Problems

Age
Difficulty
Found: 705

Catherine laid out 2016 matches on a table and invited Anna and Natasha to play a game which involves taking turns to remove matches from a table: Anna can take 5 matches or 26 matches in her turn, and Natasha can take either 9 or 23. Without waiting for the start of the game, Catherine left, and when she returned, the game was already over. On the table there are two matches, and the one who can not make another turn loses. After a good reflection, Catherine realised which person went first and who won. Figure it out for yourself now.

At a round table, there are 10 people, each of whom is either a knight who always speaks the truth, or a liar who always lies. Two of them said: “Both my neighbors are liars,” and the remaining eight stated: “Both my neighbors are knights.” How many knights could there be among these 10 people?

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: 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.

An abstract artist took a wooden \(5\times 5\times 5\) cube and divided each face into unit squares. He painted each square in one of three colours – black, white, and red – so that there were no horizontally or vertically adjacent squares of the same colour. What is the smallest possible number of squares the artist could have painted black following this rule? Unit squares which share a side are considered adjacent both when the squares lie on the same face and when they lie on adjacent faces.

At the power plant, rectangles that are 2 m long and 1 m wide are produced. The length of the objects is measured by the worker Howard, and the width, irrespective of Howard, is measured by the worker Rachel. The average error is zero for both, but Howard allows a standard measurement error (standard deviation of length) of 3 mm, and Rachel allows a standard error of 2 mm.

a) Find the mathematical expectation of the area of the resulting rectangle.

b) Find the standard deviation of the area of the resulting rectangle in centimetres squared.

At a factory known to us, we cut out metal disks with a diameter of 1 m. It is known that a disk with a diameter of exactly 1 m weighs exactly 100 kg. During manufacturing, a measurement error occurs, and therefore the standard deviation of the radius is 10 mm. Engineer Gavin believes that a stack of 100 disks on average will weigh 10,000 kg. By how much is the engineer Gavin wrong?

At a conference there were 18 scientists, of which exactly 10 know the eye-popping news. During the break (coffee break), all scientists are broken up into random pairs, and in each pair, anyone who knows the news, tells this news to another if he did not already know it.

a) Find the probability that after the coffee break, the number of scientists who know the news will be 13.

b) Find the probability that after the coffee break the number of scientists who know the news will be 14.

c) Denote by the letter \(X\) the number of scientists who know the eye-popping news after the coffee break. Find the mathematical expectation of \(X\).

A high rectangle of width 2 is open from above, and the L-shaped domino falls inside it in a random way (see the figure).

a) \(k\) \(L\)-shaped dominoes have fallen. Find the mathematical expectation of the height of the resulting polygon.

b) \(7\) \(G\)-shaped dominoes fell inside the rectangle. Find the probability that the resulting figure will have a height of 12.

Two hockey teams of the same strength agreed that they will play until the total score reaches 10. Find the mathematical expectation of the number of times when there is a draw.