Prove that rational numbers from \([0; 1]\) can be covered by a system of intervals of total length no greater than \(1/1000\).
A city in the shape of a triangle is divided into 16 triangular blocks, at the intersection of any two streets is a square (there are 15 squares in the city). A tourist began to walk around the city from a certain square and travelled along some route to some other square, whilst visiting every square exactly once. Prove that in the process of travelling the tourist at least 4 times turned by \(120^{\circ}\).
The positive irrational numbers \(a\) and \(b\) are such that \(1/a + 1/b = 1\). Prove that among the numbers \(\lfloor ma\rfloor , \lfloor nb\rfloor\) each natural number occurs exactly once.
A convex polygon on a plane contains no fewer than \(m^2+1\) points with whole number co-ordinates. Prove that within the polygon there are \(m+1\) points with whole number co-ordinates that lie on a single straight line.
Does a continuous function that takes every real value exactly 3 times exist?
To transmit messages by telegraph, each letter of the Russian alphabet () ( and are counted as identical) is represented as a five-digit combination of zeros and ones corresponding to the binary number of the given letter in the alphabet (letter numbering starts from zero). For example, the letter is represented in the form 00000, letter -00001, letter -10111, letter -11111. Transmission of the five-digit combination is made via a cable containing five wires. Each bit is transmitted on a separate wire. When you receive a message, Cryptos has confused the wires, so instead of the transmitted word, a set of letters is received. Find the word you sent.
A rectangular billiard with sides 1 and \(\sqrt {2}\) is given. From its angle at an angle of \(45 ^\circ\) to the side a ball is released. Will it ever get into one of the pockets? (The pockets are in the corners of the billiard table).
Let \(E\) and \(F\) be the midpoints of the sides \(BC\) and \(AD\) of the parallelogram \(ABCD\). Find the area of the quadrilateral formed by the lines \(AE, ED, BF\) and \(FC\), if it is known that the area \(ABCD\) is equal to \(S\).
Prove that a convex quadrilateral \(ICEF\) can contain a circle if and only if \(IC+EH = CE+IF\).
Let \(O\) be the center of the rectangle \(ABCD\). Find the geometric points of \(M\) for which \(AM \geq OM, BM \geq OM\), \(CM \geq OM\), and \(DM \geq OM\).