1. Find all differentiable functions $f:\mathbb{R}\to\mathbb{R}$ such that
\[f'(x)=\frac{f(x+n)-f(x)}{n}\]
for all real numbers $x$ and all positive integers $n.$
https://artofproblemsolving.com/community/c7h381033p2109503

2. Prove that for each positive integer $n,$ the number $10^{10^{10^n}}+10^{10^n}+10^n-1$ is not prime.
https://artofproblemsolving.com/community/c7h381036p2109507

3. Let $ d_n$ be the determinant of the $ n\times n$ matrix whose entries, from left to right and then from top to bottom, are $ \cos 1,\cos 2,\dots,\cos n^2.$ (For example, $ d_3 = \begin{vmatrix}\cos 1 & \cos2 & \cos3 \\
\cos4 & \cos5 & \cos 6 \\
\cos7 & \cos8 & \cos 9\end{vmatrix}.$ The argument of $ \cos$ is always in radians, not degrees.)

Evaluate $ \lim_{n\to\infty}d_n.$

https://artofproblemsolving.com/community/c7h316668p1703556

4. Show that every positive rational number can be written as a quotient of products of factorials of (not necessarily distinct) primes. For example, $ \frac{10}{9}=\frac{2!\cdot 5!}{3!\cdot 3!\cdot 3!}.$

https://artofproblemsolving.com/community/c7h316672p1703568

5. Is there a strictly increasing function $f:\mathbb{R}\to\mathbb{R}$ such that $f'(x)=f(f(x))$ for all $x?$

https://artofproblemsolving.com/community/c7h381045p2109524

6. Let $T$ be the set of all triples $(a,b,c)$ of positive integers for which there exist triangles with side lengths $a,b,c.$ Express \[\sum_{(a,b,c)\in T}\frac{2^a}{3^b5^c}\] as a rational number in lowest terms.

https://artofproblemsolving.com/community/c7h1171039p5624386

7. A hilarious integer is a positive integer whose digits in base $10$ are all ones. Find all polynomials $f$ with real coefficients such that if $n$ is hilarious, then so is $f(n)$.

https://artofproblemsolving.com/community/c7h177229p978388

8. Four points are chosen uniformly and independently at random in the interior of a given circle. Find the probability that they are the vertices of a convex quadrilateral.

https://artofproblemsolving.com/community/c7h122451p694692

9. https://artofproblemsolving.com/community/c7h513433p2882551

10. https://artofproblemsolving.com/community/c7h122452p694694

11. https://artofproblemsolving.com/community/c7h413599p2325325

12. https://artofproblemsolving.com/community/c7h466470p2612317

13. https://artofproblemsolving.com/community/c7h530478p3027460

14. https://artofproblemsolving.com/community/c7h20497p134592

15. https://artofproblemsolving.com/community/c7h469262p2626897

16. https://artofproblemsolving.com/community/c7h469256p2626877

17. https://artofproblemsolving.com/community/c7h429342p2429218

18. https://artofproblemsolving.com/community/c7h504348p2832941