The question asks the student to prove the existence of an object such as a sequence, function, polynomial, constant, or solution satisfying given conditions, often using compactness, fixed-point theorems, or constructive arguments.
There are three cities each of which has exactly the same number of citizens, say $n$. Every citizen in each city has exactly a total of $n + 1$ friends in the other two cities. Show that there exist three people, one from each city, such that they are friends. We assume that friendship is mutual (that is, a symmetric relation).
If a given equilateral triangle $\Delta$ of side length $a$ lies in the union of five equilateral triangles of side length $b$, show that there exist four equilateral triangles of side length $b$ whose union contains $\Delta$.
Suppose $f : \mathbb { R } \rightarrow \mathbb { R }$ is differentiable and $\left| f ^ { \prime } ( x ) \right| < \frac { 1 } { 2 }$ for all $x \in \mathbb { R }$. Show that for some $x _ { 0 } \in \mathbb { R } , f \left( x _ { 0 } \right) = x _ { 0 }$.
Consider a ball that moves inside an acute-angled triangle along a straight line, until it hits the boundary, which is when it changes direction according to the mirror law, just like a ray of light (angle of incidence $=$ angle of reflection). Prove that there exists a triangular periodic path for the ball, as pictured below.