Found problems: 39
2009 Tournament Of Towns, 1
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.
2019 Romanian Master of Mathematics Shortlist, N1
Let $p$ and $q$ be relatively prime positive odd integers such that $1 < p < q$. Let $A$ be a set of pairs of integers $(a, b)$, where $0 \le a \le p - 1, 0 \le b \le q - 1$, containing exactly one pair from each of the sets $$\{(a, b),(a + 1, b + 1)\}, \{(a, q - 1), (a + 1, 0)\}, \{(p - 1,b),(0, b + 1)\}$$ whenever $0 \le a \le p - 2$ and $0 \le b \le q - 2$. Show that $A$ contains at least $(p - 1)(q + 1)/8$ pairs whose entries are both even.
Agnijo Banerjee and Joe Benton, United Kingdom
2015 Danube Mathematical Competition, 2
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
2019 Silk Road, 4
The sequence $ \{a_n \} $ is defined as follows: $ a_0 = 1 $ and $ {a_n} = \sum \limits_ {k = 1} ^ {[\sqrt n]} {{a_ {n - {k ^ 2 }}}} $ for $ n \ge 1. $
Prove that among $ a_1, a_2, \ldots, a_ {10 ^ 6} $ there are at least $500$ even numbers.
(Here, $ [x] $ is the largest integer not exceeding $ x $.)
1996 Greece Junior Math Olympiad, 4a
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).
2015 Brazil Team Selection Test, 1
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.
2021 China National Olympiad, 5
$P$ is a convex polyhedron such that:
[b](1)[/b] every vertex belongs to exactly $3$ faces.
[b](1)[/b] For every natural number $n$, there are even number of faces with $n$ vertices.
An ant walks along the edges of $P$ and forms a non-self-intersecting cycle, which divides the faces of this polyhedron into two sides, such that for every natural number $n$, the number of faces with $n$ vertices on each side are the same. (assume this is possible)
Show that the number of times the ant turns left is the same as the number of times the ant turn right.
1989 All Soviet Union Mathematical Olympiad, 508
A polyhedron has an even number of edges. Show that we can place an arrow on each edge so that each vertex has an even number of arrows pointing towards it (on adjacent edges).
1956 Moscow Mathematical Olympiad, 322
A closed self-intersecting broken line intersects each of its segments once. Prove that the number of its segments is even.
1978 Bundeswettbewerb Mathematik, 1
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.
2000 Czech And Slovak Olympiad IIIA, 1
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.
2013 Grand Duchy of Lithuania, 4
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.
1947 Moscow Mathematical Olympiad, 139
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.
1993 Swedish Mathematical Competition, 3
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.