We solve problems

Problem #PRU-97956

Problems Geometry Plane geometry Polygons Regular polygons Discrete Mathematics Algorithm Theory Game theory Symmetric strategies

12–14 2.0

a) The vertices (corners) in a regular polygon with 10 sides are colored black and white in an alternating fashion (i.e. one vertex is black, the next is white, etc). Two people play the following game. Each player in turn draws a line connecting two vertices of the same color. These lines must not have common vertices (i.e. must not begin or end on the same dot as another line) with the lines already drawn. The winner of the game is the player who made the final move. Which player, the first or the second, would win if the right strategy is used?

b) The same problem, but for a regular polygon with 12 sides.

Problem #PRU-100454

Problems Discrete Mathematics Set theory and logic Mathematical logic Paradoxes Geometry Plane geometry Quadrilateral Parallelogram Special cases Rectangles and squares. Criteria and properties

The area of a rectangle is 1 cm\(^2\). Can its perimeter be greater than 1 km?

Problem #PRU-57301

Problems Geometry Plane geometry Geometrical inequalities Geometric inequalities (other)

13–14 3.0

Let \(ABC\) be a triangle, prove that \(\angle ABC > 90^{\circ}\) if and only if the point \(B\) lies inside a circle with diameter \(AC\).

Problem #PRU-57633

Problems Geometry Plane geometry Triangles Relationship of side lengths and angles of a triangle. Solving triangles. Relations between sides and angles of a triangle (other)

13–15 2.0

Two intersecting circles of radius \(R\) are given, and the distance between their centers is greater than \(R\). Prove that \(\angle ECD = 3\angle CAD\).

Problem #PRU-57808

Problems Geometry Plane geometry Plane transformations Parallel translation

13–15 4.0

Two circles of radius \(R\) touch at point \(B\). On one of them, point \(D\) is chosen and on the other point \(E\) is chosen. These points have a property of \(\angle DBE = 90^{\circ}\). Prove that \(DE = 2R\).

Problem #PRU-57809

Problems Geometry Plane geometry Plane transformations Parallel translation

13–15 3.0

Two circles of radius \(R\) intersect at points \(B\) and \(D\). Consider the perpendicular bisector of the segment \(BD\). This line meets the two circles again at points \(F\) and \(G\), both chosen on the same side of \(BD\). Prove that \[BD^2 + FG^2 = 4R^2.\]

Problem #PRU-57810

Problems Geometry Plane geometry Plane transformations Parallel translation

13–14 2.0

Inside the rectangle \(ABCD\), the point \(E\) is taken. Prove that there exists a convex quadrilateral with perpendicular diagonals of lengths \(AB\) and \(BC\) whose sides are equal to \(AE\), \(BE\), \(CE\), \(DE\).

Problem #PRU-58082

Problems Geometry Plane geometry Geometrical inequalities Inequalities in a triangle A segment inside the triangle is smaller than the largest side ,Problems Geometry Plane geometry Triangles Types of triangles Right-angled triangles Pythagorean theorem and its converse

12–14 4.0

A \(3\times 4\) rectangle contains 6 points. Prove that amongst them there will be two points, such that the distance between them is no greater than \(\sqrt5\).

Problem #PRU-58085

Problems Discrete Mathematics Combinatorics Dissections, partitions, covers and tilings Covers Geometry Plane geometry Polygons Inscribed and circumscribed polygons Methods Pigeonhole principle Pigeonhole principle (finite number of poits, lines etc.)

13–15 4.0

A unit square contains 51 points. Prove that it is always possible to cover three of them with a circle of radius \(\frac{1}{7}\).

Problem #PRU-58089

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

12–15 4.0

What is the minimum number of points necessary to mark inside a convex \(n\)-gon, so that at least one marked point always lies inside any triangle whose vertices are the vertices of the polygon?