Problems

There are 23 students in a class. During the year, each student of this class celebrated their birthday once, which was attended by some (at least one, but not all) of their classmates. Could it happen that every two pupils of this class met each other the same number of times at such celebrations? (It is believed that at every party every two guests met, and also the birthday person met all the guests.)

Author: G. Zhukov

The square trinomial $f(x)=a{x}^2+bx+c$ that does not have roots is such that the coefficient $b$ is rational, and among the numbers $c$ and $f(c)$ there is exactly one irrational.

Can the discriminant of the trinomial $f(x)$ be rational?

Author: A. Khrabrov

Do there exist integers $a$ and $b$ such that

a) the equation $x^2+ax+b=0$ does not have roots, and the equation $\lfloor x^2\rfloor +ax+b=0$ does have roots?

b) the equation $x^2+2ax+b=0$ does not have roots, and the equation $\lfloor x^2\rfloor +2ax+b=0$ does have roots?

Note that here, square brackets represent integers and curly brackets represent non-integer values or 0.

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.

Author: A.K. Tolpygo

12 grasshoppers sit on a circle at various points. These points divide the circle into 12 arcs. Let’s mark the 12 mid-points of the arcs. At the signal the grasshoppers jump simultaneously, each to the nearest clockwise marked point. 12 arcs are formed again, and jumps to the middle of the arcs are repeated, etc. Can at least one grasshopper return to his starting point after he has made a) 12 jumps; b) 13 jumps?

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: $A/B/C+D/E/F+G/H/I=1$.