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R3

Part of 2021 Stanford Mathematics Tournament

Problems(1)

2021 SMT Guts Round 3 p9-12 - Stanford Math Tournament

Source:

2/10/2022
p9. The frozen yogurt machine outputs yogurt at a rate of 55 froyo3^3/second. If the bowl is described by z=x2+y2z = x^2+y^2 and has height 55 froyos, how long does it take to fill the bowl with frozen yogurt?
p10. Prankster Pete and Good Neighbor George visit a street of 20212021 houses (each with individual mailboxes) on alternate nights, such that Prankster Pete visits on night 11 and Good Neighbor George visits on night 22, and so on. On each night nn that Prankster Pete visits, he drops a packet of glitter in the mailbox of every nthn^{th} house. On each night mm that Good Neighbor George visits, he checks the mailbox of every mthm^{th} house, and if there is a packet of glitter there, he takes it home and uses it to complete his art project. After the 2021th2021^{th} night, Prankster Pete becomes enraged that none of the houses have yet checked their mail. He then picks three mailboxes at random and takes out a single packet of glitter to dump on George’s head, but notices that all of the mailboxes he visited had an odd number of glitter packets before he took one. In how many ways could he have picked these three glitter packets? Assume that each of these three was from a different house, and that he can only visit houses in increasing numerical order.
p11. The taxi-cab length of a line segment with endpoints (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2) is ∣x1−x2∣+∣y1−y2∣|x_1 - x_2| + |y_1- y_2|. Given a series of straight line segments connected head-to-tail, the taxi-cab length of this path is the sum of the taxi-cab lengths of its line segments. A goat is on a rope of taxi-cab length 72\frac72 tied to the origin, and it can’t enter the house, which is the three unit squares enclosed by (−2,0)(-2, 0),(0,0)(0, 0),(0,−2)(0, -2),(−1,−2)(-1, -2),(−1,−1)(-1, -1),(−2,−1)(-2, -1). What is the area of the region the goat can reach? (Note: the rope can’t ”curve smoothly”-it must bend into several straight line segments.)
p12. Parabola PP, y=ax2+cy = ax^2 + c has a>0a > 0 and c<0c < 0. Circle CC, which is centered at the origin and lies tangent to PP at PP’s vertex, intersects PP at only the vertex. What is the maximum value of a, possibly in terms of cc?
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.
algebrageometrycombinatoricsnumber theoryStanford Math TournamentSMT