Given a $m \times n$ ($m, n \geq 1$) grid, place a pawn at the center of each square and one at each vertex of the grid (in the example shown alongside with $n=4$ and $m=3$, 32 pawns are placed).
a. Find all tables that have exactly 500 checkers.
b. Prove that there exist infinitely many positive integers $k$ such that there are no grids with exactly $k$ checkers.
On an island there are two categories of people: the knights, who always tell the truth, and the villains, who always lie. On a certain day the Grand Council meets, consisting of 2003 members. They sit randomly around a round table, and during the meeting each of them declares: "Both of them $i$ My neighbors are rogues. "The next day the council meets again, but one of the members is sick, so there are only 2002 members present. They re-arrange themselves at random again around the round table and each of them says:" Both of my neighbors belong to the category opposite to mine ". Was the patient a villain or a knight?
For how many pairs $(p, q)$ of (positive) prime numbers does the polynomial $x^{2}+p x+q$ have two integer roots? NOTE: Please remember that 1 is not a prime number.
In a rectangular parallelepiped $P$ the length of the diagonal is $\sqrt{133}$ and the total surface area is $228$. Knowing that one of the sides is average proportional between the other two, the volume of $P$ is
Un'azienda dolciaria produces two tipi di torrone, using the stessa bianca pasta and the stesse nocciole, but in diverse proportions. In the ring of type $A$ the number shows the $30\%$ of the weight and the $40\%$ of the volume; In the $B$ type, the number indicates the $60\%$ weight. What percentage of the volume represents the noise in the tower of type $B$?
You want to give seven gift packages to seven children, one to each. We want to ensure that in each package there are three different games and that, however two children are chosen, they receive at most one game in common. What is the minimum number of distinct game types you need to play?
In a square $ABCD$ with side $2$, a segment $MN$ of length $1$ is constrained to have endpoint $M$ on side $AB$ and endpoint $N$ on side $BC$. This segment divides the square into a triangle $T$ and a pentagon $P$. What is the maximum value that the ratio of the area of $T$ to that of $P$ can take?
In the country of $\mathrm{Oz}$, in addition to normal people (who can lie or tell the truth), there live knights (who always tell the truth) and knaves (who always lie). A strange law dictates that in every marriage the spouses are either both normal, or one is a knight and the other a knave. Once you arrive at the semi-detached house of the Allegri and Bianchi spouses, ask if Mr. Bianchi, who is absent, is a knight, obtaining an affirmative response from both Mrs. Bianchi and both Mr. and Mrs. Allegri. How many spouses can you determine the type of?
A 6-pointed star is drawn by constructing six isosceles triangles with a vertex angle of $30$ degrees on the sides of a regular hexagon. Knowing that the circumference passing through the points of the star has radius $1$, calculate the area of the star itself.
A game consists of 10 throws of a normal cubic die with the faces numbered from 1 to 6. At the end the scores obtained are added, with the rule that if you get a 6 in one throw the points of the next throw are counted doubled and that if you get a 6 on the last throw you are entitled to one (only) additional throw to which the score (not doubled) is added. Which of these final scores cannot be obtained?
Il piccolo Marco comes out and climbs on a piano all'altro the mobile scale of a shopping center, one scale there. Proceed in the direction of the scale at constant speed according to this (the interval of time between one step and the other is constant), calpesta 15 steps, proceed in the opposite direction (with the interval of time between one step and the other) and calpesta 35. Quanti scalini calpesterebbe Marco in the passare Does a piano all'altro se la scala mobile fosse ferma?
Let $\mathcal{R}$ be the finite region of the plane that is bounded by the $x$ axis and the graph of the $2 x^{2}+5 y=10$ equation curve. Given three points of $\mathcal{R}$, which of the following statements is always true?
at least two of the three points have distance $\geq \frac{\sqrt{5}}{2}$
at least two of the three points have distance $\leq 3$
the sum of the distances between the points is $\leq \frac{9}{2} \sqrt{5}$
the sum of the squares of the distances between the points is $\leq 38$
the product of the distances between points is $\leq 16 \sqrt{5}$.
A dodecahedron is a regular solid with 12 pentagonal faces. A diagonal of a solid is a segment whose ends are two vertices of the solid that do not belong to the same face. How many diagonals are there in the dodecahedron?
Let $x_{0}, x_{1}, x_{2}, \ldots$ be the sequence defined by $x_{0}=2$ and $x_{n+1}=5+\left(x_{n}\right)^{2}$ for each $n \geq 0$. Prove that no prime numbers other than 2 appear in this sequence.
A square sided numeric keypad $4$ has $16$ keys numbered $1$ to $16$. Agent $007$ must press two buttons at the same time to enter the enemy base, and if he gets it wrong he will sound the alarm. However, it only knows that the two keys are not contiguous (i.e. they do not have a common edge or vertex). What is the probability that he will be able to infiltrate?
Determine the number of quadruples of integers (not necessarily distinct) between $1$ and $12$ (extremes included) that satisfy all the following conditions:
- the sum of the first two numbers is even
- the sum of the first three numbers is a multiple of $3$
- the sum of the four numbers is a multiple of $4$.
(Two quadruples that differ even only in the order of the addends are to be considered distinct).
