HMMT 2013 #0jfz

HMMT 2013Hard

From module Generating Functions (Intro)

Community difficultyNo ratings yetBe the first to rate this problem.

Understand the scale

Scale 0.5: Easiest math competition problems, often solvable even for early middle school students without experience. 1: Problems strictly for beginners, often at a moderate middle school level. 1.5: Problems for stronger beginner students, on the level of the middling problems in most middle school contests. 2: For motivated beginners, harder questions from the previous categories. 2.5: More advanced beginner problems, hardest questions from previous categories. 3: Early intermediate problems that require more creative thinking. 3.5: Tougher early intermediate problems that consistently stretch into higher-level creative thinking or conceptual knowledge. 4: Intermediate-level problems. 4.5: Upper intermediate problems approaching the upper bound for non-invitational sprint math competitions. 5: More difficult AIME problems or simple proof-based Olympiad-style problems. 6: High-level AIME-styled questions or introductory Olympiad-level questions. 7: Tougher Olympiad-level questions, may require more technical knowledge. 8: High-level Olympiad-level questions. 9: Expert Olympiad-level questions. 9.5: The hardest problems appearing on Olympiads which the strongest students could reasonably solve. 10: Historically notorious problems, generally unsuitable for even very hard competitions because they are exceedingly tedious, long, and difficult.

Generating functionsEnumeration with symmetry

Problem

Problem:

For how many unordered sets $\{a, b, c, d\}$ of positive integers, none of which exceed $168$ , do there exist integers $w, x, y, z$ such that $(-1)^{w} a + (-1)^{x} b + (-1)^{y} c + (-1)^{z} d = 168$ ? If your answer is $A$ and the correct answer is $C$ , then your score on this problem will be $\left\lfloor 25 e^{-3 \frac{|C-A|}{C}}\right\rfloor$ .

← Back to all problems