We define $u_{1}=17 \times(1+2)$, $u_{2}=17^{2} \times(2+2)$, and more generally $u_{n}=17^{n} \times(n+2)$ for all integers $n \geqslant 1$. Let $a$ and $b$ be real numbers such that $u_{n+2}=a u_{n+1}+b u_{n}$ for all $n$. Determine $a^{2}-b$.

## 4 Junior Eliminators: Solutions

Translate the above text into English, preserving the original text's line breaks and format, and output the translation result directly.
Answer: 1445. Indeed, we have $17^{n+2}(n+4)=a\left(17^{n+1}(n+3)\right)+b\left(17^{n}(n+2)\right)$, so $17^{2}=17 a+b$ and $17^{2} \times 4=51 a+2 b$. We deduce that $17 a=(51 a+2 b)-2(17 a+b)=$ $17^{2} \times(4-2)$, hence $a=34$. From this, we get $b=17^{2}-17 \times a=-289$, and then $a^{2}-b=1445$.

## 6 Statements of the Animath Cup - June 6, 2017

## Instructions

Write the different problems on separate sheets. On each sheet, write your name and first name in capital letters at the top left and 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 containing significant results for the problem, neatly. Do not submit your drafts: they will not be considered.

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

$\triangleright$ 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 3 to 7. Each exercise is graded out of 7 points.

Thank you for respecting the numbering of the exercises. Write the different problems on separate sheets. On each sheet, write your name and first name in capital letters at the top left and your class, and the problem number at the top right.

## Middle School Statements