We solve problems

Problems Geometry Solid geometry Visual geometry in space

How many times greater is the sum of the numbers in the hundred and first line of the Pascal triangle than the sum of the numbers in the hundredth line?

Problem #PRU-32903

Problems Geometry Solid geometry Visual geometry in space

12–12 2.0

Let’s put plus and minus signs in the 99th line of Pascal’s triangle. Between the first and second number there is a minus sign, between the second and the third there is a plus sign, between the third and the fourth there is a minus sign, then again a plus sign, and so on. Find the value of the resulting expression.

Problem #PRU-32904

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

Can you find

a) in the 100th line of Pascal’s triangle, the number \(1 + 2 + 3 + \dots + 98 + 99\)?

b) in the 200th line the sum of the squares of the numbers in the 100th line?

Problem #PRU-32987

Problems Algebra and arithmetic Algebraic equations and systems of equations

11–13 2.0

Prove there are no integer solutions for the equation \(3x^2 + 2 = y^2\).

Problem #PRU-33134

Problems Algebra and arithmetic Number theory. Divisibility Odd and even numbers

11–13 2.0

Prove that the sum of

a) any number of even numbers is even;

b) an even number of odd numbers is even;

c) an odd number of odd numbers is odd.

Problem #PRU-33135

Problems Algebra and arithmetic Number theory. Divisibility Odd and even numbers

11–13 2.0

Prove that the product of

a) two odd numbers is odd;

b) an even number with any integer is even.

Problem #PRU-34834

Problems Algebra and arithmetic Sequences Progressions Arithmetic progression Methods Pigeonhole principle Pigeonhole principle (angles and lengths)

12–14 3.0

A road of length 1 km is lit with streetlights. Each streetlight illuminates a stretch of road of length 1 m. What is the maximum number of streetlights that there could be along the road, if it is known that when any single streetlight is extinguished the street will no longer be fully illuminated?

Problem #PRU-34840

Problems Set theory and logic Mathematical logic Mathematical logic (other)

12–13 2.0

Several guests are sitting at a round table. Some of them are familiar with each other; mutually acquainted. All the acquaintances of any guest (counting himself) sit around the table at regular intervals. (For another person, these gaps may be different.) It is known that any two have at least one common acquaintance. Prove that all guests are familiar with each other.

Problem #PRU-34851

Problems Methods Pigeonhole principle Pigeonhole principle (other)

12–14 3.0

15 MPs take part in a debate. During the debate, each one criticises exactly \(k\) of the 14 other contributors. For what minimum value of \(k\) is it possible to definitively state that there will be two MPs who have criticised one another?

Problem #PRU-34854

Problems Methods Pigeonhole principle Pigeonhole principle (other) Algebra and arithmetic Sequences Recurrent relations Recurrent relations (other)

12–14 3.0

In the number \(1234096\dots\) each digit, starting with the 5th digit is equal to the final digit of the sum of the previous 4 digits. Will the digits 8123 ever occur in that order in a row in this number?