One way of encouraging students to explore mathematics can be letting them to use different approaches to solve the same problem. If students can find alternatives from different areas of mathematics in the future they can develop that skill to use different areas outside of mathematics to solve global challenges in mathematics that was already achieved by great mathematicians in the past. What are the examples you have relevant to fulfill this requirement?
When you talk about geometry, you can find problems which can be solved using Geometry, Coordinate geometry, Trigonometry, Vectors and Argand diagram. In combinatorics you can use formula to simplify or combinatorial arguments. What would you suggest as problems to give this exposure to advanced level students ?

One way of encouraging students to explore mathematics can be letting them to use different approaches to solve the same problem. If students can find alternatives from different areas of mathematics in the future they can develop that skill to use concepts from different areas outside of mathematics to solve global challenges in mathematics that was already achieved by great mathematicians in the past. As an example famous squaring the square problem was solved by applying Kirchhoff's circuit laws in physics .What are the examples you have relevant to fulfill this requirement not in mathematicians level but within the scope of students level ?
When you talk about geometry, you can find problems which can be solved using Geometry, Coordinate geometry, Trigonometry, Vectors and Argand diagram. In combinatorics you can use formula to simplify or combinatorial arguments where students have much freedom to think creatively. What would you suggest as problems to give this exposure to advanced level students ?