real numbers Integer and fractional parts. Archimedean property Methods Pigeonhole principle Pigeonhole principle (angles and lengths) Algebra Sequences Sequences Algebraic methods Symmetry and involutions

14–16 5.0

For each pair of real numbers \(a\) and \(b\), consider the sequence of numbers \(p_n = \lfloor 2 \{an + b\}\rfloor\). Any \(k\) consecutive terms of this sequence will be called a word. Is it true that any ordered set of zeros and ones of length \(k\) is a word of the sequence given by some \(a\) and \(b\) for \(k = 4\); when \(k = 5\)?

Note: \(\lfloor c\rfloor\) is the integer part, \(\{c\}\) is the fractional part of the number \(c\).

Problem #PRU-108604

Problems Geometry Plane geometry Triangles Relationship of side lengths and angles of a triangle. Solving triangles. Ceva's theorem and Menelaus's theorem Calculus Integral Existence of a definite integral Vectors Law of polygon of vectors Similar triangles

13–14 3.0

On the sides \(AB\), \(BC\) and \(AC\) of the triangle \(ABC\) points \(P\), \(M\) and \(K\) are chosen so that the segments \(AM\), \(BK\) and \(CP\) intersect at one point and \[\vec{AM} + \vec{BK}+\vec{CP} = 0\] Prove that \(P\), \(M\) and \(K\) are the midpoints of the sides of the triangle \(ABC\).

Problem #PRU-109748

Problems Geometry Plane geometry Convex and non-convex figures Convex polygons Combinatorics Painting problems Methods Pigeonhole principle Pigeonhole principle (finite number of poits, lines etc.) Polygons Polygons (other)

13–15 4.0

We are given a convex 200-sided polygon in which no three diagonals intersect at the same point. Each of the diagonals is coloured in one of 999 colours. Prove that there is some triangle inside the polygon whose sides lie some of the diagonals, so that all 3 sides are the same colour. The vertices of the triangle do not necessarily have to be the vertices of the polygon.

Problem #PRU-110157

Problems Geometry Plane geometry Plane transformations Similarity Homothetic transformations, rotations of homothetic transformations A homothetic transformation helps solve the problem Circles Connection between the arc length, the angle, and the chord Calculus Derivative Derivative and multiple roots Homothetic circles Tangent lines and tangent circles Tangent lines to circles Properties and criteria of a secant

14–16 5.0

The circles \(\sigma_1\) and \(\sigma_2\) intersect at points \(A\) and \(B\). At the point \(A\) to \(\sigma_1\) and \(\sigma_2\), respectively, the tangents \(l_1\) and \(l_2\) are drawn. The points \(T_1\) and \(T_2\) are chosen respectively on the circles \(\sigma_1\) and \(\sigma_2\) so that the angular measures of the arcs \(T_1A\) and \(AT_2\) are equal (the arc value of the circle is considered in the clockwise direction). The tangent \(t_1\) at the point \(T_1\) to the circle \(\sigma_1\) intersects \(l_2\) at the point \(M_1\). Similarly, the tangent \(t_2\) at the point \(T_2\) to the circle \(\sigma_2\) intersects \(l_1\) at the point \(M_2\). Prove that the midpoints of the segments \(M_1M_2\) are on the same line, independent of the positions of the points \(T_1, T_2\).