A knight is modified so that it moves $p$ fields horizontally or vertically and $q$ fields in the perpendicular direction. It is placed on an infinite chessboard. If the knight returns to the initial field after $n$ moves, show that $n$ must be even.

Call a year [i]ultra-even[/i] if all of its digits are even. Thus $2000,2002,2004,2006$, and $2008$ are all [i]ultra-even[/i] years. They are all $2$ years apart, which is the shortest possible gap. $2009$ is not an [i]ultra-even[/i] year because of the $9$, and $2010$ is not an ultra-even year because of the $1$. (a) In the years between the years $1$ and $10000$, what is the longest possible gap between two [i]ultra-even[/i] years? Give an example of two ultra-even years that far apart with no [i]ultra-even[/i] years between them. Justify your answer. (b) What is the second-shortest possible gap (that is, the shortest gap longer than $2$ years) between two [i]ultra-even[/i] years? Again, give an example, and justify your answer.

There are $K$ boys placed around a circle. Each of them has an even number of sweets. At a command each boy gives half of his sweets to the boy on his right. If, after that, any boy has an odd number of sweets, someone outside the circle gives him one more sweet to make the number even. This procedure can be repeated indefinitely. Prove that there will be a time at which all boys will have the same number of sweets. (A Andjans, Riga)

Let the product of two polynomials of a variable $x$ with integer coefficients be a polynomial with even coefficients not all of which are divisible by $4$. Prove that all the coefficients of one of the polynomials are even and that at least one of the coefficients of the other polynomial is odd.

In the numerical triangle $................1..............$ $...........1 ...1 ...1.........$ $......1... 2... 3 ... 2 ... 1....$ $.1...3...6...7...6...3...1$ $...............................$ each number is equal to the sum of the three nearest to it numbers from the row above it; if the number is at the beginning or at the end of a row then it is equal to the sum of its two nearest numbers or just to the nearest number above it (the lacking numbers above the given one are assumed to be zeros). Prove that each row, starting with the third one, contains an even number.

Given $n \in N$, let $A$ be a family of subsets of $\{1,2,...,n\}$. If for every two sets $B,C \in A$ the set $(B \cup C) -(B \cap C)$ has an even number of elements, find the largest possible number of elements of $A$ .

Let $n > 6$ be a perfect number. Let $p_1^{a_1} \cdot p_2^{a_2} \cdot ... \cdot p_k^{a_k}$ be the prime factorisation of $n$, where we assume that $p_1 < p_2 <...< p_k$ and $a_i > 0$ for all $ i = 1,...,k$. Prove that $a_1$ is even. Remark: An integer $n \ge 2$ is called a perfect number if the sum of its positive divisors, excluding $ n$ itself, is equal to $n$. For example, $6$ is perfect, as its positive divisors are $\{1, 2, 3, 6\}$ and $1+2+3=6$.

Let $n$ be a positive integer and $(x_1, x_2, ..., x_{2n})$, $x_i = 0$ or $1, i = 1, 2, ... , 2n$ be a sequence of $2n$ integers. Let $S_n$ be the sum $S_n = x_1x_2 + x_3x_4 + ... + x_{2n-1}x_{2n}$. If $O_n$ is the number of sequences such that $S_n$ is odd and $E_n$ is the number of sequences such that $S_n$ is even, prove that $$\frac{O_n}{E_n}=\frac{2^n - 1}{2^n + 1}$$

For a natural number $n$, denote by $s(n)$ the sum of all positive divisors of n. Prove that for every $n > 1$ the product $s(n - 1)s(n)s(n + 1)$ is even.

The product $a_1 \cdot a_2 \cdot ... \cdot a_{100}$ is written on the board , where $a_1$, $a_2$, $ ... $, $a_{100}$, are natural numbers. Let's consider $99$ expressions, each of which is obtained by replacing one of the multiplication signs with an addition sign. It is known that the values of exactly $32$ of these expressions are even. What is the largest number of even numbers among $a_1$, $a_2$, $ ... $, $a_{100}$ could it be?

