A pharmacist has three weights, with which he measured out and gave 100 g of iodine to one buyer, 101 g of honey to another, and 102 g of hydrogen peroxide to the third. He always placed the weights on one side of the scales, and the goods on the other. Could it be that each weight used is lighter than 90 grams?
Gary drew an empty table of \(50 \times 50\) and wrote on top of each column and to the left of each row a number. It turned out that all 100 written numbers are different, and 50 of them are rational, and the remaining 50 are irrational. Then, in each cell of the table, he wrote down a product of numbers written at the top of its column and to the left of the row (the “multiplication table”). What is the largest number of products in this table which could be rational numbers?
On the school board a chairman is chosen. There are four candidates: \(A\), \(B\), \(C\) and \(D\). A special procedure is proposed – each member of the council writes down on a special sheet of candidates the order of his preferences. For example, the sequence \(ACDB\) means that the councilor puts \(A\) in the first place, does not object very much to \(C\), and believes that he is better than \(D\), but least of all would like to see \(B\). Being placed in first place gives the candidate 3 points, the second – 2 points, the third – 1 point, and the fourth - 0 points. After collecting all the sheets, the election commission summarizes the points for each candidate. The winner is the one who has the most points.
After the vote, \(C\) (who scored fewer points than everyone) withdrew his candidacy in connection with his transition to another school. They did not vote again, but simply crossed out \(B\) from all the leaflets. In each sheet there are three candidates left. Therefore, first place was worth 2 points, the second – 1 point, and the third – 0 points. The points were summed up anew.
Could it be that the candidate who previously had the most points, after the self-withdrawal of \(B\) received the fewest points?
Four outwardly identical coins weigh 1, 2, 3 and 4 grams respectively.
Is it possible to find out in four weighings on a set of scales without weights, which one weighs how much?
Author: A.K. Tolpygo
An irrational number \(\alpha\), where \(0 <\alpha <\frac 12\), is given. It defines a new number \(\alpha_1\) as the smaller of the two numbers \(2\alpha\) and \(1 - 2\alpha\). For this number, \(\alpha_2\) is determined similarly, and so on.
a) Prove that for some \(n\) the inequality \(\alpha_n <3/16\) holds.
b) Can it be that \(\alpha_n> 7/40\) for all positive integers \(n\)?
Author: L.N. Vaserstein
For any natural numbers \(a_1, a_2, \dots , a_m\), no two of which are equal to each other and none of which is divisible by the square of a natural number greater than one, and also for any integers and non-zero integers \(b_1, b_2, \dots , b_m\) the sum is not zero. Prove this.
Author: V.A. Popov
On the interval \([0; 1]\) a function \(f\) is given. This function is non-negative at all points, \(f (1) = 1\) and, finally, for any two non-negative numbers \(x_1\) and \(x_2\) whose sum does not exceed 1, the quantity \(f (x_1 + x_2)\) does not exceed the sum of \(f (x_1)\) and \(f (x_2)\).
a) Prove that for any number \(x\) on the interval \([0; 1]\), the inequality \(f (x_2) \leq 2x\) holds.
b) Prove that for any number \(x\) on the interval \([0; 1]\), the \(f (x_2) \leq 1.9x\) must be true?
The triangle \(C_1C_2O\) is given. Within it the bisector \(C_2C_3\) is drawn, then in the triangle \(C_2C_3O\) – bisector \(C_3C_4\) and so on. Prove that the sequence of angles \(\gamma_n = C_{n + 1}C_nO\) tends to a limit, and find this limit if \(C_1OC_2 = \alpha\).
What has a greater value: \(300!\) or \(100^{300}\)?
There are 13 weights, each weighing an integer number of grams. It is known that any 12 of them can be divided into two cups of weights, six weights on each one, which will come to equilibrium. Prove that all the weights have the same weight.