MathDB

5

Part of 2023 Harvard-MIT Mathematics Tournament

Problems(4)

weird square( 2023 HMMT G5)

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2/21/2023
Let ABCABC be a triangle with AB=13,BC=14,AB = 13, BC = 14, andCA=15 CA = 15. Suppose PQRSPQRS is a square such that PP and RR lie on line BC,QBC, Q lies on line CACA, and SS lies on line ABAB. Compute the side length of this square.
HMMT
HMMT Feb 2023 Team p5

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2/20/2023
Let SS be the set of all points in the plane whose coordinates are positive integers less than or equal to 100100 (so SS has 1002100^2 elements), and let LL be the set of all lines ℓ\ell such that ℓ\ell passes through at least two points in SS. Find, with proof, the largest integer N≥2N \geq 2 for which it is possible to choose NN distinct lines in LL such that every two of the chosen lines are parallel.
2023 Algebra/NT #5

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4/18/2023
Suppose EE, II, LL, VV are (not necessarily distinct) nonzero digits in base ten for which
[*] the four-digit number E‾ V‾ I‾ L‾\underline{E}\ \underline{V}\ \underline{I}\ \underline{L} is divisible by 7373, and
[*] the four-digit number V‾ I‾ L‾ E‾\underline{V}\ \underline{I}\ \underline{L}\ \underline{E} is divisible by 7474.
Compute the four-digit number L‾ I‾ V‾ E‾\underline{L}\ \underline{I}\ \underline{V}\ \underline{E}.
2023 Combinatorics #5

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4/17/2023
Elbert and Yaiza each draw 1010 cards from a 2020-card deck with cards numbered 1,2,3,…,201,2,3,\dots,20. Then, starting with the player with the card numbered 11, the players take turns placing down the lowest-numbered card from their hand that is greater than every card previously placed. When a player cannot place a card, they lose and the game ends.
Given that Yaiza lost and 55 cards were placed in total, compute the number of ways the cards could have been initially distributed. (The order of cards in a player’s hand does not matter.)