Found problems: 39
Determine all even positive integers that can be written as the sum of odd composite positive integers.
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 $a_1 , a_2 , ..., a_n$ be non-zero integers that satisfy the equation
$$a_1 +\dfrac{1}{a_2+\dfrac{1}{a_3+ ... \dfrac{1}{a_n+\dfrac{1}{x}} } } = x$$
for all values of $x$ for which the lefthand side of the equation makes sense.
(a) Prove that $n$ is even.
(b) What is the smallest n for which such numbers $a_1 , a_2 , ..., a_n$ exist?
(M Skopenko)
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.
It is known that the sum of the digits of the natural number $N$ is $100$, and the sum of the digits of the number $5N$ is $50$. Prove that $N$ is even.
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 $x$.
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.
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.
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.
Show that the edges of a connected simple (no loops and no multiple edges) finite graph can be oriented so that the number of edges leaving each vertex is even if and only if the total number of edges is even
Let $a_1, a_2, ...,a_n$ represent an arbitrary arrangement of the numbers $1, 2, ...,n$. Prove that, if $n$ is odd, the product $$(a_1 - 1)(a_2 - 2) ... (a_n -n)$$ is an even number.
Find the number of even integers n such that $0 \le n \le 100$ and $5 | n^2 \cdot 2^{{2n}^2}+ 1$.
Assume that $a$ and $b$ are integers. Prove that the equation $a^2 +b^2 +x^2 = y^2$ has an integer solution $x,y$ if and only if the product $ab$ is even.
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 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 $$\frac{G(n)}{U(n)}= \frac{2^n + 1}{2^n - 1}.$$
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.
Let $n$ be a positive integer. Prove that at least one of the integers $[2^n \cdot \sqrt2]$, $[2^{n+1} \cdot \sqrt2]$, $...$, $[2^{2n} \cdot \sqrt2]$ is even, where $[a]$ denotes the integer part of $a$.
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.
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.
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.
If the fraction $\frac{an + b}{cn + d}$ may be simplified using $2$ (as a common divisor ), show that the number $ad - bc$ is even. ($a, b, c, d, n$ are natural numbers and the $cn + d$ different from zero).
A closed self-intersecting broken line intersects each of its segments once. Prove that the number of its segments is even.
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.
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.