LEAKED 2035 IMO questions

Word has spread that a genius yet evil mathematician has leaked the problems that will be contained in the 2035 International Math Olympiad. This mathematician is reportedly a diehard fan of Gottfried Leibniz and will stop at nothing to sabotage math competitions which don’t include calculus. (They also claim to have burned over four hundred copies of Voltaire’s Candide.) Seeing as I once got a 120 on the AMC 10 but was not once selected for my country’s IMO team, I have decided to join the evil mathematician in their quest and publish all of the leaked questions.

Day 1

Question 1

What is 3471239847981237489021374897 times 7807123894710928375490128508734657801634?

Question 2

Suppose there exists a blue guy $P$ at the center of circle $Q$ (the “blue world”) with radius 7. Suppose also that $Q$ has two parallel chords $\overline{AB}$ and $\overline{CD}$. The region $R$ is a blue tree defined by $\overline{AB}$, $\overline{CD}$, and $Q$. If triangle $ACP$ is isosceles with $AP = PC = 8$, what is the area of the blue tree?

Question 3

Prove that every even number greater than 2 is the sum of two primes.

Day 2

Question 4

An island contains 100 goblins, labeled $g_1, g_2, \ldots, g_{100}$. Each goblin has either a red or a blue hat. The island also contains two gates, one red and one blue. Each day at noon, the red gate opens with probability $2/5$ and the blue gate opens with probability $7/10$. The goblins are not allowed to speak or hear or look at anything, except that they are allowed to look at the hats of exactly two other goblins of their choice. Each goblin has 100 days to go through a gate, but if they go through the gate that matches their color of hat they will be killed.

Question 5

Ok now get this one: Is $x$ bigger than $y$?

Question 6

How many numbers are there