We solve problems

Problem #PRU-34905

Problems Geometry Plane geometry Area Area of a trapezium Area ratio Area Ratio Methods Pigeonhole principle Pigeonhole principle (finite number of poits, lines etc.) Quadrilateral Parallelogram Special cases Rectangles and squares. Criteria and properties

13–14 4.0

9 straight lines each divide a square into two quadrilaterals, with their areas having a ratio of \(2:3\). Prove that at least three of the nine lines pass through the same point.

Problem #PRU-34924

2.0

There is a counter on the chessboard. Two in turn move the counter to an adjacent on one side cell. It is forbidden to put a counter on a cell, which it has already visited. The one who can not make the next turn loses. Who wins with the right strategy?

Problem #PRU-34931

Problems Algebra and arithmetic Algebraic equations and systems of equations Higher order equations. Palindromic polynomial equations Equations of higher order (other) Calculus Real numbers Integer and fractional parts. Archimedean property

12–14 2.0

On the dining room table, there is a choice of six dishes. Every day Valentina takes a certain set of dishes (perhaps, she does not take a single dish), and this set of dishes should be different from all of the sets that she took in the previous days. What is the maximum number of days that Valentina will be able to eat according to such rules and how many meals will she eat on average during the day?

Problem #PRU-34943

Problems Combinatorics Geometry on grid paper

3.0

The city plan is a rectangle of \(5 \times 10\) cells. On the streets, a one-way traffic system is introduced: it is allowed to go only to the right and upwards. How many different routes lead from the bottom left corner to the upper right?

Problem #PRU-34945

Problems Set theory and logic Algorithm Theory Balance puzzles

2.0

27 coins are given, of which one is a fake, and it is known that a counterfeit coin is lighter than a real one. How can the counterfeit coin be found from 3 weighings on the scales without weights?

Problem #PRU-34962

Problems Methods Pigeonhole principle Pigeonhole principle (other) Combinatorics Systems of points and line segments Systems of points, lines, and line segments

12–14 2.0

Is it possible to arrange 1000 line segments in a plane so that both ends of each line segment rest strictly inside another line segment?

Problem #PRU-34968

Problems Algebra and arithmetic Number systems Decimal number system Number theory. Divisibility Division with remainders. Arithmetic of remainders Division with remainder

13–15 4.0

Prove that for any number \(d\), which is not divisible by \(2\) or by \(5\), there is a number whose decimal notation contains only ones and which is divisible by \(d\).

Problem #PRU-34970

Problems Combinatorics Partitions Covers Methods Pigeonhole principle Pigeonhole principle (angles and lengths)

13–14 3.0

Some open sectors – that is sectors of circles with infinite radii – completely cover a plane. Prove that the sum of the angles of these sectors is no less than \(360^\circ\).

Problem #PRU-34972

Problems Geometry Solid geometry Visual geometry in space

11–13 2.0

For which \(n > 3\), can a set of weights with masses of \(1, 2, 3, ..., n\) grams be divided into three groups of equal mass?

Problem #PRU-34976

Problems Probability and statistics Probability theory Discrete distribution Methods Algebraic methods Partitions into pairs and groups; bijections

12–16 4.0

Two people toss a coin: one tosses it 10 times, the other – 11 times. What is the probability that the second person’s coin showed heads more times than the first?