Competition problem — Number theory

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Number theory Difficulty 5.4 AIME, harder Find the answer

# Assignment 1. (10 points)

Each of the 2017 middle school students studies English or German. English is studied by $70 \%$ to $85 \%$ of the total number of students, and both languages are studied by $5 \%$ to $8 \%$ . What is the maximum number of students who can study German.

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

# Solution.

Let $A$ be the number of people who study English, $N$ be the number of people who study German, and $AN$ be the number of people who study both languages.

Then $N=2017-A+AN$ .

It is known that $A \geq 2017 \cdot 0.7=1411.9$ and $AN \leq 0.08 \cdot 2017=161.36$ .

Therefore, $N \leq 2017-1412+161=766$ .

Note that $N=766$ , if $A=1412, AN=161$ .

Answer: 766.

Criterion	Evaluation	Points
Complete solution.	+	10
A minor computational error was made. Main estimates are provided. Lack of necessary rigor in obtaining the answer. The answer is correct.	.+	8
Main estimates are provided. Lack of necessary rigor in obtaining the answer. The answer is incorrect.	$+/ 2$	7
Not all estimates are provided. The answer is not obtained or is incorrect.	$\mp$	2
The answer is correct, but the solution is missing.	+	$-/ 0$
The solution does not meet any of the criteria described above.	0
Maximum score		10

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.