Olympiad problems

2005 Gheorghe Vranceanu, 2

Three natural numbers $ a,b,c $ with $ \gcd (a,b) =1 $ define in the Diophantine plane a line $ d: ax+by-c=0. $ Prove that: [b]a)[/b] the distance between any two points from $ d $ is at least $ \sqrt{a^2+b^2} . $ [b]b)[/b] the restriction of $ d $ to the first quadrant of the Diophantine plane is a finite line having at most $ 1+\frac{c}{ab} $ elements.

2000 Romania National Olympiad, 3

Tags: function , complex number , cartesian plane , geometry , analytic geometry , algebra

A function $ f:\mathbb{R}^2\longrightarrow\mathbb{R} $ is [i]olympic[/i] if, any finite number of pairwise distinct elements of $ \mathbb{R}^2 $ at which the function takes the same value represent in the plane the vertices of a convex polygon. Prove that if $ p $ if a complex polynom of degree at least $ 1, $ then the function $ \mathbb{R}^2\ni (x,y)\mapsto |p(x+iy)| $ is olympic if and only if the roots of $ p $ are all equal.

2019 Belarusian National Olympiad, 10.8

Tags: parallelogram , cartesian plane , combinatorial geometry , analytic geometry

Call a polygon on a Cartesian plane to be[i]integer[/i] if all its vertices are integer. A convex integer $14$-gon is cut into integer parallelograms with areas not greater than $C$. Find the minimal possible $C$. [i](A. Yuran)[/i]

2020 Junior Macedonian National Olympiad, 5

Tags: combinatorics , nice , geometrical , cartesian plane , geometry , analytic geometry

Let $T$ be a triangle whose vertices have integer coordinates, such that each side of $T$ contains exactly $m$ points with integer coordinates. If the area of $T$ is less than $2020$, determine the largest possible value of $m$.

2025 Philippine MO, P6

Tags: combinatorics , path , cartesian plane , knight

An ant is on the Cartesian plane. In a single move, the ant selects a positive integer $k$, then either travels [list] [*] $k$ units vertically (up or down) and $2k$ units horizontally (left or right); or [*] $k$ units horizontally (left or right) and $2k$ units vertically (up or down). [/list] Thus, for any $k$, the ant can choose to go to one of eight possible points. \\ Prove that, for any integers $a$ and $b$, the ant can travel from $(0, 0)$ to $(a, b)$ using at most $3$ moves.

1978 Romania Team Selection Test, 1

Tags: number theory , analytic geometry , cartesian plane

Associate to any point $ (h,k) $ in the integer net of the cartesian plane a real number $ a_{h,k} $ so that $$ a_{h,k}=\frac{1}{4}\left( a_{h-1,k} +a_{h+1,k}+a_{h,k-1}+a_{h,k+1}\right) ,\quad\forall h,k\in\mathbb{Z} . $$ [b]a)[/b] Prove that it´s possible that all the elements of the set $ A:=\left\{ a_{h,k}\big| h,k\in\mathbb{Z}\right\} $ are different. [b]b)[/b] If so, show that the set $ A $ hasn´t any kind of boundary.

2021 Bangladeshi National Mathematical Olympiad, 12

Tags: counting , cartesian plane , bijection , combinatorics

Two toads named Gamakichi and Gamatatsu are sitting at the points $(0,0)$ and $(2,0)$ respectively. Their goal is to reach $(5,5)$ and $(7,5)$ respectively by making one unit jumps in positive $x$ or $y$ direction at a time. How many ways can they do this while ensuring that there is no point on the plane where both Gamakichi And Gamatatsu land on?