Problems

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 $(k{x}_0,k{y}_0,k{z}_0)$ lies strictly inside some octahedron from the partition.

One hundred cubs found berries in the forest: the youngest managed to grab 1 berry, the next youngest cub – 2 berries, the next – 4 berries, and so on, until the oldest who got $2^{99}$ berries. The fox suggested that they share the berries “fairly.” She can approach two cubs and distribute their berries evenly between them, and if this leaves an extra berry, then the fox eats it. With such actions, she continues, until all the cubs have an equal number of berries. What is the largest number of berries that the fox can eat?

Hannah Montana wants to leave the round room which has six doors, five of which are locked. In one attempt she can check any three doors, and if one of them is not locked, then she will go through it. After each attempt her friend Michelle locks the door, which was opened, and unlocks one of the neighbouring doors. Hannah does not know which one exactly. How should she act in order to leave the room?

There are 30 students in a class: excellent students, mediocre students and slackers. Excellent students answer all questions correctly, slackers are always wrong, and the mediocre students answer questions alternating one by one correctly and incorrectly. All the students were asked three questions: “Are you an excellent pupil?”, “Are you a mediocre student?”, “Are you a slacker?”. 19 students answered “Yes” to the first question, to the second 12 students answered yes, to the third 9 students answered yes. How many mediocre students study in this class?

100 switched on and 100 switched off lights are randomly arranged in two boxes. Each flashlight has a button, the button of which turns off an illuminated flashlight and switches on a turned off flashlight. Your eyes are closed and you can not see if the flashlight is on. But you can move the flashlights from a box to another box and press the buttons on them. Think of a way to ensure that the burning flashlights in the boxes are equally split.

Replace the letters with numbers (all digits must be different) so that the correct equality is obtained: $A/B/C+D/E/F+G/H/I=1$.

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?

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).

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?