There are scales and 100 coins, among which several (more than 0 but less than 99) are fake. All of the counterfeit coins weigh the same and all of the real ones also weigh the same, while the counterfeit coin is lighter than the real one. You can do weighings on the scales by paying with one of the coins (whether real or fake) before weighing. Prove that it is possible with a guarantee to find a real coin.
Author: I.S. Rubanov
On the table, there are 7 cards with numbers from 0 to 6. Two take turns in taking one card. The winner is the one is the first person who can, from his cards, make up a natural number that is divisible by 17. Who will win in a regular game the person who goes first or second?
Henry wrote a note on a piece of paper, folded it two times, and wrote “FOR MOM” on the top. Then he unfolded the note, added something to it, randomly folded the note along the old folding lines (not necessarily in the same way as he did it before) and left it on the table with random side up. Find the probability that “FOR MOM” is still on the top.
Author: Shapovalov A.V.
Let \(A\) and \(B\) be two rectangles. From rectangles equal to \(A\), a rectangle similar to \(B\) was created.
Prove that from rectangles equal to \(B\), you can create a rectangle similar to \(A\).
Author: A. Glazyrin
In the coordinate space, all planes with the equations \(x \pm y \pm z = n\) (for all integers \(n\)) were carried out. They divided the space into tetrahedra and octahedra. Suppose that the point \((x_0, y_0, z_0)\) with rational coordinates does not lie in any plane. Prove that there is a positive integer \(k\) such that the point \((kx_0, ky_0, kz_0)\) lies strictly inside some octahedron from the partition.
There is an elastic band and glass beads: four identical red ones, two identical blue ones and two identical green ones. It is necessary to string all eight beads on the elastic band in order to get a bracelet. How many different bracelets can be made so that beads of the same colour are not next to each other? (Assume that there is no buckle, and the knot on the elastic is invisible).
Authors: B. Vysokanov, N. Medved, V. Bragin
The teacher grades tests on a scale from 0 to 100. The school can change the upper bound of the scale to any other natural number, recalculating the estimates proportionally and rounding up to integers. A non-integer number, when rounded, changes to the nearest integer; if the fractional part is equal to 0.5, the direction of rounding can be either up or down and it can be different for each question. (For example, an estimate of 37 on a scale of 100 after recalculation in the scale of 40 will go to \(37 \cdot 40/100 = 14.8\) and will be rounded to 15).
The students of Peter and Valerie got marks, which are not 0 and 100. Prove that the school can do several conversions so that Peter’s mark becomes b and Valerie’s mark becomes a (both marks are recalculated simultaneously).
Author: A.A. Egorov
Calculate the square root of the number \(0.111 \dots 111\) (100 ones) to within a) 100; b) 101; c)* 200 decimal places.
The segment \(OA\) is given. From the end of the segment \(A\) there are 5 segments \(AB_1, AB_2, AB_3, AB_4, AB_5\). From each point \(B_i\) there can be five more new segments or not a single new segment, etc. Can the number of free ends of the constructed segments be 1001? By the free end of a segment we mean a point belonging to only one segment (except point \(O\)).
Given \(n\) points that are connected by segments so that each point is connected to some other and there are no two points that would be connected in two different ways. Prove that the total number of segments is \(n - 1\).