Mark on a cellular paper four nodes forming a convex quadrilateral with the sidelengths equal to four different primes.
                                                                                                                                             [i](Proposed by A.Zaslavsky)[/i]
1. **Identify the vertices of the quadrilateral:**
   Let the vertices of the quadrilateral be \( A = (0,0) \), \( B = (5,0) \), \( C = (5,17) \), and \( D = (0,29) \).

2. **Calculate the side lengths:**
   - \( AB \) is the distance between points \( A \) and \( B \):
     \[
     AB = \sqrt{(5-0)^2 + (0-0)^2} = \sqrt{25} = 5
     \]
   - \( BC \) is the distance between points \( B \) and \( C \):
     \[
     BC = \sqrt{(5-5)^2 + (17-0)^2} = \sqrt{0 + 289} = 17
     \]
   - \( CD \) is the distance between points \( C \) and \( D \):
     \[
     CD = \sqrt{(5-0)^2 + (17-29)^2} = \sqrt{25 + 144} = \sqrt{169} = 13
     \]
   - \( DA \) is the distance between points \( D \) and \( A \):
     \[
     DA = \sqrt{(0-0)^2 + (29-0)^2} = \sqrt{0 + 841} = 29
     \]

3. **Verify the side lengths are all different primes:**
   - \( AB = 5 \)
   - \( BC = 17 \)
   - \( CD = 13 \)
   - \( DA = 29 \)
   All these numbers are prime and different from each other.

4. **Check if the quadrilateral is convex:**
   - A quadrilateral is convex if all its interior angles are less than \( 180^\circ \).
   - Since \( A, B, C, D \) form a trapezium (a type of quadrilateral where at least one pair of opposite sides are parallel), and the vertices are given in a specific order, it ensures that the quadrilateral is convex.

Therefore, the quadrilateral \( ABCD \) with the given vertices and side lengths is a convex quadrilateral with four different prime side lengths.

The final answer is the quadrilateral \( \boxed{ ABCD } \) with vertices \( A = (0,0) \), \( B = (5,0) \), \( C = (5,17) \), and \( D = (0,29) \).