We solve problems

Problem #PRU-78661

Problems Methods Pigeonhole principle Pigeonhole principle (angles and lengths) Set theory and logic Set theory Union, intersection, and set difference

12–14 3.0

In a corridor of length 100 m, 20 sections of red carpet are laid out. The combined length of the sections is 1000 m. What is the largest number there can be of distinct stretches of the corridor that are not covered by carpet, given that the sections of carpet are all the same width as the corridor?

Problem #PRU-78768

Problems Methods Pigeonhole principle Pigeonhole principle (angles and lengths) Geometry Plane geometry Polygons Sum of internal and external angles of polygons

12–14 4.0

It is known that a camera located at $O$ cannot see the objects $A$ and $B$ , where the angle $A O B$ is greater than $179^{\circ}$ . 1000 such cameras are placed in a Cartesian plane. All of the cameras simultaneously take a picture. Prove that there will be a picture taken in which no more than 998 cameras are visible.

Problem #PRU-79285

Problems Algebra and arithmetic Number theory. Divisibility Division with remainders. Arithmetic of remainders Arithmetic of remainders Methods Pigeonhole principle Pigeonhole principle (other)

12–14 4.0

The sum of 100 natural numbers, each of which is no greater than 100, is equal to 200. Prove that it is possible to pick some of these numbers so that their sum is equal to 100.

Problem #PRU-79293

Problems Graph theory

12–14 3.0

A conference was attended by a finite group of scientists, some of whom are friends. It turned out that every two scientists, who have an equal number of friends at the conference, do not have friends in common. Prove that there is a scientist who has exactly one friend among the conference attendees.

Problem #PRU-79426

Problems Geometry Plane geometry Triangles Types of triangles Equilateral triangle Combinatorics Painting problems Methods Pigeonhole principle Pigeonhole principle (finite number of poits, lines etc.)

12–14 3.0

A white plane is arbitrarily sprinkled with black ink. Prove that for any positive $l$ there exists a line segment of length $l$ with both ends of the same colour.

Problem #PRU-79505

Problems Methods Pigeonhole principle Algebraic methods Proof by exhaustion

12–14 3.0

In March 2015 a teacher ran 11 sessions of a maths club. Prove that if no sessions were run on Saturdays or Sundays there must have been three days in a row where a session of the club did not take place. The 1st March 2015 was a Sunday.

Problem #PRU-79506

Problems Methods Pigeonhole principle Pigeonhole principle (other)

12–14 4.0

Prove that from any 27 different natural numbers less than 100, two numbers that are not coprime can be chosen.

Problem #PRU-79603

Problems Algebra and arithmetic Word problems Percentages and ratios Methods Pigeonhole principle Pigeonhole principle (other)

12–14 4.0

A council of 2,000 deputies decided to approve a state budget containing 200 items of expenditure. Each deputy prepared his draft budget, which indicated for each item the maximum allowable, in his opinion, amount of expenditure, ensuring that the total amount of expenditure did not exceed the set value of $S$ . For each item, the board approves the largest amount of expenditure that is agreed to be allocated by no fewer than $k$ deputies. What is the smallest value of $k$ for which we can ensure that the total amount of approved expenditures does not exceed $S$ ?

Problem #PRU-79636

Problems Set theory and logic Mathematical logic Paradoxes

10–12 2.0

Izzy wrote a correct equality on the board: $35 + 10 - 41 = 42 + 12 - 50$ , and then subtracted 4 from both parts: $35 + 10 - 45 = 42 + 12 - 54$ . She noticed that on the left hand side of the equation all of the numbers are divisible by 5, and on the right hand side by 6. Then she took 5 outside of the brackets on the left hand side and 6 on the right hand side and got $5 (7 + 2 - 9) 4 = 6 (7 + 2 - 9)$ . Having simplified both sides by a common multiplier, Izzy found that $5 = 6$ . Where did she go wrong?

Problem #PRU-79639

Problems Methods Pigeonhole principle Pigeonhole principle (finite number of poits, lines etc.)

11–12 2.0

A carpet of size 4 m by 4 m has had 15 holes made in it by a moth. Is it always possible to cut out a 1 m $\times$ 1 m area of carpet that doesn’t contain any holes? The holes are considered to be points.