Olympiad problems

2025 Kosovo National Mathematical Olympiad`, P1

Find all real numbers $a$, $b$ and $c$ that satisfy the following system of equations: $$\begin{cases} ab-c = 3 \\ a+bc = 4 \\ a^2+c^2 = 5\end{cases}$$

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 $m$.

2013 India Regional Mathematical Olympiad, 3

Tags: easy , nice , number theory

Find all primes $p$ and $q$ such that $p$ divides $q^2-4$ and $q$ divides $p^2-1$.

2023 Philippine MO, 8

Tags: combinatorial geometry , functional equation , function , geometry , nice

Let $\mathcal{S}$ be the set of all points in the plane. Find all functions $f : \mathcal{S} \rightarrow \mathbb{R}$ such that for all nondegenerate triangles $ABC$ with orthocenter $H$, if $f(A) \leq f(B) \leq f(C)$, then $$f(A) + f(C) = f(B) + f(H).$$

2020 Junior Macedonian National Olympiad, 4

Tags: geometry , nice , collinearity , circumcircle , parallelogram

Let $ABC$ be an isosceles triangle with base $AC$. Points $D$ and $E$ are chosen on the sides $AC$ and $BC$, respectively, such that $CD = DE$. Let $H, J,$ and $K$ be the midpoints of $DE, AE,$ and $BD$, respectively. The circumcircle of triangle $DHK$ intersects $AD$ at point $F$, whereas the circumcircle of triangle $HEJ$ intersects $BE$ at $G$. The line through $K$ parallel to $AC$ intersects $AB$ at $I$. Let $IH \cap GF =$ {$M$}. Prove that $J, M,$ and $K$ are collinear points.

II Soros Olympiad 1995 - 96 (Russia), 10.4

Tags: algebra , floor function , nice , system of equations

Solve the system of equations $$\begin{cases} x^2+ [y]=10 \\ y^2+[x]=13 \end{cases}$$ ($[x]$ is the integer part of $x$, $[x]$ is equal to the largest integer not exceeding $x$. For example, $[3,33] = 3$, $[2] = 2$, $[- 3.01] = -4$).