Let $n$ be a positive integer. A square $ABCD$ is divided into $n^2$ identical small squares by drawing $(n-1)$ equally spaced lines parallel to the side $AB$ and another $(n- 1)$ equally spaced lines parallel to $BC$, thus giving rise to $(n+1)^2$ intersection points. The points $A, C$ are coloured red and the points $B, D$ are coloured blue. The rest of the intersection points are coloured either red or blue. Prove that the number of small squares having exactly $3$ vertices of the same colour is even.

If you take a subset of $4002$ numbers from the whole numbers $1$ to $6003$, then there is always a subset of $2001$ numbers within that subset with the following property: If you order the $2001$ numbers from small to large, the numbers are alternately even and odd (or odd and even). Prove this.

Determine all even positive integers that can be written as the sum of odd composite positive integers.

In a convex $2009$-gon, all diagonals are drawn. A line intersects the $2009$-gon but does not pass through any of its vertices. Prove that the line intersects an even number of diagonals.

Let $n$ be a positive integer. Let $G (n)$ be the number of $x_1,..., x_n, y_1,...,y_n \in \{0,1\}$, for which the number $x_1y_1 + x_2y_2 +...+ x_ny_n$ is even, and similarly let $U (n)$ be the number for which this sum is odd. Prove that $$\frac{G(n)}{U(n)}= \frac{2^n + 1}{2^n - 1}.$$

The systems of equations \[\left\{ \begin{array}{l} 2x_1 - x_2 = 1 \\ -x_1 + 2x_2 - x_3 = 1 \\ -x_2 + 2x_3 - x_4 = 1 \\ -x_3 + 3x_4 - x_5 =1 \\ \cdots\cdots\cdots\cdots\\ -x_{n-2} + 2x_{n-1} - x_n = 1 \\ -x_{n-1} + 2x_n = 1 \\ \end{array} \right. \] has a solution in positive integers $x_i$. Show that $n$ must be even.

In a pyramid, the base is a right-angled triangle with integer sides. The height of the pyramid is also integer. Show that the volume of the pyramid is even.

Let $m, n > 1$ be coprime odd integers. Show that $$\big \lfloor \frac{m^{\phi (n)+1} + n^{\phi (m)+1}}{mn} \rfloor$$ is an even integer, where $\phi$ is Euler’s totient function.

Let $a, b$ and $c$ be positive integers such that $a^2 + b^2 + 1 = c^2$ . Prove that $[a/2] + [c / 2]$ is even. Note: $[x]$ is the integer part of $x$.

A positive integer $n \ge 2$ is called [i]peculiar [/i] if the number $n \choose i$ + $n \choose j $ $-i-j$ is even for all integers $i$ and $j$ such that $0 \le i \le j \le n$. Determine all peculiar numbers.

Let $P_1(x) = x^2 + a_1x + b_1$ and $P_2(x) = x^2 + a_2x + b_2$ be two quadratic polynomials with integer coeffcients. Suppose $a_1 \ne a_2$ and there exist integers $m \ne n$ such that $P_1(m) = P_2(n), P_2(m) = P_1(n)$. Prove that $a_1 - a_2$ is even.

Let $n$ be a natural number. Prove that the number $4 \cdot 3^{2^n}+ 3 \cdot4^{2^n}$ is divisible by $13$ if and only if $n$ is even.

A closed self-intersecting broken line intersects each of its segments once. Prove that the number of its segments is even.

Let's call a function $f : R \to R$ [i]cool[/i] if there are real numbers $a$ and $b$ such that $f(x + a)$ is an even function and $f(x + b)$ is an odd function. (a) Prove that every cool function is periodic. (b) Give an example of a periodic function that is not cool.

Prove that for a natural number $n$ greater than 1, the following conditions are equivalent: a) $ n $ is an even number, b) there is a permutation $ (a_0, a_1, a_2, \ldots, a_{n-1}) $ of the set $ \{0,1,2,\ldots,n—1\} $ with the property that the sequence of residues from dividing by $ n $ the numbers $ a_0, a_0 + a_1, a_0 + a_1 + a_2, \ldots, a_0 + a_1 + a_2 + \ldots a_{n-1} $ is also a permutation of this set.