Determine all pairs \((a,b)\) of positive integers for which there exist positive integers \(g\) and \(N\) such that

\[ \gcd(a^n + b, b^n + a) = g \]

holds for all integers \(n \ge N\).

Let \(N\) be a positive integer. Geoff and Ceri play a game in which they start by writing the numbers \(1,2,\dots,N\) on a board. They then take turns to make a move, starting with Geoff.

Each move consists of choosing a pair of integers \((k,n)\), where \(k \ge 0\) and \(n\) is one of the integers on the board, and then erasing every integer \(s\) on the board such that \(2^k \mid n-s\).

The game continues until the board is empty. The player who erases the last integer on the board loses.

Determine all values of \(N\) for which Geoff can ensure that he wins, no matter how Ceri plays.

Let \(\mathcal{F}\) be the set of all functions \(f: \mathbb{R} \to \mathbb{R}\) such that

\[ f(x + f(y)) = f(x) + f(y) \]

for every \(x, y \in \mathbb{R}\). Find all rational numbers \(q\) such that for every \(f \in \mathcal{F}\), there exists some \(z \in \mathbb{R}\) satisfying \(f(z) = qz\).

Let \(a, b, c, d\) be real numbers such that \(a^2 + b^2 + c^2 + d^2 = 1\). Determine the minimum value of \((a - b)(b - c)(c - d)(d - a)\) and find all values of \(a, b, c, d\) where this minimum is achieved.

Let \((a_n)_{n \ge 1}\) and \((b_n)_{n \ge 1}\) be sequences of positive real numbers such that \(a_1, b_1 < 5\) and for any positive integer \(n\),

\[ a_{n + 1} = \frac{b_n + \sqrt{a_n b_n}}{2} \qquad \text{and} \qquad b_{n + 1} = \sqrt{\frac{a_n(a_n + b_n)}{2}}. \]

Prove that

\[ |a_{20} - b_{20}| < \frac{1}{2025}. \]

Let \( f : \mathbb{R} \to \mathbb{R} \) be a function that satisfies

• \( f(x+f(y)) = f(x) + f(y) \) for all \( x,y \in \mathbb{R} \).
• There exists a real number \( x \) such that \( f(x) = \frac{1}{2025}\).
• There does not exist a real number \( x \) such that \( 0 < f(x) < \frac{1}{2025} \).

Determine all possible images of \( f \).

Let \( S \) be the set of all triples of positive real numbers \( (a,b,c) \) such that \( a + b + c = ab + bc + ca \).

1. Prove that \[ \min \{ a+b,\; b+c,\; c+a \} > 1. \]
2. Does there exist a triple \( (a,b,c) \in S \) such that \[ \min \{ a+b,\; b+c,\; c+a \} < 1 + \frac{1}{20^{25}} ? \]

Find the number of positive integers \( (a,b) \) where \( 1 \le a,b \le 19^2 \) such that \( a^4 + b^3 \) is divisible by \( 19^2 \).

Find the number of positive integer pairs \(1 \le a, b \le 2027\) that satisfy

\[ 2027 \mid a^6 + b^5 + b^2. \]

Find all functions \(f: \mathbb{N} \to \mathbb{N}\) such that for every prime number \(p\) and natural number \(x\),

\[ \{x, f(x), \dots, f^{p-1}(x)\} \]

is a complete residue system modulo \(p\).

Determine all functions \(f: \mathbb{R} \to \mathbb{R}\) such that for every real \(x, y\),

\[ f(f(x) + y) = \lfloor x + f(f(y)) \rfloor. \]

Let \(a, b, c\) be three distinct positive integers. Define \(S(a, b, c)\) as the set of all rational roots of \(px^2 + qx + r = 0\) for every permutation \((p, q, r)\) of \((a, b, c)\).

For example, \(S(1, 2, 3) = \{-1, -2, -1/2\}\) because the equation \(x^2 + 3x + 2 = 0\) has roots \(-1\) and \(-2\), the equation \(2x^2 + 3x + 1 = 0\) has roots \(-1\) and \(-1/2\), and other permutations yield no rational roots.

Determine the maximum number of elements in \(S(a, b, c)\).