Coffee Hour Problems

(How many mathematicians does it take to solve interesting math problems when they are being fueled by coffee?)

Since November 2017, I regularly leave a math problem on the table where the departmental coffee hour is held. I usually do it on days when I'm in charge of coffee and cookies, and I think this should become a departmental tradition!

Here you can find a list of these problems and their solutions. They come from various sources, such as math olympiads, book problems, communication with others, and sometimes my own head. The difficulty and prerequisites are also varied. The only constants are that the problem and solution should always fit a single page and should be related to coffee and cookies, preferably using food emoji instead of letter variables because that makes it look really annoying.

Problem 1: 1/28/17. Coffee monster filling and emptying cups. (Solution)
Problem 2: 12/5/17. Infinite sum involving perfect powers. (Solution)
Problem 3: 12/12/17. 13 real numbers. (Solution)
Problem 4: 1/22/18. Function of natural numbers that composes with itself to give translation by 2018. (Solution)
Problem 5: 1/29/18. System of 4 equations with many emojis. (Solution)
Problem 6: 2/5/18. Alice and Bob and a line of cookies. (Solution)
Problem 7: 2/12/18. Shadow of a unit cube. (Solution)
Problem 8: 2/19/18. Infinitely caffeinated sum. (Solution)
Problem 9: 2/26/18. Sum of 3 inverse trig functions. (Solution)
Problem 10: 3/5/18. 3 numbers and an inequality. (Solution)
Problem 11: 3/19/18. 42 Biscotti forming an equation. (Solution)
Problem 12: 4/2/18. Coffee cup passing game at a round table. (Solution)
Problem 13: 4/9/18. Angles in a triangle. (Solution)
Problem 14: 4/16/18. Flipping 1000 and 1001 cookies. (Solution)
Problem 15: 4/23/18. Cookies forming polygons. (Solution)
Special problem: 4/30/18.
Problem 16: 9/4/18. Largest power of 2 dividing a sum with binomials. (Solution)
Problem 17: 9/11/18. Sums of powers of three numbers having sum zero. (Solution)
Problem 18: 9/18/18. Alice and Bob's two player game. (Solution)
Problem 19: 9/25/18. 7x7 matrix with given eigenvalues. (Solution)
Problem 20: 10/2/18. Area of trapezoid. (Solution)
Problem 21: 10/9/18. Rectangular boxes with integer lengths. (Solution)
Problem 22: 10/16/18. Write 512 using 2, 3 and 5. (Solution)
Problem 23: 10/23/18. Divisibility of summation involving lcm and gcd. (Solution)
Problem 24: 10/30/18. Pairs of positive integers satisfying quadratic Diophantine equation. (Solution)
Problem 25: 11/6/18. Irrational number with long streak of consecutive nines. (Solution)
Problem 26: 11/13/18. Equilateral triangle with point inside. (Solution)
Problem 27: 11/20/18. Uncountable increasing family of subsets of the naturals. (Solution)
Problem 28: 12/4/18. Sequence without arithmetic progressions. (Solution)
Problem 29: 1/25/19. Cookies in a grid becoming contaminated. (Solution)
Problem 30: 2/1/19. Continuous functions mapping irrationals to rationals. (Solution)
Problem 31: 2/8/19. Two numbers that must be perfect cubes. (Solution)
Problem 32: 2/15/19. Uncountable family of infinite subsets of the naturals with finite intersections. (Solution)
Problem 33: 3/1/19. A colored hat guessing game. (Solution)
Problem 34: 3/15/19. Compute the integral (logarithms). (Solution)
Problem 35: 4/5/19. Complete the sequence. (Solution)