Four points $A, O, B, O^{\prime}$ are aligned in this order on a line. Let $C$ be the circle with center $O$ and radius 2015, and $C^{\prime}$ be the circle with center $O^{\prime}$ and radius 2016. Suppose that $A$ and $B$ are the intersections of two common tangents to the two circles. Calculate $A B$ given that $A B$ is an integer $<10^{7}$ and that $A O$ and $A O^{\prime}$ are integers.

## 3 Junior Eliminatory: Solutions

Translate the above text into English, retain the original text's line breaks and format, and output the translation result directly.
8124480. Let $x$ be the distance $O O^{\prime}$. If a common tangent to the two circles passing through $A$ touches these circles at $M$ and $M^{\prime}$, it is easy to see by Thales' theorem that: $\frac{A O^{\prime}}{2016}=\frac{A O}{2015}=O O^{\prime}=x$, so $A O=2015 \times x$. By a similar calculation, $\frac{B O^{\prime}}{2016}=\frac{B O}{2015}$, hence: $B O=\frac{2015 \times x}{2016+2015}$. Therefore, $A B=(2015 x)\left(1+\frac{1}{4031}\right)=\frac{4032 \times 2015 \times x}{4031}$. Since 4031 is prime with 2015 and 4032, 4031 divides $x: x=4031 k$, so $A B=4032 \times 2015 \times k$. Since $4032 \times 2015=8124480$, $k=1$ otherwise $A B$ would be greater than $10^{7}$, which is excluded by hypothesis.

## 5 Statements of the Animath Cup - June 2, 2015

## Instructions

$\triangleright$ Write the different problems on separate sheets. At the top left of each sheet, write your name and first name in capital letters, as well as your class, and the problem number at the top right.

$\triangleright$ Complete solutions are required (except for exercise 1), where every statement is carefully justified. Clarity and precision of the copy will be taken into account in the grading.

Work first in draft, and then write your solution, or a significant attempt at a solution, neatly. Do not submit your drafts: they will not be considered.

$\Delta$ A complete solution will earn more points than several incomplete attempts. It is better to finish a small number of problems than to tackle all of them.

$\Delta$ Rulers, set squares, and compasses are allowed. Protractors are prohibited.

Calculators are prohibited, as are all electronic devices.

Middle school students work on exercises 1 to 5. High school students work on exercises 4 to 8. Each exercise is graded out of 7 points.

Please respect the numbering of the exercises. Write the different problems on separate sheets. At the top left of each sheet, write your name and first name in capital letters, as well as your class, and the problem number at the top right.

## Middle School Statements