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 \(n \geq N\).

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 \( n \),

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

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 exist 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 the inequality \[ \min \{ a + b, b + c + c + a \} > 1 \] holds for every triple \( (a,b,c) \in S \).
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 \(1 \leq a, b \leq 2027\) 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 \(S(a, b, c)\).